[    {        "title": "2312.11293v1.Enhancing_thermal_stability_of_optimal_magnetization_reversal_in_nanoparticles.pdf",        "content": "Enhancing thermal stability of optimal magnetization reversal in nanoparticles\nMohammad H. A. Badarneh,1Grzegorz J. Kwiatkowski,1and Pavel F. Bessarab1,2,∗\n1Science Institute of the University of Iceland, 107 Reykjavík, Iceland\n2Department of Physics and Electrical Engineering, Linnaeus University, SE-39231 Kalmar, Sweden\n(Dated: December 19, 2023)\nEnergy-efficient switching of nanoscale magnets requires the application of a time-varying mag-\nnetic field characterized by microwave frequency. At finite temperatures, even weak thermal fluc-\ntuations create perturbations in the magnetization that can accumulate in time, break the phase\nlocking between the magnetization and the applied field, and eventually compromise magnetization\nswitching. It is demonstrated here that the magnetization reversal is mostly disturbed by unstable\nperturbations arising in a certain domain of the configuration space of a nanomagnet. The insta-\nbilities can be suppressed and the probability of magnetization switching enhanced by applying an\nadditional stimulus such as a weak longitudinal magnetic field that ensures bounded dynamics of\nthe perturbations. Application of the stabilizing longitudinal field to a uniaxial nanomagnet makes\nit possible to reach a desired probability of magnetization switching even at elevated temperatures.\nThe principle of suppressing instabilities provides a general approach to enhancing thermal stability\nof magnetization dynamics.\nIntroduction — Identification of energy-efficient meth-\nods for controlling magnetization is both fundamentally\ninteresting and technologically relevant, e.g., in the de-\nvelopment of magnetic memory devices. While magne-\ntization switching in magnetic recording is convention-\nally achieved by applying a static external magnetic field\nopposite to the initial magnetization direction, previous\nstudies have demonstrated that the energy cost of this\nprocess can be reduced by applying time-varying stimuli,\nsuch as a microwave magnetic field [1–5]. For a uniaxial\nmonodomainparticle, theoptimalmagnetizationreversal\nisachievedbyarotatingmagneticfieldsynchronizedwith\nthe precessional dynamics of the magnetic moment [6–9].\nThe assessment of the stability of energy-efficient\nswitching protocols with respect to ever-present thermal\nfluctuations is an important problem. The thermal fluc-\ntuations perturb the phase locking between the magneti-\nzation and the external stimulus. As a result, the magne-\ntization switching can be compromised unless the energy\nbarrier between the initial and final states is much larger\nthan the thermal energy, and the switching time does\nnot exceed a few periods of Larmor precession [9]. This\nposes a challenge for the realization of energy-efficient\nswitching protocols at elevated temperatures, such as a\ncombination of a microwave and heat-assisted technique.\nEven at low temperatures, the perturbations in the dy-\nnamics can accumulate in time potentially leading to de-\ncoherence between the magnetization and the microwave\npulse for relatively slow switching which is required for\nthe autoresonance-based protocols [10]. In general, the\nassessmentandcontrolofdynamicalstabilityofmagnetic\nsystems is a crucial problem [11, 12].\nIn this work, we demonstrate that thermal stability of\nmagnetization switching in nanoparticles is mostly de-\nfined by unstable perturbations arising in a certain do-\nmain of the configuration space of the system. The in-\nstabilities can be suppressed by application of a longi-tudinal magnetic field, which provides a mechanism for\nenhancing the thermal stability of optimal magnetization\nswitching induced by rotating magnetic field. We show\nthat the success rate of the switching for a given temper-\nature and switching time can be tuned by adjusting the\nstrength of the stabilizing field. Our results provide a\nperspective on the control of dynamical stability of mag-\nnetic systems subject to thermal fluctuations.\nModel and spin dynamics simulations — We con-\nsider energy-efficient magnetization switching of a uni-\naxial monodomain nanoparticle characterized by nor-\nmalized magnetic moment ⃗ sand internal energy E=\n−K(⃗ s·⃗ ez)2/2, with unit vector ⃗ ezbeing the direction\nandK > 0beingthestrengthofthemagneticanisotropy.\nThe switching is induced by an optimal pulse of a rotat-\ning magnetic field ⃗B0(t)that, for a given switching time,\nminimizes the energy cost of switching [see Ref. [9] for\nthe exact time dependence of ⃗B0(t)as a function of pa-\nrameters of the nanoparticle]. The switching dynamics\nis simulated by the time integration of the the Landau-\nLifshitz-Gilbert (LLG) equation:\n(1 +α2)˙⃗ s=−γ⃗ s×⃗Beff−αγ⃗ s×\u0010\n⃗ s×⃗Beff\u0011\n,(1)\nwhere the dot denotes the time derivative, αis the\nGilbert damping factor, γis the gyromagnetic ratio, and\n⃗Beff≡⃗B0+⃗b+⃗ξis the effective field that, in addi-\ntion to the switching pulse, includes the internal field\n⃗b=−µ−1∂E/∂⃗ s =µ−1K(⃗ s·⃗ ez)⃗ ez, with µbeing the\nmagnitude of the magnetic moment, and the stochastic\nterm ⃗ξmimicking interaction of the system with the heat\nbath[13]. Eachsimulationinvolvesthreestages[9,14]: i)\nInitialization of the magnetic moment close to the energy\nminimum at sz= 1and equilibration of the system at\nzero applied magnetic field to establish local Boltzmann\ndistribution at the initial state; ii) Switching where the\noptimal magnetic field is applied; iii) Final equilibrationarXiv:2312.11293v1  [cond-mat.mes-hall]  18 Dec 20232\nFigure 1. Calculated dynamics of the magnetic moment\nfor a uniaxial nanoparticle induced by the optimal switching\nmagnetic field. The black line shows the zero-temperature\ntrajectory of the magnetic moment which corresponds to the\noptimal control path ⃗ s0(t)for the magnetization switching.\nThe green (red) line shows the trajectory for successful (un-\nsuccessful) switching at finite temperature corresponding to\nthe thermal stability factor ∆ = 20. Labels A and B show\npositions of the magnetic moment for which the dynamics of\nlocal perturbations in the magnetization is illustrated in the\ncorresponding insets of Fig. 2. The light red (blue) shaded\narea marks the domain where the perturbation dynamics is\nunstable (stable). The damping factor αis0.2, the switching\ntime Tis5τ0.\nat zero applied magnetic field. The switching is consid-\nered successful if the system is close to the reversed state\natsz=−1at the end of the simulation. Proper statistics\nof switching is obtained by repeating simulations multi-\nple times.\nAt zero temperature, the switching trajectory corre-\nsponds to optimal control path (OCP) ⃗ s0(t)between\nthe energy mimima of the system (see the black line in\nFig. 1): magnetic moment rotates steadily from the ini-\ntial state at sz= 1to the final state at sz=−1and si-\nmultaneouslyprecessesaroundtheanisotropyaxis,where\nthe sense of precession changes at the top of the energy\nbarrier. Magnetization dynamics is synchronized with\nthe switching field so that ⃗ s0is always perpendicular to\n⃗B0.\nAt nonzero temperature, thermal fluctuations perturb\nthemagnetizationdynamicsmakingtheswitchingtrajec-\ntory deviate from the OCP (see the green line in Fig. 1).\nThe deviation can become so large that the phase locking\nbetween the switching pulse and the magnetic moment\nis lost which may eventually prevent the magnetizationreversal (see the red line in Fig. 1).\nThe success rate of switching depends strongly on the\nswitching time Tand the strength of thermal fluctua-\ntions, which can be quantitatively described by the ther-\nmal stability factor ∆– the ratio between the energy bar-\nrier separating the stable states and the thermal energy.\nFor∆≳70, which is a standard case for magnetic mem-\nory elements [15, 16], and relatively fast switching with\nT≲10τ0, the switching success rate is close to unity [9].\nHowever, the success rate becomes 0.85for∆ = 20and\nT= 10τ0, and further decreases with decreasing ∆. Fur-\nthermore, an increase in the switching time leads to a\nhigher chance for perturbations in the magnetization dy-\nnamics to accumulate, thereby increasing the likelihood\nof unsuccessful switching even for large thermal stabil-\nity factors. For example, the switching success rate is\n0.7for∆ = 70 andT= 30 τ0. These effects make it\nproblematic to realize energy-efficient protocols involv-\ning multiple precessions around the anisotropy axis, es-\npecially at elevated temperature [9]. In the following,\nwe analyze the local dynamics of perturbations to gain\ninsight into the mechanism of decoherence between the\nswitching pulse and the magnetic moment. This anal-\nysis ultimately reveals a method to control the thermal\nstability of magnetization switching.\nLocal dynamics of perturbations — The interaction of\nthe nanoparticle with the heat bath results in the per-\nturbed trajectory: ⃗ s(t) =⃗ s0(t) +δ⃗ s(t). If the perturba-\ntionbecomestoolarge,thecoherencebetweentheswitch-\ning pulse and the magnetic moment will be lost resulting\nin a failed switching attempt (see red trajectory in Fig.\n1). Therefore, the dynamical stability of the system can\nbe investigated by analyzing the time evolution of the\nperturbation δ⃗ s(t). Linearization of Eq. (1) leads to the\nfollowing equation of motion for the perturbation:\n1 +α2\nγ˙⃗ ϵ(t) =\u0014−α−1\n1−α\u0015\n·\u0014w10\n0w2\u0015\n·⃗ ϵ(t).(2)\nHere, ⃗ ϵ(t) = (ϵ1, ϵ2)Tisthetwo-dimensionalvectorwhose\ncomponentsarethecoordinatesof δ⃗ sinthetangentspace\nof⃗ s0(t)defined by the eigenvectors of the Hessian of the\nenergy of the system [17], and w1,w2are the Hessian’s\neigenvalues given by the following equations:\nw1=Br+K\nµcos (2 θ), (3)\nw2=Br+K\nµcos2θ, (4)\nwhere θis the polar angle of ⃗ s0andBris the component\noftheexternalmagneticfieldparallelto ⃗ s0. Interestingly,\nlocal dynamics of the perturbations does not depend ex-\nplicitly on the optimal switching pulse, for which Br= 0.\nFor zero damping, Eq. (2) predicts two types of dy-\nnamical trajectories for the perturbation depending on3\nFigure 2. Diagram classifying dynamics of perturbations in\nthe magnetization. The green, blue, and red lines show how\nthe Hessian’s eigenvalues w1andw2change along the zero\ntemperature reversal trajectory (see the black line in Fig. (1)\nfor three values of the longitudinal magnetic field as indicated\nin the legend. The right end of the lines correspond to the\ninitial and the final states at the energy minima, while the\nleft end of the lines corresponds to the top of the energy bar-\nrier. The gray shaded area marks the domain of possible w1,\nw2. Lables A-D indicate pairs of the eigenvalues for which\nthe velocity diagrams illustrating the perturbation dynamics\nare shown in the insets. The background color in the insets\nsignify whether the amplitude of the perturbation is increas-\ning (blue), decreasing (red), or constant (gray). The damping\nfactor αis0.2.\nthe sign of w1w2. The trajectories are elliptic, bound for\nw1w2>0. For the optimal switching pulse with Br= 0,\nthis regime is realized in the vicinity of the energy min-\nima for θ < π/ 4andθ > 3π/4(see the blue regions\nin Fig. 1). However, the perturbation trajectories be-\ncome hyperbolic, divergent for π/4≤θ≤3π/4where\nw1w2≤0(see the red region in Fig. 1). It is important\ntorealizethatfor α= 0thetrajectoriesareequallystable\nregardless of whether both w1andw2are positive or neg-\native. Situation changes with non-zero damping: for pos-\nitivew1,w2, theperturbationstendtorelaxtoward ⃗ s0(t),\nwhile for negative w1,w2, the relaxation amplifies the\nperturbations. In principle, the latter case is unstable.\nHowever, this instability is expected not to significantly\naffect the magnetization switching if the switching time\nis short on the time scale of relaxation dynamics which is\ndefined by the damping parameter α. We conclude that\nthe hyperbolic instabilities in the perturbation dynamics\nare the primary reason for the decoherence between the\nmagnetization and the switching pulse. These instabili-\nties ultimately define thermal stability of magnetization\ndynamics.\nThe diagram in Fig. 2 shows evolution of w1andw2during magnetization switching. For zero Br, a signif-\nicant part of the switching trajectory lies in the region\nof unstable perturbations corresponding to the second\nquadrant of the diagram where the eigenvalues w1and\nw2have different signs. However, the values of w1and\nw2can be controlled by application of the longitudinal\nfield Br. In particular, the hyperbolic instabilities can\nbe removed by shifting w1andw2either to the first\n(Br> K/µ) or to the third ( Br<−K/µ) quadrant\nof the diagram in Fig. 2. Therefore, the longitudinal ex-\nternal magnetic field can be used as a control parameter\nto improve thermal stability of magnetization switching.\nThis conclusion is confirmed in the following by direct\nsimulations of magnetization dynamics at elevated tem-\nperature ( ∆ = 20), where the switching is induced by a\nmodified pulse ⃗B(t):\n⃗B(t) =⃗B0(t) +Br⃗ s0(t). (5)\nEffect of longitudinal magnetic field on the success rate\nof magnetization switching — Figure 3 shows calculated\nsuccess rate of switching as a function of Brfor vari-\nous values of the switching time and damping param-\neter. As predicted, the switching success rate reaches\nunity for Br> K/µregardless of the damping factor α\nand switching time T. Longer switching times require\nstronger longitudinal field to reach a certain value of the\nsuccess rate, as expected, but the threshold value of Br\nis not very sensitive to the damping parameter. Interest-\ningly, the success rate as a function of the longitudinal\nfield exhibits a minimum at Br≈0.5that becomes more\npronounced for longer switching times. At Br= 0.5,\nthe ratio between the eigenvalues becomes w1/w2=−1\nat the top of the energy barrier. This corresponds to\nparticularly unstable perturbations in the magnetization\ndynamics, therefore explaining the drop in the success\nrate of switching. The longer the switching time, the\nmore time the system spends in the vicinity of the en-\nergy barrier [9]. This increases the chances of decoher-\nence between the magnetization and the switching pulse,\nand lowers the switching probability.\nApplication of the longitudinal field opposite to ⃗ s0\n(Br<0) renders both of the eigenvalues w1,w2nega-\ntive near the energy barrier, thus altering the hyperbolic\ncharacter of the perturbation dynamics. As a result, the\nsuccessrateofswitchinginitiallyincreaseswithrising Br.\nHowever, further increases in Brlead to the success rate\nreaching a maximum value before eventually declining\n(see Fig. 3). The drop in the success rate is a conse-\nquence of divergent dynamics due to relaxation, which\nbecomes more prominent for larger damping parameters\nand longer switching times, as expected.\nThe switching dynamics is further illustrated by Fig. 4\nshowing the calculated distribution of the copies of the\nsystem in the statistical ensemble at t=T/2forα= 0.1,\nT= 10 τ0, and various values of Br. For the unper-4\nFigure 3. Calculated success rate of magnetization reversal as a function of the longitudinal magnetic field Brfor various\nvalues of the switching time T(a) and the damping parameter α(b). In (a), α= 0.1; In (b), T= 10τ0. The thermal stability\nfactor ∆ = 20. The shaded areas around the curves indicate the statistical error.\nFigure 4. Calculated distribution of the copies of the system in the statistical ensemble at t=T/2and various values of the\nlongitudinal magnetic field, superimposed on the Lambert azimuthal projection [18] of the energy surface of the system. The\ngreen dots correspond to the copies that will eventually reach the final state at −Z(successful switching), while the red dots\nmark the copies that will end up at the initial state at +Z(unsuccessful switching). The black line shows the calculated OCP\nfor the reversal. The damping factor αis0.1, the thermal stability factor ∆is20, and the switching time Tis10τ0.\nturbed OCP, the system is at the top of the energy bar-\nrier. Thermal fluctuations make the system deviate from\nthe OCP. For zero longitudinal field, the system copies\nspread quite far, with those corresponding to unsuccess-\nful switching trajectories grouped closer to the initial\nstate. For Br= 0.5K/µ, the distribution of the copies\nbecomes more elongated – the result of the hyperbolic\ncharacter of the perturbation dynamics at the energy\nbarrier – and the number of the unsuccessful trajecto-\nries increases. As Brincreases beyond K/µ, a progres-\nsively tighter grouping of the copies around the OCP isobservedduetotheconvergentdynamicsoftheperturba-\ntions, resulting in the switching probability approaching\nunity (see Fig. 3).\nFor negative Br, the copies of the system are grouped\nin an ellipse around the OCP even for Br=−0.5K/µ.\nFor stronger anti-parallel fields, the spread of the distri-\nbution increases due to relaxation, resulting in a decrease\nin the success rate of switching.\nFigure 5 shows the calculated dependencies of the suc-\ncess rate on the damping constant αand switching time\nTforBr= 0andBr=±K/µ. Bothcaseswithfinitelon-5\nFigure 5. (a) Calculated success rate of magnetization reversal as a function of damping parameter αfor switching time\nT= 10τ0. (b)-(c) Calculated success rate as a function of Tforα= 0.1andα= 0.2. The red, blue, and black lines correspond\nto the three values of the longitudinal magnetic field Bras indicated in the legend. The thermal stability factor ∆ = 20. The\nshaded areas around the curves indicate the statistical error.\ngitudinal field ensure w1w2≥0for the whole switching\ntrajectory. Positive (negative) Brcorrespond to conver-\ngent (divergent) relaxation of the perturbation dynam-\nics, which explains monotonic increase (decrease) of the\nswitchingprobabilitywithincreasing α. However, forlow\ndampingandshortswitchingtimes, applyingthelongitu-\ndinal field opposite to the magnetic moment ( Br<0) is\nmore efficient than applying the longitudinal field along\nthe magnetic moment ( Br>0), as it requires lower fields\nto achieve high success rates (see also Fig. 3). Longer\nswitching times result in lower success rate in all con-\nsidered cases, as expected. The decrease in the success\nrate with Tbecomes more (less) pronounced for negative\n(positive) Bras damping increases, which is a result of\ndestabilizing (stabilizing) effect of relaxation.\nConclusions — In this work, we uncovered that the\ninstability of energy-efficient protocols for magnetization\nreversal in nanoparticles with respect to thermal fluctu-\nations originates from the divergent magnetization dy-\nnamics arising around the top of the energy barrier of\nthe system. We demonstrated that these instabilities can\nbe eliminated by applying an additional magnetic field\neither aligned or opposed to the magnetic moment’s di-\nrection, consequently enhancing the thermal stability of\nmagnetization switching. We examined the success rate\nof switching at elevated temperatures as a function of\nvarious control parameters, such as the switching time,\nGilbert damping, and the magnitude of the longitudi-\nnal field. The application of a longitudinal field along\nthe magnetic moment consistently increases the success\nrate of switching, provided that the field magnitude sur-\npasses the characteristic anisotropy field. However, for\nshorter switching times and weaker damping, employing\na smaller field opposed to the magnetic moment can also\naugment the success rate. Our results warrant a general\nprinciple for improved control of magnetization dynamicsby suppressing divergent perturbations.\nAcknowledgments — This work was supported by\nthe Icelandic Research Fund (Grant Nos. 217750 and\n217813), the University of Iceland Research Fund (Grant\nNo. 15673), and the Swedish Research Council (Grant\nNo. 2020-05110).\n∗Corresponding author: pavel.bessarab@lnu.se\n[1] C. Thirion, W. Wernsdorfer, and D. Mailly, Nature Ma-\nterials 2, 524 (2003).\n[2] Z. Z. Sun and X. R. Wang, Phys. Rev. B 74, 132401\n(2006).\n[3] K. Rivkin and J. B. Ketterson, Appl. Phys. Lett. 89,\n252507 (2006).\n[4] G. Woltersdorf and C. H. Back, Phys. Rev. Lett. 99,\n227207 (2007).\n[5] L. Cai, D. A. Garanin, and E. M. Chudnovsky, Phys.\nRev. B 87, 024418 (2013).\n[6] Z. Z. Sun and X. R. Wang, Phys. Rev. Lett. 97, 077205\n(2006).\n[7] N.Barros, M.Rassam, H.Jirari,andH.Kachkachi,Phys.\nRev. B 83, 144418 (2011).\n[8] N. Barros, H. Rassam, and H. Kachkachi, Phys. Rev. B\n88, 014421 (2013).\n[9] G. J. Kwiatkowski, M. H. A. Badarneh, D. V. Berkov,\nand P. F. Bessarab, Phys. Rev. Lett. 126, 177206 (2021).\n[10] G. Klughertz, P.-A. Hervieux, and G. Manfredi, Journal\nof Physics D: Applied Physics 47, 345004 (2014).\n[11] D. V. Berkov, IEEE transactions on magnetics 38, 2489\n(2002).\n[12] X. Wang, Y. Zheng, H. Xi, and D. Dimitrov, Journal of\nApplied Physics 103, 034507 (2008).\n[13] D. V. Berkov, Magnetization dynamics including ther-\nmal fluctuations: Basic phenomenology, fast remagneti-\nzation processes and transitions over high-energy barri-\ners, in Handbook of Magnetism and Advanced Magnetic\nMaterials , Vol. 2 (John Wiley & Sons, Ltd, 2007) pp.6\n795–823.\n[14] M. H. Badarneh, G. J. Kwiatkowski, and P. F. Bessarab,\nPhysical Review B 107, 214448 (2023).\n[15] H. Richter, Journal of Magnetism and Magnetic Materi-\nals321, 467 (2009), current Perspectives: Perpendicular\nRecording.\n[16] M. Krounbi, V. Nikitin, D. Apalkov, J. Lee, X. Tang,\nR. Beach, D. Erickson, and E. Chen, ECS Transactions69, 119 (2015).\n[17] A. S. Varentcova, S. von Malottki, M. N. Potkina,\nG. Kwiatkowski, S. Heinze, and P. F. Bessarab, npj Com-\nputational Materials 6, 193 (2020).\n[18] J. P. Snyder, Map projections: A working manual , Tech.\nRep. (U.S. Geological Survey, Washington, D.C., 1987)\nreport."    },    {        "title": "1508.01532v1.Near_Sun_Speed_of_CMEs_and_the_Magnetic_Non_potentiality_of_their_Source_Active_Regions.pdf",        "content": "GEOPHYSICAL RESEARCH LETTERS, VOL. ???, XXXX, DOI:10.1002/,\nNear-Sun Speed of CMEs and the Magnetic Non-potentiality of\ntheir Source Active Regions\nSanjiv K. Tiwari1, David A. Falconer1,2, Ronald L. Moore1,2, P. Venkatakrishnan3,\nAmy R. Winebarger1, and Igor G. Khazanov2\nWe show that the speed of the fastest coronal mass\nejections (CMEs) that an active region (AR) can produce\ncan be predicted from a vector magnetogram of the AR.\nThis is shown by logarithmic plots of CME speed (from\nthe SOHO LASCO CME catalog) versus each of ten AR-\nintegrated magnetic parameters (AR magnetic \rux, three\ndi\u000berent AR magnetic-twist parameters, and six AR free-\nmagnetic-energy proxies) measured from the vertical and\nhorizontal \feld components of vector magnetograms (from\ntheSolar Dynamics Observatory's Helioseismic and Mag-\nnetic Imager ) of the source ARs of 189 CMEs. These plots\nshow: (1) the speed of the fastest CMEs that an AR can\nproduce increases with each of these whole-AR magnetic\nparameters, and (2) that one of the AR magnetic-twist pa-\nrameters and the corresponding free-magnetic-energy proxy\neach determine the CME-speed upper-limit line somewhat\nbetter than any of the other eight whole-AR magnetic pa-\nrameters.\n1. Introduction\nOne of the most challenging tasks in the \feld of modern\nspace research is the prediction of the severity of geomag-\nnetic storms and solar energetic particle storms caused by so-\nlar \rares and coronal mass ejections (CMEs). Active regions\n(ARs) on the Sun are the main sources of the biggest \rares\nand most energetic CMEs [ Zirin and Liggett , 1987; Subra-\nmanian and Dere , 2001; Falconer et al. , 2002; Venkatakrish-\nnan and Ravindra , 2003; Guo et al. , 2007; Wang and Zhang ,\n2008; Gopalswamy et al. , 2010]. The initial speed of CMEs\nis one of the most important parameters that (among oth-\ners e.g., the direction, width and mass of CMEs, orientation\nand strength of magnetic \feld therein) can help forecast the\nseverity of geomagnetic storms and particle storms [see e.g.,\nSrivastava and Venkatakrishnan , 2002; Gopalswamy et al. ,\n2010; Dumbovi\u0013 c et al. , 2015, and references therein]. The\nmagnetic non-potentiality of an AR, inferred by, for in-\nstance, free magnetic-energy proxies and magnetic-twist pa-\nrameters, is most likely to determine the initial speed of\nCMEs emanating from the AR. Several other unexplored\nparameters e.g., AR lifetime, \rux emergence/cancellation\n1NASA Marshall Space Flight Center, ZP 13, Huntsville,\nAL 35812, USA.\n2Center for Space Plasma and Aeronomic Research,\nUniversity of Alabama in Huntsville, Huntsville, AL 35805,\nUSA.\n3Udaipur Solar Observatory, Physical Research\nLaboratory, Udaipur 313001, India.\nCopyright 2015 by the American Geophysical Union.\n0094-8276/15/$5.00[e.g., Subramanian and Dere , 2001] might be important as\nwell. Therefore, study of the relationship between properties\nof the photospheric magnetic \feld of an AR and the physi-\ncal properties of the CMEs produced by the AR, e.g., their\ninitial speed, is of great importance for forecasting severe\nspace weather.\nVenkatakrishnan and Ravindra [2003] estimated the\npotential-magnetic-\feld energy of 37 ARs from their line-of-\nsight (LOS) magnetograms and found it to be a reasonable\npredictor of the speed of CMEs arising from the ARs. The\npresent paper reports a similar but more extensive investiga-\ntion based on vector magnetograms instead of LOS magne-\ntograms. Liu[2007] studied 21 halo CMEs and found a pos-\nitive correlation of free magnetic energy of ARs with CME\nspeed. CME speed is also found to be correlated with the\nGOES X-ray magnitude of the co-produced \rare [ Ravindra ,\n2004; Burkepile et al. , 2004; Vr\u0014 snak et al. , 2005; Gopalswamy\net al. , 2007; Bein et al. , 2012]. Tiwari et al. [2010] found a\ngood correlation between a twist parameter [spatially aver-\naged signed shear angle: Tiwari et al. , 2009a] of ARs and the\nGOES X-ray magnitude of \rares produced by the ARs. A\ncomparison of results from Tiwari et al. [2010] and Jing et al.\n[2010] suggests that this global twist parameter is strongly\ncorrelated with the free magnetic energy of ARs. From the\nabove, one expects twist parameters and free-energy proxies\nto be determinants of the speed of the CMEs from an AR,\nand therefore determinants of the severity of the resultant\ngeomagnetic storms, based on the results of Srivastava and\nVenkatakrishnan [2002]. In the present analysis, we investi-\ngate the relationship between magnetic parameters of ARs\n(mainly various twist parameters and free-energy proxies)\nand the initial speed of CMEs arising from the ARs.\nMajor CMEs emanating from ARs are co-produced with a\n\rare [ Yashiro et al. , 2008; Wang and Zhang , 2008; Schrijver ,\n2009]. Although several investigations have focused on pre-\ndicting the \rares from ARs by measuring various magnetic\nnon-potentiality parameters [see e.g., Hagyard et al. , 1984;\nCan\feld et al. , 1999; Falconer et al. , 2002; Georgoulis and\nRust, 2007; Leka and Barnes , 2007; Falconer et al. , 2009;\nMoore et al. , 2012; Falconer et al. , 2014; Bobra et al. , 2014;\nBobra and Couvidat , 2015], a direct link of any of these\nmagnetic parameters to CME parameters have not been es-\ntablished thus far. To establish such a relationship requires\n1. a careful manual inspection of which CME comes from\nwhich AR, 2. analysis of vector magnetograms of source\nARs within 45 heliocententric degrees of disk center. In the\npresent work, we \frst generated a list of a large number of\nCMEs that were observed by the SOHO LASCO/C2 coro-\nnagraph and were identi\fed with \rares in ARs observed by\nthe Solar Dynamics Observatory (SDO). We then manually\ninspected those CMEs to \fnd the CMEs that came from\na clearly identi\fed source AR or sometimes two neighbor-\ning ARs. We then calculated di\u000berent twist parameters and\nfree-energy proxies using vector magnetograms from SDO's\nHelioseismic and Magnetic Imager [HMI: Schou et al. , 2012;\nHoeksema et al. , 2014], and studied their relationships to ini-\ntial CME speeds collected from the LASCO/CME catalog\n[Gopalswamy et al. , 2009].\n1X - 2 TIWARI ET AL.: SPEED OF CMES VS NON-POTENTIALITY OF ACTIVE REGIONS\n2. Event Selection and Data Analysis\nFirst, we determined from the online LASCO/CME cat-\nalog ( http://cdaw.gsfc.nasa.gov/CME_list/ ) all CMEs\nthat took place between the start of the SDO mission (May\n2010) through March 2014 (as far as the LASCO/CME cat-\nalog covered at the time of our analysis). We identi\fed all\nCMEs that had a plane-of-sky width greater than 30 de-\ngrees, and had a co-produced \rare in an AR identi\fed by\nNOAA. Further, the \raring AR had to be between 45E to\n45W, and the \rare occurred ( tflare) up to 2 hours before the\nrecorded start time till half an hour after the recorded start\ntime of the CME ( tcme) in images from the LASCO/C2\ncoronagraph ( tcme\u00002hrs< tflare< tcme+ 30 min). The\nbroad window for automatic selection was chosen, so as to\nnot accidentally eliminate a CME/\rare combination before\nwe manually checked it. We found 946 CMEs following our\ncriteria during the given time period of observation by SDO.\nFor each of the 946 automatically selected CMEs, we manu-\nally veri\fed that: (1) the CME was not seen in the LASCOC2 before the \rare took place, (2) the CME occurred in the\nsame quadrant as the source AR, and (3) there was no sec-\nond \rare occurring in another AR at nearly the same time.\nIf there was a second \raring AR, we further veri\fed that it\nwas not the source of the CME under investigation.\nBy looking at LASCO-C2 movies and GOES X-ray \rux\nplots, we made sure that the prospective \raring source AR\nwas present on the frontside of the Sun. By looking at\nSTEREO A & B movies we ensured that the CME was\ndirected towards Earth; it did not come from a source on\nthe back of the Sun. Far-side CMEs were discarded. We\nthen used AIA 193 \u0017A movies to determine which AR \rare\n(out of sometimes several listed) was co-produced with the\nCME under investigation. The selection procedure for an\nexample CME is illustrated in Figure 1. The movies (both\nfor STEREO A & B and AIA 193) for the example event,\nshown in Figure 1, can be found at: http://cdaw.gsfc.\nnasa.gov/CME_list/daily_movies/2012/07/12/ .\n11520\t\r \nFigure 1. Images illustrating how we verify that a CME comes from an AR or group of ARs. The top two images are\nLASCO C2 images, the \frst one during the rise of the source AR \rare, and the second one when the CME was clearly\nvisible outside the C2 occulting disk. A corresponding GOES X-ray plot is shown in top right. STEREO-A (center) and\nB (left) images in the bottom row verify that the CME is Earthward directed. Last panel is an image taken from the AIA\n193\u0017A movie, verifying that the position and NOAA number of the AR responsible for the CME are correct.\nThis careful manual selection procedure left a sample\nof 252 CMEs, with known \raring source ARs. The sam-\nple was further reduced by the requirement that there\nwas available a de\fnitive HMI AR Patch (HARP) vec-tor magnetogram that covered the source AR, which\nwas taken within 12 hours of the CME \rare, and which\nhad its magnetic \rux centroid (de\fned below) within\n45 heliocentric degrees of disk center. We also required\nthat the source AR (1) be the only NOAA AR in theTIWARI ET AL.: SPEED OF CMES VS NON-POTENTIALITY OF ACTIVE REGIONS X - 3\nHARP tile, (2) the values of parameters are mostly\n(\u001590%) from the AR and only negligibly ( \u001410%) from\nthe other parts of the tile, or (3) if many ARs exist in\nthe HARP, they are closely merged together and can\nbe treated as one AR. This left a sample of 189 CMEs\nthat we \fnally kept for our study.\nWe use the HARP vector magnetograms, which\nhave better azimuthal disambiguation than the Space-\nWeather HMI AR Patches [SHARPs Bobra et al. , 2014].\nFrom the HARP vector magnetograms, we measured\nthe magnetic parameters described in the next section.\nThe magnetograms have a pixel size of 0.500and a ca-\ndence of 12 minutes. We prefer HARP over SHARP\nbecause our purpose in this paper is to look for any\nrelationship between magnetic parameters and speedof CMEs that might lead to improvements in ongo-\ning/future forecasting tools e.g., MAG4 [ Falconer et al. ,\n2014], in contrast to the aim of devising a near real time\ntool for forecasting the speed of CMEs.\nEach HARP was deprojected to disk center, i.e., LOS\nand transverse vector components were transformed to\nvertical and horizontal vector components and resam-\npled to square pixels. Noise from transverse \feld and\nforeshortening is prohibitive when ARs are far from\ndisk center. Therefore, we limit our sample to HARPs\nwithin 45 heliocentric degrees. Falconer et al (2015,\nin preparation) lays out the center-to-limb increase in\ndeprojection errors in detail and shows that the param-\neters studied here have acceptably small projection er-\nrors out to 45 heliocentric degrees.\nFigure 2. An example de-projected HARP-tile vector magnetogram containing NOAA AR 11520, which produced the\nX-class \rare and CME shown in Figure 1. The size and direction of red vectors, overplotted on the grey-scaled vertical-\feld\nmagnetogram, show the magnitude and direction of the horizontal \feld. The longest/shortest vector is for 500/100 G \feld\nstrength.\nIn Figure 2, we display an example of a deprojected\nvector magnetogram tile. We have reduced noise in the\nmeasured parameters by using only pixels where the\n\feld components are above certain threshold values (see\nnext Section).\nThe CME speeds have been obtained from the online\nLASCO/CME catalog [ Gopalswamy et al. , 2009]. We\nuse the speeds that are obtained from the linear \fts to\nthe height-time plot of the CME front in the plane ofthe sky [ Yashiro et al. , 2004; Gopalswamy et al. , 2009].\nThe uncertainty in this measurement of the speed is\nless than 10% [ Yashiro et al. , 2004; Gopalswamy et al. ,\n2012, Yashiro, 2015, private communication]. This is\napart from the basic de\fciency of using 2-D images\nin contrast to using speeds calculated from 3-D recon-\nstruction of the CMEs [e.g., Joshi and Srivastava , 2011;\nMishra and Srivastava , 2013]. The di\u000berence between\nthe true speed and the measured plane-of-sky speed is\nfound to be more for the CMEs with smaller widthsX - 4 TIWARI ET AL.: SPEED OF CMES VS NON-POTENTIALITY OF ACTIVE REGIONS\n[e.g., Gopalswamy et al. , 2012, Yashiro, 2015, private\ncommunication]. As mentioned before, our CMEs are\nwider than 30\u000e, therefore not exposed to larger error\nfrom projection on the plane-of-sky. We have included\nboth halo and non-halo CMEs to investigate the gen-\neral correspondence between the speed of CMEs and the\nmagnetic non-potentiality of the source ARs. Because\nthe CME speeds used are plane-of-sky speeds, they are\nsmaller than the true speeds of the CME fronts. It\nis worth mentioning here that the estimation of true\nspeeds of Earth-directed CMEs is di\u000ecult. In a case\nstudy, by using stereoscopic observations, Gopalswamy\net al. [2012] found the plane-of-sky speed measured by\nLASCO to be smaller by only 7.6% and 3.4% than\nthe plane-of-sky speeds measured by STEREO-A and\nSTEREO-B, respectively.\n3. AR Magnetic Parameters Studied\n3.1. AR Size Parameters\nWe use two AR size parameters. Both are integrals\nof all pixels that have absolute vertical magnetic \feld\nstrengthBzgreater than 100 G. The \frst is the total\nmagnetic area A,\nA=Z\ndA; (1)\nand the second is the total magnetic \rux \b,\n\b =Z\nBzdA: (2)\n3.2. Length of Strong-Field Neutral Line\nThe strong-\feld neutral-line length of an AR is de-\n\fned by\nLS=Z\ndl; (3)\nwhere the integral is over all intervals of neutral lines\nin which the horizontal component of the potential \feld\nis greater than 150 G, and the interval separates oppo-\nsite polarities of at least 20 G \feld strength [ Falconer\net al. , 2008]. These neutral-line intervals are used for\nthe two other neutral-line-length parameters, which are\nfree-energy proxies, described in Section 3.4.\nTo avoid magnetic parameters being dominated by\nnoise, in this study we measure only ARs that are what\nwe de\fne to be strong-\feld ARs. Our de\fnition follows\nFalconer et al. [2009]: a strong-\feld AR is one for which\nthe ratio of L Sto the square root of the magnetic area\nA is greater than 0.7.\n3.3. Global Twist Parameters\nGlobal Alpha ( \u000bg):The magnetic twist parameter\n\u000bmeasures the vertical gradient of magnetic twist (ra-\ndians of twist per unit length of height) in each pixel of\na deprojected AR vector magnetogram [see AppendixA of Tiwari et al. , 2009b]; see also Leka and Skumanich\n[1999]. A global value of \u000bcan be calculated using the\nfollowing formula [e.g., Tiwari et al. , 2009b]:\n\u000bg=P(@By\n@x\u0000@Bx\n@y)BzPB2z(4)\nWe use this direct way of obtaining global \u000bbe-\ncause the singularities at neutral line are automatically\navoided in this method by using the second moment\nof minimization. Only pixels with absolute B zgreater\nthan 100 G are included in \u000bg.\nSigned Shear Angle (SSA): Motivated by the pres-\nence of oppositely directed twists at small-scales in\nsunspot penumbrae, Tiwari et al. [2009a] proposed\nSSA, which measures magnetic twist in ARs irrespec-\ntive of their force-free nature [ Tiwari et al. , 2009a] and\nshape [ Venkatakrishnan and Tiwari , 2009]. It can be\ncomputed for each pixel of the deprojected vector mag-\nnetograms from the following formula:\nSSA = tan\u00001(ByoBxp\u0000BypBxo\nBxoBxp+ByoByp) (5)\nwhereBxo,ByoandBxp,Bypare the observed and\npotential horizontal components of sunspot magnetic\n\felds, respectively. The potential \feld is calculated\nfrom the vertical magnetic \feld using the method of\nAlissandrakis [1981]. Only pixels with absolute B z\ngreater than 100 G are used.\nSpatially averaged SSA (SASSA) and the median of\nSSA (MSSA) are each a global magnetic twist param-\neter of an AR. The di\u000berence between the two is the\nfollowing: noisy pixels contribute directly to SASSA,\nwhereas they are least weighted for MSSA. Therefore\nwe have treated MSSA as a third global twist parame-\nter here.\nThe SASSA and MSSA are both signed parameters;\nhowever in the present study only magnitude is taken\ninto account.\n3.4. AR Free-energy Proxies\nGradient-weighted Neutral Line Length (WL SG):\nintroduced by Falconer et al. [2008], this neutral line\nlength measure is de\fned as\nWLSG=Z\njrBzjdl (6)\nwherejrBzjis the horizontal gradient of the vertical\nmagnetic \feld. The integral is computed for all neutral-\nline intervals that separate opposite polarities of at least\nmoderate \feld strength of 20 G and have horizontal po-\ntential \feld greater than 150 G. Please note that these\ncuto\u000b values are based on those taken by Falconer et al.\n[2008] for MDI data, and smaller numbers can be chosenTIWARI ET AL.: SPEED OF CMES VS NON-POTENTIALITY OF ACTIVE REGIONS X - 5\nfor the HMI data. However, to be on the safe side, we\nhave kept the same cuto\u000b values in our present study.\nShear-weighted Neutral Line Length (WL SS):also\nintroduced in Falconer et al. [2008], this parameter is\ngiven by\nWLSS=Z\nj\u001e\u0000\u001epjdl (7)\nwhere\u001eis the azimuth angle of the observed horizontal\nmagnetic \feld, and \u001epis the azimuth angle of the poten-\ntial horizontal magnetic \feld computed from the verti-\ncal magnetic \feld. The two free-energy proxies (WL SG\nand WLSS) are strongly correlated, and are being ex-\nplored in another work to determine which parameter\nis better for \rare prediction.\nSchrijver's-R: Schrijver [2007] developed a free-\nenergy proxy that measures the amount of \rux near\nneutral line pixels. To obtain Schriver's-R, which we\ndenote as R Schr, \frst a neutral-line pixel map is deter-\nmined. This is done by determining all pixels are near\na neutral line and that have positive or negative \rux\ngreater than 150 G. This step identi\fes strong-gradient\nneutral lines. This strong-gradient neutral-line-pixel\nmap is then convolved with a 15 Mm Gaussian (as de-\n\fned in Schrijver [2007] for MDI resolution). R Schr is\nthe unsigned \rux in that area divided by that area, giv-\ning RSchr a unit of G (Gauss). See Schrijver [2007], for\nmore detail.\nNet Current: The vertical current density Jzcan\nbe measured from a deprojected vector magnetogram\nusing the following formula:\nJz=1\n\u00160\u0012@By\n@x\u0000@Bx\n@y\u0013\n: (8)\nAn integration of Jzover all strong-\feld pixels ( jBzj>\n100 G orBh>200 G) of an AR provides the net current\nfor that AR. Following Ravindra et al. [2011], we use\nthe sign convention that positive current \rows upward\nin positive polarity regions, and downward in negative\npolarity regions, with negative current having the oppo-\nsite \row. To obtain the net current Iz, the net current\nin the positive polarity pixels, and the net current in\nthe negative polarity pixels (see Figure 2), are added\nand divided by 2. This is the net current and not the\ntotal current since ARs can easily have, in one part of a\npolarity domain, positive current \rowing and in other\nparts have negative current [e.g., Ravindra et al. , 2011].\nIn addition to the above free-energy proxies, by mul-\ntiplying by AR magnetic \rux \b, we converted the twist\nparameters \u000bg, SASSA and MSSA to the AR free-\nenergy proxies \u000bg\u0002\b, SASSA\u0002\b, MSSA\u0002\b. This is\nmeaningful because an AR with large twist but little\ntotal \rux plausibly does not have enough free-energy\nto produce fast CMEs, but an AR with the same large\ntwist and large-enough \rux plausibly does have enough\nfree-energy to produce fast CMEs.4. Results and Discussion\nIn Figure 3, total unsigned magnetic \rux, three twist\nparameters and six free-energy proxies (two of which\nare combinations of twist and \rux) of ARs are plotted\nagainst the plane-of-sky speed of CMEs emanating from\nthe ARs. For all the ten plots in Figure 3 most data\npoints \fll a triangle portion of the phase space. Dashed\nred lines, drawn by eye following Venkatakrishnan and\nRavindra [2003] (see dashed line in their Figure 3), out-\nline the triangle area in each panel to roughly trace the\nupper bound of the speeds of CMEs.\nThree important features in the plots that determine\nhow well the speed of the fastest CMEs arising from an\nAR can be predicted are 1. the y-intercept of the red\ndashed line, 2. number of outliers above the line, and\n3. how far above in y-direction the outliers are from the\nline. By the y-intercept of the red dashed line in each\nplot of Figure 3, we mean the y value at the point of\nintersection of the dashed line with the y-axis of that\nplot (the left side of the box).\nThe triangular shape of the clouds of plotted points\nshows that the ARs with large non-potentiality and\nlarge total \rux produce both fast and slow CMEs,\nwhereas ARs with the lower non-potentiality and less\n\rux produce only slower CMEs. This behavior is sim-\nilar to the behavior that the most non-potential ARs\ncapable of producing large X-class \rares also produce\nmany smaller M-, and C- class \rares, and ARs with\nrelatively small non-potentiality rarely if ever produce\nlarger \rares [e.g., Tiwari et al. , 2010].\nFor most plots there are a few outliers that are above\nthe line. For two plots, \u000bgand\u000bg\u0002\b, the upper limit\nline is not strongly violated and the upper-limit CME\nvelocity for the smallest of these two parameters (y-\nintercept) is\u0018300 km s\u00001. The number of outliers and\ntheir distance in y-direction from the line is least for\nthese two parameters. The magnetic \rux plot shows a\nlimit line of similar low y-intercept, but has more out-\nliers, which are relatively farther in y-direction above\nthe limit line.\nFor the two better performing parameters, \u000bgand\n\u000bg\u0002\b, the red dashed lines in Figure 3 give:\nv= 10(2:48+0:42\u0002log10(\u000bg=2\u000210\u000010))km s\u00001;and (9)\nv= 10(2:48+0:31\u0002log10(\u000bg\u0002\b=2\u00021012))km s\u00001;respectively :\n(10)\nBecause the lines are drawn by eye, they are not the\nonly ones that could be drawn. By drawing di\u000berent\nlimit lines however we \fnd no improvements in the pre-\ndictive capabilities of the other eight parameters. For\nexample, a less steep slope on FEP4 (Figure 3) can re-\nduce the number of outliers but it also increases the\ny-intercept of the limit line signi\fcantly. Similarly the\nsteepness of line can be increased in plots of the otherX - 6 TIWARI ET AL.: SPEED OF CMES VS NON-POTENTIALITY OF ACTIVE REGIONS\nFigure 3. Scatter plots, in logarithmic scales on both x- and y-axes, of CME speed vs 10 di\u000berent magnetic parameters of\nthe source ARs. The \frst plot is for the AR's magnetic size (total magnetic \rux \b), the next three plots are for whole-AR\nmagnetic twist parameters ( \u000bg, SASSA, MSSA), and the last six plots are for AR free-energy proxies (WL SG, WL SS,\nRSchr, Iz, MSSA \u0002\b,\u000bg\u0002\b).\nseven of the eight parameters but that increases the\nnumber of outliers.\nThe\u000bglimit line has a lower y-intercept and all of\noutliers are as close or closer to the limit line in y-\ndirection, than for other two twist parameters, of which\nMSSA does better than SASSA. The fact that the \u000bg\nis weighted by strong magnetic \feld values and not af-\nfected by singularities at neutral lines [ Tiwari et al. ,\n2009b, a] might be responsible for its superior behavior\nover the other two twist parameters. The MSSA does\nbetter than the SASSA probably because while taking\nmedian, a few noisy pixels with extremely high values\nof SSA are suppressed whereas they contribute more to\nSASSA.The neutral-length free-energy proxies do not directly\ninclude the full area of the ARs [ Falconer et al. , 2008],\nand display some outliers. The same is true for R Schr.\nThe limit line for net current shows a number of out-\nliers. The net current varies from zero to nonzero val-\nues [Venkatakrishnan and Tiwari , 2009; Ravindra et al. ,\n2011; Vemareddy et al. , 2015] in di\u000berent phases of AR's\nlifetime. The evolution of net current could possibly ex-\nplain why this free-energy proxy is not the best for pre-\ndicting the upper speed limit of CMEs in a statistical\nsense.\nThe curent solar cycle has been weak and we do not\nhave many CMEs faster than 1000 kms\u00001in our sample.\nBy extending the sample as more data becomes avail-\nable in the LASCO/CME catalog we will determine ifTIWARI ET AL.: SPEED OF CMES VS NON-POTENTIALITY OF ACTIVE REGIONS X - 7\nthis result is robust, or if the speed limit edge for \u000bg\nand\u000bg\u0002\b in Figure 3 becomes less sharp.\nFrom the results of Venkatakrishnan and Ravindra\n[2003] and Liu[2007], we expect the free-energy prox-\nies to better determine the upper speed limit of CMEs\nthat an AR can produce than twist parameters do. The\nfact that the twist parameter \u000bgdisplays nearly similar\nlimit line as its corresponding free-energy proxy \u000bg\u0002\b,\nis surprising and remains to be explained.\nIn line with our observations, numerical simulations\nalso suggest that the same ARs can produce both fast\nand slow CMEs, with the most complex ones produc-\ning the fastest CMEs [see e.g., T or ok and Kliem , 2007].\nThe origin of slow CMEs from ARs with large non-\npotentiality can probably be explained by the fact that\noftentimes only a part of AR takes part in the eruption\nleading to a CME and the full non-potentiality of the\nAR does not drive those CMEs. However, identifying\nthe exact part of the ARs that produces a \rare/CME is\nnot an easy task due to their complex magnetic struc-\nturing.\nIn the present analysis, we have used free-energy\nproxies instead of computing free magnetic energy it-\nself that requires vector magnetograms measured in the\nforce-free \feld above the photosphere, which are not\navailable owing to instrumental limitations, and also\nto the lack of reliable STOKE's pro\fles inversion codes\nfor NLTE atmospheres. This limitation can be partially\novercomed by reliable non-linear force-free \feld model-\ning [Wiegelmann and Sakurai , 2012; Wiegelmann et al. ,\n2014] based on the photospheric vector \feld measure-\nments of ARs, which are not entirely force-free on the\nAR photosphere [ Metcalf et al. , 1995; Tiwari , 2012] but\ncan be preprocessed to make them force-free under cer-\ntain circumstances. Future studies should make use of\nsuch techniques to improve the accuracy of the predic-\ntion of the upper speed limit of the CMEs that an AR\ncan produce.\n5. Conclusions\nIn this Letter, we investigated the correspondence be-\ntween the speed of CMEs and non-potentiality of their\nsource ARs by using a total of 189 CMEs.\nPlane-of-sky speed of CMEs were taken from the\nSOHO/LASCO CME Catalog. In addition to total un-\nsigned magnetic \rux, various magnetic twist parame-\nters and free energy proxies of the source ARs were\nmeasured to gauge their non-potentiality. To measure\nthese parameters, HARP vector magnetograms from\nHMI were used after deprojection onto the solar disk\ncenter.\nWe \fnd a general trend among all parameters that\nthe ARs with larger non-potentiality and total magnetic\n\rux can produce both fast and slow CMEs, whereas\nthe ARs with smaller non-potentiality and \rux can onlyproduce slower CMEs. There are exceptions present for\nall of the parameters. Out of all the parameters studied,\n\u000bgand\u000bg\u0002\b show the best triangular pattern with\nleast outliers, and lowest y-intercept of the limit line,\nthus conveying their better performance over the other\nparameters for predicting the upper limit of the speed\nof CMEs that an AR can produce.\nSince fast CMEs tend to be a greater threat for se-\nvere space weather than slower ones, knowing that an\nAR can not produce a fast CME would be a useful fore-\ncast. Thus, our results can be incorporated in near\nreal time forecasting tools e.g., MAG4 [ Falconer et al. ,\n2014]. Expanding the data set of CMEs having mea-\nsured speeds and measureable source ARs in future will\nimprove statistics and con\frm or modify our results.\nAcknowledgments. SKT would like to thank Tibor T or ok\nand Yang Liu for useful discussion on this work during the AGU-\n2014 meeting. We acknowledge Phyllis Whittlesey and Malte\nBroese, students from Joint Space Weather Summer Camp 2014\n(sponsored by the University of Alabama in Huntsville and its\nCenter for Space Plasma and Aeronomic Research, The Univer-\nsity of Rostock and its Leibniz-Institute of Atmospheric Physics,\nand the German Aerospace Center (DLR)), for initially identify-\ning some of the CMEs and their source ARs. SKT is supported\nby an appointment to the NASA Postdoctoral Program at the\nNASA MSFC, administered by ORAU through a contract with\nNASA. RLM and ARW are supported by funding from the LWS\nTRT Program of the Heliophysics Division of NASAs SMD. Sup-\nport for MAG4 development comes from NASA's Game Chang-\ning Development Program, and Johnson Space Center's Space\nRadiation Analysis Group (SRAG). Use of SOHO LASCO CME\ncatalog, and data from AIA and HMI (SDO), STEREO, SOHO is\nsincerely acknowledged. 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TIWARI, NASA Marshall\nSpace Flight Center, ZP 13, Huntsville, AL 35812, USA (san-\njiv.k.tiwari@nasa.gov)"    },    {        "title": "1712.01825v1.Relativistic_Dynamics_of_Point_Magnetic_Moment.pdf",        "content": "arXiv:1712.01825v1  [physics.class-ph]  1 Dec 2017EPJ manuscript No.\n(will be inserted by the editor)\nRelativistic Dynamics of Point Magnetic Moment\nJohann Rafelski, Martin Formanek, and Andrew Steinmetz\nDepartment of Physics, The University of Arizona, Tucson, A rizona, 85721, USA\nSubmitted:December 1, 2017 / Print date: January 18, 2018\nAbstract. The covariant motion of a classical point particle with magn etic moment in the presence of\n(external) electromagnetic fields is revisited. We are inte rested in understanding Lorentz force extension\ninvolving point particle magnetic moment (Stern-Gerlach f orce) and how the spin precession dynamics is\nmodified for consistency. We introduce spin as a classical pa rticle property inherent to Poincare´ e symmetry\nof space-time. We propose a covariant formulation of the mag netic force based on a ‘magnetic’ 4-potential\nand show how the point particle magnetic moment relates to th e Amperian (current loop) and Gilbertian\n(magnetic monopole) description. We show that covariant sp in precession lacks a unique form and discuss\nconnection to g−2 anomaly. We consider variational action principle and find that a consistent extension\nof Lorentz force to include magnetic spin force is not straig htforward. We look at non-covariant particle\ndynamics, and present a short introduction to dynamics of (n eutral) particles hit by a laser pulse of\narbitrary shape.\nPACS.21.10.Ky Electromagnetic moments 03.30.+pSpecial relati vity 13.40.EmElectric and magnetic mo-\nments\n1 Introduction\nThe (relativistic) dynamics of particle magnetic moment\nµ,i.e.the proper time dynamics of spin sµ(τ), has not\nbeen fully described before. Our interest in this topic orig-\ninates in a multitude of current research topics:\ni) the ongoing effort to understand the magnetic moment\nanomaly of the muon [1,2];\nii) questions regarding how elementary magnetic dipoles\n(e.g.neutrons) interact with external fields [3,4];\niii) particle dynamics in ultra strong magnetic fields cre-\nated in relativistic heavy ion collisions [5,6];\niv)magnetars,stellarobjectswithextreme O(1011)Tmag-\nnetic fields [7,8];\nv) the exploration of particle dynamics in laser generated\nstrong fields [9];\nvi) neutron beam guidance and neutron storagerings [10];\nand\nvii) the finding of unusual quantum spin dynamics when\ngyromagnetic ratio g∝ne}ationslash= 2 [11,12].\nThe results we present will further improve the under-\nstanding of plasma physics in presence of inhomogeneous\nmagnetic fields, and improve formulation of radiation re-\naction forces, topics not further discussed in this presen-\ntation.\nIn the context of the electromagnetic (EM) Maxwell-\nLorentz theory we learn in the classroom that\n1. The magnetic moment µhas an interaction energy\nwith a magnetic field B\nEm=−µ·B. (1)The corresponding Stern-Gerlach force FSGhas been\nwritten in two formats\nFSG≡/braceleftBigg\n∇(µ·B),Amperian Model ,\n(µ·∇)B,Gilbertian Model .(2)\nThe name ‘Amperian’ relates to the loop current gen-\neratingthe force. The Gilbertian model invokesa mag-\nnetic dipole made of two magnetic monopoles. These\ntwo forces written here in rest frame of a particle are\nrelated [3,4]. We will show that a internal spin based\nmagnetic dipole appears naturally; it does not need to\nbe made of magnetic monopoles or current loops. We\nfind that both force expressions in Eq.(2) are equiv-\nalent, this equivalence arises from covariant dynamics\nwe develop and requires additional terms in particle\nrest frame complementing those shown in Eq.(2).\n2. The torque Tthat a magnetic field Bexercises on\na magnetic dipole µin a way that tends to align the\ndipole with the direction of a magnetic field B\nT≡ds\ndt=µ×B=gµBs\n/planckover2pi1/2×B, µB≡e/planckover2pi1\n2m.(3)\nThe magnetic moment is defined in general in terms of\nthe product of Bohr magneton µBwith the gyromag-\nnetic ratio g,|µ| ≡gµB. In Eq.(3) we used |s|=/planckover2pi1/2\nfor a spin-1/2 particle, a more general expression will\nbe introduced in subsection 3.1.1.\nWe used the same coefficient µto characterize both\nthe Stern-Gerlach force Eq.(2) and spin precession force2 Johann Rafelski, Martin Formanek, and Andrew Steinmetz: R elativistic Dynamics of Point Magnetic Moment\nEq.(3). However,thereis nocompellingargumentto doso\nand we will generalize this hypothesis — it is well known\nthat Dirac quantum dynamics of spin-1/2 particles pre-\ndicts both the magnitude g= 2 and identity of magnetic\nmoments entering Eq.(2) and Eq.(3).\nWhile the conservation of electrical charge is rooted\nin gauge invariance symmetry, the magnitude of electrical\ncharge has remained a riddle; the situation is similar for\nthe the case of the magnetic moment µ: spin properties\nare rooted in the Poincar´ e symmetry of space-time, how-\never, the strength of spin interaction with magnetic field,\nEq.(1)andEq.(3), isarbitrarybut uniqueforeachtypeof\n(classical) particle. Introducing the gyromagnetic ratio g\nwe in fact create an additional conserved particle quality.\nThis becomes clearer when we realize that the appearance\nof ‘e’ does not mean that particles we study need to be\nelectrically charged.\nFirst principle considerations of point particle rela-\ntivistic dynamics experience some difficulties in generat-\ning Eqs.(2,3), as a rich literature on the subject shows\n– we will cite only work that is directly relevant to our\napproach; for further 70+ references see the recent nu-\nmerical study of spin effects and radiation reaction in a\nstrong electromagnetic field [9].\nFor what follows it is important to know that the spin\nprecession Eq.(3) is a result of spatial rotational invari-\nance which leads to angular and spin coupling, and thus\nspin dynamics can be found without a new dynamical\nprinciple has been argued e.g.by Van Dam and Ruij-\ngrok [13] and Schwinger [14]. Similar physics content is\nseen in the work of Skagerstam and Stern [15,16], who\nconsidered the context of fiber bundle structure focusing\non Thomas precession.\nCovariant generalization of the spin precession Eq.(3)\nis often attributed to the 1959 work by Bergmann-Michel-\nTelegdi [17]. However we are reminded [18,19,20] that\nthis result was discovered already 33 years earlier by L.H.\nThomas [23,24] at the time when the story of the elec-\ntron gyromagnetic ratio g= 2 was unfolding. Following\nJackson [18] we call the corresponding equation TBMT.\nJ. Frenkel, who published [21,22] at the same time with\nL.H.ThomasexploredcovariantformoftheStern-Gerlach\nforce, a task we complete in this work.\nThere have been numerous attempts to improve the\nunderstanding of how spin motion back-reacts into the\nLorentz force generating the the Stern-Gerlach force. In\nthe 1962 review Nyborg [25] summarized efforts to for-\nmulate the covariant theory of electromagnetic forces in-\ncluding particle intrinsic magnetic moment. In 1972Itzyk-\nson and Voros [26] proposed a covariant variational ac-\ntion principle formulation introducing the inertia of spin\nI, seeking consistent variational principle but they found\nthat no new dynamical insight resulted in this formula-\ntion.\nOur study relates most to the work of Van Dam and\nRuijgrok [13]. This work relies on an action principle and\nhence there are in the Lorentz force inconsistent terms\nthat violate the constraint that the speed of light is con-\nstant, see e.g.their Eq.3.11 and remarks: ‘ The last twoterms are O(e2) and will be omitted in what follows.’\nOther authors were proposing mass modifications to com-\npensate these terms, a step which is equally unacceptable.\nFor this reasonour approachis intuitive, without insisting\non ‘in principle there is an action’. Once we have secured\na consistent, unique covariant extension of the Lorentz\nforce, we explore the natural variational principle action.\nWe find it is not consistent and we identify the origin of\nthe variational principle difficulties.\nWe develop the concept of the classical point parti-\ncle spin vector in the following section 2. Our discussion\nrelates to Casimir invariants rooted in space-time symme-\ntry transformations. Using Poincar´ e group generators and\nCasimir eigenvalues we construct the particle momentum\npµand particle space-like spin pseudo vector sµ. In sec-\ntion 3 we present a consistent picture of the Stern-Gerlach\nforce (subsection 3.1) and generalize TBMT precession\nequation (subsection 3.2) linear in both, the EM field and\nEM field derivatives. We connect the Amperian form of\nSG force (3.1.1) with the Gilbertian force (3.1.2). We dis-\ncuss non-uniqueness of spin dynamics (3.2.3) in consider-\nation of impact on muon g−2 experiment. We show in\nsection 4 that the natural choice of action for the consid-\nered dynamical system does not lead to a consistent set of\nequations; in this finding we align with all prior studies of\nStern-Gerlach extension of the Lorentz force.\nIn the final part of this work section 5 we show some\nof the physical content of the theoretical framework. In\nsubsection5.1we presentamoredetailed discussionofdy-\nnamical equations for the case of particle in motion with\na givenβ=v/candE,Bfields in laboratory. In sub-\nsection 5.2 we study solution of the dynamical equations\nfor the case of an EM light wave pulse hitting a neutral\nparticle.Wehaveobtainedexactsolutionsofthisproblem,\ndetailwillfollowunderseperatecover[27].Theconcluding\nsection 6 is a brief summary of our findings.\n1.1 Notation\nFor most of notation, see Ref.[28]. Here we note that we\nuse SI unit system and the metric:\ndiaggµν={1,−1,−1,−1}, pµpµ=gµνpµpν=E2\nc2−p2,\nWe further recognize the totally antisymmetric covariant\npseudo tensor ǫ:\nǫµναβ=√−g/braceleftbigg(−1)perm,if all indices are distinct\n0, otherwise ,\nwhere ‘perm’ is the signature of the permutation. It is\nimportant to remember when transiting to non-covariant\nnotation in laboratory frame of reference that the analog\ncontravariant pseudo-tensor due to odd number of space-\nlike dimensions is negative for even permutations and pos-\nitive for odd permutations. The Appendix B of Ref.[29]\npresents an introduction to ǫ.\nWewillintroduceanelementarymagneticdipolecharge\nd— the limitations of the alphabet force us to adopt theJohann Rafelski, Martin Formanek, and Andrew Steinmetz: Re lativistic Dynamics of Point Magnetic Moment 3\nletterdotherwise used to describe the electric dipole to\nbe the elementary magnetic dipole charge. The magnetic\ndipole charge of a particle we call dconverts the spin vec-\ntorsto magnetic dipole vector µ,\nsdc=µ, d≡|µ|\nc|s|. (4)\nThe factor cis needed in SI units since in the EM-tensor\nFµνhas as elements E/candB. It seems natural to in-\ntroduce also sµd=µµ— an object µµcan confuse and\nwe stick to the product sµd, however we always replace\nsd→µ/c. Note that we place dto right of pertinent\nquantities to avoid confusion such as dx.\nWe cannot avoid appearance in the same equation of\nboth magnetic moment µand vacuum permeability µ0.\n2 Spin Vector\nA classical intrinsic covariant spin has not been clearly\ndefined or even identified in prior work. In some work ad-\ndressingcovariantdynamicsofparticleswith intrinsicspin\nandmagneticmomentparticlespinisbyimplicationsolely\na quantum phenomenon. Therefore we describe the pre-\ncise origin of classical spin conceptually and introduce it\nin explicit terms in the following.\nConsidering the Poincare group of space-time symme-\ntry transformations [30,31], it has been established that\nelementary particles have to be in a representation that is\ncharacterized by eigenvalues of two Casimir operators (a\n‘bar’ marks operators)\n¯C1≡¯pµ¯pµ= ¯p2≡m2c2,¯C2≡¯wα¯wα.(5)\nAllphysicalpoint‘particles’havefixedeigenvaluesof C1,C2.\nThe quantities (with a bar) ¯ pµand ¯wαare differential\noperators constructed from generators of the symmetry\ntransformations of space-time; that is 10 generators of the\nPoincar´ e group of symmetry transformations of 4-space-\ntime: ¯pµfortranslations, Jforrotationsand Kforboosts.\nOnce we construct suitable operator valued quantities we\nwilltransitiontothephysicsof‘c-number’valued(without\nbar)variablesasusedinclassicaldynamicswhereallquan-\ntities will be normal numbers and rely on the eigenvalues\nof Casimir operators C1,C2for each type of particle.\nIn Eq.(5) the first of the space-time operators based\non generators of the four space-time translations pµguar-\nantees that a point particle has a conserved inertial mass\nm(with a value specific for any particle type). The sec-\nond Casimir operator C2is obtained from the square of\nthe Pauli-Luba´ nski 4-vector\n¯wα=M⋆\nαβ¯pβM⋆\nαβ≡1\n2ǫαβµνMµν.(6)\nHereMµνis the antisymmetric tensor (operator) created\nfrom three Lorentz-boost generators Kand three space\nrotation generators Jsuch that\n1\n2MµνMνµ=K2−J2,1\n4M⋆\nµνMµν=J·K.(7)These relations help us see that\nFµν(E→K,B→J) =Mµν.\nThe generators J,Kof space-time transformations are\nrecognized by their commutation relations. They are used\nin a well known way to construct representations of the\nLorentz group.\nIn terms of the generator tensor Mνµthe covariant\ndefinition of the particle spin (operator) vector is\n¯sµ≡¯wµ√C1=M⋆\nµν¯uν\nc,¯uµ≡c¯pµ\n√C1=¯pµ\nm.(8)\nAccording to Eq.(8), spin ¯ sµis a pseudo vector, as re-\nquired for angular dynamics. The dimension of ¯ sµis the\nsame as the dimension of the generator of space rotations\nJ. We further find that ¯ sµis orthogonal to the 4-velocity\n(operator) ¯ uµ\nc¯s·¯u= ¯uνM⋆\nνµ¯uµ= 0, (9)\nby virtue of the antisymmetry of M⋆evident in the def-\ninition Eq.(6). The definition of the particle spin (oper-\nator) is unique: no other space-like (space-like given the\northogonality ¯ s·¯u= 0) pseudo vector associated with the\nPoincar´ e group describing space-time symmetry transfor-\nmations can be constructed.\nWe now transition to c-numbered quantities (dropping\nthe bar): an observer ‘(0)’ co-moving with a particle mea-\nsures the 4-momentum and 4-spin sµ\npµ\n(0)≡ {/radicalbig\nC1,0,0,0}, sµ\n(0)≡ {0,0,0,/radicalbig\n|C2|/C1}(10)\nwhere according to convention ˆ z-axis of the coordinate\nsystem points in direction of the intrinsic spin vector s. In\nthe particle rest frame we see that\n0 =pµ\n(0)s(0)\nµ=pµsµ|(0)=m(uµsµ)|any frame ,(11)\nconsistent with operator equation Eq.(9); more generally,\nany space-like vector is normal to the time-like 4-velocity\nvector. For the magnitude of the spin vector we obtain\n−s2≡sµ\n(0)s(0)\nµ≡sµsµ|(0)=s2|any frame =−|C2|\nC1.(12)\nWe keep in mind that s2must always be a constant of\nmotion in any frame of reference. Its value s·s=−s2is\nalways negative, appropriate for a space-like vector. Sim-\nilarly\npµ\n(0)p(0)\nµ=p2|any frame =C1≡m2c2,(13)\nmust be a constant of motion in any frame of reference\nand the value p2is positive, appropriate for a time-like\nvector.\nAs long as forces are small in the sense discussed in\nRef.[28] we can act as if rules of relativity apply to both\ninertial and (weakly) accelerated frames of reference. This\nallows us to explore the action of forces on particles in\ntheir rest frame where Eq.(10) defines the state of a par-\nticle. By writing the force laws in covariant fashion we\ncan solve for dynamical evolution of pµ(τ), sµ(τ) classical\nnumbered variables.4 Johann Rafelski, Martin Formanek, and Andrew Steinmetz: R elativistic Dynamics of Point Magnetic Moment\n3 Covariant dynamics\n3.1 Generalized Lorentz force\n3.1.1 Magnetic dipole potential and Amperian force\nWe have gone to great lengths in section 2 to argue for the\nexistenceofparticleintrinsicspin.Forallmassiveparticles\nthis implies the existence of a particle intrinsic magnetic\ndipole moment, without need for magnetic monopoles to\nexist or current loops. Spin naturally arises in the con-\ntext of symmetries of Minkowski space-time, it is not a\nquantum property.\nIn view of above it is appropriate to study classical\ndynamics of particles that have both, an elementary elec-\ntric charge e, and an elementarymagnetic dipole charge d.\nThe covariant dynamics beyond the Lorentz force needs\nto incorporate the Stern-Gerlach force. Thus the exten-\nsion has to contain the elementary magnetic moment of\na particle contributing to this force. To achieve a suitable\ngeneralization we introduce the magnetic potential\nBµ(x,s)d≡F⋆\nµν(x)sνd ,F⋆\nµν=1\n2ǫµναβFαβ.(14)\nWe use dual pseudo tensor since sµis a pseudo vector; the\nproduct in Eq.(14) resultsin apolar 4-vector Bµ. We note\nthat the magnetic dipole potential Bµby construction in\nterms of antisymmetric field pseudo tensor F⋆\nµνsatisfies\n∂µBµ= 0, s·B= 0,→B·ds\ndτ=−s·dB\ndτ.(15)\nThe additional potential energy of a particle at rest\nplaced in this magnetic dipole potential is\nU(0)≡B0cd=cF⋆\n0ν(x)sνd=−|µ|B·s\n|s|≡ −µ·B.\n(16)\nThis shows Eq.(14) describes the energy content seen in\nEq.(1); all factors are appropriate.\nThe explicit format of this new force is obtained when\nwe use Eq.(14) to define a new antisymmetric tensor\nGµν=∂µBν−∂νBµ=sα[∂µF⋆να−∂νF⋆µα].(17)\nEquation(17) allows us to add to the Lorentz force\nm˙uµ=Hµνuν, Hµν=eFµν+Gµνd .(18)\nIn theG-tensor we note appearance in the force of the\nderivative of EM fields, required if we are to see the Am-\nperian model variant of the Stern-Gerlach force Eq.(2) as\na part of generalized Lorentz force.\nThe Amperian-Stern-Gerlach (ASG) force 4-vector is\nobtainedmultiplyingwith uνdtheG-tensorEq.(17).Thus\nthe total 4-forcea particle ofcharge eand magnetic dipole\nchargedexperiences is\nFµ\nASG=eFµνuν−u·∂F⋆µνsνd+∂µ(u·F⋆·sd).\n(19)In the particle rest frame we have\nuν|RF={c,0}, csνd|RF={0,µ}.(20)\nWe can use Eq.(20) to read-off from Eq.(18) the particle\nrest frame force to be\nFµ\nASG|RF=/braceleftbigg\n0, eE−1\nc2µ×∂E\n∂t+∇(µ·B)/bracerightbigg\n,(21)\nwhere two contributions ∂(µ·B)/∂ttoF0cancel. Each of\nthe three terms originates in one of the covariant terms in\nthesequenceshown.TheresultiswhatonecallsAmperian\nmodeloriginatingin dipolescreatedbycurrentloops.This\nis, however, not the last word in regard to the form of the\nforce.\n3.1.2 Gilbertian model Stern-Gerlach force\nWerestatetheStern-Gerlach-LorentzforceEq.(18),show-\ning the derivative terms explicitly,\nm˙uµ=eFµνuν+(∂µ(u·F⋆·s)−sαu·∂F⋆µα)d .(22)\nMultiplying with sµthe last term vanishes due to anti-\nsymmetry of F⋆and we obtain\ns·˙u=1\nms·(eF−s·∂F⋆d)·u . (23)\nThis equation suggests that we explore\neFµν→/tildewideFµν=eFµν−s·∂F⋆µνd,(24)\nas the generalized Lorentz force replacing the usual field\ntensoreFby/tildewideFin a somewhat simpler way compared to\nthe original HµνEq.(18) modification.\nWe demonstrate now that the field modification seen\nin Eq.(24) leads to a different and fully equivalent for-\nmat of the force. We replace in the first term in Eq.(22)\nF→/tildewideFand add the extra term from Eq.(24) to the two\nreminder terms. Changing the index naming these we can\nwrite symmetrically\nm˙uµ=/tildewideFµνuν (25)\n+sα/parenleftbig\n∂αF⋆µβ+∂µF⋆βα+∂βF⋆αµ/parenrightbig\nuβd .\nThe tensor appearing in the parentheses in the 2nd line of\nEq.(25)isantisymmetricunderanyofthethreeexchanges\noftheindices.Itisthereforeproportionaltothetotallyan-\ntisymmetric tensor ǫαµβγwhich must be contracted with\nsome 4-vector Vγcontaining a gradient of the EM dual\nfield tensor, there are two such available 4-vectors ∂κF⋆\nκγ\nwhich vanishes by virtue of Maxwell equations, and\nVγ=1\n2ǫγκηζ∂κF⋆ηζ=∂κFκγ=µ0jγ.\nThus we introduce the Gilbertian form of the 4-force\nFµ\nGSG=/tildewideFµνuν−µ0jγǫγαβνuαsβgνµd .(26)Johann Rafelski, Martin Formanek, and Andrew Steinmetz: Re lativistic Dynamics of Point Magnetic Moment 5\nNotethatinourformulationtheAmperianandtheGilber-\ntian 4-forces are identical\nFµ\nASG=Fµ\nGSG, (27)\nthey are just written differently.\nIn the rest frame of a particle, see Eq.(20) the Gilber-\ntian force Eq.(27) is\nFµ\nGSG|RF={0, eE+(µ·∇)B+µ0µ×j}.(28)\nIt is interesting to see the mechanism by which the two\nformats of the forces equal to each other in the particle\nrest frame. With\n∇(µ·B)−(µ·∇)B=µ×(∇×B),\nwe form the force difference between Eq.(21) and Eq.(28)\n[FASG−FGSG]RF=µ×/parenleftbigg\n−1\nc2∂E\n∂t+∇×B−µ0j/parenrightbigg\n= 0.\n(29)\nThe terms in parenthesis cancel according to Maxwell\nequation confirming that both the Amperian and the Gil-\nbertian forces are equal taking as an example the instan-\ntaneous rest frame. From now one we will use Gilbertian\nform of the force and in later examples we will focus on\nparticle motion in vacuum, jµ= 0.\nIn this discussionofforceswe keptthe electricalcharge\neand the elementary magnetic moment ‘charge’ dEq.(4)\nas independent qualities of a point particle. As noted in\nthe introduction it is common to set |µ| ≡gµB, see be-\nlow Eq.(3). Hence we can have both, charged particles\nwithout magnetic moment, or neutral particles with mag-\nnetic moment, aside from particles that have both charge\nand magnetic moment. For particles with both chargeand\nmagnetic moment we can write, using Gilbertian format\nof force\nm˙uµ=/tildewideFµνuν=e/parenleftbigg\nFµν−(1+a)¯λs·∂\n|s|F⋆µν/parenrightbigg\nuν,\n(30)\nwherea= (g−2)/2 is the gyromagnetic ratio anomaly.\nThe Compton wavelength ¯ λ=/planckover2pi1/mcdefines the scale at\nwhichthespatialfieldinhomogeneityisrelevant;notethat\ninhomogeneities of the field are boosted in size for a parti-\ncle in motion, a situation which will become more explicit\nin section 5.1.3.\n3.2 Spin motion\n3.2.1 Conventional TBMT\nFor particles with m∝ne}ationslash= 0 differentiating Eq.(11) with re-\nspect to proper time we find\n˙u·s+u·˙s= 0, (31)\nwhere we introduced proper time derivative ˙ sµ=dsµ/dτ.\nSchwinger observed [14] that given Eq.(31) one can usecovariant form of the dynamical Lorentz force equations\nforduµ/dτto obtain\nuµ/parenleftbiggdsµ\ndτ−e\nmFµνsν/parenrightbigg\n= 0. (32)\nHereFµνis the usual EM field tensor. Equation(32) has\nthe general TBMT solution\ndsµ\ndτ=e\nmFµνsν+/tildewideae\nm/parenleftbigg\nFµνsν−uµ\nc2(u·F·s)/parenrightbigg\n,(33)\nwhere we used the notation u·F·s≡uµFµνsν.\nIn Eq.(33)/tildewideais an arbitrary constant considering that\nthe additional term multiplied with uµvanishes. On the\nother hand we can read off the magnetic moment enter-\ning Eq.(3): the last term is higher order in 1 /c2. Hence\nin the rest frame of the particle we see that 2(1+ /tildewidea) =g\ni.e.Eq.(33) reproducesEq.(3) with the magnetic moment\ncoefficient when /tildewidea=a. Therefore, as introduced, /tildewidea=a\nis theg∝ne}ationslash= 2 anomaly. However, in Eq.(33) we could for\nexample use/tildewidea= (g2−4)/8 =a+a2/2, which classical\nlimit of quantum dynamics in certain specific conditions\nimplies [12]. In this case /tildewidea→aup to higher order correc-\ntions. This means that measurement of /tildewideaas performed in\nexperiments [1,2] depends on derivation of the relation of\n/tildewideawithaobtained from quantum theory. These remarks\napply even before we study gradient in field corrections.\n3.2.2 Gradient corrections to TBMT\nThe arguments by Schwinger, see Eqs.(31,32,33), are ide-\nallypositionedtoobtaininaconsistentwaygeneralization\nof the TBMT equations including the gradient of fields\nterms required for consistency. We use Eq.(24) in Eq.(33)\nto obtain\ndsµ\ndτ=1+/tildewidea\nm(eFµν−s·∂ F⋆µνd)sν(34)\n+/tildewidea\nmc2(s·eF·u−s·∂s·F∗·ud)uµ.\nThe dominant gradient of field correction arises for an\nelementary particle from the 2nd term in the first line\nin Eq.(34), considering the coefficient of the second line\na=α2/2π+...= 1.2×10−3. One should remember that\ngiven the precision of the measurement [1,2] of /tildewideawhich\nis driven by the first term in the second line in Eq.(34)\nwe cannot in general neglect the new 2nd term in first\nline in Eq.(34) even if the characteristic length defining\nthe gradient magnitude is the Compton wavelength ¯ λ, see\nEq.(30).\n3.2.3 Non-uniqueness of gradient corrections to TBMT\nIt is not self evident that the form Eq.(34) is unique. To\nsee that a family of possible extensions TBMT arises we\nrecall the tensor Eq.(18) Hµνmade of the two potentials\nAµandBµ. We now consider the spin dynamics in terms6 Johann Rafelski, Martin Formanek, and Andrew Steinmetz: R elativistic Dynamics of Point Magnetic Moment\nof the two field tensors, FandGreplacing the usual EM-\ntensorFµνin the Schwinger solution, Eq.(33). In other\nwords, we explore the dynamics according to\ndsµ\ndτ=1\nmeFµνsν+/tildewideae\nm/parenleftbigg\nFµνsν−uµ\nc2(u·F·s)/parenrightbigg\n(35)\n+Gµνsνd\nm+/parenleftbigg\nGµνsν−uµ\nc2(u·G·s)/parenrightbigg/tildewidebd\nm.\nTwo different constants /tildewideaand/tildewidebare introduced now since\nthe two terms shown involving FandGtensors could be\nincluded in Schwinger solution independently with differ-\nent constants. Intuition demands that /tildewidea=/tildewideb. However,\naside from algebraic simplicity we do not find any com-\npelling argument for this assumption.\nWereturnnowtothedefinitionofthe GtensorEq.(17)\nto obtain\nGµνsν=(sνsα∂µF⋆να−s·∂F⋆µαsα) (36)\n=−s·∂F⋆µνsν.\nThe first term in the first line vanishes by antisymmetry\nofF⋆tensor. We also have\nu·G·s=−s·∂u·F⋆·s . (37)\nUsingEq.(36)andEq.(37)wecancombinein Eq.(35) the\nfirst two terms in both lines, and the last terms in both\nlines to obtain\ndsµ\ndτ=1+/tildewidea\nm/parenleftBigg\neFµν−1+/tildewideb\n1+/tildewideas·∂F⋆µνd/parenrightBigg\nsν(38)\n−/tildewideauµ\nmc2/parenleftBigg\nu·/parenleftBigg\neF−/tildewideb\n/tildewideas·∂F⋆d/parenrightBigg\n·s/parenrightBigg\n.\nThis equation agrees with Eq.(34) only when /tildewidea=/tildewideb. How-\never,thisrequirementisneithermathematicallynorphysi-\ncallynecessary.Forexample usingEq.(26) we easilycheck\ns·˙u+u·˙s= 0 without any assumptions about /tildewidea,/tildewideb.\nAs Eq.(35) shows the physical difference between fac-\ntors/tildewideaand/tildewidebis related to the nature of the interaction:\nthe ‘magnetic’ tensor Gis related to/tildewidebonly. Thus for a\nneutral particle e→0 we see in Eq.(38) that the torque\ndepends onlyon /tildewideb.Conversely,whenthe effect ofmagnetic\npotential is negligible Eq.(38) becomes the textbook spin\ndynamics that depends on /tildewideaalone.\nTomakefurthercontactwithtextbookphysicswenote\nthat the coefficient of the first term in Eq.(38)\n1+/tildewidea\nme= 2(1+/tildewidea)e/planckover2pi1\n2m1\n/planckover2pi1=/tildewidegµB1\n/planckover2pi1,/tildewideg= 2(1+/tildewidea),(39)\nshould reproduce in leading order the torque coefficient in\nEq.(3) as is expected from study of quantum correspon-\ndence. However,quantum correspondencecould mean /tildewidea=\na+a2/2, which follows comparing exact solutions of the\nDirac equation with spin precession for the case we ex-\nplored [12] and which is not exactly the motion of a muonin storage ring. However, this means that in order to com-\npare the measurement of magnetic moment of the muon\ncarried out on macroscopic scale [1,2] with quantum com-\nputations requires a further step, the establishment of\nquantum correspondence at the level of precision at which\nthe anomaly is measured.\n4 Search for variational principle action\nAt the beginning of earlier discussions of a covariant ex-\ntension to the Lorentz force describing the Stern-Gerlach\nforce was always a well invented covariant action. How-\never, the Lorentz force itself is not a consistent comple-\nment of the Maxwell equations. The existence of radia-\ntion means that an accelerated particle experiences radi-\nation friction. The radiation-reaction force has not been\nincorporated into a variational principle [28,32]. Thus we\nshould not expect that the Stern-Gerlach force must orig-\ninate in a simple action.\nWe seek a path xµ(τ) in space-time that a particle\nwill take considering an action that is a functional of the\n4-velocity uµ(τ) =dxµ/dτand spin sµ(τ). Variational\nprinciple requires an action I(u,x;s). When Irespects\nspace-time symmetries the magnitudes of particle mass\nand spin are preserved in the presence of electromagnetic\n(EM) fields. We also need to assure that u2=c2which\nconstrains the form of force and thus Ithat is allowed.\nMoreover, we want to preserve gauge invariance of the\nresultant dynamics.\nThe component in the action that produced the LHS\n(inertia part) of the Lorentz force remains in discussion.\nTo generate the Lorentz force one choice of action is\nILz(u,x) =−/integraldisplay\ndτ mc√\nu2−e/integraldisplay\ndτ u(τ)·A(x(τ)).(40)\nWe note that reparametrization of τ→kτconsidering\nu=dx/dτhas no effect on value of ILz.\nVariation with respect to path lead to\nd\ndτmcuµ\n√\nu2=Lµ\nLz=uν∂µeAν−deAµ\ndτ,(41)\nwheretheRHSproducesupondifferentiationof eAµ(x(τ))\nthe usual Lorentz force\nLµ\nLz=e(∂µAν−∂νAµ)uν=eFµνuν.(42)\nMultiplying Eq.(41) with mcuµ/√\nu2we establish by an-\ntisymmetry of the tensor FµνEq.(42) that also the prod-\nuct with the LHS in Eq.(41) vanishes. This means that\n(mcuν/√\nu2)2=m2c2≡p2=Const.Henceforth\npµ≡mcuµ\n√\nu2. (43)\nThere is a problem when we supplement in Eq.(40)\nthe usual action ILzby a term Imbased on our prior con-\nsideration of Aµ→Aµ+Bµ, see subsection 3.1.1. TheJohann Rafelski, Martin Formanek, and Andrew Steinmetz: Re lativistic Dynamics of Point Magnetic Moment 7\nproblem one encounters is that the quantity Bµcontains\nadditional dependence on sµ(τ) which adds another term\nto the force. Let us look at the situation explicitly\nI(u,x;s) =ILz+Im, Im≡ −/integraldisplay\ndτ u·B(u,x;s)d .(44)\nHere the dependence on sµ(τ) is akin to a parameter de-\npendence; some additional consideration defines the be-\nhavior, in our case this is the TBMT equations.\nVarying with respect to the path the modified action\nEq.(44) we find the modified covariant force\ndpµ\ndτ=Lµ\nL+Lµ\nS1+Lµ\nS2, (45)\nwith two new contributions\nLµ\nS1= (∂µBν−∂νBµ)uν=Gµνuν,(46)\nLµ\nS2=−F⋆µνdsν\ndτd . (47)\nWe applied here with A→Bthe result seen in Eq.(41),\nand the additional term Lµ\nS2follows by remembering to\ntake proper time derivative of sµ. The first term Eq.(46)\nis as we identified previously in Eq.(18). We note that\nanother additional term arises if and when an additional\npower of√\nu2to accompany u·Bas was done in [13].\nAn unsolved problem is created by the torque-like term,\nEq.(47).\nIf we replace in our thoughts dsν/dτin Eq.(47) by the\nTBMT equation Eq.(33) or as would be more appropri-\nate by its extended version Eq.(35), we see that the force\nLµ\nS2would be quadratic in the fields containing also field\nderivatives. However, by assumption we modified the ac-\ntion limiting the new term in Eq.(44) to be linear in the\nfields and derivatives. Finding non linear terms we learn\nthat this assumption was not justified. However, if we add\nthe quadratic in fields term to the action we find follow-\ning the chain of arguments just presented that a cubic\nterm is also required and so on; with derivatives of fields\nappearing along.\nWe have searched for some time for a form that avoids\nthis circular conundrum, but akin to previous authors we\ndid not find one. Clearly a ‘more’ first principle approach\nwould be needed to create a consistent variational princi-\nplebasedequationsystem.Ontheotherhandwehavepre-\nsented before a formulation of spin dynamics which does\nnot require a variational principle in the study the parti-\ncle dynamics: as is we haveobtained a dynamical equation\nsystem empirically. Our failing in the search for an under-\nlying action is not critical. A precedent situation comes to\nmind here: the radiation emitted by accelerated charges\nintroduces a ‘radiation friction’which must be studied [28,\n32] without an available action, based on empirical knowl-\nedge about the energy loss arising for accelerated charges.5 Experimental consequences\n5.1 Non covariant form of dynamical equations\n5.1.1 Laboratory frame\nIn most physical cases we create a particle guiding field\nwhich is at rest in the laboratory. Particle motion occurs\nwith respect to this prescribed field and thus in nearly all\nsituations it is practical to study particle position zµ(τ) in\nthe laboratoryframe of reference. Employing the Lorentz-\ncoordinate transformationsfrom the particle rest frame to\nthe laboratory frame we obtain\ndzµ\ndτ≡uµ|L=cγ{1,β},β≡dz\ndct=v\nc,(48)\nsµ|L=/braceleftbigg\nγβ·s,/parenleftbiggγ\nγ+1γβ·s/parenrightbigg\nβ+s/bracerightbigg\n,(49)\nwhere as usual γ= 1//radicalbig\n1−β2and one often sees the spin\nwritten with γ2/(γ+1) = (γ−1)/β2.\nOneeasilychecksthatEq.(48)andEq.(49)alsosatisfy\nEq.(11): uµsµ= 0. A classic result of TBMT reported in\ntextbooks is that the longitudinal polarization ˆβ·sfor\ng≃2 andβ→1 is a constant of motion. This shows that\nfor a relativistic particle the magnitude of both time-like\nand space-like components of the spin 4-vector Eq.(49)\ncan be arbitrarily large, even if the magnitude of the 4-\nvector is bounded sµsµ=−s2. This behavior parallels\nthe behavior of 4-velocity uµuµ=c2.\nWe remind that to obtain in the laboratory frame the\nusual Lorentz force we use the 4-velocity with respect to\nthe Laboratory frame Eq.(48), with laboratory defined\ntensorF,i.e.with laboratory given E,BEM-fields\nd(muµ|L)\ndτ= (eFµνuν)|L=eFµν|Luν|L.(50)\nSometimes it is of advantage to transform Eq.(50) to the\nparticlerestframe.Suchatransformation LwithLu|rest=\nuLwhen used on the left hand side in Eq.(50) produces\nproper time differentiation of the transformation opera-\ntor, see also [33]. Such transformation into a co-rotating\nframe of reference originates the Thomas precession term\nin particle rest frame for the torque equation. This term is\nnaturally present in covariant formulation when we work\nin the laboratory reference frame.\nFor the full force Eq.(26) we thus have\nd(muµ|L)\ndτ=eFµν|Luν|L (51)\n−d sα|L(∂αF⋆µν)|Luν|L.(52)\nWe see that in laboratory frame of reference a covariant\ngradient of the fields is prescribed, i.e.that some appara-\ntus prescribes the magnitude\nQαµν|L≡∂αF∗µν|L, (53)8 Johann Rafelski, Martin Formanek, and Andrew Steinmetz: R elativistic Dynamics of Point Magnetic Moment\nwhich allows for a moving particle with uµ|LEq.(48) and\nsµ|LEq.(49) to experience the Stern-Gerlach force Fµ\nSG\nFµ\nSG|L≡ −dsα|LQαµν|Luν|L. (54)\nWe have gone to extraordinary length in arguing Eq.(54)\nto make sure that the forthcoming finding of the Lorentz\nboost of field inhomogeneity is not questioned.\n5.1.2 Magnetic potential in the laboratory frame\nWe evaluate in the laboratory frame the form of Eq.(14).\nThe computation is particularly simple once we first recall\nthe laboratory format of the Lorentz force Fµ\nL\nFµ\nL|L=Fµν(x)uν|L=cγ{β·E/c,E/c+β×B}(55)\nThe magnetic part of the action will be evaluated (see\nsecond line below) in analogy to above. We now consider\nB·u|L=u·F⋆·s|L=−sµ|L(F⋆\nµνuν)|L (56)\n=−sν|Lcγ{−β·B,B−β×E/c}\n=cγ/parenleftbigg\nβ·s β·Bγ\nγ+1−s·(B−β×E/c)/parenrightbigg\nwhere we used in 2nd line i) F⋆\nµνfollows from the usual\nFµνupon exchange of E/c↔Band ii) flip β→ −β\nto account for contravariant and not covariant 4-velocity.\nIn the 3rd line we used γ(γ/(γ+ 1)−1) =−γ/(γ+ 1).\nNotable in Eq.(56) is the absence of the highest power γ2\nas all terms cancel, the result is linear in (large) γ.\nFor the magnetic action potential energy of a particle\nin lab frame we obtain\nU≡B·u|Ld=γ/parenleftBig\nKˆβ·µˆβ·B−µ·(B−β×E/c)/parenrightBig\n,\n(57)\nK=β2γ\nγ+1= 1−/radicalbig\n1−β2=/braceleftBigg1\n2β2,forβ→0\n1,forβ→1\nEquation(57) extends the rest frame β= 0 Eq.(16) and\nrepresents covariant generalization of Eq.(1). In ultrarel-\nativistic limit all terms in Eq.(57) have the same magni-\ntude.\n5.1.3 Field to particle energy transfer\nWe now consider the energy gain by a particle per unit of\nlaboratory time, that is we study the 0th component of\nEq.(26)\ndE\ndt=cdτ\ndtd(mu0|L)\ndτ=cγ−1/tildewideF0νuν|L (58)\n=eE·v+cd sα|L(∂αB)|L·v\ndE\ndt=(eE+(µ·∇)B)·v (59)\n+γβ·µ/parenleftbigg∂B\nc∂t+γ\nγ+1(β·∇)B/parenrightbigg\n·v.A further simplification is achieved considering\n∂B\nc∂t+(β·∇)B=∂B\nc∂t+3/summationdisplay\ni=1dxi\ncdt∂B\n∂xi=dB\ncdt,(60)\nwhere the total derivative with respect to time accounts\nfor both, the change in time of the laboratory given field\nB, and the change due to change of position in the field\nby the moving particle. We thus find two parts\ndE\ndt=v·/parenleftBig\neE+(µ·∇)B−Kˆβ·µ(ˆβ·∇)B/parenrightBig\n(61)\n+β·dB\ndtγβ·µ,\nwhere the 2nd line is of particular interest as it is propor-\ntional to γ. Focusing our attention on this last term: we\ncan useβ=cp/Eandγβ=p/mc. Upon multiplication\nwithEand remembering that c2pdp=EdEwe obtain\np·/parenleftbiggdp\ndt−dB\ndtµ·p\nmc2/parenrightbigg\n= 0. (62)\nwhichin qualitativetermsimplies anexponentialresponse\nof particle momentum as it crosses a magnetic field\n|p| ≃mc e±(|B|−B0))|µ|/mc2. (63)\nHowever, even a magnetar magnetic field of up 1011T will\nnot suffice to impact electron momentum decisively in\nview of the smallness of the electron magnetic moment\n5.810−11MeV/T. However, in ultrarelativistic heavy ion\ncollisionsatLHCa10,000strongerverynon-homogeneous\nB-fields arise.\n5.2 Neutral particle hit by a light pulse\n5.2.1 Properties of equations\nThe dynamical equations developed here have a consid-\nerably more complex form compared to the Lorentz force\nandTBMTspin precessioninconstantfields[33].We need\nfield gradients in the Stern-Gerlach force, and in the re-\nlated correction in the TBMT equations. Since the new\nphysics appears only in the presence of a particle mag-\nnetic moment, we simplify by considering neutral parti-\ncles. We now show that the external field described by\na light wave (pulse) lends itself to an analytical solution\neffort. This context could be of practical relevance in the\nstudy of laser interaction with magnetic atoms, molecules,\nthe neutron and maybe neutrinos.\nFore= 0 our equations Eq.(26) and Eq.(38) read\n˙uµ=−s·∂F∗µνuνd\nm, (64)\n˙sµ=−s·∂F∗µνsν1+/tildewideb\nmd+uµu·(s·∂)F∗·s/tildewidebd\nmc2.\n(65)Johann Rafelski, Martin Formanek, and Andrew Steinmetz: Re lativistic Dynamics of Point Magnetic Moment 9\nThe external light wave field is a pulse with\nAµ=εµf(ξ), ξ=k·x , k·ε= 0.(66)\nThe derivative of the dual EM tensor for linear fixed in\nspace pulse polarization εµis\n(s·∂)F∗µν=(k·s)ǫµναβkαεβf′′(ξ),(67)\nprime ‘′’ indicates derivative with respect to the phase ξ.\nNotice that if we contract Eq.(67) with kµorεµwe\nget zero because Levi-Civita tensor ǫµναβis totally anti-\nsymmetric. Therefore contracting Eq.(64) with either kµ\norεµwe find\n0 =k·˙u→k·u=k·u(0), uµ(0) =uµ(τ0) (68)\n0 =ε·˙u→ε·u=ε·u(0). (69)\nWefurthernotethattheargumentofthelightpulseEq.(66)\nsatisfies\nξ=k·x→˙ξ=k·˙x=k·u=k·u(0).(70)\nwhereweusedEq.(68).Thusweconcludethattheparticle\nfollows the pulse such that\nξ=k·x=τ k·u(0)+ξ0, ξ0=k·x(0).(71)\nThe two conservation laws Eq.(68) and Eq.(69) along\nwith Eq.(70) make the light pulse an interesting exam-\nple amenable to an analytical solution.\nWe now evaluate several invariants in the laboratory\nframe seeking understanding of their relevance. A parti-\ncle moving in the laboratory frame in consideration of\nEq.(48) experiences in its rest frame a plane wave with\nthe Doppler shifted frequency\nk·u(0) =γ0(1−n·β0)ω (72)\nwhich is unbounded as it grows with particle laboratory\nLorentz-γ0.However, k·s,theprojectionofspinontoplane\nwave4-momentum kµisbounded. Toseethis werecallthe\nconstraint Eq.(11) which in the laboratory frame reads\nS0\nL−β·SL= 0. (73)\nWe thus obtain\nk·s(τ) =k·s(τ)|L=|k|/parenleftbig\nS0\nL−n·SL/parenrightbig\n=|k|(β−n)·SL,\n(74)\nwhere we used Eq.(73) in last equality. Since βandn=\nk/|k|are unit-magnitude vectors we find\n(k·s(τ))2≤4k2S2\nL. (75)\nThe magnitude of the spin vector in the lab frame is con-\nstrained by Eq.(12)\n−s2=S02\nL−S2\nL= (β·SL)2−S2\nL=−sin2θS2\nL,(76)\nwhere we again used Eq.(73). Combining Eq.(75) and\nEq.(76) we see that except when particle is moving ex-\nactly in direction of SL(sin2θ= 0), the magnitude of\n(k·s(τ))2is bounded.5.2.2 Invariant acceleration and spin precession\nEven without knowing the explicit form for uµ(τ), sµ(τ)\nwe were able to obtain [27] the invariant acceleration\n˙u2(τ) =−/parenleftbiggd\nmf′′(ξ(τ))k·s(τ)k·u(0)/parenrightbigg2\n.(77)\nThis result follows using the usual trick oftaking a further\n(proper) time derivative of Eq.(64) (multiplied by a suit-\nable factor) and on RHS eliminating ˙ uby using Eq.(64).\nMultiplying the result with uµand eliminating u·¨uusing\nthe 2nd differential of u2=c2produces Eq.(77).\nWe see in Eq.(77) that the magnitude of the 4-force\ncreated by a light pulse and acting on an ultrarelativistic\nparticle is dependent on square of the product of the 2nd\nderivative of pulse function with respect to ξ,f′′(ξ), with\nthe Doppler shifted frequency Eq.(72). The value Eq.(77)\nis negative since acceleration is a space-like vector.\nAs we discussed below Eq.(76) the spin precession fac-\ntork·sseenin Eq.(77) is bounded. We wereable toobtain\na soluble formulation of the spin precession dynamics de-\nscribed by the dimensionless variable\ny=k·s(τ)/tildewidebd\nmcC1(78)\nwhich satisfies the differential equation\n/parenleftbiggdy(s)\nds/parenrightbigg2\n=y2(1−y2)s= (f′(ξ(τ))−f′(ξ0))C1\n(79)\nobtained performing suitable manipulations of dynamical\nequationspriorto solvingfor uµ(τ), sµ(τ). We areseeking\nbounded periodic solutions of nonlinear Eq.(79) no mat-\nter how large the constant C1determined by the initial\nconditions\nC1≡/tildewidebd\nmck·s(0)C2, C 2≥1, (80)\nC2≡/radicalBigg\n|(k·u)2|s2|−[(k·u)(ε·s)−(ε·u)(k·s)]2|\nc2(k·s)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nτ=0.\n(81)\nC2contains the initial particle Lorentz- γfactor. One can\nsee severalpossible solutions of interest of Eq.(79); for ex-\nampley= sin(φ(s)) satisfies all constraints.It leads to the\npendulum type differential equationandwerecognizethat\nhigh intensity light pulses can flip particle spin. However,\nthere are other relevant solutions, e.g.y∝1/coshz.\nUponsolutionofEq.(79) k·s(τ)isknown,givenEq.(71)\nwe also know the dependence of Eq.(67) on proper time\nτ. Hence Eq.(64) can be solved for uµand Eq.(65) can\nbe solved for sµresulting in an analytical solution of the\ndynamics ofa neutral magnetic dipole moment in the field\nof a light pulse of arbitrary shape. The full description of\nthe dynamics exceeds in length this presentation and will\nfollow [27].10 Johann Rafelski, Martin Formanek, and Andrew Steinmetz: Relativistic Dynamics of Point Magnetic Moment\n6 Conclusions\nTheStern-GerlachcovariantextensionoftheLorentzforce\nhas seen considerable interest as there are many immedi-\nate applications listed in first paragraph. Here we have:\n1)introduced in Eq.(10) the covariantclassical4-spinvec-\ntorsµin a way expected in the context of Poincare sym-\nmetry of space-time;\n2) presented a unique linear in fields form of the covari-\nant magnetic moment potential, Eq.(14), which leads to\na natural generalization of the Lorentz force;\n3)shownthattheresultantAmperian,Eq.(19),andGilber-\ntian, Eq.(26), forms of the magnetic moment force are\nequivalent;\n4) extended the TBMT torque dynamics, Eq.(35), mak-\ning these consistent with the modifications of the Lorentz\nforce;\n5) demonstrated the need to connect the magnetic mo-\nment magnitude entering the Stern-Gerlach force with\nthe one seen in the context of torque dynamics, subsec-\ntion 3.2.3;\n6) shown that variational principle based dynamics has\nsystemic failings when both position and spin are ad-\ndressed within present day conceptual framework, see sec-\ntion 4;\n7) reduced the covariant dynamical equations to labora-\ntoryframeofreferenceuncoveringimportantfeaturesgov-\nerning the coupled dynamics, see section 5.1;\n8) obtained work done by variations of magnetic field in\nspace-time on a particle, Eq.(61);\n9) shown salient features of solutions of neutral particles\nwith non-zero magnetic moment hit by a laser pulse, see\nsection 5.2;\nReferences\n1. J. Grange et al.[Muon g-2 Collaboration], “Muon (g-2)\nTechnical Design Report,” FERMILAB-FN-0992-E, 666\npages, arXiv:1501.06858 [physics.ins-det].\n2. G. W. Bennett et al.[Muon g-2 Collaboration], “Final\nReport of the Muon E821 Anomalous Magnetic Moment\nMeasurement at BNL,” Phys. Rev. D 73(2006) 072003\ndoi:10.1103/PhysRevD.73.072003\n3. Kirk T. McDonald, “Forces on Magnetic Dipoles”; Joseph\nHenry Laboratories, Princeton University, Princeton,\nUSA; Retrieved November 2017; document dated October\n26, 2014; updated February 17, 2017\n4. F. Mezei “La nouvelle vague in polarized neu-\ntron scattering,” Physica B+C 137(1986) 295\ndoi:10.1016/0378-4363(86)90335-9\n5. X. G. Huang, “Electromagnetic fields and anomalous\ntransports in heavy-ion collisions — A pedagogical\nreview,” Rept. Prog. Phys. 79(2016) no.7, 076302\ndoi:10.1088/0034-4885/79/7/076302 [arXiv:1509.04073\n[nucl-th]].\n6. M. Greif, C. Greiner and Z. Xu, “Magnetic field\ninfluence on the early time dynamics of heavy-\nion collisions,” Phys. Rev. C 96(2017) 014903\ndoi:10.1103/PhysRevC.96.014903 [arXiv:1704.06505 [hep -\nph]].7. R. Turolla, S. Zane and A. Watts, “Magne-\ntars: the physics behind observations. A re-\nview,” Rept. Prog. Phys. 78(2015) no.11, 116901\ndoi:10.1088/0034-4885/78/11/116901 [arXiv:1507.02924\n[astro-ph.HE]].\n8. A. Sedrakian, X. G. Huang, M. Sinha and J. W. Clark,\n“From microphysics to dynamics of magnetars,”\nJ. Phys. Conf. Ser. 861(2017) no.1, 012025\ndoi:10.1088/1742-6596/861/1/012025 [arXiv:1701.00895\n[astro-ph.HE]].\n9. M. Wen, C. H. Keitel and H. Bauke, “Spin-one-half\nparticles in strong electromagnetic fields: Spin effects\nand radiation reaction,” Phys. Rev. A 95(2017) 042102\ndoi:10.1103/PhysRevA.95.042102 [arXiv:1610.08951\n[physics.plasm-ph]].\n10. K. J. Kugler, W. Paul and U. Trinks, “A Magnetic Stor-\nage Ring For Neutrons,” Phys. Lett. 72B(1978) 422.\ndoi:10.1016/0370-2693(78)90154-5\n11. J. Rafelski and L. Labun, “A Cusp in QED at g=2,”\narXiv:1205.1835 [hep-ph].\n12. A. Steinmetz, et. al, in preparation\n13. H. Van Dam and T. W. Ruijgrok, “Classical Rel-\nativistic Equations for Particles With Spin Mov-\ning in External Fields,” Physica A 104(1980) 281.\ndoi:10.1016/0378-4371(80)90088-6\n14. J. S. Schwinger, “Spin precession – A dynamical dis-\ncussion,”American Journal of Physics 42, (1974) 510,\ndoi:10.1119/1.1987764\n15. B. S. Skagerstam and A. Stern, “Lagrangian Descriptions\nof Classical Charged Particles With Spin,” Phys. Scripta\n24(1981) 493. doi:10.1088/0031-8949/24/3/002\n16. A. P. Balachandran, G. Marmo, B.-S. Skagerstam and\nA. Stern, “Gauge Theories and Fibre Bundles - Ap-\nplications to Particle Dynamics,” Lect. Notes Phys.\n188, 1 (1983) doi:10.1007/3-540-12724-0 1 2nd. Edition:\n[arXiv:1702.08910 [quant-ph]].\n17. V. Bargmann, L. Michel and V. L. Telegdi, “Precession\nof the polarization of particles moving in a homogeneous\nelectromagnetic field,” Phys. Rev. Lett. 2(1959) 435.\ndoi:10.1103/PhysRevLett.2.435\n18. J.D. Jackson, “Examples of the zeroth theorem of the his-\ntory of science,” Am. J. Phys 76(2008) 704 (see case VI).\ndoi:10.1119/1.2904468\n19. V. Hushwater, “On the discovery of the classical equatio ns\nfor spin motion in electromagnetic field,” Am. J. Phys 82\n(2014) 6. doi:10.1119/1.4821347\n20. J.D. Jackson,“Jackson responds,”Am. J. Phys 82(2014) 7.\n21. J. Frenkel,(no title letter to editors) Nature 117(1926)\n653 (Issue of 8 May)\n22. J. Frenkel, “Die Elektrodynamik des rotierenden Elek-\ntrons,” (translated: The dynamics of spinning electron) Z.\nPhys.37(1926) 243. doi:10.1007/BF01397099\n23. L. H. Thomas, “The motion of a spinning electron,” Na-\nture117(1926) 514 (Issue of April 10, discussion of spin\nprecession for g= 2) doi:10.1038/117514a0\n24. L. H. Thomas, “The Kinematics of an electron\nwith an axis,” Phil. Mag. Ser. 7 3(1927) 1.\ndoi:10.1080/14786440108564170\n25. P. Nyborg “On classical theories of spinning particles, ” Il.\nNuovo Cimento 23(1962) 47. doi:10.1007/BF02733541\n26. C. Itzykson and A. Voros, “Classical Electrodynam-\nics of Point Particles,” Phys. Rev. D 5(1972) 2939.\ndoi:10.1103/PhysRevD.5.2939Johann Rafelski, Martin Formanek, and Andrew Steinmetz: Re lativistic Dynamics of Point Magnetic Moment 11\n27. M. Formanek, et. al, in preparation.\n28. J. Rafelski, Relativity Matters: From Einstein’s EMC2\nto Laser Particle Acceleration and Quark-Gluon Plasma,\nXXV+468 pages, ISBN: 978-3-319-51230-3, Springer (Hei-\ndelberg, New York 2017) doi:10.1007/978-3-319-51231-0\n29. R. M. Wald, “General Relativity,” XIII+491 pages,\nISBN: 0-226-87033-2, Chicago U.Press (Chicago 1984)\ndoi:10.7208/chicago/9780226870373.001.0001\n30. Walter Greiner and Johann Rafelski, Spezielle Relativit¨ ats-\ntheorie: Ein Lehr- und ¨Ubungsbuch f¨ ur Anfangssemester\n(translated: Special Theory of Relativity: A Text and ex-\nercise book for undergraduates); appeared as Volume 3a of\nWalter Greiner Series in Theoretical Physics; H. Deutsch\n(Frankfurt 1984, 1989, 1992) ISBN3-87144-711-0; 1989:\nISBN 3-97171-1063-4; and 1992: ISBN 3-8171-1205-X.\n31. For contemporary account see T. Ohlsson, “Relativis-\ntic quantum physics: From advanced quantum mechan-\nics to introductory quantum field theory,” 297p,ISBN:\n9781139210720 (eBook), 9780521767262 (Print) (Cam-\nbridge University Press 2011)\n32. Y. Hadad, L. Labun, J. Rafelski, N. Elkina, C. Klier\nand H. Ruhl, “Effects of Radiation-Reaction in Relativis-\ntic Laser Acceleration,” Phys. Rev. D 82(2010) 096012\ndoi:10.1103/PhysRevD.82.096012 [arXiv:1005.3980 [hep-\nph]].\n33. A. E. Lobanov and O. S. Pavlova, “Solutions of the clas-\nsical equation of motion for a spin in electromagnetic\nfield,” Theoretical and Mathematical Phys. 121(1999)\n1691 doi:10.1007/BF02557213; translated from Russian:\n121(1999) 509"    },    {        "title": "1502.06505v1.Table_top_Measurement_of_Local_Magnetization_Dynamics_Using_Picosecond_Thermal_Gradients__Toward_Nanoscale_Magnetic_Imaging.pdf",        "content": "Page 1 \n  \n \n \n  \nTable-top Measurement of Local Magnetiza tion Dynamics Using Picosecond Thermal \nGradients: Toward Nanoscale Magnetic Imaging                                                               \nJ. M. Bartell\n1,†, D.H. Ngai1,†, Z. Leng1, and G. D. Fuchs1                                                     \n1Cornell University, Ithaca, NY 14853                                                                        \n†These authors contributed equally to this work \n \n  Page 2 \n Recent advances in nanoscale magnetism have  demonstrated the potential for spin-\nbased technology including magnetic random access memory1,2, nanoscale microwave \nsources3,4, and ultra-low pow er signal transfer5. Future engineering advances and new \nscientific discoveries will be enabled by research tools capable of examining local magnetization dynamics at length and time  scales fundamental to spatiotemporal \nvariations in magnetic systems\n6 – typically 10-200 nm7,8 and 5 – 50 ps.  A key problem is \nthat current table-top magnetic microscop y cannot access both of these scales \nsimultaneously. In this letter, we introduce a spatiotemporal magnetic microscopy that uses \nmagneto-thermoelectric interact ions to measure local magnet ization via the time-resolved \nanomalous Nernst effect (TRANE). By generati ng a short-lived, local thermal gradient, the \nmagnetic moment is transduced into an elect rical signal. Experiment ally, we show that \nTRANE microscopy has time resolution below 30 ps and spatial resolution limited by the \nthermal excitation area. Furthermore, we p resent numerical simulation s to show that the \nthermal spot size sets the limits of the spatia l resolution, even at 50 nm. The thermal effects \nused for TRANE microscopy have no fundame ntal limit on their spatial resolution, \ntherefore a future TRANE micro scope employing a scanning plasmon antenna could \nenable measurements of nano scale magnetic dynamics.  \nMagnetic microscopy has played a fundamental role in the study of magnetic behavior \nsuch as domain wall motion9,10, skyrmion formation11, magnetic switching12, and spin wave \npropagation13, spurring interest in the dynami cs of nanoscale magnetic features14,15. This \nmotivates a method of spatiotemporal magnetic  microscopy capable of measuring picosecond \nchanges in the magnetic moment with spatial resolution of less than 200 nm7,8. Magneto-optical Page 3 \n measurements are currently the only table-top appr oach to measure spatially varying dynamics in \nthe time-domain. Unfortunately, the spatial reso lution available to optical measurements is \nlimited by diffraction to approximately /ߣሺ2\tܣܰሻ , where λ is the wavelength of light and ܣܰ is \nthe effective numerical aperture of the focusi ng optics. Therefore, opt ical techniques including \nthe time-resolved magneto-optical Kerr effect (TRMOKE), have a diffraction limited resolution \nof roughly 200 nm. One solution is to use nanomet er-scale wavelength ra diation, as in X-ray \nmagnetic circular dichroism (XMCD) experiments which provide spatial re solution of 30 nm and \ntime-domain resolution of less than 100 ps16.  Unfortunately, spatiotemporal XMCD requires \nsynchrotron-based sources which limits its wide-spread use. \nTo circumvent the spatial limitation impos ed by optical diffraction, we propose a new \ntechnique for magnetic spatiotemporal microsco py that is based on th e interaction between \nmagnetization and heat rather than light. Our method is based on the time-resolved anomalous \nNernst effect (TRANE). The geometry for TRANE is depicted in Fig. 1a.  The anomalous Nernst \neffect (ANE) is a magnetization dependent, thermoelectric effect17–19, in which a thermal \ngradient, transverse to the film’s magnetic moment, generates an el ectric field given by20, \nEሬሬറሺxሬറ,tሻൌെ N \t ሬሬറTሺxሬറ,tሻൈμ୭Mሬሬሬറሺxሬറ,tሻ, where N is the anomalous Nernst coefficient. Previous \nstudies have demonstrated that by confining ሬሬറTሺxሬറ,tሻ to a micron-scale region in a thin-film \nferromagnetic metal, an anomalous Nernst voltage is generated that is proportional to the local \nmagnetic moment21,22. This has inspired proposals for app lications that include microscopy and \nspectroscopy21–24, yet, a fully developed spatiotemporal microscope  with nanoscale resolution \nhas not yet been realized. Here, we show that using a pulsed thermal source with a short duty Page 4 \n cycle from a focused laser, E ANE can be localized in both time and space to generate a TRANE \nsignal.  \nTRANE microscopy is a viable strategy for hi gh spatiotemporal resolution because the \nANE interaction time and the electron thermal carri er wavelength are both short in comparison to \nthe scales of magnetic dynamics and the spa tial variation of magneti zation. Because thermal \ngradients are not fundamentally limited by op tical diffraction, micros copy based on magneto-\nthermal interactions has no fundame ntal barrier to decr easing the spatial re solution. Therefore, \nthe spatiotemporal resolution of TRANE microscopy is predom inantly limited by the generation \nand evolution of the loca lized thermal gradient.  \nFigure 1b shows a schematic for the measur ement setup. We focus a pulsed laser to \ngenerate the short-lived, local temperature grad ient for each optical pulse, thus creating a \ncorresponding voltage pulse. Using a homodyne tec hnique by electrically mixing the generated \nvoltage pulse and lock-i n measurement, we measure the TRANE signal, V TRANE , which is \nproportional to the stroboscopically sampled local magnetization. In this work, we used optical \npulses with a fluence of 2.3 mJ/cm2 that created vertical thermal gradients of 3.3×108 K/m and a \ntemperature increase of 30 K at the hotte st point (see supporting online information)25. The first \nstructure we use to demonstrate TRANE micros copy is a 30 nm cobalt film patterned in 18 µm \nwide cross-structures. Fig. 1c shows a hysteresis curve of this sample measured using TRANE, \nwhich demonstrates the proportionality between V TRANE  and the local magnetic moment. \nTo show that the optically generated therma l gradients are short-lived, we time-resolved \nVTRANE  by mixing it with a short, 75 ps electrical probe pulse. Fi g. 1d shows our measurement of \nVTRANE  as a function of electrical pulse delay.  Th ese data can be unders tood as the temporal Page 5 \n convolution of V TRANE with the electrical probe pulse25. We find that the convolution also has a \n75 ps width, which we interpret as the upper limit of  the thermal gradient’s lifetime.  As we show \nwith subsequent magnetic resonance experiments, thermal gradients produced in our microscope \nare actually much  shorter-lived.  \nThe sensitivity of TRANE microscopy is dependent on several factors, including the \nNernst coefficient, the geometry and the impedance. For the 18 µm cross structure, we calculate \nthe magnetization angle sensitivity to be 0.73/√ݖܪ 25. The electrical TRANE signal scales as \n݀ଶ/ݓ for a probe diameter, d, and a channel width, w21.   For comparison, the signal scaling of \nmagneto-optical microscopy is essentially indepe ndent of the device geometry above the optical \ndiffraction limit of ݀ ∼ /ߣሺ2\tܣܰሻ ∼  200 nm. It is below this fundamental limit that resolution of \nfar-field magneto-optical microscopy is sharply reduced.  In contrast, a TRANE signal collected \nwith a nanoscale probe diameter (<200 nm) can remain large provided that w is also scaled. \nBecause of its picosecond duration, the TRANE signal is also sensitive to the microwave \nimpedance of the sample, with the strongest signal occurring when the sample impedance \nmatches the 50 Ω impedance of the measurement circuit.  \nNext we experimentally demonstrate that lateral thermal diffusion does not limit spatial \nresolution at the scale of a tightly focused laser by imaging the local magnetic moment of the cobalt cross. Scanning th e laser across the sample, a map of  the magnetization is created (shown \nin Fig. 2a) in which, domain walls are visible wher e the projected moment is zero. For the cobalt \nfilms studied here, the doma in walls are 150-200 nm wide\n26, which is far below the 440 nm \nAbbe resolution limit we calculate for our apparatus. We use this fact to evaluate the resolution \nof our TRANE microscope by fitting spatial line  cuts across a magnetic domain wall with a Page 6 \n convolution of a step function and a Gaussian function of width, 2 δ. The fit yields δ =460 ± 90 \nnm25. These results suggest that the main limitation to the spatial resolution is the size of the \nthermal gradient spot.  \nTo gain a deeper understanding of thermal di ffusion in our magnetic thin film samples, \nwe performed time-dependent, finite element si mulations of the picosecond heating dynamics. \nUsing numerical simulations, we show in Fig. 2d that  when the laser pulse is at its maximum, the \ngradient’s vertical component of the thermal gr adient does not spread laterally beyond the pulsed \nheat source. Interestingly, our simulations show this statemen t is true even for nanoscale \ndiameter thermal sources. Although the thermal spot size used in this dem onstration is limited by \noptical diffraction, the thermal gradient diameter  could be reduced below the far-field optical \ndiffraction limit using a light -confining plasmon antenna27,28. \nWe study TRANE’s temporal resolution by stroboscopically meas uring ferromagnetic \nresonance (FMR) in Ni 20Fe80 (permalloy) wires using the apparatu s depicted in Fig. 3a. The wire \naxis of the sample is aligned parallel to the applied magnetic field.  To excite magnetization dynamics, a microwave frequency current is passed through a nearby copper wire to generate an \nout-of-plane magnetic field on the permalloy.  We choose a drive frequency that is \ncommensurate with the laser repetition rate to create a constant phase relationship between the \nV\nTRANE  measurement of magnetization and the exc itation field. Starting with the detection \nscheme as before, we also modulate the external magnetic field to distinguish V TRANE  from other \nvoltages due to inductive coupling betw een the two wires (see methods).  \nIn Fig. 3b we plot FMR as a function of ma gnetic field.  The magnetization is excited by \na 5.00 GHz stimulation to a maximum oscillation angle of angle of 0.07o.  The measurement Page 7 \n sensitivity is 0.093/√ݖܪ ,which is improved from the cobalt cross sample chiefly because of \nthe reduced sample width that increases the ratio ݀ଶ/ݓ   .By electrically shifting the relative time \ndelay between the microwave magnetic field drive and the laser probe by 50 ps, these data also \ndemonstrate that TRANE microscopy is sensit ive to the phase of magnetic precession.   \nThe FMR data are analyzed by fitting to linear combinations of symmetric and anti-\nsymmetric Lorentzian functions modified to ac count for the magnetic field modulation (see SI). \nFrom the fits, we extract a phase difference between the two of 64o ± 24o. The discrepancy from \nour expectation of a 90 ° shift might be because our simple model accounts only for a single, \nuniform FMR mode.  Close inspecti on of the two data sets in Fig. 3b reveals additional features \nthat are anti-correlated between measurement phase s.  This suggests more complicated magnetic \nbehavior than we model, including the exis tence of additional magnetic modes that may \ninfluence the accuracy of the phase we extrac t from fitting.  Although full imaging and analysis \nof these modes is a capability of TRANE microscopy, their detailed  study is beyond this scope of \nthe present demonstration.   \nAs we increase the frequency of the magnetic  excitation, we find th at (as expected) the \nFMR resonance field evolves as described by the Kittel equation, \nఊ\nଶ\tగඥሺܪ ܰ ௭\tܯ௦ሻሺܪ ܰ ௬\tܯ௦ሻ, as shown in Fig. 3d. Here, we use Ny = 0.015, and Nz = 0.985, \nwhich are determined separately with measuremen ts of the hard axis magnetic saturation. From \nthese fits we find an effective magnetic moment 4 πMs = 840 emu/cm3 and a Gilbert damping \nparameter, α = 0.009 ± 0.001. The damping in this sample is consistent with separate FMR \nmeasurements that we made by electrically m onitoring the DC rectific ation voltage.  These \nresults are also in excelle nt agreement with literature values for permalloy29,30. The consistency Page 8 \n among our various measurements and prior reports supports the proposal that the local, transient \nheating of the sample during measurement does not  significantly alter its dynamical properties as \nprobed by TRANE microscopy.   \nTo experimentally determine the time scale of  the vertical thermal gradient decay, we \nmeasure FMR as we increase the stimulation fr equency.  Assuming that the thermal pulses \nsample the mean magnetic projection over their duration, observation of FMR at a particular \nfrequency sets an upper limit on the thermal decay tim e.  In the inset to Fig. 3c we plot the FMR \nspectra at 16.4 GHz, which is the highest fre quency that we can produce with our microwave \nelectronics. From these data, we conclude that th e thermal gradient decays in under 30 ps. This is \nsupported by our time-dependent finite elemen t modeling (Fig. 3c) which shows the thermal \ngradient pulse has a full width at  half-maximum of 10 ps for these samples. Therefore, this \ntechnique could potentially be extended to measure FMR dynamics up to 50 GHz.  \nIn conclusion, we demonstrated that TRAN E microscopy enables a pathway to resolving \nmagnetic dynamics on fundamental length and time scales. We have demonstrated that for the \nthin film samples studied here, TRANE micros copy has temporal resolution below 30 ps and \nspatial resolution at the limit of focused light . As a magneto-thermoelectric technique, TRANE \nmicroscopy is not subject to the fundamental diffr action limit of spatial re solution that constrains \nfar-field optical methods. Crucially, the numerical simulations indicate that  the spatial resolution \nis only determined by the lateral diameter of th e thermal gradient, which we have verified down \nto a diameter of 50 nm. Applying these capabilities in a table-top imaging platform can enable \nnew access to time-resolved magnetization dynam ics which supports the burgeoning field of \nhigh-speed magnetoelectronics. Page 9 \n Methods \nSample Preparation \nFor measurements of spatial resolution, 30 nm thick cobalt films were deposited by \nelectron beam evaporation onto sapphire substr ates. Photolithography and ion milling was used \nto pattern the films into square crosses as pict ured. For the spatial images presented, the cross \narms were 18 µm wide. Electrical contact wa s made by wire bonding to evaporated copper \ncontacts.  \nThe samples used for ferromagnetic resonance measurements were 30 nm thick Ni 20Fe80 \n(permalloy) films deposited by DC magnetron sputtering at a base pressure below 10-7 Torr. The \nfilms were patterned with e-beam lithography and ion milled into wires 2 µm wide and 950 µm long. Evaporated copper contacts 1 µm wide were fabricated to c ontact the permalloy wire with \na range of separations to enable a DC impedance match close to 50 Ω. The contacts chosen for \nthe measurement were 3 µm apart and had a DC resistance of 74 Ω. The wire we used as a \nmicrowave antenna to excite magnetization dynamics was fabricated in a lift-off process to be 2 \nµm wide, 50 µm long, and 102 nm thick.  It was positioned 1 µm away from the permalloy wire \nand had a DC resistance of 48 Ω.  \nThermal Gradient Generation \nLocal thermal gradients were generated by fo cusing light from a Titanium:Sapphire laser \ntuned to 794 nm with 3 ps pulses and a fluence at the sample of 2.3 mJ/cm\n2. The repetition rate \nwas controlled with an electro-optic modulator/p ulse picker. We used a repetition rate of 76 \nMHz for measurements of the spatial imagi ng and 25.33 MHz for the ferromagnetic resonance Page 10 \n measurements. An optical chopper was used to m odulate the optical pulse train at 9.7 kHz. To \nscan the beam, we used a 4-F optical path in combination with a voice-coil controlled fast-\nsteering mirror. The light was focused into a diffraction-limited spot  using a 0.90 numerical \naperture air objective.  \nDetection \nTo detect the TRANE voltage pulses, we c onnect the voltage contact to a microwave \ntransmission line through a coplan ar waveguide soldered to a t ype-K connector.  The signal is \npassed through a low-pass filter with a 4 GHz break frequency to attenuate GHz frequency artifacts from inductive electri cal coupling between the copper antenna and permalloy wire. \nAfter the filter, the signal is amplified by 40 dB  with a 0-1 GHz bandwidth. The amplified pulse \ntrain is sent to the RF port of an electrical mixer, where it is mi xed with a 1.5 ns pulse train from \na pulse/pattern generator that is referenced to th e laser repetition rate. When the two pulse trains \ntemporally overlap, a voltage modulated by th e optical chopper (and, for FMR, the field \nmodulation) is passed to a low-fr equency preamplifier before being sent to a lock-in amplifier.  \nTo determine the amplitude of V\nTRANE  prior to amplification and electrical mixing, we \ncalibrated the transfer function of the “collect ion” circuit by measuri ng the signal produced by \nelectrically generated re ference pulses and systematically vary ing their widths. We find that our \ncollection circuit tr ansfer coefficient is 0.47 ± 0.04 for a 10 ps signal pulse (see SI).  Using this \ncalibration, we measure that the anomalous  Nernst coefficient in permalloy is 2.4േ0.2ൈ10ି \nVT-1K-1, which agrees with a previous report20.  \n2D Imaging Page 11 \n Imaging the static magnetic moment  is performed by measuring the V TRANE  along a \nchannel perpendicular to the applied magnetic field so that the maxi mum signal was obtained \nduring saturation of the magnetic moment. The multi-domain state was prepared by saturating \nthe cross with a 130 Oe field and decreasing the fiel d to 32 Oe. For the data in Fig. 2a, we used a \n250 nm step and a lock-in time constant of 500 ms.   \nFMR Excitation \nFMR was excited in the samples using a microwave signal produced by an arbitrary \nwaveform generator (AWG) with a clock referenced  to the laser repetition rate. This clock is \nmultiplied up within the AWG to a sampling ra te of 19.98 GS/s derived from the 25.3 MHz laser \npulse repetition rate. The waveforms from the AWG can be delayed in steps of 50 ps with \nrespect to the laser pulse s without re-triggering, a llowing resonant behavior of different phases to \nbe observed. For excitation frequencies above 5.7 GHz, the output frequency of the AWG was \ndoubled or quadrupled with electrical frequenc y multipliers to achieve frequencies up to 16.4 \nGHz. This excitation signal was then amplifie d to a power between 13-20 dBm and coupled to \nthe copper stimulation wire. \nTRANE detection of FMR \nFerromagnetic resonance was detected by usi ng a second lock-in with dual demodulation. \nIn this technique, two modulation sources at different frequencies are used. The signal is \nextracted by first demodulating the input referen ced to the optical chopper . The resulting signal \nis then sent to a second demodulator (time consta nt of 1 s) that is referenced to a 5-10 Hz \nmodulation of the magnetic field.   Page 12 \n Acknowledgements \nThis work was supported by AFOSR.  The authors would like to thank Isaiah Gray for his help \nmeasuring the temperature dependence of permall oy. This work made use of the Cornell Center \nfor Materials Research Shared Facilities wh ich are supported through the NSF MRSEC program \n(DMR-1120296) as well as the Cornell NanoSca le Facility, a member of the National \nNanotechnology Infrastructure Network, s upported by the NSF (Grant ECCS-0335765). \nAuthor Contributions \nGDF, DHN, and JMB developed the concept and procedure for the experiment. JMB \nassembled the optical apparatus, DHN fabricated  the samples. ZL and JMB performed numerical \nsimulations. JMB and DHN performed the expe rimental measurements. GDF, JMB, and DHN \nanalyzed the data and wrote the manuscript.  \nReferences \n1. Katine, J., Albert, F., Buhrman, R., Myers,  E. & Ralph, D. Current-Driven Magnetization \nReversal and Spin-Wave Excitations in Co /Cu /Co Pillars. Phys. Rev. Lett.  84, 3149 \n(2000). \n2. Mangin, S. et al.  Current-induced magnetization reversal  in nanopillars with perpendicular \nanisotropy. Nat. Mater.  5, 210 (2006). \n3. Deac, A. M. et al.  MgO-based tunnel magnetoresistance devices. Nat. Phys.  4, 803 (2008). \n4. Kiselev, S. I. et al.  Microwave oscillations of a nanom agnet driven by a spin-polarized \ncurrent. Nature  425, 380 (2003). Page 13 \n 5. Kajiwara, Y. et al.  Transmission of electrical signal s by spin-wave inte rconversion in a \nmagnetic insulator. Nature  464, 262 (2010). \n6. Freeman, M. R. & Choi, B. C. Advances in magnetic microscopy. Science  294, 1484 \n(2001). \n7. Trunk, T., Redjdal, M., Kákay, a., Ruane, M. F. & Humphrey, F. B. Domain wall structure \nin Permalloy films with decreasing thic kness at the Bloch to Néel transition. J. Appl. Phys.  \n89, 7606 (2001). \n8. Allenspach, R., Stampanoni, M. & Bischof, A.  Magnetic domains in thin epitaxial Co/Au \n(111) films. Phys. Rev. Lett.  65, 3344 (1990). \n9. Beach, G. S. D., Nistor, C., Knutson, C., Tsoi , M. & Erskine, J. L. Dynamics of field-\ndriven domain-wall propagati on in ferromagnetic nanowires. Nat. Mater.  4, 741 (2005). \n10. Yamaguchi, a. et al.  Real-Space Observation of Curre nt-Driven Domain Wall Motion in \nSubmicron Magnetic Wires. Phys. Rev. Lett.  92, 077205 (2004). \n11. Yu, X. Z. et al.  Real-space observation of a two-dimensional skyrmion crystal. Nature  \n465, 901 (2010). \n12. Acremann, Y. et al.  Time-Resolved Imaging of Spin Transfer Switching: Beyond the \nMacrospin Concept. Phys. Rev. Lett.  96, 217202 (2006). \n13. Madami, M. et al.  Direct observation of a propagating spin wave induced by spin-transfer \ntorque. Nat. Nanotechnol.  6, 635 (2011). Page 14 \n 14. O’Brien, L. et al.  Near-Field Interaction between Do main Walls in Adjacent Permalloy \nNanowires. Phys. Rev. Lett.  103, 077206 (2009). \n15. Ruotolo, a et al.  Phase-locking of magnetic vortices mediated by antivortices. Nat. \nNanotechnol.  4, 528 (2009). \n16. Acremann, Y., Chembrolu, V., Strachan, J. P., Tyliszczak, T. & Stöhr, J. Software defined \nphoton counting system for time resolved x-ray experiments. Rev. Sci. Instrum.  78, \n014702 (2007). \n17. Guo, G. Y., Niu, Q. & Nagaosa, N. Anomal ous Nernst and Hall effects in magnetized \nplatinum and palladium. Phys. Rev. B  89, 214406 (2014). \n18. Mizuguchi, M., Ohata, S., Uchida, K. I., Sa itoh, E. & Takanashi, K. Anomalous nernst \neffect in an L1 0-ordered epitaxial FePt thin film. Appl. Phys. Express  5, 093002 (2012). \n19. Weischenberg, J., Freimuth, F., Blügel, S. & Mokrousov, Y. Scattering-independent \nanomalous Nernst effect in ferromagnets. Phys. Rev. B  87, 060406 (2013). \n20. Slachter, A., Bakker, F. L. & van Wees, B. J. Anomalous Nernst and anisotropic \nmagnetoresistive heating in a lateral spin valve. Phys. Rev. B  84, 020412 (2011). \n21. Weiler, M. et al.  Local Charge and Spin Currents in Magnetothermal Landscapes. Phys. \nRev. Lett.  108, 106602 (2012). Page 15 \n 22. Von Bieren, A., Brandl, F., Grundler, D. & Ansermet, J.-P. Space- and time-resolved \nSeebeck and Nernst voltages in laser-h eated permalloy/gold microstructures. Appl. Phys. \nLett. 102, 052408 (2013). \n23. Boona, S. R., Myers, R. C. & He remans, J. P. Spin caloritronics. Energy Environ. Sci.  7, \n885 (2014). \n24. Schultheiss, H., Pearson, J. E., Bader, S. D. & Hoffmann, a. Thermo electric Detection of \nSpin Waves. Phys. Rev. Lett.  109, 237204 (2012). \n25. Supporting Online Material. \n26. Suzuki, T. Domain Wall Width Measuremen t in Cobalt Films by Lorentz Microscopy. J. \nAppl. Phys.  40, 1216 (1969). \n27. Kryder, M. H. et al.  Heat assisted magnetic recording. Proc. IEEE  96, 1810 (2008). \n28. Challener, W. A. & Amit, V. in Modern Aspects of Electrochemistry No. 44  (ed. \nSchlesinger, M.) 53 (Springer, 2009). doi:10.1007/b113771 \n29. Yamaguchi, A. et al.  Broadband ferromagnetic resonance of Ni81Fe19 wires using a \nrectifying effect. Phys. Rev. B  78, 104401 (2008). \n30. Mecking, N., Gui, Y. & Hu, C.-M. Microwav e photovoltage and photoresistance effects in \nferromagnetic microstrips. Phys. Rev. B  76, 224430 (2007).  \n Page 16 \n Figures\nFigure 1| The Time Resolve Anomalous N ernst Effect for Magnetic Measurement.  a, \nSchematic diagram of the anomalous Nernst effect in our measurements b, Schematic of the \nexperimental setup. 792 nm pulsed laser light is focused to a diffraction limited spot on the \nmagnetic film patterned on top of a thermally c onductive, electrically insulating substrate. \nBonding pads enable detection of the TRANE volta ge proportional to the perpendicular magnetic \nmoment. c, TRANE measurement of local hys teresis in the cobalt cross. d, Signal from mixing \nthe TRANE voltage with a 75 ps elect rical pulse. We observe that th e width of the initial peak is \nPage 17 \n 75 ps, indicating that the TRANE pulse is < 75 ps. The other, br oader features are electrical \nreflections due to the impedance mismatch of the device and the 50 Ω transmission line. \n  Page 18 \n  \nFigure 2 | Spatial Resolution of Magnet ic Imaging Using a Thermal Gradient.  a, Image of \nmagnetic structure taken with TRANE. b & c,  Line cuts of the image showing the TRANE \nsignal in blue and the simultaneously measured re flected light in red. Top: line cut from top to \nbottom. Bottom: line cut from left to right we note  that the reflected signal drops off at the edge \nof the cross while the TRANE signal goe s to zero without edge artifacts. d, Finite element, time \ndependent simulation of the vertic al component of the thermal gr adient at peak applied power \nalong the x-axis as a function of distance from the spot center. The blue and red temperature \nprofiles correspond to a thermal generation spot  size of 440 nm and 50 nm respectively. The \ninset shows the x-z profile of the temperature at  the center of the optical pulse. We note that, \nbecause of radial symmetry, the radial (in-plane) gradient gives no signal.  \nPage 19 \n Figure 3 | Measurement of Magnetic Dynamics using TRANE.  a, Schematic of the \nexperimental setup used to measure FMR in pe rmalloy wires. The 2 µm wide permalloy wire \n(blue) is stimulated by a 2 µm Cu wire 1 µm aw ay from the permalloy wire. The contacts used to \nmeasure the TRANE signal were separated by 3 µm. b, FMR spectra of taken at 5 GHz and two \ndifferent phases of the stimulation. The solid lines are fits to the Lo rentzian curves after \naccounting for the modulated lock-in technique. c, Time dependent numerical simulation of the \nthermal gradient as a function of time. d, Plot of the resonant fre quencies as a function of the \nresonant field determined by fitting. The solid lin e is a fit to the Kittel equation. FMR spectra \nPage 20 \n taken at 16.4 GHz is shown in the inset. For the FMR spectra, the points show the data after \nsmoothing over 3 neighboring points. The solid li ne is a fit to the linear combination of \nsymmetric and anti-symmetric Lorentzians after accounting fo r the modulation frequency.  \n  Page 21 \n Supplemental Information \n1. Determination of Laser Induced Temperature Change \nThe increase in local sample temperature due to laser heating can be measured from the \nresulting increase in local resist ance. Using the same detection sc heme as depicted in Fig. 1b, we \nmeasure the voltage change due to the heating as a function of  an applied DC current. The \nvoltage measured is given by ܸൌܫ\tߚ \t∆ܴ ,where ߚ is the collection circuit transfer coefficient, \nܫ is the applied DC current and ∆ܴ is the resistance change due to laser heating. Note that as \ndiscussed in a later section, ߚ varies for the pulse width. Thus  we have different transfer \ncoefficients for the overall temperature and the te mperature gradient due to the different decay \ntimes. By relating the resistance change to a temperature increase, we can quantitatively \ndetermine the heating induced by the laser.  \nTo relate the resistance change to the temperature profile, we first measure the \ntemperature dependence of the 4-terminal sample  resistivity. For our permalloy sample in the \ntemperature range of interest, the resistivity is given by a linear temperature dependence given \nby ߩሺܶሻൌߩሺ1\tߙሺܶെܶሻሻ with ߙ ൌ 0.0025\tܭିଵ, ߩൌ 30\tΩ\t݉ܿିଶ and ܶൌ 293\tܭ . Using \nthe temperature profile calculated from numerical  simulations described in  the following section, \nwe measure the maximum temperature increase at various fluences as shown in Fig. S1. For the \ntypical fluence of 2.3 mJ/cm2 used throughout the measurement, we have a maximum \ntemperature increase of 30 K. Using the simula ted temperature profile and maximum signal from \nthe hard-axis hysteresis, we measure an anomalous Nernst coefficient of 2.4േ0.2ൈ10ି V/(T \nK). This measurement is in agreem ent with a recently reported value1. Page 22 \n  \nFigure S1 | Maximum Temperatu re Increase due to the Laser.  The experimentally measured \ntemperature increase as a function of laser fluence. For TRANE measurements, we used a \nfluence of 2.3 mJ/cm2, which corresponds to 30 K increase in temperature.  \n2. Finite Element Simulations of Thermal Evolution \nFinite element modeling of the thermal gr adient evolution was performed using the \nCOMSOL Multiphysics Heat Transfer Module. We consider a single temperature diffusive \nmodel in which the laser is treated only as a he at source, rather than considering different the \nphonon and electrons temperatures. This is justified by the fact  that the optically excited \nelectrons are thermalized on time scales comp arable to the laser pulse width of 3 ps2.  \nThe spatiotemporal evolution of the thermal gradient in our system is calculated \nnumerically with the Fourier diffusion equation usi ng the material parameters given in Table S1. \nThe heat source ܳሺݔԦ,ݐሻ, is given by,  \nPage 23 \n  ܳሺݔԦ,ݐሻൌொ\nଶ\tగ\tఋೣ\tఋௗ݁ିೣమ\nమഃೣమିమ\nమഃమ݁ି\n݁ିమ\nమഓమ      (S1)\nwhere, δx and δy are the Gaussian widths in the x and y di rection of the laser s pot (440 nm), d is \nthe skin depth (12nm), Q o is the incident peak power of  a single pulse (2.19 W), and τ is the \npulse Gaussian temporal wi dth of the 3 ps pulse.  \nTable S1 | Material Parameters used for Simulation \nMaterial Thermal Conductivity (W/m K) Specific Heat (J/g K) Density (g/cm3)\nSapphire3 30.3 0.764 3.98 \nPermalloy 46.44 0.435 8.76 \n \nThe results of the simulations yield spatiotemp oral profiles of the temperature and thermal \ngradient shown in Fig. S2. To apply the simulatio n for quantitative analysis we need a sample \nspecific scaling factor determined experimenta lly (See supplemental sections 1 and 6). For the \npermalloy samples presented in this letter the scaling factor was found to be 0.47 ± 0.04.  Page 24 \n  \nFigure S2 | Simulated Spatial a nd Temporal Temperature Profiles.  a, Spatial profile of the \nheat source, ܳሺݔԦ,ݐሻ, for the 440 nm spot size.  b-c, Temperature profiles across an axial slice of \nthe thermal source of the 440 nm Gaussian wi dth. The dashed line indicates the interface \nbetween the permalloy wire and the sapphire. b, is the temperature at th e peak of the pulse and c, \nis the temperature 982 ps after the peak. d, Time dependence of the laser induced temperature \nincrease for 440 nm spot size. The dashed line sh ows the temporal profile of the heat source in \narbitrary y – axis for reference.   \nPage 25 \n 3. Fitting Lateral Resolution \nWe measure the value for the lateral resolution by taking vertical line cuts of the 2D scan \nacross a portion of the domain wall (Fig. 2a in th e main text and Fig. S3a). The 4 µm region of \nthe domain wall used for fitting is shown boxed in  Fig. S3a. This region was chosen because it \nwas the portion of the image with the clearest step  function behavior. Fits of the line scans where \ndone using a least means squared method to find th e Gaussian width, amplitude, and center of a \nfunction derived by convolution of a Gaussian with  a –1 to 1 step function. The results of the \nindividual fits are shown in Fig. S3b. The mean of  the fits is 460 nm with  a standard deviation of \n90 nm, the standard deviation is used as the uncertainty, as it was larger than the uncertainty of \nthe individual fits.  \n \nFigure S3 | Region and Results of Spatial Fitting.  a, Spatial map of the static magnetic \nmoment showing the region used for the lateral resolution measurement. b, One-half the \nGaussian-width of the pulse that  was convolved with the unit step  determined by fitting. The X-\naxis is the horizontal coordinate of the line cu t used and the dashed line indicates the mean.  \nPage 26 \n 4. Modification of the Resonant Li ne-shapes due to Field Modulation \nTo measure the FMR of the permalloy wires we detect the projected magnetic moment \nperpendicular to the wire. The magnetic moment  of the wire precesses about the externally \napplied magnetic field when driven by an exte rnal microwave field from a microwave antenna \npatterned parallel to the magnetic wire. The FMR precession angle of a ferromagnetic in the \nlinear response regime can be modeled as a driv en damped oscillator. The projection amplitude \nof this motion is the linear combination of even and odd Lorentzian functions. \n ݊݅ܵሺ߮ሻሺܪ െܪ ሻ/ߴ\n1ሺܪെܪ ሻଶ/ߴଶ ݏܥሺ߮ሻ1\n1ሺܪെܪ ሻଶ/ߴଶ  (S2)\nIn addition to the magnetic signal due to FMR,  we also detect an induced electrical \nresponse from coupling between the microwave an tenna and the magnetic bar. To separate the \ntwo signals, a cascaded lock-in technique was used in which the first demodulation was \nreferenced to a square modulated 9.7 kHz signal from an optical  chopper and the second \ndemodulation was referenced to a 14 Oe sinusoi dal field modulated at 10 Hz (5 Hz for FMR \nfrequencies above 10 GHz). The TRANE signal de tected by the second lock-in can be modeled \nby Eq. S3.  \n݊݅ܵሺ߮ሻනሺܪܪ ௗݏܥሺݐ\t߱ሻെܪ ሻ/ߴ\n1ሺܪܪ ௗݏܥሺݐ\t߱ሻെܪ ሻଶ/ߴଶ∗ݏܥሺݐ߱ሻݐ݀\n\tݏܥሺ߮ሻන1\n1ሺܪܪ ௗݏܥሺݐ߱ሻെܪሻଶ/ߴଶ∗ݏܥሺݐ\t߱ሻ\tݐ݀( S3)Page 27 \n The resulting analytical equation is then used  to fit the resonance data obtained with \nTRANE to quantify the values of the linewidth, amplitude, phase, and cen ter frequency. We note \nthat the modification to the Lorentzian shape does not add free parameters to the fitting function \nbecause the modulation amplitude is a known value.  The modulation does impact the \nuncertainty and it reduces the ove rall signal amplitude, but at th e benefit of increased angular \nsensitivity. \n \nFigure S4 | Modification of Lorentzian Functions.   The blue curve in each plot shows the \nmodeled normalized - Lorentzian response func tion for the projected amplitude of the FMR \nprecession for a resonant field, H r = 180 Oe and line-width of 80 Oe. The red curves show the \ncorresponding signal as detected by the lock-in wh en using a modulation amplitude of 20 Oe.   \n \n \n \n \nPage 28 \n 5. Sensitivity  \nThe sensitivity is calculated using the fi eld-dependent magnetization measurements \nshown in Fig. 1c in the main text and Fig. S5. This measurement is done in the transverse \ngeometry – the saturated moment is perpendi cular to the voltage pick-ups – so that \nሺ்ܸோோି௫்ܸோோሻ corresponds to a 180o rotation. The standard devia tion of points at saturation is \ntaken as the detected voltage uncertainty, δTRANE . As a longer sampling time will reduce the \nvalue of δTRANE  regardless of the sample, it is desirable to have a sensitivity figure of merit \nindependent of the sampling time. Thus, the signal -to-noise ratio by the measurement rate, in the \ncase of a lock-in measurement this is the time co nstant. This yields an equation for the minimum \ndetectable angle, ߠ, with respect to the angle of highest sensitivity, ߠൌ9 0, measurable \nwith the TRANE technique. \n ߠൌߜ்ோோ\n݊݅ܵሺߠሻ൫்ܸோோି௫்ܸோோ൯/2√ܥܶ( S4)\n \nPage 29 \n Figure S5 | Comparison of Hyst eresis Measurement Sensitivity. In this graph we plot \nTRANE – measured hysteresis loops for two different cross sizes. We observe that the sensitivity \nis ߠൌ\t4.6୭/√ݖܪ for the cross with the 52 µm arms and ߠൌ\t0.73/√ݖܪ for the 18 µm \nwide cross.  \n6. Collection Circuit Transfer Coefficient  \nTo determine the transfer coefficient of the collection circuit depicted in Fig. 1b and Fig. \n3a in the main text, we measure the collection voltage from a calibration pulse. Numerical simulations suggests the voltage pulse from TRAN E has a width of 10 ps. With the electronics \navailable, we cannot create a 10 ps pulse to direc tly measure the transfer coefficient. Instead, we \nextrapolate it through measuring the gain of s quare pulses of wider widths. Fig. S6 shows the \ntotal gain in the collection circuit as a function of the square pulse width and the fit with our \nmodel.  \nTo model the gain, we calculate the collection voltage from the amplified calibration \nsquare wave and pulse pattern generator. We treat the amplified calibration voltage into the \nmixer as a square wave with only its spectra l components below 1 GHz due to the bandwidth \nlimitations of the amplifier. The pulse pattern gene rator voltage is modeled as a triangle function \nwith a width of 1.5 ns. The measured voltage after the mixer is the DC component of the \nmultiplication of these two voltages. We fit this m odel to the data in Fig. S6 with the amplitude \nas the only free parameter. Extrapolating the fit,  our model estimates a transfer coefficient of \n0.47 ± 0.04 for a 10 ps pulse.   Page 30 \n  \nFigure S6 | Collection Circuit Gain. We plot the dependence of th e collection circuit gain on \nthe width of a calibrating square pulse. The mode l used to fit the data estimates a transfer \ncoefficient of 0.47 ± 0.04 for a 10 ps TRANE pulse.   \n \nSupplementary References \n1. Slachter, A., Bakker, F. L. & van Wees , B. J. Anomalous Nernst and anisotropic \nmagnetoresistive heating in a lateral spin valve. Phys. Rev. B  84, 020412 (2011). \n2. Eesley, G. L. Generation of nonequilibrium electron and lattice temp eratures in copper by \npicosecond laser pulses. Phys. Rev. B  33, 2144 (1986). \n3. Dobrovinskaya, E. R., Lytvynov, L. A. & Pishchik, V. Sapphire: Material, \nManufacturing, Applications . (Springer, 2009). doi:10.1007/978-0-387-85695-7 \nPage 31 \n 4. Ho, C., Ackerman, M., Wu, K., Oh, S. & Hav ill, T. Thermal conductivity of ten selected \nbinary alloy systems. J. Phys. Chem. Ref. Data  7, 959 (1978). \n5. Bonnenberg, D., Hempel, K. A., Wijn, H.P.J.: 1.2.1.2.10 Thermomagnetic properties, \nthermal expansion coefficient, specific he at, Debye temperature, thermal conductivity . \nWijn, H.P.J. (ed.). Spring erMaterials - The Landolt- Börnstein Database DOI: \n10.1007/10311893_32 \n6. Owen, E. A., Yates, E. L. & Sully, A. H. An X-ray investigation of pure iron-nickel \nalloys. Part 4: the variation of lattice-parameter with composition. Proc. Phys. Soc.  49, \n315 (1937).  \n "    },    {        "title": "1405.6551v1.Magneto_Optical_Spectrum_Analyzer.pdf",        "content": "Magneto-Optical Spectrum Analyzer\nM. Helsen,1,a)A. Gangwar,2A. Vansteenkiste,1and B. Van Waeyenberge1\n1)Department Solid State Sciences, Ghent University, Krijgslaan 281/S1, 9000 Ghent,\nBelgium\n2)Department of Physics, Universit at Regensburg, Universit atsstrasse 31, 93040 Regensburg,\nGermany\n(Dated: 31 August 2021)\nWe present a method for the investigation of gigahertz magnetization dynamics of single magnetic nano\nelements. By combining a frequency domain approach with a micro focus Kerr e\u000bect detection, a high\nsensitivity to magnetization dynamics with submicron spatial resolution is achieved. It allows spectra of single\nnanostructures to be recorded. Results on the uniform precession in soft magnetic platelets are presented.\nI. INTRODUCTION\nMagnetism at the sub-micrometer and nano scale at-\ntracts a great deal of interest for both fundamental rea-\nsons and for their prospective use in logic and memory\napplications. As not only the static properties, such as\nthe magnetic magnetic domain con\fguration, but also\nthe dynamics properties on the sub-nanosecond timescale\n(e.g. the resonances and magnetization switching) are\nstrongly determined by the reduced dimensionality, ap-\npropriate characterization techniques are required. The\nconventional method for high frequency characterisation\nof magnetic systems is the cavity based ferromagnetic\nresonace technique. However, nanostuctures can not be\nstudied in remanence as the \fxed frequency operation\nrequires the bias \feld to be swept. To address this prob-\nlem, di\u000berent techniques have been developed. Depen-\ndent on how the magnetic response is detected they can\nbe divided in two categories. Vector Network Analyzer\nFerromagnetic Resonance (VNA-FMR3,4) and Pulsed In-\nductive Microwave Magnetometry (PIMM5,6) detect the\nresonance electrically and in Time Resolved Magneto-\nOptical Kerr E\u000bect (TR-MOKE7) and Time Resolved\nScanning Transmission X-Ray Microscopy experiments\n(STXM8{10) optical detection is used. The optical meth-\nods have a high sensisitivity to detect the signal of a\nsingle magnetic microstructure, but have a much more\ncomplicated set-up than the electrical methods, and re-\nquire a femtosecond laser or pulsed X-ray source. On the\nother hand, detection in the frequency domain, like con-\nventional FMR and VNA-FMR can achieve much higher\nsignal-to-noise ratios3. Here we present an approach\nwhich combines the frequency domain method with op-\ntical detection: the Magneto-Optical Spectrum Analyzer\n(MOSA).\nThis method is a hybrid method between VNA-FMR\nand TR-MOKE in the context of measurement abilities\nand construction. TR-MOKE makes use of a pulsed laser\nto probe the magnetization ( ~M) through the magneto-\noptical Kerr e\u000bect11{15at regular intervals, while the\na)Electronic mail: Mathias.Helsen@UGent.besample is excited using e.g. another laser pulse or a mi-\ncrowave signal generator. By shifting the arrival time\nof the probe pulses with respect to the excitation, time\ndomain information can be gathered. This method, on\nthe other hand probes the magnetization using a contin-\nuous wave laser and measures (again through the Kerr\ne\u000bect) the fast change of magnetization with an ultrafast\nphotodiode in the frequency domain.\nII. DESCRIPTION OF THE SET-UP\nShown in Fig.1 is the optical layout of our set-up. Light\n(660nm) from a laser diode (LD) is linearly polarized us-\ning a polarizer (Pol) and passes through a non-polarizing\nbeamsplitter (BS). An objective lens (L1) focusses it to a\ndi\u000braction limited spot ( \u0019500nm) on the sample, where\nmicrowave interconnects provide RF current for the ex-\ncitation and an electromagnet can provide a bias \feld of\nup to 50mT.\nThe re\rected light is collected by the same objective\nlens and redirected by the beamsplitter onto an analyzer\n(An). The tranmission axis of this analyzer is set at\nan angle of 45\u000ewith respect to the transmission axis of\nthe \frst polarizer. The beam is then focussed using an\naspheric lens (L2) onto a multimode optical \fber (MM\nFiber), transporting it over a large distance (50 m in our\ncase) to an ultrafast photodiode (PD) with a bandwidth\nof approximately 12GHz. At this point intensity varia-\ntions are measured.\nThe objective lens is mounted on a piezo stage to allow\nscanning of the probe beam over the sample surface. By\nrecording the DC re\rected light intensity, we can image\nthe sample and correctly focus and position the probe\nbeam on the sample. For this purpose the light is de-\n\rected with a mirror towards a conventional photodiode\n(both not shown for clarity).\nThe out-of-plane magnetization dynamics are mea-\nsured via the polar Kerr e\u000bect. When re\recting o\u000b the\nsample, linearly polarized lights gains both an elliptic-\nity (\u000fK) and a rotation of the major polarization axis,\nknown as the Kerr angle ( \u0012K). For out-of-plane saturated\nPermalloy, the polar Kerr angle is typically 1 mrad16.\nBoth the sign and magnitude of this angle depend on the\nsign and magnitude of the out-of-plane magnetization ofarXiv:1405.6551v1  [physics.ins-det]  26 May 20142\nFIG. 1. A basic sketch of the optical part of the set-up, used\nfor polar Kerr detection. LD is a laser diode, Pol a polarizer,\nBS a non-polarizing beamsplitter, L1 an objective lens, An\nan analyzer, L2 and L3 aspheric lenses and PD an ultrafast\nphotodiode. The angle between the transmission axis of Pol\nand An is 45\u000e.\nthe probed area.\nThe re\rected light is analyzed with the polarizer at an\nangle of 45\u000e, thus yielding an intensity of I=I0(1=2 +\n\u0012K), whereI0is the intensity before the analyzer. Placing\nthe analyzer at 45\u000emaximizes the signal.\nThe electrical part of the set-up is shown in Fig. 2.\nOne signal generator produces the RF current at fre-\nquencyfused for exciting the sample. The RF power\nthrough the sample is kept level by using a diode detec-\ntor measuring the power coming out of the sample. The\nbias tee provides the necessary bias voltage and shunts\nthe DC photocurrent. The DC photocurrent \rowing out\nof the bias tee is measured for monitoring purposes.\nAssuming a linear dependence of the light intensity on\nthe magnetization11,14, we can estimate the AC current\ninduced in the photodiode due to magnetization dynam-\nics as\u000eI\u0019IDC\u0012K,max\u000emz, where\u0012K,max is the Kerr angle\nat saturation ( Mz=MS),\u000emthe reduced out-of-plane\nmagnetization ( Mz=MS) andIDCthe DC photocurrent.\nThe AC photocurrent is passed on to the low noise\npreampli\fer, which increases the signal level by 30dB.\nAfter this preampli\fer, a mixer downconverts this high\nfrequency signal to a frequency in the audio range. To\nthis end a second signal generator, phase locked to the\n\frst one, produces a high frequency signal at a frequency\no\u000bseted by several kilohertz (\u0001 f) with respect to the\nexcitation frequency f. Thus, the signal that the mixer\nproduces is at the frequency \u0001 f. It is further ampli\fed\nby a second ampli\fer and \fnally sampled using a high\nend ADC. A computer records the incoming data from\nthe ADC and performs an FFT to compute the signal\nstrength at \u0001 f.\nFIG. 2. A simpli\fed schematic of the electrical part of the\nset-up. A diode detector connected to the sample helps in\nkeeping the power transmitted through the sample level when\nthe frequency is varied. Also shown are the photodiode (PD),\ntermination resistor ( RTerm), bias tee, RF preampli\fer, fre-\nquency mixer, low frequency ampli\fer and ADC. Both signal\ngenerators produce a tone in the microwave range ( f), but\nare slightly o\u000bset by frequency \u0001 fin the kHz range.\nIII. COMPARISON WITH OTHER METHODS\nTo estimate the detection limit of the set-up, we com-\npare the typical signal levels to the fundamental noise\npresent. There are three main contributions to this\nnoise17:\n\u000fa contribution from the DC photocurrent, known\nas shot noise, with a current noise density given by\ninoise=p2qIDC;\n\u000fa contribution from the noise intrinsic to the diode,\nquanti\fed by the Noise Equivalent Power (NEP);\n\u000fand a contribution from the terminating resistor,\nRTerm (50\n), known as Johnson noise, with a volt-\nage noise density given by vnoise=p4kTR Term.\nWe assume that these are sources of white noise and that\nall other possible sources generate much less noise in the\nfrequency range of interest.\nIf we compare this noise with the signal strength, we\narrive at the following expression for the Signal-to-Noise\nRatio (SNR) at the input of the preamp:\nSNR = 10log10I2\nDC\u00122\nKhm2\nzi\nB(2qIDC+ 4kT=R Term + (R\u0001NEP)2);\n(1)\nwhereBis the measurement bandwidth and Rthe re-\nsponsivity of the photodiode (A/W (at a speci\fc wave-\nlength)). Typical values for our set-up would involve\nIDC= 100:0\u0016A,B= 1Hz andhm2\nzi= 10\u00004. The John-\nson noise is the largest contribution (1 :8\u000110\u000011A/p\nHz),\nfollowed by the shot noise (5 :7\u000110\u000012A/p\nHz). In com-\nparision, the NEP for the photodiode we have used is\nnegligible (only 4 :5\u000110\u000017A/p\nHz). Because the contri-\nbution from shot noise is still smaller than the Johnson3\nFIG. 3. Shown is a FFT when both a magnetic signal and a\ndirect coupling are present. In this particular case a Permal-\nloy disc with a diameter of 20 \u0016m was excited at 4GHz with\n+10dBm of power and biased with a \feld of 20mT. The leak-\nage was caused by a microstrip, which was properly termi-\nnated and carried the same amount of power as used for the\nexcitation, but at a slightly di\u000berent frequency.\nnoise, the SNR can be signi\fcantly improved by increas-\ning the light intensity incident on the photodiode. Only\nwhen the photocurrent reaches 1 mA (equivalent to 10\nmW optical power on the diode) does the shot noise ex-\nceed the Johnson noise. However increasing light inten-\nsity also entails heating up to sample, even to above the\nCurie temperature. A further increase in SNR could be\nobtained by cooling the detector, lowering the Johnson\nnoise.\nAdding these terms we \fnd a SNR of approximatly\n34dB. In addition, the \frst ampli\fer adds a noise \fgure\nof 2dB, thus the \fnal SNR is about 32dB. But as the\nsignal itself is on the order of -130dBm or 5 \u000110\u000014mW,\nabsolute care in handling the microwave signals is still\nrequired.\nTo illustrate this, we have excited a sample with\nenough RF power (+10dBm) so that the it would pre-\ncess with\u000emz\u00190:01. The same amount of RF power\nbut at a slightly o\u000bseted frequency was sent through a\nmicrostrip next to the receiver, to allow for a comparison\nbetween the magnetic signal and the direct coupling from\nthe excitation to the receiver. The result in Fig. 3 shows\nthat direct coupling is much stronger than the magnetic\nsignal and that our estimate of the SNR is quite accurate.\nTo this end, the detection has been separated by a large\ndistance from the excitation. Low frequency 1 =fnoise\nis not showing due to a low frequency cut-o\u000b character-\nistic of the low frequency ampli\fer, but the noise \roor\nincreases below 5kHz.\nOur method is complementary to both VNA-FMR and\nTR-MOKE. To the former, we add the advantage of spa-\ntial selectivity. This means that several magnetic struc-\ntures can be fabricated in each others vicinity and we\nare still able to probe each one separately, or that the\nspatial variation of the magnetization dynamics can be\nanalyzed, in contrast to VNA-FMR.\n100µmFIG. 4. A SEM micrograph of a sample used for measure-\nments. The dots are 75 \u0016m and 20\u0016m in diameter.\nWhere TR-MOKE is a time domain method, we are\nmeasuring directly in frequency domain, thus it is eas-\nier to measure resonance curves directly and not have to\nrely on Fourier transforms of time domain data. In com-\nparison to pulsed methods, we have independent control\nof frequency and amplitude, resulting in non-ambiguous\nspectra. Further, our method allows us to probe the sig-\nnal at any arbitrary frequency, something that is di\u000ecult\nto do with stroboscopic time domain measurement meth-\nods.\nOne last advantage of our method is that it does not\nrequire a femtosecond pulsed laser, high end oscilloscope\nor vector network analyzer, thus eliminating a large cost.\nIV. EXPERIMENTAL DETAILS\nAs an illustration we have measured the uniform pre-\ncession of magnetization in Permalloy discs with a 20 \u0016m\ndiameter. The samples were fabricated on a silicon sub-\nstrate. Structure de\fnitions were made with electron\nbeam lithography and the lift-o\u000b technique. After the\nlast metal deposition an ALD coating of Al 2O3was de-\nposited. The samples were then wire bonded to a high\nfrequency substrate. An example of such a sample is\nshown in Fig. 4.\nAn example of a resonance curve where the frequency\nis swept at a \fxed \feld of 45mT, is shown in Fig. 5.\nThe peak was \ftted to a Lorentz curve and the result-\ning resonance frequency was determined to be 6120 \u00062\nMHz, with a linewidth of 188 \u00064 MHz. This illustrates\nthat the linewidth can be accuratly measured on single\nmicroscopic elements.\nThe detector itself has a frequency dependence, mak-\ning the recorded spectrum a product of the magnetic\nresponse and the detector response. To investigate\nthis e\u000bect, a frequency sweep was performed for a dif-\nferent number of bias \felds and the resonance fre-\nquency was determined for each. In Fig. 6 the re-\nsulting datapoints are compared with the Kittel equa-4\nFIG. 5. The resonance curve for a 20 \u0016m diameter, 50nm\nthickness Permalloy disc in a \feld of 45mT. The solid line is\nthe Lorentzian \ft.\nFIG. 6. The evolution of the resonance peak of the uniform\nprecession when the \feld is increased. The Kittel equation\nfor a thin \flm is shown as a black line.\ntion, which for an in\fnitly thin \flm is given by fRes=\n28:0p\n\u00160HBias(\u00160HBias+\u00160Ms) GHz/T. Literature val-\nues have been used for calculating the Kittel equation\n(\u00160Ms= 1:04T)18. The datapoints are in good agree-\nment with the Kittel equation, ruling out strong fre-\nquency variations in the detection which might interfere\nwith measurements.\nFinally, the magnetic signal can also be used for imag-\ning purposes; when the laser beam is scanned over the\nsample using the piezo stage and the magnetic signal at\neach point is recorded. This can be compared with the\nre\rectivity, which is acquired simultaneously. When the\nsample is excited at resonance, the contrast is highest.\nAn example is shown in Fig. 7. Here we compare the\nre\rectivity (clearly showing the Au CPW, Si substrate\nand Permalloy discs) with the magnetic signal showing\nonly the Permalloy disc.\nFIG. 7. At the top an image generated using the magnetic\nresponse of a uniform resonance in a 3 \u0016m dot at 6GHz. Shown\nat the bottom is an image generated using re\rectivity data\nthat was collected simultaneously.\nV. CONCLUSION\nWe have developed a method of probing magnetization\ndynamics at the multiple-GHz range using a frequency\ndomain method that o\u000bers spatial sensitivity. The set-up\nis relatively simple, yet allows for high quality measure-\nments, thus enabling a fast exploration of excitation pa-\nrameters. We have illustrated that our method can yield\nquantitative results using uniform excitation on single\nmicroscopic elements.\nACKNOWLEDGMENTS\nMathias Helsen, Arne Vansteenkiste and Bartel Van\nWaeyenberge acknowledge funding by FWO-Vlaanderen\nand BOF-UGent.\n1\n2\\-1.\"\n3I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno, G. Gub-\nbiotti, and C. Back, \\Comparison of frequency, \feld, and time\ndomain ferromagnetic resonance methods,\" J. Magn. Magn.\nMater. , vol. 307, p. 148, 2006.\n4S. Kalarickal, P. Krivosik, M. Wu, C. Patton, M. Schneider,\nP. Kabos, T. Silva, and J. Nibarger, \\Ferromagnetic resonance\nlinewidth in metallic thin \flms: Comparison of measurement\nmethods,\" J. Appl. Phys. , vol. 99, p. 093909, 2006.\n5A. Kos, T. Silva, and P. Kabos, \\Pulsed inductive microwave\nmagnetometer,\" Rev. Sci. Instrum. , vol. 73, p. 3563, 2002.\n6T. Silva, C. Lee, T. Crawford, and C. Rogers, \\Inductive\nmeasurement of ultrafast magnetization dynamics in thin-\flm\npermalloy,\" J. Appl. Phys. , vol. 85, p. 7849, 1999.\n7Y. Acremann, M. Buess, C. Back, M. Dumm, G. Bayreuther, and\nD. Pescia, \\Ultrafast generation of magnetic \felds in a Schottky\ndiode,\" Nature , vol. 414, p. 51, 2001.5\n8M. Bolte, G. Meier, B. Kr uger, A. Drews, R. Eiselt, L. Bocklage,\nS. Bohlens, T. Tyliszczak, A. Vansteenkiste, B. Van Waeyen-\nberge, K. W. Chou, A. Puzic, and H. Stoll, \\Time-resolved x-ray\nmicroscopy of spin-torque-induced magnetic vortex gyration,\"\nPhys. Rev. Lett. , vol. 100, p. 176601, Apr 2008.\n9A. Vansteenkiste, K. W. Chou, M. Weigand, M. Curcic, V. Sack-\nmann, H. Stoll, T. Tyliszczak, G. Woltersdorf, C. H. Back,\nG. Schutz, and B. Van Waeyenberge, \\X-ray imaging of the\ndynamic magnetic vortex core deformation,\" Nat Phys , vol. 5,\npp. 332{334, May 2009.\n10M. Kammerer, M. Weigand, M. Curcic, M. Noske, M. Sproll,\nA. Vansteenkiste, B. Van Waeyenberge, H. Stoll, G. Woltersdorf,\nC. H. Back, and G. Schuetz, \\Magnetic vortex core reversal by\nexcitation of spin waves,\" Nat Commun , vol. 2, pp. 279{, Apr.\n2011.\n11P. N. Argyres, \\Theory of the Faraday and Kerr E\u000bects in Fer-\nromagnetics,\" Phys. Rev. , vol. 97, no. 2, pp. 334{345, 1955.12J. Zak, E. Moog, C. Liu, and S. Bader, \\Universal approach to\nmagneto-optics,\" J. Magn. Magn. Mater. , vol. 89, p. 107, 1990.\n13S. Polisetty, J. Sche\u000fer, S. Sahoo, Y. Wang, T. Mukherjee,\nX. He, and C. Binek, \\Optimization of magneto-optical kerr\nsetup: Analyzing experimental assemblies using Jones matrix\nformalism,\" Rev. Sci. Instrum. , vol. 79, p. 055107, 2008.\n14A. Zvezdin and V. Kotov, Modern Magnetooptics and Mange-\ntooptic Materials . Institute of Physics Publishing, 1997.\n15Z. Qiu and S. Bader, \\Surface magneto-optic Kerr e\u000bect,\" Rev.\nSci. Instrum. , vol. 71, no. 3, p. 1243, 2000.\n16M. Veis and R. Antos, \\Advances in Optical and Magnetooptical\nScatterometry of Periodically Ordered Nanostructured Arrays,\"\nJ. Of Nanomaterials , 2012.\n17P. Horowitz and W. Hill, The art of electronics . Cambridge\nUniversity Press, 1989.\n18J. Coey, Magnetism and Magnetic Materials . Cambridge Uni-\nversity Press, 2010."    },    {        "title": "1601.05521v1.Vortex_Dynamics_Mediated_Low_Field_Magnetization_Switching_in_an_Exchange_Coupled_System.pdf",        "content": "Zhou et al. Page 1     Vortex Dynamics-Mediated Low-Field Magnetization Switching in  an Exchange-Coupled System    Weinan Zhou,1 Takeshi Seki,1,2* Hiroko Arai,2,3 Hiroshi Imamura,3 Koki Takanashi1   1Institute for Materials Research, Tohoku University, Sendai, 980-8577, Japan 2JST PRESTO, Saitama, 332-0012, Japan 3National Institute of Advanced Industrial Science and Technology, Tsukuba, 305-8568, Japan    *Corresponding author: go-sai@imr.tohoku.ac.jp     Zhou et al. Page 2   A magnetic vortex [1, 2] h as a t t r a c t e d  s i g n i f i c a n t  a t t e n t i o n  s i n c e  i t  i s a topologically stable magnetic structure i n  a  s o f t  m a g n e t i c  n a n o d i s k . Many studies have been devoted to understanding the nature of magnetic vortex in isolated systems. Here we show a new aspect o f  a  m a g n e t i c  vortex the dynamics of which strongly affects the magnetic structures of environment. We exploit a nanodot of an exchange-coupled bilayer w i t h a soft magnetic Ni81Fe19 (permalloy; Py) having a magnetic vortex a n d  a  p e r p e n d i c u l a r l y  m a g n e t i z e d  L10-FePt exhibiting a large switching field (Hsw). The vortex dynamics with azimuthal spin waves m a k e s  t h e  excess energy accumulate in the Py, which triggers the reversed-domain nucleation in the L10-FePt a t  a  l o w  m a g n e t i c  f i e l d. Our results shed light on the non-local mechanism of a reversed-domain nucleation, and provide with a route for efficient Hsw reduction that is needed for ultralow-power spintronic devices [3].  (146 words)  Zhou et al. Page 3  A magnetic vortex in a soft magnetic disk is an in-plane curling magnetic structure having a core whose magnetic moments are normal to the disk plane [1,2]. Magnetic vortices have fascinated us because of their unique functionalities [4] a n d  r i c h  p h y s i c s  [5]. Several kinds of non-equilibrium dynamical motion can be excited by applying an rf magnetic field (Hrf) [5-7] or injecting spin current [8-10], leading to promising applications such as a vortex-type magnetic random access memory and a spin torque vortex oscillator. At certain conditions, the vortex polarity (core magnetization direction) and/or the circulation of in-plane magnetic moments can be switched [11-14]. Those studies focus on the control of magnetic moments in the vortex. Although interplay of the vortex in a magnet and the magnetization in an adjacent exchange-coupled magnet was investigated in a previous paper [15], no one has t r i e d  t o  use m a g n e t i c  v o r t e x  d y n a m i c s  in a soft magnet as a route to switching t h e  magnetization of a  h a r d  m a g n e t. Here, we show Hrf-induced vortex dynamics in soft magnetic Py non-locally triggers the magnetization switching of h a r d  m a g n e t i c  L10-FePt, which can balance competing goals for reducing Hsw a n d  m a i n t a i n i n g  t h e  t h e r m a l  s t a b i l i t y  o f  m a g n e t i z a t i o n  i n  a  nanosized magnet. We used the exchange-coupled system consisting of nanodots with a hard magnetic L10-FePt layer and a soft magnetic Py layer (Figs. 1a and 1b). The 10-nm-thick L10-FePt had large uniaxial magnetocrystalline anisotropy (K) along the perpendicular direction to the disk plane (z direction), whereas the 150-nm-thick Py possessed negligible K, but the nanodot shape induced the Zhou et al. Page 4  shape magnetic anisotropy. This enabled us to saturate the magnetic moments of Py in the z direction by applying a d c  m a g n e t i c  f i e l d  (H) perpendicular to the disk plane that was lower than the demagnetizing field for the thin film form [16].  First, we performed micromagnetic simulations to reveal the equilibrium magnetic state. The simulated magnetization (M) versus H i s  s h o w n  i n  Fig. 1c, where M w a s  n o r m a l i z e d  b y  t h e  saturation value a n d  H w a s  a p p l i e d  a l o n g  t h e  z d i r e c t i o n. The M - H c u r v e e x h i b i t s  a two-step behaviour. As H is swept from -10 kOe, M starts to increase at H ~ -2.5 kOe. As depicted in Fig. 1d, at H =  0  k O e ,  the magnetic vortex is formed i n  P y  w h e r e a s  all the magnetic moments in L10-FePt saturate along the -z direction. The vortex structure has a small deformation, in which the magnetic moments are slightly tilted from the azimuthal d i r e c t i o n  o f  t h e  d i s k  t o  t h e  r a d i a l  d i r e c t i o n .  As H increases to 2.6 kOe, the magnetic moments are tilted to the +z direction (mz in Fig. 1e), although the vortex structure is still maintained (mx i n  Fig. 1e). Increasing H t o  5 kOe compresses the vortex structure in Py to t he i nt er face, in which the core polarity is switched (Fig. 1f). By comparing the cross-sectional x - y i m a g e s  near t h e  i n t e r f a c e  (Figs. 1e and 1f), one can see that there is no remarkable change i n  t h e  mx c o m p o n e n t  e v e n  f o r  t h e  c o m p r e s s e d  v o r t e x  s t r u c t u r e .  These m a g n e t i c  states a r e  t o t a l l y  d i f f e r e n t  f r o m  t h e  s p a t i a l l y  t w i s t e d  m a g n e t i c  s t r u c t u r e s  o b s e r v e d  i n  the in-plane magnetized L10-FePt | Py bilayers [17].  Next, we experimentally examined the question of whether t h e  vortex d y n a m i c s  i n  P y  Zhou et al. Page 5  affect Hsw i n  L10-FePt. Fig. 2a d i s p l a y s  the full M - H c u r v e e x h i b i t i n g  t w o-step magnetization reversal behaviour similar to the simulation. W h e n  H was s w e p t  f r o m  p o s i t i v e  t o  n e g a t i v e ,  the magnetization switching of the L10-FePt o c c u rred i n  t h e  r a n g e  f r o m  -6 kOe to -9 kOe. The minor magnetization curves showing spring-back behaviour also suggest t h a t t h e  m a g n e t i c  m o m e n t s  o f  L10-FePt (mFePt) switched in the hatched H region in Fig. 2a (see Supplementary Fig. S1). In order to evaluate Hsw of L10-FePt under the vortex dynamics excitation in Py, we measured the anisotropic magnetoresistance (AMR) effect for the nanodot array located on a coplanar waveguide (CPW) (see Methods and Supplementary Fig. S2). Fig. 2b shows the electrical resistance (R) as a function of H without Hrf being applied. At large positive H, e.g. H = 9 kOe, all the magnetic moments in the bilayer were aligned with H, giving a low R value. As H decreased, mFePt maintained the positive value while the magnetic moments in Py (mPy) rotated gradually, forming a spatially non-uniform magnetic structure. The part of mPy directed along the signal line of the CPW increased the value of R due to the AMR effect. As H decreased further, mFePt started to switch and eventually all the magnetic moments saturated again at H = - 9 kOe. As indicated by the hatched areas in Figs. 2a and 2b, the H regions showing the switching of mFePt in the R - H curve are in good agreement with those in the full M - H curve. In case of no Hrf, i.e. n o  excitation of vortex d y n a m i c s, Hsw is obtained to be 8.6 kOe as indicated by the green arrows.  To excite the vortex dynamics in Py, we applied Hrf transverse to the signal line of the CPW Zhou et al. Page 6  by injecting rf power of 22 dBm, which c o r r e s p o n d e d  t o Hrf = 200 Oe. The representative ΔR - H curve is shown in Fig. 2c, where ΔR is the resistance change from R at H = -11 kOe. The frequency (f) of Hrf was 11 GHz. The shape of ΔR - H curve for f = 11 GHz (solid circles) is rather different from that without Hrf ( open ci r cl es). R sharply dr ops t o t he l ow R st at e at  H = ±2.8 kOe, indicating that applying Hrf with f = 11 GHz significantly reduced Hsw. Fig. 2d summarizes Hsw and R as a function of f. Compared to the value of Hsw with no Hrf applied, one can see a small decrease in Hsw in the whole f region when 22 dBm of rf power was injected. This f-independent decrease is attributable to Joule heating caused by the high rf power injection. In addition to the f-independent decrease, a strong reduction of Hsw is evident in the range of 11 ≤ f ≤ 17 GHz. In this f range, Hsw gradually increases, and the values of Hsw in f ≥  1 8  G H z  a r e  a l m o s t  t h e  s a m e  a s those in 6  G H z  ≤ f ≤  1 0  G H z. This reduction is not due to the Joule heating because R reflecting the device temperature does not show any correlation with Hsw. We can also exclude the possibility that excitation of uniform magnetization dynamics in L10-FePt could lead t o  t h e  l o w Hsw b e c a u s e  t h e  r e s o n a n c e  f r e q u e n c y  o f  L10-FePt is estimated to be about 170 G H z .  The numerical simulation reproduces the experimental results of f dependence of Hsw as shown by the open circles in Fig. 2d.   Figs. 3a - 3e show snapshots of the time evolution of mz for f = 11 GHz and H = 3.4 kOe (see also Supplementary Movie 1). The sliced planes of Py (top panels) and FePt (bottom panels) are 2 nm and 5 nm, respectively, away from the L10-FePt | Py interface. Inhomogeneous dynamics are Zhou et al. Page 7  excited in the Py (Figs. 3b - 3d). At t = 0.525 nsec, reversed-domain nucleation occurs in L10-FePt beneath the vortex core in Py (Fig. 3c). The reversed domain expands coherently in Py and L10-FePt (Fig. 3d). Similar inhomogeneous dynamics are excited for all conditions of f when Hsw is reduced (see Supplementary Fig. S3). Now let us discuss the magnetization switching process induced by vortex dynamics. Several excitations such as gyrotropic motion of vortex core, azimuthal spin waves, and radial spin waves have been observed in Hrf-induced vortex dynamics for a single soft magnetic disk [7, 18-20]. In order to assign which dynamical mode is responsible for the switching we observed, we calculated the deviation of the magnetization (dm) from the equilibrium state. The time evolution of the z-component dm (dmz) at f =  1 1  G H z  are displayed i n  Fig. 4. The borders between the red region (positive dmz) and blue one (negative dmz) correspond to nodes of spin waves. One sees that there are several n o d e s  e x h i b i t i n g c l o c k w i s e  r o t a t i o n, in which the wave vectors of spin waves are along the azimuthal direction. In addition, we have the node surrounding the vortex core. This node is attributable to the standing spin wave along the radial direction. Consequently, we consider that the eigenmode is the azimuthal spin wave having the node in the radial direction. Near the vortex core located at the center, dmz shows the steep spatial change in the narrow region.  The azimuthal r o t a t i o n  o f  s p i n  w a v e  e n l a r g e s t h e  area o f  v o r t e x  c o r e  o r  produces the multiple vortex cores in some cases (see Supplementary Fig. S3). When the area of vortex core reaches a critical size, the reversed-domain nucleation occurs in L10-FePt. We quantitatively evaluate Zhou et al. Page 8  the nucleation volume in L10-FePt from the measurement temperature (T) dependence of Hsw without Hrf (Fig. 5a). According to the Néel-Arrhenius law, the T dependence of Hsw can be fitted by [21] \n€ HswT()=Hsw,0T()1−kBTE0(T)lnkBTE0(T)Hsw,0T()f0R⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ ,  (1) where Hsw,0 is the switching field without thermal agitation, kB is the Boltzmann constant, and E0 is the energy barrier given by the product of K and the magnetic volume (V). f0 is the attempt frequency (109 Hz). R i s  t h e  r a t e  o f  H s w e e p (10 Oe/sec). We assume no remarkable T d e p e n d e n c e  o f  Hsw,0 as previously reported for FePt [22]. Hsw,0 and E0 are obtained to be 14.4 ± 0.7 kOe and (0.9 ± 0.3) × 10-18 J, respectively. Using K = 3.1 × 106 J/m3 evaluated experimentally from the M - H curves for the FePt single layer, we obtain V =  3 0 0 ± 100 nm3, which corresponds to the nucleation volume under the static H (Vnuc, H). On the other hand, the nucleation volume under the excitation of vortex dynamics (Vnuc, D) is estimated numerically. Let us once again look at the case of FePt switching under Hrf of f = 11 GHz and H = 3.4 kOe. Fig. 5b displays simulated mz as a function of the position in the dot along the in-plane y direction (Dy) at various t. At t = 0.51 nsec, the region having mz > 0.9 appears, which is defined Vnuc, D. T h u s Vnuc, D is estimated to be ~ 800 n m3. T h i s  Vnuc, D i s  c o m p a r a b l e  t o  Vnuc, H, suggesting that both nucleation processes have comparable E0. We calculated the t dependence of total mz (mz,total) and total energy (Etotal), which is the sum of Zeeman energy (EZ), demagnetizing energy (Ed), exchange energy (Eex) and anisotropy energy (Eani) (Fig. 5c). Etotal gradually increases just after starting the vortex dynamics excitation, and then Etotal decreases whereas mz increases. As shown in the Zhou et al. Page 9  middle panel of Fig. 5c, the time dependent energy difference (ΔE) indicates that only Eex contributes to the increase of Etotal. At t = 0.51 nsec, the excess ΔEex = 5 × 10-17 J is accumulated mainly in Py (bottom panel of Fig. 5c), which is large enough to overcome E0 = (0.9 ± 0.3) × 10-18 J. Consequently, the accumulated Eex in Py due to the vortex dynamics excitation becomes a non-local trigger for the reversed-domain nucleation in L10-FePt. Again we emphasize that mere vortex core switching does not nucleate the reversed domain in L10-FePt (Fig. 1c) and the vortex dynamics excitation is essential for the reversed domain nucleation. Our results lead to not only insight into the nucleation phenomena but also a new way f o r  i n f o r m a t i o n  w r i t i n g  of m a g n e t i c  s t o r a g e a n d  s p i n t r o n i c a p p l i c a t i o n s u s i n g  topologically unique magnetic structures.  METHODS Device Fabrication An ultrahigh vacuum magnetron sputtering system and an ion beam sputtering (IBS) system were used f o r  t h i n  f i l m  p r e p a r a t i o n .  T h e  thin films w e r e  g r o w n  o n  a n  M g O ( 1 0 0 )  s i n g l e  c r y s t a l  substrate with the stack of MgO subs. || Fe (1) | Au (60) | FePt (10) | Py (150) | Au (5) | Pt (3) (in nanometer). All the layers except FePt were grown at ambient temperature. The FePt (001) layer was epitaxially grown on the Au (100) buffer layer using the magnetron sputtering system. The substrate temperature was set at 550 ºC. The high temperature deposition promoted the L10 ordering in FePt, Zhou et al. Page 10  resulting to a large K. As a result, the FePt layer had the easy magnetization axis normal to the film plane. After the FePt deposition, the sample was transferred to the IBS chamber to deposit the Py | Au | Pt layers. In order to induce the perpendicular component of magnetization in the soft magnetic Py at small H, the thin films were microfabricated into circular nanodots through the use of electron beam lithography and Ar ion milling to reduce the demagnetization field of Py. The scanning electron microscope (SEM) image shows that the dots are about 260 nm in diameter. The details concerning the thin film preparation and the microfabrication process are described in [16].  Device Characterization The magnetization curves for the dot arrays were measured using a superconducting quantum interference device magnetometer at room temperature. On the other hand, R was measured for the dot array located on the CPW using a lock-in amplifier. The Au buffer layer was patterned into the CPW with the signal line of 4 µm × 50 µm, and an array of more than 1000 dots of the FePt | Py bilayer was placed on the signal line. A signal generator was connected to the device through the RF port of a bias-Tee to apply the rf power to the CPW, which generated Hrf along the transverse direction to the signal line. H w a s  a p p l i e d  p e r p e n d i c u l a r  t o  t h e  s a m p l e . The AMR curves were averaged for three measurements, and a linear change in R due to the resistance drift was calibrated. Micromagnetic simulation Zhou et al. Page 11  We have used the mumax3 package [23] for full micromagnetic simulations of FePt | Py magnetic layers. The simulated structure was a cylindrical d o t  200 nm in diameter a n d  10 nm in thickness for the FePt layer and 150 nm in thickness for the Py layer. The sample was divided into discrete computational cells, and the size of each cell was 3.125 × 3.125 × 2.5 nm3. The saturation magnetization and the uniaxial anisotropy constant for FePt were Ms=1.15 × 106 A/m and Ku = 3.2 × 106 J / m3, respectively. The easy axis of the FePt was in the z-direction as shown in Fig. 1a. The material parameters for Py were Ms = 0.8×106 A/m, and Ku = 0 J/m3. We chose the stiffness constant of A = 1.3 × 10-11 J/m for whole the system. In the M - H curve simulation, H was applied along the z-direction from -10 kOe to 10 kOe and from 10 kOe and -10 kOe in 200 Oe stepwise increments. In this calculation, the damping parameter of α = 0.5 was used for both materials in order to expedite relaxation to the equilibrium orientation. The magnetization dynamics under Hrf were c a l c u l a t e d  b y  applying Hext(t) = (0, H'sin(2πft), H), where H' a n d  f a r e  t h e  a m p l i t u d e  a n d  t h e  f r e q u e n c y  o f  Hrf, respectively. The amplitude H' was set to be 200 Oe. For the magnetization dynamics calculation, the values of α were set at 0.1 for FePt and 0.01 for Py. Thermal effects were neglected throughout the simulation for simplicity. Zhou et al. Page 12  References 1. Cowburn, R. P., Koltsov, D. K., Adeyeye, A. O., Welland, M. E., & Tricker, D. M. Single-Domain Circular Nanomagnets. Phys. Rev. Lett. 83, 1042-1045 (1999). 2. Shinjo, T., Okuno, T., Hassdorf, R., Shigeto, K. & Ono, T. Magnetic vortex core observation in circular dots of permalloy. Science 289, 930-932 (2000). 3. Kent, A. D. & Worledge, D. C. A New spin on Magnetic memories. Nature Nanotech. 10, 187-191 (2015). 4. Bohlens, S., Krüger, B., Drews, A., Bolte, M., Meier, G. & Pfannkuche, D. Current controlled random-access memory based on magnetic vortex handedness. Appl. Phys. Lett. 93, 142508 (2008). 5. Choe, S.-B., Acremann, Y., Scholl, A., Bauer, A., Doran, A., Stöhr, J. & Padmore, H. A. Vortex core-driven magnetization dynamics. Science 304, 420-422 (2004). 6. Buess, M., Höllinger, R., Haug, T., Perzlmaier, K., Krey, U., Pescia, D., Scheinfein, M. R., Weiss, D. & Back C. H. Fourier Transform Imaging of Spin Vortex Eigenmodes. Phys. Rev. Lett. 93, 077207 (2004). 7. Novosad, V., Fradin, F. Y., Roy, P. E., Buchanan, K. S., Guslienko, K. Y. & Bader, S. D. Magnetic vortex resonance in patterned ferromagnetic dots. Phys. Rev. B 72, 024455 (2005). Zhou et al. Page 13  8. Kasai, S. Nakatani, Y., Kobayashi, K., Kohno, H. & Ono T. Current-Driven Resonant Excitation of Magnetic Vortices. Phys. Rev. Lett. 97, 107204 (2006). 9. Pribiag, V. S., Krivorotov, I. N., Fuchs, G. D., Braganca, P. M., Ozatay, O., Sankey, J. C., Ralph, D. C. & Buhrman, R. A. Magnetic vortex oscillator driven by d.c. spin-polarized current. Nature Phys. 3, 498-503 (2007). 10. Dussaux, A., Georges, B., Grollier, J., Cros, V., Khvalkovskiy, A. V., Fukushima, A., Konoto, M., Kubota, H., Yakushiji, K., Yuasa, S., Zvezdin, K. A., Ando, K. & Fert, A. Large microwave generation from current-driven magnetic vortex oscillators in magnetic tunnel junctions. Nature Commun. 1:8. doi: 10.1038/ ncomms1006 (2010). 11. Van Waeyenberge, B., Puzic, A., Stoll, H., Chou, K. W., Tyliszczak, T., Hertel, R., Fähnle, M., Brückl, H., Rott, K., Reiss, G., Neudecker, I., Weiss, D., Back, C. H., & Schütz, G. Magnetic vortex core reversal by excitation with short bursts of an alternating field. Nature 444, 461-464 (2006). 12. Kammerer, M., Weigand, M., Curcic, M., Noske, M., Sproll, M., Vansteenkiste, A., Van Waeyenberge, B., Stoll, H., Woltersdorf, G., Back, C. H., & Schütz, G. Magnetic vortex core reversal by excitation of spin waves. Nature Commun. 2:279 doi: 10.1038/ncomms1277 (2011). 13. Yamada, K., Kasai, S., Nakatani, Y., Kobayashi, K., Kohno, H., Thiaville, A .  &  O n o ,  T .  Electrical switching of the vortex core in a magnetic disk. Nat. Mater. 6, 269-273 (2007). Zhou et al. Page 14  14. Uhlíř, V., Urbánek, M., Hladík, L., Spousta, J., Im, M-Y., Fischer, P . , Eibagi, N . , Kan, J .  J . ,  Fullerton, E. E. & Šikola, T. Dynamic switching of the spin circulation in tapered magnetic nanodisks. Nat. Nanotech. 8, 341-346 (2013). 15. Wohlhüter, P., Bryan, M. T., Warnicke, P., Gliga, S., Stevenson, S. E., Heldt, G., Saharan, L., Suszka, A. K., Moutafis, C., Chopdekar, R. V., Raabe, J., Thomson, T., Hrkac, G. & Heyderman L. J. Nanoscale switch for vortex polarization mediated by Bloch core formation in magnetic hybrid systems. Nature Commun. 6:7836 doi: 10.1038/ncomms8836 (2015). 16. Zhou, W., Seki, T., Iwama, H., Shima, T. & Takanashi, K. Perpendicularly magnetized L10-FePt nanodots exchange-coupled with soft magnetic Ni81Fe19. J. Appl. Phys. 117, 013905 (2015). 17. Seki, T., Utsumiya, K., Nozaki, Y., Imamura, H. & Takanashi, K. Spin wave-assisted reduction in switching field of highly coercive iron-platinum magnets. Nat. Commun. 4:1726 doi: 10.1038/ncomms2737 (2013). 18. Neudecker, I., Perzlmaier, K., Hoffmann, F., Woltersdorf, G., Buess, M., Weiss, D. & Back, C. H. Modal spectrum of permalloy disks excited by in-plane magnetic fields. Phys. Rev. B 73, 134426 (2006). 19. Aliev, F. G., Sierra, J. F., Awad, A. A., Kakazei, G. N., Han, D.-S., Kim, S.-K., Metlushko, V., Ilic, B. & Guslienko, K. Y. Spin waves in circular soft magnetic dots at the crossover between vortex and single domain state. Phys. Rev. B 79, 174433 (2009). Zhou et al. Page 15  20. Zhu, Z., Liu, Z., Metlushko, V., Grütter, P. & Freeman, M. R. Broadband spin dynamics of the magnetic vortex state: Effect of the pulsed field direction, Phys. Rev. B 71, 180408 (2005). 21. El-Hilo, M., de Witte, A. M., O’Grady, K. & Chantrell, R. W. The sweep rate dependence of coercivity in recording media, J. Magn. Magn. Mater. 117, L307-L310 (1992). 22. Kikuchi, N., Okamoto, S., Kitakami, O., Shimada, Y. & Fukamichi, K. Sensitive detection of irreversible switching in a single FePt nanosized dot, Appl. Phys. Lett. 82, 4313-4315 (2003). 23. Vansteenkiste, A., Leliaert, J., Dvornik, M., Helsen, M., Garcia-Sanchez, F. & Van Waeyenberge, B. The design and verification of MuMax3. AIP Advances 4, 107133 (2014).   Acknowledgements This work was partially supported by a Grant-in-Aid for Young Scientists A (25709056) and Grant-in-Aid for Scientific Research S (23226001). The device fabrication was partly performed at Cooperative Research and Development Center for Advanced Materials, IMR, Tohoku University.  Competing financial interests Zhou et al. Page 16  The authors declare that they have no competing financial interests.  Author contributions T.S. and K.T. planned and supervised the study. W.Z. prepared the thin films, fabricated the devices, and carried out the measurements on the magnetic and electrical properties. W.Z. and T.S. analyzed the experimental data. H.A. and H.I. performed the micromagnetic simulation. All of the authors contributed to the physical understanding and manuscript preparation.   Figure 1 | L10-FePt | permalloy (Py) exchange-coupled system. a, Schematic illustration of a microfabricated dot. The coordinate axes are also shown. b, Scanning electron microscope image for the array of dots. c, Simulated result of magnetization (M) versus the dc magnetic field (H), where M was normalized by the saturation value and H was applied along the z direction. d, Simulated structures of magnetic moments (m) of the x component (mx, upper panel) and the z component (mz, lower panel) at H = 0 Oe. e and f, Simulated magnetic structures of mx in the y - z plane (left panel), mz in the y - z plane (middle panel), and mx in the x - y plane (right panel) at H = 2.6 kOe and 5 kOe.   Zhou et al. Page 17  Figure 2 | Evaluation of switching field (Hsw) for microfabricated dots with L10-FePt | Py. a, M as a function of H, where the value of M was normalized by the saturation magnetization. b, Electrical resistance of device (R) as a function of H. The rf magnetic field (Hrf) was not applied. The red (blue) circles denote the results when H was swept from positive (negative) to negative (positive). c, ΔR - H curves with Hrf (solid circles) and without Hrf being applied (open circles). ΔR is the resistance change from R at H = -11 kOe. 22 dBm of rf power was applied to the device, which corresponded to Hrf = 200 Oe. The frequency (f) of Hrf was set at 11 GHz. For both M - H and R - H curves, H was applied perpendicular to the device plane. The green arrows denote Hsw of L10-FePt. d, Hsw ( t o p  p a n e l )  a n d  R ( b o t t o m  p a n e l )  a s  a  function of f. The red dotted line denotes Hsw without Hrf, and the solid (open) circles represent the experimental (simulated) results. R is the value at H = -11 kOe.  Figure 3 | Time evolution of magnetic structures under excitation of vortex dynamics. a - e, Snapshots for mz at H = 3.4 kOe under the application of Hrf with f = 11 GHz. Top and bottom panels are x - y plane images for Py and FePt, respectively. The planes of the Py and FePt slices are 2 nm and 5 nm, respectively, away from the FePt | Py interface.   Figure 4 | Time evolution of deviated magnetic structures from the equilibrium state. a Zhou et al. Page 18  - e, Snapshots for deviation of mz (dmz) for Py at H = 3.4 kOe under the application of Hrf with f = 11 GHz. The plane of the Py slice was 2 nm away from the FePt | Py interface.   Figure 5 |  Volume and energy for reversed-domain nucleation. a, Hsw as a function of measurement temperature (T), which was experimentally obtained from the temperature dependence of M - H curves without Hrf being applied. The red solid line denotes the result of fitting. b, mz a s  a  f u n c t i o n  o f  t h e  p o s i t i o n  i n  t h e  d o t  a l o n g  t h e  i n-plane y d i r e c t i o n  (Dy) at various times (t). Dy is denoted in the inset. H was set at 3.4 kOe. The plane of the FePt slice was 5 nm away from the FePt | Py interface. c, (Top panel) Calculated t dependence of total mz (mz,total) and total energy (Etotal). Since H was set at 3.4 kOe, mz,total shows the value of ~ 0.7 before the reversed-domain nucleation. (Middle panel) t dependent energy difference (ΔE) for Zeeman energy (EZ), demagnetizing energy (Ed), exchange energy (Eex) and anisotropy energy (Eani). (Bottom panel) t dependence of Eex in Py and L10-FePt.  Zhou et al. Page 19      \n  Figure 1 \nZhou et al. Page 20    \n  Figure 2 \nZhou et al. Page 21    \n   Figure 3 \nZhou et al. Page 22     \n  Figure 4  \nZhou et al. Page 23     \n   Figure 5   \nZhou et al. Page 24  Vortex Dynamics-Mediated Low Field  Magnetization Switching  in an Exchange-Coupled System -Supplementary Information-  Weinan ZHOU, Takeshi SEKI, Hiroko ARAI, Hiroshi IMAMURA, Koki TAKANASHI  This Supplementary Information includes 3 figures (Figures S1 - S3).  \n   Supplementary Figure S1 | Spring-back behavior in nanodots with exchange-coupled L10-FePt | Py. a, Minor magnetization curve, where the perpendicular magnetic field (H) direction was reversed at Hsp = -6 kOe. H was swept from 10 kOe to -6 kOe, and then H was swept to 10 kOe. b, M i n o r  m a g n e t i z a t i o n  c u r v e  w i t h  Hsp =  -8 kOe. The gray solid circles denote the full magnetization curve. The reversible minor magnetization curve was obtained for Hsp =  -6 kOe whereas the hysteresis appeared for Hsp = -8 kOe. This indicates that the magnetic moments of L10-FePt are switched in a part of dots when H was swept to -8 kOe. \nZhou et al. Page 25       Supplementary Figure S2 |  Setup of electrical measurement for evaluating the switching field. The electrical r e s i s t a n c e  o f  t h e  d e v i c e w a s  m e a s u r e d  f o r  t h e  d o t  a r r a y  l o c a t e d  o n  t h e  coplanar waveguide (CPW) using a lock-in amplifier. A signal generator was connected with the device through the RF port of a bias-Tee to apply the rf power to the CPW, which generated the rf magnetic field along the transverse direction to the signal line. The magnetic field was applied perpendicular to the device plane. \nZhou et al. Page 26       Supplementary Figure S3 | Snapshots of simulated mz at the nucleation of switched domain in L10-FePt. The frequency was varied in the range from 1 GHz to 24 GHz. Top and Bottom panels are the x - y plane images for Py and FePt, respectively. The sliced planes of Py and FePt are 2 nm and 5 nm away from the Py | FePt interface. \n"    },    {        "title": "1604.00626v1.Spin_relaxation_signature_of_colossal_magnetic_anisotropy_in_platinum_atomic_chains.pdf",        "content": "Spin relaxation signature of colossal magnetic\nanisotropy in platinum atomic chains\nAnders Bergman1, Johan Hellsvik2, Pavel F. Bessarab2, and Anna Delin1,2,3,*\n1Department of Physics and Astronomy, Materials Theory Division, Uppsala University, Box 516, SE-75120\nUppsala, Sweden\n2Department of Materials and Nano Physics, School of Information and Communication Technology, KTH Royal\nInstitute of Technology, Electrum 229, SE-16440 Kista, Sweden\n3Swedish e-Science Research Center (SeRC), KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden\n*annadel@kth.se\nABSTRACT\nRecent experimental data demonstrate emerging magnetic order in platinum atomically thin nanowires. Furthermore, an\nunusual form of magnetic anisotropy – colossal magnetic anisotropy (CMA) – was earlier predicted to exist in atomically thin\nplatinum nanowires. Using spin dynamics simulations based on first-principles calculations, we here explore the spin dynamics\nof atomically thin platinum wires to reveal the spin relaxation signature of colossal magnetic anisotropy, comparing it with other\ntypes of anisotropy such as uniaxial magnetic anisotropy (UMA). We find that the CMA alters the spin relaxation process\ndistinctly and, most importantly, causes a large speed-up of the magnetic relaxation compared to uniaxial magnetic anisotropy.\nThe magnetic behavior of the nanowire exhibiting CMA should be possible to identify experimentally at the nanosecond time\nscale for temperatures below 5 K. This time-scale is accessible in e.g., soft x-ray free electron laser experiments.\nLate 4d and 5d transition metals such as palladium and platinum are paramagnetic in the bulk, but at the same time exhibit\nenhanced magnetic susceptibility. Thus, perturbations such as reduced dimensionality may result in emerging magnetism in\nthese metals. The magnetic state might only exist at very low temperatures, or have other features making it difficult to observe\nexperimentally. Recently, Strigl et al.1demonstrated emerging magnetic order in platinum atomic contacts and chains by\nmeasuring the magnetoconductance. Here, we take an alternative route and address the time evolution of magnetic order in\nplatinum nanowires. The underlying idea is that the unusual anisotropy predicted to exist in these systems, where the magnetic\nmoments of the wires depend strongly on the angle of deviation from the easy-axis2, could affect the dynamics in such a way\nthat it the dynamical behavior could function as a measurable signature for the emergent magnetism and its associated colossal\nmagnetic anisotropy.\nUnderstanding spin relaxation and long-range order in low-dimensional systems are questions of fundamental interest.\nRecently, they have also become core technological issues in the quest for ever-smaller nanosized magnetism-based information\nstorage systems. Generally, as the dimensionality of a system is reduced, fluctuations become larger and more important and the\ntendency toward magnetic ordering decreases. According to the Mermin-Wagner theorem3, infinite 1D chains with sufficiently\nshort range magnetic interactions should spontaneously break up into segments with different spin orientation. This in turn\nimplies that long-range order would be impossible in these systems. However, these early spin-lattice models assume the\nabsence of kinetic barriers as well as anisotropies. Thus, by introducing such barriers one might hope to build 1D magnetic\nsystems with long-range magnetic order and even zero-dimensional magnetic systems with the capability to store magnetic\ninformation on a macroscopic time scale3–5.\nIn practice, barriers can be introduced by growing 1D systems on a substrate or by using magnetic species with substantial\norbital moments. Such kinetic barriers may result in long-lived stable states creating ordered magnetic structures below a\ncertain threshold temperature even in 1D systems. However, magnetic order can be destroyed by thermally activated transitions\nbetween magnetic states available in the system. 1D nanowires may exhibit many different types of magnetic arrangements\ndepending on the exchange coupling between the spins, the atomic geometry, the shape of the nanowire and the size and type of\nanisotropy. Recent experimental studies of chains of Fe atoms on a Cu 2N substrate showed evidence of both ferromagnetic6\nand antiferromagnetic7ordering at low temperatures, depending on the relative positioning of the atoms. Gambardella et al,8\nobserved both long- and short-range magnetic order in Co chains arranged on a Pt(997) surface, with a blocking temperature\nof 15 K for the long-range order. In meandered Fe nanowires grown on Au(788), Shiraki et al.9confirmed the theoretical\nexpectation10, 11that the average size of the ferromagnetic domains in the nanowire decreases exponentially with temperature.\nStrigl et al.1showed experimentally that even nanowires of platinum, which is paramagnetic in bulk, demonstrate signatures ofarXiv:1604.00626v1  [cond-mat.mtrl-sci]  3 Apr 2016local magnetic order.\nThe effect of temperature and magnetic anisotropy have been the subject of previous theoretical studies12–14where the\nrelaxation dynamics was found to depend substantially on the description of the magnetic anisotropy.\nIn this work, we explore the spin dynamics in a platinum atomic wire using atomistic spin dynamics simulations where the\ninteractions have been calculated from first principles15. Specifically, we analyze how the dynamics is altered when we include\nan energy barrier against relaxation in the form of magnetic anisotropy. In this context, colossal magnetoanisotropy (CMA)2–\na new type of magnetic anisotropy where the magnetic moments become zero for large enough angles between the wire and the\nmagnetic moment – is of special interest.\nWe have employed atomistic spin dynamics (ASD) simulations16as implemented by Skubic et al.15In brief, the ASD\nmethod is based on solving the equations of motion for the atomic moments, mi, as expressed by the Landau-Lifshitz-Gilbert\n(LLG) equation\n¶mi\n¶t=\u0000g\n1+a2mi\u0002[Bi+bi(t)]\u0000g\n1+a2a\nmmi\u0002fmi\u0002[Bi+bi(t)]g: (1)\nThe time evolution described by the LLG equation comes about through a combination of precessional motion around the\nquantization axis and dissipation. The dissipative part was originally introduced phenomenologically. It is however intrinsic\nand can be derived by calculating the time evolution of the spin observable in the presence of the full spin-orbital coupling17.\nThe gyromagnetic ratio is denoted by g,Biis the effective magnetic field on atom iandbiis a stochastic magnetic field with\na Gaussian distribution, the magnitude of which is related to the temperature. The Gilbert damping parameter is denoted by\na. We have used the semi-implicit solver by Mentink et al.18to treat the time evolution in the LLG equations. The effective\nmagnetic field is formally defined as the functional derivative of the Gibbs free energy of the magnetization and is taken as\nBi=\u0000¶H=¶miin our simulations, where His the hamiltionian of the system. The Hamiltonian we consider consists of\ntwo terms – describing Heisenberg exchange and magnetic anisotropy, respectively. The Heisenberg Hamiltonian is given by\nH=\u0000åi6=jJi jmi\u0001mj, where Ji jis the strength of the exchange interaction between the moments on site iand site j. Magnetic\nanisotropy is modeled in two different ways, i.e., in the form of uniaxial anisotropy (UMA) and CMA. In both cases, the\nanisotropy axis, eK, is chosen to be along the nanowire axis. The UMA is introduced as HUMA=\u0000Kåi(ˆmi\u0001eK)2, with K\nbeing the strength of the anisotropy along eKand ˆmibeing the unit vector pointing in the direction of ith magnetic moment. The\nCMA, in turn, is treated as a combination of a modified uniaxial anisotropy energy term and a dependence of the magnitude of\nthe magnetic moments as a function of the angle of deviation fiof the moments from the nanowire axis. Both effects have\nbeen modeled in the spin dynamics simulations by parametrization of the calculations by Smogunov et al.2who found an\nanisotropy energy of 1.8 meV and reported a monotonic decrease in the magnitude of magnetic moments of platinum atoms,\nfrom 0:4mBfor zero deviation angle to zero for fi\u001945\u000e. It is found that the modified uniaxial anisotropy can actually be quite\nwell described by HCMA=åiH(i)\nCMA, whereH(i)\nCMAis given by\nH(i)\nCMA=(\n0; if 45\u000e\u0014fi\u0014135\u000e;\n\u0000Kcos2(2fi);otherwise,\nto be contrasted with the cos2(fi)behavior of the UMA. The expressions above for the anisotropy energies present a simple\nway of including the effects of anisotropy on the dynamics in this case and will provide us with an understanding of the effect of\nCMA on the spin dynamics of platinum wires. We note however that with non-constant magnitudes of the magnetic moments –\nthe case in CMA – the equation of motion itself will in principle be modified. First steps in this direction were recently taken in\nconnection to modeling of longitudinal and transversal fluctuations of magnetic moments in bcc Fe19. Here we employ a simple\nre-scaling scheme where for each time step, the magnitude miof each magnetic moment miis determined by its deviation from\nthe nanowire axis. In addition, we also perform, for comparison, simulations for the corresponding system with no magnetic\nanisotropy (NMA).\nThe interatomic exchange interactions Ji jentering the spin Hamiltonian have been calculated from first principles by\nmeans of the \"frozen magnon\" approximation20– i.e. by inverse Fourier transform of the q-dependence on the total energy\nE(q)for a large number of spin-spirals with wave vector q. The spin spirals were calculated with a full potential linearised\naugmented-plane wave (FP-LAPW) method21using the magnetic force theorem22, starting from the ferromagnetic ground\nstate.\nFrom our frozen magnon approach we have extracted values for the eight nearest neighbour exchange couplings. The\nexchange interactions are found to be strongly dominated by the nearest-neighbour interaction J1with a strength of 0:49mRy.\nFor both types of anisotropy modeled, the anisotropy constant Kwas set to K=0:13 mRy2.\nThe Gilbert damping parameter ain Eq. (1) can be calculated from first principles17, 23–27. However, in the present study,\nwe have chosen to vary aover an order of magnitude (from a=0:01toa=0:1) in order to investigate the effect of dissipation\n2/9in further detail. We find that changing the value of the damping parameter acts essentially as a time rescaling, and thus\naffects the behavior of the dynamics in a very simple way. This result is in agreement with Néel-Brown relaxation theory for\nmagnetic systems with axial symmetry, where the relaxation time is inversely proportional to the damping parameter10, 11.\nUnless otherwise stated, we have used a=0:05 in our simulations.\nThe time evolution of the average magnetization of ensembles of 1000 atom long platinum wires with UMA, CMA and\nNMA is shown in Fig. 1. Such a large number of atoms has been chosen in order to get good statistics and clear, smooth curves.\nHowever, as long as the chain length is larger than the correlation length, variation of the chain length does not change the\nresults significantly.\nThe relaxation times for the CMA wires (blue curves) are significantly shorter than for the UMA wires (red curves) over\nthe whole temperature range studied (3-15 K). However, the relaxation mechanisms in both cases appear to be similar and\ninvolve two steps as an inflection point can be seen on all curves corresponding to the CMA and UMA wires. The first step is\nassociated with the small-angle precession around the anisotropy axis and establishment of local equilibrium, while the second\nstep involves nucleation of reversed-magnetization domains. The first relaxation step is very rapid for both CMA and UMA and\noccurs on a sub-picosecond timescale for all studied temperatures. The second step is slower, and, therefore, it is the timescale\nof domain nucleation that defines the relaxation time in UMA and CMA wires. For both UMA and CMA, the duration of the\nsecond relaxation step is strongly temperature dependent. For example, for temperatures below 7 K (data not shown) the spin\nflip relaxation is not even noticeable for the wires with UMA, during the entire simulation time of 1 ns. In contrast, at 9 K one\ncan clearly see how the spin-flips, on a time scale of about 1 ns, contribute significantly to the total decrease of the average\nmagnetic moment and destruction of long-range order.\nThe NMA wires (green curves) do not show the two-step decrease in the average magnetization and the temperature\ndependence of the relaxation time is not as pronounced as in the other two cases which is a sign of the fact that the relaxation\nprocess is fundamentally different in this case compared to when anisotropy is present in the system. In the absence of\nanisotropy, the concept of spin flip is not suitable since excitations of an isotropic Heisenberg system have the form of collective\nspin wave formation.\nIn the low temperature regime, the CMA wires need longer times to relax than the anisotropy-free wires (see the upper\npanel of Fig. 1). On the other hand, the relaxation time changes more with temperature for the CMA case, compared to the\nNMA case. As a consequence, there is a cross-over temperature around 5 K (see the middle panel of Fig. 1) above which the\nCMA wires relax faster than the NMA wires.\nThe relaxation times as a function of inverse temperature for UMA, CMA and NMA wires are shown in Fig. 2. Here we\nhave considered chain lengths of 100 atoms since they give indistinguishable changes compered with the 1000 atom chains\nused in Fig. 1 but requires less computational effort.\nHere, we have defined the relaxation time as the time it takes for the average magnetization to reach 1=eof its maximum\n(i.e. initial) value. It is seen that for the wires with UMA and CMA, the temperature dependence of relaxation time, tr, follows\nthe Arrhenius law\ntr=n0\u00001eEa=kBT; (2)\nimplying thermal activation as a mechanism of the relaxation process. In Eq. (2), Eais interpreted as an activation energy, n0is\nthe attempt frequency, Tis the absolute temperature, and kBis the Boltzmann constant.\nIf anisotropy is present, thermal magnetic relaxation in the nanowire involves nucleation of domains with the reversed\nmagnetization. Each nucleation event requires overcoming an energy barrier, Ea, and the time scale is defined by Eq. (2) in the\nhigh-barrier limit. This mechanism is similar to the Néel-Brown relaxation scenario for an ensemble of non-interacting spins\nwith the activation energy defined by the magnetic anisotropy of each spin10, 11. In magnetic nanowires, the activation energy,\nEa, as well as the pre-exponential factor, n0, are affected by exchange interaction between atomic moments and, in particular,\nby the anisotropy type (see Fig. 2), as explained below. If the anisotropy is removed, the relaxation behavior of the wire cannot\nbe fitted successfully to the Arrhenius formula, which is a sign of a fundamentally different relaxation mechanism, as explained\nearlier.\nIt seems clear that our simulated results agree very well with the Arrhenius law not only for the UMA case but also for\nthe CMA wires. This is an interesting result in itself considering that in the CMA case, the potential landscape is altered\nas a function of the magnetic moment rotation. However, the activation energies are different. A least-squares fit of the\nspin-dynamics data shows that the activation energy for the UMA wires, 0.51 mRy, is more than three times larger than for the\nCMA wires, for which the value of 0.15 mRy is found. This result is consistent with the much more rapid relaxation in the\nCMA case compared to the UMA case.\nIn order to shed light on the microscopic mechanism of spin relaxation in UMA and CMA wires and gain a better\nunderstanding of why the effective activation energy is significantly lower for the latter, we present an illustrative visualization\n3/9of the relaxation process using color-coded spin mapping of individual trajectories of the atomic moments, see Fig. 3. Here all\nwires start from the ferromagnetic ground state.\nWe now go through the maps starting with the uppermost row, i.e. the UMA case. At 9 K (the leftmost panel), only a few\nshort sections of flipped spins – which also relax back to the un-flipped state after a quite short time, on the order of tens of\nps – can be observed during the entire simulation time of 1 ns. As the temperature is increased, the number of streaks with\nflipped spins increases (see middle and rightmost maps in the top row), and the flipped regions have a much longer lifetimes, as\nonly a few regions can be seen relaxing back to the unflipped state. The initial streak width remains roughly constant over the\nentire simulation time but as more and more regions lump together, the flipped regions become wider, forming a clear domain\nstructure.\nIn contrast, the CMA wires do not exhibit the wide domain formation as the one observed for the uniaxial anisotropy\ncase. The spin map for the CMA case at 3 K, shown in the leftmost column in the middle row in Fig. 3, resembles partly the\nmaps for the UMA case with the difference that the flipped domains are much narrower and with the existence of clear sharp\ngreen/yellow lines, signifying atoms where the local moments have vanished. As the temperature increases, more and more\natoms lose their moments but it is also seen that due to the thermal fluctuations, a moment with a magnitude close to zero might\nflip towards the anisotropy axis, regaining the magnetic moment in the process. Even at 5 K (middle panel, middle row) there is\nno visible long range order despite the very short simulation time of 50 ps. At higher temperatures the wire becomes even more\ndisordered with life times of small domains in the sub-picosecond range.\nThe very narrow stripes in the middle row of Fig. 3 indicate that the minimum size of a stable reversed-magnetization\ndomain in the CMA wires is significantly smaller than in the UMA wires. This seems to be the main reason for the lower\nactivation energy in the CMA wires, because the energy cost for the critical domain nucleation is, to a first approximation,\nproportional to the domain size. Furthermore, Fig. 3 demonstrates that even atomically thin domains represent relatively\nlong-lived metastable states in the CMA wires. This is expected since the magnitude of the magnetic moments decreases\nquickly as they rotate away from the easy axis, thus lowering the effective exchange interaction and making spins less connected\nto each other in the CMA wires. Direct calculations of minimum energy paths for the magnetization switching in the UMA\nwires using the geodesic nudged elastic band (GNEB) method28show that the minimum stable domain size is 3 spins with\ncorresponding activation energy of 0.64 mRy, which is in a good agreement with Arrhenius fits to the spin dynamics data (see\nFig. 2). Although the GNEB method accounts for the change in the magnitude of magnetic moments, it is problematic to apply\nit to the CMA wires because there are large regions in the configuration space where magnetic moments vanish. However, a\nsingle spin-flip scenario for the magnetization reversal in CMA wires is supported by the fact that the activation energy derived\nfrom the spin dynamics simulations agrees very well with the anisotropy energy given by the anisotropy constant K.\nFinally, turning to the wires lacking anisotropy (third row in Fig. 3) it is clear that time-stable domains are not formed in\nthe absence of anisotropy. Long range orde is not apparent even at 3 K. The lack of anisotropy also introduces an oscillatory\nbehavior of the magnetism in the evolution of the different domains, adding to the disorder.\nIn conclusion, we find that the CMA wires relax much faster than the wires with UMA, and we attribute this to the\ndecreasing magnitude of the magnetic moments: the decrease in effective exchange interactions makes the size of a critical\nreversed-magnetization domain smaller, thus lowering the energy cost for the domain nucleation. We also find that for both\nthese types of anisotropy, the spin relaxation times can be described quite well with the Néel-Brown model of magnetic\nrelaxation. According to our relaxation-time calculations, the magnetic behavior of the CMA wire should be possible to resolve\nexperimentally at the nanosecond time scale for temperatures below 1 K. This time scale should be accessible for soft x-ray\nfree electron lasers29and even pump-probe x-ray transition microscopy30even though the lateral resolution needed might prove\nvery difficult to achieve.\nReferences\n1.Strigl, F., Espy, C., Bückle, M., Scheer, E., Pietsch, T. Emerging magnetic order in platinum atomic contacts and chains.\nNature Commun. 6, 6172 (2015).\n2.Smogunov, A., Dal Corso, A., Delin, A., Weht, R., Tosatti, E. Colossal magnetic anisotropy of monatomic free and\ndeposited platinum nanowires. Nature Nanotech. 3, 22 (2008).\n3.Mermin, N.D., Wagner, H. Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic\nHeisenberg Models. Phys. Rev. Lett. 17, 1133 (1966).\n4.Ising, E., Beitrag zur Theorie des Ferromagnetismus. Z. Phys. 31, 253 (1925).\n5.Landau, L.D., Lifshitz, E.M. Statistical Physics (Pergamon, 1959).\n6.Spinelli, A., Bryant, B., Delgado, F., Fernándes-Rossier, J., Otte, A.F. Imaging of spin waves in atomically designed\nnanomagnets. Nature Mater. 13, 782 (2014).\n4/97.Loth, S., Baumann, S., Lutz, C.P., Eigler, D.M., Heinrich, A.J. Bistability in atomic-scale antiferromagnets. Science 335,\n196 (2012).\n8.Gambardella, P., et al. Ferromagnetism in one-dimensional monatomic metal chains. Nature 416, 301 (2002).\n9.Shiraki, S., et al. Magnetic structure of periodically meandered one-dimensional Fe nanowires. Phys. Rev. B 78, 115428\n(2008).\n10.Brown, W.F. Thermal Fluctuations of a Single-Domain Particle. Phys. Rev. 130, 1677 (1963).\n11.Néel, L. Théorie du traînage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites. Ann.\nGeophys. 5, 99 (1949).\n12.Rózsa, L., Udvardi, L., Szunyogh, L. Langevin spin dynamics based on ab initio calculations: numerical schemes and\napplications. J. Phys. Condens. Matter 26, 216003 (2014).\n13.Bauer, D.S.G., Mavropoulos, P., Lounis, S., Blügel, S. Thermally activated magnetization reversal in monatomic magnetic\nchains on surfaces studied by classical atomistic spin-dynamics simulations. J. Phys. Condens. Matter 23, 394204 (2011).\n14.Beaujouan, D., Thibaudeau, P., Barreteau, C. Anisotropic magnetic molecular dynamics of cobalt nanowires. Phys. Rev. B\n86, 174409 (2012).\n15.Skubic, B., Hellsvik, J., Nordström, L., Eriksson, O. A method for atomistic spin dynamics simulations: implementations\nand examples. J. Phys.: Condens. Matter 20, 315203 (2008).\n16.Antropov, V .P., Katsnelson, M.I., Harmon, B.N., van Schilfgaarde, M., Kusnezov, D. Spin dynamics in magnets: Equation\nof motion and finite temperature effects. Phys. Rev. B 54, 1019 (1996).\n17.Hickey, M.C., Moodera, M.S. Origin of Intrinsic Gilbert Damping. Phys. Rev. Lett. 102, 137601 (2009).\n18.Mentink, J.H., Tretyakov, M.V ., Fasolino, A., Katsnelson, M.I., Rasing, Th. Stable and fast semi-implicit integration of the\nstochastic Landau–Lifshitz equation. J. Phys. Condens. Matter 22, 176001 (2010).\n19.Ma, P.-W., Dudarev, S.L. Longitudinal magnetic fluctuations in Langevin spin dynamics. Phys. Rev. B 86, 054416 (2012).\n20.Halilov, S.V ., Perlov, A.Y ., Oppeneer, P.M., Eschrig, H. Magnon spectrum and related finite-temperature magnetic\nproperties: A first-principle approach. Europhys. Lett. 39, 91 (1997).\n21.,The Elk FP-LAPW Code . Available at: http://elk.sourceforge.net.\n22.Macintosh, A.R., Andersen, O.K. Electrons at the Fermi Surface (Cambridge University Press, 1980).\n23.Mankovsky, S., Ködderitzsch, D., Woltersdorf, G., Ebert, H. First-principles calculation of the Gilbert damping parameter\nvia the linear response formalism with application to magnetic transition metals and alloys. Phys. Rev. B 87, 014430 (2013).\n24.Dürrenfeldt, P., et al. Tunable damping, saturation magnetization, and exchange stiffness of half-Heusler NiMnSb thin\nfilms. Phys. Rev. B 92, 214424 (2015).\n25.Yin, Y ., et al. Tunable permalloy-based films for magnonic devices. Phys. Rev. B 92, 024427 (2015).\n26.Starikov, A. A., Kelly, P. J., Brataas, A., Tserkovnyak, Y ., Bauer, G.E.W. Unified first-principles study of Gilbert damping,\nspin-flip diffusion and resistivity in transition metal alloys. Phys. Rev. Lett. 105, 236601 (2010).\n27.Kapetanakis, M.D., Perakis, I.E. Spin dynamics in (III,Mn)V ferromagnetic semiconductors: the role of correlations. Phys.\nRev. Lett. 101, 097201 (2008).\n28.Bessarab, P.F., Uzdin, V .M., Jónsson, H. Method for finding mechanism and activation energy of magnetic transitions,\napplied to skyrmion and antivortex annihilation. Comput. Phys. Commun. 196, 335 (2015).\n29.Gutt, C., et al. Single-pulse resonant magnetic scattering using a soft x-ray free-electron laser. Phys. Rev. B 81, 100401\n(2010).\n30.Stoll, H., et al. High-resolution imaging of fast magnetization dynamics in magnetic nanostructures. App. Phys. Lett. 84,\n3328 (2004).\nAcknowledgements\nWe acknowledge financial support from Vetenskapsrådet (VR), The Royal Swedish Academy of Sciences (KV A), the Knut\nand Alice Wallenberg Foundation (KAW), Swedish Energy Agency (STEM), Swedish Foundation for Strategic Research\n(SSF), Carl Tryggers Stiftelse (CTS), eSSENCE, and Göran Gustafssons Stiftelse (GGS). The computations were performed on\nresources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Center (NSC),\nLinköping University and at the PDC center for high-performance computing, KTH.\n5/9Author contributions statement\nA.D. and A.B. initially designed the project; A.B. and P.F.B. performed the calculations; all authors contributed to analysing the\ndata and writing the paper.\nAdditional information\nCompeting financial interests: The authors declare no competing financial interests.\n6/9✵ \u0000 ✁\n✶✸ ✂\n❯✄☎❈ ✄☎◆\n✄☎✵ \u0000 ✁\n✶❆✆✝✞✟✠✝✡✟✠☛✝☞✌✍✟☞✌✎☛\n✁ ✂\n❯✄☎❈ ✄☎◆\n✄ ☎✵ \u0000\n✶ ✶ ✶✵\n✶✵ ✵\n✶✵ ✵ ✵❚✏ ✑✒ ✓ ✔ ✕ ✖\n✵ \u0000 ✁\n✶✾ ✂\n❯✄ ☎❈ ✄☎\n◆\n✄☎Figure 1. Average magnetization as a function of time for atomically thin platinum wires containing 1000 atoms at three\ndifferent temperatures: 3, 5 and 9 K. CMA stands for colossal magnetic anisotropy (blue curves), UMA stands for uniaxial\nmagnetic anisotropy (red curves), and NMA stands for \"no magnetic anisotropy\" (green curves).7/9✥\u0000 ✁\n✁\n✁ ✥\n✁ ✥✥\n✁ ✥✥✥✥\u0000 ✁ ✥\u0000 ✂ ✥\u0000 ✄\n❘☎✆✝✞✝✟✠✡☛✟✠☞☎✌✍✎✏■ ✑✒✓ ✔ ✕ ✓ ✖ ✓ ✗✘✓ ✔ ✙ ✖ ✚✔ ✓ ✛ ✁✜ ✢✣\n❯✤✦◆✤✦❈✤✦Figure 2. Simulated relaxation time as a function of inverse temperature for platinum wires. The symbols represent simulated\ndata, and the drawn lines are least-squares fits. The results for a 100 atom long chain are shown.\n8/9Figure 3. Maps of the magnetic moment per site as a function of time. The atomic positions in the 100-atom long wires are\nalong the x-axis and the time evolution is shown along the y-axis, starting from the top of each map. Total simulation time is\n1 ns for UMA and 50 ps for CMA and NMA. Red areas correspond to spin-up magnetization (i.e. the initial state) while blue\nareas show spin-down magnetization. As the spin deviates from the easy axis, the color turns yellow and when the spin is fully\northogonal to the easy axis, it is colored green. The top row maps show simulations assuming UMA at (from left to right) 9, 11,\nand 15 K, respectively. The middle row maps show simulations assuming CMA at (from left to right) 3, 5, and 9 K, respectively.\nThe bottom row maps show simulations assuming NMA at (from left to right) 3, 5, and 9 K, respectively.\n9/9"    },    {        "title": "1503.08036v1.Spin_superradiance_by_magnetic_nanomolecules_and_nanoclusters.pdf",        "content": "arXiv:1503.08036v1  [cond-mat.mes-hall]  27 Mar 2015Spin superradiance by magnetic nanomolecules and\nnanoclusters\nV I Yukalov1, V K Henner2,3and E P Yukalova4\n1Bogolubov Laboratory of Theoretical Physics, Joint Instit ute for Nuclear Research,\nDubna 141980, Russia\n2Department of Physics, Perm State University, Perm 614190, Russia\n3Department of Physics, University of Louisville, Louisvil le, Kentucky 40292, USA\n4Laboratory of Information Technologies, Joint Institute f or Nuclear Research,\nDubna 141980, Russia\nE-mail:yukalov@theor.jinr.ru\nAbstract. Spin dynamics of assemblies of magnetic nanomolecules and n anoclusters can be\nmade coherent by inserting the sample into a coil of a resonan t electric circuit. Coherence is\norganized through the arising feedback magnetic field of the coil. The coupling of a magnetic\nsample with a resonant circuit induces fast spin relaxation and coherent spin radiation, that\nis, superradiance. We consider spin dynamics described by a realistic Hamiltonian, typical of\nmagnetic nanomolecules and nanoclusters. The role of magne tic anisotropy is studied. A special\nattention is paid to geometric effects related to the mutual o rientation of the magnetic sample\nand resonator coil.\n1. Introduction\nThere exists a large class of magnetic nanomolecules and mag netic nanoclusters that can be\nconsideredas nanoparticles possessinghightotal spins(s ee review articles [1–9]). Below blocking\ntemperature, thespinof such magnetic nanoparticles isfro zen. For instance, thetypical blocking\ntemperature of magnetic nanomolecules is of order 1 −10 K. The blocking temperature for\nnanoclustres is 10 −100 K.\nMagnetic properties of nanomolecules and nanoclusters are similar to each other. There are\ntwo main features distinguishing them. Magnetic molecules of the same chemical composition\nare identical and they can form crystals with almost ideal pe riodic lattice. While magnetic\nnanoclusters, even being made of the same element, say Fe, Ni , or Co, differ by their sizes,\nand they do not form periodic structures. Otherwise, the spi n Hamiltonian for an ensemble of\nmagnetic nanoparticles is of the same form for nanomolecule s as well as for nanoclusters.\nIn the usual case, spin relaxation is due to spin-phonon inte ractions and, below the blocking\ntemperature, is very slow. Thus for nanomolecules, the spin -phonon relaxation time is T1∼\n(105−107) s. But the spin relaxation time can be drastically shortene d, if the magnetic sample\nis inserted into a coil of a resonant electric circuit. This i s termed the Purcell effect [10]. In\nthat case, the relaxation is caused by the resonator feedbac k field collectivizing moving spins\nand forcing them to move coherently. Coherent spin dynamics have been studied in several\npublications, e.g., [11–25].Coherentlymovingspinsproducecoherent radiation, which , whenitisself-organized, iscalled\nsuperradiance. It is worth stressing that spin superradian ce is rather different from atomic\nsuperradiance. The latter is caused by the Dicke effect [26], w hile the cavity Purcell effect is\nsecondary [27–29]. Contrary to this, spin superradiance is completely due to Purcell effect, with\nthe Dicke effect playing no role [30]. The Purcell effect also enh ances the signals of nuclear\nmagnetic resonance [31,32] and of spin echo [33–35].\nHere we consider the peculiarities of spin superradiance by magnetic nanomolecules and\nnanoclusters having strong magnetic anisotropy. We shall p ay attention to the role of geometric\neffects related tothefinitenessoftheconsideredsamples. Fi nitesystems, asisknown[36,37], can\nexhibit properties different from those of bulk systems. In th e present case, we are interested in\nthe geometric effects due to the mutual orientation of a finite m agnetic sample and the resonator\ncoil.\n2. Spin Hamiltonian\nAn ensemble of magnetic nanomolecules or nanoclusters is de scribed by the Hamiltonian\nˆH=/summationdisplay\niˆHi+1\n2/summationdisplay\ni/negationslash=jˆHij, (1)\nconsisting of single-spin terms ˆHiand spin-interaction terms ˆHij, with the index i= 1,2,...,N\nenumerating nanoparticles. The single-spin Hamiltonian\nˆHi=−µ0B·S−D(Sz\ni)2+D2(Sx\ni)2+D4/bracketleftbig\n(Sx\ni)2(Sy\ni)2+(Sy\ni)2(Sz\ni)2+(Sz\ni)2(Sx\ni)2/bracketrightbig\n(2)\nis a sum of the Zeeman energy and single-site magnetic anisot ropy terms. The total magnetic\nfield, acting on each spin,\nB=B0ez+Hex, (3)\nincludes an external field B0and a resonator feedback field H. Spins interact with each other\nthrough dipolar forces characterized by the Hamiltonian\nˆHij=/summationdisplay\nαβDαβ\nijSα\niSβ\nj, (4)\nwith the dipolar tensor\nDαβ\nij=µ2\n0\nr3\nij/parenleftBig\nδαβ−3nα\nijnβ\nij/parenrightBig\n,\nwhere\nrij≡ |rij|,nij≡rij\nrij,rij≡ri−rj.\nThe resonator feedback field is given by the Kirchhoff equatio n\ndH\ndt+2γH+ω2/integraldisplayt\n0H(t′)dt′=−4πηdmx\ndt, (5)\nin which γis resonator damping, ωis resonator natural frequency, ηis filling factor, and\nmx≡µ0\nVN/summationdisplay\nj=1/angbracketleftSx\nj/angbracketright (6)\nis the transverse magnetization density of the sample havin g volume V.In addition to the resonator natural frequency ω, there are the following characteristic\nfrequencies. The Zeeman frequency\nω0≡ −µ0\n/planckover2pi1B0=2\n/planckover2pi1µBB0 (7)\nand the anisotropy frequencies\nωD≡(2S−1)D\n/planckover2pi1, ω 2≡(2S−1)D2\n/planckover2pi1, ω 4≡(2S−1)D4\n/planckover2pi1S2. (8)\nThe resonator natural frequency has to be close to the Zeeman frequency, in order to satisfy the\nresonance condition /vextendsingle/vextendsingle/vextendsingle/vextendsingleω−ω0\nω/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1. (9)\nAnd the Zeeman frequency has to be larger than the anisotropy frequencies that freeze spin\nmotion, /vextendsingle/vextendsingle/vextendsingle/vextendsingleωD\nω0/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1,/vextendsingle/vextendsingle/vextendsingle/vextendsingleω2\nω0/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1,/vextendsingle/vextendsingle/vextendsingle/vextendsingleω4\nω0/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1. (10)\nAmong the anisotropy frequencies, the most important are ωDandω2that are close to each\nother. The frequency ω4, up to spins S∼103, is much smaller than ωD.\nWe analyze the Heisenberg equations of motion for spins in tw o ways, by employing the scale\nseparation approach [4,5] and by directly solving the spin e volution equations in semiclassical\napproximation. Both ways give close results. Finding the av erage spins as functions of time, we\ncan calculate the radiation intensity.\n3. Radiation intensity\nThe intensity of radiation, induced by moving spins, can be c alculated in two ways. One\npossibility is the classical formula\nI(t) =2µ2\n0\n3c3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nj/angbracketleft¨Sj/angbracketright/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n(11)\nthat should provide good approximation for high spins S≫1. The other way is to use the\nquantum formula [5,38,39], according to which the radiatio n intensity\nI(t) =Iinc(t)+Icoh(t) (12)\nis the sum of the incoherent radiation intensity\nIinc(t) = 2ω0γ0SN[1+s(t)] (13)\nand the coherent radiation intensity\nIcoh(t) = 2ω0γ0S2N2ϕ0w(t), (14)\nwhere the natural width is\nγ0≡2\n3|/vector µ|2k3\n0=1\n3µ2\n0k3\n0/parenleftBig\nk0≡ω0\nc/parenrightBig\n,ϕ0is a form-factor, and\ns(t)≡1\nNSN/summationdisplay\nj=1/angbracketleftSz\nj(t)/angbracketright, w(t)≡1\nN2S2N/summationdisplay\ni/negationslash=j/angbracketleftS+\ni(t)S−\nj(t)/angbracketright. (15)\nIf the wavelength is larger than the sample linear size, then ϕ0≃1. But, when the wavelength\nis shorter than the system linear size, then the form-factor essentially depends on the sample\nshape [39,40].\nWe have accomplished computer simulation for Nmagnetic nanomolecules possessing spin\nS= 10, such as Mn 12or Fe8, employing the parameters typical of these nanomolecules, for\nwhichD2andD4are negligible. The spin system is prepared in a nonequilibr ium initial state,\nwith the external magnetic field directed along the initial s pin polarization, so that the spins\ntend to reverse to the opposite direction. It is convenient t o consider a dimensionless radiation\nintensity, expressed through the units of\nI0≡2µ2\n0\n3c3γ4\n2/parenleftbigg\nγ2≡1\nT2/parenrightbigg\n,\nmultiplied by the number of nanomolecules squared, N2, whereµ0=−2µB= 1.855×10−20\nerg/G and γ2= 10101/s, which gives I0= 0.852×10−38W. All frequencies are measured in\nunits ofγ2. The resonance condition ω=ω0is assumed.\nWe have considered the influence of different factors on the rad iation intensity. Thus, the\nrole of the Zeeman frequency is exemplified in Fig. 1, showing that the larger the Zeeman\nfrequency, the higher the radiation intensity. Figure 2 ill ustrates that the larger the initial spin\npolarization, the larger the radiation intensity. Figure 3 shows that increasing the magnetic\nanisotropy suppresses the radiation intensity. The role of dipole interactions is described in\nFig. 4, demonstrating that they suppress the radiation inte nsity by a factor of 1 .5. In Fig. 5,\nwe study the role of the sample shape and its orientation, fro m where it follows that, under\nthe same number of nanomolecules, the most favorable situat ion, with the highest radiation\nintensity, corresponds to the chain of nanomolecules along the resonator axis.\nCalculations for the coherent radiation intensity (14) red uces to the solution of the evolution\nequation for the coherence function w(t). Numerical solution yields the resultsclose tothe quasi-\nclassicalcase, illustratedinFigs. 1to5. Thisisnotsurpr ising,sincethecoherentregimeisknown\nto be well represented by a quasi-classical approximation. The maximal number of coherently\nradiating spins can be estimated as Ncoh∼ρVcoh, whereρis the density of nanomolecules and\nVcohis the coherence volume. The latter, for a cylindrical sampl e, isVcoh∼πR2\ncohL, whereLis\nthe cylinder length and Rcohis a coherence radius [39], which is of order 0 .3√\nλL. This gives\nNcoh∼ρλL2.\nThe typical density of magnets, formed by nanomolecules, is ρ≈0.4×1021cm−3. For the\nZeeman frequency ω∼2×10131/s, the wavelength is λ∼10−2cm. Ifλ∼L, thenNcoh∼1014.\nThe typical time of a superradiant pulse is 10−11s.\nIn this way, magnetic nanomolecules and nanoclusters can be described by a similar\nmacroscopic Hamiltonian. In the process of spin reversal fr om an initially prepared non-\nequilibrium state, there appears spin superradiance, due t o the Purcell effect of the resonator\nfeedback field. The radiation is mainly absorbed by the reson ant coil surrounding the sample.\nAcknowledgments\nThe authors are grateful for financial support to the Russian Foundation for Basic Research\n(grant 13-02-96018) and to the Perm Ministry of Education (g rant C-26/628).0 0.25 0.5 0.75 1\nt00.2 0.4 0.6 0.8 1I(t)\n0 0.25 0.5 0.75 1\nt00.2 0.4 0.6 0.8 1I(t)\n0 0.25 0.5 0.75 1\nt00.2 0.4 0.6 0.8 1I(t)\nFigure 1. Radiation intensity (11) from N= 125 nanomolecules, with molecular spin S= 10,\nfor a cubic sample. Initial reduced polarization is s0= 0.9, the anisotropy frequency is ωD= 20,\nand the resonator damping is γ= 10. The Zeeman frequency is ω0= 1000 (solid line), with the\nintensity in units of 3 .2×1013N2I0;ω0= 2000 (long-dashed line), in units of 0 .8×1015N2I0,\nandω0= 5000 (short-dashed line), in units of 5 .1×1016N2I0.0 0.25 0.5 0.75 1\nt00.2 0.4 0.6 0.8 1I(t)\nFigure 2. Radiation intensity (11) for a cubic sample of N= 125 nanomolecules, with spin\nS= 10, under the Zeeman frequency ω0= 2000, anisotropy frequency ωD= 20, and the\nresonator damping γ= 10, for different initial polarizations: s0= 0.9 (solid line), s0= 0.7\n(long-dashed line), and s0= 0.5 (short-dashed line). All intensities arein unitsof 0 .8×1015N2I0.\n0 0.25 0.5 0.75 1\nt00.2 0.4 0.6 0.8 1I(t)\nFigure 3. Radiation intensity (11) for a cubic sample of N= 125 nanomolecules, with spin\nS= 10, under the Zeeman frequency ω0= 2000, resonator damping γ= 10, and the initial\nspin polarization s0= 0.9, for varying anisotropy frequency: ωD= 20 (solid line), ωD= 50\n(long-dashed line), and ωD= 100 (short-dashed line). The values of the radiation inten sity are\nin units of 0 .8×1015N2I0.0 0.2 0.4 0.6 0.8 1\nt00.2 0.4 0.6 0.8 1I(t)\nFigure 4. Radiation intensity (11) for a cubic sample of N= 125 nanomolecules, with spin\nS= 10, the Zeeman frequency ω0= 2000, anisotropy frequency ωD= 20, resonator damping\nγ= 10, and the initial spin polarization s0= 0.9, for the cases with dipole interactions (solid\nline), in units of 0 .8×1015N2I0, and without these interactions (dashed line), in units of\n1.2×1015N2I0.\n0 0.25 0.5 0.75 1\nt00.25 0.5 0.75 1I(t)\nFigure 5. Radiation intensity (11) for a cubic sample of N= 144 nanomolecules, with spin\nS= 10, the Zeeman frequency ω0= 2000, anisotropy frequency ωD= 20, resonator damping\nγ= 30, and the initial spin polarization s0= 0.9, for different sample shapes and orientations:\nthe chain of molecules along the z- axis (solid line), the chain along the x- axis (long-dashed\nline), the y−zplane of molecules (short-dashed line), and the x−yplane of molecules (dotted-\ndashed line). The intensities are in units of 1 .2×1015N2I0.[1] Barbara B, Thomas L, Lionti F, Chioresku I, and Sulpice A 1 999J. Magn. Magn. Mater. 200167\n[2] Wernsdorfer W 2001 Adv. Chem. Phys. 11899\n[3] Ferr´ e J 2002 Top. Appl. Phys. 83127\n[4] Yukalov V I 2002 Laser Phys. 121089\n[5] Yukalov V I and Yukalova E P 2004 Phys. Part. Nucl. 35348\n[6] Bedanta S and Kleemann W 2009 J. Phys. D 42013001\n[7] Berry C C 2009 J. Phys. D 42224003\n[8] Beveridge J S, Stephens J R, and Williams M E 2011 Annu. Rev. Annal. 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Lett. 2302\n[39] Yukalov V I 2014 Laser Phys. 24094015\n[40] Allen L and Eberly J H 1975 Optical Resonance and Two-Level Atoms (New York: Wiley)"    },    {        "title": "2101.08868v1.Effects_of_the_dynamical_magnetization_state_on_spin_transfer.pdf",        "content": "E\u000bects of the dynamical magnetization state on spin transfer\nNeil Tramsen,\u0003Alexander Mitrofanov,\u0003and Sergei Urazhdin\nDepartment of Physics, Emory University, Atlanta, GA, USA\nWe utilize simulations of electron scattering by a chain of dynamical quantum spins, to analyze\nthe interplay between the spin transfer e\u000bect and the magnetization dynamics. We show that the\ncomplex interactions between the spin-polarized electrons and the dynamical states of the local spins\ncan be decomposed into separate processes involving electron re\rection and transmission, as well\nas absorption and emission of magnons - the quanta of magnetization dynamics. Analysis shows\nthat these processes are substantially constrained by the energy and momentum conversation laws,\nresulting in a signi\fcant dependence of spin transfer on the electron's energy and the dynamical state\nof the local spins. Our results suggest that exquisite control of spin transfer e\u000eciency and of the\nresulting dynamical magnetization states may be achievable by tailoring the spectral characteristics\nof the conduction electrons and of the magnetic systems.\nI. I. INTRODUCTION\nSpin transfer e\u000bect (ST) - the transfer of spin angu-\nlar momentum from spin-polarized conduction electrons\nto magnetic systems [1{4] - is one of the most exten-\nsively studied e\u000bects in modern nanomagnetism, thanks\nto the unique fundamental insights it provides into elec-\ntron spin physics, a plethora of related magnetoelectronic\nand dynamical e\u000bects, as well as viable applications in\ninformation technology [5{9]. ST can result in magne-\ntization reversal [10, 11], precession [12{14] and other\ndynamical e\u000bects [15{17]. Magnetic switching driven by\nST is \fnding applications in memories and biomimet-\nics, while its ability to generate magnetization dynamics\nprovides unique opportunities for magnonics - the infor-\nmation and telecommunication technology utilizing the\nquanta of magnetization dynamics (magnons) as infor-\nmation carriers [18, 19].\nST is a consequence of spin angular momentum con-\nservation in the process of scattering of spin-polarized\nconduction electrons by magnetic systems. This pro-\ncess has been extensively analyzed in the semiclassical\napproximation for the magnetic systems, in which the\nmagnetic order is approximated as a continuous classical\nvector \feld with \fxed magnitude, or as an array of lo-\ncalized classical magnetic moments. The latter approx-\nimation is commonly utilized in micromagnetic simula-\ntions. The magnetization dynamics of ferromagnets is\nusually described by the semiclassical Landau-Lifshitz-\nGilbert (LLG) equation with an additional Slonczewski's\nterm arising from ST [1]. The possibility to include ST in\nthe LLG equation, instead of jointly solving the dynam-\nical equations for the conduction electrons and the mag-\nnetization coupled by the exchange interaction, requires\nan adiabatic approximation, i.e., it is assumed that the\nrelevant magnetization dynamics are signi\fcantly slower\nthan the spin dynamics of conduction electrons involved\nin ST.\nWe now discuss recent developments in the studies of\n\u0003N.T. and A.M. contributed equally to this work.ST, relevant to the present work, that transcend these\napproximations. In the semiclassical approximation, the\nmagnitude of magnetization in ferromagnets is \fxed, so\nST is forbidden by angular momentum conservation if\nthe electron is spin-polarized collinearly with the mag-\nnetic order. However, recent experimental measure-\nments [20, 21] and theoretical studies [22{26] revealed\na contribution to ST, termed the quantum ST, which\npersists in the collinear geometry. In ferromagnets, this\ne\u000bect becomes noticeable only at cryogenic temperatures.\nHowever, it may be dominant in antiferromagnets, and\nis the only expected contribution to ST in spin liquids,\nwhose magnetic state cannot be described semiclassi-\ncally [26, 27].\nTheoretical studies have shown that ST leads to quan-\ntum entanglement between conduction electrons and\nmagnetization [23, 25], and can also mediate entangle-\nment within the magnetic system [26], with possible ap-\nplications in quantum information technologies [28, 29].\nFurthermore, it was shown that the conservation of the\ntotal energy and linear momentum of the quasiparticles\ninvolved in ST, the electrons and the magnons, can im-\npose substantial constraints on the magnetization dy-\nnamics induced by ST, and on scattering of electrons\nby magnetic systems [30]. While the roles of energy and\nlinear momentum in ST were analyzed already in the\nsemiclassical models [31{33], the constraints imposed by\nthe magnon energy and momentum, in the de Broglie\nsense, were not captured by these models.\nIn this work, we utilize quantum simulations of elec-\ntron scattering by a quantum spin chain initially popu-\nlated with one magnon, to analyze the e\u000bects of magne-\ntization dynamics on ST. In principle, such e\u000bects can\nbe introduced already in the semiclassical approach, by\njointly solving the coupled dynamical equations for the\nconduction electrons and the magnetization [34]. How-\never, our quantum simulations show that the conserva-\ntion of the total energy and linear momentum (in the de\nBroglie sense) of the quasiparticles involved in ST, the\nelectrons and the magnons, plays a central role in the\nstudied phenomena. Thus, our results provide further\nevidence for the signi\fcance of non-classical aspects of\nST, and suggest a new route for controlling its e\u000eciency.arXiv:2101.08868v1  [cond-mat.mtrl-sci]  21 Jan 20212\nThe rest of this paper is organized as follows. In Sec-\ntion II, we describe the model and provide the compu-\ntational details. In Section III, we classify and analyze\ndi\u000berent scattering processes involved in ST, for a spin\nchain populated with one magnon. In Section IV, we con-\nsider ST for the electron spin-polarized orthogonally to\nthe equilibrium magnetization, and show that this case,\ncommon in the ST studies, includes all the contributions\nanalyzed in Section III. We summarize our observations\nin Section V.\nII. II. MODEL AND SIMULATION DETAILS\nWe consider an electron initially propagating in a non-\nmagnetic medium, and subsequently scattered by a fer-\nromagnet (FM) modeled as a chain of n= 10 localized\nspins-1/2. In the tight-binding approximation, this sys-\ntem can be described by the Hamiltonian [23, 30]\n^H=\u0000X\nibjiihi+ 1j\n\u0000X\nj(J^Sj\u0001^Sj+1+Jsdjjihjj\n^Sj\u0001^s);(1)\nwhere the \frst term on the right describes hopping of the\nitinerant electron, the second - its exchange interaction\nwith the local spins, and the last term - the exchange\nbetween the local spins. Indices i= 1:::180 in Eq. (1)\nenumerate the tight-binding sites of the entire considered\nsystem, including the FM and the non-magnetic medium\nsurrounding it, while indices j= 70\u000080 enumerate the\nsites occupied by the localized spin-1/2 chain represent-\ning the FM. ^Sj,^sare the spin operators of the electron\nand of the local spins, bis the electron hopping parame-\nter,Jis the exchange sti\u000bness of the local spins, and Jsd\nis their exchange interaction with the electron. The cal-\nculations below use b= 1 eV,J=Jsd= 0:1 eV, unless\nspeci\fed otherwise.\nWe use periodic boundary conditions for both the elec-\ntron and the spin chain, to avoid artifacts associated with\nre\rections at the boundaries. Spin-orbit interactions are\nneglected in our model. Nevertheless, we expect our re-\nsults to be broadly relevant to spin-orbit torques, thanks\nto the general validity of conservation laws governing\nelectron-magnon scattering. For typical experimental\nmagnetic \felds, the e\u000bects of the Zeeman interaction are\nnegligible on the considered time scales, aside from de\fn-\ning the quantization axis for the local spin dynamics. In\nthe following, we assume that the local spins are aligned\nwith the z-axis in their ground state, with hSzi= 5\u0016h.\nThe same value aof the lattice constant, which de\fnes\nthe possible values of quasiparticle wavevectors, is used\nthroughout the entire system.\nTo analyze ST, the system is initialized with the elec-\ntron forming a Gaussian wavepacket centered around the\nwavevector k(i)\neand polarized in the + z,\u0000z, orxdirec-\ntion. The local spins are populated with one magnonwith wavevector k(i)\nm. In this state, hSzi= 4\u0016h. The\nsystem is then evolved using the Schrodinger equation\nwith the Hamiltonian Eq. (1). We choose the width of\nthe electron wavepacket so that it remains well-de\fned\nthroughout the scattering process, allowing us to clearly\nidentify the time intervals corresponding to its propaga-\ntion in the non-magnetic medium before and after ST,\nand enabling us to unambiguously determine the associ-\nated changes of physical quantities.\nTo analyze the evolution of the two subsystems, we in-\ntroduce the density matrices ^ \u001ae= Tr m^\u001aand ^\u001am= Tr e^\u001a\nfor the electron and the local spins, respectively, by trac-\ning out the full density matrix ^ \u001awith respect to the other\nsubsystem [23]. The expectation value of a physical quan-\ntity ^Aassociated with the electron isD\n^AE\n= Tr( ^A^\u001ae),\nwhile the probability of its value aisPa=h aj^\u001aej ai,\nwhere ais the corresponding eigenstate. Similar re-\nlations are used to analyze the observables associated\nwith the local spins. For instance, the expectation val-\nues of di\u000berent contributions to the system's energy are\nobtained by using the corresponding terms in the Hamil-\ntonian Eq. (1) as ^A. The distribution of electron mo-\nmentumpeis obtained by projecting onto the plane-\nwave eigenstates j ki\u0018eikex. Here,ke=pe=\u0016his the\nwavevector describing the corresponding Fourier compo-\nnent of the electron wave. For brevity, we interchange-\nably use the terms \"wavevector\" (or \"wavenumber\") and\n\"momentum\" (in the de Broglie sense), since the two\nquantities are simply related by the Planck's constant\n\u0016h. For magnons, we utilize the Bethe ansatz to classify\nthe eigenstates, as described below, and project ^ \u001amonto\nthese states to determine the distribution of magnon pop-\nulations and their energies/momenta.\nIII. III. CLASSIFICATION AND ANALYSIS OF\nTHE SCATTERING PROCESSES INVOLVED IN\nST\nIn this Section, we identify and characterize several\nscattering processes involved in ST, for a spin chain ini-\ntially populated with one magnon, and demonstrate that\nthese processes lead to distinct outcomes. For clarity,\nFig. 1 illustrates these processes separately for the spin-\nup and spin-down polarization (i.e., polarization in the\n+zand\u0000zdirections) of the incident electron. In the\nnext Section, we show that scattering of the incident\nelectron polarized in the x-direction can be interpreted\nas a superposition of all these processes, demonstrating\ntheir relevance for the generic ST geometries involving\nspin currents non-collinear with the magnetization.\nThe incident electron can be either transmitted or re-\n\rected, and its polarization can either change or remain\nthe same. Spin-up is parallel to the local spins in their\nground state. Since each magnon carries spin 1, angular\nmomentum conservation requires that electron scattering\neither results in the absorption of the initially present\nmagnon (panel a), or one magnon remains in the system3\n(a)Before scattering After scattering\n(b)\n(c)0 magnons\n2 magnons1 magnon\n1 magnon\n1 magnon\n1 magnon\nFigure 1. (Color online) Processes involved in ST, for the\nlocal spins populated with one magnon. (a) For a spin-up\nincident electron, the magnon can be absorbed, \ripping elec-\ntron spin to down. (b) For a spin-down incident electron, an\nadditional magnon can be emitted, \ripping electron spin to\nup. (c) The electron can be scattered without spin-\ripping\nor changing the number of magnons in the system, but nev-\nertheless exchange energy and momentum with the existing\nmagnon. In all cases, the electron can be either re\rected or\ntransmitted.\n(panel c). In the former case, the electron must spin-\rip\nto spin-down, while in the latter its spin must remain\nthe same. We note that even if the magnon population\ndoes not change, energy and linear momentum can be ex-\nchanged between the electron and the magnon as a result\nof scattering.\nBy the same spin angular momentum conservation ar-\ngument for the incident spin-down electron, ST can result\nin the generation of a second magnon accompanied by\nelectron spin-\ripping into the spin-up state [Fig. 1(b)].\nThe electron can be also scattered without ST, but\nnevertheless exchange energy and momentum with the\nmagnon, similarly to the spin-up electron.\nWe con\frmed the scenarios identi\fed in Fig. 1 by pro-\njecting the results of the simulations onto the eigenstates\nof the system, as described in Section II. The dependen-\ncies of the probabilities of di\u000berent scattering outcomes\non the initial magnon wavevector are shown for spin-up\nand spin-down incident electrons in Figs. 2 (a) and (b),\nrespectively.\nWe note several important features of scattering. First,\ndi\u000berent scattering processes contribute di\u000berently to\nST. For instance, absorption by a spin-up electron of a\nmagnon with wavenumber k(i)\nm= 0 is accompanied by\nelectron transmission (labeled \\t. m. abs.\"). Second,\nthe probabilities of di\u000berent processes are strongly de-\npendent on the magnitude of the magnon wavevector.\nFor instance, for the spin-up polarization of the incident\nelectron, the probability of electron transmission with-\nout ST (labeled \\t. no ST\" in Fig. 2(a)) is close to 1\nr. no STr . m. em.r\n. no STr . m. abs.t. m. abs.t. no ST (b)( a)0\nπ - πPk\nma10\n.50\nt. m. em.t. no ST 0\n.50\n1Pk\nma-π0 π Figure 2. (Color online) Probabilities of di\u000berent scattering\noutcomes, as classi\fed in Fig. 1, for spin-up (a) and spin-down\n(b) incident electron vs k(i)\nma, fork(i)\nea= 0:6. In the abbre-\nviated labels, electron transmission and re\rection is denoted\nas \\t.\" and \\r.\", respectively, while \\m. abs.\", \\m. em.\",\nand \\no ST\" denote the cases of magnon absorption, magnon\nemission, and the absence of ST, respectively.\nfor most values of km>0, except for a pronounced dip\naroundkm= 0. Meanwhile, the probability of re\rection\nwithout ST, labeled \\r. no ST\", remains negligible at all\nk(i)\nm. As a consequence, the total probability of scatter-\ning without ST, and conversely the magnitude of ST, is\ndependent on k(i)\nm. A clear asymmetry of these results\nwith respect to the sign of k(i)\nmindicates that both the\nmagnitude and the direction of the magnon momentum\nplay important roles in the scattering processes.\nThe dominance of transmission without ST among the\nscattering processes for spin-up electrons can be quali-\ntatively understood as a consequence of weak e\u000bects of\nthes-dinteraction between the conduction electron's spin\nand the local spins that are almost parallel to each other.\nThese e\u000bects are signi\fcantly stronger for the spin-down\nincident electron. In particular, while electron transmis-\nsion without ST is still dominant, the probability of ST\naccompanied by both electron transmission and re\rection\nis signi\fcantly larger [Fig. 2(b)]. We also note that the\ndependencies on k(i)\nmfor the spin-down electron are sub-\nstantially di\u000berent from those for the spin-up electron,\nsuggesting that ST involves a complex interplay between\nthe spin and the orbital degrees of freedom of the mag-\nnetic system and of the electron.\nIII.1. A. Classi\fcation of the dynamical states of\nthe magnetic system and of the electron\nThe results of Fig. 2 demonstrate that electron scatter-\ning and ST are strongly a\u000bected by the dynamical state\nof the magnetic system. We now quantitatively analyze\nthese e\u000bects, and show that they are governed by the\nconservation of energy and linear momentum, in the de\nBroglie sense, of quasiparticles involved in ST - the elec-\ntrons and the magnons [30].\nTo determine the characteristics of magnons generated\nby ST, we classify the eigenstates of the magnetic system\nusing the Bethe ansatz [35, 36]. The one-magnon states4\nare plane waves with wave numbers km:\nj i=nX\nx=11pnexp (ikmax)jxi; (2)\nwherenis the number of the local spins, and ais the lat-\ntice constant. Inserting this ansatz into the Hamiltonian,\nwe obtain the dispersion relation:\nEm= 4J(1\u0000coskma)\u0000E0: (3)\nwhereE0=\u0000Jnis the ground state energy of FM.\nThe two-magnon states are characterized by two\nwavenumbers k1,k2that are either real or complex-\nconjugates, and satisfy k1+k2\u0011k2m= 2\u0019=n(\u00151+\u00152),\nwhere\u0015iare integer Bethe numbers [36]. The values of\nk1,k2are obtained numerically by plugging the Bethe\nansatz for the two-magnon states into the Hamiltonian.\nThe eigenenergies have the form\nE2m(k) = 4JX\ni(1\u0000coskia)\u0000E0; (4)\nwherei= 1;2. Forn= 10 spins in the chain, there are\n45 two-magnon states.\nFor the electron wavepacket centered at the wavevector\nkeand localized outside the magnetic system before or\nafter scattering, the dispersion is\nEe= 2b(1\u0000coskea); (5)\nas determined from the hopping term in the Hamiltonian.\nWe will now utilize this classi\fcation of the dynam-\nical states to separately analyze the distinct scattering\nprocesses involved in ST.\nIII.2. B. Magnon absorption\nAn incident spin-up electron can absorb a magnon,\n\ripping its spin, and bringing the magnetic system to\nits ground state [Fig. 1(a)]. As Fig. 2(a) illustrates, for\nthe transmitted electron, the probability of this process\nis maximized for the initial magnon momentum k(i)\nm= 0.\nMeanwhile, for the re\rected electron, the probability is\nmaximized at large negative k(i)\nm. These observations can\nbe explained by the constraints imposed by the conser-\nvation of energy and linear momentum, in the de Broglie\nsense, as follows.\nSince the Hamiltonian Eq. (1) is time-independent, the\ntotal energy of the system comprising the electron and\nthe spin chain must be conserved. This conservation law\ntakes the simplest form when the electron is localized in\nthe non-magnetic medium before or after scattering and\nthe two subsystems do not interact,\nE(i)\ne+E(i)\nm=E(f)\ne; (6)\nwhereEe(Em) is the electron (magnon) energy, in the de\nBrogile sense, and superscripts ( i), (f) denote the charac-\nteristics of the quasiparticle before and after scattering,\nrespectively.\n02-\n2k(i)e\nak(i)e\na−\nπ0\nJ=0.2J\n=0.1ππ\n0k(i)m\nak\n(i)e\na(b)( a)0\nπ - πk\naE, eVk\n(f)m\nak(i)m\nak(f)e\naFigure 3. (Color online) Analysis of the magnon absorption\nprocess. (a) Electron dispersion (solid curve) and magnon dis-\npersion (dashed curve). The states of the subsystems before\nand after scattering are shown with open and \flled symbols,\nrespectively, for k(i)\nm=\u00002:3=a,k(i)\ne= 1=aallowing magnon\nabsorption. (b) Relationship between the electron momen-\ntum and the magnon momentum allowing magnon absorp-\ntion, for the labeled values of J. Curves: numerical solution\nof Eqs. (6), (7), symbols: results of simulations. Dashed lines\nshowk(i)\nm=\u00002k(i)\neandk(i)\nm=\u00002k(i)\ne+ 2\u0019=a.\nFor the momentum (or more precisely, quasi-\nmomentum for the discrete lattice), the situation is more\ncomplicated, because the spin chain breaks the transla-\ntion symmetry of the electron's spatial domain, so the\nmomentum needs not be conserved. Nevertheless, the\nconservation of linear momentum can be interpreted as a\nconsequence of the constructive wave interference among\nthe quasiparticles involved in the scattering process. This\ncondition can be expressed by [30]\nk(i)\ne+k(i)\nm=k(f)\ne+ 2\u0019l=a; (7)\nwhereke(km) is the electron (magnon) wavenumber, and\nthe last term with integer laccounts for the umklapp\nprocesses.\nWe note that interference takes place inside the spin\nchain, while k(i)\neandk(f)\neare de\fned outside the chain.\nThe s-d exchange may be expected to result in the shift\nof the electron's momentum as it crosses the boundary of\nthe spin chain. It is common to account for such e\u000bects\nby using the approximation of s-d exchange-induced con-\nduction band splitting. However, the s-d exchange term\nin the Hamiltonian Eq. (1) is itself the mechanism of ST\ndiscussed in this work, i.e., at small Jsdit can be consid-\nered as a perturbation for an electron whose dispersion\nis not modi\fed by s-d exchange. In quantum-mechanical\nterms, the complexity of this problem stems from the\nquantum entanglement between the conduction electron\nand the local spins, such that single-quasiparticle disper-\nsion relations become insu\u000ecient to describe the dynam-\nics of the system [23, 25]. However, these e\u000bects are small\nif the electron energy is substantially larger than the s-d\nexchange energy, and will be neglected here.\nEquations (6) and (7), with the quasiparticle energies\nrelated to their momenta by dispersions Eqs. (3), (5), give\ntwo possible values for k(i)\nmfor a givenk(i)\ne(or vice versa)5\nthat permit magnon absorption. Since magnon disper-\nsion is gapless [37], the magnon with k(i)\nm= 0,E(i)\nm= 0\ncan be absorbed, while the electron is transmitted with-\nout changing its energy or momentum.\nTo analyze the second possibility, we consider the\nelectron and the magnon dispersions [Fig. 3(a)]. Elec-\ntrons are signi\fcantly more dispersive than magnons. If\nmagnon dispersion were negligible, the electron could\nbe elastically re\rected while absorbing a magnon with\nk(i)\nm=\u00002k(i)\ne+ 2\u0019l=a, where integer laccounts for the\numklapp processes. Non-negligible magnon dispersion\nresults in an increase of the re\rected electron's energy,\nwhile the absorbed magnon's momentum shifts in the\nnegative direction. Symbols in Fig. 3(a) show the mo-\nmenta and energies of quaisparticles before and after\nscattering, obtained from the simulations for k(i)\ne= 1=a,\nk(i)\nm=\u00002:3=aallowing magnon absorption. The rela-\ntions among the momenta and energies are consistent\nwith our analysis.\nFigure 3(b) shows the dependence of the wavenum-\nber of the absorbed magnon on the wavenumber of the\nincident electron, both for the transmitted and for the\nre\rected electrons. Calculations based on the conserva-\ntion laws [curves], are in agreement with the results of\nquantum simulations [symbols]. The transmitted elec-\ntron absorbs a zero-momentum magnon, regardless of k(i)\ne\nor the exchange sti\u000bness J, as expected from energy and\nmomentum conservation [horizontal line in Fig. 3(b)].\nFor the re\rected electron, the dependence k(i)\nm(k(i)\ne) is\nslightly shifted below the linear form k(i)\nm=\u00002k(i)\ne+\n2\u0019l=a expected for negligible magnon dispersion [dashed\nlines in Fig. 3(b)], with an abrupt jump close to k(i)\ne<\u0018\n\u0019=2adue to umklapp. The downward shift increases with\nincreasingJthat controls magnon dispersion, consistent\nwith the discussed mechanisms.\nThe two main takeaways from our analysis of magnon\nabsorption are i) this process is selective with respect to\nthe magnon characteristics. The selectivity is controlled\nby the electron's energy and momentum, and ii) re\rected\nelectrons contribute to this process di\u000berently from the\ntransmitted electrons.\nIII.3. C. Scattering without spin transfer\nStudies of the e\u000bects of spin-polarized currents on mag-\nnetic systems usually focus on the transfer of spin. From\nthis perspective, no e\u000bects on the state of the magnetic\nsystem would be expected in the absence of ST. We show\nbelow that this is not the case. While the magnon pop-\nulation does not change in the absence of ST, magnon\nenergy can be modi\fed by electron scattering. The re-\nsults discussed below were obtained for scattering of the\nspin-up electron. They were similar for the spin-down\nincident electron.\nIn the absence of ST, the energy and momentum con-\nk(i)m\na= -1.25k(i)m\na= -1.25(b)( a)−\nπ0π0\nπ Δkmak\n(i)e\na−π0π0\nπ k(f)e\nak\n(f)e\n= -k(i)e\nk (i)m\na=1.25k(i)m\na=1.25k\n(i)e\naFigure 4. (Color online) E\u000bects of electron scattering with-\nout ST. (a) The change of magnon's wavenumber vs the\nwavenumber of the incident electron, (b) The wavenumber of\nthe electron after scattering vs its initial wavenumber. Curves\nare calculations based on the conservation laws Eqs. (8), sym-\nbols are the results of simulations. Dashed dotted curves are\nfork(i)\nm=\u00001:25=aandk(i)\nm= 1:25=a, respectively, the solid\nline isk(f)\ne=\u0000k(i)\ne. The trivial forward-scattering solution\nk(f)\ne=k(i)\neis not shown.\nservation relations are\nE(i)\ne+E(i)\nm=E(f)\ne+E(f)\nm;\nk(i)\ne+k(i)\nm=k(f)\ne+k(f)\nm+ 2\u0019l=a;(8)\nwith integer laccounting for umklapp. For given k(i)\nmand\nk(i)\ne, Eqs. (8) give two solutions for the \fnal state.\nOne of the two solutions is trivial - the wavenumbers of\nboth the electron and the magnon remain the same, i.e.,\nit is an elastic forward-scattering process. Naively, one\nmay expect that the second solution must correspond to a\nsimilar elastic electron re\rection. However, the reversal\nof the sign of electron's wavevector must be associated\nwith the exchange of momentum, and consequently of\nenergy, between the electron and the magnetic system,\nresulting in the modi\fcation of the magnon properties\neven without ST.\nFigure 4(a) shows the change of the magnon's\nwavenumber as a function of the incident electron's\nwavenumber, for two opposite values of the initial\nmagnon wavenumber. These results demonstrate that\nnon-ST electron scattering results in a large magnon drag\ne\u000bect - a shift of the magnon momentum determined by\nthe initial momentum of the electron. At small k(i)\ne, the\nmagnon's wavenumber is shifted in the direction of in-\ncident electron's momentum. At large k(i)\ne, the shift\nswitches to the opposite direction, due to the onset of\numklapp process at k(i)\neclose to\u0019=2a.\nThe reciprocal e\u000bect on the scattered electron is il-\nlustrated in Fig. 4(b), which shows the dependence of\nthe electron's wavenumber after scattering on its initial\nwavenumber. At k(i)\ne< \u0019= 2a, the wavenumber of the\nscattered electron is shifted in the direction opposite to\nthe wavenumber of the magnon, relative to the depen-\ndencek(f)\ne=\u0000k(i)\neexpected for the elastic electron back-6\n0.00.1k(i)e\nak(i)e\na, k(f)2\nmak\n(i)m\nak(f)2\nma(b)π0\n−\nπ(a)−\nπ0 π reflectedtransmittedPk\n(f)2\nma−π0 π \nFigure 5. (Color online) Excitation of the two-magnon\nstate by the spin-down electron. (a) Probability distribu-\ntion of the two-magnon wavenumber k2mfor the re\rected\n(squares connected by solid lines) and transmitted electron\ncomponents (circles connected by dashed lines), at k(i)\ne= 1=a,\nk(i)\nm= 1:88=a. (b) The resonant value of k(i)\nevsk(i)\nm, and the\nresulting values of the two-magnon wavevector k(f)\n2m.\nscattering. At larger k(i)\ne, the sign of the shift is reversed\ndue to the onset of magnon umklapp.\nFor most values of k(i)\ne, the electron is backscattered,\nk(f)\ne<0. However, for ki\neclose to the center or the\nboundary of the Brillouin zone and ki\nm<0, the electron\nbecomes forward-scattered. This e\u000bect can be described\nas suppression of electron backscattering by the magnetic\nmaterial due to the constraints imposed by the conserva-\ntion laws.\nFor parameters corresponding to the transition be-\ntween forward- and back-scattering, the velocity of the\nscattered electron vanishes. This outcome may be partic-\nularly useful for current-driven phenomena, for two rea-\nsons. First, all the initial kinetic energy of the electron\nis transferred to the magnetic system. Second, the scat-\ntered electron remains in the magnetic system for a long\nperiod of time, increasing the probability of magnon gen-\neration due to the s-d exchange.\nIII.4. D. Excitation of two-magnon states\nAn incident electron polarized in the \u0000zdirection can\nexcite two-magnon states [Fig. 1(b)]. In this case, one can\nexpect the conservation laws to be less restrictive than in\nmagnon absorption or magnon number-conserving pro-\ncesses, because of the additional degrees of freedom of\ntwo magnons in the \fnal state. Indeed, the probability\nof two-magnon excitation remains \fnite for all k(i)\nmat a\ngivenk(i)\ne, both for the re\rected and for the transmitted\nelectron component, as illustrated by the curves labeled\n\"r. m. em.\" and \"t. m. em.\" in Fig. 2(b). Neverthe-\nless, these curves exhibit sharp peaks resulting from the\nconservation laws, as follows.\nThe energy conservation relation for the two-magnon\nstate excitation is\nE(i)\ne+E(i)\nm=E(f)\ne+E(f)\n2m; (9)whereE(f)\n2mis the energy of the two-magnon state given\nby the Bethe ansatz Eq. (4).The wavefunction of the two-\nmagnon state can be characterized by the two-magnon\nwavenumber k2m=k1+k2= 2\u0019=n(\u00151+\u00152), where\nn= 10 is the size of the spin chain, and \u0015iare the in-\nteger Bethe numbers [36]. The values of k1,k2can be\ncomplex due to the magnon-magnon interaction, so for\nsome two-magnon states they cannot be interpreted as\nsingle-magnon wavevectors. The momentum conserva-\ntion relation is\nk(i)\ne+k(i)\nm=k(f)\ne+k(f)\n2m+ 2\u0019l=a: (10)\nThe conservation relations Eqs. (9), (10) do not pre-\nvent excitation of the two-magnon state for any given\nk(i)\nm,k(i)\ne. For instance, for the transmitted electron, there\nis always a solution k(f)\n2m=k(i)\nmwithk1= 0 andk2=k(i)\nm,\nwithE(k(f)\n2m) =E(k(i)\nm). Nevertheless, these relations re-\nsult in well-de\fned characteristics of scattered electron\nand of the excited two-magnon state, as illustrated in\nFig. 5(a) for the two-magnon momentum, for a generic\nset of initial conditions. As expected, for the transmitted\nelectron component, the two-magnon momentum is the\nsame as the initial magnon momentum. For the re\rected\nelectron, the \fnal-state momentum is determined by the\ndispersions of the involved quasiparticles.\nThe probability of two-magnon excitation is strongly\ndependent on the initial state. For the transmitted elec-\ntron, it is maximized for k(i)\nm= 0, i.e. when k1=k2=\nk(i)\nm= 0 - the additional excited magnon has the same\nwavevector as the initial magnon wavevector, which re-\nmains unchanged [see the curve labeled \"t. m. em.\" in\nFig. 2(b)]. This result can be interpreted as stimulated\nmagnon emission, i.e., electron spin \rip-driven emission\nof an additional magnon with the same characteristics\nas the magnon(s) initially in the system. Stimulated\nmagnon emission is the quantum-mechanical picture for\nthe semi-classical spin torque [2, 20, 22].\nThe situation is more complicated for the re\rected\nelectron. Figure 5(b) shows the dependence of k(i)\neon\nk(i)\nmmaximizing the probability of two-magnon excita-\ntion by the re\rected electron. The \"resonant\" condition\nisk(i)\ne\u0019k(i)\nm=2 +\u0019l=a, such that the linear momentum\nconservation relation Eq. (10) gives k(f)\n2m= 2k(i)\nm+2\u0019l0=a,\nwherel0is an integer accounting for umklapp [circles and\ndashed lines in Fig. 5(b)].\nThe relation k(f)\n2m= 2k(i)\nmseems to suggest that the\nresonant condition is associated with stimulated magnon\nemission by the re\rected electron, i.e., k1=k2=k(i)\nm.\nHowever, because of magnon interactions, bound two-\nmagnon states with complex k2=k\u0003\n1become formed\ninstead of real k1=k2[36]. Analysis of the simulation\nresults reveals that the resonantly excited two-magnon\nstates are not such bound states, but rather unbound\ntwo-magnon states with k1,k2shifted with respect to k(i)\nm\nin the opposite directions. For instance, for the resonant\nvaluesk(i)\ne= 1:1=a,k(i)\nm= 1:88=a, we obtain k1= 1:51=a,7\n-0.060.000.060\n.10.20.30.40\n10\n.00.10.2ΔE, eVk\nma−ΔSzΔSx(\nd)( c)(b)( a)−\nΔSz, ΔSx(hbar)k\nma−π0 π −\nπ0 π − π0 π −π0 π m\nagnon emissionmagnon absorptionno z-STPk\nmaReflectedT ransmitted P\nk\nmamagnon absorptionmagnon emission\nFigure 6. (Color online) Scattering of the electron polarized\nin thexdirection vs k(i)\nm, forJ= 0:2 eV,kea= 0:8. (a)\nEnergy transfer \u0001 E=E(i)\ne\u0000E(f)\nefrom the electron to the\nlocal spins. (b) Transfer of the z(open symbols) and x(solid\nsymbols) electron spin components to the local spins. Lines\nconnecting symbols are guides for the eye. (c),(d) Probabil-\nities of di\u000berent scattering scenarios for the transmitted (c)\nand re\rected (d) electron. Re\rection without ST is negligible\nand not shown.\nk2= 2:26=afor the re\rected electron. This \fnding pro-\nvides insight into the problem of stimulated scattering in\nnonlinear systems, warranting further studies beyond the\nscope of this work.\nIV. IV. SCATTERING OF ELECTRON\nPOLARIZED IN THE X DIRECTION\nIn this Section, we analyze the scattering of an electron\npolarized in the xdirection, perpendicular to the local\nspins in their ground state. Absorption of the spin cur-\nrent component perpendicular to the magnetization plays\na central role in the semiclassical theories of ST. Naively,\nthis process is unrelated to magnon absorption or emis-\nsion, which requires transfer of the z-component. Never-\ntheless, we show that scattering of the x-polarized elec-\ntron can be interpreted as a superposition of scattering\nof spin-up and spin-down electrons, as may be expected\nfrom the spin decomposition jsxi= (j\"i+j#i)=p\n2. As a\nconsequence, the outcomes of this process are determined\nby the conservation laws discussed above.\nEnergy transfer between the electron and the local\nspins exhibits complex variations with k(i)\nm, Fig. 6(a).\nOverall, at small k(i)\nmenergy \rows from the electron to\nthe local spins, while at large k(i)\nmit \rows from the lo-\ncal spins to the electron. The transfer of the x- and\nz-components of spin also exhibit a complex dependenceonk(i)\nm, Fig. 6(b). The transferred x-component is always\npositive, i.e., the initial electron spin is always partially\nabsorbed by the local spins, consistent with the usual\napproximations of semiclassical ST theories. In contrast,\nthe transferred zcomponent of spin is always negative,\nindicating that the magnon emission process is dominant\nover magnon absorption. We now demonstrate that the\ndependencies in Fig. 6(a),(b) result from the constraints\nimposed by the conservation laws.\nFigures 6(c),(d) show the probabilities of di\u000berent scat-\ntering scenarios identi\fed in Fig. 1, determined from the\nquantum simulations by projecting the \fnal state of the\nsystem onto the corresponding eigenstates. The magnon\nabsorption probability becomes \fnite at k(i)\nm= 0 for the\ntransmitted electron component, and at large negative\nk(i)\nmfor the re\rected electron component, due to the en-\nergy and momentum constraints discussed in Section III.\nThe magnon emission probability remains \fnite for all\nk(i)\nm, and exhibits peaks at k(i)\nm= 0 for the transmit-\nted electron component, and at k(i)\nm= 1:3=afor the re-\n\rected component, in accordance with the magnon emis-\nsion mechanisms discussed in Section III.\nThe variations of the energy and spin transfer are ex-\nplained in terms of these contributions, as follows. The\nminimum in the transferred energy [Fig. 6(a)], coincid-\ning with the maximum of the transferred z-component\nof spin [Fig. 6(b)], is explained by the resonant absorp-\ntion of the initial magnon as well as a broad minimum\nin magnon emission for the re\rected electron component\n[Fig. 6(d)]. On the other hand, a peak in the energy\ntransfer coinciding with a minimum in the transfer of\nthe z-component of spin, is explained by the resonant\nmagnon emission. The negative energy transfer for large\ninitial magnon momentum can be understood as a con-\nsequence of the availability of many two-magnon states\nwith energies smaller than that of the initial one-magnon\nstate, which is not the case for small k(i)\nmdue to the con-\nstraints imposed by momentum conservation.\nThe transfer of the xspin component [solid symbols\nin Fig.6(b)] does not follow the trends discussed above.\nTo analyze this e\u000bect in terms of the conservation laws,\nthe states of the magnetic system must be expanded in\nthexbasis. In the absence of anisotropy or a Zeeman\n\feld, a magnon in the xbasis is a superposition of two\nmagnons polarized in zand\u0000zdirections. Addition-\nally, the ground state of the z-basis is transformed into\na highly excited state in the x-basis. Analysis of ST in\nthese highly excited states is beyond the scope of the\npresent work.\nV. V. CONCLUSIONS\nIn this work, we utilized simulations of electron scatter-\ning by a chain of quantum spins, to reveal the quantum\nconstrains on ST that may not be captured by the semi-\nclassical approximation for the magnetic system. Our8\nmain results are:\n•The generally complex process of ST can be de-\ncomposed into a superposition of several distinct\nprocesses with qualitatively di\u000berent characteris-\ntics. These processes are magnon absorption, emis-\nsion, and the magnon number-conserving process,\naccompanied by electron transmission or re\rection,\n•In addition to angular momentum conservation\nusually considered in the analysis of ST, energy\nand momentum conservation laws, in the de Broglie\nsense of energy and momentum of quasiparticles in-\nvolved in ST, the electrons and the magnons, sub-\nstantially constrain the energy and the momentum\nof magnons that can be absorbed or emitted in\nthe ST process. These constraints may enable the\nrealization of selective cooling of speci\fc magnon\nmodes, or laser-like emission of speci\fc modes not\nlimited to the lowest-frequency dynamical states\nexcited in typical ST experiments,\n•Even in the absence of ST, spin-polarized electri-\ncal current can change the dynamical state of the\nmagnetic system. In particular, we demonstrated\nthe magnon drag e\u000bect - the change of the magnon\nmomentum due to the electron current. Generally,\nST processes are highly asymmetric with respect to\nthe direction of magnon momentum relative to the\nelectron current,\n•The magnon emission process, which is central to\nthe ST-induced generation of coherent dynamical\nmagnetization states, involves a complex interac-\ntion with the existing magnon in the system, which\nbears some signatures of the usual stimulated emis-\nsion of bosons, but is more complex due to the non-\nlinear magnon interactions, which warrants further\ntheoretical and experimental studies.\nThe presented analysis was based on numerical simu-\nlations of the simplest Heisenberg Hamiltonian and the\ns-dexchange approximation. Because of the exponential\nscaling of the dimensionality of the Hilbert space with the\nsize of the quantum systems, our simulations were lim-\nited to a chain of 10 local spins representing the magneticsystem. We expect these results to be directly relevant\nto quasi-1D systems. However, some of our conclusions\nwill likely become signi\fcantly modi\fed in higher dimen-\nsions. For instance, in 2D and 3D, the momentum con-\nservation will not limit the possible scattering processes\nonly to re\rection or transmission, expanding the range\nof accessible dynamical states. Interface roughness and\ndefects braking the translational invariance should fur-\nther relax the constraints imposed by momentum con-\nservation. Nevertheless, based on our analysis, we can\nconclude that the spatial characteristics of the magnetic\nsystems, such as the quality of their interfaces and the\nspatial homogeneity, which do not play any role in the\nsemiclassical limit, must signi\fcantly a\u000bect the e\u000eciency\nof ST and the spectral characteristics of the dynamical\nstates induced by this e\u000bect.\nAnother demonstrated e\u000bect, expected to be relevant\nto ST regardless of the system dimensionality, is the\nmagnon drag e\u000bect - the directionality of the magnon \row\ninduced by ST and controlled by the direction of electron\n\row. This e\u000bect is highly attractive both for magnon-\nics and for spin-caloritronics. The demonstrated e\u000bects\nare also relevant for the reciprocal phenomena associated\nwith electron transport. For instance, using conserva-\ntion laws as selection rules, the electron's dynamics can\nbe controlled, enabling almost re\rectionless \row of elec-\ntrons through magnetic systems. Furthermore, certain\ndynamical magnetization states can serve as electron ac-\ncelerators, increasing the transmitted electron's velocity\ndue to energy and momentum transfer from the magnetic\nsystem. Yet another possibility highlighted by our sim-\nulations is electron stopping due to the interaction with\nmagnetic systems, enabling the formation of entangled\nelectron-magnon state that can be useful for the quantum\ninformation technologies [28, 29]. 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Muller, \\Introduction to the\nbethe ansatz i,\" (1998), arXiv:cond-mat/9809162 [cond-\nmat.stat-mech].\n[37] For ferromagnets, the magnon dsipersion gap associated\nwith the magnetic \feld and anisotropy in real materials is\nnegligible on the scale of characteristic electron energies\nor temperature."    },    {        "title": "2310.16880v2.Post_dynamical_inspiral_phase_of_common_envelope_evolution__The_role_of_magnetic_fields.pdf",        "content": "Astronomy &Astrophysics manuscript no. aanda ©ESO 2023\nDecember 18, 2023\nPost-dynamical inspiral phase of common envelope evolution\nThe role of magnetic fields\nDamien Gagnier and Ond ˇrej Pejcha\nInstitute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, V Holešovi ˇckách 2, Praha 8, 180 00, Czech\nRepublic, e-mail: damien.gagnier@matfyz.cuni.cz\nReceived 2023\nABSTRACT\nDuring common envelope evolution, an initially weak magnetic field may undergo amplification by interacting with spiral density\nwaves and turbulence generated in the stellar envelope by the inspiralling companion. Using 3D magnetohydrodynamical simulations\non adaptively refined spherical grids with excised central regions, we studied the amplification of magnetic fields and their e ffect on\nthe envelope structure, dynamics, and the orbital evolution of the binary during the post-dynamical inspiral phase. About 95% of\nmagnetic energy amplification arises from magnetic field stretching, folding, and winding due to di fferential rotation and turbulence\nwhile compression against magnetic pressure accounts for the remaining ∼5%. Magnetic energy production peaks at a scale of 3 ab,\nwhere abis the semimajor axis of the central binary’s orbit. Because the magnetic energy production declines at large radial scales, the\nconditions are not favorable for the formation of magnetically collimated bipolar jet-like outflows unless they are generated on small\nscales near the individual cores, which we did not resolve. Magnetic fields have a negligible impact on binary orbit evolution, mean\nkinetic energy, and the disk-like morphology of angular momentum transport, but turbulent Maxwell stress can dominate Reynolds\nstress when accretion onto the central binary is allowed, leading to an α-disk parameter of ≃0.034. Finally, we discovered accretion\nstreams arising from the stabilizing e ffect of the magnetic tension from the toroidal field about the orbital plane, which prevents\noverdensities from being destroyed by turbulence and enables them to accumulate mass and eventually migrate toward the binary.\nKey words. binaries: close - magnetohydrodynamics (MHD) - methods: numerical – stars: magnetic field\n1. Introduction\nCommon envelope evolution (CEE) is a phase in the evolution of\nbinary star systems where one star undergoes significant expan-\nsion and engulfs its companion within a shared envelope. The\ndrag experienced by the companion star initiates its rapid spiral-\nin through the envelope, which leads to the transfer of energy\nand angular momentum to the envelope gas (Paczynski 1976).\nCEE leads to two potential outcomes: either the stars merge into\na single object or the rapid inspiral slows down. The exact causes\nof the inspiral deceleration remain elusive, but factors such as re-\nduced gas density or corotation of gas with the companion likely\nplay a major role (Roepke & De Marco 2022). Simulations sug-\ngest that the post-dynamical inspiral phase associated with weak\ndrag from the shared envelope can persist for at least hundreds\nof orbits (e.g., Ricker & Taam 2012; Passy et al. 2012; Ohlmann\net al. 2016a; Ivanova & Nandez 2016; Gagnier & Pejcha 2023).\nIt is expected that such a weak drag gradually brings the binary\ncloser together and ejects the envelope, ultimately resulting in a\npost-CEE binary (e.g., Ivanova et al. 2013; Clayton et al. 2017;\nGlanz & Perets 2018).\nAccurately modeling CEE presents significant challenges\ndue to the complex interplay of hydrodynamics, magnetohydro-\ndynamics, radiative processes, and stellar evolution. While sig-\nnificant progress has been made through all-encompassing end-\nto-end simulations, interpreting these simulation results presents\nsignificant challenges due to a high degree of complexity result-\ning from a combination of multiple physical processes. A com-\nplementary approach, focusing on specific phases of CEE or dis-\ntinct physical processes, is essential to achieve a deeper under-standing. For instance,“wind tunnel” simulations conducted by\nMacLeod & Ramirez-Ruiz (2015), MacLeod et al. (2017), Cruz-\nOsorio & Rezzolla (2020), and De et al. (2020) characterized\nthe flow properties around objects embedded within common\nenvelopes to determine drag and accretion coe fficients relevant\nfor the rapid inspiral.\nIn Gagnier & Pejcha (2023), we carried out 3D hydrody-\nnamical numerical simulations focusing exclusively on the post-\ndynamical inspiral phase of CEE. To have control over the sim-\nulation and to achieve a longer time step, we excised the cen-\ntral region containing the binary cores and we emulated the pre-\nceding dynamical plunge-in by depositing angular momentum\nin the envelope. This approach enabled us to perform a compre-\nhensive analysis of the envelope structure and evolution, angular\nmomentum transport, the short-term variability of accretion, and\nthe estimation of the orbital contraction timescale. We found that\nthe orbital contraction timescale is long, ∼103to≳105orbits\nof the central binary, but it can become shorter than the ther-\nmal timescale of many envelopes. Furthermore, provided that\nthere is gas remaining around the binary, the orbital contrac-\ntion timescale never reaches zero, because the gas outside of the\nbinary orbit cannot be kept in corotation and it develops spiral\nwaves, which back-react on the orbit. Furthermore, our study of\nthe variability of accretion exhibited remarkable similarities with\nwhat is seen in circumbinary disk simulations.\nSo far, the vast majority of CEE simulations have been per-\nformed only in the limit of hydrodynamics. It is possible that\nadditional physical processes may introduce further complexity\nand potentially impact the results. In particular, in the field of\nArticle number, page 1 of 24arXiv:2310.16880v2  [astro-ph.SR]  15 Dec 2023A&A proofs: manuscript no. aanda\nstellar physics, magnetic fields play a central role in governing\nthe dynamics of various systems, including protoplanetary and\ncircumbinary disks. Within these disks, magnetic fields have a\nsignificant impact on crucial processes such as angular momen-\ntum transport (e.g., Papaloizou & Lin 1995; Balbus & Haw-\nley 1998; Sano et al. 2004; Ji et al. 2006; Pessah et al. 2007;\nShi et al. 2012), acceleration of winds (e.g., Bai & Stone 2013;\nLesur 2021; Wa fflard-Fernandez & Lesur 2023), and the launch-\ning of jets (e.g., Ferreira 1997; Gold et al. 2014; Qian et al. 2018;\nV ourellis et al. 2019; Saiki & Machida 2020). This raises the fun-\ndamental question of whether the influence of magnetic fields\nextends to CEE, where the highly turbulent environment could\nbe conducive to their amplification. There exists a possibility that\nmagnetic fields could significantly reshape the envelope’s struc-\nture and dynamics and could impact the binary’s orbital evolu-\ntion. In fact, jet activity has been argued as a fundamental feature\nof CEE and shaping of planetary nebulae (e.g., Soker & Livio\n1994; Soker 2016; Shiber et al. 2019; Hillel et al. 2022).\nDue to numerical challenges of 3D magnetohydrodynamic\n(MHD) simulations of CEE, there are only a select number of\npapers addressing the potential e ffects of magnetic fields in this\ncontext. The first 3D MHD simulation of CEE was performed\nby Ohlmann et al. (2016b) who found that the amplification of\nmagnetic energy was insu fficient for magnetic fields to become\ndynamically significant during the dynamical inspiral phase of\nCEE. Schneider et al. (2019) conducted 3D MHD simulations\nof the merger of two massive stars, resulting in strong magnetic\nfields and yielding a rejuvenated merged star appearing younger\nand bluer than stars of its age. The most recent 3D MHD simula-\ntion of CEE was performed by Ondratschek et al. (2022). Their\nfindings indicate that magnetic fields may have a substantial in-\nfluence on shaping the morphology of emerging planetary neb-\nulae by launching jet-like outflows from the immediate vicinity\nof the two central cores. Similar jet-like polar outflows were also\nobserved in the 2D MHD post-common envelope circumbinary\ndisk simulations of García-Segura et al. (2021). This scarcity of\nresearch presents an opportunity for conducting more thorough\ninvestigations into the role of magnetic fields in CEE.\nIn this paper, we aim to investigate the amplification of an\ninitially weak seed magnetic field as it interacts with spiral den-\nsity waves and hydrodynamical turbulence arising in the enve-\nlope from the time-changing gravitational force of the central bi-\nnary system. We examined the influence of the resulting Lorentz\nforce feedback on the structure and dynamics of the envelope\nand studied the impact on the orbital evolution of the central bi-\nnary during the post-dynamical phase of CEE. Building upon\nour previous work in Gagnier & Pejcha (2023), we conducted\n3D magnetohydrodynamical numerical simulations dedicated to\nthe post-dynamical inspiral phase of the CEE. To emulate the\noutcome of the preceding dynamical inspiral phase, we injected\nangular momentum into the envelope following the methodology\ndescribed in Morris & Podsiadlowski (2006, 2007, 2009), Hirai\net al. (2021), and Gagnier & Pejcha (2023). We excised a cen-\ntral sphere containing the binary and we applied inner boundary\nconditions controlling the presence or absence of accretion. With\nthese simulations, we performed a detailed analysis of the en-\nergy transfer both within and between the kinetic and magnetic\nenergy reservoirs. Subsequently, we used similar techniques and\ndiagnostics as those used by Gagnier & Pejcha (2023) to assess\nthe impact of magnetic fields on the binary separation evolution\ntimescale, the short-term variability of mass accretion onto the\nbinary, the formation of overdensities, and angular momentum\ntransport within the shared envelope.This work is structured as follows: in Sect. 2, we introduce\nour physical model and describe the numerical setup used in our\ncommon envelope simulations. In Sect. 3, we present the results\nof our simulations. In particular, we study how kinetic and mag-\nnetic energy reservoirs are interconnected, and what contributes\nto their evolution. We then determine the scales at which the en-\nergy reservoirs interact using energy transfer analysis. By com-\nparing our results with those of Gagnier & Pejcha (2023), we\nassess the impact of magnetic fields on the binary separation evo-\nlution timescale, on the short-term variability of mass accretion\nonto the binary, the formation of overdensities, and on angular\nmomentum transport within the shared envelope. In Sect. 4, we\nsummarize and discuss our results.\n2. Physical model and numerical setup\nWe construct our post-dynamical inspiral model in the inertial\nframe at rest with the center of mass of the binary. We do not\nfollow the previous evolution of the inspiraling binary, instead,\nwe emulate its outcome following a procedure similar to Mor-\nris & Podsiadlowski (2006, 2007, 2009), Hirai et al. (2021),\nwhere the envelope is spun-up until a satisfactory amount of\ntotal angular momentum is injected. We describe the details\nof our implementation of this methodology in Gagnier & Pe-\njcha (2023). We set the gravitational constant G, the total bi-\nnary mass M=M1+M2, the primary’s initial radius R, and\nthe angular velocityp\nGM/R3to unity. The orbital velocity is\nfixed to Ωorb=»\nGM/a3\nb, where ab=r1+r2is the fixed bi-\nnary separation, M1is the mass of the primary’s core located at\n{r1,θ1,φ1}, and M2is the mass of the secondary object (either a\nmain-sequence star or a compact object) located at {r2,θ2,φ2}.\nThe orbital period is Porb=2π/Ωorb. The two objects are not\nresolved and are considered as point masses. To simplify our\nmodel and to make connection with our previous work, we con-\nsider an equal mass binary, q≡M2/M1=1, on a fixed circular\norbit. The mass of the envelope is Menv=2 in our units. Our\nchoice of initial parameters for the binary and envelope broadly\nrepresents results of ab initio simulations for a range of progen-\nitor types, as we detailed in Gagnier & Pejcha (2023). Because\nwe are most concerned with the angular momentum transport\nwithin the common envelope in the two extreme regimes of mass\nand angular momentum accretion onto the binary rather than the\nspecific details of the individual cores, we excise a central region\nencompassing the binary with a radius rin=0.625ab=R/10. In\nthe rest of this Section, we describe the equations used for solv-\ning the problem (Sect. 2.1) and the initial conditions (Sect. 2.2).\nWe then present our polar averaging implementation (Sect. 2.3)\nand the mesh structure (Sect. 2.4).\n2.1. Equations of magnetohydrodynamics\nWe use Athena ++(Stone et al. 2020) to solve the equations of\ninviscid and ideal magnetohydrodynamics\n∂ρ\n∂t+∇·ρu=0, (1)\n∂ρu\n∂t+∇·(ρu⊗u−B⊗B+P∗I)=−ρ∇Φ, (2)\n∂E\n∂t+∇·((E+PI)u−B(B·u))=−ρ∇Φ·u, (3)\n∂B\n∂t−∇×(u×B)=0, (4)\nArticle number, page 2 of 24Gagnier & Pejcha: Post-dynamical inspiral of common envelope\nwhere P∗Iis a diagonal tensor with components P∗=P+B2/2=\nP+PB,E=e+ρu2/2+B2/2 ,eis the internal energy density,\nP=(Γ−1)e,Γ = 5/3 is the adiabatic index, and Φ(r) is the\ngravitational potential of the binary,\nΦ(r)=−2X\ni=1GM i\n|r−ri|. (5)\nHeaviside–Lorentz units are used for electromagnetic quantities\nso that the magnetic permeability µ0=1. In Athena ++(and in\nalmost all higher order Godunov codes), the internal energy den-\nsity is inferred from the di fference between the total energy den-\nsityEand the kinetic and magnetic energy densities. In the vicin-\nity of the inner boundary, the plasma βparameter,β=P/PB,\nmay become very small locally, implying that the internal en-\nergy density is locally much smaller than the magnetic and ki-\nnetic energy densities. As a result, the MHD solver can locally\nreturn unphysically small or even negative internal energy den-\nsities. We circumvent this issue by solving the internal energy\ndensity equation (e.g., Stone & Norman 1992; Bryan et al. 2014;\nTakasao et al. 2015)\n∂ˇe\n∂t+∇·(uˇe)+(Γ−1)ˇe∇·u=0, (6)\nin addition to the total energy density equation Eq. (3). Eq. (6)\nensures internal energy density positivity. The selection criterion\noperates on a cell by cell basis as follows\ne=®\nE−ρu2/2−B2/2,E−ρu2/2−B2/2<(1−ϵe)e\nˇe,otherwise.(7)\nWe choose ϵe=0.05. The system is completed by the same\nboundary conditions as in Gagnier & Pejcha (2023) for hydro-\ndynamic variables to model the two regimes of accretion. Either\nthe inner boundary supports the weight of the primary’s enve-\nlope preventing the fluid to flow through it, or the inner bound-\nary is open to angular momentum and mass flow by imposing\nzero radial gradient of ρ,P,uθ, and angular momentum in ghost\nzones. The horizontal components of the magnetic field are set\nto zero in inner ghost zones, and are copied from the last radial\ncell in the domain in outer ghost zones. The component normal\nto the boundaries is calculated by imposing the divergence-free\nconstraint.\n2.2. Initial conditions\nAs in Gagnier & Pejcha (2023), we assume that the gas in the\nenvelope is initially in hydrostatic equilibrium and that it can be\ndescribed by a polytropic equation of state ignoring the gas self-\ngravity and considering purely radial initial profiles. The radial\ndensity profiles are\nρ(r)\nρ(rin)=Ç\n1+CÇ\nB\n3Ç\n1\nr3−1\nr3\ninå\n−AÅ1\nr−1\nrinãåån\n, (8)\nP(r)=KρΓ, (9)\nwhere n≡1/(Γ−1)=3/2 is the polytropic index and\nA′=AÅ1\nR−1\nrinã\n,B′=B\n3Ç\n1\nR3−1\nr3\ninå\n,\nC=κ−1\nB′−A′, K=(1−Γ)(B′−A′)\nΓ(κ−1)ρ(rin)1/n,\nκn=ρ(R)/ρ(rin).(10)\n4\n2\n024\nx4\n2\n024z\n1010\n109\n108\n107\n106\n105\n×/parenleftBig100R\nR∗/parenrightBig3/parenleftBigM1+M2\nM/parenrightBig\ngcm3 \nFig. 1. Zoomed-in snapshot of density cross section and initial poloidal\nmagnetic field lines in the xzplane after the end of spin-up.\nDensity at the inner boundary ρ(rin) is calculated from the pre-\nscribed total mass of the envelope. The primary star is initially\nembedded in a low-density medium to which we apply our outer\ndiode-type boundary conditions and in which the envelope will\nexpand later on. To model this low-density medium, we consider\nan atmosphere in hydrostatic equilibrium with constant ambi-\nent sound speed cs,ambassuming P=ρc2\ns,amb/Γ, which yields\nρext∝expÄ\nΓ/c2\ns,amb\u0000\nA/r−B/(3r3)\u0001ä\n. More details on the hy-\ndrodynamical initial conditions can be found in Gagnier & Pe-\njcha (2023).\nAs is commonly done in (GR)MHD simulations of circumbi-\nnary disks (e.g., Noble et al. 2012; Shi et al. 2012; Lopez Ar-\nmengol et al. 2021), the magnetic field is initialized as a single\npoloidal loop within the main body of the envelope at the end of\nthe spin-up phase, with a vector potential A={0,0,Aφ}where\nAφ=A0max (ρ−ρcut,0), (11)\nandρcut=10−5ρmax. Here,ρmaxis the maximum density in the\nenvelope at the moment when magnetic field is introduced and\nA0is computed to achieve the prescribed initial volume-averaged\nβparameter\nβm,i=R\nρ>ρ cutPdV\nR\nρ>ρ cutPBdV. (12)\nSpecifically,\nA0=Ñ R\nρ>ρ cut2PdV\nβm,iR\nρ>ρ cut\r\r\r∇×max ((ρ−ρcut),0)eφ\r\r\r2dVé1/2\n.(13)\nIn this work, we consider βm,i=103in the magnetized re-\ngion, which corresponds to a field initially too weak to a ffect\nthe dynamics of the envelope. In Fig. 1, we present a zoomed-in\nsnapshot of density cross section in the xzplane as well as the\npoloidal magnetic field lines at the end of spin-up. In Table 1, we\nsummarize parameters and outcomes of our runs.\nArticle number, page 3 of 24A&A proofs: manuscript no. aanda\nTable 1. Run parameters and simulations outcome at quasi-steady state.\nRun βm,iAccretion αQSS\nPαQSS\nKαQSS\nMRQSS\nMPQSS\nM\nA (Gagnier & Pejcha 2023) ∞ yes 0 .016 0.034... ......\nA’ (Gagnier & Pejcha 2023) ∞ no−0.026−0.042... ......\nB (This work) 103yes 0 .034 0.043 0.371 939 0 .643\nB’ (This work) 103no−0.007−0.020 0.060 899 1 .071\nNotes. RQSS\nM=VL/ηnumis computed using a lengthscale Lcorresponding to the typical scale of magnetic energy amplification (see Sect. 3.2.3),\nandVis taken as the averaged Alfvén speed in the region where 90% of magnetic energy amplification by line stretching occurs, averaged between\nt=100Porbandt=160Porb. Runs B and B’ in this study di ffer from the ones in Gagnier & Pejcha (2023).\n2.3. Polar averaging\nThe clustering of cells in the azimuthal direction near the polar\naxis makes the time step given by the Courant–Friedrichs–Lewy\ncondition extremely small. To mitigate this issue, we follow\nGagnier & Pejcha (2023) and use a polar averaging technique\nbased on the Ring Average technique (Lyon et al. 2004; Zhang\net al. 2019). The r- andθ-directed face-centered magnetic fields\nare also averaged in the azimuthal direction near the polar axis\nand theφ-directed field is updated from the Faraday’s law of in-\nduction, accounting for the electric field perturbations resulting\nfrom the reconstructed r- andθ-directed face-centered magnetic\nfields (e.g., Zhang et al. 2019). This procedure ensures ∇·B=0\nwithin each cell and each averaged chunk of cells.\nAt some point in our simulations, the flow ends up stagnating\nnear the polar axis, which leads to the formation of magnetic\nloops around the axis. If left unattended, the magnetic field can\ngrow to large magnitude. To prevent this, we apply a resistive\nelectric field along the axis after the reconstruction of the face-\ncentered magnetic fields (see also Lyon et al. 2004). The resistive\nfield is given by\n∆Eaxis\nr∆r=cosθaxis\nNφτdampNφX\nk=1ΦB\nφ(k), (14)\nwhere\nθaxis=®\n0 in the northern hemisphere,\nπin the southern hemisphere.(15)\nHere, Nφis the number of cells in the φdirection adjacent\nto the polar axis at a given radius, ∆ris the radial extent of\nthe cell along the polar axis, ΦB\nφ(k) is the azimuthal magnetic\nflux through the kthcell face, and τdamp is an arbitrary damping\ntimescale which we choose to be a few time steps, O(10−5Porb).\nBy construction, the application of such a resistive electric field\nleaves ∇·Bunchanged. While primarily serving magnetic field\nstabilization, the application of such a resistive electric field may\nalso result in the suppression of jets.\n2.4. Mesh structure and convergence\nAt the root level, we set 512 ×256×256 active cells in {r,θ,φ}.\nThe radial domain extends from r=0.1 tor=10 with geomet-\nric grid spacing so that the aspect ratio of the cells is approxi-\nmately constant over the entire range of radial scales. The grid\nspacing inθandφdirections is uniform with 0 < θ < π and\n0<φ< 2π. We use polar boundary conditions in the θdirection\nallowing free-flow through the pole (Zhu & Stone 2018; Stone\net al. 2020), and periodic boundary conditions in the φdirection.On top of the root level, we add one level of adaptive refine-\nment with criteria based on the azimuthal average of the second\nderivative error norm χof a function σ(e.g., Lohner et al. 1987;\nMignone et al. 2012). The norm χis defined as\nχ=ÃP\nd|∆d,+1/2σ−∆d,−1/2σ|2\nP\nd\u0000\n|∆d,+1/2σ|+|∆d,−1/2|+ϵσd,ref\u00012≥χr. (16)\nHere, ∆r,±1/2=±(σi±1−σi) andσr,ref=|σi+1|+2|σi|+|σi−1|. The\nvalue of the threshold χrdepends on the specifics of the problem\nand on the chosen refinement variable σ. Here, we take σ=|B|.\nWe ensure that magnetorotational instability (MRI), which\ncould occur in the simulation, is well resolved. We compute the\nquality factors in the three directions (e.g., Noble et al. 2010;\nHawley et al. 2013) defined as\nQi=λMRI,i\n∆i=2π|vA,i|\nΩ∆ i, (17)\nwhere vA,i=»\nB2\ni/ρis the i-component of the Alfvén speed in\ncode units, and ∆iis the size of individual cells in the i-direction.\nUsing Qi, we measure the resolvability Ri, which is the frac-\ntion of cells with a quality factor larger than that associated with\nmarginal resolvability Qmin=8 in the direction i(e.g., Sorathia\net al. 2012; Parkin & Bicknell 2013). In Figure 2, we show evo-\nlution of resolvability, volume-averaged quality factor, and the\nproduct QθQφ, for simulation run B. A value exceeding 200–\n250 for the product QθQφis commonly regarded as a reliable in-\ndicator of convergence in MRI simulations (e.g., Narayan et al.\n2012; Porth et al. 2019; Dhang et al. 2023). We find that while\ninitially small due to limited envelope extent within the numer-\nical domain, these quantities increase as the envelope expands\nand reaches the outer boundary, and surpass the threshold for\nconvergence at t≃55Porb. This behavior can be attributed to the\nzero magnetic field amplitude we initially set in the low-density\nambient medium encompassing the envelope. We conclude that\nour simulations have achieved numerical convergence.\n3. Results\nWe use a total of 4 million CPU hours on the Karolina cluster at\nIT4Innovations to perform our simulations. The parameters used\nfor our simulation runs are outlined in Table 1. In Figs. 3 and 4,\nwe present a zoomed-in snapshot of density and magnetic field\namplitude cross section in the xzplane at t=140Porbfor simula-\ntion B. In the rest of this section, we investigate the approach to\nquasi-steady state (Sect. 3.1), kinetic and magnetic energy bud-\nget (Sect. 3.2), evolution of the binary orbit (Sect. 3.3), angular\nmomentum transport (Sect. 3.4), value of the αdisk parameter\n(Sect. 3.5), time variability of accretion (Sect. 3.6), and behavior\nof the lump (Sect. 3.7).\nArticle number, page 4 of 24Gagnier & Pejcha: Post-dynamical inspiral of common envelope\n0.00.51.0Resolvability(a)\nRrR\nR\n103\n100103Quality factor(b)\nQr\nQmin=8Q\nQ\n255075100125150\ntime[Porb]101\n102105QQ\n(c)\nQQ=250\nFig. 2. Temporal evolution of resolvability (panel a), quality factor\n(panel b), and QθQφ(panel c) in simulation run B. As the envelope ex-\npands within the numerical domain, these quantities progressively reach\nvalues that exceed the threshold values commonly associated with nu-\nmerical convergence (horizontal dashed lines).\n3.1. Quasi-steady state\nAfter an initial transient period, the envelope settles into a quasi-\nsteady state (QSS) during which the global properties of the en-\nvelope’s dynamics change only slowly. During this phase, matter\ncontinues to slowly accrete onto the central binary (if accretion\nis allowed) and simultaneously escapes from the numerical do-\nmain by the large scale flow. To asses the turbulent fluxes of an-\ngular momentum and their quasi-steadiness, we apply Reynolds\ndecomposition to the density, the gravitational potential of the\nbinary, and the magnetic field\nρ=ρ+ρ′, (18)\nΦ =Φ + Φ′, (19)\nBi=Bi+B′\ni, (20)\nwhere overlines indicate azimuthal averages of quantity q,\nq=1\n2πZ2π\n0qdφ. (21)\nWe also apply Favre decomposition to the velocity field (Favre\n1965, 1969)\nui=eui+u′′\ni, (22)\nwhereeq=ρq/ρis the density-weighted Reynolds average also\nknown as Favre average. Taking the cross product of rwith\nthe momentum equation (2), multiplying by ezand applying\nReynolds average to the result yields the Reynolds averaged an-\ngular momentum evolution equation,\n∂ρs‹uφ\n∂t=−∇·\u0000\ns\u0000\nWrφ+Wθφ\u0001\u0001\n−ρs∂Φ′\n∂φ, (23)\nwhere s=rsinθis the radial cylindrical coordinate,\nWiφ=Wiφ+wiφ (24)is the total stress and\nWiφ=ρeui‹uφ−BiBφ, (25)\nwiφ=ρfiu′′\niu′′φ−B′\niB′φ. (26)\nWe assume that a quasi-steady state is reached when the\nvolume-averaged turbulent stress normalized to gas pressure\n(Shakura & Sunyaev 1973),\nαP=⟨wrφ⟩\n⟨P⟩, (27)\nand the normalized rφ–component of the Reynolds and Maxwell\nstresses, respectively\nαK=⟨2ρu′′\nru′′\nφ⟩\n⟨ρ|u′′|2⟩, (28)\nαM=−⟨2B′\nrB′\nφ⟩\n⟨|B′|2⟩, (29)\nbecome statistically time-independent (e.g., Simon et al. 2012;\nParkin & Bicknell 2013). We show the time evolution of αP,αK\nandαMin Fig. 5 for runs B and B’. We find that these three\nquantities reach quasi-steady values from t≃100Porbfor sim-\nulation run B. αKandαMalso reach quasi-steady values from\nt≃100Porbfor simulation run B’, however, as αKis small, it\ncan change sign and lead to a substantial relative variation in αP.\nStill, we assume that the system has reached a quasi-steady state\nfrom t≃100Porbfor the two simulation runs. The time-averaged\nvalues ofαK,αPandαMat QSS are summarized in Table 1. The\nvalues ofαPandαMat QSS for run B are in agreement with\nglobal and local magnetohydrodynamic simulations of accretion\ndisks (e.g., Simon et al. 2012; Hawley et al. 2013; Parkin & Bick-\nnell 2013) and indicate globally outward turbulent transport of\nangular momentum. These results, however, do not imply that\nthe angular momentum turbulent flux cannot be locally directed\ninward nor that Maxwell stress dominates Reynolds stress. The\nradial and latitudinal dependence of angular momentum radial\ntransport is investigated in Sect. 3.4.\n3.2. Energy budget and dynamical relevance of magnetic\nfields\nIn this section, we determine how kinetic and magnetic energy\nreservoirs are interconnected, what contributes to their evolution,\nand the dynamical relevance of magnetic fields during the post-\ndynamical phase of CEE. In particular, we identify the source\nof magnetic energy amplification in Sects. 3.2.1 and 3.2.2, and\ndetermine the scales on which energy is transferred within and\nbetween energy reservoirs in Sect. 3.2.3. For the sake of brevity,\nmost of this analysis is restricted to simulation run B, but results\nfor B’ are analogous.\n3.2.1. Magnetic energy budget\nWe show the evolution of the magnetic energy\n⟨EB⟩=Z1\n2B·BdV (30)\nand kinetic energy\n⟨EK⟩=Z1\n2ρu·udV, (31)\nArticle number, page 5 of 24A&A proofs: manuscript no. aanda\n2\n1\n012\nx2.0\n1.5\n1.0\n0.5\n0.00.51.01.52.0z\n108\n107\n×/parenleftBig100R\nR∗/parenrightBig3/parenleftBigM1+M2\nM/parenrightBig\ngcm3 \nFig. 3. Zoomed-in snapshot of density cross section in the xzplane at t=140Porbfor simulation B. The texture, computed using the publicly\navailable Line Integral Convolution Knit python package (Wa fflard-Fernandez & Robert 2023), indicates the meridional streamlines.\nfor simulation B in Fig. 6. We find that, after a rapid amplifica-\ntion phase, the total magnetic energy saturates and then slowly\ndecays. The initial rapid amplification of the magnetic energy re-\nsults from the stretching, folding, and winding of the initial weak\npoloidal field by di fferential rotation and turbulence.\nAs the magnetic field strengthens, the Lorentz force begins\nto act back on the fluid, promoting more ordered and stable flow\nand leading to magnetic energy saturation. The magnetic ten-\nsion associated with the strong toroidal field resulting from the\nwinding up of the initial poloidal field, maintains the spiral den-\nsity waves’ structure by resisting their deformation against ra-\ndial perturbation. This can be seen in Fig. 7 where we compare\ndensity snapshots taken at t=140Porbfor models A and B, re-\nvealing that model B exhibits sharper spiral waves and a more\nuniform density distribution compared to model A. Additionally,\nmagnetic tension tends to align fluid motion with magnetic field\nlines, resulting in a preference for azimuthal kinetic energy over\nradial kinetic energy during the saturated phase, in contrast to\nnonmagnetic simulations (Fig. 6). Hence, the presence of mag-\nnetic fields, even if they are relatively weak, has a significant im-\npact on the envelope’s structure and dynamics. This is discussed\nin more details later in this subsection, as well as in the following\nsubsections, 3.2.2 and 3.2.3. Finally, the magnetic energy at sat-uration, approximately 1044erg, is similar to the findings from\nprevious ab initio MHD simulations by Ohlmann et al. (2016b)\nand Ondratschek et al. (2022). However, in our case, the ratio be-\ntween kinetic and magnetic energy is 10, which is significantly\nsmaller than in their results, where it was roughly 1000.\nIn Fig. 8, we show the kinetic and magnetic horizontally in-\ntegrated energy spectra averaged over twenty spectra spanning\nthree orbital periods between t=137Porbandt=140Porbdur-\ning the saturated phase (see Appendix A and e.g., Baddour 2010;\nParkin & Bicknell 2013). We find that kinetic energy is dominant\non all scales with a diminishing ratio towards smallest scales\nwhere equipartition is approached. To assess the relevant physics\nassociated with the amplification and saturation of the magnetic\nenergy, we write the total magnetic energy evolution equation\n(see Appendix B),\n⟨˙EB⟩=Z\n(B⊗B):∇udV−Z\nEB∇·udV−I\nEBu·ndS,\n=⟨˙EB,stretch⟩+⟨˙EB,exp⟩+⟨˙EB,adv⟩,\n(32)\nwhere nis the outward-pointing unit vector at the domain bound-\naries surface,⟨˙EB,stretch⟩is associated with the change in mag-\nnetic energy within the domain resulting from the stretching of\nArticle number, page 6 of 24Gagnier & Pejcha: Post-dynamical inspiral of common envelope\n2\n1\n012\nx2.0\n1.5\n1.0\n0.5\n0.00.51.01.52.0z\n101102103104\nB×/parenleftBig100R\nR∗/parenrightBig2/parenleftBigM1+M2\nM/parenrightBig\n G\nFig. 4. Zoomed-in snapshot of magnetic field amplitude cross section at t=140Porbfor simulation B. The texture indicates meridional magnetic\nfield lines.\n20406080100120140160\ntime[Porb]0.06\n0.04\n0.02\n0.000.020.040.06P\n0.1\n0.00.10.20.30.40.50.60.7\nK,M\nP\nK\nM\nFig. 5. Evolution of the normalized volume-averaged stresses for sim-\nulation runs B (full lines) and B’ (dashed lines). We consider a quasi-\nsteady state to be reached after t≃100Porb.the magnetic field lines by the fluid elements, ⟨˙EB,exp⟩corre-\nsponds to changes in magnetic energy due to expansion or com-\npression e ffects, and⟨˙EB,adv⟩corresponds to changes in magnetic\nenergy from field advection through the domain boundaries.\nThe di fference between the direct measurement of ⟨˙EB⟩and\nof the sum⟨˙EB,stretch⟩+⟨˙EB,exp⟩+⟨˙EB,adv⟩corresponds to the\nnumerical dissipation rate of magnetic energy by Joule heating\n⟨˙EB,diss⟩. Assuming numerical dissipation to be of the Ohmic\nform (e.g., Parkin & Bicknell 2013), we estimate the numerical\nresistivity as\nηnum=⟨˙EB,diss⟩\n⟨B∆B⟩=−⟨˙EB,diss⟩H\n(∇·σ)·ndS+R\n|∇×B|2dV. (33)\nHere, σ=B⊗B−EBIis the total Maxwell stress tensor and\nwe used integration by parts and the identity B×(∇×B)=\n∇(B·B)/2−(B·∇)B. We show the di fferent contributions to\nthe evolution of the total magnetic energy density in Fig. 9 and\nour measure of the numerical resistivity ηnumin Fig. 10. We find\na magnetic Reynolds number ReQSS\nM=O(103) at quasi-steady\nstate for our models (see Table 1). However, we emphasize that\nthe Ohmic dissipation formalism might not be valid and the mea-\nsured resistivity likely depends on grid resolution, time integra-\ntion method, spatial reconstruction scheme and Riemann solver\nArticle number, page 7 of 24A&A proofs: manuscript no. aanda\n255075100125150\ntime[Porb]104410451046Energy×/parenleftBig100R\nR∗/parenrightBig/parenleftBigM1+M2\nM/parenrightBig2 erg\n/angbracketleftbig\nErK/angbracketrightbig\n/angbracketleftbig\nEK/angbracketrightbig\n/angbracketleftbig\nEK/angbracketrightbig\n/angbracketleftbig\nErB/angbracketrightbig\n/angbracketleftbig\nEB/angbracketrightbig\n/angbracketleftbig\nEB/angbracketrightbig\n/angbracketleftbig\nEK/angbracketrightbig\n/angbracketleftbig\nEB/angbracketrightbig\nFig. 6. Evolution of the total kinetic and magnetic energy and of their\nradial, latitudinal and azimuthal components for Gagnier & Pejcha\n(2023)’s simulation A (dashed lines) and this work’s simulation B (solid\nlines).\n(e.g., Rembiasz et al. 2017). In Fig. 9, we see that the dominant\nsource of magnetic energy during both the growth and satura-\ntion phases is the stretching of the magnetic field lines by the\nvelocity shear, which accounts for 95% of magnetic energy gen-\neration during QSS. The remaining 5% is produced by compres-\nsion against magnetic pressure. The stretching of the magnetic\nfield lines and the convergent transport of magnetic energy trans-\nfer kinetic energy into magnetic energy and e ffectively main-\ntains dynamo action against dissipation and advection through\nthe boundaries.\n3.2.2. Kinetic energy budget\nTo assess the relevant physics associated with the evolution of\nthe kinetic energy budget and the dynamical relevance of mag-\nnetic fields in our simulations, we decompose the kinetic energy\nevolution equation into mean and turbulent contributions in Ap-\npendix C (see also Favre 1965, 1969). The Reynolds averaged\nmean kinetic energy evolution equation reads\n˙Emean\nK(r,θ)=1\n2∂ρ(eu·eu)\n∂t=−∇·Å\nρeu(eu·eu)\n2ã\n−∇·\u0010\nρ·\u0000u′′⊗u′′·eu\u0011\n+ρ·\u0000u′′⊗u′′:∇eu−eu·∇P+eu·\u0000\n∇·σ\u0001\n−eu·ρ∇Φ +eu·\u0000\n∇·τ\u0001\n,\n(34)\nand the Reynolds averaged turbulent kinetic energy evolution\nequation reads\n˙Eturb\nK(r,θ)=1\n2∂ρ(‡u′′·u′′)\n∂t=−∇· \nρeu(‡u′′·u′′)\n2!\n+∇·Å\nσ·u′′\n−ρu′′(u′′·u′′)\n2−P′u′′ã\n−ρ·\u0000u′′⊗u′′:∇eu−σ:∇u′′−u′′·∇P\n+P′∇·u′′−ρu′′·∇Φ′+u′′·(∇·τ),\n(35)\n2\n 0 2\nx3\n2\n1\n0123y\n108\n107\n×/parenleftBig100R\nR∗/parenrightBig3/parenleftBigM1+M2\nM/parenrightBig\ngcm3 \n2\n 0 2\nx3\n2\n1\n0123y\n108\n107\n×/parenleftBig100R\nR∗/parenrightBig3/parenleftBigM1+M2\nM/parenrightBig\ngcm3 \nFig. 7. Zoomed-in snapshot of density cross section in the xyplane at\nt=140Porbfor Gagnier & Pejcha (2023)’s simulation A (top) and this\nwork’s simulation B (bottom).\nwhere τ=νnumρcis the (numerical) viscous stress tensor, c=\n∇u+(∇u)⊺−(2/3)(∇·u)Iis the shear tensor and νnumis the\nnumerical viscosity that we assume spatially constant. Finally,\nwe integrate Eqs. (34) and (35) over the meridional plane, and\nmultiply the result by 2 πto obtain the mean and turbulent kinetic\nenergy equations.\n⟨˙Emean\nK⟩=⟨˙Emean\nK,adv⟩+⟨˙Emean\nK,R⟩+⟨˙Emean\nK,turb⟩+⟨˙Emean\nK,P⟩\n+⟨˙Emean\nK,L⟩+⟨˙Emean\nK,Φ⟩+⟨˙Emean\nK,diss⟩,(36)\nArticle number, page 8 of 24Gagnier & Pejcha: Post-dynamical inspiral of common envelope\n101\n100101\nk/kR10511052105310541055105610571058E(k)×/parenleftBigM1+M2\nM/parenrightBig2ergcm\nk2\nk5/2\nEB\nEK\nFig. 8. Kinetic and magnetic horizontally integrated energy spectra dur-\ning the saturation phase obtained with ℓmax=128 for simulation run\nB. Opaque lines represent the spectra excluding ℓ=0 components,\nwhile transparent lines include them. The oscillations visible at high\nk/kRwhen including ℓ=0 components result from the our direct in-\ntegration of Eq. (A.5) which involves spherical Bessel functions jℓ(kr)\nthat are very oscillatory for kr≫ℓ.\n255075100125150\ntime[Porb]1.0\n0.5\n0.00.51.0˙EB×1039/parenleftBig100R\nR∗/parenleftBigM1+M2\nM/parenrightBig/parenrightBig5/2erg·s1 \n/angbracketleftbig˙EB/angbracketrightbig\n/angbracketleftbig˙EB,stretch/angbracketrightbig\n/angbracketleftbig˙EB,exp/angbracketrightbig/angbracketleftbig˙EB,adv/angbracketrightbig\n/angbracketleftbig˙EB,diss/angbracketrightbig\nFig. 9. Evolution of the rate of change of magnetic energy ⟨˙EB⟩and of\nits components defined in Eq. (32), for simulation run B. The numeri-\ncal dissipation rate of magnetic energy by Joule heating is measured as\n⟨˙EB,diss⟩=⟨˙EB⟩-⟨˙EB,stretch⟩-⟨˙EB,exp⟩-⟨˙EB,adv⟩.\nwhere\n⟨˙Emean\nK,adv⟩=−¨\n∇·Ä\nEmean\nKeuä∂\n, (37a)\n⟨˙Emean\nK,R⟩=−¨\n∇·\u0010\nρeu··\u0000u′′⊗u′′\u0011∂\n, (37b)\n⟨˙Emean\nK,t⟩=¨\nρ·\u0000u′′⊗u′′:∇eu∂\n, (37c)\n⟨˙Emean\nK,P⟩=−¨\neu·∇P∂\n, (37d)\n⟨˙Emean\nK,L⟩=¨\neu·\u0000\n∇·σ\u0001∂\n, (37e)\n⟨˙Emean\nK,Φ⟩=−¨\neu·ρ∇Φ∂\n, (37f)\n⟨˙Emean\nK,diss⟩=νnum¨\neu·\u0000\n∇·ρc\u0001∂\n. (37g)\n255075100125150\ntime[Porb]0.00.20.40.60.81.01.2Diffusivity×1016/parenleftBigR∗\n100R/parenrightBig1/2/parenleftBigM1+M2\nM/parenrightBig1/2cm2s1\n num\nnum\nFig. 10. Time evolution of the measured numerical resistivity (Eq. (33))\nand viscosity (Eq. (40)) for simulation run B. At quasi-steady state, we\nfindνnum≃4.35×1015cm2s−1andηnum≃7.11×1015cm2s−1.\nHere,⟨˙Emean\nK,adv⟩corresponds to the changes of mean kinetic energy\nfrom its advection through the domain boundaries, ⟨˙Emean\nK,R⟩is as-\nsociated with the transport of mean kinetic energy by Reynolds\nstress,⟨˙Emean\nK,t⟩measures the destruction rate of mean kinetic en-\nergy into turbulence, ⟨˙Emean\nK,P⟩is the work done by pressure on the\nmean flow,⟨˙Emean\nK,L⟩is the work done by the Lorentz force on the\nmean flow, and⟨˙Emean\nK,Φ⟩is the production rate of mean kinetic\nenergy from the binary’s gravitational potential energy. The tur-\nbulent kinetic energy evolution equation in turn reads\n⟨˙Eturb\nK⟩=⟨˙Eturb\nK,adv⟩+⟨˙Eturb\nK,trans⟩+⟨˙Eturb\nK,t⟩+⟨˙Eturb\nK,P⟩\n+⟨˙Eturb\nK,σ⟩+⟨˙Emean\nK,exp⟩,(38)\nwhere\n⟨˙Eturb\nK,adv⟩=−¨\n∇·\u0010\nEturb\nKeu\u0011∂\n, (39a)\n⟨˙Eturb\nK,trans⟩=≠\n∇·Å\nσ·u′′−ρu′′(u′′·u′′)\n2−P′u′′ã∑\n, (39b)\n⟨˙Eturb\nK,t⟩=−⟨˙Emean\nK,t⟩, (39c)\n⟨˙Eturb\nK,P⟩=−¨\nu′′·∇P∂\n, (39d)\n⟨˙Eturb\nK,σ⟩=−¨\nσ:∇u′′∂\n, (39e)\n⟨˙Eturb\nK,exp⟩=¨\nP′∇·u′′∂\n, (39f)\n⟨˙Eturb\nK,Φ⟩=−¨\nρu′′·∇Φ′∂\n, (39g)\n⟨˙Eturb\nK,diss⟩=νnum¨\nu′′·(∇·ρc)∂\n. (39h)\nHere,⟨˙Eturb\nK,adv⟩corresponds to the changes in turbulent kinetic en-\nergy from its advection through the domain boundaries, ⟨˙Eturb\nK,trans⟩\nis the turbulent kinetic energy transport rate from the work done\nby the Lorentz and pressure forces and by turbulent velocity\nfluctuations,⟨˙Eturb\nK,t⟩measures the production of turbulent kinetic\nenergy from the interaction between mean shear and Reynolds\nArticle number, page 9 of 24A&A proofs: manuscript no. aanda\nstress,⟨˙Eturb\nK,P⟩is the work done by pressure on the turbulent flow,\n⟨˙Eturb\nK,σ⟩is the rate of production or destruction of turbulent ki-\nnetic energy from the interaction between turbulent shear and\nMaxwell stress, and ⟨˙Emean\nK,exp⟩is the rate of change of turbulent ki-\nnetic energy due to pressure dilatation. Finally ⟨˙Eturb\nK,Φ⟩is the pro-\nduction rate of turbulent kinetic energy from the work done by\nthe multipole moments of the binary’s gravitational force on the\nturbulent flow. We note that because we apply Favre decompo-\nsition to the velocity field, the rate of change of the total kinetic\nenergy is simply the sum of its mean and turbulent counterparts,\nthat is,⟨˙EK⟩=⟨˙Emean\nK⟩+⟨˙Eturb\nK⟩.\nWe show the evolution of the mean and turbulent kinetic en-\nergy change rates and of their components at QSS in Figs. 11\nand 12. At QSS, we find that the binary orbital energy is the\nsole contributor to the envelope’s mean kinetic energy produc-\ntion, while other processes act as sinks of mean kinetic en-\nergy. The primary sinks for mean kinetic energy are the losses\nthrough the boundaries and the work done by pressure on the\nmean flow. Additionally, but to a lesser extent, mean kinetic en-\nergy is dissipated into turbulence and lost due to the advection\nof mean flow by the Reynolds stress and by the work done by\nthe Lorentz force. Turbulent kinetic energy production is evenly\nsplit between the work done by the multipole moments of the\nbinary’s gravitational force on the turbulent flow and the inter-\naction between the mean shear and Reynolds stress. Conversely,\nthe loss of turbulent kinetic energy mainly comes from its advec-\ntion through the domain boundaries. Still, numerical dissipation,\npressure dilatation, and interactions between turbulent shear and\nMaxwell stress significantly contribute to the loss of turbulent\nkinetic energy.\nSimilarly to the way we measured numerical Ohmic resistiv-\nity, we estimate the numerical kinematic viscosity, assuming it\nto be spatially constant, as\nνnum=⟨˙Emean\nK,diss⟩+⟨˙Eturb\nK,diss⟩\n¨\neu·\u0000\n∇·ρc\u0001∂\n+¨\nu′′·(∇·ρc)∂, (40)\nwhere⟨˙Emean\nK,diss⟩and⟨˙Eturb\nK,diss⟩are obtained by substracting the var-\nious nondissipative components to the measured ⟨˙Emean\nK⟩and\n⟨˙Eturb\nK⟩. We show our measurement of the numerical kinematic\nviscosity for simulation B in Fig. 10. We estimate the magnetic\nPrandtl number to be PM≃0.6, a value identical to that ob-\ntained by Parkin (2014) in the context of global simulations of\nmagnetorotational turbulence using the PLUTO code (Mignone\net al. 2012). However, astrophysical bodies are characterized by\na wide range of magnetic Prandtl numbers which often sub-\nstantially deviate from unity(e.g., Brandenburg & Subramanian\n2005; Balbus & Henri 2008; Brandenburg 2009). Our measured\nPMmay thus not be realistic for common envelopes and cast\ndoubts on the actual saturation values of αMandαP, that is,\non angular momentum transport e fficiency. Still, an asymptotic\nregime RM>Rcin whichαPbecomes PM–independent may be\nreached in our simulations (e.g., Brandenburg 2009; Käpylä &\nKorpi 2011; Oishi & Mac Low 2011). Here, RM=VL/ηnumis\nthe magnetic Reynolds number quantifying the relative impor-\ntance of magnetic induction to magnetic di ffusion, Rcis a criti-\ncal Reynolds number of order 103, and VandLare respectively a\ntypical velocity and lengthscale of the flow. We reiterate here that\nour measure of the Prandtl number is only indicative and should\nbe taken with caution. It is the result of strong approximations\non the spatial uniformity of the numerical di ffusivities and that\nthe numerical di ffusivity is equivalent to an Ohmic resistivity.\n100110120130140150160\ntime[Porb]1.0\n0.5\n0.00.51.01.52.0˙EmeanK ×1039/parenleftBig100R\nR∗/parenleftBigM1+M2\nM/parenrightBig/parenrightBig5/2ergs1 \n/angbracketleftbig˙EmeanK/angbracketrightbig\n/angbracketleftbig˙EmeanK,adv/angbracketrightbig\n/angbracketleftbig˙EmeanK,R/angbracketrightbig\n/angbracketleftbig˙EmeanK,t/angbracketrightbig/angbracketleftbig˙EmeanK,P/angbracketrightbig\n/angbracketleftbig˙EmeanK,L/angbracketrightbig\n/angbracketleftbig˙EmeanK,/angbracketrightbig\n/angbracketleftbig˙EK,diss/angbracketrightbigFig. 11. Evolution of the rate of change of mean kinetic energy ⟨˙Emean\nK⟩\nand of its components defined in Eqs. (37a)–(37g), for simulation run\nB.\n100110120130140150160\ntime[Porb]4\n3\n2\n1\n0123˙EturbK ×1038/parenleftBig100R\nR∗/parenleftBigM1+M2\nM/parenrightBig/parenrightBig5/2ergs1 \n/angbracketleftbig˙EturbK/angbracketrightbig\n/angbracketleftbig˙EturbK,adv/angbracketrightbig\n/angbracketleftbig˙EturbK,trans/angbracketrightbig/angbracketleftbig˙EturbK,t/angbracketrightbig\n/angbracketleftbig˙EturbK,P/angbracketrightbig\n/angbracketleftbig˙EturbK,/angbracketrightbig\n/angbracketleftbig˙EturbK,/angbracketrightbig\n/angbracketleftbig˙EturbK,exp/angbracketrightbig\n/angbracketleftbig˙EturbK,diss/angbracketrightbig\nFig. 12. Evolution of the rate of change of turbulent kinetic energy\n⟨˙Emean\nK⟩and of its components defined in Eqs. (39a)–(39h), for simu-\nlation run B.\n3.2.3. Energy transfer analysis\nHere we analyze the scales at which the energy sources described\nin sections 3.2.1 and 3.2.2 operate and we focus attention to the\nscales at which magnetic energy is amplified. In order to do so,\nwe perform a spectral energy transfer analysis, which is a com-\nmonly used method for understanding turbulent processes (e.g.,\nKraichnan 1967; Debliquy et al. 2005; Alexakis et al. 2007; Si-\nmon et al. 2009; Pietarila Graham et al. 2010; Lesur & Longaretti\n2011; Rempel 2014; Grete et al. 2017, 2021; Du et al. 2023).\nWe show the energy transfer equations and the resulting trans-\nfer functions in Appendix D. We emphasize that our approach\ninvolves horizontally integrated Fourier transformation (see Ap-\nArticle number, page 10 of 24Gagnier & Pejcha: Post-dynamical inspiral of common envelope\npendix A) for computing the transfer functions. Consequently,\nwe are unable to assess the anisotropy of energy transfer in this\nanalysis.\nIn Fig. 13, we show the energy transfer quantities averaged\nover 137≤t/Porb≤140. In panel (a), we display the mag-\nnetic energy transfer spectra associated with the stretching of\nmagnetic field lines TMT(k), the advection of magnetic energy\nwithin the magnetic energy reservoir TMA(k), and with compres-\nsion e ffectsTMC(k). In panel (b), we show the spectra associ-\nated with the magnetic cascade TMA(k)+0.5TMC(k) and the ki-\nnetic to magnetic energy transfer TMT(k)+0.5TMC(k). Positive\nvalues of transfer functions at a given wavenumber kindicate\ngain of energy at a scale 2 π/k. Conversely, negative values in-\ndicate a loss of energy at this scale. In panel (a), we find that\nTMT(k) is positive for all wavenumbers k, indicating that mag-\nnetic energy production occurs through line stretching across\nall scales. Conversely, TMA(k) is only positive on scales smaller\nthan∼ab/10, suggesting a forward magnetic energy cascade\nfrom larger to smaller scales through advection. Remarkably,\nboth TMT(k) and TMA(k) peak at a scale of R∗/2≈3ab, which\nis approximately the wavelength of the binary-driven spiral den-\nsity waves, with almost equal but opposite values. This indicates\nthat the bulk of magnetic energy production occurs at that scale\nthrough line stretching and that the newly generated magnetic\nenergy is promptly advected towards the smallest scales. We find\nthat the energy transfer rate by compression e ffects, TMC(k), is\noverall weaker compared to TMT(k) and TMA(k). Unlike TMT(k)\nandTMA(k), transfer by compression e ffectsTMC(k) alternates\nbetween positive and negative values with varying k. This can be\nattributed to the compression–rarefaction pattern of spiral den-\nsity waves. Although weaker, TMC(k) plays a crucial role in the\nnet magnetic energy production on the R∗/2 scale due to the\nnear-perfect balance between TMT(k) and TMA(k) at this scale.\nIn panel (b), we see that, on scales larger than 2 R∗,TMT(k)+\n0.5TMC(k) is in balance with TMA(k)+0.5TMC(k). Such balance\nindicates no net gain of magnetic energy at scales larger than 2 R∗\nand thus implies the absence of large-scale magnetic field pro-\nduction. On smaller scales however, TMT(k)+TMA(k)+TMC(k)\nis positive. Despite this, ⟨˙EB⟩≲0 due to numerical dissipation\nof magnetic energy by Joule heating (see Figs. 6 and 9).\nIn panel (a) of Fig. 13, we also show the time-averaged spec-\ntra associated with the gain and loss of kinetic energy from the\nwork of the Lorentz force via magnetic tension and magnetic\npressure, TKL. The kinetic energy reservoir’s perspective o ffers a\ndifferent view. Indeed, we see that kinetic energy is lost to mag-\nnetic energy on scales down to ∼0.4ab. Approximately 20% of\nthe energy drawn from the kinetic reservoir on these larger scales\nreturns as kinetic energy on smaller scales. Remarkably, we find\nthat most of the kinetic energy is lost to magnetic energy on the\nlargest scales. However, the gain of magnetic energy on these\nscales is notably lower, which hinders the generation of signifi-\ncant large-scale magnetic fields during QSS. This highlights the\nnonlocality of interactions between magnetic and kinetic energy\nin spectral space (Alexakis et al. 2007). Unfortunately, we can-\nnot pinpoint the specific origin or destination scales in spectral\nspace. This would require a more complex shell-to-shell energy\ntransfer analysis (e.g., Alexakis et al. 2007; Lesur & Longaretti\n2011; Grete et al. 2017, 2021) that is beyond the scope of this\nwork.\nThe lack of magnetic energy production on large radial\nscales implies that the amplified magnetic field may not be able\nto funnel and collimate a radial polar outflow, potentially imped-\ning the formation of well-defined bipolar jets or jet-like outflows\nthat often give rise to the characteristic bipolar shapes commonlyobserved in planetary nebulae (e.g., García-Segura et al. 1999,\n2018, 2020; Zou et al. 2020; Ondratschek et al. 2022). Nonethe-\nless, the presence of a strong toroidal magnetic field about the\norbital plane introduces magnetic tension. This tension acts to\ndecelerate the expansion of the envelope in directions perpendic-\nular to the polar axis. Additionally, centrifugal forces, along with\nturbulent mixing, jointly contribute to the formation of inter-\nmittent, jittering low-density regions near the poles (see Fig. 3)\nwhich may facilitate the e fficient channeling of outflows. Hence,\nthe combined e ffects of magnetic tension, centrifugal forces, and\nturbulent mixing may ultimately contribute to the formation of\nnonspherical planetary nebulae.\n3.3. Binary evolution and angular momentum conservation\nHere, we address the impact of magnetic fields on the orbital\nevolution of the central binary. The commonly accepted model\nof common envelope evolution assumes a near-monotonic de-\ncrease in binary separation over time. In Gagnier & Pejcha\n(2023), we showed that this process is characterized by an orbital\ncontraction timescale that can be much shorter than the thermal\ntimescale of the envelope, particularly when the central binary is\nable to accrete mass and angular momentum from the surround-\ning envelope. In our setup, and as in Gagnier & Pejcha (2023),\nwe fix the orbital parameters and predict the orbital evolution\nby measuring the rate of angular momentum transfer between\nthe binary and the envelope without self-consistently consider-\ning any reciprocal influence from the envelope to the binary or-\nbit. To assess the influence of magnetic fields on the secular evo-\nlution of binary separation, it is crucial to evaluate the torques\nacting within the shared envelope. These torques originate from\nboth the quadrupolar moment of the gravitational potential and\nthe angular momentum fluxes resulting from stresses across the\ndomain boundaries. The conservation equation for angular mo-\nmentum can be expressed as\n˙Jz=˙Jz,adv+˙Jz,grav+˙Jz,mag, (41)\nwhere ˙Jz,advis the advective torque associated with the loss of\nangular momentum through the boundaries,\n˙Jz,adv=−I\n∂Rρsuφu·n⊥dS, (42)\n˙Jz,gravis the gravitational torque exerted by the binary,\n˙Jz,grav=−Z\nρs∂Φ\n∂φdV, (43)\nand ˙Jz,magis the magnetic torque,\n˙Jz,mag=I\n∂RsBφB·n⊥dS. (44)\nHere, n⊥is the outward-pointing unit vector at the boundaries’\nsurface. In Fig. 14, we show the evolution of these torques for\nruns B and B’. We find that at QSS, as in the nonmagnetic simu-\nlations of Gagnier & Pejcha (2023), the total angular momentum\nevolution is primarily driven by the outflow at the outer bound-\nary, which is a consequence of envelope expansion and the finite\nradial extent of our numerical domain. Mass accretion hinders\nmaterial buildup at the inner boundary and facilitates e fficient\nhorizontal turbulent mixing (as discussed in Gagnier & Pejcha\n2023). Such e fficient mixing weakens the gravitational torque,\nresulting in a modest transfer rate of angular momentum from\nArticle number, page 11 of 24A&A proofs: manuscript no. aanda\n100101\nk/kR10461047104810491050Energy transfer ×/parenleftBigM1+M2\nM/parenrightBig5/2/parenleftBig100R\nR∗/parenrightBig3/2ergcms1\n2(a)\nTMT\nTMA\nTMC\nTKL\n100101\nk/kR2\n1\n012Energy transfer ×1049/parenleftBigM1+M2\nM/parenrightBig5/2/parenleftBig100R\nR∗/parenrightBig3/2ergcms1\n0.52(b)\nTMT+0.5TMC\nTMA+0.5TMC\nTMT+TMA+TMC\nFig. 13. Results of the energy transfer analysis. Panel (a): Magnetic en-\nergy transfer as a function of the normalized wavenumber with kR=\n2π/R∗averaged over twenty spectra spanning three orbital periods dur-\ning the saturated phase, 137 ≤t/Porb≤140. Positive values of transfer\nfunctions at a given wavenumber k(full line) indicate gain of energy at\nthe scale 2π/k, negative values (dashed line) indicate a loss of energy at\nthis scale. Panel (b): Same as panel a, except splitting the magnetic en-\nergy transfer rate into magnetic cascade contribution TMA(k)+0.5TMC(k)\nand kinetic to magnetic energy transfer TMT(k)+0.5TMC(k).\nthe binary’s orbit to the envelope at QSS. Conversely, when ac-\ncretion onto the binary is prevented (simulation run B’), mate-\nrial accumulation at the inner boundary and its stabilizing e ffect\nleads to an injection rate of angular momentum by the gravi-\ntational torque that is an order of magnitude larger than when\naccretion is allowed. The influence of magnetic fields on angu-\nlar momentum evolution remains marginal in our two simula-\ntion runs. This is particularly evident in simulation run B’ where\n107\n106\n105\n104\n103\n102\n101\n100(a)\n˙Jz,adv,in\n˙Jz,adv,out\n˙Jz,mag,in\n˙Jz,mag,out\n˙Jz,grav\n˙Jz\n406080100120140160\ntime[Porb]107\n106\n105\n104\n103\n102\n101\n100˙Jz×1045/parenleftBigM1+M2\nM/parenrightBig2/parenleftBig100R\nR∗/parenrightBig\ngcm2s2\n(b)Fig. 14. Moving average of the time evolution of the advective, mag-\nnetic, and gravitational torques for run B (panel a) and B’ (panel b).\nThe measured total torque is indicated by a black line, positive values\nare denoted by solid lines and negative values by dashed lines.\nmagnetic fields cannot be advected through the inner boundary\nby the fluid flow, unlike in simulation run B. Nevertheless, this\nimpact very much depends on our selection of boundary condi-\ntions for the magnetic field.\nAs in Gagnier & Pejcha (2023), we set the orbital eccentric-\nityeb=0 and enforce the binary mass ratio to q=1. Conse-\nquently, we assume an equal distribution of mass and angular\nmomentum accretion through the inner boundary between the\ntwo cores, resulting in ˙ q=0. Additionally, we make the as-\nsumption that ˙ eb=0. The time derivative of the binary’s angular\nmomentum can then be expressed using the orbital separation\nevolution equation\n˙ab\nab=˙M\nMÇ\n2M˙Jz,b\n˙MJ z,b−3å\n=1\nτab, (45)\nwhereτabis the orbital separation evolution timescale. If ac-\ncretion onto the binary is disabled, ˙M=0 and ˙Jz,b=−˙Jz,grav,\notherwise ˙M=−R\n∂RinρurdSand ˙Jz,b=−˙Jz,adv\f\f\nr=rin−˙Jz,grav−\n˙Jz,mag\f\f\nr=rin. We show the orbital separation evolution timescale\nfor our two MHD runs B and B’, along with the non-MHD runs\nArticle number, page 12 of 24Gagnier & Pejcha: Post-dynamical inspiral of common envelope\n255075100125150\ntime[Porb]101\n100101102103104ab×/parenleftBigR∗\n100R/parenrightBig3/2/parenleftBigM\nM1+M2/parenrightBig1/2yr\n Run A\nRun B\nRun A'\nRun B'\nFig. 15. Time evolution of the orbital separation evolution timescale for\nour two MHD runs B and B’, along with the non-MHD runs A and A’\nfrom Gagnier & Pejcha (2023). The black dashed line represents τab=\n50 yr, while the black dashdotted line corresponds to τab=λ10γt/Porb\nwithλ=0.42 andγ=0.0115.\nA and A’ from Gagnier & Pejcha (2023) in Fig. 15. As expected\nfrom the previous section, magnetic fields play little to no role\nin the binary separation evolution. Hence, as in the nonmagnetic\nsimulations of Gagnier & Pejcha (2023), we find that the or-\nbital separation evolution timescale reaches a quasi-steady value\nof the order 10–100 years when accretion onto the binary is al-\nlowed, and increases with time as\nτab∝10γt/Porb, (46)\nwithγ≃0.0115, when accretion onto the binary is prevented.\n3.4. Angular momentum transport\nRadial transport of angular momentum within the common en-\nvelope can either facilitate or hinder the binary’s post-dynamical\ninspiral through the generation of accretion flows. This process\nalso shapes the envelope’s structure by introducing density dis-\ntribution asymmetries that enable angular momentum exchange\nbetween the binary and the envelope via the gravitational torque.\nThese interactions further contribute to the shaping of emerging\nplanetary nebulae resulting from the common envelope outflow.\nMagnetic fields are also well known to enhance or impede angu-\nlar momentum transport within various astrophysical systems,\nas well as to generate torques, instabilities, and outflows, such\nas winds or jets. Here, we analyze angular momentum trans-\nport within the shared envelope, evaluating the morphology and\nstrength of this process while assessing the influence of each\nphysical mechanism.\nIn Fig. 16, we show the radial profile of the gravitational\ntorque and mean-flow and turbulent contributions to the local\nangular momentum transfer rate across the common envelope,\naveraged over 140 ≤t/Porb≤160 for simulations B (panel a)\n10381039104010411042104310441045(a)˙Jz,adv(r,t)\n˙Jz,R(r,t)\n˙Jz,mag(r,t)˙Jz,(r,t)\n˙Jz,grav(r,t)\n˙Jz(r,t)\n101\n100101\nr/R∗10381039104010411042104310441045˙Jz(r,t)×/parenleftBigM1+M2\nM/parenrightBig2/parenleftBig100R\nR∗/parenrightBig\ngcm2s2\n(b)Fig. 16. Gravitational torque and mean-flow and turbulent contributions\nto the local angular momentum transfer rate across the common en-\nvelope for model B (panel a) and B’ (panel b), averaged in time from\nt=140Porbtot=160Porb. Dashed lines depict negative values, whereas\nsolid lines represent positive values.\nand B’ (panel b). The transfer rates are defined as (see Eq. (23))\n˙Jz,adv(r,t)=−I\n∂rsρeuφeurdS, (47)\n˙Jz,R(r,t)=−I\n∂rsρfiu′′φu′′rdS, (48)\n˙Jz,mag(r,t)=I\n∂rsBφBrdS, (49)\n˙Jz,σ(r,t)=I\n∂rsB′φB′rdS, (50)\n˙Jz,grav(r,t)=ZRdomain\nrÇZ\n∂rsρ∂Φ′\n∂φdSå\ndr. (51)\nWhen accretion onto the binary is allowed (simulation run\nB), we find that angular momentum is transported inward for\nr≲1.4R∗, and outward further out in the envelope. It is\ndominated by the contribution from the mean-flow advection\n˙Jz,adv(r,t) which is directed inward for r≲1.8R∗, and outward\nfurther out. The transport of angular momentum associated with\nArticle number, page 13 of 24A&A proofs: manuscript no. aanda\nthe mean magnetic field ˙Jz,mag(r,t) is directed outward across\nthe envelope, with negligible amplitude. Within r≲0.75R∗,\nturbulent Reynolds ˙Jz,R(r,t) and Maxwell stresses ˙Jz,σ(r,t) con-\ntribute equally and significantly to outward angular momentum\ntransport. Moving outward, the Reynolds stress changes direc-\ntion intermittently, while the turbulent Maxwell stress maintains\nan outward direction with comparable magnitude. Overall, we\nfind the combined e ffect of turbulent Reynolds and Maxwell\nstresses on radial transport of angular momentum to be signif-\nicant. The combined e ffect damps the inward transport of an-\ngular momentum driven by the mean flow by a factor ∼2 for\nr≲1.4R∗, it reverses the direction of angular momentum trans-\nport for 1.4≲r/R∗≲1.8, and the turbulent Maxwell stress\nslightly enhances outward transport for 1 .8≲r/R∗≲4. Con-\nversely, the binary’s gravitational torque impact on the radial\nangular momentum transport is negligible throughout the enve-\nlope. However, a notable feature is the presence of global radial\noscillations about zero, which arise due to the gravitational cou-\npling between the binary potential and the spiral density waves\noriginating near the binary and propagating out in the envelope\n(Cimerman & Rafikov 2023).\nWhen accretion onto the binary is prevented (simulation run\nB’), angular momentum is transported outward across the enve-\nlope. Notably, the positive values of ˙Jzwithin the very inner en-\nvelope result from the injection of angular momentum by the bi-\nnary’s gravitational torque and they do not indicate inward trans-\nport. We further note that this injection of angular momentum\nis more pronounced in simulation B’ compared to simulation B\nowing to material accumulation at the reflecting inner boundary.\nIn simulation B’, and contrary to simulation B, magnetic fields\nhave a negligible impact on radial angular momentum transport.\nThe primary driver remains advection by the mean flow, with\nthe Reynolds stress also making a significant contribution. The\nReynolds stress alternates between dampening and enhancing\noutward transport across varying radii from the central binary.\nBecause the time-periodic gravitational body force exerted\nby the binary system on its surrounding envelope is zero at the\npoles and maximum in the orbital plane, the dynamics of com-\nmon envelopes is highly dependent on latitude. It is therefore\nessential to analyze the latitudinal dependence of angular mo-\nmentum transport. To do so, we first show in Figs. 17 and 18\nthe total angular momentum radial advective flux Fadv=sWrφ\n(see Eq. 23). We see that in simulations B and B’ and as in the\nnonmagnetic simulations of Gagnier & Pejcha (2023), the angu-\nlar momentum advective flux is directed outward in an equato-\nrial disk-like structure. Above and below such disk, the angular\nmomentum advective flux is directed inward. We assess the rel-\native significance and latitudinal variation of each component\nof the total angular momentum radial advective flux in Figs. 19\nand 20. The top panel shows that the morphology of the angular\nmomentum radial flux is dominated by the nonmagnetic com-\nponent. The radial transport of angular momentum by magnetic\nfields is directed outward at all latitudes and radii for simulation\nB. Consequently, it enhances outward transport in the equatorial\ndisk-like structure, and damps the inward transport elsewhere.\nThe morphology of magnetic fields driving radial transport of\nangular momentum is slightly more complex in simulation B’\nas its direction and amplitude vary with randθ. From the sec-\nond and third panels of Figs. 19 and 20 showing the mean and\nturbulent components of the total flux, we see that the turbulent\ntransport associated with the Reynolds stress is less e fficient than\ntransport by the mean flow. This result is similar to the nonmag-\nnetic simulations of Gagnier & Pejcha (2023). Still, the complex\nFig. 17. Reynolds average of the total radial angular momentum flux\nsWrφ, averaged over twenty orbital periods for 140 ≤t/Porb≤160 for\nsimulation B.\nmorphology and nonnegligible amplitude of the turbulent trans-\nport may significantly damp or enhance the angular momentum\nradial transport locally and may even reverse the latitudinally\nintegrated angular momentum radial transport. We see that the\nmagnetic radial transport of angular momentum is dominated by\nthe turbulent Maxwell stress while the transport by mean (ax-\nisymmetric) field is negligible.\n3.5. The value of the disk αparameter\nRecently, Tuna & Metzger (2023) have developed an α-disk\nmodel for the long-term evolution of post-common envelope cir-\ncumbinary disks. To check their assumptions and to facilitate\nsimilar studies in the future, it is important to constrain the value\nofαmore accurately. In our analysis, presented in Sect. 3.1 and\nsummarized in Table 1, we measured the volume-averaged tur-\nbulent stress normalized to the gas pressure, αP, as well as the\nnormalized rφcomponent of the Reynolds and Maxwell stresses.\nThese measurements indicate that at QSS, angular momentum is\nglobally transported outward when accretion is not prevented.\nThe values we obtain are in agreement with previous global and\nlocal magnetohydrodynamic simulations of accretion disks (e.g.,\nSimon et al. 2009; Hawley et al. 2013; Parkin & Bicknell 2013).\nHowever, while many simulations of circumbinary disks are\ndominated by MRI with αPmainly driven by the Maxwell stress\nthat is predominantly positive throughout the disk (e.g., Shi et al.\n2012; Shi & Krolik 2015), our simulations reveal a contrast-\ning behavior. Magnetic fields in our accreting simulations are\nweak compared to the strength of the mean flow driven by the\nbinary and have consequently limited direct dynamical impact\n(in agreement with Ondratschek et al. 2022). Yet, as we have\nseen in Sect. 3.2.1, magnetic fields have a considerable e ffect on\nthe envelope’s density structure and therefore indirectly on an-\nArticle number, page 14 of 24Gagnier & Pejcha: Post-dynamical inspiral of common envelope\nFig. 18. Same as Fig. 17 but for simulation B’.\ngular momentum transport through the mean flow and through\nReynolds stress. The Reynolds stress is the main contributor to\nthe local turbulent flux of angular momentum. However, because\nit exhibits significant variations in sign and amplitude through-\nout the shared envelope (see also Gagnier & Pejcha 2023), its\ncontribution to αPis of the same order as that of the turbu-\nlent Maxwell stress. The turbulent Maxwell stress enhances the\noutward and damps the inward turbulent transport of angular\nmomentum without changing its global morphology. We obtain\nsimilar results when accretion through the inner boundary is pre-\nvented by reflecting boundary conditions (Figs. 18, 20). Angular\nmomentum transport is dominated by the mean flow and consists\nof an outward transport in a disk-like structure and an inward\ntransport along the polar axis. The main di fference comes from\nthe more complex morphology of the Maxwell and Reynolds\nstress and their overall weaker amplitude resulting in near-zero\nnet radial turbulent transport of angular momentum. The global\nmorphology of the turbulent transport of angular momentum is\nrather complex as its direction varies with latitude and distance\nto the central binary (see Figs. 21 and 22). Hence the relevance\nof employing a spatially constant value of αPin post-CE cir-\ncumbinary disks models may be questionable.\nOur findings also reveal that the radial transport of angular\nmomentum is predominantly driven by the mean flow, which es-\nsentially consists of spiral density waves. This poses a signif-\nicant challenge to the use of the conventional α–viscosity for-\nmalism in 1D models, as it cannot directly account for the non-\ndiffusive nature of angular momentum transport associated with\nthese waves. Alternative models are needed to capture the intri-\ncate dynamics and angular momentum transport involving spiral\ndensity waves in post-CE circumbinary disks.\nFig. 19. Reynolds average of the mean and turbulent contribution of the\nof the total radial angular momentum flux (see Eqs. (25) and (26)), aver-\naged over twenty orbital periods for 140 ≤t/Porb≤160 for simulation\nB.\n3.6. Time variability of accretion\nIn Fig. 23, we show a space-time diagram of the mass flux cross-\ning the inner boundary normalized by its maximum value in the\ntime interval 140≤t/Porb≤160 for our accreting model B. As\nin the nonmagnetic simulations of Gagnier & Pejcha (2023), we\nsee that mass accretion exhibits periodic variability at all colat-\nitudes. On the orbital plane, a variability frequency of 2 Ωorbis\nclearly visible. This frequency is associated with the quadrupo-\nlar moment contribution to the binary potential. In addition to\nthis expected variability frequency, Gagnier & Pejcha (2023) fur-\nther found an additional low-frequency modulation of accretion\nArticle number, page 15 of 24A&A proofs: manuscript no. aanda\nFig. 20. Same as Fig. 19 but for simulation B’.\nω≃0.2Ωorb, which they attribute to the presence of an eccentric\nand tilted overdensity, successively feeding the individual binary\ncomponents through accretion streams. Remarkably, such mass\naccretion variability frequency are identical to those measured in\ncircumbinary disks simulations where it is attributed to the orbit-\ning frequency of a nonaxisymmetric overdensity (or “lump”).We\nshow the power spectral density of the total mass accretion rate\n˙M, as well as the mass accretion rates measured in various open-\ning angles about the orbital plane in Fig. 23, panel c. We find the\n0.2Ωorbfrequency is present, suggesting that magnetic fields do\nnot prevent the formation of the overdensity observed in the non-\nmagnetic simulations of Gagnier & Pejcha (2023). It is however\nlikely that higher magnetization would weaken the lump and its\ninduced low-frequency accretion modulation (Noble et al. 2021).\nFig. 21. Reynolds-averaged turbulent stress normalized to gas pressure,\nαP, averaged over twenty orbital periods for 140 ≤t/Porb≤160 for\nsimulation B.\nIn addition to the 0 .2Ωorbfrequency, we find a clear and domi-\nnating peak at ω≃0.1Ωorb. We investigate its origin in Sect. 3.7.\nWe note that the 0 .1Ωorband 0.2Ωorbfrequencies are not visi-\nble in the power spectral density of the total mass accretion rate\nbecause of the asynchronicity of mass accretion between colat-\nitudes. These results suggest that local analyses are necessary\nwhen studying short-term variability of accretion in CEE (Gag-\nnier & Pejcha 2023).\n3.7. The lump\nIn the context of circumbinary disks, the appearance of a lump\nis commonly thought to be the result of the interaction between\ngas streams inside the cavity and the cavity edge (e.g., Shi et al.\n2012; Noble et al. 2012). However, recent work by Mignon-\nRisse et al. (2023) and Rabago et al. (2023) suggest an alterna-\ntive lump formation channel consisting of the merging of Rossby\nWave Instability (RWI, Lovelace et al. 1999; Li et al. 2000)\ninduced large-scale vortices into a single lump. Cimerman &\nRafikov (2023) further show that such large-scale vortices likely\nalso play a major role in angular momentum transport in CBDs\nby launching vortex-driven spiral density waves (see also Huang\net al. 2019). In circumbinary disks, RWI can be triggered from\nan axisymmetric bump in the inverse vortensity associated with\nthe strong radial gradient of density at the cavity edge. In com-\nmon envelopes however, there is likely no cavity, and such strong\nvortensity (or magnetized vortensity) maxima may not naturally\nemerge. Still, Rossby wave instability can be triggered by the\nvortensity production caused by the nonlinear damping of spiral\ndensity waves (Coleman et al. 2022). Here, we investigate the\nexistence of nonaxisymmetric overdensities in our simulations\nand we explore their behavior and origin.\nArticle number, page 16 of 24Gagnier & Pejcha: Post-dynamical inspiral of common envelope\nFig. 22. Same as Fig. 21 but for simulation B’.\n123\n(a)\n140.0142.5145.0147.5150.0152.5155.0157.5160.0\ntime [Porb]123˙M[103]\n(b)\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0˙m(,t)/max{˙m(,t):,t}\n0.10.2 0.5 1.0 1.5 1.841.94\n [orb]\n012Power spectral density(c)\nFig. 23. Detailed view on the variability of mass flux for model B. Panel\n(a): Space-time diagram of the local mass flux through the inner bound-\nary. Panel (b): Time evolution of the mass accretion rate onto the binary.\nPanel (c): Power spectral density of the total mass accretion rate onto the\nbinary (black line) and of the mass accretion rate in various colatitude\nranges about the orbital plane (colored lines). Green lines correspond to\nthe rangeπ/4≤θ≤3π/4, blue lines to the range 3 π/8≤θ≤5π/8, and\norange lines to the range 7 π/16≤θ≤9π/16.In Fig. 24, we show a snapshot of the relative surface density\nperturbation ( Σ−Σ)/Σatt=140Porbfor simulation run B. The\nsurface density is defined as\nΣ =Z9π/16\n7π/16ρrsinθdθ. (52)\nWe find strong and structured nonaxisymmetric overdensities\ntaking the form of spiral density waves with a large pitch angle.\nThe presence of a single well-defined spiral arm reaching the bi-\nnary in Fig. 24 suggests that m=1 modes also play a major role\nin the shaping of overdensities. In Fig. 25, we investigate the ori-\ngin of such overdensities by first showing the nonaxisymmetric\ndensity perturbation on the orbital plane as well as the vorten-\nsity perturbation early in our simulation at t=25Porb. In the\ninner part of the envelope, we find small scale turbulence to be\nfully developed, which is likely a consequence of stratified shear\ninstability emerging from the interaction between spiral density\nwaves generated by the binary’s torque and the background ve-\nlocity shear. We further note the presence of four large anticy-\nclonic vortices located at r≃1. These vortices likely emerge\nfrom Rossby wave instability, triggered by the vortensity pro-\nduction at this radius due to the nonlinear damping of spiral den-\nsity waves Coleman et al. (2022). The propagation of spiral den-\nsity wavefronts through such vortices induces their deformation,\nas is evident in panel b. This deformation of the wavefront en-\ntails the emergence of crests where the deformed density waves\nconverge. Notably, two out of the four vortices exhibit su fficient\nstrength to cause the associated wavefront crests to give rise to\nthe formation of persistent, slowly orbiting overdensities taking\nthe form of m=2 spiral density waves with large pitch angle.\nThe existence of such overdensities prevent the spiral density\nwaves generated by the binary’s torque from propagating freely\noutward, resulting in material accumulation and the formation of\na nonaxisymmetric lump (see Gagnier & Pejcha 2023, for more\ndetails).\nEventually, the inner parts of the two spiral arms become suf-\nficiently dense for the gravitational force exerted on them by the\ncentral binary to overcome the pressure gradient driving the en-\nvelope’s expansion, leading to their inward migration. In Fig. 26,\nwe see that the two overdensities exhibit slightly di fferent inward\nmigration speeds due to the imperfect mass distribution symme-\ntry. Such inward migration of overdensities within the expand-\ning envelope induces strong and unstable horizontal shear, giv-\ning rise to two prominent vortices. As the faster-moving over-\ndensity approaches the binary, it experiences substantial torque\nand ultimately reaches the inner boundary as a singular accretion\nstream. This event locally disrupts the m=2 symmetry, favor-\ning instead m=1 symmetry in the inner part of the envelope\n(Fig. 27). Over time, the density of this single spiral arm dimin-\nishes as material is accreted by the binary and is eventually de-\nstroyed by the m=2 spiral density waves and by tidal disruption\nof the overdense stream, thereby restoring the m=2 symmetry\nin the binary’s vicinity. This cycle repeats every ∼10Porband\nit therefore associated with the ω≃0.1Ωorbfrequency observed\nin Fig. 23. We note that this process and thus this characteristic\nfrequency were not present in Gagnier & Pejcha (2023)’s non-\nmagnetic runs. In simulation B, the formation and persistence\nofm=1 accretion streams can be attributed to the stabilizing\neffect of the toroidal magnetic fields preventing the destruction\nof overdensities through magnetic tension, allowing these struc-\ntures to progressively increase in density. As these overdensities\nbecome denser, their gravitational attraction to the inner binary\nsystem grows stronger, leading to their inward migration. Their\nArticle number, page 17 of 24A&A proofs: manuscript no. aanda\nFig. 24. Snapshot of relative surface density perturbation ( Σ−Σ)/Σin\nthe xy plane, at t=140Porb.\nFig. 25. Snapshot of density perturbation ( ρ−ρ)/ρ(panel a) and of\nvortensity deviation ζ−ζ, whereζ=ρ−1∇×u, in the rφplane at\nt=25Porb.\nrelatively high density enables them to reach the inner boundary\nand feed the binary before being destroyed.\n4. Discussions and conclusions\nIn this paper, we have significantly extended our previous work\nin Gagnier & Pejcha (2023) on the post-dynamical inspiral phase\nof CEE by including magnetic fields in 3D simulations. Our first\naim was to determine the sources of magnetic energy amplifi-\ncation, and the relative importance of magnetic to kinetic en-\nergy reservoir size. We found that magnetic energy amplifica-\ntion arises primarily from the stretching, folding, and winding\nof the initial weak poloidal field due to di fferential rotation and\nturbulence. As the magnetic field strengthens, it stabilizes fluid\nmotion, favoring azimuthal kinetic energy over radial kinetic\nenergy during saturation. Magnetic fields significantly impact\nthe envelope’s structure and dynamics, with the magnetic en-\nergy reaching levels similar to the previous MHD simulations of\nOhlmann et al. (2016b) and Ondratschek et al. (2022), but with\n0.51.01.52.0r(a)\n0123456\n0.51.01.5r(b)\n100\n101\n102\n0102\n101\n100\n( )/\n104\n103\n102\n0102103104\nFig. 26. Same as Fig. 25 but at t=45Porb.\n0.51.01.52.0r(a)\n0123456\n0.51.01.5r(b)\n100\n101\n102\n0102\n101\n100\n( )/\n104\n103\n102\n0102103104\nFig. 27. Same as Figs. 25 and 26 but at t=140Porb.\na much lower kinetic-to-magnetic energy ratio. Energy spectra\nshow kinetic energy dominance at all scales, but decreasing to-\nwards smaller scales. Our analysis identifies stretching of mag-\nnetic field lines by velocity shear as the dominant source of mag-\nnetic energy, contributing 95% during quasi-steady state with\nthe remaining 5% resulting from compression against magnetic\npressure.\nOur second aim was to determine how kinetic and magnetic\nenergy reservoirs are interconnected and what contributes to\ntheir evolution during the post-dynamical phase of CEE. We first\ndecomposed the kinetic energy evolution equation into mean and\nturbulent contributions to determine the relative contribution of\nthe di fferent processes. We found the impact of magnetic fields\non the mean kinetic energy evolution to be negligible, and thus\nnot to directly a ffect the mean flow. However, the interaction be-\ntween turbulent shear and Maxwell stress leads to a nonnegli-\ngible sink of turbulent kinetic energy. We then employed en-\nergy transfer analysis to investigate the scales at which energy\nis transferred between kinetic and magnetic energy reservoirs.\nArticle number, page 18 of 24Gagnier & Pejcha: Post-dynamical inspiral of common envelope\nThis analysis revealed that magnetic energy production by the\nstretching of magnetic field lines occurs at all scales and that the\nadvection of magnetic energy within the magnetic energy reser-\nvoir indicates a forward cascade from larger to smaller scales.\nNotably, both field line stretching and magnetic advection peak\nat a critical scale of approximately 3 ab, that is, approximately\nthe wavelength of the binary-driven spiral density waves. Com-\npression e ffects, though weaker, play a significant role in net\nmagnetic energy production due to the balance between stretch-\ning and advection at this scale. Our analysis also shows that the\nscales at which kinetic energy is lost to magnetic energy and the\nscales at which magnetic energy is received from kinetic energy\nare di fferent. This highlights the nonlocality of interactions be-\ntween magnetic and kinetic energy in spectral space.\nOur analysis indicates the absence of magnetic energy pro-\nduction on large radial scales. This suggests potential challenges\nin funneling and shaping radial polar outflows in planetary neb-\nulae, possibly hindering the formation of well-defined bipolar\nstructures. However, the presence of a strong toroidal magnetic\nfield on the orbital plane, coupled with centrifugal forces and\nturbulent mixing, could respectively slow down envelope equa-\ntorial expansion and channel jittery and irregular outflows near\nthe poles, thus leading to the emergence of highly nonspherical\nplanetary nebulae.\nBy comparing our results to the nonmagnetic simulations\noutcomes of Gagnier & Pejcha (2023), our third aim was to\nassess the impact of magnetic fields on the binary separation\nevolution timescale, on angular momentum transport within the\nshared envelope, on the short-term variability of mass accretion\nonto the binary, and on the formation of overdensities. We found\nthat magnetic fields play little to no role in the binary separa-\ntion evolution. This result may, however, be closely tied to our\nchoice of boundary conditions for the magnetic field and may\nthus call for further investigation through numerical simulations\non the scale of binary interaction that can accurately resolve the\naccretion flow and the interaction between magnetic fields and\nthe cores. This will be explored in future works.\nWe found magnetic fields do not change the disk-like mor-\nphology of radial angular momentum transport found by Gag-\nnier & Pejcha (2023). However, when accretion onto the central\nbinary is allowed, we found that turbulent Maxwell stress has\na comparable or even larger net contribution to the radial an-\ngular momentum transport than the Reynolds stress. Turbulent\nMaxwell stress transport of angular momentum points outward\nat all radii and it locally reverses the direction of angular mo-\nmentum transport that is otherwise dominated by the mean-flow\nadvection. When accretion is prevented by imposing reflecting\nboundary conditions, the net contribution of Maxwell stress to\nthe radial transport of angular momentum is negligible compared\nto the contribution from Reynolds stress and mean-flow.\nOur investigation of the intricate dynamics of nonaxisym-\nmetric overdensities within the common envelope of binary star\nsystems reveals their pivotal role in modulating mass accre-\ntion onto the binary components. Similarly to Gagnier & Pejcha\n(2023), we find mass accretion to exhibit distinct frequencies, in-\ncluding the 2 Ωorbfrequency stemming from the quadrupolar mo-\nment of the binary’s potential and a lower-frequency ≃0.2Ωorb\nmodulation associated with overdensities known as “lumps” in\nthe context of circumbinary disks. Our analysis uncovers the ori-\ngins of these various overdensities: they form as a result of the\nintricate interplay between spiral density waves launched by the\ncentral binary and spiral density waves with a much larger pitch\nangle, which arise from interaction with vortices. Additionally,\nwe unveiled the emergence of m=1 accretion streams associ-ated with the≃0.1Ωorbfrequency. These streams owe their ex-\nistence to the stabilizing e ffect of the magnetic tension from the\nstrong toroidal field about the orbital plane, which prevents over-\ndensities from being destroyed by turbulence and enables them\nto accumulate mass and eventually migrate towards the binary.\nUltimately, these denser structures are destroyed by spiral waves\nand tidal forces in the binary’s vicinity.\nFinally, Figs. 3 and 4 show a feature that looks like an “S-\nshaped” cavity. We saw this cavity already in our previous work\n(Gagnier & Pejcha 2023). This cavity results from the interplay\nof centrifugal force, which “pushes” material perpendicularly\naway from the rotation axis, and stochastic convection motion\ndisrupting the symmetry of the resulting low-density chimney.\nAlthough our simulations do not exhibit the launching of jets or\njet-like outflows, the presence of the cavity suggests the possi-\nbility of collimation of a “wobbling” jet, which could originate\naround one of the cores. It is exciting to speculate that “wob-\nbling” of jets seen in many planetary nebulae might not arise\ndue to the precession from the second core, but instead due to\ntime-variable S-shaped cavity in the circumbinary envelope. Re-\nsolving the inner region, specifically inside the orbit, is probably\nnecessary to determine whether outflows are actually launched.\nWe plan to study this aspect in our future work.\nThis study has provided a comprehensive examination of the\nimpact of magnetic fields on the post-dynamical inspiral phase\nof common envelope evolution. We have revealed the intricate\ninterplay between magnetic energy amplification, energy reser-\nvoirs, and the dynamics of the envelope, shedding light on the\nmechanisms responsible for magnetic energy amplification and\ntransfer. Our findings have also elucidated the role of magnetic\nfields in angular momentum transport and the formation of non-\naxisymmetric overdensities within the common envelope. How-\never, further investigation is needed to fully comprehend the\nrole of magnetic fields in this context and their wider implica-\ntions. In particular, recent circumbinary disk simulations have\nshowed that resolving the cavity within the computational do-\nmain is essential for accurately measuring torques, orbital evo-\nlution timescales, and orbital eccentricity excitation (e.g., Tang\net al. 2017; Moody et al. 2019; Muñoz et al. 2019; Muñoz &\nLithwick 2020; Du ffell et al. 2020; Tiede et al. 2020; Dittmann\n& Ryan 2021; Combi et al. 2022), and to self-consistently launch\njets (e.g., Gold et al. 2014). Adopting this approach is likely even\nmore crucial in the context of CEE because of the probable ab-\nsence of a low-density cavity. Instead, we anticipate that the re-\ngion inside the binary orbit exhibits highly complex gas dynam-\nics and substantial magnetic energy amplification. We plan to\nexplore these phenomena in future works.\nAcknowledgements. We thank the anonymous referee for comments that im-\nproved this paper. We acknowledge fruitful discussions with K. Tomida, S.\nTakasao, G. 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M. 2018, ApJ, 857, 34\nZou, Y ., Frank, A., Chen, Z., et al. 2020, Monthly Notices of the Royal Astro-\nnomical Society, 497, 2855\nArticle number, page 20 of 24Gagnier & Pejcha: Post-dynamical inspiral of common envelope\nAppendix A: Horizontally integrated energy spectrum\nThe Fourier transform of a function f(r)=f(r,θ,φ) reads\nbf(k)=bf(k,θk,φk)=Z2π\n0Zπ\n0Z∞\n0f(r)eik·rr2sinθdrdθdφ. (A.1)\nWe expand both fand the Fourier kernel eik·ron the spherical harmonics basis\nf(r)=∞X\nℓ=0ℓX\nm=−ℓfℓ\nm(r)Ym\nℓ(θ,ϕ), (A.2)\nand\neik·r=4π∞X\nℓ=0ℓX\nm=−ℓiℓjℓ(kr)Ym\nℓ(θ,ϕ)∗Ym\nℓ(θk,ϕk), (A.3)\nwhere Ym\nℓis the usual scalar spherical harmonics function and jℓ(z) is the spherical Bessel function of the first kind of order ℓ.\nInjecting (A.2) and (A.3) into (A.1) yields\nbf(k)=∞X\nℓ=0ℓX\nm=−ℓFm\nℓ(k)Ym\nℓ(θk,ϕk), (A.4)\nwhere Fm\nℓ(k) is theℓthorder spherical Bessel transform of the spherical harmonics expansion coe fficients\nFm\nℓ(k)=4πiℓZ∞\n0jℓ(kr)fℓ\nm(r)r2dr. (A.5)\nThe horizontally integrated (in Fourier space) energy spectrum of f(r) finally reads\nEf(k)=k2Z2π\n0Zπ\n0bf(k)bf(k)∗sinθkdθkdφk\n=k2∞X\nℓ=0ℓX\nm=−ℓFm\nℓ(k)Fm\nℓ(k)∗.(A.6)\nThis quantity relates to the total energy through Parseval’s theorem\n⟨Ef⟩=Z\n∂V|f(r)|2dV=1\n(2π)3Z∞\n0Ef(k)dk. (A.7)\nAppendix B: Magnetic energy evolution equation\nThe magnetic energy evolution equation (32) is derived from the dot product of Bwith the induction equation (4), that is\n1\n2∂(B·B)\n∂t−B·(B·∇u)+B·B(∇·u)+B·(u·∇B)=0. (B.1)\nUsing the fact that B·(u·∇B)=u·(∇B·B)and∇B·B=B·∇B+B×(∇×B), yields\n1\n2∂(B·B)\n∂t−B·(B·∇u)+EB∇·u+∇·(EBu)=0, (B.2)\nwhich finally yields the magnetic energy evolution equation (32)\n1\n2∂(B·B)\n∂t−(B⊗B):∇u+EB∇·u+∇·(EBu)=0. (B.3)\nArticle number, page 21 of 24A&A proofs: manuscript no. aanda\nAppendix C: Mean and turbulent kinetic energy evolution equations\nThe mean kinetic energy evolution equation (34) is derived from the Reynolds averaged momentum and mass conservation equations\n∂ρeu\n∂t+∇·(ρeu⊗eu)+∇·(ρu′′⊗u′′)=−∇P+∇·σ−ρ∇Φ +∇·τ, (C.1)\n∂ρ\n∂t+∇·(ρeu)=0. (C.2)\nTaking the dot product of −euwith Eq. (C.2) and adding it to Eq. (C.1), making use of the divergence of a dyad formula ∇·(a⊗b)=\n(∇·a)b+a·∇b, yields\nρ∂eu\n∂t+(ρeu·∇)eu+∇·(ρu′′⊗u′′)=−∇P+∇·σ−ρ∇Φ +∇·τ. (C.3)\nWe then take the dot product of euwith Eq. (C.3) by and we multiply Eq. (C.2) by ( eu·eu)/2, respectively yielding\n1\n2ρ∂(eu·eu)\n∂t+ρeu·∇(eu·eu)\n2+eu·Ä\n∇·(ρu′′⊗u′′)ä\n=−eu·∇P+eu·\u0000\n∇·σ\u0001\n−ρeu·∇Φ +eu·\u0000\n∇·τ\u0001\n, (C.4)\nand\n(eu·eu)\n2∂ρ\n∂t+∇·Å\nρeu(eu·eu)\n2ã\n−ρeu·∇(eu·eu)\n2=0, (C.5)\nwhere we have used the vector identity ∇(a·a)/2=(a·∇)a+a×(∇×a). Finally, we add Eqs. (C.4) and (C.5) together to obtain\nthe Reynolds averaged mean kinetic energy evolution equation (34)\n1\n2∂ρ(eu·eu)\n∂t+∇·Å\nρeu(eu·eu)\n2ã\n+∇·\u0010\nρ·\u0000u′′⊗u′′·eu\u0011\n=ρ·\u0000u′′⊗u′′:∇eu−eu·∇P+eu·\u0000\n∇·σ\u0001\n−eu·ρ∇Φ +eu·\u0000\n∇·τ\u0001\n. (C.6)\nHere, we have used the identity ∇·(T·a)=T:∇a+(∇·T)·a, where Tis a rank 2 tensor. The colon symbol indicates a Frobenius\ninner product T:G=Ti jGi j=Tr(TG⊺). To obtain the turbulent kinetic evolution equation (35), we first expand the di fferentiated\nscalar and vector fields in the convective form of Eq. (2), except for the directional derivative ( u·∇), and we take the dot product of\nu′′with the result. This yields\n1\n2ρ∂(u′′·u′′)\n∂t+ρu′′·∂eu\n∂t+ρu·∇(u′′·u′′)\n2+ρu⊗u′′:∇eu=−u′′·∇Ä\nP+P′ä\n+u′′·(∇·σ)−ρu′′ρ∇\u0000\nΦ + Φ′\u0001\n+u′′·(∇·τ).(C.7)\nHere, we have used the same identities as for the derivation of the mean kinetic energy equation. We then multiply Eq. (1) by\n(u′′·u′′)/2 and we add the result to Eq. (C.7) to obtain\n1\n2∂ρ(u′′·u′′)\n∂t+ρu′′·∂eu\n∂t+∇·Å\nρu(u′′·u′′)\n2ã\n+ρu⊗u′′:∇eu=−u′′·∇Ä\nP+P′ä\n+u′′·(∇·σ)−ρu′′ρ∇\u0000\nΦ + Φ′\u0001\n+u′′·(∇·τ).(C.8)\nFinally, taking the Reynolds average of Eq. (C.8) and making use of the Favre average properties\nρu′′=0, (C.9)\nfu′′=0, (C.10)\nρu⊗u=ρeu⊗eu+ρu′′⊗u′′, (C.11)\nyields the turbulent kinetic evolution equation (35)\n1\n2∂ρ(‡u′′·u′′)\n∂t+∇· \nρeu(‡u′′·u′′)\n2!\n−∇·Ç\nσ·u′′−ρu′′(u′′·u′′)\n2−P′u′′å\n=−ρ·\u0000u′′⊗u′′:∇eu−σ:∇u′′−u′′·∇P\n+P′∇·u′′−ρu′′·∇Φ′+u′′·(∇·τ),(C.12)\nwhere we have expanded the pressure perturbation term into an advective term and a pressure dilatation term, and the magnetic term\ninto an advective term and a term associated with the interaction between turbulent shear and Maxwell stress.\nArticle number, page 22 of 24Gagnier & Pejcha: Post-dynamical inspiral of common envelope\nAppendix D: Transfer functions\nAppendix D.1: Magnetic energy transfer equation\nThe magnetic energy transfer equation can be simply derived by taking the dot product of bBwith the complex conjugate of the\nhorizontally integrated Fourier transformed induction equation\nd∂tB∗+⁄\u0000∇·(u⊗B)∗\n−⁄\u0000∇·(B⊗u)∗\n=0, (D.1)\nadding the conjugate of the result and dividing by two,\n˙EB(k)=TMT(k)+TMP(k), (D.2)\nwhere, from Eq. (A.6),\n˙EB(k)=k2\n2Z2π\n0Zπ\n0∂tÄbB·bB∗ä\nsinθkdθkdφk. (D.3)\nThe first term on the right-hand side of Eq. (D.2) represents the rate of energy transfer from the kinetic energy reservoir to the\nk–component of the magnetic energy reservoir by the stretching of the field lines against magnetic tension force (e.g., Pietarila\nGraham et al. 2010)\nTMT(k)=k2\n2Z2π\n0Zπ\n0\u0010\nbB·\u0010\n÷B·∇u∗\u0011\n+bB∗·Ä÷B·∇uä\u0011\nsinθkdθkdφk. (D.4)\nThe second term on the right-hand side of Eq. (D.2) represents the rate of energy transfer to the k–component of the magnetic energy\nreservoir by advection and compression against magnetic pressure\nTMP(k)=TMC(k)+TMA(k), (D.5)\nwhere\nTMC(k)=−k2\n2Z2π\n0Zπ\n0\u0010\nbB·Ä÷B∇·uä∗+bB∗·Ä÷B∇·uä\u0011\nsinθkdθkdφk, (D.6)\nTMA(k)=−k2\n2Z2π\n0Zπ\n0\u0010\nbB·Ä÷u·∇Bä∗+bB∗·Ä÷u·∇Bä\u0011\nsinθkdθkdφk. (D.7)\nAs Rempel (2014) and Grete et al. (2017), we note that we can split TMC(k) in two, such that the terms underlying TMA(k)+0.5TMC(k)\nin the real space may be identified with a magnetic energy advective transport to other scales within the magnetic energy reservoir\n(i.e., magnetic cascade)\n−B·Å\nu·∇B+B\n2∇·uã\n=−∇·Å\nu(B·B)\n2ã\n. (D.8)\nSimilarly, the terms underlying TMT(k)+0.5TMC(k) in the real space may be identified with energy transfer from kinetic to magnetic\nenergy reservoir via Lorentz force and the remaining nonadvective terms of the Poynting flux (e.g., Rempel 2014; Grete et al. 2017),\nB·Å\nB·∇u−B\n2∇·uã\n=∇·Å\nB(u·B)−u(B·B)\n2ã\n−u·(∇·σ). (D.9)\nThe magnetic transfer functions are related to the magnetic energy rate of change in real space as follows\n⟨˙EB,stretch⟩=1\n(2π)3Z∞\n0TMT(k)dk,\n⟨˙EB,exp⟩=0.5\n(2π)3Z∞\n0TMC(k)dk,\n⟨˙EB,adv⟩=1\n(2π)3Z∞\n0TMA(k)dk+0.5\n(2π)3Z∞\n0TMC(k)dk.(D.10)\nArticle number, page 23 of 24A&A proofs: manuscript no. aanda\nAppendix D.2: Kinetic energy transfer equation\nFollowing Kida & Orszag (1990) and Grete et al. (2017), we introduce a new variable\nw=√ρu, (D.11)\nwhich, combining mass and momentum conservation equations follows\n∂w\n∂t=−u·∇w−1\n2w∇·u+1√ρ∇·(B⊗B)−1√ρ∇P−1\n2√ρ∇(B·B)−√ρ∇Φ. (D.12)\nThe kinetic energy transfer equation is derived by taking the dot product of bwwith the complex conjugate of the horizontally\nintegrated Fourier transform of Eq. D.12, adding the conjugate of the result and dividing by two. This tields\n˙EK(k)=TKK(k)+TKL(k)+TKP(k), (D.13)\nwhere\n˙EK(k)=k2\n2Z2π\n0Zπ\n0∂t\u0000bw·bw∗\u0001\nsinθkdθkdφk. (D.14)\nFurthermore,\nTKK(k)=TKKa(k)+TKKb(k) (D.15)\ncorresponds to a kinetic energy transport to other scales within the kinetic energy reservoir (kinetic cascade), and\nTKKa(k)=−k2\n2Z2π\n0Zπ\n0\u0010\nbw·Ä÷u·∇wä∗+bw∗·Ä÷u·∇wä\u0011\nsinθkdθkdφk (D.16)\nTKKb(k)=−k2\n4Z2π\n0Zπ\n0\u0010\nbw·Ä÷w∇·uä∗+bw∗·Ä÷w∇·uä\u0011\nsinθkdθkdφk. (D.17)\nTKL(k)=TKLa(k)+TKLb(k) (D.18)\nrepresents the rate of energy transfer from the magnetic energy reservoir to the k–component of the kinetic energy reservoir by the\nwork of the Lorentz force via magnetic tension and magnetic pressure, with\nTKLa(k)=k2\n2Z2π\n0Zπ\n0 \nbw· \n¤\u00001√ρ∇·(B⊗B)!∗\n+bw∗· \n¤\u00001√ρ∇·(B⊗B)!!\nsinθkdθkdφk (D.19)\nTKLb(k)=−k2\n4Z2π\n0Zπ\n0 \nbw·¤\u00001√ρ∇(B·B)∗\n+bw∗·¤\u00001√ρ∇(B·B)!\nsinθkdθkdφk. (D.20)\nFinally,\nTKP(k)=TKPa(k)+TKPb(k) (D.21)\nis the energy transfer from pressure forces and energy injection from the binary potential, with\nTKPa(k)=−k2\n2Z2π\n0Zπ\n0 \nbw·÷1√ρ∇P∗\n+bw∗·÷1√ρ∇P!\nsinθkdθkdφk (D.22)\nTKPb(k)=−k2\n2Z2π\n0Zπ\n0\u0010\nbw·÷√ρ∇Φ∗\n+bw∗·÷√ρ∇Φ\u0011\nsinθkdθkdφk. (D.23)\nArticle number, page 24 of 24"    },    {        "title": "2211.05479v1.Electron_dynamics_in_planar_radio_frequency_magnetron_plasmas__III__Comparison_of_experimental_investigations_of_power_absorption_dynamics_to_simulation_results.pdf",        "content": "arXiv:2211.05479v1  [physics.plasm-ph]  10 Nov 2022Electron dynamics in planar radio frequency\nmagnetron plasmas: III. Comparison of\nexperimental investigations of power absorption\ndynamics to simulation results\nB Berger1, D Eremin2, M Oberberg1, D Engel2, C W¨ olfel3,\nQ-Z Zhang4, P Awakowicz1, J Lunze3, R P Brinkmann2,\nand J Schulze1,4\n1Chair of Applied Electrodynamics and Plasma Technology, Ruhr Univer sity\nBochum, Universitaetsstrasse 150, 44801 Bochum, Germany\n2Institute of Theoretical Electrical Engineering, Ruhr University B ochum,\nUniversitaetsstrasse 150, 44801 Bochum, Germany\n3Institute of Automation and Computer Control, Ruhr University Bo chum,\nUniversitaetsstrasse 150, 44801 Bochum, Germany\n4Key Laboratory of Materials Modification by Laser, Ion, and Electr on Beams\n(Ministry of Education), School of Physics, Dalian University of Tec hnology,\nDalian 116024, People’s Republic of China\nE-mail:berger@aept.rub.de\nAbstract.\nIn magnetized capacitively coupled radio-frequency discharges op erated at\nlow pressure the influence of the magnetic flux density on discharge properties\nhasbeenstudiedrecentlybothbyexperimentalinvestigationsand insimulations.\nIt was found that the Magnetic Asymmetry Effect allows for a contr ol of\nthe DC self-bias and the ion energy distribution by tuning the magnet ic field\nstrength. In this study, we focus on experimental investigations of the electron\npower absorption dynamics in the presence of a magnetron-like mag netic field\nconfigurationinalowpressurecapacitiveRF dischargeoperatedina rgon. Phase\nResolved Optical Emission Spectroscopy measurements provide ins ights into the\nelectron dynamics on a nanosecond-timescale. The magnetic flux de nsity and\nthe neutral gas pressure are found to strongly alter these dyna mics. For specific\nconditions energetic electrons are efficiently trapped by the magne tic field in a\nregion close to the powered electrode, serving as the target surf ace. Depending\non the magnetic field strength an electric field reversal is observed that leads\nto a further acceleration of electrons during the sheath collapse. These findings\nare supported by 2-dimensional Particle in Cell simulations that yield d eeper\ninsights into the discharge dynamics.Experimental study of electron power absorption dynamics i n RF magnetrons 2\nKeywords : Magnetic Asymmetry Effect, Non-Linear Electron Resonance Hea ting,\nDC self-bias voltage, magnetized plasma, capacitively coupled RF disc harge\nPACS numbers: 52.25.-b,52.25.Xz,52.27.-h,52.27.Aj,52.50.Dg,52.50.Qt,52.5 5.-\ns,52.70.Ds,52.77.-j,52.80.Pi\nSubmitted to: Plasma Sources Sci. Technol.Experimental study of electron power absorption dynamics i n RF magnetrons 3\n1. Introduction\nThe process of depositing thin films is of high importance in a variety of technolog-\nical applications. A high quality of these films is needed for biomedical a nd optical\napplications as well as for the manufacturing of microelectronics [1– 4]. Commonly\nused in this context is the process of Physical Vapor Deposition (PV D) [5]. Here,\na target is positioned in contact with a plasma at comparably low press ure. This\nleads to the development of a sheath between the plasma and the ta rget that ac-\ncelerates positive ions to the target surface. These impinging ions c an then sputter\natoms from the target that condensate on the walls in the chamber . Putting a sub-\nstrate in the chamber leads to the deposition of a thin layer of the ta rget material\nor a compound with the used gas on the substrate surface.\nThis process can be enhanced by adding a magnetic field that is stron g close the\ntarget surface. This leads to an increase of the charged particle d ensity in this\nregion, which, in turn, leads to an increased ion flux to the target. B y doing so,\nthe sputter rate of the target material can be increased, which le ads to a shorter\nprocess time. A typical magnetic field structure is a torus shaped fi eld that has\nstrong magnetic flux components perpendicular to the target sur face at the target\nedges and at its center while having a strong parallel component in th e region\nin between. Such so-called planar magnetron discharges have the a dvantage of a\nhigher sputter rate compared to discharges without a magnetic fie ld, but suffer\nfrom a decreased degree of target material utilization, since the s putter process\npredominantly takes place in the magnetized torus, which forms the typical race-\ntrack on the target surface. In order to overcome this disadvan tage a cylindrical\ntarget or rotating magnets for planar targets can be used [6–8].\nDepending on the desired process, a wide range of different driving v oltage wave-\nforms can be used. A DCpower source is used for sputtering of con ducting materi-\nals, but cannot be used to sputter dielectric material, since the sur face charges up\nwhen getting in contact with a plasma, which leads to arcing at the tar get. Even\nwhen a metallic target is used, process gas admixtures can form a no n-conducting\nlayer on the surface, which results in similar problems [9–13]. This prob lem can be\novercome by pulsing the DC voltage [9,14–16]. Another approach is t o use higher\nfrequencies to avoid arcing effects, which allows for the usage of die lectric targets\nin the deposition process [17–19]. Commonly used is a radio frequenc y (RF) of\n13.56MHz. In such a discharge operated at a low neutral gas press ure of a few\npascal or less, the energy of the ions impinging upon the target is de termined by\nthe voltage drop across the sheath and typically corresponds to t he time averagedExperimental study of electron power absorption dynamics i n RF magnetrons 4\nsheath voltage, since the mean free path of the ions is larger than t he sheath thick-\nness. The voltage drop, in turn, is dominated by the DC self-bias, wh ich can be\ncontrolled by changing the applied voltage amplitude.\nRecently, the Magnetic Asymmetry Effect (MAE) in Capacitively Coup led RF\n(CCRF) discharges has been investigated both numerically and expe rimentally\n[20–24]. It was found that process relevant plasma parameters su ch as the DC\nself-bias and the ion energy distribution function at the electrodes , which affects\nthe sputter rate at the target as well as characteristics of the d eposited films at the\nsubstrate [25], can be controlled by changing the applied magnetic fie ld strength\nat the powered electrode. This control can be conceptually compa red to the Elec-\ntrical Asymmetry Effect (EAE) in CCRF discharges that allows for a c ontrol of\nplasma parameters by applying a tailored voltage waveform [26–32].\nIn order to understand the control mechanisms in these discharg es a detailed un-\nderstanding of the electron power absorption dynamics is needed, since these dy-\nnamics predominantly affect relevant plasma parameters, namely th e energy dis-\ntribution functions, flux, and density distributions of charged par ticles and radi-\ncals in the discharge. Recent studies presented results of simulatio ns and experi-\nments on how electrons gain energy by the applied power in unmagnet ized CCRF\ndischarges [33–35]. In the α-mode, electrons are accelerated by the expanding\nsheath [36–40]. The γ-mode is characterized by strong ionization by ion-induced\nsecondary electrons inthe plasma sheath [41–43]. Inelectronegat ive discharges the\nDrift-Ambipolar (DA) mode becomes important [44,45], as well as the formation\nof striations [46,47]. Additionally, the electron power absorption dy namics can\nbe affected by the plasma series resonance (PSR) [48–50] and non- linear electron\nresonance heating (NERH) [51].\nThe electronheating dynamics inamagnetized capacitively coupled pla sma arefar\nless well understood than in the unmagnetized case. Lieberman et al.described\nthe enhanced acceleration of electrons in the presence of a magne tic field due to\nmultiple interactions of the particle with the expanding sheath as a co nsequence\nof the Lorentz force [52–54]. A study by Turner et al.showed that applying a\nweak magnetic field leads to a heating mode transition from a pressur e-heating\ndominated to an Ohmic-heating dominated discharge [55]. More recen t simulation\nresults by Zheng et al.showed that this strong Ohmic heating is decisively en-\nhanced by the Hall current in the azimuthal direction of the dischar ge [56]. Wang\net al.recently presented simulation results for a magnetized CCRF discha rge that\ninvestigate the electron power absorption dynamics as a function o f the magnetic\nfield strength in oxygen. It was shown that the power absorption m ode changesExperimental study of electron power absorption dynamics i n RF magnetrons 5\nfrom DA- to α-mode when increasing the magnetic flux density [57]. Another pub-\nlication by Wang et al.showed that the excitation of the PSR and the NERH at\nhigh excitation frequencies is reduced when a magnetic field is applied t o a CCP,\nwhich leads to a reduction of the averaged plasma density for low mag netic flux\ndensities. When a stronger magnetic field is applied, the plasma densit y increases\nagain due to a longer interaction time between the electrons and the expanding\nsheath and an additional acceleration of the electrons in a reverse d electric field\nduring the collapsing sheath phase [58]. The aforementioned publicat ions, just as\nmost other publications, assume the magnetic field to be applied solely parallel to\nthe powered electrode’s surface.\nIn another recent publication by Zheng et al.2-dimensional-Particle in Cell (PIC)\nsimulations were conducted to investigate the electron dynamics in a magnetron\ndischarge operated in argon. It was found that when using a dielect ric target a\nradially dependent charging is building up at the target surface leadin g to a re-\nduction of the electric field in the region of the plasma bulk. This, in tur n, leads\nto the formation of a spatially dependent ion energy distribution fun ction at the\ntarget [59].\nIn the framework of the companion papers the electron power abs orption\ndynamics in a geometrically asymmetric magnetized CCRF discharge op erated in\nargonareinvestigatedinthepresence ofaconductingtarget. Wh ilethecompanion\npapers [60,61] focus on the fundamental heating mechanisms bas ed on 1d3V and\n2d3v PIC/MCC simulations of a single set of typical conditions, in this w ork\nthe plasma discharge is studied as a function of the neutral gas pre ssure and the\nmagnetic flux density and based on experiments and 2d3v-PIC simula tions. The\napplied magnetic field has a magnetron-like torus shape that provide s closed field\nlines at the powered electrode. An aluminum disk is attached to the ele ctrode\nserving as a conducting target. It is found that the magnetic flux d ensity as well\nas the neutral gas pressure strongly alter the electron dynamics . Firstly, at a\nhigher magnetic flux density and/or higher pressure the electron b eam accelerated\nby the expanding sheath is confined to a region close to the target s urface, which\nleads to stronger ionization in this region, while such a confinement is n ot observed\nat low magnetic field and/or low pressure. Secondly, there is a large p opulation\nof electrons trapped above the racetrack region and remaining in t he discharge for\na long time. Such electrons are strongly energized due to the gener ation of the\n/vectorE×/vectorBdrift in the azimuthal direction. With an increase of the magnetic field this\npopulation grows and shifts closer to the powered electrode.Experimental study of electron power absorption dynamics i n RF magnetrons 6\nThe PIC simulations show that this effect is the strongest in the regio n where\nthe magnetic field is parallel to the target electrode. A reversal of the electric field\nis found to be induced by the magnetic field in regions where the magne tic field is\nperpendicular to the surface.\nThe manuscript is structured in the following way: In section 2, the e xperimental\nset-up is introduced, followed by a description of the PIC simulations in section\n3. In section 4, the results from both the experiment and the PIC s imulations are\npresented and discussed. Finally, conclusions are drawn in section 5 .\n2. Experimental set-up\nmatching\nnetworkVI\nprobe\nmagnetsgrounded\nshieldgrounded\nmeshx\nzy\nwered\nectrode\ngrounde\nelectrodedigital delay\ngenerator\nICCD\noptical filter\nFigure 1: Schematic of the experimental set-up.\nThe experimental set-up used in this work is shown schematically in fig ure 1 and\nconsists of a cylindrical vacuum chamber with a height of 400mm and a diameter\nof 318mm. The powered aluminum electrode serves as target, has a diameter of\n100mm, and is mounted to the top flange of the chamber. A grounde d shield\nand a grounded mesh prevent parasitic plasma ignition towards the g rounded\nchamber walls. A grounded counter electrode is positioned at a gap d istance\nof 70mm. The powered electrode system includes NdFeB permanent magnets\ninstalled in two concentric rings to create an azimuthally symmetric, b alanced,\ntorus-shaped magnetic field. Different magnetic flux densities can b e used by\nusing different stacked magnets located behind the target surfac e. In this way\ndifferent maximum magnetic flux densities at a reference position 8mm below\nthe racetrack, where the magnetic field is parallel to the powered e lectrode, ofExperimental study of electron power absorption dynamics i n RF magnetrons 7\n0mT (no magnets used), 7mT, 11mT, 18mT, and 20mT can be realized . By\nincreasing the distance of the magnets from the electrode surfac e a magnetic flux\ndensity of approximately 5mT at the reference position can be facilit ated as well.\nIn the direction perpendicular to the target surface the B-field st rength decreases\nexponentially. A detailed description of the configuration as well as m easurements\nof the magnetic flux density can be found in ref. [22].\nThe discharge is driven by a sinusoidal voltage waveform with a frequ ency of\n13.56MHz. A VI probe (Impedans Octiv Suite) is used to measure the driving\nvoltage amplitude V0.\nAll measurements are performed in pure argon (25sccm flow rate) as a function of\nthe radial magnetic field strength ( B0= 0−11mT) at the reference position and\nas a function of the neutral gas pressure ( p= 0.5−3Pa).\nAn ICCD (intensified charge-coupled device) camera (Stanford Co mputer Optics\n4 Picos) is used for Phase Resolved Optical Emission Spectroscopy ( PROES)\nmeasurements. The used objective is equipped with an interferenc e filter with\na central wavelength of 750.4nm and a full width at half maximum of 1.0 nm to\nmeasure the emission of a transition from the Ar2p1state. This transition has\nbeen used because of the relatively short natural lifetime of the ex cited state of\n22.2ns that resolves the period of the RF voltage of 73.7ns. Addition ally, the\nthreshold energy for the electron impact excitation of this state f rom the ground\nstate is 13.5eV and hence only the excitation by highly energetic elect rons is\nobserved. A Digital Delay Generator is used to synchronize the cam era with the\ndriving voltage waveform and to allow for a precise control of the ca mera trigger\nsignal within the applied RF period. The ICCD camera is positioned at a w indow\nand measures the emission line-integrated along the line-of sight. Ad ditionally, all\nimages are binned across the radial direction. By doing this all inform ation about\nradial components of the emission is lost and only the axial dimension is resolved.\nConsidering the radial dependence of the magnetic flux density alon g the target\nsurface, one should mention that this is quite a severe method of da ta processing.\nHowever, by comparing the experimental results to 2d-PIC simulat ions all radial\neffects are investigated in detail by the simulations. Using an Abel tr ansform to\ncalculate radially resolved emission from the line-integrated experime ntal data was\nfound to lead to a strongly reduced signal-to-noise ratio. The emiss ion is measured\nwithaspatial resolutionofabout1mminaxialdirection andatempora l resolution\nof 3ns. Based on a simplified rate equation model the electron impact excitation\nrate from the ground state into the observed excited state is calc ulated. Each plot\nof the calculated excitation rate is normalized to its respective maxim um value.Experimental study of electron power absorption dynamics i n RF magnetrons 8\nIn Ref. [62] a detailed description of the diagnostic can be found.\n3. PIC/MCC code\nHere, onlyabriefdescriptionisgiven. Foramoredetailedaccount of thenumerical\ntechniques used refer to [61].\nFigure 2: Modeled geometry in (a) the lab coordinates and (b) the normalized logical\ncoordinates. The blue area denotes the reactor’s chamber, w hile the grounded metal\nareas are shown in black. The light and dark gray areas mark th e powered electrode\nand the dielectric separator, respectively.\nBecause of the low pressure used to operate the device, the mode l should take\ninto account kinetic and nonlocal effects. A suitable algorithm is the p article-in-\ncell (PIC) method [63–65] enhanced with the Monte-Carlo method to model the\ncollisions of plasma particles with neutral particles of the working gas [66].\nTo reduce the computational time, a recently proposed implicit ener gy-\nconserving algorithm [67, 68] of the PIC method was employed, in a va riant\ndiscussed in [69]. Using the energy-conserving PIC algorithm for non uniform\nmapped grids [70], one can efficiently reduce the number of computat ional grid\ncells even further by allocating fewer grid cells in the areas where no s ignificant\nphysics is expected, and do so without causing any numerical heatin g. For the\nmodel geometry considered in the present work (see Fig. 2(a)), m any essential\nphenomena take place in a small region above the racetrack, where as the rest\nof the reactor chamber, by far occupying most of the space, is ne eded only for a\nproperaccount oftheDCself-bias. Thelatterisdemanded forapr operdescription\nof the plasma series resonance excitation [23,61] and is modeled by in cluding anExperimental study of electron power absorption dynamics i n RF magnetrons 9\nexternal network model [71–73] modified for the implicit energy-co nserving PIC\nmethodassuggestedin[61,69]. Furthermore, sinceforhighermag neticfields(cases\nwithB0= 7mT and 10mT in section 4) the powered electrode sheath’s width\nbecomes rather small compared to the electrode gap, we have fur ther modified the\naxial coordinate transformation to ensure an increased resolutio n in the vicinity of\nthe electrodes. The computational domain in the transformed logic al coordinates\n(ηin the axial direction z and ξin the radial direction r) is depicted in Fig.\n2(b). Forthenumerical treatment ofcylindrical coordinates, ar adiallynonuniform\ncomputational grid was utilized, where only a few computational cells close to the\nradial axis were discretized uniformly with respect to r2, whereas the rest of the\nradial grid was discretized uniformly with respect to r. In the logical coordinates\nwe used a uniform grid with (161 ×258) cells in the radial and the axial directions,\nrespectively. The corresponding Poisson equation in the logical coo rdinates [60]\nwas solved using the geometrical multigrid algorithm. The nonuniform grid was\naccompanied by an adaptive particle management algorithm [74–76] d esigned for\nthe energy-conserving PIC method [61]. Such an algorithm was nee ded to ensure\na balanced resolution of the computational phase space with super particles and it\nkept onaverage approximately 500superparticles per cell inthepla sma-filled areas\nin all simulations at the converged state. The time step for the field in tegration\nwaschosen tobe2 .5×10−11sandthesub-stepping algorithmwasused fortheorbit\nintegration to ensure the energy-conservation and the numerica l plasma response\naccuracy [67].\nDue to the large computational cost the code was parallelized on GPU using\na two-dimensional version of the fine-sorting algorithm described in [77]. The\nmethod has been benchmarked in 1d [69] for a CCP rf discharge in heliu m [78] and\nin 2d for magnetized discharges in the ( θ,z) and (r,θ) geometries in [79] and [80],\nrespectively. The method with all the techniques described above r esulted in the\nthe 2d3v energy-conserving implicit electrostatic ECCOPIC2S-M mo dification of\nthe ECCOPIC code family based on the algorithm of [69] and was develo ped in-\nhouse.\nThe PIC simulations were conducted for argon at T= 450K. Due to the\nlow pressures used, the modeled reactions included elastic scatter ing, ionization,\nand excitation [81] for the electron-neutral collisions with the elast ic scattering\nand the charge exchange [82] for the ion-neutral collisions. The co llisions were\nimplemented using the null-collision Monte-Carlo algorithm [83] modified f or\nGPUs [77] using the Marsaglia xorshift128 pseudorandom number ge nerator [84],\nrandomly initialized for each thread. The plasma-surface interactio n was modeledExperimental study of electron power absorption dynamics i n RF magnetrons 10\nas follows. The ion-induced secondary electron emission was taken in to account\nwith the energy-dependent yield adopted from [81] for clean metals under the\nassumption that the conducting target is sufficiently cleaned by the sputtering\nprocess. The electron-induced secondary electron emission was m odeled in the\ncode after [85–87], albeit with non-uniform energy distribution adop ted from [88]\nand implemented employing the acceptance-rejection method [89]. T he model\nparameters were fitted to the measurements of the electron-ind uced secondary\nelectron emission yield measurements made for Al, which were report ed in [90] for\nthe mid- and high-energy range and in [91] for the low-energy range . However, it\nis worth noting that, unlike in dcMS, in rfMS the secondary electron e mission is\ntypically not essential for the discharge sustainment [59–61]. The e lectron heating\nmechanisms are much more versatile in rfMS compared to dcMS [61], bu t the\ndominant contribution typically comes from the Hall heating [59–61].\n4. Results\nIn Fig. 3 the results of the PROES measurements at 1Pa are shown f or two cases:\n(a) no magnetic field and (b) B0= 5mT. These plots show the spatio-temporally\nresolved electron impact excitation rate fromthe ground state int o theAr2p1state\ntime resolved within the RF period as a function of the distance from t he powered\nelectrode. For comparison the excitation rate obtained from the P IC simulations\nunder the same conditions as in the experiment is presented in Fig. 3 ( c) and\n(d). When no magnetic field is applied to the discharge the sheath at t he powered\nelectrode expands for one half of the RF cycle until it reaches its ma ximum width\nof approximately 20mm. In the second half of the period the sheath collapses\nagain. According to the experimental results and during the expan sion phase an\nelectron beam seems to be accelerated by the sheath that traver ses through the\ndischarge and reaches the opposite electrode where it is reflected back into the\nplasma bulk. This behavior is well known for low pressure capacitively c oupled\nplasmasinvestigatedhere[92]. Thesimulationrevealsthatactuallym ultiplebeams\nare accelerated during a single sheath expansion phase as previous ly observed in\nPIC simulations and in experiments [50,93]. This behavior cannot be re solved\ntemporally in the present experiment due to the temporal resolutio n of 3ns.\nWhen a magnetic field is applied to the discharge, the excitation dynam ics\nchange, as shown in Fig. 3(b) and (d). The oscillation of the sheath is still\nvisible but the maximum sheath width is reduced to approximately 10mm due\nto the higher plasma density caused by the magnetic electron confin ement. TheExperimental study of electron power absorption dynamics i n RF magnetrons 11\nB0=0 mT B 0=5 mTvertical distance from powered electrode [mm]\nFigure 3: Spatio-temporal plots of the line-integrated exc itation rate from the ground\nstate into the Ar2p1state with (a) and (c) no magnetic field and (b) and (d) a magnet ic\nflux density of B0=5mT at the reference position obtained in (top) the experim ent and\n(bottom)thesimulation. Dischargecondition: Ar,1Pa, 900 Vdrivingvoltageamplitude.\nexcitation is still strong during the expansion phase of the sheath b ut no beam-like\nstructure is visible anymore. The excitation mainly occurs close to th e powered\nelectrode in a region where the magnetic field is strong. This can be ex plained by\nthe reduced electron mobility across the magnetic field lines. One of t he electron\nheating mechanisms is due to the fact that the accelerated electro ns gyrate around\nthe magnetic field lines, which guide them back to the sheath region [52 ], as\ninvestigated before in PIC simulations [53,57]. In the companion pape rs [60]\nand[61] weshow thatthedominant electronheating mechanism istyp ically caused\nby the emergence of a strong and time-dependent /vectorE×/vectorBdrift in the azimuthal\ndirection, leading to a force in the azimuthal direction. The resulting Hall heating\nmechanism [56,60] involves an enhanced time-dependent electric fie ld, which isExperimental study of electron power absorption dynamics i n RF magnetrons 12\ngenerated predominantly during the sheath collapse and the sheat h expansion and\nis required to ensure a sufficient level of electron transport acros s the magnetic\nfield lines [60]. This mechanisms creates a large population of energetic electrons\nabove the racetrack, where they are trapped and move back and forth along the\nmagnetic field lines due to the mirror effect and can remain in the discha rge for a\nlong time [61].\nThe comparison between the experimental and the computational data shows\nthe good agreement between both results, which allows for a more d etailed\ninvestigation of fundamental processes by analyzing the simulation data.\nFigure 4: 2-dimensional plots of the (a) time averaged excit ation rate from the ground\nstate into the Ar2p1state and (b) time averaged density of electrons above the io nization\nenergy for argon of 15.8eV obtained from the PIC simulation. The boxes in (a) mark\nthree different regions of interest and the arrows mark the mag netic field used in the\nsimulation. Conditions: Ar, 1Pa, 900V driving voltage ampl itude, 5mT.\nFig. 4 shows simulation results of the time-averaged excitation rate into theAr2p1\nstate andthe time-averaged density of highly energetic electrons insubplot (a)and\n(b), respectively, under the same discharge conditions as before . In contrast to the\nline-integrated PROES measurements, PIC simulations allow a two-dim ensional\ninvestigation and, hence, a deeper understanding of the observe d effects. The\nzones indicated in Fig. 4(a) mark three important regions in the disch arge. In\nzones 1 (0-10mm of the electrode radius) and 3 (40-50mm of the ele ctrode radius)\nthe magnetic field mainly consists of components perpendicular to th e powered\nelectrode, whilezone 2(20-35mmof theelectrode radius) isthereg ionwith mainly\nparallelcomponents. Onecanseethatmostoftheexcitationoccu rsinzone2while\nin zones 1 and 3 almost no excitation is observed. The same holds true for the\ndensity of highly energetic electrons as shown in Fig. 4(b). This mean s that mostExperimental study of electron power absorption dynamics i n RF magnetrons 13\nof the ionization by electrons in a discharge with such a magnetron-lik e magnetic\nfield configuration happens in this region as well.\nzone 2\nzone 3zone 1\nFigure 5: Spatio-temporal plots of the electron impact exci tation rate from the ground\nstate into the Ar2p1state time-resolved within one RF cycle in the three regions defined\nin Fig. 4(a) obtained from the simulations. Conditions: Ar, 1Pa, 900V driving voltage\namplitude, 5mT.\nIn Fig. 5 simulation results for the excitation dynamics averaged rad ially over\nthe assigned regions of interest are shown as a function of time with in the RF\nperiod and of the axial position between the powered and the groun ded electrode.\nComparing zone 2 with the results obtained in the experiment one can see a very\nsimilar behavior of the accelerated electrons. When the sheath exp ands starting\nat t/T RF=0.9 a strong excitation is visible, which lasts until the maximum sheathExperimental study of electron power absorption dynamics i n RF magnetrons 14\nwidthofapproximately8mmisreachedinthefollowingperiodatt/T RF=0.3. This\npattern can be associated to electrons being accelerated by the e xpanding sheath,\nmoving back towards the expanding sheath due to their gyromotion , and hitting\nthe sheath again. As discussed before the limited electron mobility pe rpendicular\nto the magnetic field hinders the accelerated electrons from moving across the\nplasma bulk as it was the case without applied magnetic field. In this way the\ninteraction time of electrons with the expanding sheath is extended due to the\npresence of the magnetic field. Another strong effect acting on th e electrons plays\nan essential role and is described in detail in the companion papers [6 0,61]: Due to\nthe large electric and magnetic fields in the region close to the powere d electrode,\nelectrons experience a strong E×Bdrift in the azimuthal direction. With an\nelectric field growing in time, this leads to a rapid energization of the ele ctrons,\nwhich, in turn, leads to the strong ionization/excitation in zone 2 visib le in 5(b).\nIn zones 1 and 3, however, the excitation pattern changes. In ge neral, the\nexcitation rate in these regions is reduced compared to zone 2. Wea k electron\nbeams can be observed when the sheath expands and energetic ele ctrons move\ninto the plasma bulk. The magnetic field here points in the same directio n as the\nelectron beam’s movement and electrons are not trapped by the ma gnetic field\nanymore (see also [61]). Another interesting feature is the increas ed maximum\nsheath width of approximately 12mm due to the reduced local electr on density in\nthese regions. Due to the reduced magnetic field at positions farth er away from\nthe electrode, any electron located at the sheath edge is less influe nced by the\napplied magnetic field in zones 1 and 3 compared to zone 2.\nIn order to further investigate the effect of the magnetic field on t he electron\ndynamics in a CCP, the neutral gas pressure is varied while keeping th e magnetic\nfield constant. Fig. 6 shows the excitation rateobtained by PROES m easurements\nand from the simulation. The applied voltage amplitude and the magnet ic field\nare the same as in Fig. 3(b). If the pressure is reduced to 0.5Pa, th e excitation\ndynamics are strongly altered compared to the case discussed bef ore. Now, the\nexpansion of the sheath leads to an electron beam traversing the b ulk and hitting\nthe sheath at the grounded electrode, similar to the dynamics obta ined without\nexternal magneticfield, asshown inFig. 3(a). Whenfurther increa sing theneutral\ngas pressure to 3Pa, see Fig. 6(b), the excitation is localized in a reg ion even\ncloser to the powered electrode compared to 1Pa. This behavior is a ssociated to\nthe reduced sheath thickness in case of an increased neutral gas pressure, which\nis characteristic for CCP discharges. As previously discussed a red uced sheath\nthickness leads to a higher number of electrons in regions closer to t he poweredExperimental study of electron power absorption dynamics i n RF magnetrons 15\np=0.5 Pa p=3 Pavertical distance from powered electrode [mm]\nFigure 6: Spatio-temporal plots of the radially averaged el ectron impact excitation rate\nfrom the ground state into the Ar2p1state at a neutral gas pressure of 0.5Pa [(a) and\n(c)] and3Pa [(b) and (d)] obtained in (top) the experiment an d (bottom) thesimulation.\nDischarge conditions: Ar, 900V driving voltage amplitude, 5mT\nelectrode where the magnetic flux density is high and the electrons’ Larmor radius\nis reduced. Hence, the electrons are trapped more efficiently by th e magnetic field.\nOntheother hand, when thesheath isthicker at lowpressure, elec trons arepushed\nout of the region of strong magnetic field, which allows them to move a cross the\nplasma bulk.\nA similar trend can be observed when increasing the magnetic flux den sity in the\ndischarge. Fig. 7 shows the excitation rate for three different cas es at 1Pa, i.e. no\nmagnetic field [(a) and (d)], B0=7mT [(b) and (e)], and B0=11mT [(c) and (f)].\nIn the experiment a higher power is needed for higher magnetic flux d ensities in\norder to ensure a constant applied voltage amplitude. For this reas on, the voltage\namplitude is reduced to 300V in this investigation, both in the experime nt and inExperimental study of electron power absorption dynamics i n RF magnetrons 16\nB0=0 mT B 0=7 mT B 0=11 mT\nFigure 7: Spatio-temporal plots of the radially averaged el ectron impact excitation rate\nfrom the ground state into the Ar2p1state for a magnetic flux density at the reference\nposition of 0mT [(a) and (d)], 7mT [(b) and (e)], and 11mT [(c) and (f)] obtained in the\nexperiment (top) and the simulation (bottom). Discharge co nditions: Ar, 1Pa, 300V\ndriving voltage amplitude.\nthe simulation, to protect the target from thermal damage. As in t he previous\ncase, there are two different mechanism contributing to the obser ved pattern.\nFirstly, a highly energetic electron beam forms due to the expanding sheath in\nthe unmagnetized scenario. When increasing the magnetic flux dens ity to 7mT,\nthe accelerated electrons are trapped by the magnetic field and st rong excitation\nformsclosetothepoweredelectrode. Secondly, theelectrons tr appedbythemirror\neffect above the racetrack region and energized due to the Hall he ating also reside\ncloser to the powered electrode with the growing magnetic field due a reduced\nwidth of the magnetized sheath [61].\nCompared to the electron dynamics at 5mT, the sheath width is furt her\ndecreased. Also, the temporal modulation of the sheath is reduce d and the\nexcitationishighforalongerperiodoftheRFcycle. Thiseffectisfurt herenhanced\nwhen the magnetic flux density is increased to 11mT. This leads to the conclusion\nthat, for most of the RF period, energetic electrons are trapped in a region close\nto the powered electrode where the magnetic field is strong.\nThese results are further investigated with the help of the PIC simu lationsExperimental study of electron power absorption dynamics i n RF magnetrons 17\nFigure 8: 2-dimensional plots of the (a) time averaged elect ron impact excitation rate\nfrom the ground state into the Ar2p1state within one RF period and (b) time averaged\ndensity of electrons above the ionization energy for argon o f 15.8eV obtained from the\nPIC simulation. The arrows mark the magnetic field used in the simulation. Conditions:\nAr, 1Pa, 300V driving voltage amplitude, 11mT.\nfor the case of the strongest magnetic field. The two-dimensional time averaged\nelectron impact excitation from the ground state into the experime ntally observed\nargon state is shown in Fig. 8(a). The pattern is similar to the one sho wn in Fig.\n4. However, the excitation occurs much closer to the electrode su rface due to the\nincreased plasma density as a consequence of the enhanced electr on confinement\nin the stronger magnetic field. Even though the voltage amplitude is r educed to\none third compared to the case with lower magnetic flux density, the maximum\nexcitation rate is approximately four times higher. The same trend h olds true for\nthe time averaged density of energetic electrons shown in Fig. 8(b) . Both obser-\nvations show how efficiently the energetic electrons are trapped by the magnetic\nfield in a magnetized CCP discharge.\nFig. 9 shows the 2d space resolved electric field obtained from the PI C\nsimulations at three different time steps within the RF period again for the\nstrongest magnetic field (11mT). The plots only show the region clos e to the\npowered electrode up to a distance from the electrode of 20mm. It should also be\nnoted that the color scale is chosen to only show values between -15 kVm−1and\n+15kVm−1to focus on the variations at the sheath edge in contrast to the st rong\nelectric field inside the sheath. At t/T RF=0.67 the sheath at the powered electrode\nis collapsing, but the electric field is still strong preventing most of th e electrons\nfrom moving to the electrode surface. In general, the discharge c an be dividedExperimental study of electron power absorption dynamics i n RF magnetrons 18\nt/TRF=0.77\nt/TRF=0.93t/TRF=0.67\nFigure 9: 2-dimensional plots of the electric field componen t perpendicular to the\nelectrode surface at three different times within one RF perio d: (a) t/T RF=0.67\n(collapsing sheath), (b) t/T RF=0.77 (sheath collapse), (c) t/T RF=0.93 (expanding\nsheath). The data are obtained from the PIC simulation. The b oxes in (a) mark three\ndifferentregionsofinterestandthearrowsmarkthemagnetic fieldusedinthesimulation.\nConditions: Ar, 1Pa, 300V driving voltage amplitude, 11mT.\ninto three zones again, as marked in Fig. 9(a). In the center and at the edge of\nthe target electrode, zone 1 and zone 3, respectively, the region of strong negative\nelectric field reaches several millimeters into the discharge, while in th e racetrack\nregion, zone 2, the electric field develops in a small region of approxim ately 1.4mm\nwidth, which isdueto thehighlocal iondensity and, hence, theshort sheath width\nin this zone.Experimental study of electron power absorption dynamics i n RF magnetrons 19\nWhen the sheath expands again at a later time frame as shown in Fig. 9 (c) a\nsimilar behavior can be observed. Investigating the time frame when the sheath is\ncollapsed (t/T RF=0.77), the electric field strongly alters. In zones 1 and 3, where\nthe magnetic field component perpendicular to the electrode surfa ce is strong, a\npositive electric field is formed. Such a behavior corresponds to an e lectric field\nreversal and leads to the acceleration of electrons towards the e lectrode surface.\nThiseffect istypically observedinelectronegative discharges [94], ine lectropositive\ndischarges using tailored voltage waveforms [95], as well as in previo us simulations\nof CCPs with an externally applied magnetic field parallel to the electro de [57,96]\n(see also [60]). It usually occurs, if the electron mobility is reduced an d the\npositive ion flux to the surface cannot be compensated by the diffus ive flux of\nthermal electrons anymore. This leads to the generation of an elec tric field that\nadditionally accelerates electrons to the surface.\nFigure 10: Time averaged fluxes of charged particles to the po wered electrode surface\nwithin one RF period as a function of the radial position. The black solid line shows\nthe flux of electrons and the red dashed line shows the flux of po sitive argon ions. The\ndata are obtained from the PIC simulation. Conditions: Ar, 1 Pa, 300V driving voltage\namplitude, 11mT.\nIn Fig. 10 the time averaged fluxes of charged particles to the powe red\nelectrode areshown asa function ofthe radial positiononthe surf ace. As expected\nthe ions mainly reach the electrode in zone 2, where the charged par ticle densityExperimental study of electron power absorption dynamics i n RF magnetrons 20\nis high. In this zone the magnetic field is parallel to the electrode surf ace. The\nions are almost unaffected by the magnetic field and are accelerated by the electric\nfield before reaching the electrode surface. Most of the electron s are not able to\nreach the surface in this zone due to their reduced mobility across t he magnetic\nfield lines, but reach the electrode in zones 1 and 3, where the electr on flux is\nmuch higher than the ion flux. This means that flux compensation of e lectrons\nand ions on time average is happening globally for the target surface with locally\nvery different electron and ion fluxes. This is only possible in the prese nce of a\nconducting target, for which a current can flow inside the electrod e. In case of\na non-conducting target a local flux compensation would be require d. In zones 1\nand 3, another important mechanism needs to be considered. When an electron\nenters an area with an increasing density of the magnetic field lines th e magnetic\nmirror force acts on the electron along the magnetic field lines, which leads to a\ndeceleration and ultimately to a reflection of the electron. Here, su ch a magnetic\nfield configuration can be found in the regions outside of the racetr ack above\nthe used magnets, which keeps the electrons from reaching the su rface here. In\norder to compensate the ion flux to the electrode on time average [9 7], an electric\nfield reversal is, thus, generated in zones 1 and 3 where the magne tic field is\nperpendicular to the surface, as shown in Fig. 9(b), to counterac t the strong\nmirror force that acts in the direction away from the target. The e ffect of the\nmirror force on magnetized RF discharges is discussed in detail in the companion\npaper [61].\n5. Conclusion\nA capacitively coupled argon plasma with an externally applied magnetr on-like\nmagnetic field configuration was investigated experimentally by Phas e Resolved\nOptical Emission Spectroscopy measurements to study the electr on dynamics in\nsuch plasmas. The results are compared to 2d3v-Particle in Cell simu lations that\noffer further insights into the physics of such magnetically enhance d CCPs. It\nwas found that depending on the neutral gas pressure the excita tion dynamics of\nhighly energetic electrons differ from those induced by ’classical’ she ath expansion\nacceleration in unmagnetized CCPs. Whereas the free propagation of energetic\nelectron beams generated by sheath expansion heating is found at low neutral\ngas pressure, comparable to unmagnetized low pressure CCPs, a s trong excitation\nlocalized close to the target surface is observed when the neutral gas pressure is\nincreased. This is explained by the magnetic confinement of energet ic electronsExperimental study of electron power absorption dynamics i n RF magnetrons 21\ndue to a reduced sheath thickness that confines electrons in regio ns of strong\nmagnetic field close to the powered electrode. The PIC simulations sh ow that this\nexcitationismainlyduetoelectronsbeingconfinedintheregionofstr ongmagnetic\nfields parallel to the target surface, while in the regions of strong p erpendicular\ncomponents electrons are able to leave the magnetized zone. This le ads to an\nacceleration of electrons beams traversing the gap between the e lectrodes in these\nregions. A more detailed description of the relevant mechanisms can be found in\nthe companion paper [61].\nAdditionally, it was found that due to the reduced flux of electrons t o the target\nsurface in regions of a strong parallel magnetic field an electric field r eversal is\ngenerated in regions of a strong perpendicular magnetic field to bala nce the fluxes\nof positive and negative particles to the surface globally on time aver age. This\nelectric field reversal is necessary to overcome the magnetic mirro r force acting\non the electrons in the regions outside of the racetrack. No local fl ux balance is\nobserved in the presence of a conducting target, i.e. the ion flux is h igher than the\nelectron flux in regions of parallel magnetic field and the electron flux is higher in\nregions of perpendicular magnetic field.\nThis paper in conjunction with the two companion papers [60] and [6 1] aids\nto comprehend the heating mechanisms in RF magnetron discharges . The\npower absorption dynamics are now understood based on PIC simula tions and\nexperimental investigations ranging from fundamental studies in o ne-dimensional\nsimulations to application-oriented analyses in a real discharge. 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Czarnetzki, “Cha rge dynamics in capacitively\ncoupled radio frequency discharges,” Journal of Physics D: Applied Physics , vol. 43,\np. 225201, may 2010.zone 1 t/TRF=0.52"    },    {        "title": "1604.06262v2.Non_equilibrium_magnetic_fields_in_ab_initio_spin_dynamics.pdf",        "content": "Ultrafast demagnetizing \felds from \frst principles\nJacopo Simoni,1,\u0003Maria Stamenova,1and Stefano Sanvito1\n1School of Physics, AMBER and CRANN Institute, Trinity College, Dublin 2, Ireland\nWe examine the ultrafast demagnetization process of iron-based materials, namely Fe 6clusters\nand bulk bcc Fe, with time-dependent spin-density functional theory (TDSDFT). The magnetization\ncontinuity equation is reformulated and the torque due to the spin-current divergence is written\nin terms of an e\u000bective time-dependent kinetic magnetic \feld, an object already introduced in\nliterature before. Its time evolution, as extracted from the TDSDFT simulations, is identi\fed as\none of the main sources of the local out-of-equilibrium spin dynamics and plays a major role in\nthe demagnetization process. Such demagnetization is particularly strong in \\hot spots\" where the\nkinetic torque is maximized. Finally, we \fnd the rate of demagnetization in Fe 6to be strongly\ndependent on the direction of polarization of the exciting electric \feld and this can be linked to the\nout of equilibrium distribution of the kinetic \feld in two comparative cases.\nPACS numbers: 75.75.+a, 73.63.Rt, 75.60.Jk, 72.70.+m\nI. INTRODUCTION\nThe search for practical solutions for increasing the\nspeed of manipulation of magnetic bits is essential for\nthe progress of modern information and communication\ntechnology. It has been shown that there is an upper\nlimit to the speed of the magnetization switching process\nwhen this is driven by a magnetic \feld1,2. An increase in\npower absorption beyond this limit and for higher mag-\nnetic \feld amplitudes push a spin system out of equilib-\nrium into a chaotic behaviour, and the switching speed\ndecreases. For this reason the discovery made by Beaure-\npaire et al. [3] in 1996 that a ferromagnetic Ni \flm could\nbe demagnetized by a 60 femtosecond optical laser pulse\nattracted a lot of interest and was the seed to a new \feld,\nnow called femto-magnetism.\nIn a standard pump-probe experiment the system is\ninitially excited by an optical pulse (pump) and then\nthe magnetization dynamics is monitored by analysing\na second signal (probe)4,5. Depending on the minimal\ndelay between the pump and the probe, one can anal-\nyse the demagnetization process at di\u000berent timescales\nand thus observe the dissipation mechanisms active at\nthat particular time. The interpretation of the results is,\nhowever, a complicate matter. In general for demagne-\ntization processes observed on a timescale ranging from\nnanoseconds to 100 picoseconds one considers an empiri-\ncal three temperature model6, where electrons, spins and\nphonons de\fne three energy baths, all interacting with\neach other. In contrast, ultrafast spin dynamics, tak-\ning place within a few hundreds femtoseconds, is yet not\ndescribed in terms of a single uni\fed scheme and var-\nious models for the demagnetization process have been\nadvanced. These include fully relativistic direct trans-\nfer of angular momentum from the light to the spins7,8,\ndynamical exchange splitting9, electron-magnon spin-\rip\nscattering10, electron-electron spin-\rip scattering11and\nlaser-generated superdi\u000busive spin currents12.\nGiven the complexity of the problem ab initio meth-\nods, resolved in the time domain, provide a valuable\ntool to probe the microscopic aspects of the ultrafast\nspin dynamics of real magnetic materials by means\nof time-dependent simulations. In this work we ap-ply time-dependent spin density-functional-theory (TD-\nSDFT)13,14in its semi-relativistic, non-collinear, spin-\npolarized version to analyse the ultrafast laser-induced\ndemagnetization of two ferromagnetic transition metal\nsystems: a Fe 6cluster (see Fig. 1) and bulk bcc Fe.\nRecently, within a similar theoretical description, it has\nbeen demonstrated that the spin-orbit (SO) interaction\nplays a central role in the demagnetization process15{17.\nFurthermore, it was showed by us18that the laser-\ninduced spin dynamics can be understood as the result\nof the interplay between the SO coupling potential and\nan e\u000bective magnetic \feld. The so-called kinetic mag-\nnetic \feld19,20,Bkin(r;t), originates from the presence of\nnon-uniform spin currents in the system. In this work\nwe focus on the anatomy of Bkin(r;t) and we analyze in\ndetail its role in the highly non-equilibrium process of\nultrafast demagnetization.\nThe \frst formulation of the spin dynamics problem in\ntransition metal systems was given in Refs. [19,20] by\nAntropov and Katsnelson, who laid down the founda-\ntion of DFT-based spin dynamics, by deriving a set of\nequations of motion for the local magnetization vector.\nIn those seminal works the magnetization dynamics was\nanalyzed at the level of the adiabatic local spin-density\napproximation (ALSDA), but actual applications to real\nout-of-equilibrium systems were not described. Our pur-\npose is to clarify and quantify, through TDSDFT simu-\nlations at the level of the non-collinear ALSDA, the role\nplayed by Bkin(r;t) in the laser-induced ultrafast spin\ndynamics of transition metal ferromagnets.\nThe paper is divided into four main sections. In Sec-\ntion II we de\fne the various \felds that couple to the\nspins by isolating in the continuity equation only the\nterms that play a major role in the dynamical process.\nIn Section III we present the results of the calculations\nfor Fe 6clusters and show that \\hot spots\" for demag-\nnetization are associated with larger misalignment of the\nkinetic magnetic \feld and the local spin density. This be-\ncomes more clear through evaluation of material deriva-\ntives. A demonstration of the e\u000bect of the polarization\nof the electric \feld on the rate of demagnetization of Fe 6\nis discussed in Section IV. In Section V we show that\nprevious observations for Fe 6are valid for bulk bcc Fe\nas well. Finally we conclude. The paper is supplementedarXiv:1604.06262v2  [cond-mat.mes-hall]  18 Nov 20162\nwith an Appendix where we present a detailed derivation\nof the spin continuity equation (A).\nII. THEORY\nWe consider the TDSDFT problem within the ALSDA\nfor a spin-polarized system excited by an electric \feld\npulse. If one neglects second-order contributions arising\nfrom the solution of the coupled Maxwell-Schr odinger\nsystem of equations, the dynamics will be governed by\nthe usual set of time-dependent Kohn-Sham (KS) equa-\ntions\ni~d\ndt KS\nj(r;t) =HKS(r;t) KS\nj(r;t): (1)\nIn Eq. (1)  KS\nj(r;t) are the KS orbitals and the KS\nHamiltonian, HKS(r;t), can be expressed in the velocity\ngauge formulation and the minimal coupling substitution\nas,\nHKS(r;t) =1\n2m\u0010\n\u0000i~r\u0000q\ncAext(t)\u00112\n\u0000\n\u0000\u0016B^\u001b\u0001Bs[n;m](r;t) +vs[n](r;t);(2)\nwhere\nvs[n](r;t) =Z\nd3r0n(r0)\njr\u0000r0j+vALSDA\nxc [n](r;t) +\n+X\nIVI\nPP(jr\u0000RIj) (3)\nand\nBs[n;m](r;t) =BALSDA\nxc [n;m](r;t) +Bext(r;t):(4)\nHerevs(r;t) represents the usual non-interacting KS\npotential and the full non-interacting magnetic \feld,\nBs(r;t), consists of the external one, Bext(r;t), and the\nexchange-correlation (XC) magnetic \feld, BALSDA\nxc (r;t).\nIn the equations above mis the electron mass, qthe elec-\ntron charge, cthe speed of light, Aext(t) the vector poten-\ntial associated to the external magnetic \feld, ^\u001bthe spin\noperator,\u0016Bthe Bohr magneton, nthe electron density\nandmthe magnetization density. Then, vs(r;t) is de-\ncomposed into a Hartree contribution, an XC correlation\none,vALSDA\nxc [n](r;t), and into an ionic pseudo-potential\nVI\nPP(jr\u0000RIj). For a fully relativistic, norm-conserving\npseudopotential the SO coupling enters into the KS equa-\ntions in the form21\nVI\nPP(jr\u0000RIj) =X\nl\u0010\n\u0016VI\nl(r) +1\n4VI;SO\nl(r)+\n+lX\nm=\u0000lVI;SO\nl(r)^LI\u0001^SjI;l;mihI;l;mj\u0011\n:\n(5)\nIn Eq. (5) the orbital momentum operator associated to\ntheI-th atomic center is ^LI, while the vectors fjI;l;mig\nare the associated set of spherical harmonics centered on\nthat given atomic position. In Eq. (5) VI;SO\nl(r) de\fnes ageneralized space-dependent SO coupling parameter pro-\nviding a measure of the SO interaction strength close to\nthe atomic site, while \u0016VI\nl(r) includes all the ionic rel-\nativistic corrections like the Darwin and the mass cor-\nrection term. Within the ALSDA vxc(r;t) and Bxc(r;t)\nare local functions in time of the electron density and\nmagnetization, which in turn are written in terms of the\ntime-dependent KS orbitals\nn(r;t) =X\nj2occ:X\n\u001b KS\nj\u001b(r;t)\u0003 KS\nj\u001b(r;t); (6)\nm(r;t) =X\nj2occ:X\n\u000b;\f KS\nj\u000b(r;t)\u0003\u001b\u000b;\f KS\nj\f(r;t):(7)\nStarting from the set of time-dependent KS equations in\n(1) it is possible to derive an equation of motion for the\nmagnetization, or a spin-continuity equation, in terms of\nthe non-interacting KS observables. This reads\nd\ndtm(r;t) =\u0000r\u0001JKS(r;t) +\u0016Bm(r;t)\u0002Bs(r;t)+\n+TSO(r;t); (8)\nwhere JKS(r;t) represents the non-interacting KS spin-\ncurrent rank-2 tensor\nJKS(r;t) =~\n2miX\nj2occ:\u0000\n KSy\nj^\u001br KS\nj\u0000h:c:\u0001\n;(9)\nand the SO torque contribution reads\nTSO(r;t) =X\nIX\nl;m1;m2occupiedX\nj;\u000b;\fVSO\nl(jr\u0000RIj)\u0001\n\u0001h KS\nj\u000bjl;m1;Iihl;m1;IjLIjl;m2;Ii\u0002\u001b\u000b\f\u0001\n\u0001hl;m2;Ij KS\nj\fi: (10)\nThe KS magnetic \feld Bs(r;t) is taken as in Eq. (4),\nwhich in absence of an external magnetic \feld reduces\ntoBxc(r;t). In DFT there are a set of zero-force the-\norems stating that the interaction between the parti-\ncles cannot generate a net force22. In the case of the\nexchange-correlation magnetic \feld we have the exact\nconditionR\nd3rm(r;t)\u0002Bxc(r;t) = 0, which is satis-\n\fed by the ALDA. Combining this equality with the as-\nsumption that the currents at the system boundary are\nnegligible allows us to conclude that the only source of\nglobal spin loss is the SO coupling torque, TSO, and that\nthe spin lost during the temporal evolution is transferred\nto the orbital momentum of the system, which in turn\nis partially damped into the lattice (we consider frozen\nions). Hence we have the relation,\nd\ndtZ\n\nd3rm(r;t) =Z\n\nd3rTSO(r;t); (11)\nwhere the integration extends over the entire volume \n.\nWithin the ALDA, the exchange-correlation functional\nsatis\fes also a local variant of the zero-torque theorem23,\nwhich is not a property of the exact DFT functional38{40.\nAccording to this condition m(r;t)\u0002Bxc(r;t) = 0 and\ntherefore the exchange-correlation magnetic \feld cannot\ncontribute, even locally, to the magnetization dynamics.\nThis leads us to conclude that the local magnetization3\ndynamics is solely the result of the interplay between\nthe spin-polarized currents and the SO torque (in reality\nBxccan still contribute indirectly to the spin dynamics\nthrough a dynamical modi\fcation of the gap between up\nand down spin polarized bands, which in turn determines\nan enhancement of the spin dissipation via the spin orbit\ncoupling channel). In order to elucidate this view further\nwe make use of the hydrodynamical formalism applied\nto spin systems, which has been already introduced in\nReferences [24,28]. This approach needs to be slightly\nmodi\fed in view of the fact that we are considering an\ne\u000bective Kohn-Sham system and not a set of independent\nspin particles. In fact, as it was already pointed out in\nRefs. [19,20], Eq. (8) can be written in a di\u000berent form\n(the details of the derivation are shown in the appendix\nA)\nD\nDtm(r;t) +X\nj2occ:r\u0001vj(r;t)mj(r;t) =\u0000r\u0001D (r;t)+\n+\u0016Bm(r;t)\u0002Be\u000b(r;t) +TSO(r;t);(12)\nwhere a couple of new terms appear. In the equation\nD\nDt=d\ndt+v\u0001ris a material derivative, vj(r;t) repre-\nsents a single Kohn-Sham state velocity \feld (see ap-\npendix A), and mj(r;t) = KSy\nj^\u001b KS\nj. On the right\nhand-side of Eq. (12) in addition to the spin-orbit cou-\npling torque, TSO(r;t), we have a new term, \u0000r\u0001D (r;t),\nthat describes the spin dissipation in the system due to\nthe internal motion of the spin currents. It can be in-\nterpreted as an e\u000bective spin-current divergence object\ninvolving only transitions among di\u000berent Kohn-Sham\nstates [inter-band transitions, see Eq. (A16)]. Finally\nthe e\u000bective \feld Be\u000bis given by the sum of two terms,\nBe\u000b=Bxc+Bkin, with Bxcexchange-correlation \feld\nandBkinde\fned as [see Eq. (A25)]\nBkin(r;t) =1\n\u0016Fe\u0014rn\u0001rs\nn+r2s\u0015\n; (13)\nwith spin vector \feld s(r;t) =m(r;t)\nn(r;t).\nSuch Bkin(r;t) \feld has only an instrumental r^ ole in\nthe equations of motion for the spin density, a very sim-\nilar expression was already introduced in some previ-\nous work. In Ref. [19] it was expressed in the form\n@k1\nn(m\u0002@km), while in Ref. [20] it appears asrnrm\nn.\nThe interpretation of Bkinmay look quite obscure at a\n\frst sight, however, in Ref. [25,26] it was identi\fed as a\npossible source of spin wave excitations in the form of a\nspin-spin interaction potential.\nIn order to clarify this point, let us consider the Heisen-\nberg interaction between two spins centered on atoms\nplaced at a distance d=jdj. We can assume naively, but\nreasonably, that the spin-spin interaction among the two\nspin distributions, computed at an arbitrary point rin\nspace could be expressed in the following form\nHe\u000b(r)'s(r\u0000d=2)\u0001s(r+d=2); (14)\nwhere it is more convenient for us to employ a spin \feld,\ns(r), which describes the spin distribution in space, in-\nstead of an atom localized spin vector. Hence, He\u000bde-\n\fnes an e\u000bective single-particle Hamiltonian. By averag-\ning over the number of electrons in the entire space weobtain\nS1\u0001S2'Z\n\nd3rn(r)s(r\u0000d=2)\u0001s(r+d=2):(15)\nThen, by expanding the spin density in Taylor series up\nto second order in the distance dand by neglecting the\nzeroth-order contribution (we focus our attention on the\nnon-local term appearing in the expansion) after some\nstraightforward rearrangement we arrive at\nS1\u0001S2'\u0000d2\n4Z\n\nd3rn(r)rs(r)\u0001rs(r); (16)\nwhich in turn becomes\nS1\u0001S2'd2\n4Z\n\nd3r\u0014\n\u0000r\u0001\u0000\nn(r)s(r)\u0001rs(r)\u0001\n+\n+m(r)\u0001\u0010rn(r)\u0001rs(r)\nn(r)+r2s(r)\u0011\u0015\n:(17)\nFinally, by considering a su\u000eciently large integration vol-\nume, the use of the divergence theorem allows to neglect\nall the boundary terms with consequent \fnal expression\nS1\u0001S2'd2\n4Z\n\nd3rm(r)\u0001hrn(r)\u0001rs(r)\nn(r)+r2s(r)i\n;(18)\nwhich remarkable resembles the result in Eq. (13) for the\nkinetic magnetic \feld. We can therefore tentatively in-\nterpret Bkin(r;t) as an e\u000bective mean-\feld internal mag-\nnetic \feld, which plays a r^ ole in coupling the spins at dif-\nferent locations in the system in the spirit of the Heisen-\nberg spin-spin interaction.\nIII. ANALYZING SPIN DYNAMICS FROM\nTDSDFT SIMULATIONS IN Fe 6CLUSTER\nHere we present the results of TDSDFT calculations,\nperformed with the Octopus code31, where we simulate\nthe ultrafast demagnetization process in iron-based fer-\nromagnetic systems. In all those, at time t= 0 the sys-\ntem is in its ground state. Then we apply an intense\nelectric \feld pulse with a duration of less than 10 fs,\nwhich initiates the dynamics. The pseudo-potentials for\nFe used in the calculations are fully relativistic, norm-\nconserving and are generated using a Multi-Reference-\nPseudo-Potential (MRPP) scheme32at the level imple-\nmented in APE33,34, which takes directly into account the\nsemi-core states. For the XC functional we employ the\nALSDA with parameterization from Perdew and Wang35.\nOur simulations then consist in evolving in time the KS\nwave functions, i.e. in solving numerically the set of equa-\ntions (1). The results are then interpreted through the\nmagnetization continuity equation (12).\nIn Fig. 1 the extracted magnetization dynamics of\na Fe 6magnetic cluster is presented. We use the\nLDA ground-state geometry of Fe 6as extracted from\nRef. [36,37] for which we reproduce the reported therein\nspin stateS= 20 ~=2. The nuclei are kept stationary\nduring the dynamics. In panel (c) we observe that the\ntotal loss of the zcomponent of the total magnetiza-\ntion,Stot\nz(t), is exactly equal to the variation in value of4\nFIG. 1: (Color online) (a) Typical electric \feld pulse used\nto excite the Fe 6cluster with the black arrow indicating the\ndirection of the \feld. The \ruence of this pulse is 580 mJ =cm2.\n(b) Time evolution of the z-component of BkinandBx(ex-\nchange component of the \feld), with respect to their values\natt= 0 integrated over the system volume, \u0016BBtot(t) =\n\u0016BP\nIBI(t). (c) Time evolution of the variation of the total\nmagnetization \u0001 Stot\nz(t) =P\nI\u0001SI\nz(t) with respect to its ini-\ntial value. (d) Time evolution on atomic site 6 of the magne-\ntization variation along zand of the electron density variation\nwith respect to its value at t= 0 integrated inside a sphere\nof radiusR= 0:9\u0017A.\nits module,jStotj, since the global non-collinear contri-\nbution is negligible. This indicates that the spin is not\nexchanged globally between the di\u000berent components of\nthe magnetization vector, but, according to Eq. (11), it\nis, at least, partially transfered into the orbital momen-\ntum of the system. We note that due to the electrostatic\ninteractions with the nuclei and due to the interaction\nwith the laser \feld the rotational invariance of the elec-\ntronic system is broken and the total orbital momentum\nis not conserved.\nIn Fig. 1(b) we observe that the average kinetic mag-\nnetic \feld (over the entire simulation box, for \u0016F= 1)\nis comparable in magnitude to the exchange component.\nAt the same time, Btot\nkin;zshows a much more oscillatory\nbehaviour compared to Btot\nx;z. In particular, While Btot\nx;z\nevolves smoothly in time following the action of the opti-\ncal excitation, Btot\nkin;zpresents an abrupt variation at the\non-set of the electrical pulse. This is due to the fact that\nthe laser pulse directly excites currents, through the term\n\u0000r\u0001D (r;t), which, in turn, produces a modi\fcation of\nthe gradients of the charge/spin density, even on a global\nscale since they are not conserved. Thus we observe huge\nvariations of Btot\nkin;z.Bx;zcan also oscillate very strongly\nlocally, following the temporal variation of the densities,\nbut when we measure Btot\nx;zthese oscillations are averaged\nout given that the densities are approximately conserved\nover the entire simulation box. During the action of the\npulse we see a tendency of the two \felds to compensate\neach other, an e\u000bect strongly resembling the Lenz law.\nAfter the pulse, Bkincontinues to oscillate dramatically\nwith its average value that slowly increases. In contrast\nBtot\nx;zdecreases (in absolute value) due the net dissipation\nof spin angular momentum.Moving from an analysis of global quantities to prob-\ning locally the spin dynamics, in Fig. 1(d) we compare\nthe magnetization and the electron density around the\natomic site 6 at the tip of the cluster (see inset of Fig. 1(a)\nfor the numbering labels of all the cluster atoms). We\nde\fne local magnetization and charge associated to the\nparticular atomic site I as\nSI(t) =Z\nSI\nRd3rm(r;t); QI(t) =Z\nSI\nRd3rn(r;t);(19)\nwhere the integration volume SI\nRis a sphere of radius\nR centered at site I. Our results show that the loss of\nS6\nzis not taking place just during the action of the ex-\nternal pulse, but it is rather distributed over the entire\ntime evolution. This suggests that the spin-sink mecha-\nnism is not directly related to the coupling of the system\nto the laser \feld, but is rather intrinsic to the electron\ndynamics following the pulse. Furthermore, close to the\natomic site, the temporal variation of the charge, Q6, is\nmuch smaller in magnitude and smoother than that of\nS6\nz. In addition for long times Q6settles close to an av-\nerage value, while S6\nzcontinues to decrease. Hence the\nlong-term spin dynamics is not the result of a net charge\ndisplacement from the region close to the ions to the in-\nterstitial space. These observations are valid for all the\natomic sites in the cluster.\nIf we now consider the continuity equation for the elec-\ntron density (see the Appendix A for further explana-\ntions)\nD\nDtn(r;t) =\u0000n(r;t)r\u0001v(r;t); (20)\nwhereD\nDtn(r;t) is the material derivative of the electron\ndensity\nD\nDtn(r;t) =\u0010d\ndt+v\u0001r\u0011\nn(r;t): (21)\nFrom Fig. 1(d) we observe that during the action of the\npulse the density variation in the vicinity of the atoms\nappears to be very small compared to the magnetization\nvariation. We can therefore safely assume that in this\nspatial region _ n(r;t)'0, with at the same time n(r;t)6=\n0. From these considerations we deduce that v(r;t)'0\nis a reasonably good approximation for the velocity \feld\nin the vicinity of the atoms (this does not imply that the\nvelocity \feld is exactly zero, but only that its e\u000bect on\nthe spin dynamics in this particular case is negligible).\nThe same argument is valid also for the state resolved\ndensitynj(r;t), given that _ n(r;t) =P\nj2occ:_nj(r;t), the\ncontribution of the local time derivative of the Kohn-\nSham state density can be neglected. By applying the\nlatter into Eq. (12) we \fnally obtain a relation that could\nbe considered approximately valid in this spatial region\nof the simulation box,\nd\ndtm(r;t)'\u0000r\u0001D +\u0016Bm\u0002Bkin+TSO; (22)\nwhere the contribution to the spin dynamics due to the\nvelocity \feld term has been neglected. Note that here we\nhave also used the condition m(r;t)\u0002Bxc(r;t) = 0, that5\n0 10 20 30\nt [ fs ]1215Sz [ h_ /2 ]Site 1\nSite 6\n0 10 20 30\nt [ fs ]-11-10-9-8-7\nµΒBz[ eV ](a) (b)\nFIG. 2: (Color online) Local spin dynamics of the Fe 6clus-\nter: (a) Time evolution of the magnetization SI\nz(t) around\nthe atomic centers; (b) time evolution of the zcomponent of\nBI\nkin(t). All the quantities are integrated inside a sphere of\nradiusR= 0:9\u0017A centered on the two atomic sites, where we\nhave usedBI\nz(t) =R\nSI\nRd3rBz(r;t).\nis consequential to the local density approximation. In\naddition, the decay of Bxc, during the evolution, is not\nso relevant to justify a dynamical modi\fcation of the gap\nbetween up and down spin states.\nIn Fig. 2 we compare the behaviour of the kinetic \feld\nand of the local magnetization at two atomic sites, re-\nspectively 1 (one of the atoms in the base plane of the\nbi-pyramid) and 6 (an atom at one of the apexes). It can\nbe seen from panel (a) that these two sites present di\u000ber-\nent rates of demagnetization. In particular, at site 6 the\nspin decay is considerably more prominent with respect\nto that observed at site 1. In contrast the \ructuations\ninSI\nzare signi\fcantly more pronounced for site 1 than\nfor site 6. This can be understood from the fact that\nwe have chosen here an electric pulse with polarization\nvector in the basal plane of the bi-pyramid. As such,\nthe charge \ructuations for the atoms in the basal plane\nare expected to be much larger than those of the api-\ncal atoms. Finally, we note that BI\nkin;z(t) follows similar\nqualitative trends as SI\nz(t) [see Fig. 2(b)]. In fact, the av-\nerage change following the excitation pulse is larger for\nsite 6 (the one experiencing the larger demagnetization),\nbut the \ructuations are more pronounced for site 1 (the\none experiencing the larger \ructuations in SI\nz(t)).\nThe correlation between the kinetic \feld and the mag-\nnetization loss is also rather evident in Fig. 3. There the\ntime-averaged variations in the x-component of the two\n\felds m\u0002Bkinand_s(r;t) are clearly comparable in mag-\nnitude and localized over the same regions of the simula-\ntion box. This demonstrates that the kinetic \feld can be\nconsidered as the main force driving the non-collinearity\nduring the spin evolution. The fact that the contrast\nis stronger at the apex atoms (\\hot spots\" for demag-\nnetization) agrees with Fig. 2(a), while the dipole-type\npatterns indicate how the longitudinal spin decays pre-\nserving global collinearity. The correlation between the\nzcomponents of m\u0002Bkinand _s(r;t) is not as evident\nas that for the transverse component x. This is due tothe fact that the xandycomponents of the \feld are\nmuch smaller compared to the zone. Furthermore, the\ncontribution to the spin dynamics along zof the SO cou-\npling, together with the internal dissipative term due to\nthe spin currents, cannot be neglected.\nFIG. 3: (Color online) Contour plots of the time- and space-\naveraged (in direction perpendicular to the plane spanned by\natoms 1, 3, 5 and 6, as indicated on the plot) observables\nevaluated only within spheres of radius R= 1:0\u0017A around\neach atom: (a) and (b) the temporal variation of the spin\ndensity \u0001 s(x;z)(r;t)=\u0001tfor \u0001t= 0:1fs; (c) and (d) the x\nandzcomponents of the second term on the right-hand side\nof Eq. (22).\nIn order to quantify the local non-collinearity we exam-\nine the evolution of the misalignment angle, \u0012, between\nthez-axis and the direction of the magnetic \felds (av-\neraged over spheres). It can be seen in Fig. 4 that at\nsite 1 the averaged kinetic \feld and the local spin de\rect\nvery little from the quantization axis and remain rather\nparallel to each other. The angle that B1\nkin(t) forms with\nthe magnetization direction (the m1direction) is sub-\nstantially negligible. Instead, at site 6, B6\nkin(t) shows a\nsigni\fcant de\rection from the z-axis after the \frst 5 fs\nof the evolution and so does the spin, without the two\nbeing parallel to each other. It is important to notice\nthat the angle between magnetization mandBkinstarts\nto grow only after the action of the pulse. These results\nfor atom 1 and 6 are representative for all the other sites\nin the base plane or outside of it, respectively. The sites\nlocated in the plane, where BI\nkinis mostly collinear, lose\nless magnetization with respect to the ones at the apices\nwhere, instead, the kinetic \feld shows a signi\fcant de\rec-\ntion from the magnetization axis and provides additional\ntorque driving further demagnetization.\nAnalogous conclusions arise from the introduction of\nthe concept of parallel transport, commonly used in dif-\nferential geometry. This requires a proper de\fnition of\nthe covariant derivative obtained by comparing s(r+dr)\nnot with s(r), but with the value that the spin vector\nwould have if it was translated from rtor+drwhile6\nkeeping the axes in the isospin space \fxed,\nDis(r;t) =dis(r;t) +Ai(r;t)\u0002s(r;t): (23)\nThe connection \feld A(r;t) provides a measure of the\namount of non collinearity accumulated in the translation\nof the spin vector from rtor+dr\n-0.0100.01 ∆θ [ deg ]Bkin \nS\n036 S\nB kin \n0 10 20 30\nt [ fs ]00.30.6 | A | [ 1 / Å ]| A  xy  |\nA z \n0 10 20 30\nt [ fs ]024| Axy  |\n| A z  |(a) (b) (d) (c)\nFIG. 4: (Color online) Evolution of the spin non-collinearity\nin the Fe 6cluster. (a) \u0001 \u0012=\u0012(t)\u0000\u0012(0) at site 1, for the\nBxc[orS(t)] direction (black curve) and the Bkindirection\n(red dashed curve). (b) The same quantities of panel (a)\nbut calculated at the atomic site 6. (c)p\u0016A2\n1+\u0016A2\n2where\n\u0016A=qP3\ni=1A2\niandAiis introduced in Eq. (23), compared\nto\u0016A3at the atomic site 1. (d) The same quantities of panel (c)\nbut calculated at the atomic site 6. The \felds are measured\nwithin a sphere of radius R= 0:8\u0017A centered on the atom\ncenter.\nBy using the previous expression to rewrite the \frst\nand second order spatial derivatives, the kinetic \feld of\nEq. (13) can be divided in two components\nBkin(r;t) =B0\nkin(r;t) +\u000eBkin(r;t): (24)\nHere we have introduced\nB0\nkin(r;t) =1\n\u0016Fehrn(r;t)\nn(r;t)\u0001Ds(r;t) +D2s(r;t)i\n;(25)\nwhich has no e\u000bects on the dynamics, having locally the\nsame direction of the spin vector by construction, and\n\u000eBkin(r;t) =1\n\u0016Fe3X\ni=1hdin\nn\u0000\n\u0000s2Ai+ (s\u0001Ai)s\u0001\n\u0000\n\u00002\u0000\ns2diAi\u0000s(s\u0001diAi)\u0001\n\u00002\u0000\nAi(s\u0001Dis)\u0000\n\u0000Dis(s\u0001Ai)\u0001\n+ 4(s\u0002Ai)(s\u0001Ai)i\n:(26)\nAccording to Fig. 4, in the case of Fe 6, the direction in\nthe isospin space of the connection tensor Aifor every\nicomponent can be considered in \frst approximation\northogonal to the direction of the spin vector s(r;t), since\nits component along zis considerably smaller than thecomponents along xandy. From this we could assume s\u0001\nAi'0 and we obtain the following simpli\fed expression\nfor\u000eBkin(r;t)\n\u000eBkin(r;t) =1\n\u0016Fe3X\ni=1\u0014\u0010\n\u0000din\nns2\u00002s\u0001Dis\u0011\n\u0001Ai\u00002s2diAi\u0015\n:\n(27)\nThis represents the part of Bkinwhich gives rise to a\nnon-zero torque in Eq. (22).\nIV. DIRECTIONALITY OF THE\nDEMAGNETIZATION IN Fe 6CLUSTER\nIn the previous sections we have revisited the concept\nof kinetic \feld, its derivation within DFT and its prop-\nerties as a major source of torque for the spin dynamics\nwithin ALSDA. We have provided supporting evidence\nfor the latter from TDSDFT calculations of the ultrafast\ndemagnetizing Fe 6cluster under the e\u000bect of a single\nfs electric \feld pulse. Despite the conceptual clarity of\nBkinas an instrumental object, very little useful physi-\ncal intuition can be drawn from its de\fnition in Eq. (13).\nClearly, it is an intrinsic dynamic \feld that depends on\nthe spin texture and its response to the external stimuli.\nIt also feeds back into the dynamics of this same spin den-\nsity, clearly a non-linear process. In this section we seek\nto extend the evidential base for the connection between\nthe torque due to Bkinand the rate of demagnetization.\nTogether with that we report a situation, where the di-\nrection of the polarization vector of the electric \feld of\nthe laser pulse alone has a signi\fcant e\u000bect on the de-\nmagnetization of a material (the Fe 6cluster).\nFIG. 5: (Color online) Global (a) spin and (b) energy varia-\ntion in Fe 6for two di\u000berent excitations di\u000bering only by the\ndirection of polarization of the electric \feld pulse. Cartoon of\nthe cluster with labels of the relevant atoms and a reference\nframe are depicted as insets. The contour plots represent the\ndistribution of the angle (c, e) between mandBkinin a plane\nthrough atoms 1, 3, 5 and 6 (as in Fig. 3) and the perpendicu-\nlar component of mwith respect to Bkin(d, f) averaged over\nthe time of the simulation, for the two di\u000berent excitations\nEjjxandEjjz, respectively the top and bottom panels.\nFigure 5 shows a comparison between two simulations\ndi\u000bering only by the direction (but notably not the mag-\nnitude) of the electric \feld applied. In one case this is in\nthex-direction, which is oriented along the slightly longer7\nside of the base of the bi-pyramid36, and in the other\nsimulation it is along z, the direction connecting the two\napex atoms 5 and 6. The case Ejjxshows nearly 3 times\nfaster demagnetization compared to the Ejjzone. Evi-\ndently, in the former situation more energy is absorbed\nby the cluster (an excess of about 8 :6 eV), we show a\ncomparison of the total energy shift due to the pulse in\nFig. 5(b).\nFrom panel (c) we observe that the amount of non-\ncollinearity enclosed in a relatively small radius around\nthe apex atoms does not signi\fcatively change in the two\ncases, suggesting that the intra-site non-collinear compo-\nnent of the spin vector is already present in the ground-\nstate con\fguration. What really di\u000bers is the amount\nof inter-sites non collinearity concentrated in the out-of-\nplane region. In the Ejjxcase, if we examine the angle\nbetween Bkin(r;t) and m(r;t) averaged in time over the\nentire simulation along one particular cross-section plane\n(vertical through the base diagonal of the cluster as de-\npicted in the inset), one can observe, in particular in\nthe out-of-plane interstitial regions, a signi\fcant larger\namount of non-collinearity, of the spin vector density, ac-\ncumulating in the case of faster demagnetization.\nNotably, there is a change in the symmetry between\npanels (c) and (e) in Fig. 5 { in both cases no signi\fcant\nspin non-collinearity arises in the plane parallel to the\n\feld connecting the atomic centers. It is, however, im-\nportant to notice that the amount of intra-site spin non-\ncollinearity for the in-plane atoms is strongly dependent\non the polarization direction of the applied laser \feld.\nIn fact, in the case of Ejjxthe temporal averaged angle\nbetween sandBkinappears much higher with respect\nto the one computed in the Ejjzcase. This quantity is\nmostly averaged out by the integration procedure but it\nis clearly visible in the panel (c) of the \fgure.\nAlthough Bkinis not the only torque generator and\nthe SO contribution is signi\fcant too, the former plays\na r^ ole in de\recting the spins in the system in a manner\nthat correlates with the rate of global demagnetization.\nImportantly, our simulations clearly show that the de-\nmagnetization process is very anisotropic and particular\ndirections of the exciting electric \feld may enhance the\nrate of demagnetization (the study of these e\u000bects is be-\nyond the scope of this paper and it will be explored in\nmore details in a coming publication).\nV. DEMAGNETIZATION OF bcc Fe\nFinally, we present results of analogous simulations in\nbulk materials, namely in bcc Fe, with the aim of demon-\nstrating the qualitative universal r^ ole played by Bkinin\nthe ultrafast demagnetization process. We consider bcc\nFe in its ferromagnetic phase with total spin in the unit\ncellS= 14:97~=2 and with 2 atoms in it.\nWe employ a lattice parameter a= 2:9\u0017A, with a\n4\u00024\u00024k-points grid. In Fig. 6(c) we show the de-\nmagnetization rate of the single unit cell after it has\nbeen excited with the electric \feld pulse [panel (a)]. The\ngreen curve represents the demagnetization computed in-\nside the unit cell and resembles in shape the sum of the\nFIG. 6: (Color online) Demagnetization of bcc Fe: (a) ap-\nplied external electric \feld, (b) local value of \u0001 BI\nkin;zand\n\u0001BI\nx;zaround atom 1; (c) comparison between the value of\nthe local magnetization \u0001 Sz;I(t) around atom 1 (black curve),\nthe total magnetization integrated around the two sites (red\ncurve), \u0001Sz;I(t), and the total magnetization integrated in-\nside the unit cell \u0001 Sz(t) (green curve); (d) local value ofP\nI\u0001BI\nkin;xy(t) (red curve) and of \u0001 SI\nxy(t), non collinear mag-\nnetization component for atom 1. All the local quantities are\ncalculated inside a sphere of radius R= 0:8\u0017A centered on site\n1.\nmagnetization variation,P\nI\u0001Sz;I(t), calculated in the\nvicinity of the two Fe atoms, even if it is di\u000berent in\nmagnitude. This suggests that a large amount of spin is\ndriven outside from the atomic integration region during\nthe evolution. Similarly to the cluster, the dynamics of\nthe onsite magnetization can be described in terms of a\ntwo-step process with an initial fast decay during the ac-\ntion of the external pulse, followed by a slower and noisy\ndecrease in magnitude. The \frst fast decay may be at-\ntributed to the e\u000bect of the SO enhanced by the collapse\nof the e\u000bective \feld Be\u000bfollowing the action of the laser\npulse. In Fig. 6(b) the collapse after the \frst 5 fsof\nthezcomponent of the e\u000bective \feld is quite clear, even\nif it appears to be more pronounced for the kinetic \feld\nBI\nkin;z(t) with respect to the exchange \feld BI\nx;z(t). Sim-\nilarly to the case of Fe 6the role played by r\u0001D (r;t) is\ndominant during the action of the pulse, but after this\ninitial phase the dynamics is dominated by intra-band\ntransitions and the interplay between the spin-orbit cou-\npling and the e\u000bective \feld Be\u000bbecomes dominant.\nFig. 6(d) shows the evolution of the non-collinearity\nof the spin vector,P\nISI\nxy(t), and of the kinetic \feld,P\nIBI\nkin;xy(t). The level of correlation among the two\nquantities con\frms the importance of the kinetic \feld in\nthe evolution of the spin non-collinearity. The long tail of\nspin dissipation may be explained in terms of intra-band\nspin-up/spin-down transitions through an Elliott-Yafet\ntype of mechanism triggered by the scattering with the\ne\u000bective \feld Be\u000b,\nAi!f=h\tn;k1j^\u001b\u0001Be\u000bj\tn;k2i: (28)\nAi!frepresents the transition amplitude between two\nstates with di\u000berent kvector and in presence of SO with\ndi\u000berent mixing of up and down spin components.8\nVI. CONCLUSIONS\nIn conclusion, we remark the central result of our work,\nnamely that the equation of motion for the spin dynam-\nics within the ALSDA of TDSDFT [see Eq. (8)] can be\nrewritten in the form of Eq. (12), by using a formalism\nborrowed from magneto-hydrodynamics. Subsequently\nwe have analyzed the properties of the so-called kinetic\nmagnetic \feld Bkinand its r^ ole in the ultrafast demag-\nnetization process in two di\u000berent systems: a ferromag-\nnetic Fe 6cluster and bulk bcc Fe. The r^ ole of this \feld\nis particularly signi\fcant for processes far from equilib-\nrium, such as the ultrafast demagnetization observed in\ntransition metals.\nIn both the systems studied the spin dynamics is the\nresult of the interplay between the SO coupling and\nBkin(r;t), which, in general, is strongly coupled to the\nexternal pulse and highly non-uniform in space. We have\nshown that the spin loss locally correlates with Bkin(r;t).\nThrough the concept of parallel transport and the def-\ninition of a connection tensor \feld Ai, we have gained\nfurther insight into the evolution of the spin texture. As\nAidescribes the degree of spin rotation per in\fnitesimal\nspatial translation, it also provides a measure for the mis-\nalignment between the kinetic \feld and the spin texture.\nThe regions with higher kAkcorrespond to stronger local\ndemagnetization.\nFinally, the e\u000bect of the direction of the polarization\nvector of the electric \feld pulse has been studied for Fe 6.\nWe have found that clusters will demagnetize about twice\nas fast, if the polarization vector is in the base plane and\nnot vertical (through the apex atoms). Our analysis has\nshown a signi\fcant increase in the non-collinearity be-\ntween Bkin(r;t) and the spin density in the fast demag-\nnetizing case. Such anisotropy, due to the electric dipole\nmatrix elements for the valence electrons, is likely to oc-\ncur in crystalline systems as well. During the application\nof the laser pulse, the rise of spin non-collinearity may\nbe enhanced by the particular polarization direction of\nthe laser pulse through the spin orbit coupling and this\ne\u000bect combined with the collapse of the kinetic \feld may\nexplain the initial spin loss. However, in both Fe 6and Fe\nbcc the magnetization loss is more prominent after that\nthe laser pulse has been set to zero. During this second\nphase of spin dissipation we need to distinguish between\nthe spin decay observed in bcc Fe due to intra-band tran-\nsitions among states with di\u000berent spin up/down mixing\nand the spin dynamics observed in correspondence of the\napex atoms in Fe 6that is, instead, driven by Bkinand di-\nrectly related to the onsite intrinsic spin non-collinearity\nnear the atomic sites.Acknowledgments\nThis work has been funded by the European Commis-\nsion project CRONOS (grant no. 280879) and by Science\nFoundation Ireland (grant No. 14/IA/2624). We grate-\nfully acknowledge the DJEI/DES/SFI/HEA Irish Cen-\ntre for High-End Computing (ICHEC) for the provision\nof computational facilities. We also thank Prof. E.K.U.\nGross for valuable discussions.\nAppendix A: Derivation of the hydrodynamic\ncontinuity equation\nIn this appendix we show in some detail how Eq. (12)\ncan be obtained by starting from the standard TDSDFT\ncontinuity equation (8). Here we follow the hydrody-\nnamical formalism of quantum mechanics, where a sin-\ngle particle with spin is considered equivalent to a non-\nlinear vector \feld. In this type of hydrodynamics the\nquantum e\u000bects are separated as non-linear terms and\nare described through e\u000bective quantum potentials (see\nRef. [25]).\nThe formalism is based on the assumption that it is\npossible to describe the dynamical evolution of a sin-\ngle particle immersed in an external vector potential,\nA(r;t), through the so-called Madelung decomposition\n(see Ref. [27]) of the system wave function, in which the\namplitude is translated into the probability density and\nthe gradient of the phase determines the velocity \feld.\nA hydrodynamical description of the wave function was\nalso obtained in Ref. [30] starting from the ordinary inter-\npretation of quantum mechanics and by introducing an\noperator for the charge density and the current density.\nThe formalism was also later extended to the semi-\nrelativistic description (Pauli approximation) of a single\nparticle in an external electro-magnetic \feld in Ref. [24].\nHowever, while in all the previous studies the main ob-\njective was to derive the single-particle dynamics of the\nspin 1=2 plasma, only recently the study of the collective\ndynamical properties of the quantum plasma started to\nattract some interests (see Ref. [28] and [29]).\nIn deriving the Eq. (12) for the spin density in the\nKohn-Sham system we introduce also of the electron den-\nsity,n(r;t), and the velocity \feld, v(r;t). The equation\nof motion for the velocity \feld are not explicitly written\nhere for the reasons explained in section III.\nThe electron density is written in terms of the Kohn-\nSham wave functions  KS\nj(r;t) as\nn(r;t) =X\nj2occ: KS\nj(r;t)y KS\nj(r;t); (A1)\nwhile the spin density is\ns(r;t) =P\nj2occ: KS\nj(r;t)y\u001b KS\nj(r;t)\nn(r;t); (A2)9\nand the covariant velocity \feld appears as\nv(r;t) =~\n2mi\u0001P\nj2occ:( KS\nj(r;t)yr KS\nj\u0000 KS\njr KS\nj(r;t)y)\nn(r;t)\u0000e\nmcA(r;t): (A3)\nBy making use of the Kohn-Sham equations (1) the charge continuity equation can be written straightforwardly as\nd\ndtn(r;t) =\u0000~\n2mir\u0001X\nj2occ:\u0002\n KS\nj(r;t)y(\u0000 !r\u0000 \u0000r) KS\nj(r;t)\u0003\n+e\nmcr\u0001\u0002\nnA(r;t)\u0003\n; (A4)\nwhile for spin it is written in terms of n(r;t) and s(r;t) as\nd\ndt(ns) =\u0000~2\n4mir\u0001X\nj2occ:\u0002\n KS\nj(r;t)y^\u001br KS\nj\u0000r KS\nj(r;t)y^\u001b KS\nj\u0003\n+e\nmcX\nj@j\u0002\nAjns\u0003\n+\u0016Bn(s\u0002Bxc) +TSO:(A5)\nBy evaluating explicitly the spatial partial derivative of the spin vector and by multiplying it with the component si\nwe obtain the following equality\nnsi@lsk=siX\nj2occ:\u0002\n@l KS\nj(r;t)y^\u001bk KS\nj+ KS\nj(r;t)y^\u001bk@l KS\nj\u0003\n\u0000@lnsi\u0001sk: (A6)\nWe now need to focus our attention on the \frst term on the right-hand-side of Eq. (A6), by using the following\nnotation\nFik(r;t) =siX\nj2occ:\u0002\nr KS\nj(r;t)y^\u001bk KS\nj+ KS\nj(r;t)y^\u001bkr KS\nj\u0003\n; (A7)\nthat leads to\nFik(r;t) =1\nnX\nj;r2occ:\u0002\n KSy\nr^\u001bi KS\nrr KSy\nj^\u001bk KS\nj+ KSy\nr^\u001bi KS\nr KSy\nj^\u001bkr KS\nj\u0003\n: (A8)\nThe anti-symmetric part, Kik(r;t), of the tensorFik(r;t), de\fned asKik=Fik\u0000(i$k) may be written as\nKik(r;t) =1\nnX\nj;r2occ:X\n\u000b;\f;\u000b0;\f0\u0002\n KS\u0003\nr;\u000b KS\nr;\f\u001b[i;\n\u000b;\f\u001bk]\n\u000b0;\f0r KS\u0003\nj;\u000b0 KS\nj;\f0+ KS\u0003\nr;\u000b KS\nr;\f\u001b[i;\n\u000b;\f\u001bk]\n\u000b0;\f0 KS\u0003\nj;\u000b0r KS\nj;\f0\u0003\n; (A9)\nBy making use of the following relation between Pauli matrices [see Ref. 25]\n\u001b[i;\n\u000b;\f\u001bk]\n\u000b0;\f0=iX\ns\u000fiks[\u001bs\n\u000b;\f0\u000e\u000b0;\f\u0000\u000e\u000b;\f0\u001bs\n\u000b0;\f]; (A10)\nwe obtain the following \fnal expression for Kikthat can be splitted in two parts as follows\nKik(r;t) =X\nj;r2occ:K(j;r)\nik(r;t)\u000ej;r+X\nj2occ:X\nr6=j2occ:K(j;r)\nik(r;t); (A11)\nwhere we have introduced the tensor\nK(j;r)\nik(r;t) =i\nnX\ns\u000fiksh\n KSy\nr KS\nj\u0000\n KSy\nj^\u001bsr KS\nr\u0000r KSy\nj^\u001bs KS\nr\u0001\n+ KSy\nr^\u001bs KS\nj\u0000\nr KSy\nj KS\nr\u0000 KSy\njr KS\nr\u0001i\n:(A12)\nThe procedure that we have followed up to now is formally exact. Then, in order to simplify the previous expression we\nsubstitute the Kohn-Sham ratio Fj= KSy\nj KS\nj\nn(r;t)with its average over the various occupied states Fj'\u0016F=h KSy\nj KS\njij\nn(r;t).\nFrom the fact that, to a good degree of approximation, \u0016F'1\nNwithNtotal number of particles in the system, we\nwill consider from now on \u0016Fto be spatially homogeneous and constant in time. Then Eq. (A11) becomes\nKik(r;t) =i\u0016FX\ns\u000fiksX\nj2occ:\u0002\n KSy\nj^\u001bsr KS\nj\u0000r KSy\nj^\u001bs KS\nj\u0003\n+2m\u0016F\n~X\ns\u000fiksX\nj2occ: KSy\nj^\u001bs KS\njh\nvj(r;t) +e\nmcA(r;t)i\n+\n+X\nj2occ:X\nr6=j2occ:K(j;r)\nik(r;t): (A13)10\nIn order to simplify the formalism we employ the notation Kik;l(r;t) =n(si@lsk\u0000sk@lsi). Immediately from Eq. (A13)\nfollows that\n~\n2\u0016Fmn\u0000\nsi@lsk\u0000sk@lsi\u0001\n=\u0000X\ns\u000fiksJsl\nKS(r;t)+X\ns\u000fiksX\nj2occ:ms\nj(r;t)h\nvl\nj(r;t)+e\nmcAl(r;t)i\n+~\n2\u0016FmX\nj2occ:\nr6=j2occ:K(j;r)\nik;l(r;t);\n(A14)\nwhere mjandvjde\fne, respectively, the single Kohn-Sham state magnetization and velocity \feld. By employing the\nproperties of the Levi-Civita tensor we have\n~\n2\u0016Fm\u0000\nns\u0002@ls\u0001n=\u0000Jnl\nKS(r;t) +X\nj2occ:mn\nj(r;t)h\nvl\nj(r;t) +e\nmcAl(r;t)i\n+Dnl(r;t); (A15)\nwhere we have introduced the new tensor quantity\nD(r;t) =\u0000X\nj2occ:X\nr6=j2occ:\u0014\nFrjJ(j;r)(r;t)\u0000Fjrm(r;j)(r;t)\n\u0012\nv(j;r)(r;t) +e\nmcA(r;t)\u0013\u0015\n: (A16)\nHereFrj= KSy\nr KS\nj\nn(r;t)and the other many-particle objects are de\fned as\nJ(j;r)(r;t) =\u0000i~\n2m\u0002\n KSy\nj^\u001br KS\nr\u0000r KSy\nj^\u001b KS\nr\u0003\n; (A17)\nv(j;r)(r;t) =~\n2mi KSy\njr KS\nr\u0000r KSy\nj KS\nr\n KSy\nj KSr\u0000e\nmcA(r;t); (A18)\nm(j;r)(r;t) = KSy\nj^\u001b KS\nr: (A19)\nFinally, from Eq. (A15) the divergence of the spin current tensor may be rewritten as\n\u0000r\u0001JKS(r;t) =~\n2\u0016Fmr\u0001\u0000\nns\u0002rs\u0001\n\u0000X\nj2occ:X\nl@l\u0002\nmj(r;t)\u0001vl\nj(r;t)\u0003\n\u0000e\nmcX\nl@l\u0002\nnsAl(r;t)\u0003\n\u0000r\u0001D (r;t):(A20)\nThen, by substituting Eq. (A20) into Eq. (A5) we obtain\nd\ndtm(r;t) =\u0000r\u0001D (r;t)\u0000X\nj2occ:X\nl@l\u0002\nmj(r;t)\u0001vl\nj(r;t)\u0003\n+~\n2\u0016Fmr\u0001\u0000\nns\u0002rs\u0001\n+\u0016Bns\u0002Bxc(r;t) +TSO(r;t):(A21)\nFinally, by decomposing the magnetization into its single-particle components, mj, we can de\fne the magnetization\nmaterial derivative as follows\nD\nDtm(r;t) =d\ndtX\nj2occ:mj(r;t) +X\nj2occ:\u0000\nvj\u0001r\u0001\nmj(r;t); (A22)\nwith the spin continuity equation that becomes\nD\nDtm(r;t) =\u0000r\u0001D (r;t)\u0000X\nj2occ:r\u0001vj(r;t)mj(r;t) +~\n2\u0016Fmr\u0001\u0000\nns\u0002rs\u0001\n+\u0016Bns\u0002Bxc(r;t) +TSO(r;t);(A23)\nor\nD\nDtm(r;t) +X\nj2occ:r\u0001vj(r;t)mj(r;t) =\u0000r\u0001D (r;t) +\u0016Bm(r;t)\u0002Be\u000b(r;t) +TSO(r;t); (A24)\nwhere we have introduced an e\u000bective magnetic \feld\nBe\u000b[n;s](r;t) =Bxc[n;s](r;t) +1\n\u0016Fe\u0014rn\u0001rs\nn+r2s\u0015\n: (A25)\nThe continuity equation for the electron density instead follows immediately from Eq. (A4) through the de\fnition of\nvelocity \feld\nDn\nDt=\u0000nr\u0001v: (A26)11\nIt should be noted that the kinetic \feld,1\n\u0016Fe\u0002rn\u0001rs\nn+r2s\u0003\n, written in Eq. (A25) is expressed in term of the density\nand spin density, that are observables of the many-body system. This means that the kinetic \feld obtained within\nthe Kohn-Sham formalism is identical to its many-body counterpart.\n\u0003Contact email address: simonij@tcd.ie\n1I. Tudosa, C. Stamm, A.B. Kashuba, F. King, H.C. Sieg-\nmann, J. Stohr, G. Ju, B. Lu and D. Weller, Nature 428,\n831 (2004).\n2Th. Gerrits, H.A.M. Van den Berg, J. Hohlfeld, L. B ar and\nTh. Rasing, Nature 418, 509 (2002).\n3E. Beaurepaire, J.-C. Merle, A. Daunois and J.-Y. Bigot,\nPhys. Rev. Lett. 76, 4250 (1996).\n4A.V. Kimel, A. Kirilyuk and Th. Rasing, Laser & Photon.\nRev.1, 275 (2007).\n5A. Kirilyuk, A.V. Kimel and Th. Rasing, Rev. Mod. Phys.\n82, 2731 (2010).\n6B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf,\nM. F ahnle, T. 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B 76,\n100401 (2007)."    },    {        "title": "2103.10616v1.Spin_Injection_Generated_Shock_Waves_and_Solitons_in_a_Ferromagnetic_Thin_Film.pdf",        "content": "CE-07 1\nSpin-Injection-Generated Shock Waves and Solitons\nin a Ferromagnetic Thin Film\nMingyu Hu1, Ezio Iacocca2, and Mark Hoefer1\n1Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO 80309 USA\n2Department of Mathematics, Physics, and Electrical Engineering,\nNorthumbria University, Newcastle upon Tyne NE1 8ST, United Kingdom\nUnsteady nonlinear magnetization dynamics are studied in an easy-plane ferromagnetic channel subject to spin injection at one\nedge. The model Landau-Lifshitz equation is known to support steady-state solutions, termed dissipative exchange flows (DEFs)\nor spin superfluids. In this work, by means of numerical simulations and theoretical analysis, we provide a full description of the\ninjection-induced, large-amplitude, nonlinear magnetization dynamics up to the steady state. The dynamics prior to reaching steady\nstate are driven by spin injection, the perpendicular applied magnetic field, the exchange interaction, and local demagnetizing fields.\nWe show that the dynamics take well-defined profiles in the form of rarefaction waves (RW), dispersive shock waves (DSW), and\nsolitons. The combination of these building blocks depends on the interplay between the spin injection strength and the applied\nmagnetic field. A solitonic feature at the injection boundary, signaling the onset of the magnetic \"supersonic\" condition at the\ninjection edge, rapidly develops and persists in the steady-state configuration of a contact soliton DEF. We also demonstrate the\nexistence of sustained soliton-train dynamics in long time that can only arise in a nonzero applied magnetic field scenario. The\ndynamical evolution of spin-injection-induced magnetization dynamics presented here may help guide observations in long-distance\nspin transport experiments.\nIndex Terms —ferromagnet, spin injection, nonlinear dynamics, dispersive spin shock wave, soliton\nI. I NTRODUCTION\nAPROMISING means for long-distance transport of an-\ngular momentum is so-called spin superfluidity [1]–[4].\nThis type of spin transport extends the fluid-like behavior\nof small-amplitude spin waves, first proposed by Halperin\nand Hohenberg [5], into a large-amplitude regime capable\nof exhibiting nonlinear waves conveniently analyzed within\na dispersive hydrodynamic (DH) framework [6]. Here, the\nmagnetization vector m= (mx;my;mz)is recast in terms of\nthe longitudinal spin density n=mzand the magnetic fluid\nvelocity u=\u0000rarctan(my=mx)that is proportional to the\nspin current, revealing an analogy between magnetodynamics\nand fluid dynamics. This has been found to be especially\nbeneficial for theoretical studies in the context of spin super-\nfluids and their instabilities [7]. The DH representation of the\nLandau-Lifshitz (LL) equation is an exact transformation and\ndescribes the essential physics of a ferromagnet: exchange,\nanisotropy, and damping, manifested as wave dispersion, non-\nlinearity, and viscous effects, respectively. In a dispersion-\ndominated fluid-like medium, large gradients in a physical\nquantity (e.g. fluid density) can give rise to dispersive shock\nwaves (DSWs) [8]. DSWs are expanding, highly oscillatory,\nnonlinear excitations that realize a coherent transition between\ntwo states, the superfluidic, dispersive counterpart to viscous\nshock waves. Ferromagnets are rich in dispersive phenomena\nso these dispersive nonlinear wave patterns are expected to\narise under the appropriate conditions. Indeed, DSWs have\nbeen experimentally observed in the envelope of weakly\nnonlinear spin waves excited in Yttrium Iron Garnet (YIG)\n[9].\nCorresponding author: M. Hu (email: mingyu.hu@colorado.edu).In this work, we consider the spin transport dynamics\nexcited by spin injection at a material boundary. This can\nbe realized, for example, by the spin Hall effect, which has\nbeen experimentally used to detect, e.g., spin waves at long\ndistances [10] and to observe signatures of spin superfluidity\n[11], [12]. The initial condition is a uniform ferromagnetic\nstate with zero velocity u= 0 everywhere. Subsequently,\nspin injection at the left boundary is initiated and gradually\nincreases in magnitude, modeled as a hydrodynamic boundary\ncondition (BC) [13] with jujrising smoothly. In this paper,\nwe present the temporal evolution of magnetization dynamics,\ndescribed in DH variables, in an easy-plane anisotropic fer-\nromagnetic channel subject to spin injection at one end. Our\nmodel incorporates the easy-plane shape anisotropy induced\nby a thin-film ferromagnetic sample, for which variations in\nthe directions transverse to wave propagation are assumed neg-\nligible. This somewhat idealized model has been shown to be\nquantitatively accurate in realistic micromagnetic simulations\nin the absence of an externally applied field [14] where the\nsteady-state solutions, dissipative exchange flow (DEF) and\ncontact soliton DEF (CS-DEF), were robustly observed in the\npresence of nonlocal dipole fields and transverse variations.\nA DEF, sustained by spin injection, is a stable noncollinear\nmagnetization state that demonstrates spatially diffusing trans-\nport of angular momentum, that can be interpreted as a spin\ncurrent. The emergence of a CS corresponds to the magnetic\nsonic condition [6], [14], a valuable concept introduced by\nthe analogy between the DH framework and fluid dynamics.\nIn this work, we identify the stages of development–from spin-\ninjection ramp-up to the steady state–of the magnetization\nstates in the absence and presence of an externally applied\nmagnetic field. The corresponding solution structures within\neach stage are found to be rarefaction waves (RWs), DSWs,arXiv:2103.10616v1  [cond-mat.mes-hall]  19 Mar 2021CE-07 2\nand solitons, depending on the spin injection strength and the\napplied magnetic field magnitude. In particular, DSWs can\nonly arise when the applied field is nonzero. Additionally, a\nnonzero applied field can give rise to a persistent, propagating,\nself-interacting soliton-train dynamical solution in long time.\nII. M ODEL\nIn this work, the magnetization dynamics are effectively\nmodeled by one-dimensional (1D) variations in a planar fer-\nromagnetic channel oriented in the ^xdirection. The governing\nequation is the non-dimensional LL equation in 1D, given by\n@tm=\u0000m\u0002he\u000b\u0000\u000bm\u0002(m\u0002he\u000b); x2(0;L); t> 0;\n(1)\nwhere\nhe\u000b=@xxm\u0000mz^z+h0^z: (2)\nHere, mis the magnetization vector normalized by the\nsaturation magnetization Ms. The effective field is he\u000b=\n\u0001m\u0000mz^z+h0^z, also normalized by Ms, consisting of\nexchange, easy-plane anisotropy, and a constant externally\napplied magnetic field along the perpendicular-to-plane ( z)\naxis, respectively. The Gilbert damping parameter is \u000b > 0.\nThe non-dimensionalization is achieved by scaling time by\nj\rj\u00160Msand scaling space by \u0015\u00001\nex, where\ris the gyro-\nmagnetic ratio, \u00160is the vacuum permeability, and \u0015exis\nthe exchange length. All dimensional quantities quoted in\nthis work are for Permalloy (Py) in which \r= 28 GHz/T,\n\u00160= 4\u0019\u000210\u00007N/A2,Ms= 790 kA/m,\u0015ex= 5 nm, and\n\u000b= 0:005. The DH form is obtained by recasting the LL\nequation in terms of the hydrodynamic variables\nspin density: n=mz;\nfluid velocity: u=\u0000@x\b =\u0000@xarctan(my=mx):\nThe spin injection at x= 0 is modeled as a perfect spin\nsource. Atx=L, we assume a perfect spin sink with no spin\npumping, modeled as a free spin BC. Thus, the BCs are given\nby\n@xn(x= 0;t) = 0; @xn(x=L;t) = 0; (3a)\nu(x= 0;t) =ub(t); u(x=L;t) = 0; (3b)\nwhereub(t)is the time-dependent spin injection strength\nwhose magnitude increases from 0 at t= 0 to the fi-\nnal intensityju0jmonotonically and smoothly. In the sim-\nulations, we adopt a hyperbolic tangent profile ub(t) =\nu0\n2h\ntanh\u0010\nt\u0000t0=2\nt0=10\u0011\n+ 1i\nto model a smooth change in the fluid\nvelocity and thus the rise time t0is defined as the time where\nthe injection magnitude reaches 99.99% of its extremum ju0j.\nIn addition, we consider only modulationally stable dynamics\n[6], [15] by restricting the injection to ju0j<1, so there are\nno long-wave instabilities. The initial condition (IC) in the DH\nvariables is given by\nn(x;t= 0) =h0; (4a)\nu(x;t= 0) = 0; (4b)\nwithjh0j<1.The long-wave phase velocities can be derived from the\nspin-wave dispersion of waves on a uniform hydrodynamic\nstate (UHS), described by spatially uniform spin density and\nfluid velocity \u0016nand\u0016u[6], and are given by\ns\u0006= 2\u0016n\u0016u\u0006p\n(1\u0000\u0016n2)(1\u0000\u0016u2): (5)\nThe current system is identified to be subsonic when s\u0000<\n0< s +and supersonic when s+<0. In a supersonic\nsystem,jujis larger than the magnetic sound speed jusonicj=p\n(1\u0000\u0016n2)=(1 + 3\u0016n2). In addition, we use the long-wave\nvelocities to predict the dynamical structures that arise for\ngiven spin injection and applied field. The temporal evolution\nof spin-injection-induced dynamics involves three stages,\n1: Injection rise. If the injection is supersonic, that is when\ns+jx=0<0, a CS at the injection end is developed\nwithin the rise time. The emergence of the CS can be\nunderstood as an accumulation of spin current at the\ninjection boundary because the long waves propagate to\nthe left and encounter the boundary. We show in the\nsimulation section that the CS developed at this stage\ntypically persists throughout the dynamical evolution\nand the steady state. The remaining dynamics can be\nmodeled as effectively damping-free. There are two\npossible solution types:\n(a) Ifs+jx=0< s+jx=L,expansion dynamics occur,\nfor example a RW. The solution of this type can be\nfurther approximated by the long-wavelength limit.\n(b) Ifs+jx=0> s +jx=L,compression dynamics oc-\ncur and self-steepening in physical variables takes\nplace. Within the injection time, the self-steepening\nresults in wave-breaking that leads to the formation\nof a dispersive shock.\nOne mixed case is also possible with s+jx=0>s+jx=L\nat an early time during injection rise and s+jx=0<\ns+jx=Llater.\n2: Pre-relaxation when the spin injection is maintained\nat its maximum strength. The dynamics in this stage\ncan be approximated by the conservative limit on times\nt0< t\u00181\n\u000bas we will show in simulation. Besides\nthe CS, the rest of the solution structure continues to\ndevelop. This stage marks the temporal range where\nthe dispersive hydrodynamics become fully developed,\ndissipation has not diminished prominent features in\nthe solution structures, and the right boundary has not\ninteracted with the developed dynamics.\n3: Relaxation. In this stage, damping is essential and drives\nthe system to a long-time configuration.\nAn example of the entire time evolution exhibiting all dy-\nnamical structures in terms of the hydrodynamic variables n\nanduis shown in Fig. 1. The detailed numerical scheme\nused is presented in the next section. In the example, the\nferromagnet length is L= 300 (1.65\u0016m for Py), the injection\nrise time is t0= 80 (2.8 ns for Py), the maximum injection\nstrength is u0= 0:9, and the applied field is h0= 0:5.\nDuring the injection rise in Stage 1, the injection induces\nboth compression and expansion, giving rise to a variety of\nstructures. First, the injection satisfies case 1(b) (compression)CE-07 3\nStage 1: Injection riseStage 2: Pre-relaxationStage 3: Relaxation and steady-state-0.500.5\n030000.51-0.500.5\n0300-0.500.51-0.500.5\n0300-0.500.51\nFig. 1. An example time evolution of 1D magnetization dynamics excited by spin injection in a planar ferromagnetic channel of length L= 300 (1.5\u0016m):\ninjection rise time t0= 80 (2.8 ns), injection strength u0= 0:9, applied magnetic field h0= 0:5. Blue solid line: damping coefficient \u000b= 0; orange dotted\nline:\u000b= 0:005. Green shade: CS; yellow shade: RW; blue shade: DSW.\nwhich leads to self-steepening (shaded in blue). Then, the\ninjection satisfies case 1(a) (expansion) so that the structure is\nslowly varying (shaded in yellow). Finally, s+jx=0becomes\nnegative and a CS develops at the injection boundary (shaded\nin green). Within the injection rise, the damped and the\nundamped solution are almost identical, indicating that the\ndynamics are dominated by dispersion. In Stage 2 at t= 200\n(\u00197ns for Py), the compression and expansion dynamics\ndevelop further into well-defined structures. Reading from\nright to left in the middle panel of Fig. 1: Compression\nleads to a DSW (shaded in blue), an expanding, highly-\noscillatory, and rank-ordered structure with large amplitude\nat one edge and diminishing wave amplitude at the other\nedge [8]; Expansion gives rise to a RW (shaded in yellow),\nwhich is an expanding, non-oscillatory, slowly-varying wave.\nThe CS (shaded in green) remains pinned to the injection site\nand an intermediate constant state develops between it and\nthe RW. It is worth pointing out that DSWs and RWs have\nbeen identified in a two-component Bose-Einstein condensate\nwhose approximate governing dynamical equations are the\nsame as the DH formulation of the dissipationless LL [16],\n[17]. Here, both DSW and RW persist in the presence of\ndamping (dashed red curves) while the intermediate state\nbetween the RW and the CS is lost. Finally, in Stage 3 the\ndamping-driven relaxation process dissipates all oscillations\nand relaxes the system to the steady-state solution, a CS-DEF.\nIII. N UMERICAL SIMULATIONS\nIn this section, we present the simulation results of the\nmagnetization dynamics induced by spin injection in a planar\nferromagnetic channel. We solve the LL equation (1) subject\nto the BCs\n@xm(x= 0;t) =ub(t)m(x= 0;t)\u0002^z; @xm(x= 0;t) = 0;\n(6)\nwhere the spin injection BC is of a Robin (mixed) type and\nthe free spin BC is of the Neumann type [2]. The IC is given\nby\nmx(x;t= 0) =q\n1\u0000h2\n0; (7a)\nmy(x;t= 0) = 0; (7b)\nmz(x;t= 0) =h0: (7c)This initial-boundary value problem (IBVP) is solved using\nthe method of lines: the right-hand-side of Eq. (1) is spatially\napproximated by the sixth-order centered finite difference\nmethod and the result serves as an approximation to the time\nderivative at the current time; then the IC is time-stepped\ndiscretely using the MATLAB built-in initial value problem\n(IVP) solver ode23 . The spin injection and free spin BCs are\nimplemented using 6th-order local extrapolation polynomials\nwith ghost points [18] to maintain the order of accuracy and\nsmoothness in the solution near the boundaries. During the\nrise time, the spin injection changes slowly enough so as to\nreduce additional oscillations. With this numerical model, we\nexplore the unsteady magnetization dynamics in the absence\nand presence of an externally applied magnetic field. We\nshow results at Stage 2, where the nonlinear solutions are\nfully developed, and Stage 3, where dissipation dominates the\nsolution.\nA. Zero Applied Field h0= 0\nFig. 2 shows the numerical simulation results for the zero\nfield case. We only discuss the negative injection results here\nsince the solutions for positive injection strength have the\nopposite signs, due to the symmetry x!\u0000xandu!\u0000u\nwhenh0= 0.\nWe start our discussion with u0=\u00000:7solutions, shown in\nFig. 2(a). In Stage 1, it is found that the long-wave velocities\nsatisfys+jx=0<s+jx=L(case 1(a)) throughout the injection\nrise time. Hence, only expansion dynamics arise. In addition,\ns+jx=0<0and thus the system becomes supersonic and a\nCS is formed during this stage. In Stage 2, shown in the left\npanels of Fig. 2(a), a RW (shaded in yellow) develops and\nexpands while the CS (shaded in green) stays stationary at the\ninjection end. We term the undamped solution a CS|RW, with\n\"|\" denoting the intermediate constant state in between. We\npoint out that the small-amplitude oscillations at the leading\nedge of the RW are caused by the injection rise dynamics and\nare not a DSW solution [8], [16]. At the end of Stage 3, the\ndamped solution reaches a CS-DEF steady-state configuration.\nNotice that the RW has decayed into the DEF.\nTheu0=\u00000:3solution is shown in Fig. 2(b). Throughout\nStage 1 of the injection rise, it is found that s+jx=0<s+jx=LCE-07 4\n(a)\n(b)-0.100.10.3\n0100200300-0.3-0.20-0.100.30.7\n0100200300-0.7-0.4-0.1-0.100.30.7\n0100200300-0.7-0.4-0.1-0.100.30.7\n0100200300-0.7-0.4-0.1\n-0.100.10.3\n0100200300-0.3-0.20\nFig. 2. Simulation results of Stage 2 (left panels) and Stage 3 (right panels,\n52.5 ns) of the time evolution of spin-injection-induced dynamics in a planar\nferromagnetic channel of length L= 300 (1.5\u0016m) with zero applied field\nh0= 0. The spin injection intensities are (a) u0=\u00000:7, (b)u0=\u00000:3.\n(case 1(a)). Hence, only expansion dynamics (shaded in yel-\nlow) are identified in Stage 2, shown in the left panels in\nFig. 2(b). The steady-state solution after relaxation is identified\nto be a DEF for this subsonic injection.\nB. Uniform Perpendicular Applied Field h06= 0\nFig. 1 and 3 show the numerical simulation results when\nh0= 0:5. It is found that the dynamical and steady-state\nsolutions with negative injections have the same solution\nstructures as the zero field case discussed earlier: a RW\nfor subsonic injection relaxes to a DEF, and a CS|RW for\nsupersonic injection relaxes to a CS-DEF. Therefore, we only\nfurther discuss the positive injection results here. In addition,\nwe only present the solutions for a positive field (in the\n+zdirection) because there exhibits an odd symmetry in the\nu0\u0000h0plane, a generalization of the symmetry about u0= 0\nin the zero field case. We also point out that relaxation (Stage\n3) requires longer time to achieve a long-time configuration\nbecause of the large amplitude in a DSW.\nWe start the discussion with the small injection strength\nu0= 0:3. During Stage 1, it is found that s+jx=0>s+jx=L\n(case 1(b)) throughout. Thus compression dynamics are gen-\nerated immediately and self-steepening is expected to lead to\nwave-breaking that results in a highly-oscillatory DSW. The\nsimulation shown in the left panel of Fig. 3(a) confirms this\nprediction (shaded in blue). The DSW reveals the dispersion-\ndominated dynamics that are characteristic of ferromagnets on\nshort enough time scales. In addition, s\u0000jx=0<0< s+jx=0\nforu0= 0:3, and hence the magnetic \"flow\" is always sub-\nsonic without a CS. After relaxation, the steady-state solution\nis a DEF, shown in the right panel of Fig. 3(a).\nWith moderate injection u0= 0:7, during Stage 1, it\nis found that s+jx=0> s +jx=L(case 1(b)) at first and\n(a)\n(b)00.40.8\n0100200300-0.400.4\n00.51\n0100200300-10100.51\n0100200300-10100.40.8\n0100200300-0.400.4Fig. 3. Simulation results of Stage 2 (left panels) and Stage 3 (right panels,\n114 ns) of the time evolution of spin-injection-induced dynamics in a planar\nferromagnetic channel of length L= 300 (1.5\u0016m) with constant applied\nfieldh0= 0:5along the perpendicular z-axis. The spin injection intensities\nare (a)u0= 0:3, (b)u0= 0:7. An additional supersonic simulation result\nwithu0= 0:9is shown in Fig. 1.\nFig. 4. Space-time contour of long-time dynamics for u0= 0:7,h0= 0:5,\na traveling soliton train on DEF. This contour plot demonstrates a full cycle\nof the soliton train traveling left, being reflected by the injection boundary,\nself-interacting, traveling right, and being reflected by the spin sink boundary.\nthens+jx=0< s +jx=L(case 1(a)) thereafter. Therefore,\ncompression dynamics are induced immediately and a DSW\nwill emerge as a result of self-steepening and wave-breaking.\nThis is followed by expansion dynamics, manifesting in a\nRW. Both of these structures are fully developed by Stage\n2, shown in the left panel of Fig. 3(b). The DSW (shaded\nin blue) is observed directly adjacent to the RW (shaded in\nyellow). The left edge of the DSW travels at the same speed\nas the right edge of the RW. Hence, this DSW is the analog\nof a contact discontinuity in classical fluids and is termed\na contact DSW (cDSW) [16], [17]. Thus, we term the pre-CE-07 5\nrelaxation solution a RW-cDSW composite wave. Similarly,\nthe supersonic solution with u0= 0:9shown in Fig. 1 is a\nCS|RW-cDSW. Throughout Stage 1, s\u0000jx=0<0< s+jx=0\nso the solution remains subsonic. In Stage 3, a new long-\ntime configuration is observed. In Fig. 3(b) right panel and\nFig. 4, the background mean flow can still be identified as a\nDEF steady-state, but there is additionally a train of solitons\non top. These solitons are dynamic, as they travel back and\nforth and interact with each other within the ferromagnet. A\ncomplete cycle of the periodic motion of the soliton train\nis shown in Fig. 4. It is also observed that vacuum states,\nwherejnj= 1 anduswitches sign, can be reached after\nthe solitons are reflected by the spin injection boundary. The\nsimulation in Fig. 4 indicates that these solitons are amplified\nto reach vacuum after being reflected by the spin injection left\nboundary and then decrease in amplitude to fall back from\nvacuum through the spin sink at the right boundary. These\ndistinct long-time dynamics call for a different analytical\ndescription than those for the DEF/CS-DEF steady states.\nIV. D ISCUSSION AND CONCLUSION\nIn this paper, we described the time evolution of magne-\ntization dynamics induced by spin injection at one edge of\nan effective easy-plane ferromagnetic channel. Our analysis\nutilizes the DH framework, which provides a fluid analogy of\nferromagnetism. We use the long-wave velocities during the\ninjection rise stage to predict the solution structures that are\nverified qualitatively by numerical simulations. If the injection\nis supersonic, a CS at the injection site completes development\nby the end of the rise time and lives through the entire\ntime evolution. This signature feature indicates that there is a\nsaturation limit of angular momentum a thin-film ferromagnet\ncan support through nonlinear textures. During pre-relaxation\n(Stage 2), highly-oscillatory dispersive wave structures, such\nas a DSW and a RW-cDSW, arise only when the applied\nmagnetic field is non-zero. A more detailed theoretical de-\nscription is needed to clarify the interplay between the spin\ninjection strength and the externally applied field magnitude\nthat gives rise to these structures in order to identify their\nsalient features. After the relaxation process, other than a\nDEF or a CS-DEF steady-state configuration, our numerical\nsimulation reveals a dynamical long-time solution of a train of\ntraveling, interacting solitons on a DEF profile in a subsonic\nscenario. The conditions required for this novel long-time\nbehavior and its mechanism are under further investigation. We\nalso point out that the presented simulation results are obtained\nunder ideal conditions: a defect-free ferromagnet with only\nlocal dipole fields, a perfect spin source at one boundary, and a\nperfect spin sink at the other. Nevertheless, the predicted time\nevolution of magnetization dynamics suggests new features\nto look for in an experimental realization of microscopic\nspin transport in a ferromagnet that can be detected in the\nnanosecond regime.\nACKNOWLEDGMENT\nThe authors acknowledge funding from the National In-\nstitute of Standards and Technology Professional ResearchExperience Program and the U.S. Department of Energy,\nOffice of Science, Office of Basic Energy Sciences under\nAward No. DE-SC0018237.\nREFERENCES\n[1] E. Sonin, “Spin currents and spin superfluidity,” Advances in Physics ,\nvol. 59, no. 3, pp. 181–255, 2010.\n[2] S. Takei and Y . Tserkovnyak, “Superfluid spin transport through easy-\nplane ferromagnetic insulators,” Physical review letters , vol. 112, no. 22,\np. 227201, 2014.\n[3] H. Chen, A. D. Kent, A. H. MacDonald, and I. Sodemann, “Nonlocal\ntransport mediated by spin supercurrents,” Physical Review B , vol. 90,\nno. 22, p. 220401, 2014.\n[4] M. Evers and U. 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Wu, “Observation of\nself-cavitating envelope dispersive shock waves in yttrium iron garnet\nthin films,” Physical Review Letters , vol. 119, no. 2, p. 024101, 2017.\n[10] R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh, L. Baldrati,\nJ. Cramer, A. Brataas, R. A. Duine, and M. Kläui, “Tunable long-\ndistance spin transport in crystalline antiferromagnetic iron oxide,”\nNature , vol. 561, pp. 222–225, 2018.\n[11] P. Stepanov, S. Che, D. Shcherbakov, J. Yang, R. Chen, K. Thilahar,\nG. V oigt, M. W. Bockrath, D. Smirnov, K. Watanabe et al. , “Long-\ndistance spin transport through a graphene quantum hall antiferromag-\nnet,” Nature Physics , vol. 14, no. 9, pp. 907–911, 2018.\n[12] W. Yuan, Q. Zhu, T. Su, Y . Yao, W. Xing, Y . Chen, Y . Ma, X. Lin, J. Shi,\nR. Shindou et al. , “Experimental signatures of spin superfluid ground\nstate in canted antiferromagnet cr2o3 via nonlocal spin transport,”\nScience advances , vol. 4, no. 4, p. eaat1098, 2018.\n[13] E. Iacocca, T. Silva, and M. A. Hoefer, “Symmetry-broken dissipative\nexchange flows in thin-film ferromagnets with in-plane anisotropy,”\nPhysical Review B , vol. 96, no. 13, p. 134434, 2017.\n[14] E. Iacocca and M. A. Hoefer, “Hydrodynamic description of long-\ndistance spin transport through noncollinear magnetization states: Role\nof dispersion, nonlinearity, and damping,” Physical Review B , vol. 99,\nno. 18, May 2019. [Online]. Available: https://link.aps.org/doi/10.1103/\nPhysRevB.99.184402\n[15] C. Law, C. Chan, P. Leung, and M.-C. Chu, “Critical velocity in a binary\nmixture of moving bose condensates,” Physical Review A , vol. 63, no. 6,\np. 063612, 2001.\n[16] T. Congy, A. Kamchatnov, and N. Pavloff, “Dispersive hydrodynamics\nof nonlinear polarization waves in two-component Bose-Einstein\ncondensates,” SciPost Physics , vol. 1, no. 1, Oct. 2016. [Online].\nAvailable: https://scipost.org/10.21468/SciPostPhys.1.1.006\n[17] S. K. Ivanov, A. M. Kamchatnov, T. Congy, and N. Pavloff, “Solution\nof the Riemann problem for polarization waves in a two-component\nBose-Einstein condensate,” Physical Review E , vol. 96, no. 6, Dec.\n2017. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevE.\n96.062202\n[18] F. Gibou and R. Fedkiw, “A fourth order accurate discretization for the\nlaplace and heat equations on arbitrary domains, with applications to\nthe stefan problem,” Journal of Computational Physics , vol. 202, no. 2,\npp. 577–601, 2005."    },    {        "title": "2303.15837v1.Role_of_intersublattice_exchange_interaction_on_ultrafast_longitudinal_and_transverse_magnetization_dynamics_in_Permalloy.pdf",        "content": "Role of intersublattice exchange interaction on ultrafast longitudinal and transverse\nmagnetization dynamics in Permalloy\nA. Maghraoui, F. Fras, M. Vomir, Y. Brelet, V. Halt\u0013 e, J. Y. Bigot, and M. Barthelemy\u0003\nUniversit\u0013 e de Strasbourg, CNRS, Institut de Physique et Chimie des\nMat\u0013 eriaux de Strasbourg, UMR 7504, F-67000 Strasbourg, France\n(Dated: March 29, 2023)\nWe report about element speci\fc measurements of ultrafast demagnetization and magnetization\nprecession damping in Permalloy (Py) thin \flms. Magnetization dynamics induced by optical pump\nat 1:5eV is probed simultaneously at the M2;3edges of Ni and Fe with High order Harmonics for\nmoderate demagnetization rates (less than 50%). The role of the intersublattice exchange interaction\non both longitudinal and transverse dynamics is analyzed with a Landau Lifshitz Bloch description\nof ferromagnetically coupled Fe and Ni sublattices. It is shown that the intersublattice exchange\ninteraction governs the dissipation during demagnetization as well as precession damping of the\nmagnetization vector.\nPACS numbers: 71.20.Be, 75.40.Gb, 78.20.Ls, 78.47.+p\nI. INTRODUCTION\nUltrafast demagnetization of ferromagnets induced by\nfemtosecond laser pulses [1{3] promises novel applica-\ntions in data storage and processing technologies. Since\nits discovery, several microscopic mechanisms such as\nthe spin-orbit interaction [4{6], Elliott-Yafet scattering\ninduced spins-\rips [7], non-thermal excitations [8, 9],\nsuper-di\u000busive [10, 11] or ballistic spin-transport have\nbeen identi\fed to play a key role and their relative weight\ncan be element dependent [12]. Depending on magnetic\nanisotropies, such transient modi\fcation of the e\u000bective\nmagnetic \feld can trigger a coherent precession motion\nof the magnetization vector with a Gilbert damping [13]\nresulting from dissipation of energy to an external bath.\nThose longitudinal and transverse relaxation processes\nset a natural limit to optical manipulation of magneti-\nzation from femtosecond to nanosecond time scales. If\none aims to study the dynamics over such a large tempo-\nral scale, the Landau-Lifshitz Bloch (LLB) model [14] in\nwhich the e\u000bective \feld contains the essential microscopic\nmechanisms is well adapted. Among them, the exchange\ninteraction appears to be critical, but several aspects re-\nmain to be explored. In particular, in multi-compound\nmaterials, the intersublattice exchange interaction plays\na crucial role on the resulting global dynamics, acting\nas a spin momentum transfert between sublattices dur-\ning the demagnetization [15]. Over the last decade, it\nhas been investigated experimentally thanks to chemi-\ncal selectivity of XUV resonant probe of core levels of\ntransition metals (TM) and rare earths (RE). Time re-\nsolved Xray magnetic circular dichroism (XMCD) [16{20]\nand table-top high order harmonics (HH) probed time re-\nsolved magneto optical Kerr experiments (TMOKE)[21{\n29] have proven to o\u000ber a unique opportunity to study\nsublattice magnetization dynamics governed by dissipa-\n\u0003Corresponding author: barthelemy@unistra.frtion and momentum transfert mechanisms in all optical\nmagnetization switching in alloys [18{20]. In particular,\nthe demagnetization of each sublattice in a binary alloy\ncan be either accelerated or decelerated compared to the\npure element demagnetization [21, 22, 24]. This e\u000bect is\ndependent on the value of elemental magnetic momenta\nand on the ferro or antiferromagnetic nature of the ex-\nchange coupling [30, 31]. The case of Permalloy (Py) has\nattracted attention since various dynamical behaviors of\nsublattices magnetic momenta have been observed de-\npending on the photon energy range of the probe. On\none side, XMCD studies performed at L2;3-edges have\nshown a faster demagnetization of Ni momenta compared\nto Fe [31]. This observation is supported by the strong\ne\u000bective exchange coupling sustained by Fe momenta in\nPy so that Ni sublattice momenta are more submitted to\nthermal dissipation [32]. On the other side, HH TMOKE\nmeasurements show that during the early demagnetiza-\ntion of ferromagnetic Py, the momenta of Fe starts to\nrandomize before Ni momenta until a time of scale 10 fs\nafter which both sublattice relax together due to inter-\nsublattice exchange interaction (IEI) [24]. The origin of\na stronger coupling of Fe spins to the electronic system\ncompared to Ni remains unknown and deserves further\nexploration. In the present work, the magnetization dy-\nnamics of Fe and Ni sublattices of a 10 nm Permalloy thin\n\flm is studied with chemical selectivity over a wide tem-\nporal range as a function of excitation density. A table\ntop HH TMOKE con\fguration is used to measure both\ndemagnetization and precession at the M edges of Fe and\nNi. The role of strong intersublattice exchange interac-\ntion on longitudinal and transverse ultrafast magnetiza-\ntion dynamics is discussed for moderate demagnetization\namplitudes.\nII. EXPERIMENT\nIn our experiment, XUV sub 10 fs pulses are produced\nby HH generation in a Ne-\flled gas cell driven by 795arXiv:2303.15837v1  [cond-mat.mtrl-sci]  28 Mar 20232\nnm, 3 mJ, 1 kHz, 25 fs laser pulses. The resulting XUV\nprobe photons energies cover the 30 eV - 72 eV range and\nspan theM2;3-edges of Fe and Ni centered respectively at\n66 eV and 54 eV. Ultrafast demagnetization is induced\nby 795 nm, 25 fs pump with variable \ruence in a 10 nm\nthick Ni 80Fe20(Py) thin \flm with an in-plane anisotropy\ndeposited on a crystalline Al 2O3substrate by ion beam\nsputtering.\nTG sample \nIR pump HH probe \nz \nx y o H+ H- \nt \nFIG. 1: Principle of XUV probe IR pump transverse HH\nTMOKE experimental con\fguration with static magnetic\n\feld H along zaxis. TG: toroidal grating. CCD: Charge\nCoupled Device camera. Inset: example of re\rected spectra\non Py for the antiparallel orientations of the applied magnetic\n\feld +/-H (respectively blue solid line and red dotted line).\nFigure 1 illustrates the transverse time resolved\nmagneto-optical Kerr con\fguration used in this work.\nAn external static magnetic \feld ( H= 450 Oe) is ap-\nplied on the sample along the transverse axis, i:e:along\nz direction in \fgure 1 and perpendicularly to the plane\nof incidence xOy of the p-polarized IR pump and VUV\nprobe. The angle of incidence of the probe was set to\n45\u000ewith respect to the sample normal in order to maxi-\nmize the magnetic contrast obtained from spectrally re-\nsolved re\rectivity measurements [21]. In the inset of \fg-\nure 1, the re\rected XUV probe spectra Istat\nH+andIstat\nH\u0000is\nshown for two antiparallel orientations of the transverse\nmagnetic \feld H. The maximum intensity di\u000berence be-\ntween the two re\rected spectra is seen at the harmonics\nh45(centered at 66 eV) and h35(centered at 54 eV) corre-\nsponding to the M-edges of Ni and Fe respectively. Both\nspectra are further measured as a function of pump probe\ndelay by varying the optical path of the pump with a me-\nchanical delay line.\nIII. ULTRAFAST DEMAGNETIZATION IN\nPERMALLOY PROBED AT M-EDGES OF NI\nAND FE\nWe \frst consider the short time scale corresponding\nto demagnetization process in Permalloy. The elementalmagnetization dynamics of Ni and Fe elements m( q;\u001c)\nmeasured as a function of the pump-probe delay \u001cis\nthen integrated over each resonant qthharmonic:\n\u0001m\nm(q;\u001c) =Idyn\nH+(q;\u001c)\u0000Idyn\nH\u0000(q;\u001c)\nIstat\nH+(q)\u0000Istat\nH\u0000(q)(1)\nforq= 45 andq= 35 with Idyn\nH\u0006=Iwith\nH\u0006\u0000Istat\nH\u0006and\nIwith\nH\u0006being the intensity of signal with pump and Istat\nH\u0006\nwithout pump. Figure 2 shows demagnetization\u0001m\nm(q;\u001c)\n-0.4-0.20.0Δm/m\n8006004002000\n delay (fs)Ni\nFe\n-0.4-0.20.0Δm/m\n8006004002000\n delay (fs) Ni\n  Fe\n-0.4-0.20.0Δm/m\n8006004002000\n delay (fs)  Ni\n  Fe(a)\n(b)\n(c)\nFIG. 2: Demagnetization dynamics in permalloy probed at\nthe Ni (full blue dots) and Fe (empty red dots) M-edges and\n\fts (grey lines: Fe, blue lines: Ni) for incident pump \ruences\nof a) 2:5 mJ/cm2. b) 3:8 mJ/cm2. c) 4:7 mJ/cm2.\nin Py at the M2;3edges of Fe (\u0001mFe\nmFe) and Ni (\u0001mNi\nmNi)\nintegrated over harmonics h35andh43respectively for\nthree increasing pump \ruences. Ni and Fe sublattices\nappear to demagnetize simultaneously. The demagneti-\nzation amplitude of both sublattices increases from 25\n% to 40 %. Contrary to reference [24], no reproducible\ndelay between the two sublattices demagnetizations is\nobserved with our pump duration of 25 fs. A possible\nexplanation could be a slight variation of intersublattice\nexchange interaction (sample dependent due to change\nof crystallinity or grating vs alloy) that may induce a\nchange of the temporal shift value. Moreover the di\u000ber-\nent conditions of HH generation could lead to a di\u000berent3\n-0.6-0.4-0.20.0|Δm/m|\n86420\npump fluence (mJ/cm2)  Fe ; τmax\n   Ni ; τmax\n  Fe ; τ=5 ps\n   Ni ; τ=5 ps\n \nFIG. 3: Demagnetization amplitudes measured at M-edges of\nNi and Fe in Py as a function of incident pump \ruence. For\nmaximum demagnetization (full symbols) and at a 5 ps pump\nprobe delay (empty symbols).\ntime duration of our probe resulting to a lower tempo-\nral resolution, or a delay between h35 and h45 due to a\npossible chirp.\nThe framework of analysis and data \ftting is described\nin the following, where details about linearized LLB are\npresented. This approach is based on the knowledge of\nthe laser induced temperature of the system. In order to\nde\fne such laser induced temperature, let us \frst explore\nthe demagnetization amplitude behavior with incident\npump \ruence.\nThe demagnetization amplitude of Ni and Fe in\npermalloy with respect to the pump \ruence is plotted\nin \fgure 3. For each point, two delays have been cho-\nsen. The \frst delay corresponds to maximal demagneti-\nzation level (the corresponding time delay range is 300\nfs - 500 fs from low to large pump \ruences), the second\ndelay is \fxed at 5 ps and corresponds to a thermal quasi-\nequilibrium between electrons and lattice baths (remag-\nnetization). Within the \ruence range of this experiment\nthe demagnetization amplitude is always found identical\nfor both sublattices. Moreover, up to 40% demagnetiza-\ntion given by the damage threshold of our sample, a lin-\near increase of demagnetization amplitude with the pump\n\ruence is observed, with the same slope for both delays.\nSuch linear behavior can be attributed to a coupling with\na thermalized bath[2], characterized by an energy of kBT,\nheated by the laser pulse. The vertical o\u000bset between\nthe two curves arises due to the magnetization recovery.\nIn LLB approach, it can be described as a temperature\nstep between maximal bath temperature (demagnetiza-\ntion) and cooling down of the bath due to coupling with\nlattice (remagnetization). One can notice that the linear\n\fts labelled \\maximum\" does not cross the zero variationat zero \ruence. At a delay of 5 ps a slight deviation from\nzero is also observed. This range of \ruence is either dif-\n\fcult to access experimentally or usually not considered.\nA hypothesis to explain such a behavior is a change of\nthe regime of interaction, at the origin of the demagneti-\nzation or remagnetization processes, with pump \ruence.\nThis could lead to a di\u000berent power dependent law in the\nrange of very low \ruences, but this aspect goes beyond\nthe scope of the present work. After thermalization of\nelectrons, the temperature dependent magnetization can\nbe approximated with a molecular \feld model as :\nm\u000f(T) = (1\u0000T\nTC)\u0014\u000f(2)\nwhereTCis the Curie temperature, \u0014\u000fis the critical ex-\nponent. In the following, the maximum amplitude of\ndemagnetization in our experiments is used to evaluate\nthe laser induced spin temperature Tthat is related to\nthe amplitude of \ructuations to which the spins are sub-\nmitted [34] in the 300-500 fs temporal range. As in ref\n[32], in our approach at short time scale, only the de-\nmagnetization process is considered, justi\fed by a slower\nrate of the re-magnetization process. The magnetic sys-\ntem can be considered initially in thermal equilibrium\nat a temperature of Ti= 300K, then for t=0 the bath\ntemperature is instantaneously changed to Tf. Thus, the\nmagnetization of the two sublattices will evolve towards\na new thermal equilibrium value given by Tf. Therefore,\nin this approach, the equivalent spin temperature is de-\n\fned at the maximum of demagnetization. By taking a\nCurie Weiss law ( \u0014= 1=2) andTC= 850 K in Ni 80Fe20,\nthe temperature range can be calibrated. In \fgure 3,\nthe evaluated spin temperature Tranges from 300 K to\n600 K, and the maximum demagnetization amplitudes of\n\fgure 2 correspond to T=TC= 0.35, 0.43 and 0.52.\nWe now analyze our experimental data in the frame of\nthe linearized LLB model. It considers an ensemble of\nrigid spins submitted to exchange interaction and cou-\npled to a thermal bath corresponding to either charges\nor phonons at the origin of dissipation [33{37]. It gives\na consistent approach that encompasses a broad tempo-\nral scale from femtoseconds to nanoseconds during which\nspin \rips as well as the magnetization precession take\nplace. By isolating the longitudinal contribution to mag-\nnetization dynamics at short time scales, it can be used\nto simulate ultrafast demagnetization in TM-RE com-\npounds. This method was \frst applied to ferrimagnets\nknown for their high potential for all optical switching\n[34], and more recently to better understand the role of\nintersublattice exchange interaction (IEI) in ferromag-\nnetic TM alloys [32]. In particular, this model allows\nto decipher quantitatively the role played by both the\nIEI and intrinsic dissipation of each sublattices magne-\ntization on the observed dynamics. Fe and Ni momenta\ndynamics in permalloy are described with the following4\n\frst order coupled rates equation:\n\u0012\n_mFe\n_mNi\u0013\n=Ak\u0012\nmFe\nmNi\u0013\n=\u0012\n\u0000\u0000FeFe \u0000FeNi\n\u0000NiFe\u0000\u0000NiNi\u0013\u0012\nmFe\nmNi\u0013\n(3)\nwhere the matrix Akdrives the dynamics. Its ele-\nments can be written as a function of micromagnetic\nparameters[32]:\n\u0000FeFe=1\n\u001cFe\nintra+\u001fNi\nk\n\u001fFe\nk1\n\u001cFeNi\nexch(4)\n\u0000FeNi=1\n\u001cFeNi\nexch(5)\n\u0000NiFe=1\n\u001cNiFe\nexch(6)\n\u0000NiNi=1\n\u001cNi\nintra+\u001fFe\nk\n\u001fNi\nk1\n\u001cNiFe\nexch(7)\nIn this basis, the elements of Akcontain two contribu-\ntions to longitudinal damping. The \frst one corresponds\nto intrasublattice demagnetization \u001cintra:\n1\n\u001cFe,Ni\nintra=\rFe,Ni\u000bFe,Ni\nk(T)\n\u001fFe,Ni\nk(T)(8)\nwhere\u000bFe,Ni\nkis longitudinal damping and \u001fFe,Ni\nkis the\ne\u000bective magnetic susceptibility, both being element and\ntemperature dependent quantities. \rFe,Nicorresponds to\nthe gyromagnetic ratio of each element. The second one\nis the intersublattice exchange mediated demagnetiza-\ntion:\n1\n\u001cFeNi,NiFe\nexch=\rFe,Ni\u000bFe,Ni\nk(T)JFeNi,NiFe\n\u0016Fe,Ni(9)\nwhereJFeNi=JNiFeis the IEI constant and \u0016Fe,Ni is the\natomic magnetic momentum.\nFinally, diagonal terms of Ak,\u0000\u0000FeFe and\u0000\u0000NiNi,\ncorrespond to a dissipation of magnetic momenta in each\nsublattice and via IEI with the second sublattice (eq.\n4). Non diagonal terms \u0000 NiFe and \u0000FeNi lead to an\nexchange of momentum between sublattices. It should\nbe underlined that, in this approach, the overall \row of\nmomentum, mediated by the IEI, is element dependent\nand weighted by the magnetic susceptibilities ratio.\nThe di\u000berential system (3) can be easily solved in the\neigen basis after diagonalization of Ak:\nAk=\u0012\n\u0000+0\n0 \u0000\u0000\u0013\n(10)\nwhere the two eigen values \u0000\u0006= 1=\u001c\u0006can be written as:\n\u0000\u0006=1\n2(\u0000FeFe+ \u0000NiNi\n\u0006p\n(\u0000FeFe\u0000\u0000NiNi)2+ 4\u0000 FeNi\u0000NiFe) (11)Finally the measured sublattices magnetization dynamics\ncan be expressed as a linear combination of the di\u000beren-\ntial system solutions:\n\u0001mFe\nmFe=AFeexp(\u0000t\n\u001c+) +BFeexp(\u0000t\n\u001c\u0000); (12)\n\u0001mNi\nmNi=ANiexp(\u0000t\n\u001c+) +BNiexp(\u0000t\n\u001c\u0000); (13)\nwhere the coe\u000ecients AFe,BFe,ANi,BNidepend on the\neigen vector components x\u0006= \u0000 FeNi=(\u0000FeFe\u00001=\u001c\u0006) as\nfollows:\nAFe= \u0001mFe\n0x+\n(x\u0000\u0000x+)(\u0001mNi\n0\n\u0001mFe\n0x\u0000\u00001) (14)\nBFe= \u0001mFe\n0x+\n(x\u0000\u0000x+)(1\u0000\u0001mNi\n0\n\u0001mFe\n0x+) (15)\nANi= \u0001mFe\n01\n(x\u0000\u0000x+)(\u0001mNi\n0\n\u0001mFe\n0x\u0000\u00001) (16)\nBNi= \u0001mFe\n01\n(x\u0000\u0000x+)(1\u0000\u0001mNi\n0\n\u0001mFe\n0x+) (17)\nwith \u0001mFe\n0and \u0001mNi\n0corresponding to the maximum\namplitude of demagnetization. It is important to notice\nthat\u001c\u0006only corresponds to \u001cFe;Niin the very low tem-\nperature range, when 1 =\u001cNiFe\nexch and 1=\u001cFeNi\nexch are negligible.\nIn the range of temperatures explored in our experiment,\ndue to IEI, the dynamics of each sublattice is a clear bi-\nexponential decay as shown in eq.12 and 13 where the\ndemagnetization time is a composition of \u001c\u0000and\u001c+.\nHaving in hands the T=TCvalues equivalent to pump\n\ruences from amplitudes of demagnetization of our ex-\nperiment, we can analyze our results in the frame of the\nlinearized LLB model. As shown earlier, the pump \ru-\nence range used in our experiments corresponds to an\nintermediate temperature range 0 :35<T=TC<0:52.\nFigure 4 shows a comparison between theoretical val-\nues obtained from multiscale LLB approach in [32] and\nour experimental ones of (a) the subsequent components\nof the LLB matrix Akand (b)\u001c+= 1=\u0000+,\u001c\u0000= 1=\u0000\u0000,\nwhich are the characteristic times of the bi-exponential\nsolutions of di\u000berential equations that govern the two\nsublattices dynamics. For each pump \ruence and corre-\nsponding laser induced temperature, a global \ftting pro-\ncedure is used simultaneously for both Fe and Ni M-edges\ndemagnetization measurements to retrieve the Akmatrix\nelements plotted as symbols in \fgure 4(a). From those\nvalues,A\u000f;B\u000fand\u001c\u0006can be calculated and injected\nin equations (12) and (13). An example of the corre-\nsponding \u0001 mFe=mFeand \u0001 mNi=mNicurves are plotted\nas lines in the inset of \fgure 4(a). It reproduces very well\nthe observed demagnetization dynamics in Py measured\nat M-edges of FeandNisublattices. Moreover, a very\ngood agreement is obtained between the theoretically\npredicted values of the Akelements and those extracted\nfrom our experiments \fts (shown as lines in \fgure 2). The\nnon diagonal elements of Akretrieved from experiments\ncon\frm that IEI mediated dissipation is much stronger\nfor Ni sublattice compared to Fe (\u0000 NiFe>\u0000FeNi). In-\ndeed, the IEI induced modi\fcations of both sublattice5\n0.1110 Γ (ps-1)\n0.80.60.40.20.0\n T/TcΓNiFe\nΓFeNi\nΓNiNi\nΓFeFe\n1.0\n0.9\n0.8\n0.7Δm/m\n5000\n τ (fs)Fe\nNi\n 150\n100\n50τ (fs)\n0.70.60.50.40.3\n T/Tcτ+\nτ−(a)\n(c)(b)\n120\n100\n80\n60\n40τintra (fs)\n0.70.60.50.40.3\n T/TcFe\nNi\nFIG. 4: Comparison of theoretical data from [32] (lines) and\nexperimental values (dots) of a) LLB matrix components for\npump energy densities of T=TC= 0.35, 0.42 and 0.52. Ex-\nperiment (markers) and theory (lines). Inset: example of M-\nedge demagnetization global \ftting (lines) giving rise to LLB\nmatrix element for 3.8 mJ/cm\u00002pump \ruence. b) charac-\nteristic times of M edges magnetization dynamics \u001c+(empty\nred dots),\u001c\u0000(full blue dots) and c) retrieved \u001cNi\nintra(full blue\ndots),\u001cFe\nintra(empty red dots), from experiment and compari-\nson with theoretical values as a function of temperature.\ndynamics are not the same, due to element dependence\nof \u0000FeNi;NiFe (via element dependence of \u000bFe;Ni\nk,\u0016Fe;Ni\nand\rFe;Ni). In order to discuss the intrasublattice dissi-\npation (without the contribution of IEI), we extract the\nintrasublattice demagnetization time from our measure-\nments, as shown in \fgure 4(c). It is deduced from the ex-\nperimentally retrieved values of Akmatrix elements and\nfrom equations 4 and 7 by taking a constant ratio of mag-\nnetic susceptibilities \u001fFe\nk=\u001fNi\nk= 2 (valid in the interme-\ndiate temperature range i.e T=TC<0.5 [32]). Up to T=TC\n= 0.52,\u001cNi\nintra\u0000\u001cFe\nintra<15 fs. Without IEI (\fgure 4(c)),\nFe sublattice undergoes a stronger dissipation. This dis-\nparity between intra sublattice dissipations is compen-\nsated by strong IEI leading to a common dynamics of\nboth sublattices (\fgure 4(b)). The above approach has\nthe advantage to explain the observed dynamics strongly\nin\ruenced by IEI. It shows that IEI mediated dissipation\ndoesn't have necessarily the same weight on each sub-\nlattice. Moreover, the rate of intrasublattice dissipation\nmediated by IEI is related to sublattices magnetic sus-\nceptibilities ratio, that is almost constant for moderate\nlaser induced temperature and diverges close to TC. Let\nus discuss this substantial di\u000berence with the de\fnition\nof a unique sub 10 fs exchange time observed in previous\nwork. Indeed, in the pioneer study proposed by Mathias\net al [24], an exchange interaction time is introduced as\na constant parameter that couples the two subsystems\nmagnetization dynamics. The key di\u000berences are based\non the following points. In ref [24], the process of IEI\nis considered in a conservative manner with equal ratesof magnetic momentum transfer between the two sub-\nlattices. The exchange interaction times are equal for\nboth sublattices and independent of \ruence. Finally the\ndata analysis, performed in this framework, imposes a\nstrong di\u000berence between \u001cFeand\u001cNi(See supplemen-\ntary informations of [24] ). The LLB based approach\nis fundamentally di\u000berent since it considers the e\u000bect of\nIEI as related to the conservative transfer of momentum\nbetween sublattices, but also to dissipation (eq. 4 - 7).\nSecondly, with LLB approach, these contributions are\nboth found element and temperature dependent (eq. 9)\ndue to longitudinal damping and magnetic susceptibili-\nties. The outcome analysis allows retrieving the intra-\nsublattice demagnetization times: between 80 and 100\nfs for both Fe and Ni, which is consistent with earlier\nobservations [7, 38].\nIV. ULTRAFAST MAGNETIZATION\nPRECESSION AND DAMPING IN PERMALLOY\nPROBED AT M-EDGES OF NI AND FE\nWe have shown the in\ruence of intersublattice ex-\nchange interaction on longitudinal magnetization dynam-\nics occuring on the hundreds of femtoseconds time scale.\nWe now address the question of how IEI does a\u000bect the\ndamping of transverse motion of magnetization vector,\nie precessional motion over hundreds of picoseconds. In\nthe following, the IR pump XUV probed TMOKE exper-\niments are performed on a 500 ps temporal range with\na tilt of the external magnetic \feld axis with a 10\u000ean-\ngle with respect to the sample plane. This con\fguration\nallows for transverse projection measurement of magneti-\nzation precession, simultaneously at M 2;3edges of Fe and\nNi. Both Kerr rotation signals \u0001 \u0012K=\u0012Kintegrated over\nh35 (Fe) and h43 (Ni) have been \ftted using the \ftting\nfunction:ANi,Fesin(2\u0019=TNi,Fe\nprt+\u001eNi,Fe)exp(\u0000t=TNi,Fe\nd) +\nBNi,Fe, whereANi,Fe,TNi,Fe\npr,\u001eNi,Fe,TNi,Fe\ndare respec-\ntively the precession amplitudes, periods, phases and\ndamping times of each sublattice. BNi,Feis an o\u000bset that\ncorresponds to long time delay magnetization recovery\ncompared to the temporal window of our measurements.\nAs seen in \fgure 5, the Ni and Fe momenta precession\nare measured selectively in Ni 80Fe20for three incident\npump \ruences corresponding to initial 25% to 35% of\nlaser induced demagnetization. The precession motion\nat Ni edge has a slightly higher amplitude. While the\nprecession signals are increased in amplitude with pump\n\ruence, Fe and Ni momenta still precess in phase. Within\nthe range of pump \ruences used, the precession period\nstays quasi constant TNi,Fe\npr\u0018150 ps. The damping time\nremains identical for both sublattices. It is found to be of\nabout 2 ns for the intermediate \ruence and is decreased\ndown to 300 ps at the highest \ruence. For the lowest \ru-\nence the damping time is di\u000ecult to extract within our\ntemporal window and lower signal to noise ratio. It is\nis estimated higher than 2 ns. Such increase of damp-\ning can be explained by the increasing of phonon medi-6\n-60-40-200ΔθΚ/θΚ  (mrad)\n500 250 0\ndelay (ps) Ni\nFe\nFIG. 5: Precession measurements in permalloy probed with\nHigh order Harmonics. Fe M-edge (full red dots) and Ni M-\nedge (empty blue dots) magnetization dynamics of precession\nin Permalloy as a function of incident pump laser \ruence in-\ncreased from top to bottom: 1 :5 mJ/cm2; 2:5 mJ/cm2; 4\nmJ/cm2.\nated spin \rip rate with increasing \ruence. Indeed, the\ngenerated phonon density increases with pump \ruence.\nPhonons cause crystal \feld \ructuations that translates to\nthe magneto-crystalline anisotropy and leads to random\ntorques on the spins. We now analyze the Gilbert damp-\ning in the frame of the Landau Lifshitz Gilbert (LLG)\nequation. In the time scale of the transverse damping\n(three orders of magnitude longer compared to the short\ntime scale of ultrafast demagnetization), the precession\nmotion of two coupled sublattices \u000fand\u000emagnetization\ncan be written for m\u000f;\u000e(m\u000e\nsbeing the saturation magne-\ntization of the second sublattice \u000e) as [39]:\n_m\u000f\n\r\u000f=\u0000\u0000\nm\u000f\u0002H0\n\u000f\u0001\n\u0000\u000b\u000f\u0002\nm\u000f\u0002\u0000\nm\u000f\u0002H0\n\u000f\u0001\u0003\n+A\u000f\u000em\u000e\ns\b\u0000\nm\u000f\u0002m\u000e\u0001\n+\u000b\u000fA\u000f\u000em\u000e\ns\u0002\nm\u000f\u0002\u0000\nm\u000f\u0002m\u000e\u0001\u0003\t\n(18)\nThe \frst line of equation (18) corresponds to preces-\nsion of magnetization and damping related to e\u000bective\n\feldH0\n\u000f=H0+Hanis, where\r\u000fis the gyromagnetic\nratio,\u000b\u000fthe Gilbert damping, H0andHanisare the ap-\nplied and anisotropy \felds. The second line corresponds\nto the coupling of precession motion and damping via\nIEIJ\u000f\u000ebetween sublattices \u000fand\u000e. It corresponds to\na contribution to the e\u000bective \feld H00\n\u000f=\u0000A\u000f\u000em\u000e\nsn\u000eof\nthe second sublattice \u000eon the \frst sublattice \u000f. The ex-\nchange sti\u000bness parameter is de\fned by A\u000f\u000e=J\u000f\u000e=\u0016\u000e\u0016\u000f.\nNote that when A\u000f\u000eis lower than other elemental ef-\nfective \feld contributions, the two sublattices precess in-\ndependently. In our case, when A\u000f\u000edominates, the re-sulting motion corresponds to a single coupled precession\nmotion. The corresponding Gilbert damping can be eval-\nuated from the damping time TPy\ndas a single value for\neach \ruence. Considering a circular precession motion\nwith small angles, one has:\n\u000bPy= 1=(TPy\nd!Py) (19)\nwith!Py= 2\u0019=Tprbeing the precession pulsation. From\nour measurements in Py at M-edges of Ni and Fe (\fg-\nure 5) and by taking mPy\ns= 8.4 106A m\u00001, one has:\n\u000bNi,Py\u0018\u000bFe,Py\u00140:012 for the two \frst \ruences and\n\u000bNi,Py\u0018\u000bFe,Py= 0:079 for maximal \ruence. Moreover,\nin a strongly exchange coupled alloy, its Gilbert damp-\ning can be estimated as an e\u000bective damping from pure\nelements parameters [40]:\n\u000bPy\ne\u000b=mFe\ns\rNi\u000bFe+mNi\ns\rFe\u000bNi\nmFes\rNi+mNis\rFe(20)\nwhere\u000bi(i = Ni,Fe) is the pure element damping. To\ncompare the estimated value of damping \u000bPyfrom HHG\nexperiment to the one as a composition of pure elements\n\u000bPy\ne\u000b, we have performed precession measurements in\ntwo 10 nm thick \flms of pure Ni and pure Fe, using a\nTMOKE con\fguration with 25 fs, 800 nm pulses. The\nmeasured precession damping times TNi, pure\nd,TFe,pure\nd,\nat \fxed initial 20% demagnetization, allows retrieving\nthe corresponding Gilbert damping \u000bNi,pureand\u000bFe,pure\nby using equation (19). By taking mFe,pure\ns = 1.72 106A\nm\u00001; mNi,pure\ns = 4.85 106A m\u00001, the following Gilbert\ndamping values are obtained in pure thin \flms: \u000bNi,pure\n= 0.05 and \u000bFe,pure= 0.016. The e\u000bective damping\nin Py as a composition of pure elements damping\nobtained by equation (20): \u000bPy\ne\u000b= 0:041 (with\rFe=\n2.12 105m s\u00001A\u00001and\rNi= 2.03 105m s\u00001A\u00001) is\nin good agreement with values found from equation\n19. Finally, one can notice that the common Gilbert\ndamping value measured at both Fe and Ni M-edges in\nPy is close to the highest pure Ni value. It indicates\nthat the dissipation of precession is dominated by Ni\nsublattice contribution. This could be attributed to\nthe higher spin orbit coupling in Ni compared to Fe,\ngiving higher spin lattice dissipative contribution [12, 13].\nV. CONCLUSION\nIn this work, magnetization dynamics in Py induced\nby a 1.5 eV femtosecond pump pulse and probed by HH\nis investigated with chemical selectivity on Ni and Fe\nsublattices over a wide temporal scale. The role played\nby the IEI, in the sublattices damping and precession,\nhas been explored in the intermediate spin temperature\nrange. First, we show that demagnetization dynamics\nmeasured at M edge of each Fe and Ni sublattices of\npermalloy is well reproduced in the LLB framework. The7\npump \ruence dependent dynamics of each sublattice is\ncharacterized by double exponential decay with charac-\nteristics times \u001c+and\u001c\u0000both relying on elemental sus-\nceptibilities and longitudinal damping. This approach\nallows to distinguish two contributions to the demagne-\ntization time measured at M edges of each sublattice:\nthe \frst one corresponds to intrasublattice dissipation\ngoverned by longitudinal damping and magnetic suscep-\ntibilities, the second one is the IEI mediated dissipation\nresponsible of the strongly coupled response observed in\nthis study. An interesting prospective could be to study\nmagnetization dynamics beyond this range of excitation\ndensities, where both sublattices are expected to show\ndi\u000berent dynamics as predicted by LLB model.\nSecondly, we have shown that not only the longitudi-\nnal magnetization dynamics of each sublattice is dom-\ninated by IEI but also the magnetization vector orien-\ntation through precession and transverse damping. The\nstrong IEI drives the two sublattices to share a singleprecession mode of which the damping is a composition\nof pure elements damping. 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Melkov, Magnetization Oscilla-\ntions and Waves (1996)."    },    {        "title": "1807.05216v1.On_the_classical_dynamics_of_charged_particle_in_special_class_of_spatially_non_uniform_magnetic_field.pdf",        "content": "On the classical dynamics of charged particle in\nspecial class of spatially non-uniform magnetic\n\feld\nRanveer Kumar Singh\u0003\nDepartment of Mathematics\nIndian Institute of Science Education and Research Bhopal\nIndore Bypass road, Bhauri, Bhopal Madhya Pradesh, India - 462066.\nE-mail: ranveer@iiserb.ac.in\nAbstract\nMotion of a charged particle in uniform magnetic \feld has been studied in detail,\nclassically as well as quantum mechanically. However, classical dynamics of a charged\nparticle in non-uniform magnetic \feld is solvable only for some speci\fc cases. We\npresent in this paper, a general integral equation for some speci\fc class of non-uniform\nmagnetic \feld and its solutions for some of them. We also examine the supersymmetry\nof Hamiltonians in exponentially decaying magnetic \feld with radial dependence and\nconclude that this kind of non-uniformity breaks supersymmetry.\nKeywords : Classical trajectory, Landau gauge, Hamiltonian formalism, Non-uniform\nmagnetic \feld, Supersymmetry\nPACS number:45.20.-d, 45.20.Jj\n1arXiv:1807.05216v1  [quant-ph]  13 Jul 20181. Introduction\nIt is known that a charged particle in a uniform magnetic \feld exhibits circular motion\nwith speci\fed frequency of revolution, called cyclotron frequency, and speci\fed radius which\ndepends on the charge and mass of particle and magnitude magnetic \feld [1]. Quantum\nmechanically, a charged particle in uniform magnetic \feld exhibits quantised energy levels,\ncalled Landau levels [2].\nIn classical arena, non-uniform magnetic \felds of di\u000berent kinds have di\u000berent e\u000bects\non dynamics of the charged particle. For instance, a charged particle in a magnetic \feld\nwith non zero gradient undergo Grad B drift [3]. Charged particle moving along curved\nmagnetic \feld experience centrifugal force perpendicular to magnetic \feld. Hence charged\nparticles experience drift in their motion. This kind of drift is called curvature drift [3].\nThere are several applications of these e\u000bects. One of the most important application is\nmagnetic mirrors, which are used to con\fne plasmas [4]. The motion of an electron in\na magnetic \feld of constant gradient has been analyzed by Seymour et al. , where he has\nderived the xandycoordinates of electron's trajectory in terms of elliptic integrals [5].\nThe non-uniform magnetic \feld required for an electron to exhibit trochoidal trajectory has\nbeen calculated and it turns out to be a function of xcoordinate only [6]. The quantum\nmechanical treatment of charged particle in a class of non-uniform magnetic \feld has been\nstudied using Isospectral Hamiltonian and Supersymmetric quantum mechanics [7]. This\nanalysis gives the same Landau Level spectrum [8] as in case of uniform magnetic \feld.\nAlthough classical trajectory of a charged particle cannot be solved for the most general\nnon-uniform magnetic \feld i.e,non-uniform in all the three coordinates but it can still be\nsolved for some classes of non-uniform magnetic \feld. It should be noted that throughout\nthe paper, only spatially varying magnetic \feld is considered without time dependence. We\nuse Landau gauge to restrict vector potential to just one component for constant magnetic\n\feld along zdirection. To inculcate non uniformity in magnetic \feld, we introduce some\nfunction in vector potential. Using elementary classical mechanics, we obtain an integral\n2equation which can be solved to get the xandycoordinates of the particle's trajectory.\nLastly, we examine the sypersymmetry structure of non-uniform Hamiltonians and observe\nthat exponentially decaying magnetic \feld with radial dependence breaks supersymmetry.\n2. Charged Particle in Uniform Magnetic Field\nFirst, we use the quadrature to compute the trajectory of a charged particle in a uniform\nmagnetic \feld, which is known to be circular. Let us assume a constant magnetic \feld in z\ndirection i.e.~B=B^k. Vector potential for constant magnetic \feld is given by,\n~A=\u00001\n2(~ r\u0002~B) (1)\nDirect calculation for the above magnetic \feld gives Ax=\u0000yB=2 andAy=xB=2 and the\nzcomponent is 0. We can choose Landau gauge to reduce Ato one component such that\neitherAx=\u0000yBorAy=xB. Let us take Ax=\u0000yB. Thus Lagrangian for the system can\nbe written as,\nL=P\ni(1\n2m_q2\ni+qA\u0001_qi)\nwhere the symbols have their usual meaning. Here, we can assume no motion in zdirection.\nSo expanding the Lagrangian gives the following expression.\nL=1\n2m( _x2+ _y2)\u0000qyB_x (2)\nFrom Eq. 2, it is evident that xis a cyclic coordinate so px=@L\n@_xis conserved [1]. Thus we\nhavepx=@L\n@_x=m_x\u0000qBy =cor\n_x=k1+k2y (3)\nwherek1=c=m andk2=qB=m . Since Lagrangian has no explicit time dependence, thus\nthe total energy of the system is also a constant of motion. We can get the Hamiltonian by\n3using the following expression,\nH=X\ni(pi_qi\u0000L) (4)\nwherepi=@L\n@_qi. After puttingLin Eq. 4, we have,\nH=1\n2m( _x2+ _y2) (5)\nFrom Eq. 3 and Eq. 5 we get,\n_y=\u0006q\nk3\u0000k2\n1\u0000k2\n2y2\u00002k1k2y (6)\nwherek3= 2H=m. Thus we have,\n\u0006dyp\n\u000b+\fy+\ry2=dt (7)\nwhere\u000b=k3\u0000k2\n1,\f=\u00002k1k2and\r=\u0000k2\n2. Integrating both sides with appropriate\nlimits we get (one can check in integral table),\n1p\u0000\rcos\u00001[\u0000(\f+ 2\ry)=pq] =\u0006t (8)\nwithq=\f2\u00004\u000b\r. Putting all the values and after simplifying we get,\ny=pk3\nk2cosk2t\u0000k1\nk2(9)\nFrom Eq. 3 we can integrate and calculate xto get :\nx=pk3\nk2sink2t (10)\nEqs. 9 and Eq. 10 de\fne a circle (as can be checked easily by squaring and adding) with\nde\fned frequency of revolution, called cyclotron frequency !=k2=qB=m . The radius\n4of the orbit can be calculated from Eq. 9 and Eq. 10. Thus we get, r=pk3\nk2=mv=qB\nby noting that if energy of the system is conserved then H=mv2=2 wherevis the initial\nvelocity of the particle.\n3. Charged particle in a non-uniform magnetic \feld\nWe now treat the general case of non-uniform magnetic \feld. We introduce a function\ndepending on yin the Landau gauge as Ax=\u0000yBf(y) which gives us non-uniform magnetic\n\feld. The magnetic \feld can be calculated easily using B=r\u0002Awhich gives,\n~B= (yBf0(y) +Bf(y))^k (11)\nwheref0(y) =df(y)\ndy. Eq.11 represents a special kind non-uniform magnetic \feld. If we\nspecifyf(y) we obtain di\u000berent classes of non-uniform magnetic \feld. We can again, assume\nno motion along zaxis. Lagrangian for this case can be written as,\nL=1\n2m( _x2+ _y2)\u0000qyBf (y) _x (12)\nConsidering symmetry of Lagrangian along xwe havepx=@L\n@_x=m_x\u0000qByf (y) =cor :\n_x=k1+k2yf(y) (13)\nwherek1=c=m andk2=qB=m . Hamiltonian for the system can be calculated using Eq.\n4 and simple calculation gives Eq. 5. Here the Hamiltonian is a constant of motion as the\nLagrangian has no explicit time dependence. From Eqs. 5 and 13 we have,\n_y=\u0006q\nk3\u0000k2\n1\u0000k2\n2y2f2(y)\u00002k1k2yf(y) (14)\n5Eq. 14 gives us the desired integral equation which can be solved for di\u000berent f(y) and in\nturn for di\u000berent classes of non-uniform magnetic \feld to get yas a function of t. Thisy\ncan be substituted in Eq. 13 to get xas a function of t. These two equations de\fne the\ntrajectory of the particle. The integral equation is :\nZdyp\nk3\u0000k2\n1\u00002k1k2yf(y)\u0000k2\n2y2f2(y)=\u0006Z\ndt+K (15)\nwhereKis the constant of integration.\n3.1 Special Cases\nWe can check that Eq. 15 yields correct results for some special cases.\nCase 1:f(y)=1\nIn this case we get constant magnetic \feld along zdirection as can be checked by putting\nf(y) = 1 in Eq. 11. Putting f(y) = 1 in Eq. 15 gives back Eq. 7 which we have already\nsolved. Thus we get circular trajectory for constant magnetic \feld.\nCase 2:f(y) = 1=y\nFor this case, We can calculate that ~B= 0. This means particle must go undeviated i.e,the\ntrajectory should be a straight line. Putting f(y) = 1, Eq. 15 gives,\nRdyp\nk3\u0000k2\n1\u00002k1k2\u0000k2\n2=\u0006t+K\nThe denominator is just a constant (say a). So we get,\ny=at+K0whereK0=aK\nIt can be noted that both the signs + and \u0000give the same trajectory. We can calculate x\nfrom Eq. 13 which comes out to be :\nx=bt+Dwhereb=k1+k2andDis a constant of integration.\n6xandyindeed de\fne a straight line. We can check that _ x2+ _y2=v2wherevis the initial\nvelocity of the particle.\nWe can extent the same procedure for a non-uniform magnetic \feld in xcoordinate. In\nthat case we use the Landau gauge in which Ax= 0 =AzandAy=xBf(x). Magnetic \feld\nin this case takes the following form,\n~B= (xBf0(x) +Bf(x))^k (16)\nAccordingly Lagrangian assumes the form given by,\nL=1\n2m( _x2+ _y2) +qxBf (x) _x (17)\nNowyis a cyclic coordinate so that py=@L\n@_y=m_x+qBxf (x) =cor\n_y=k1\u0000k2xf(x) (18)\nwherek1=c=m andk2=qB=m . Hamiltonian of the system remains the same. Invoking\nenergy conservation and using Eqs. 17 and 5 we obtain,\n_x=\u0006q\nk3\u0000k2\n1\u0000k2\n2x2f2(x) + 2k1k2xf(x) (19)\nThis gives us another integral equation which can be solved to obtain particle's xcoordinate\nas a function of time which can be substituted in Eq. 17 to obtain particle's ycoordinate.\nThus we can obtain particle's trajectory. The integral equation is given as :\nZdxp\nk3\u0000k2\n1+ 2k1k2xf(x)\u0000k2\n2x2f2(x)=\u0006Z\ndt+K (20)\nwhereKis the constant of integration. The above cases give us the same result as it should\nbe.\n74. Exponentially decaying Magnetic \feld\nAlthough Eq. 15 can be solved for di\u000berent kinds of non uniform magnetic \felds, but let's\nconsider an exponentially decaying magnetic \feld. Let the magnetic \feld be given by,\n~B=Be\u0000y^k (21)\nwhereBis a constant. From Eq. 11 we can solve for f(y). Thus we have,\nBe\u0000y=yBf0(y) +Bf(y)\nThis is an ordinary di\u000berential equation. Solution to this is given by,\nf(y) =c\ny\u0000e\u0000y\ny(22)\nWe can \fx c= 1 so that,\nf(y) =1\ny(1\u0000e\u0000y) (23)\nmagnetic \feld still remains the same. Putting this in Eq. 15 gives,\nZdyp\nk3\u0000k2\n1\u00002k1k2(1\u0000e\u0000y)\u0000k2\n2(1\u0000e\u0000y)2=\u0006Z\ndt+K (24)\nLet us put a=k3\u0000k2\n1,b=k2\n2andc= 2k1k2then the integral has the solution given by :\narcsin\u0010\n2(c+b\u0000a)ey\u0000c\u00002bp\nc2+4ab\u0011\n(p\nc+b\u0000a)+K0=\u0006t+K (25)\nTo make equations look simpler let's put c+b\u0000a=\u000b2,c+ 2b=\fand by calculation\nc2+ 4ab= 2k2pk3. Constants of integration can be manipulated such that K0=Kso the\nsolution after rearranging the terms looks as,\ny= log\u0012pk3k2\n\u000b2sin(\u0006\u000bt) +\f\n2\u000b2\u0013\n(26)\n8From Eq. (13) we can calculate xas,\nx=Z\u0012\nk1+k2\u0000k2\nlsin(\u0006\u000bt) +m\u0013\ndt (27)\nwherel=pk3k2\n\u000b2,m=\f\n2\u000b2. Solution of this equation is,\nx= (k1+k2)t\u0000k22 arctan\u0012\nmtan(\u000bt\n2)\u0006lp\nm2\u0000l2\u0013\n\u000bp\nm2\u0000l2(28)\nSimple calculation gives,\nx= (k1+k2)t\u00002 arctan \n(m=l) tan\u0000\u000bt\n2\u0001\n\u00061p\n(m=l)2\u00001!\n(29)\nin terms of original constants we have,\nx= (k1+k2)t\u00002 arctan \n(k1+k2) tan\u0000\u000bt\n2\u0001\n\u0006pk3p\n(k1+k2)2\u0000k3!\n(30)\nEqs. 26 and 30 de\fne the trajectory of the particle. It is worth noting that if we have a\nparticle with zero energy in magnetic \feld then it will remain stationary since magnetic \feld\ndoes no work. In this case this is indeed true. If we take zero energy then k3= 0 and a bit of\ncalculation shows that x= 0 andy=log(k2)=constant. Thus the particle remains stationary\non point (0, log( k2)).\nNow to analyze particle's trajectory in this \feld let's assume the following : k3= 1\n,k2= 0:1 (we have taken speci\fc charge of the particle to be 1 and assumed a strong\nmagnetic \feld of 0 :1T!) ;k1+k2= 1:118. This gives us,\nx= 1:12t\u00002 tan\u00001(2:24 tan(t=4) + 2);\ny= log(0:08 sin(t=2) + 0:1)\nWe plot these parametric equation for t2(0;50). In this Fig. 1 and the \fgures that follow,\nthe horizontal axis is the x-axis and the vertical axis is the y-axis. The particle shows periodic\n9Figure 1: Particle's trajectory in exponentially decaying magnetic \feld ~B=Be\u0000y^k\nmotion as can be inferred from the trajectory. We plot trajectory curves for several values\nof constants. There are several forms of f(y) for which solution to Eq. 15 exists and thus\nsuch class of non-uniform magnetic \feld can be analyzed easily.\n5. Supersymmetry in Uniform and Non-Uniform\nMagnetic Field\nNow we turn to the quantum mechanical treatment of the problem. The Pauli-Hamiltonian\nfor a charged particle moving in two dimensions in a magnetic \feld [9] is given by,\n2H= (px+Ax)2+ (py+Ay)2+ (r\u0002A)z\u001bz (31)\nwhere\u001bzis the Pauli z matrix. We have used natural units with \u0016 h= 1 =m. Suppose, we\nchooseAx=\u0000Byf(r) andAy=Bxf(r) wherer=p\nx2+y2then Eq. 31 has the following\nform,\n2H=\u0000\u0010d2\ndx2+d2\ndy2\u0011\n+B2r2f2\u00002BfLz+ (2Bf+Brf0(r))\u001bz (32)\nwhereLzis the z-component of the orbital angular momentum operator. We use cylindrical\ncoordinates ( r;\u001e) to solve the corresponding Schr odinger equation. The wave function  (r;\u001e)\n10(a)k1= 1; k2= 0:1; k3= 4\n (b)k1= 2; k2= 0:2; k3= 8\n(c)k1= 2; k2= 2; k3= 8\n (d)k1= 2; k2= 4; k3= 8\nFigure 2: Plots of trajectory curves for di\u000berent values of constants\ncan be factored as,\n (r;\u001e) =R(r)eim\u001e(33)\nwherem= 0;\u00061;\u00062;\u00063;::: are the eigenvalues of the operator Lz. On substituting Eq.33\ninto Eq. 32 we obtain the following equation,\nd2R\ndr2+1\nrdR\ndr\u0000h\nB2r2f2+m2\nr2+ 2Bmf + (2Bf+Bpf0(r)\u001bz)i\nR(r) =\u00002ER(r) (34)\nIf we further substitute R(r) =prA(r) into Eq. 34 and choose the lower eigenvalue of \u001bz,\nwe obtain :\n\"\n\u0000d2\ndr2+ \nB2r2f2\u00002Bf+ 2Bmf\u0000Brf0(r) +m2\u00001\n4\nr2!#\nA(r) = 2EA(r) (35)\n11We can write the left hand side in the form ayawhere\na=d\ndr+Brf\u0000jmj+1\n2\nr(36)\nForm\u00140, the decomposition holds and thus we have that E0\u00150 [10].E0= 0 occurs if\nand only if the solution of the equation a 0(r) = 0 i.e,\n\"\nd\ndr+Brf\u0000jmj+1\n2\nr#\n 0(r) = 0 (37)\nis square integrable [10]. In that case Supersymmetry (SUSY) remains unbroken. Now\nconsider the following cases :\n1.Uniform magnetic \feld\nIf we choose f(r) = 1 above, we end up having a uniform magnetic \feld. In this case,\nwe can solve Eq. 37 to get :\nd 0(r)\ndr= \njmj+1\n2\nr\u0000Br!\n 0(r) (38)\nd 0(r)\n 0(r)= \njmj+1\n2\nr\u0000Br!\ndr (39)\nSolution to Eq. 39 is given by\n 0(r) =N0rjmj+1\n2exp\u0010\n\u00001\n2Br2\u0011\n(40)\nwhereN0is the normalization factor. It is easy to see that  0(r) is square integrable as\nthe polynomial factor is dominated by the exponential and overall integral is conver-\ngent. Thus in this case the SUSY remains unbroken. It is also known that in uniform\nmagnetic \feld, the energy spectrum is same as that of a harmonic oscillator oscillating\nwith cyclotron frequency.\n122.Non-Uniform magnetic \feld\nThere are several forms of f(r) for which SUSY remains unbroken. For instance the\nfunctionf(r) =(r\u0000a)(r\u0000b)\nr2 keeps the SUSY unbroken. Now consider the function\nf(r) =1\nr(1\u0000e\u0000r) (41)\nFor this function, \frst note that ~B=r\u0002~A= (yBf0(y) +Bf(y))^k=Be\u0000r^k. Thus\nfor this function we obtain exponentially decaying magnetic \feld, this time with radial\ndecay, which we considered in previous section for classical treatment. Now we deal\nwith the peculiarity of this form. We can solve Eq. 37 with this function and a bit of\ncomputation gives\n 0(r) =N0rjmj+1\n2exp\u0002\n\u0000B(e\u0000r+r)\u0003\n(42)\nAgain, we can easily see that  0(r) is not square integrable as the integral diverges.\nThus we have E0>0 and thus SUSY is broken . Energy spectrum still remains discretly\nquantised [10] and can be determined using\nZ1\n0p\n2m[En\u0000W2(r)]dr=\u0010\nn+1\n2\u0011\n\u0016h\u0019;n = 0;1;2;3;::: (43)\nwhereW(r) is given by\nW(r) =\u0000\u0016hp\n2m 0\n0(r)\n 0(r)(44)\nThus we see that exponentially decaying magnetic \feld is really peculiar as it breaks\nsupersymmetry. Further experiments may be designed to detect this kind of symmetry\nbreaking.\n136. Discussion\nThe analysis presented in this paper provides an important recipe to \fnd the trajectory of\na charged particle in spatially varying magnetic \feld for some special classes of non uniform\nmagnetic \feld. The more general case of solving the trajectory still remains an open problem.\nNonetheless the example presented provides a way to solve the trajectory in exponentially\nvarying magnetic \feld. We also observe that exponentially decaying magnetic \feld shows\nspecial properties in the sense that the ground state of the isospectral partner of the non-\nuniform Hamiltonian with exponential non-uniformity has non-zero ground state energy\nand thus it breaks supersymmetry. Experiments may be designed to detect this feature of\nexponentially decaying magnetic \feld. We also note that the results of the paper may be\nused in various other cases. Physical situations where non-uniform magnetic \feld appears\nis that of the earth itself. New insights can be gained by studying the trajectory of charged\nparticles released from the sun in the earth's magnetic \feld. Study of the van Allen belts\ncan be done based on this theory.\n7. Conclusion\nIn conclusion, here we have presented a method to analyze the motion of a charged particle\nin various classes of non-uniform magnetic \feld. We also presented an speci\fc non-uniform\nmagnetic \feld with peculiar properties classically as well as quantum mechanically. Although\nthe integral equation presented do not have trivial solution for many forms of f(y) or equiv-\nalentlyf(x) but for such cases numerical integration can be used to \fnd out the trajectory.\nIt is observed that many forms have exact solution and thus it is useful to further study this\nmethod and generalize it to other forms.\n148. Acknowledgements\nThis work was carried out at Panjab University Chandigarh. The author is indebted to Prof.\nC. N. Kumar whose guidance helped to complete this work. The author also thanks Ms.\nHarneet Kaur, Dr. Amit Goyal, and Mr. Shivam Pal for useful discussions.\n9. References\n[1] Goldstein, H.; Poole, C. P. and Safko, J. L. Classical Mechanics (3rd ed.). Addison-\nwesley (2001)\n\u0002\n2\u0003\nLandau, L. D. and Lifschitz, E. M.; Quantum Mechanics: Non-relativistic Theory .\nCourse of Theoretical Physics. Vol. 3 (3rd ed. London: Pergamon Press). ISBN 0750635398\n(1977)\n\u0002\n3\u0003\nBaumjohann, W. and Treumann, R. Basic Space Plasma Physics . ISBN 978-1-86094-\n079-8 (1997)\n\u0002\n4\u0003\nKrall,N. Principals of Plasma Physics . Page 267 (1973)\n\u0002\n5\u0003\nSeymour,P.W - Aust. Jour. Phys. 12; 309-14\n\u0002\n6\u0003\nRF Mathams, R.F. Aust. Jour. Phys . 17(4), 547 - 552\n\u0002\n7\u0003\nCooper,F.; Khare,A.; and Sukhatme,U. \\Supersymmetry in quantum mechanics\" ,Phys.\nRep. 251, 267-285, (1995)\n\u0002\n8\u0003\nKhare,A. and Kumar,C.N. Mod. Phys. Lett. A 8, 523-529 (1993)\n\u0002\n9\u0003\nKhare,A. and Maharana,J. Nucl. Phys. B224, 409 (1984)\n\u0002\n10\u0003\nCooper,F.; Khare,A.; and Sukhatme,U. Phys. Rep 251, 267-285, (1995)\n15"    },    {        "title": "1205.6506v1.Magnetization_dynamics_at_elevated_temperatures.pdf",        "content": "Magnetization dynamics at elevated temperatures\nLei Xu and Shufeng Zhang\nDepartment of Physics, University of Arizona, Tucson, AZ 85721, USA\n(Dated: May 30, 2022)\nAbstract\nBy using the quantum kinetic approach with the instantaneous local equilibrium approximation,\nwe propose an equation that is capable of addressing magnetization dynamics for a wide range of\ntemperatures. The equation reduces to the Landau-Lifshitz equation at low temperatures and to\nthe paramagnetic Bloch equation at high temperatures. Near the Curie temperature, the magneti-\nzation reversal and dynamics depend on both transverse and longitudinal relaxations. We further\ninclude the stochastic \felds in the dynamic equation in order to take into account \ructuation at\nhigh temperatures. Our proposed equation may be broadly used for modeling laser pump-probe\nexperiments and heat assisted magnetic recording.\nPACS numbers: 75.78.-n, 75.40.Gb\n1arXiv:1205.6506v1  [cond-mat.mes-hall]  29 May 2012I. INTRODUCTION\nThe phenomenological Landau-Lifshitz (LL) equation is the basis of powerful micromag-\nnetic codes for simulation of magnetic structure and dynamics in magnetic materials. The\nkey component of the LL equation is that magnetization relaxation during dynamic processes\nis described by a single damping parameter \u000b[1],\ndm\ndt=\u0000\rm\u0002He\u000b\u0000\r\u000b\nmm\u0002(m\u0002He\u000b) (1)\nwhere m(r;t) is the magnetization density vector which is a function of space and time,\nm=jmjis its magnitude, and He\u000bis the e\u000bective magnetic \feld including the magnetic\nanisotropic, magnetostatic and external \felds. The second term on the right side of the\nequation describes a phenomenological transverse relaxation since the magnitude of the\nmagnetization density mis conserved. Such transverse relaxation model is indeed a valid\napproximation because the magnetization m(the order parameter) of the ferromagnet is\nnearly independent of the magnetic \feld as long as the temperature is not too close to the\nCurie temperature.\nRecently, there is an emerging technological need to extend the LL equation to high\ntemperatures in order to model the dynamics near or above the Curie temperature for laser-\ninduced demagnetization (LID) [2, 3] and heat-assisted magnetic recording (HAMR) [4, 5].\nDue to the strong \ructuation of the magnetic momentum, one needs to reduce the size of\nmagnetic cells if one continues to use the LL equation to model the magnetization dynamics.\nWhen the cell size reduces to the ultimate smallest size of magnetic atoms, the so-called\natomistic LL equation which has the same form as the conventional LL equation [6] had\nbeen proposed,\ndSi\ndt=\u0000\rSi\u0002H\u0000\r\u000bSi\u0002(Si\u0002H) (2)\nwhere His an e\u000bective magnetic \feld (treated as a c-number) including a random \ructuating\n\feld and Siis the spin of ith atom which is treated classically. While the above atomistic LL\nequation might qualitatively capture some of static and dynamic properties near the Curie\ntemperature [7, 8], we point out below that the above atomistic LL equation has several\nfundamental problems.\nFirst, the spin Siin ferromagnetic metals such as Ni, Co, Fe and their alloys is usually\nsmall. The replacement of the quantum spin by the classical vector severely neglects the\n2quantum nature of the spin \ructuation of atomic spins. More importantly, the atomistic\nLL equation, Eq. (2), has no microscopic origin and it is fundamentally incompatible with\nquantum mechanics. For example, if one takes the case for Si= 1=2 (e.g., Ni). The second\n(\"damping\") term of Eq. (2) becomes \r\u000b(i\n2Si\u0002H\u00001\n2H) for a quantum spin and thus Eq. (2)\nbecomes\ndSi\ndt=\u0000\r(1 +i\u000b=2)Si\u0002H+\r\u000bH=2: (3)\nThis unphysical equation originates from the broken time-reversal symmetry inherited on\nthe atomistic LL equation.\nThe second di\u000eculty is that the atomistic LL equation is not derivable from an e\u000bective\nHamiltonian, even at the phenomenological level. If the atomistic LL equation has some\nvalidity, a microscopic or an e\u000bective Hamiltonian should exist. For example, if we construct\na spin Hamiltonian of the form H0/P\niSi\u0001(H+\u000bSi\u0002H), the equation of motion for Si\nwould bedSi=dt= (1=i~)[Si;H0] which results in an additional term compared to Eq. (2)\ndue to none-zero commutation [ Si;\u000bSi\u0002H]6= 0. On the other hand, if one takes the\nphenomenological Hamiltonian as H0/P\niSi\u0001(H+\u000bm\u0002H) where m=<Si>(note that\n<>denotes ensemble thermal averaging and thus mis a c-number), the result dynamics for\nSiwould be\ndSi\ndt=1\ni~[Si;H0] =\u0000\rSi\u0002H\u0000\r\u000bSi\u0002(m\u0002H): (4)\nThe above equation is precisely the original LL equation after the thermal averaging. Thus,\nthe macroscopic LL equation is derivable from an e\u000bective Hamiltonian while the atomistic\nLL equation is not.\nIn spite of above conceptual di\u000eculties in the atomistic LL equation, it has been shown\nthat the result derived from the atomistic LL equation with stochastic \felds is in agreement\nwith the Monte Carlo simulation [9]. We point out that this agreement is not surprising:\nfor equilibrium properties such as magnetic moment and critical exponents, the calculated\nresults are insensitive to the details of the \\damping\"; for dynamic properties such as reversal\ntime, the Monte Carlo steps are calibrated to \ft the real time in the atomistic stochastic\nLL equation [10]. Therefore, such agreement should not be interpreted as the proof of the\nvalidity of the atomistic LL equation.\nIn this paper, we propose an e\u000bective magnetization dynamic equation for a wide range\nof temperatures without assuming the presence of the atomistic LL equation for each atomic\n3spin. By using the equation of motion for the quantum density matrix within the instanta-\nneous local relaxation time approximation [11], we show that the magnetization dynamics\nfor ferromagnets can be cast in the form of the Bloch equation for paramagnetic spins [12].\nIn Sec.II, we explicitly derive the generalized Bloch equation and show that the equation is\nconsistent with the known dynamics at low and high temperatures. In Sec. III, we analyze\nthe longitudinal and transverse relaxations from our result, and apply our e\u000bective equation\nto study the magnetization reversal processes near Curie temperatures. Finally, we add\nnecessary stochastic \felds in the equation to capture the \ructuation of the dynamics.\nII. EFFECTIVE DYNAMIC EQUATION FOR FERROMAGNETS\nWe start with a density operator ^ \u001awhich may be written in the spinor form ^ \u001a=\u001a1+\u001b\u0001\u001a2\nwhere\u001a1and\u001a2are spin-independent and spin-dependent density operators, and \u001bis the\nPauli matrix vector. Within the instantaneous local relaxation time approximation, the\ndensity operator satis\fes the quantum kinetic equation [11]\nd^\u001a\ndt=1\ni~[^\u001a;^H]\u0000\u001a1\u0000\u0016\u001a1\n\u001cp\u0000\u001b\u0001\u001a2\u0000\u0016\u001a2\n\u001cs(5)\nwhere \u0016\u001a1and\u0016\u001a2are the instantaneous local equilibrium (ILE) densities; they are di\u000berent\nfrom the static equilibrium values. In electron transport theories, these ILE densities depend\non the local chemical potential \u0016(r) or the local electric \feld E(r;t) and they are in turn\nrelated to the densities themselves. For example, for spin dependent electron transport, the\ninclusion of the spin relaxation (third term of Eq. 5) leads to the well-known spin-di\u000busion\nequation for the spin dependent chemical potential (or spin density) [13]. In the present\ncase, these ILE densities are functions of the local e\u000bective magnetic \feld. At a given time,\nthe e\u000bective \feld consists of the ferromagnetic exchange, anisotropy, external, and classical\nmagnetostatic \feld; we will discuss these \felds in more details later. The two relaxation\ntimes\u001cpand\u001csrepresent the momentum and spin relaxation times; these two relaxation\ntimes control the electron charge di\u000busion (conductance) and spin di\u000busion (spin-dependent\ntransport). If we now consider an e\u000bective Hamiltonian ^H=^H0\u0000g\u0016\u001b\u0001Ht(t) where\u0016is the\nBohr magneton, ^H0is treated as an unperturbed Hamiltonian, we \fnd the self-consistent\nequation for the magnetization m\u0011g\u0016Tr(\u001b^\u001a) =g\u0016Tr\u001a2readily from Eq. (5),\ndm\ndt=\u0000\rm\u0002Ht\u0000m\u0000meq(Ht)\n\u001cs: (6)\n4where the ILE magnetization meq=g\u0016\u0016\u001a2is identi\fed as the thermal equilibrium value for\na given magnetic \feld Ht.\nAt the \frst sight, Eq. (6) is similar to the well known Bloch equation [12] that has been\nwidely used for understanding nuclear spin resonance experiments. In the Bloch equation,\nthe equilibrium magnetization meqis a known equilibrium state which is related to the\ndynamic susceptibility \u001f(!), i.e., meq=\u001fHextandmeqis independent of m(t). In the\npresent content, meqis not known a priori andmeqvaries with time. At any time t, there\nis an instantaneous equilibrium magnetization meqthat depends on the total magnetic \feld\nHt. To solve Eq. (6), one \frst needs to model the instantaneous local \feld Htand its relation\ntomeq.\nIn the conventional LL equation, the e\u000bective \feld He\u000bconsists of the external \feld, the\nanisotropy and the magnetostatic (dipole) \felds. The exchange \feld which comes from the\nferromagnetic exchange interaction between neighboring spins is included only when there\nis spatial variation in the magnetic domain structure. The uniform exchange term, Jm, is\nunimportant since it is parallel to the magnetization and it does not contribute to the LL\ndynamic equation. In the present case, however, the exchange interaction is the largest and\nmost important term in determining the instantaneous equilibrium magnetization meq. We\nthus model the total instantaneous magnetic \feld Ht=Jm+He\u000b. It is noted that He\u000b\ndepends on the instantaneous magnetization m(t) as well.\nNext, we should establish an explicit relation between the total \feld Htwithmeq. There\nare a number of approaches available to describe such relation. The simplest approach\nwould be using the molecular \feld approximation where the equilibrium magnetization can\nbe explicitly expressed by [14]\nmeq\u0011g\u0016< Si>=g\u0016SBS(\fg\u0016Ht)^Ht (7)\nwhereSis the spin of the atom, \f= (kBT)\u00001is the inverse of temperature, BS(x)\u0011\n(1=S)[(S+ 1=2) coth(S+ 1=2)x\u0000(1=2) coth(x=2)] is the Brillouin function and ^Htis the\nunit vector in the direction of Ht, i.e., ^Ht=Ht=Ht. In the time-independent case, m=meq\nand the above equation is the well-known mean-\feld result that determines the ferromagnetic\norder parameter meq. In a non-equilibrium situation where mdepends on time, we interpret\nmeqin Eq. (7), which is also dependent on time, as the instantaneous local equilibrium\nmagnetization at a given (instantaneous) \feld Ht.\n5Our proposed Eq. (6) supplemented by Eq. (7) can semi-quantitatively describe magneti-\nzation dynamics at all temperatures. Before we examine some limiting cases, we comment on\ncertain important approximations leading to these equations. The instantaneous relaxation\ntime approximation, Eq. (5), has been routinely applied to many quantum or semi-classical\nsystems for transport and magnetic properties. The accuracy of this approximation is hard\nto assess for the ferromagnetic systems. However, the instantaneous relaxation time approx-\nimation has been very successfully applied in spin di\u000busion of magnetic multilayers where\nthe semiclassical distribution function is assumed to relax to the instantaneous chemical\npotential [13]. Furthermore, the relaxation time approximation usually serves as a \frst step\nin a phenomenological theory since it gives rise an analytically closed form. The most severe\napproximation is to replace meqby the mean \feld Brillouin function, Eq. (7). Such ap-\nproximations are known to produce inaccurate critical exponents and Curie temperatures.\nThere are several much improved approaches such as the renormalization group theory [15],\nself-consistent random phase approximation [16], and Monte Carlo simulation [17]. While\nthese approaches treat the \ructuation near the critical temperature better, they are far more\ncomplicated and without an analytical form. On the other hand, the mean \feld approx-\nimation is qualitatively correct and it allows a much simpler description of magnetization\ndynamics in spite of underestimating the critical \ructuation. For the purpose of establishing\na phenomenological dynamic equation similar to the LL equation, we believe that the choice\nof the mean \feld approximation throughout this study is appropriate.\nSimilar to the LL equation, Eq. (6) contains a phenomenological parameter, \u001cs, repre-\nsenting the magnetic relaxation of paramagnetic spins. In transition metals, \u001csis related to\nthe spin-\rip time. In fact, there are a number of theoretical and experimental studies on the\nnumerical values of \u001csin di\u000berent materials [18{20]. For transition metals, the relaxation\ntime ranges from sub-picoseconds to a few picoseconds.\nIII. LONGITUDINAL AND TRANSVERSE MAGNETIZATION DYNAMICS\nBefore we proceed to solve Eq. (6) in a number of interesting examples, we examine\nseveral limiting cases. First, by using the identity\nHt=m\u00002[(m\u0001Ht)m\u0000m\u0002(m\u0002Ht)] (8)\n6we write Eq. (6) in terms of three mutually perpendicular vectors,\ndm\ndt=\u0000\rm\u0002He\u000b\u0000\r\u000btr\nmm\u0002(m\u0002He\u000b)\u0000\r\u000bl\nm(m\u0001Ht)m (9)\nwhere we have introduced the transverse and longitudinal dimensionless damping coe\u000ecients\n\u000btrand\u000bl,\n\u000btr=meq\n\r\u001csmHt(10)\nand\n\u000bl=1\n\r\u001cs\u0014m\nm\u0001Ht\u0000meq\nmHt\u0015\n: (11)\nAt low temperatures, mis close to g\u0016S and the exchange \feld Jmis much larger than\nthe other \felds He\u000b. Thus, one immediately has \u000btr= (\r\u001csJm)\u00001. In a typical transition\nferromagnet such as Co or Fe, Jis of the order of the Curie temperature (0.1-0.2 eV) and\n\u001csis a sub-picosecond, we \fnd \u000btris of the order of 10\u00003\u000010\u00001.\nTo estimate the low temperature longitudinal relaxation \u000blfrom Eq. (11), we consider\nan initialmdeviates from the equilibrium value of g\u0016SBSand from Eq. (11), \u000blwould be\nabout the same order of magnitude as \u000btr. However, the longitudinal \feld Jmis much larger\nthanHe\u000band thus the ratio of the longitudinal ( \u001cl) to the transverse ( \u001ctr) relaxation times is\nabout\u001cl=\u001ctr\u0019He\u000b=J. Even for a very high anisotropy material and a large magnetic \feld,\nJis several orders of magnitude larger than He\u000b; this justi\fes that at the low temperature\none can neglect the longitudinal relaxation in the dynamic equation, i.e., the magnitude of\nthe magnetization is always in equilibrium.\nWhen the temperature is much higher than the Curie temperature, Eq. (6) represents\nthe paramagnetic Bloch equation. In this case, the equilibrium magnetization meqmay\nbe expressed via susceptibility \u001f, i.e., meq=\u001fHe\u000b. Such dynamic equations have been\nfrequently used for understanding paramagnetic resonant phenomena where the resonance\nwidth is determined by the relaxation time \u001cs.\nThe most interesting case of Eq. (6) is for temperature close to Curie temperature where\ntransverse and longitudinal relaxation times could become comparable. To see this, we\nconsider the e\u000bective \feld is parallel to m(t) =m(t)ezand expand BS(x) = (S+ 1)x=3\u0000\n(1=90)(S+ 1)(2S2+ 2S+ 1)x3up to the third order in the small xwherex=\fg\u0016Ht. Then,\nEq. (6) for temperature close to the Curie temperature becomes\ndm\ndt=\u00001\nJ\u001cs\u0014\u0012\n1\u0000Tc\nT\u0013\nHt+3\n10J2T3\nc\nT3\u00121\nS2+1\n(1 +S)2\u0013\nH3\nt\u0000He\u000b\u0015\n(12)\n70.900.951.001.051.10101\n1\n0-11\n0-2 \n Relaxation Time τL (ns)N\normalized Temperature /s40T/Tc/s4110-3Heff=0TH\neff=3.0THeff=1.0TFIG. 1: (Color online) The longitudinal relaxation time \u001clas a function of temperature for several\nmagnetic \felds. We choose a small di\u000berence between mandmeqatt= 0 and identify t=\u001cl\nwhere the di\u000berence is reduced by the half. We have used \u001cs= 1 ps and S= 1=2.\nwhereTc=S(S+ 1)J(g\u0016)2=3kBis the mean \feld Curie temperature. In the absence of the\nmagnetic \feld He\u000b= 0 andHt=Jm, and we can immediately solve the above equation,\nm(t) =m(0)e\u0000t=\u001cl\u0002\n1 +G(1\u0000e\u00002t=\u001cl)\u0003\u00001=2(13)\nwherem(0) is the initial magnetization, G=3T3\ncm2(0)\n10T3\u0010\n1\nS2+1\n(1+S)2\u0011\n(1\u0000Tc=T)\u00001, and\n\u001cl=\u001cs\u0012\n1\u0000Tc\nT\u0013\u00001\n: (14)\nThus, the longitudinal relaxation time, j\u001clj;near Curie temperature, is associated with the\ncritical phenomenon. The relaxation time becomes very long when the temperature ap-\nproaches the Curie temperature. The dynamics slow-down at the critical temperature is in\nfact a general property of critical phenomena [21]. In the presence of the magnetic \feld, the\nphase transition becomes a smooth change and the dynamic slow-down is no more critical.\nIn Fig. 1, we show the longitudinal relaxation as the function of the magnetic \feld and\ntemperature. Clearly, the magnetic \feld suppresses the longitudinal dynamic slowdown. It\nis noted that the peak of the relaxation time in the presence of the magnetic \feld is shifted\nto higher temperatures.\nIn order to gain more quantitative insight for the interplay between the transverse and\nlongitudinal relaxations, we consider several simple cases where the numerical calculations\n80.80 .91 .01 .1123450\n.80.91.01.10.00.51.01.52.00.00.51.01.52.02.53.0-0.3-0.2-0.10.00.10.20.3 \n   \nN ormalized Temperature /s40T/Tc/s41τLL/τnew(b) \ntransverse only (LL) \nnew equation \n Reversal Time τ (ns)N\normalized TemperatureLLN\newHext=1.0THext=1.0T transverse only (LL) \nnew equation   \nMagnetizationT\nime (ns)T=0.9TcT\n=1.0TcLLN\newLLN\new(a)FIG. 2: (Color online) a) The time dependence of magnetization reversal when a reversal magnetic\n\feld is applied at t >0 for the temperature T= 0:9TcandT= 1:0Tcas indicated. The blue\n(solid) and red (dashed) curves were obtained by Eq. (6) and by the \frst two terms of Eq. (9),\nrespectively. b) the reversal times and their ratio as a function of the temperature obtained from\na). The parameters are \u001cs= 1:0 ps,S= 1=2, the anisotropy constant K= 0, and the external\n\feldH= 1:0 T.\ncan be readily performed. We assume that the magnetic particle is a single domain so that\nthere is no spatial dependence of the e\u000bective \feld and the magnetization. Furthermore, the\nlong-range magnetostatic \feld is also discarded. In the \frst case, we compare the reversal\ntimes with and without the longitudinal relaxation in a simplest case: an isotropic magnetic\nparticle (zero magnetic anisotropy) is initially magnetized at 5\u000efrom +zaxis and a reversal\nmagnetic \feld in the direction of \u0000zis applied at t>0. Figure 2(a) shows the importance\n9of the longitudinal relaxation when the temperature approaches Curie temperature. We\ncompare the magnetization dynamics with and without the last term of Eq. (9). If the\ntemperature is considerably below the Curie temperature, e.g., T= 0:9Tc, the longitudinal\nrelaxation term has a negligible e\u000bect, i.e., the result is essentially same whether the last term\nof Eq. (9) is included. This is because the magnitude of the magnetization is nearly time-\nindependent at low temperature. When the temperature is near the Curie temperature, the\nmagnitude of the magnetization is signi\fcantly reduced. More importantly, the magnitude\nis now a function of time due to its dependence on the total e\u000bective \feld. In this case, there\nis a much di\u000berence if one includes the longitudinal relaxation. In Fig. 2(b), we show the\nratio of the reversal times calculated with and without the longitudinal relaxation. Clearly,\nthe reversal time from Eq. (6) is much faster than that of the LL equation if the temperature\nis close to or higher than Curie temperature.\nNext we apply our equation to a hypothetical HAMR process when the laser heating and\nthermal di\u000busion produce a time-dependent temperature pro\fle: the temperature of the\nparticle increases linearly T(t) =Trm+(t=theat)(Tp\u0000Trm) from the room temperature Trmto\na peak value Tpfor the period of 0 <t<t heatof lasing application. After the heating process\nis completed and the laser is removed, the temperature decreases due to heat di\u000busion into\nsurroundings. We assume the temperature is T(t) =Trm+ (Tp\u0000Trm) exp[\u0000(t\u0000theat)=tcool]\nfort>t heat. While the precise temperature pro\fle should be determined via heat transport\nequations with proper boundary conditions, our hypothetical temperature is characterized\nby three parameters: the peak temperature of the particle Tp, and the heating and cooling\nrates 1=theatand 1=tcool. We choose the low-temperature magnetic anisotropy \feld much\nlarger than the external magnetic \feld so that the magnetic reversal does not occur at the\nroom temperatures. The temperature dependence of the anisotropy energy Eais modeled by\nEa=Km2(T) sin2\u0012, wherem(T) is the magnitude of the magnetization at temperature T\nand\u0012is the angle between the magnetization vector and z-axis [22, 23]. By placing the above\ntemperature pro\fle and e\u000bective magnetic \felds into Eq. (6), we have numerically calculated\nthe time dependent magnetization shown in Fig. (3). As we expected, the magnetization\nreversal requires a high peak temperature Tpto reduce the anisotropy. The rates of heating\nand cooling are also important; they should be slow enough so that the magnetization has\nsu\u000ecient time to relax to the ground state via transverse and longitudinal relaxations.\nMore quantitatively, we have made two comparisons in Fig. 3. First, we compare our\n100.00 .20 .40 .60 .8-0.4-0.20.00.20.40.00 .20 .40 .60 .8-0.4-0.20.00.20.40\n.00.51.01.52.00.00.51.01.52.0α\n=0.1K\n=8.0T  H\next=1.5T(b)T\np=1.2TcTempertureTp=1.05Tc \n MagnetizationT\nime (ns)Tp=0.95TcT\nT\npr\nmK=8.0T  H\next=1.5T(a)T\nempertureTp=1.2TcTp=1.05Tc \n MagnetizationT\nime (ns)Tp=0.95TcT\nT\npr\nm \ntime (ns) \n 0.2 \ntime (ns) \n 0\n.2FIG. 3: (Color online) Time dependence of magnetization for given temperature pro\fles after a\nreversal magnetic \feld H= 1:5T is applied for t >0. a) The results are obtained from Eq. (6)\n(solid curves) and from Eq. (15) (dashed curves). b) Comparison of the results from Eq. (6) and\nfrom the conventional LL with a constant damping parameter (dashed curves). The Inserts are\nthe hypothetical temperature pro\fles. The parameters are K= 8:0T,S= 1=2, and\u001cs= 1:0 ps.\nequation with a modi\fed LL which allows the magnitude of the magnetization varying with\nthe time due to changing temperature in HAMR,\ndm\ndt=d(meq^m)\ndt=\u0000\rm\u0002He\u000b\u0000\r\u000btr\nmm\u0002(m\u0002He\u000b) +dmeq\ndT\u0001dT\ndt^m (15)\nwhere the transverse damping parameter is given by Eq. (10). The above equation implies\nthat the magnitude of the magnetization is always in equilibrium with the instantaneous\n11temperature, i.e., the longitudinal relaxation is in\fnite fast. Fig. 3(a) shows that such ap-\nproximation is quite accurate even for the temperature Tp= 0:95Tc. However, the deviation\nbegins to show up when the peak temperature is higher than the Curie temperature. In\nFig. 3(b), we further compare our results with a constant damping parameter (i.e., taking\n\u000btrin Eq. (15) as a constant). The deviation of this conventional LL with ours becomes\nmore signi\fcant at high temperatures. For example, even for Tp= 1:2Tc, the magnetization\nreversal is not possible from the conventional LL equation, see Fig. 3(b).\nTo end this section, we should brie\ry compare our equation with the LLB equation of\nGaranin [6]. Since the LLB equation is based on the atomistic LL equation that we believe\nis questionable, we should not make extensive comparisons. We point out that the LLB\nequation also contains the transverse and longitudinal relaxations, and the essential di\u000ber-\nence is the temperature dependence of the relaxation parameters. At low temperatures,\nboth our equation and the LLB equation reduce to the conventional LL equation. At high\ntemperatures, the relaxations in the LLB equation depend explicitly on temperatures; this\nis because the longitudinal relaxation to the equilibrium magnetization is solely controlled\nby the classical random \feld which is proportional to the temperature. In our case, the de-\npendence of the relaxation time on temperature is implicit, via the temperature dependence\nof the equilibrium magnetization. More importantly, the instantaneous relaxation time \u001cs\nin our equation has microscopical meaning as the scattering lifetime of electron spins while\nthe damping parameter in the atomistic LL does not have a microscopic counterpart.\nIV. STOCHASTIC FIELDS\nOur proposed equation, Eq. (6), describes the time-dependence of the average magnetiza-\ntion. The \ructuation at the \fnite temperature, particularly at a high temperature, becomes\nimportant. To address the \ructuation, one should include stochastic \felds in the macro-\nscopic dynamic equation. Similar to Brown's method [24] for the LL equation, we introduce\nthe stochastic \felds h(t) as follows,\ndm\ndt=\u0000\rm\u0002(He\u000b+h)\u0000m\u0000meq\n\u001cs: (16)\nWe point out that the stochastic \feld does not enter in the relaxation term although the\ninstantaneous equilibrium magnetization depends on the total \feld Ht. The reason is as\n12follows. The interaction between the random \feld and the magnetization is \u0000m\u0001h(t). This\ninteraction gives arise a random torque on the magnetization \u0000\rm\u0002h(t) that is added to\nthe deterministic torque equation. As in the case of the Brownian motion, the Langevin\nrandom \feld f(t) is only included in the particle motion d2r=dt2=\u0000\u000bv+F(t) +f(t) (where\nthe friction force \u0000\u000bvand the external driven force F(t) are not changed by the random\nforce). To determine the magnitude and the correlation of the stochastic \felds, we \frst write\nthe above stochastic equation in the standard form of Langevin,\ndmi\ndt=\u0014\n\u0000\rm\u0002He\u000b\u0000m\u0000meq\n\u001cs\u0015\ni\u0000\rX\njk\"ijkmjhk: (17)\nwhere\"ijkis the Levi-Civita symbol. The corresponding Fokker-Planck equation is thus\nhave the following form,\n@P\n@t=\u0000X\ni@\n@mi\u0014\u0012\n\u0000\rm\u0002He\u000b\u0000m\u0000meq\n\u001cs\u0013\ni+D\r2m\u0002\u0012\nm\u0002@\n@m\u0013\ni\u0015\nP (18)\nwherePis the probability density and Dis the random \feld correlation constant. At\nthe equilibrium, one may assume that the probability density takes a simple Boltzmann\ndistribution, i.e., P/exp(\u0000m\u0001H). By placing this form of Pinto Eq. (18), one \fnds the\ndesired correlation of the random \feld given below,\n<hi(t)hj(0)>=2kBT\u000btr\n\rmV\u000eij\u000e(t) (19)\nV. SUMMARY\nIn this paper, we have proposed a model of magnetization dynamics for an entire range of\ntemperature based on the quantum kinetic approach with the instantaneous local relaxation\ntime approximation. The resulting equation generates a low temperature magnetization\ndynamic same as the Landau-Lifshitz equation, namely, the transverse magnetization is\nsu\u000ecient to describe dynamics. When the temperature approaches or exceeds the Curie\ntemperature, it is essential to include the longitudinal magnetization relaxation. With our\nnew dynamic equation, one can model the entire heat-assisted magnetic recording processes\nwhen the temperature are heated and cooled through the Curie temperature [25{27]. The\nstochastic \felds on the magnetization are also proposed. This work is partially supported\n13by the U.S. DOE (DE-FG02-06ER46307) and by the NSF (ECCS-1127751).\n[1] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowietunion 8, 153(1935).\n[2] E. Beaurepaire, J. C. Merle, A. Daunois and J. Y. Bigot, Phys. Rev. Lett. 76, 4250(1996).\n[3] Stanciu, C. D. and Hansteen, F. and Kimel, A. V. and Kirilyuk, A. and Tsukamoto, A. and\nItoh, A. and Rasing, Phys. Rev. Lett. 99, 047601(2007).\n[4] M. H. Kryder et al., Proc. IEEE 96, 1810 (2008).\n[5] W. A. Challener, et al., Nature Photonics 3, 220 (2009).\n[6] D. A. Garanin, Phys. Rev. B 55, 3050 (1997).\n[7] O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell, and D. A. Garanin, Phys. Rev. 74, 094436\n(2006).\n[8] U. Atxitia, et al. , Phys. Rev. B 82, 134440 (2010).\n[9] U. Nowak, R. W. Chantrell, and E. C. Kennedy, Phys. Rev. Lett. 84, 163 (2000).\n[10] X. Z. Cheng et al. , PRL 96, 067208 (2006).\n[11] R. W. Davies and F. A. Blum, Phys. Rev. B 3, 3321 (1971)\n[12] F. Bloch, Phys. Rev. 70, 460 (1946).\n[13] T. Valet and A. Fert, Phys. Rev. B 48, 7113 (1993).\n[14] N. W. Ashroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New\nYork) 1976.\n[15] K. G. Wilson, Phys. Rev. B 4, 3174 (1971); Phys. Rev. B 4, 3184 (1971).\n[16] Norberto Majlis, The Quantum Theory of Magnetism (World Scienti\fc, Singapore) 2000.\n[17] K. Binder and D. W. Heermann, Monte Carlo Simulation in Statistical Physics (Springer-\nVerlag, Berlin) 1997.\n[18] D. V. Fedorov, P. Zahn, M. Gradhand and I. Mertig, Phys. Rev. B 77, 092406 (2008).\n[19] P. Monod and S. Schultz, J. Phys. (Paris) 43, 393 (1982).\n[20] A. C. Gossard, A. J. Heeger, and J. H. Wernick, J. Appl. Phys. 38, 1251 (1967).\n[21] P. C. Hohenberg und B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977).\n[22] J. U. Thiele, K. R. Co\u000bey, M. F. Toney, J. A. Hedstrom, and A. J. Kellock, J. Appl. Phys.\n1491, 6595 (2002).\n[23] P. Asselin, R. F. L. Evans, J. Barker, R. W. Chantrell, R. Yanes, O. Chubykalo-Fesenko, D.\nHinzke, and U. Nowak, Phys. Rev. B 82, 054415 (2010).\n[24] W. F. Brown, Phys. Rev. 130, 1677 (1963).\n[25] A. Torabi, J. van Ek, E. Champion, and J. Wang, IEEE Trans. Magn. 45, 3848 (2009).\n[26] C. Bunce et al. , Phys. Rev. B 81, 174428 (2010).\n[27] J. I. Mercer, M. L. Plumer, J. P. Whitehead, and J. van Ek, Appl. Phys. Lett. 98, 192508\n(2011).\n15"    },    {        "title": "1709.09513v3.Geometric_Dynamics_of_Magnetization__Electronic_Contribution.pdf",        "content": "Geometric Dynamics of Magnetization: Electronic Contribution\nBangguo Xiong,1,\u0003Hua Chen,2, 3Xiao Li,1and Qian Niu1, 4\n1Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA\n2Department of Physics, Colorado State University, Fort Collins, CO 80523, USA\n3School of Advanced Materials Discovery, Colorado State University, Fort Collins, CO 80523, USA\n4ICQM and CICQM, School of Physics, Peking University, Beijing 100871, China\nTo give a general description of the in\ruences of electric \felds or currents on magnetization\ndynamics, we developed a semiclassical theory for the magnetization implicitly coupled to electronic\ndegrees of freedom. In the absence of electric \felds the Bloch electron Hamiltonian changes the\nBerry curvature, the e\u000bective magnetic \feld, and the damping in the dynamical equation of the\nmagnetization, which we classify into intrinsic and extrinsic e\u000bects. Static electric \felds modify\nthese as \frst-order perturbations, using which we were able to give a physically clear interpretation\nof the current-induced spin-orbit torques. We used a toy model mimicking a ferromagnet-topological-\ninsulator interface to illustrate the various e\u000bects, and predicted an anisotropic gyromagnetic ratio\nand the dynamical stability for an in-plane magnetization. Our formalism can also be applied to\nthe slow dynamics of other order parameters in crystalline solids.\nPACS numbers: 75.78.-n, 75.60.Jk, 75.76.+j\nIntroduction |Magnetization dynamics is convention-\nally described by the phenomenological Landau-Lifshitz-\nGilbert (LLG) equation, in which the e\u000bective magnetic\n\feld and the damping factor can be associated with var-\nious mechanisms such as dipolar interaction, exchange\ncoupling, electron-hole excitations, etc., through micro-\nscopic theories. The recently discovered current-induced\nspin-orbit torques emerge as current-dependent modi\f-\ncations to the LLG equation, and can be consequently\ncategorized as \feld-like and damping-like torques1{6. In\nsystems with strong spin-orbit coupling and broken inver-\nsion symmetry, e.g. GaMnAs, heavy-metal/ferromagnet\nbilayers and magnetically doped topological insulator\nheterostructures, magnetization switching using electric\ncurrent alone through the spin-orbit torque has been\nachieved experimentally6{9. Theoretical studies of spin-\norbit torques have mostly adopted s\u0000dtype couplings\nbetween transport electrons and those contributing to\nmagnetization1{4,6, or a self-consistent-\feld picture based\non the spin density functional theory5,10. Then the spin-\norbit torques can be understood as the modi\fcation to\nthe e\u000bective exchange \felds proportional to the current-\ninduced spin densities in inversion symmetry breaking\nsystems, known as the Edelstein e\u000bect2,11. However, in\ngeneral neither the size of the exchange \feld nor its de-\npendence on order parameter (magnetization) direction\nis known a priori12,13. It is thus more desirable to de-\nvelop a theoretical framework that does not explicitly\ndepend on the details of the coupling between transport\nelectrons and the magnetization.\nIn this Rapid Communication, we provide a semiclassi-\ncal framework for the dynamics of magnetization implic-\nitly coupled to electronic degrees of freedom, based on\nthe wave-packet method. We found that the Bloch elec-\ntrons yield a Berry curvature \n mm, acting as a magnetic\n\feld in the magnetization space , while the gradient of the\nelectronic free energy with respect to the magnetization\nacts as a static electric \feld in the magnetization space,in agreement with previous adiabatic theory of magne-\ntization dynamics14. These two \felds thus govern the\ndynamics of magnetization as that of Lorentz force to a\ncharged particle. In addition, we identi\fed an extrinsic\ncontribution to the magnetization dynamics, correspond-\ning to the Gilbert damping in the LLG equation, which\nis not included in the adiabatic theory. A static elec-\ntric \feld enters the magnetization equation of motion by\nmodifying the Berry curvature \n mm, the e\u000bective \feld,\nand the damping factor as a \frst-order perturbation. In\nparticular, the modi\fcation to the e\u000bective \feld includes\na part proportional to the Berry curvature \n mkand hav-\ning a geometric nature. We used a simpli\fed model for\nthe ferromagnet-topological-insulator interface to illus-\ntrate the various e\u000bects, and showed that the gyromag-\nnetic ratio is renormalized anisotropically and that an\nin-plane magnetization can be dynamically stable under\nmoderate electric \felds.\nFormulation and general results |We start from a general\nHamiltonian of Bloch electrons implicitly depending on\nthe order parameter m,^He(q;m), whereqis the crys-\ntal momentum. External electromagnetic \felds are de-\nscribed by the scalar and vector potentials ( \u001e;A) that en-\nter the Hamiltonian through minimum coupling ( ~= 1,\ne=jej),\n^H=^He(q+eA;m)\u0000e\u001e: (1)\nFollowing Ref. 15, a wave packet is constructed with cen-\nter positionxand center physical momentum kfrom\nthe Bloch eigenstates of the local electronic Hamiltonian.\nThe Lagrangian of a single wave packet reads as\nL=_x\u0001[k\u0000eA(x;t)]+_k\u0001Ak+_m\u0001Am\u0000[\"\u0000e\u001e(x;t)];(2)\nwithA\u0015=ihujr\u0015ui(\u0015=korm) the Berry connec-\ntions of the Bloch state jui, and\"the wave packet en-\nergy. For notational simplicity we have dropped the band\nindex. The Lagrangian depends on ( x;k) of the wavearXiv:1709.09513v3  [cond-mat.mes-hall]  2 Mar 20182\npackets and magnetization m. Thus a set of coupled\nequations of motion for all three variables can be derived\nfrom the Lagrangian principle16:\n_k=\u0000eE; (3)\n_x=@\"\n@k+_k\u0001\nkk+_m\u0001\nmk; (4)\n\u0002\n[dk]f\u0012\n_m\u0001\nmm+_k\u0001\nkm+@\"\n@m\u0013\n= 0;(5)\nwhere the Berry curvatures \n \u0015i\u0015j =\n\u00002Imh@u=@\u0015ij@u=@\u0015ji,\u0015=korm. Eq. 5 is ob-\ntained by summing over all occupied states, and fis\nthe distribution function for the electrons. Note the\nmagnetization dynamics enters the electron equations\nof motion through \n kkin Eq. 4, and the terms in the\nsquare brackets of Eq. 5 can be viewed as conjugates of\nthe right hand side of Eq. (4), by interchanging kand\nm. This is a manifestation of the reciprocity between\ncharge pumping due to magnetization precession and\nelectric-current-induced spin-orbit torque.\nThe nonequilibrium response of the electrons to an\nexternal electric \feld and/or a dynamical mis ac-\ncounted for using the semiclassical Boltzmann equation,\naccording to which the deviation of the distribution\nfunction from the equilibrium Fermi-Dirac distribution\nf0[\"(k;m)] is\n\u000ef=\u0000\u001c@f0\n@\"\u0012\n_k\u0001@\"\n@k+_m\u0001@\"\n@m\u0013\n; (6)\nwhere we have assumed a grand canonical ensemble with\n\fxed temperature and chemical potential. \u001cis the re-\nlaxation time which we take as a constant for simplic-\nity. Generalization to including more speci\fc scattering\nmechanisms is straightforward but involved, and does not\nnecessarily provide additional insight on the main issues\nconsidered in this work.\nThe equations (3-6) complete our semiclassical descrip-\ntion of coupled magnetization and electron dynamics in\nthe presence of external electric \felds, though they can\nbe easily extended to including magnetic \felds and other\nperturbations.\nIn the absence of electric \felds, _k= 0, and we can\nobtain from Eq. (6) and Eq. (5) the following equations\nof motion of the magnetization,\n_m\u0001(\u0016\nmm+\u0011mm)\u0000H= 0; (7)\nin getting which we have ignored higher order _m2terms\nby assuming that the magnetization dynamics is slow\ncompared to typical electronic time scales. The Berry\ncurvature \u0016\n, the damping coe\u000ecient \u0011and the e\u000bective\n\feldHin the equation above are respectively\n\u0016\nmm=\u0002\n[dk]f0\nmm; (8)\n\u0011mm=\u0000\u001c\u0002\n[dk]@f0\n@\"@\"\n@m@\"\n@m; (9)\nH=\u0000@G\n@m; (10)whereGis the free energy of the electron system.\nFor non-interacting electrons G=\u0000\f\u00001\u0001\n[dk] ln[1 +\ne\u0000\f(\"\u0000\u0016)] for a single band, where \f= 1=kBT. Interac-\ntion e\u000bects may be included in Gthrough di\u000berent levels\nof approximations, which will also modify the way mag-\nnetization appears in G. At this point we will leave Gas\na general electron free energy depending on mimplicitly.\nWe only consider the transverse modes ( _mperpendic-\nular tom) of the magnetization dynamics in this work,\nalthough Eq. 7 can be used for the longitudinal mode as\nwell. The magnetization is thus described by the polar\nangle\u0012and the azimuthal angle \u001e. Eq. (7) can then be\nconverted to the familiar form of the LLG equation,\n_m=\u0000\rm\u0002(H\u0000\u0011mm\u0001_m); (11)\nwhere the gyromagnetic ratio \ris related to the Berry\ncurvature through\n\u0016\n=m=\rm2; (12)\nwhere \u0016\ni=\"ijk\u0016\njk=2 is the vector form of the Berry cur-\nvature tensor. Expressions similar to Eq. (7), but without\nthe damping term , have been derived using the adiabatic\ntheory17. Since the damping term is explicitly depen-\ndent on the relaxation time, which is ultimately due to\ndissipative microscopic processes such as electron-phonon\nscattering and electron-impurity scattering, we call it ex-\ntrinsic contribution to the magnetization dynamics. Note\nEq. 9 suggests \u0011is positive de\fnite, which means it al-\nways leads to energy dissipation through Eq. 11. The\nremaining terms are intrinsic contributions from the elec-\ntron degrees of freedom. In particular, from Eq. (7) one\ncan see that the two intrinsic terms are formally similar\nto the Lorentz force of a charged particle, with the anti-\nsymmetric part of \n mm(or equivalently the vector form\n\nm) analogous to the magnetic \feld and Hplaying the\nrole of the electric \feld.\nElectric \felds enter our formalism through the equa-\ntion of motion for k[Eq. (3)], which makes the 2nd term\nin the integrand of Eq. 5 nonzero and also contributes to\nthe nonequilibrium distribution function \u000efin Eq. (6).\nAfter some algebra, we arrive at the same equation as\nEq. (7), but with H,\u0016\nmm, and\u0011mmacquiring the fol-\nlowing corrections proportional to the electric \feld:\nHE=eE\u0001\u0002\n[dk]\u0012\n\nkmf0\u0000\u001c@\"\n@k@\"\n@m@f0\n@\"\u0013\n;(13)\n\u0016\nE\nmimj=e\u001cE\u0001 (14)\n\u0002\n[dk]\u0014@\"\n@k\nmimj\u0000\u0012\n\nkmi@\"\n@mj\u0013\nA\u0015@f0\n@\";\n\u0011E\nmimj=e\u001cE\u0001\u0002\n[dk]\u0012\n\nkmi@\"\n@mj\u0013\nS@f0\n@\"(15)\nwhere subscript S (A) means the part of \n kmi@\"\n@m jthat is\nsymmetric (antisymmetric) under i$j. We next discuss\nthe physical meanings of these results in detail.3\nFor the correction to the e\u000bective \feld, HE, the \frst\nterm in Eq. 13 has a geometric nature and is an intrinsic\ncontribution from the Fermi sea electrons. It is of \n mt\ntype, where the time variation is due to the momentum\nchange of a single wave packet driven by E:@t=_k\u0001@k=\n\u0000eE\u0001@k. We note there is a nice identity connecting \n mt\nand the \\magnetic \feld\" in magnetization space \nm:\n@t\nm+rm\u0002\nmt= 0: (16)\nSince \nmt= \nmk\u0001(\u0000eE) is a correction to the static\ne\u000bective electric \feld H(Eq. 10) in the magnetization\nspace, above equation is a magnetic analog of the Fara-\nday's law for charged particles. The 2nd term in Eq. 13 is\nextrinsic since it is proportional to \u001c, and does not have\nan electromagnetism analog.\nHEalso provides new insights on the charge pumping\ne\u000bect of a nonzero _m18. SinceP\u0011HE\u0001_mhas the mean-\ning of power density and is proportional to E, there is an\nelectric current induced by _masjp=@(HE\u0001_m)=@E.\nThe change of the polarization density (\\pumping\") af-\ntermcompletes a closed path in its con\fguration space\nis obtained by integrating jpover this period. A \fnite\ncharge pumping thus corresponds to a nonzero work den-\nsity, and is related to the curl of HEin the magnetization\nspace through\nW=\f\njp\u0001Edt=\f\nHE\u0001dm (17)\n=\u0004\nrm\u0002HE\u0001d\u001bm;\nwhere we have used the Stokes theorem, and d\u001bmis the\nin\fnitesimal area in the magnetization space. Thus in\norder to have \fnite charge pumping, HEmust not be\nconservative, i.e., it cannot be written as a gradient of\ncertain scalar free energy.\nWe now move on to \u0016\nE\nmm and\u0011E\nmm, which are all\nFermi surface contributions due to the non-equilibrium\npart of the distribution function \u000ef. They are important\nin magnetic metals and should be discussed on an equal\nfooting asHEfor current-induced e\u000bects on magnetiza-\ntion dynamics. In the form of Eq. 11, \u0016\nE\nmm renormal-\nizes the gyromagnetic ratio as \r0=\r=(1 +\r=\rE), where\n\rE\u00111=m\u0001\u0016\nE, while\u0011E\nmmmodi\fes the damping tensor\nas\u00110=\u0011+\u0011E. It is interesting to note that \u0011Edoes\nnot have to be positive de\fnite. A negative de\fnite total\ndamping will make the free energy minima dynamically\nunstable while the maxima dynamically stable. Thus in\naddition to the potential of switching the magnetization\nbetween di\u000berent easy directions, a suitably chosen elec-\ntric \feld can in principle switch the magnetization be-\ntween easy and hard directions, which provides a new\nmechanism (though volatile) for current driven reading\nand writing processes in magnetic memory devices.\nBefore ending this section, we translate our results\nEq. (13-15) into the commonly used spin-orbit torque\nlanguage. For small electric \felds they can be converted\nto additional terms added to the right hand side of theLLG equation Eq. 11:\n_m=\u0000\rm\u0002(H\u0000\u0011mm\u0001_m)\u0000\r\u001cso; (18)\nwhere\u001cso=\u001cH\nso+\u001c\r\nso+\u001c\u0011\nsowith the separate terms being\n\u001cH\nso=m\u0002HE; (19)\n\u001c\r\nso=\u0000\r=\rEm\u0002(H+\u0011\rm\u0002H); (20)\n\u001c\u0011\nso=\r\u0011Em\u0002(m\u0002H): (21)\nFor the special s\u0000dtype coupling, HEis propor-\ntional to the spin density response to electric \felds since\n@^H=@m\u0018s, in agreement with previous studies2,5,8,11,\nthough our formalism is not limited to this coupling form.\nMorever, there are additional torques \u001c\r\nsoand\u001c\u0011\nsothat\ncannot be directly explained using spin density response\nto electric \felds. They can, however, always be classi\fed\ninto either \feld-like or damping-like torques depending\non whether there is a sign change upon m!\u0000m.\nModel example |As a concrete example, we consider a\n2D toy model that can be used to describe the interface\nbetween a ferromagnetic insulator and a 3D topological\ninsulator (TI)19{21:\n^H(m) =~v(\u0000ky\u001bx+kx\u001by) +Jm\u0001\u001b; (22)\nwheremis the 2D magnetization of the ferromagnet, \u001b\nis the Pauli matrix vector for the spin operators, vis the\nFermi velocity of the Dirac surface electrons of the TI,\nandJis the exchange coupling strength between mand\n\u001b. The exchange coupling opens a gap proportional to\nthezcomponent of m. We consider zero temperature\nand set the chemical potential \u0016= 0. The Berry curva-\nture of the lower band is calculated similarly as the ~k\u0001~ \u001b\nmodel16\n\u0016\ns\n\u0012\u001e=\u000b2jsin 2\u0012j\n8\u0019a2sgn(\u000b); (23)\nwhere\u000b=Jma= ~vis the exchange energy measured\nin typical scales of the kinetic energy \u000f0=~v=a(ais\nthe lattice constant). Using relation Eq. (12), the Berry\ncurvature gives an anisotropic gyromagnetic ratio\n\rs(\u0012) =4\u0019ma2\n~\u000b2jcos\u0012jsgn(\u000b): (24)\nWe should note that the ferromagnet by itself has a gyro-\nmagnetic ratio, denoted as \rf, and the overall gyromag-\nnetic ratio\ris corrected as\n\r\u00001=\rf\u00001+\rs\u00001; (25)\nor equivalently\n\r=\rf\u00011\n1 +\rf=\rs(\u0012): (26)\nThe variation of \rformmoving across the Bloch sphere\nis shown in Fig. 1(a). On the equator ( \u0012=\u0019=2),\n\r=\rf; at the north and south poles, \r=\rf=(1 +4\n\rf~\u000b2=4\u0019ma2sgn(\u000b)). This angular dependence of gy-\nromagnetic ratio should be able to be detected by ferro-\nmagnetic resonance experiments in such systems.\nThe free energy density at zero temperature is calcu-\nlated by integrating the energy of the lower bands. Ig-\nnoring a constant term, we get\nGs=\u0000J0m2\nz (27)\nwhereJ0=\u000f0kc\u000b2=4\u0019m2aandkcis the momentum cut-\no\u000b.Gshas two minima at the north and south poles,\nas shown in Fig. 1(b). Thus the surface states provide a\nperpendicular magnetic anisotropy for the ferromagnet.\nFor simplicity we ignored the magnetic anisotropy energy\nof the ferromagnet itself. For nonzero mzthere is no con-\ntribution from the surface state electrons to \u0011because of\nthe \fnite gap, and if the intrinsic damping of the ferro-\nmagnet is ignorable the magnetization should move along\nequal-energy lines without driving forces, along the direc-\ntions determined by \u0000\rm\u0002H[Eq. (11)], as illustrated\nin Fig. 1(b).\nFIG. 1. (Color online) (a) Renormalized gyromagnetic ratio\n\rthrough coupling to the topological surface states. (b) Con-\ntour plot of free energy Gsin the absence of electric \felds.\nThe arrows indicate the directions of magnetization motion.\nParameters: \rf= 2ma2=~;\u000b= 1\nWe now consider the e\u000bect of an electric \feld along x\ndirection on the magnetization dynamics. For nonzero\nmzall Fermi surface contributions in Eqs. (13-15) are\nzero, and the only \fnite term is the Fermi sea contribu-\ntion inHE:\nHE=\u0000eEj\u000bj\n4\u0019masgn(mz)^x: (28)\nIt has constant magnitude but opposite directions de-\npending on the sign of mz. The curl of HEis thus\nzero everywhere except on the equator, which also means\nnonzero charge is pumped by magnetization dynamics\nwhen the precession axis is in plane22. Based on our dis-\ncussion in the previous section we can only de\fne free\nenergy functions separately for the north (N) and the\nsouth (S) hemispheres as GNandGSbut not globally:\nGN=\u0000J0m2\nz+eEj\u000bj\n4\u0019mamx; (29a)\nGS=\u0000J0m2\nz\u0000eEj\u000bj\n4\u0019mamx: (29b)\nOn each hemisphere, the 2nd term in the free energy im-\nplies a magnetization-dependent polarization, which willbe interesting to detect experimentally. Moreover, since\nGN\u0000GS/mx, they cannot be connected by a constant\nenergy shift across the equator. The electric \feld thus\nshifts the two free energy minima at the north and the\nsouth poles in opposite directions, and distorts the equal\nenergy lines in the vertical direction, as shown in Fig.\n2. In addition, the opposite signs of GNandGSvery\nclose to the equator make half of the equator dynami-\ncally stable, as can be seen from the arrows pointing to\nthe equator from both above and below in Fig. 2. Specif-\nically, if we still assume a vanishing intrinsic damping of\nthe ferromagnet, when the magnetization is very close\nto the equator with \u001e2(\u0019;2\u0019), or more generally when\nit is between the two critical trajectories determined by\nGN=S=\u0000eEj\u000bj=4\u0019a, it will follow the equal energy lines\nand end up on the half equator with \u001e2(0;\u0019). Con-\nversely, for a magnetization outside of the region between\nthe two critical trajectories, i.e., GN=S<\u0000eEj\u000bj=4\u0019a, it\nwill keep precessing around one of the free energy min-\nima. When there is a small damping, the size of the\nattraction area around the half equator reduces because\nenergy is dissipated during evolution.\nFIG. 2. (Color online) Contour plot of free energies GNand\nGSin the presence of an electric \feld along ^ x. Parameters:\n\rf= 2ma2=~;\u000b= 1,eE j\u000bj=4\u0019J0= 0:4.\nIn the limiting case of strong electric \felds\neEj\u000bj=4\u0019a > 2J0m2, the critical trajectories disappear\non the Bloch sphere and the magnetization will always\nevolve to the stable half equator. Since without the\nmagnetic \feld the magnetization has a perpendicular\nanisotropy due to the topological surface states, electric\n\felds can lead to dynamical switching between easy (out-\nof-plane) and hard (in-plane) directions. This mechanism\nis unique to the FM/TI system and is independent of\nthe easy-hard-axes switching due to a negative-de\fnite\ndamping tensor discussed in the last section.\nSince the electric \feld enters our formalism only\nthrough its modi\fcation on momentum [Eq. (3)], our the-\nory can be straightforwardly generalized to other time-\nvarying perturbations that in\ruence wave-packet dynam-\nics in similar ways, which will give both Fermi-surface\ncontributions and Fermi-sea contributions through the\nBerry curvature \n mt. For example, a potential appli-5\ncation is the magnetization dynamics driven by sound\nwave23,24. Separately, our formalism can be applied to\nthe slow dynamics of other order parameters in crys-\ntalline solids, and to its dependence on electromagnetic\n\felds through the electron degrees of freedom.\nWe acknowledge useful discussions with A. H. Mac-Donald, R. Cheng, Y. Gao, H. Zhou. This work\nis supported by National Basic Research Program of\nChina (Grant No. 2013CB921900), DOE (DE-FG03-\n02ER45958, Division of Materials Science and Engineer-\ning), NSF (EFMA-1641101), and the Welch Foundation\n(F-1255).\n\u0003bgxiong@physics.utexas.edu\n1A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008).\n2I. Garate and A. H. 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S. Salasyuk, A. V. Akimov, X. Liu,\nM. Bombeck, C. Br uggemann, D. R. Yakovlev, V. F.\nSapega, J. K. Furdyna, and M. Bayer, Physical Review\nLetters 105, 117204 (2010).\n24M. Weiler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M. S.\nBrandt, and S. T. B. Goennenwein, Physical Review Let-\nters106, 117601 (2011)."    },    {        "title": "2304.14957v1.Competing_signatures_of_intersite_and_interlayer_spin_transfer_in_the_ultrafast_magnetization_dynamics.pdf",        "content": "Competing signatures of intersite and interlayer spin transfer in the ultrafast\nmagnetization dynamics\nSimon Häuser,1,∗Sebastian T.Weber,1Christopher Seibel,1Marius Weber,1Laura Scheuer,1\nMartin Anstett,1Gregor Zinke,1Philipp Pirro,1Burkard Hillebrands,1Hans\nChristian Schneider,1Bärbel Rethfeld,1Benjamin Stadtmüller,1, 2,†and Martin Aeschlimann1\n1Department of Physics and Research Center OPTIMAS,\nRheinland-Pfälzische Technische Universität Kaiserslautern-Landau, 67663 Kaiserslautern, Germany\n2Institute of Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany\n(Dated: May 1, 2023)\nOptically driven intersite and interlayer spin transfer are individually known as the fastest pro-\ncesses for manipulating the spin order of magnetic materials on the sub 100fs time scale. However,\ntheir competing influence on the ultrafast magnetization dynamics remains unexplored. In our\nwork, we show that optically induced intersite spin transfer (also known as OISTR) dominates the\nultrafast magnetization dynamics of ferromagnetic alloys such as Permalloy (Ni 80Fe20) only in the\nabsence of interlayer spin transfer into a substrate. Once interlayer spin transfer is possible, the\ninfluence of OISTR is significantly reduced and interlayer spin transfer dominates the ultrafast mag-\nnetization dynamics. This provides a new approach to control the magnetization dynamics of alloys\non extremely short time scales by fine-tuning the interlayer spin transfer.\nIncreasing the operating speed of modern spintronic\ndevices requires new strategies to manipulate, transport,\nand store digital information encoded in the spin angu-\nlar momentum of electrons on shorter time scales. One\npromising way to achieve this goal is to use ultrashort\nfemtosecond (fs) light pulses as external stimuli to mod-\nify the material properties of spintronic relevant mate-\nrials, such as ferromagnets and antiferromagnets. The\nfeasibility of this approach for the realization of ultrafast\nspintronics has been demonstrated by pioneering studies\ninthelasttwodecades, whichrevealedthe(sub-)picosec-\nond loss of magnetic order in ferro- and antiferromagnets\nafter excitation with ultrashort optical and THz pulses\n[1–11] and even reported the all optical magnetization re-\nversal by fs light pulses [12–14]. In most cases, however,\nthe time scale of the material response and the corre-\nsponding change in the spin order of the material is not\nrelatedtothedurationoftheopticalexcitationitself(i.e.,\nthe length of the fs-light pulse). Instead, the magnetiza-\ntion dynamics evolves on a significantly longer intrinsic\ntime scale that is governed by secondary angular mo-\nmentum dissipation processes such as electron-electron\nscattering [15, 16], Elliot-Yafet electron-phonon spin flip\nscattering [16–19], generation of ultrafast non-coherent\nmagnons [9, 20]. These processes limit the optical mate-\nrial response in metallic 3d ferromagnets, such as nickel,\nto≈100fs [1, 17, 18].\nFaster material responses have only been reported for\nmagnetic materials where the ultrafast magnetization\ndynamics are dominated, or at least significantly influ-\nenced, by optically induced spin transport and transfer\nprocesses. One of these processes is the superdiffusive\nspin transport [21–24] in magnetic bilayer and multilayer\nstructures. In this case, the different velocities of the op-\ntically excited minority and majority carriers in the fer-\nromagnet lead to an effective ultrafast transport of spinangular momentum out of the magnetic material into an\nadjacent layer and thus to an ultrafast demagnetization\nthat can be faster than 50fs [22]. The second spin trans-\nfer process relevant on these ultrashort time scales is the\nso-called optical intersite spin transfer (OISTR) [25]. It\nrefers to an optically induced spin transfer between dif-\nferent magnetic subsystems of a magnetic alloy or mul-\ntilayer system, which is purely mediated by the optical\ntransitions between different electronic states of the ma-\nterials [26–32]. As a result, the influence of OISTR on\nthe magnetization dynamics of materials is determined\nsolely by the pulse length of the optical excitation.\nIn reality, however, OISTR and superdiffusive spin\ntransport influence the demagnetization dynamics on\nnearly identical time scales. This is mainly due to the\nfact that the fs light pulses used to manipulate magnetic\nmaterials are typically generated with pulse durations in\nthe range between 20fs and 50fs. This could potentially\nlead to a competing influence of OISTR and superdif-\nfusive spin transport on the ultrafast magnetization dy-\nnamicsofmagneticmultilayerstructures. Understanding\nand exploiting this competition thus offers an intriguing\npathwaytoopticallyengineertheultrafastmagnetization\ndynamics on the sub 100fs time scale.\nIn this work we investigate the competing influence\nof OISTR and interlayer spin transfer via superdiffusive\nspin transport on the ultrafast magnetization dynam-\nics of the ferromagnetic alloy Permalloy (Py, Ni 80Fe20).\nTuningthesubstratematerialofthePyfilmsfromthein-\nsulator MgO to gold films of different thicknesses ( 10 nm\nand100 nm) allows us to gradually increase the role of\ninterlayer spin transfer for the ultrafast demagnetization\ndynamics. Our joint experimental and theoretical work\nprovides substantial evidence that the OISTR signature\nand thus the relevance of OISTR for the demagnetization\ndynamics of Py decreases with increasing importance ofarXiv:2304.14957v1  [cond-mat.mtrl-sci]  28 Apr 20232\nFIG. 1. (Left) Sketch of the time-resolved Kerr spectroscopy\nexperiment. An optical pump pulse ( 1.55eV, 30fs) excites\nthe sample while the element specific magnetization dynam-\nics are monitored by changes in the magnetic asymmetry at\nthe Fe M 2,3and Ni M 3absorption edges. (Right) Schematic\nrepresentation of the density of states of Fe and Ni in Py and\nthe population changes due to the OISTR. Optical excitation\nby the IR pump leads to an effective spin transfer from the\noccupied Ni minority channel to the Fe minority channel.\ninterlayerspintransferfromPyintotheadjacentmetallic\nlayer. Our results underscore the competing roles of in-\ntralayerandinterlayerspintransferfortheultrafastmag-\nnetization dynamics of magnetic multilayer structures on\nthe ultrafast, sub 100fs time scale.\nInourstudy, weinvestigatethreePermalloy(Ni 80Fe20)\nthin films ( 10 nm) deposited on different substrates: (i)\ndirectly on the insulating substrate MgO, (ii) on a 10 nm\nAu film on MgO, and (iii) on a 100 nmAu film. All\nsamples were protected from oxidation by a 2 nmAl2O3\ncapping layer.\nThe time- and element-resolved magnetization dynam-\nics of these samples were investigated by time-resolved\nKerr spectroscopy with fs- extreme UV (XUV) radiation\nin transverse geometry (T-MOKE). Using neon for the\ngeneration of the fs XUV radiation by high harmonic\ngeneration (HHG) (similar to Ref. [33]), we can cover a\nspectral range of 40−72eV that coincides with the char-\nacteristic M 2,3absorption edges of Fe and Ni at ~ 52.7 eV\nand ~ 66.2 eV[34], respectively.\nThe magnetic contrast is obtained from the absorption\nspectra by recording the reflectivity of the whole fs-XUV\nspectrum of the sample for two alternating directions of\nthe magnetic B-fieldI+andI−as shown in Figure 1.\nThen, wecalculatethemagneticasymmetry Aby[33,35]\nA=I+−I−\nI++I−∝Spin Polarization . (1)\nThe energy resolved asymmetry Ais proportional to the\nspin polarization (SP) [36, 37] and gives clear signatures\nof OISTR as shown by Hofherr et al. [26].\nFortheopticalexcitationofoursample, weusedalaser\nfluence of 28.1mJ/cm2, which resulted in a different loss\nof magnetization for each sample. Therefore, we applieda dedicated normalization procedure to all magnetization\ntraces using the data for Py/Au(10nm) as a reference.\nWe start our discussion with the magnetization dy-\nnamics of Py/MgO where the initial magnetization dy-\nnamics are dominated by OISTR and interlayer spin\ntransfer into the substrate is absent. As reported for\nanother FeNi alloy[26], the optical excitation by the IR\npump pulse leads mainly to a transfer from the minority\nelectrons of Ni into the unoccupied minority states of Fe\nclose to the Fermi energy EF, as shown in the inset of\nFigure 1. This intralayer spin transfer can be recorded\nand visualized in the T-MOKE experiment by extracting\nthetime-dependentspinpolarizationforselectedspectral\nregionsinclosevicinitytotheM 2,3absorptionedgesofNi\n(~66.2eV) and Fe (~ 52.7eV). In particular, we follow our\nprevious work [26] and evaluate the spin polarization for\nan energy range corresponding to the occupied Ni states\nbelow the Fermi energy and the unoccupied Fe states lo-\ncated slightly above EF. The corresponding traces are\nshown as blue (Ni states) and red (Fe states) curves in\nFigure 2.\n-4000 4 008 000.70.80.91.01.1d\nelay (fs)norm. asymmetry (arb. u.)N\niF\ne-\n0.4-0.3-0.2-0.10.00.1O\nISTR trace (arb. u.)\nFIG. 2. Temporal evolution of the spin polarization of the\ncorresponding Ni states (blue line) below and Fe states (red\nline) above EFfor Py/MgO. The difference between the spin\npolarization of the Ni and Fe states is shown as a green curve.\nThe SP of the Ni states increases instantaneously upon\nlaser excitation, coinciding with a decrease in the SP of\nthe Fe states. This opposite behavior has been reported\nas the spectroscopic OISTR signature and characterizes\nthe initial influence of OISTR on the ultrafast magneti-\nzation dynamics of Py. It can be further quantified by\nthe so-called OISTR trace (OT), i.e. the difference be-\ntween the SP of the Ni and Fe states, shown as a green\ncurve in Figure 2. The OT reaches its maximum after\nabout 200 fsand disappears again within the next 250 fs.\nThe rise of the OT can be directly related to the OISTR\nand the corresponding initial changes in the magnetiza-\ntion of the Fe and Ni sublattices. The decay of the OT3\nis attributed to exchange scattering [38, 39] and spin flip\nscattering processes occurring locally within the Py film.\nTo explore the competing influence of OISTR and in-\nterlayerspintransportontheultrafastmagnetizationdy-\nnamics of Py, we now turn to the OT for similar Py\nfilms on metallic Au films of 10nm and 100nm thick-\nness. We recorded similar time-resolved T-MOKE data\nsets for both samples (see Supplementary Material) and\ndetermined the OT using the identical procedure as de-\nscribed above for the Py film on MgO. The resulting OTs\nare summarized for all three sample systems in Figure 3.\nWe find a clear decrease in the magnitude of the OT for\nPy/Au( 10 nm) compared to Py/MgO. More importantly,\nthe OT disappears completely for Py on the 100 nmAu\nfilm.\n-4000 4 008 00-0.050.000.050.100.15 \nPy/MgO \nPy/Au10nm \nPy/Au100nmOISTR trace (arb. u.)d\nelay (fs)\nFIG. 3. Temporal evolution of the OT for Py on three differ-\nent substrates: insulating MgO (green curve), metallic 10 nm\nAu film, and 10 nmAu film.\nOur observation clearly demonstrates that the ini-\ntial changes in spin polarization and the magnetization\ndynamics due to OISTR are greatly reduced or even\nsuppressed by replacing the insulating substrate with\na metallic thin film. Therefore, we propose that these\nchanges in OT are due to interlayer spin transfer and\ntransport, ratherthanamodificationofthespinflipscat-\ntering rate in Py due to the hybridization of Py and Au\nat the interface. In particular, no local interfacial ef-\nfect could account for the observed thickness dependent\nchanges in the OT.\nFigure 4 outlines a possible scenario explaining the re-\nduction of OT in the Py film by interlayer spin trans-\nfer: In the absence of OISTR, interlayer spin transfer\nleads to a flow of spin-polarized charges from the ferro-\nmagnetic material into the non-magnetic substrate and a\ncorresponding backflow of unpolarized charges to main-\ntain charge neutrality. This is shown in Figure 4(a). The\nFIG. 4. Schematic illustration of the different optically in-\nduced interlayer spin transfer pathways of majority and mi-\nnority carriers across the Py/Au interface. The hypotheti-\ncal flow of spin-polarized carriers in the absence of OISTR\nis shown in panel (a), the additional interlayer spin transfer\npathways due to OISTR are highlighted in panel (b).\nmagnitude of the interlayer spin transfer for both major-\nity and minority carriers depends (apart from the inter-\nfacial transmission efficiency), on the number of carriers\nin the ferromagnetic material, i.e. in Py, and the number\nof available states in the substrate material, i.e. in Au.\nThis results in an overall majority carrier dominated spin\ntransfer from Py to Au. In the case of OISTR, the op-\ntical excitation additionally redistributes minority elec-\ntrons from Ni states below EFto Fe states above EF, as\nshown on the right side of Figure 1. Thus, OISTR leads\nto a significant change in the number of carriers and final\nstates available for interlayer spin transfer. This popu-\nlation change leads to additional interlayer spin transfer\nchannels as shown in Figure 4(b). The optically excited\nFe minority electrons can now travel through the inter-\nface, reducing the excited state spin population initially\ncreated by the OISTR. On the other hand, the backflow\nof electrons from Au to Py can repopulate the minor-\nity states in Ni that were initially depopulated by the\nOISTR. In this way, both additional transport channels\nwould substantially counteract the changes in the spin-\ndependent population created by the OISTR and thus\nreduce the OT observed in our experiment. However, our\nmodel does not yet account for the increasing reduction\nin OT magnitude with increasing film thickness.\nTo adress this final question theoretically, we model\nthe ultrafast population of states around the Fermi en-\nergy in Au with the help of a kinetic approach consider-\ning the optical excitation as well as the electron-electron\nand electron-phonon interactions, each with a full Boltz-4\nmann collision integral in dependence of time and energy\n[40]. This allows us to determine the non-equilibrium\npopulation and depopulation of states on the same time\nscales as the measured OISTR signal in the Py layer.\nThe excitation strength is determined from the experi-\nmental fluence and absorption calculations for the exper-\nimental multilayer including capping and substrate with\nrefractive indices from Ref. [41]. The resulting absorp-\ntion profiles for a 10 nmAu-film and for a 100 nmAu-\nfilm are shown in Figure 5 a). The dashed line marks\nthe backside of the 10 nmAu-film, which is responsible\nfor the back reflections, leading to a much larger total\nabsorption in the thinner Au film as compared to the\nthicker film, note the logarithmic scale in Figure 5 a).\nFurthermore, we assume a homogeneous distribution of\nthe absorbed energy for the thin film of 10nm thickness.\nThis is already roughly fulfilled by the spatial absorp-\ntion profile depicted in Figure 5a), but also because the\nrange of ultrafast homogeneous energy distribution by\nballistic electrons has been estimated to be about 100 nm\nfor Au [4], which is much larger than the thin Au sub-\nstrate. In contrast, for the 100nm Au-film, we determine\ntwo different limits of absorption strength based on the\nprofile shown in Figure 5a), as will be described later.\nWith the given amount of absorbed energy we model the\nlaser-excitationofelectronsinAu. Typicalresultingnon-\nequilibrium electron distributions in noble metals can be\nfound, e.g., in Refs. [40, 42, 43].\nWe evaluate the change of occupation in the states at\nE=EF−1.8 eVandE=EF+ 1.35 eVaccording to ex-\nperimentally measured energy regions. The width of the\nevaluated interval, ∆E= 0.7 eV, is chosen correspond-\ning to the width of the HHG peaks of the experiment.\nAs indicated in Figure 4, the energy window above the\nFermi energy EFcorresponds to Fe states providing mi-\nnority carriers for the transport into the Au substrate,\nwhile the energy below the Fermi energy is relevant for\nreceiving minority carriers from Au in Ni.\nThe time evolution of the occupation is given as\nn(t) =E+∆E/2/integraldisplay\nE−∆E/2f(/epsilon1,t)D(/epsilon1)d/epsilon1, (2)\nwherefdenotes the electronic distribution function and\nDits density of states (DOS).\nThe change of the occupation n(t)as compared to its\ninitial value at room temperature is shown in Figure 5b).\nIn the case of the 10 nmAu layer, the excitation leads to\nafastincreaseoftheoccupationabove EFandadecrease\nbelowEFwithin the time scale of the laser pulse. After\nthe laser pulse, relaxation processes lead to a decrease of\nboth signals.\nThis shows that the laser excitation is blocking the\nstates relevant for non-local transport across the inter-\nface effectively on the time scale of the pump pulse dura-\ntion and the subsequent several 100 fs, see solid lines in\n10−210−11Absorption (%\nnm)\n0 20 40 60 80 100\nPosition (nm)Au 10 nm\nAu 100 nm\nPy Au a)\n-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.5Occupation change (10271\nm3)\n0 100 200 300 400 500\nTime (fs)Fe-Au receiving states\nNi-Au providing statesb)\n10 nm Au\n100 nm AuFIG. 5. a) Structure of the samples and absorption pro-\nfiles. b) Change of the occupation due to excitation for states\nslightly above Fermi energy (red lines) and slightly below\nFermi energy (blue lines) in Au, respectively. The solid lines\nshow the occupation difference for the 10 nmAu layer. The\nshaded areas mark the occupation difference for the case of\nthe100 nmAu layer for different assumptions on laser-energy\ntransport (see text). The temporal shape of the laser pulse is\nshown in gray.\nFigure5b). Thus, thetransportscenariodepictedinFig-\nure 4 is partially blocked and while the measured OISTR\nsignal in Py is reduced as compared to an insulating sub-\nstrate, but it does not vanish completely.\nIn the case of the thicker 100 nmAu layer, the spin\ntransport from Py to Au is not blocked, leading to a\ncomplete extinction of the resolvable OISTR signal. We\nshowthisfortwolimitingcasesofAuexcitation. Inafirst\ncalculation, we assume that the fraction of pump energy\nabsorbed in Au, see Figure 5a), is homogeneously dis-\ntributed by ballistic transport over the 100 nmAu layer.\nThe occupation change in the relevant Au states is not\nresolved in Figure 5 b), see dotted lines. Thus, in this\ncase, Aucanactasaspinacceptororsource, respectively,\nand thus eliminate the OT. To evaluate the possibility of\nstate-blocking in the 100 nmAu substrate as we found\nfor the thin 10 nmAu film, we consider no energy trans-\nport within the Au layer and average just over the first\n10 nmclosest to the Py interface. The excitation of this\nvirtual layer is lower than in the case of the 10 nmAu\nfilm, because no back reflection effects are active in the\nbulk substrate. The resulting occupation changes within5\nthe energy intervals relevant for spin transport are shown\nby the dashed lines in Figure 5 b), which represent the\nmaximum occupation change caused by the given exci-\ntation parameters. While these occupation changes are\nqualitatively similar to the corresponding changes we de-\ntermined for the thin film (solid lines), their magnitude\nis much weaker. Thus, the spin current from Py into Au\nis not blocked by the excited non-equilibrium electron\ndistribution and efficiently extincts the OISTR signal.\nOur theoretical model thus explains the existence of\nan OISTR signal in Py on a metallic substrate by a\nblocked spin transport across the interface due to a tran-\nsientnon-equilibriumelectrondistributioninthemetallic\nsubstrate. This state blocking is more effective at higher\nabsorption strength in the metal. For a thin, 10 nmAu\nfilm, we find a higher absorption due to multireflection in\nthe thin film. Moreover, the absorbed energy cannot be\ndissipated to the depth by ballistic energy transport as it\nis known for gold films [4], but remains confined in this\nthin layer close to the experimentally studied Py film.\nIn conclusion, we uncover the competing influence of\nOISTR and interlayer spin transfer on the initial ultra-\nfast magnetization dynamics of the ferromagnetic alloy\nPermalloy. We show that OISTR dominates the ultra-\nfast magnetization dynamics of the alloy only in the\nabsence of interlayer spin transfer into a metallic sub-\nstrate. However, once interlayer spin transfer is possible,\nthe optically redistributed spins due to OISTR are, at\nleastpartially, transferredfromthealloyintothemetallic\nsubstrate and thus no longer dominate the initial ultra-\nfast demagnetization dynamics. Thus, our study demon-\nstrates a clear way to tune the competing influence of\nOISTR and interlayer spin transfer on the magnetization\ndynamics by controlling the efficiency of interlayer spin\ntransfer across interfaces.\nThe experimental work was funded by the Deutsche\nForschungsgemeinschaft (DFG, German Research Foun-\ndation) - TRR 173 - 268565370 Spin + X: spin in its\ncollective environment (Project A08 and B11). 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B 99, 174314\n(2019).Competing signatures of intersite and interlayer spin transfer in the ultrafast\nmagnetization dynamics - Supplemental Material\nSimon H¨ auser,1,∗Sebastian T. Weber,1Christopher Seibel,1Marius Weber,1\nLaura Scheuer,1Martin Anstett,1Gregor Zinke,1Philipp Pirro,1Burkard Hillebrands,1Hans\nChristian Schneider,1B¨ arbel Rethfeld,1Benjamin Stadtm¨ uller,1, 2,†and Martin Aeschlimann1\n1Department of Physics and Research Center OPTIMAS,\nRheinland-Pf¨ alzische Technische Universit¨ at Kaiserslautern-Landau, 67663 Kaiserslautern, Germany\n2Institute of Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany\n(Dated: May 1, 2023)\nSAMPLE PREPARATION\nPy thin films and Py/Au bilayers were grown on 0 .5 mm-thick MgO (100) by molecular beam epitaxy (MBE)\ntechnique in an ultrahigh vacuum (UHV) chamber at a base pressure of 5 ×10−9mbar. The cleaning protocol of the\nMgO substrates included ex-situ chemical cleaning with acetone and isopropanol and in-situ heating at 600 °C for 1 h.\nPy deposition was performed at room temperature with a growth rate in the range of R= 0.15˚A/s ( R= 0.15 nm/s)\ncontrolled by a quartz crystal oscillator. In the case of the bilayer structures, a 10 nm or 100 nm Au film was first\ngrown directly on the MgO substrate using similar parameters.\nEXPERIMENTAL DETAILS\nOur time-resolved experiments were performed with a high harmonic generation transversal magneto optical Kerr\neffect (HHG-T-MOKE) pump-probe setup [1, 2]. An amplified Ti:sapphire laser system (1.57 eV, 30 fs, 6 kHz, 1.7 mJ\nper pulse) was used to generate the pump and probe beams. To generate the XUV probe beam, we used the high\nharmonic generation process with a fiber based setup and neon as the excited noble gas [1, 2]. We determined the\ntemporal pulse length at the sample position to 80 fs for the 1.57 eV pump pulse using an autocorrelation method.\nBoth p-polarized pulses were focused on the sample at an angle of incidence of 45 degrees. The reflected XUV light\nis separated from the residual fundamental light of 1.57 eV by two 100 nm aluminum filters and then detected by a\nspectrometer to ensure energy and therefore element resolution. The estimated energy resolution is ≈0.8 eV.\nSPECTRAL RANGES FOR EVALUATION\n506 07 0/s8722/s48/s46/s500.0asymmetry (arb. u.)e\nnergy (eV)(a)i\nntensity\n506 07 0/s8722/s48/s46/s500.0asymmetry (arb. u.)e\nnergy (eV)(b)i\nntensity\n506 07 0/s8722/s48/s46/s500.0asymmetry (arb. u.)e\nnergy (eV)(c)i\nntensity\nFIG. S1: Detected XUV-spectra (cyan and magenta) and the resulting asymmetries of Py on (a) MgO, (b) 10 nm Au\nand (c) 100 nm Au. The red and blue shaded areas correspond to the evaluated spectral regions for Fe and Ni,\nrespectively.\nSCALING OF MAGNETIZATION DYNAMICS\nAs mentioned in the experimental details of the paper, we applied the same fluence for each Py sample with a\ndifferent substrate, resulting in different quenching. These results are shown in Fig. S2 (a). Here, the Py/Au(10 nm)arXiv:2304.14957v1  [cond-mat.mtrl-sci]  28 Apr 20232\nand Py/Au(100 nm) sample systems lost almost the same amount of spin polarization (about 30% and 28%) after\noptical irradiation, although the remagnetization is much faster in the case of Py/Au(100 nm). For Py/MgO a\nquenching of about 15% was found. The corresponding OISTR-trace (OT), as the difference of the Ni and Fe signal,\nis shown in Fig. S2 (b). While the OT for Py/Au(100 nm) is close to zero, the OT for Py/MgO and Py/Au(10 nm) is\nsimilar, but slightly higher in the case of Py/MgO. One conclusion out of this is, that although the absorbed energy\nin the spin system is almost 50% lower in Py/MgO than in Py/Au(10 nm), the OT is still higher in Py/MgO.\n01 2 0.70.80.91.01.1norm. asymmetry (arb. u.)d\nelay (ps) Fe (MgO) \nNi (MgO) \nFe (Au10nm) \nNi (Au10nm) \nFe (Au100nm) \nNi (Au100nm)(a)\n/s8722/s52/s48/s480 4 008 00/s8722/s48/s46/s48/s530.000.050.100.15OISTR trace (arb. u.)d\nelay (fs) Py/MgO \nPy/Au10nm \nPy/Au100nm(b)\n01 2 0.70.80.91.01.1norm. asymmetry (arb. u.)d\nelay (ps) Fe (MgO) \nNi (MgO) \nFe (Au10nm) \nNi (Au10nm) \nFe (Au100nm) \nNi (Au100nm)(c)\n/s8722/s52/s48/s480 4 008 00/s8722/s48/s46/s48/s530.000.050.100.15OISTR trace (arb. u.)d\nelay (fs) Py/MgO \nPy/Au10nm \nPy/Au100nm(d)\nFIG. S2: (a) Raw dynamic of the spin polarization of Py on MgO, Au(10 nm) and Au(100 nm). (b) Resulting\nOISTR trace of (a). (c) Dynamic of the spin polarization of Py on MgO, Au(10 nm) and Au(100 nm) normalized to\n≈30% quenching. (d) Resulting OISTR trace of (c).\nFor better comparison we have scaled the quenching of each sample to the value of 30%, using Py/Au(10 nm) as a\nreference (Fig. S2 (c)) to linearly extrapolate the OT of Py/MgO with ≈30% of quenching. This was done by first\nselecting a time window between 644 fs and 991 fs for Py/MgO, 340 fs and 810 fs for Py/Au(100 nm). The averaged\nspin polarization in this window was then calculated to be 0 .828 for Py/MgO and 0 .728 for Py/Au(100 nm). This\nvalue was then subtracted from the whole time-resolved curve, resulting in different normalized levels before time zero\nlT0, 0.165 for Py/MgO and 0 .265 for Py/Au(100 nm). Now the whole time resolved curve was multiplied by 0 .3/lT0\nand summed to 0.7. The result is shown in Fig. S2 (c), where all time resolved curves show similar quenching of\n≈30%. We then extract the OT of each sample, as shown in Fig. S2 (d). Now the OT of Py/MgO is about a factor\nof two larger than Py/Au(10 nm), showing the trend of decreasing OT with increasing Au-thickness.3\nTHEORETICAL MODEL\nWe used a model based on full Boltzmann collision integrals to calculate the dynamics of the electronic distribution,\nwhich we used to determine the time-dependent occupation of states as described in the main text. The full details\nof the model are described in Ref. [3]. To determine the absorption profiles for the considered scenarios, we solved\nthe Helmholtz equation for Al 2O3(2 nm)/Py(10 nm)/Au(d)/MgO(500 nm) heterostructures with d= 10 nm and d=\n100 nm, respectively. Details of the calculations can be found in the supplementary information of Ref. [4]. The\nrefractive index for Al 2O3was taken from Ref. [5], for Py from Ref. [6], for Au from Ref. [7] and for MgO from\nRef. [8]. For all values, a wavelength of 800 nm was considered in accordance with our experiments. We determined\nthe absorbed energy density by integrating the absorption profiles in the respective materials and considering the\nexperimental fluence of 28 .1 mJ cm−2. The intensity of the laser excitation was then adjusted in the model to reproduce\nthe energy absorbed in the experiment. For the two limiting cases of no transport and full ballistic transport in the\ncase of the 100 nm Au layer, we took into account only the first 10 nm of the absorption profile for the former case\nand the full 100 nm for the latter case. The resulting averaged fractions of absorbed energy are 0 .36 % nm−1for the\n10 nm Au layer, 0 .03 % nm−1for the 100 nm Au layer and 0 .15 % nm−1for the limiting case of no transport in the\n100 nm Au layer.\n∗shaeuser@rptu.de\n†b.stadtmueller@rptu.de\n[1] C. La-O-Vorakiat, E. Turgut, C. A. Teale, H. C. Kapteyn, M. M. Murnane, S. Mathias, M. Aeschlimann, C. M. Schneider,\nJ. M. Shaw, H. T. Nembach, and T. J. Silva, Phys. Rev. X 2, 011005 (2012).\n[2] S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E. Siemens,\nS. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, Proceedings of the National\nAcademy of Sciences 109, 4792 (2012).\n[3] B. Y. Mueller and B. Rethfeld, Phys. Rev. B 87, 035139 (2013).\n[4] C. Seibel, M. Weber, M. Stiehl, S. T. Weber, M. Aeschlimann, H. C. Schneider, B. Stadtm¨ uller, and B. Rethfeld, Phys.\nRev. B 106, L140405 (2022).\n[5] R. Boidin, T. Halenkoviˇ c, V. Nazabal, L. Beneˇ s, and P. Nˇ emec, Ceramics International 42, 1177 (2016).\n[6] K. K. Tikuiˇ sis, L. Beran, P. Cejpek, K. Uhl´ ıˇ rov´ a, J. Hamrle, M. Vaˇ natka, M. Urb´ anek, and M. Veis, Materials & Design\n114, 31 (2017).\n[7] S. M. Werner, K. Glantschnig, and C. Ambrosch-Draxl, J. Phys. Chem. Ref. Data 38, 1013 (2009).\n[8] R. Stephens and I. Malitson, J. Res. Natl. Bur. Stand. 49, 249 (1952)."    },    {        "title": "1706.04002v1.The_Dynamics_of_Magnetic_Vortices_in_Type_II_Superconductors_with_Pinning_Sites_Studied_by_the_Time_Dependent_Ginzburg_Landau_Model.pdf",        "content": "The Dynamics of Magnetic Vortices in Type II\nSuperconductors with Pinning Sites Studied by the\nTime Dependent Ginzburg-Landau Model\nMads Peter S\u001crensen and Niels Falsig Pedersen\nDepartment of Applied Mathematics and Computer Science, Richard Petersens Plads,\nBldg. 321, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark\nMagnus Ogren\nSchool of Science and Technology, Orebro University, SE-70182 Orebro, Sweden\nAbstract\nWe investigate the dynamics of magnetic vortices in type II superconductors\nwith normal state pinning sites using the Ginzburg-Landau equations. Simula-\ntion results demonstrate hopping of vortices between pinning sites, in\ruenced by\nexternal magnetic \felds and external currents. The system is highly nonlinear\nand the vortices show complex nonlinear dynamical behaviour.\nKeywords: Ginzburg-Landau equations ,type II superconductivity ,\nvortices ,pinning sites\n2010 MSC: 35, 37\n1. Introduction\nThe dynamics of magnetic vortices in type II superconductors at tempera-\ntures close to the critical temperature can be modelled by the time dependent\nGinzburg-Landau equations. The theory is based on a Schr odinger type equa-\ntion with a potential containing a quadratic term and a quartic term in addition\nto a kinetic term involving the momentum operator coupled to a magnetic \feld\n\u0003Mads Peter S\u001crensen\nEmail address: mpso@dtu.dk (Mads Peter S\u001crensen and Niels Falsig Pedersen)\nURL: www.compute.dtu.dk (Mads Peter S\u001crensen and Niels Falsig Pedersen)\nPreprint submitted to Journal of L ATEX Templates September 22, 2018arXiv:1706.04002v1  [cond-mat.supr-con]  13 Jun 2017governed by the Maxwell equations [1], [2], [3]. For type-II superconductors\nthe Ginzburg-Landau equations model the magnetic \feld penetration through\nquantized current vortices as the externally applied magnetic \feld exceeds a\nthreshold value. A number of variants of the Ginzburg-Landau equations have\nbeen used to investigate pattern formation in di\u000berent nonlinear media, not\nonly in superconductivity, and hence have become a popular \feld of study in\nnonlinear science [4], [5]. Our aim here is to investigate the dynamics of vor-\ntices in the presence of normal state pinning sites in the superconductor [3],\n[6]. Such pinning sites can arise from atomic impurities, magnetic impurities,\nlattice defects and defects in general. The Gibbs energy of the superconductor\nis given by a 4'th order potential in the order parameter. The sign of the coef-\n\fcient to 2'nd order term determines the phase transition between the normal\nand the superconducting state and hence this coe\u000ecient can be used to \fx the\npositions of inserted pinning sites. Secondly, we shall present a model for the\naction of the self induced magnetic \feld on vortex generation, when a net cur-\nrent is \rowing through a superconductor enforced by metal leads at the ends of\na superconducting strip.\n2. The time dependent Ginzburg-Landau model\nThe superconducting state is described by the order parameter  (r;t), where\nris the position in the superconducting volume denoted \n \u001aR3andtis time.\nIn the framework of the Ginzburg-Landau theory the Gibbs energy of the su-\nperconducting state Gsis given by\nGs=Gn\u0000\u000b0(r)\u0012\n1\u0000T\nTc\u0013\nj j2+\f\n2j j4: (1)\nHereGnis the Gibbs energy of the normal state, Tis the absolute temperature\nandTcis the critical temperature. The parameter \fis a constant and \u000b0(r) we\nchoose such that it depends on the space variable rin order to model pinning\nsites depleting the superconducting state at speci\fc positions. For T <T Cpos-\nitive values of \u000b0correspond to the superconducting state and negative values\n2model a pinning site at which the superconducting state becomes normal.\nThe order parameter is in\ruenced by the magnetic \feld B(r;t) given by the\nmagnetic potential through the relation B=r\u0002A. The Ginzburg-Landau\nparameter\u0014is introduced as the ratio between the magnetic \feld penetration\nlength\u0015and the coherence length \u0018, i.e.\u0014=\u0015=\u0018. We shall investigate the dy-\nnamics of \rux vortices penetrating the superconductor in the presence of pinning\nsites, whose positions are given by a function f(r) taking the value one in the\nsuperconducting regions and the value \u00001 at the position of a pinning site. Af-\nter scaling to normalized coordinates and using the zero electric potential gauge\nthe Ginzburg-Landau equations for the order parameter in nondimensional form\nreads [7]\n@ \n@t=\u0000\u0012i\n\u0014r+A\u00132\n +f(r) \u0000j j2 ; (2)\n\u001b@A\n@t=1\n2i\u0014( \u0003r \u0000 r \u0003)\u0000j j2A\u0000r\u0002r\u0002 A: (3)\nIn order to make the Ginzburg-Landau equations dimensionless, we have scaled\nthe space coordinates by the magnetic \feld penetration depth \u0015and time is\nscaled by\u00182=D, whereDis a di\u000busion coe\u000ecient [8]. The magnetic \feld Ais\nscaled by the factor \u0016 h=(2e\u0018), whereeis the electron charge. The wave function\n is scaled byp\n\u000b0=\f. The term in \u001bis the conductivity of the normal current\nof unpaired electrons. It is scaled by the factor 1 =(\u00160D\u00142), where\u00160is the\nmagnetic permeability of the free space. The normal current Jnand the super\ncurrent Jsread\nJn=\u0000\u001b@A\n@tand Js=1\n2i\u0014( \u0003r \u0000 r \u0003)\u0000j j2A; (4)\nwith the total current being J=Jn+Js. The coe\u000ecient f(r) of in Eq.(2)\nde\fnes the positions of the pinning sites by changing sign from +1 to \u00001 and\nthrough scaling fis related to \u000b0in Eq. (1). In solving numerically Eqs. (2) and\n(3) we need to specify appropriate boundary conditions. We seek to satisfy the\n3following three boundary conditions on the boundary of the superconducting\nregion@\nr\u0002A=Ba;r \u0001n= 0 and A\u0001n= 0: (5)\nThe \frst condition tells that the magnetic \feld at the surface of the superconduc-\ntor equals the applied external \feld Ba. The conditionr \u0001n= 0 corresponds\nto no super current crossing the boundary. Di\u000berentiating the boundary condi-\ntionA\u0001n= 0 with respect to time shows that this condition prevents normal\nconducting current to pass the boundary.\nThe Ginzburg-Landau equations (2) and (3) have been solved using the \fnite\nelement software package COMSOL Multiphysics [9], [10]. In order to model\npinning sites we have introduced the function f(r) with real values between \u00001\nand +1, where\u00001 corresponds to a normal state pinning site and +1 corre-\nsponds to regions of the superconducting state. We have chosen fto take the\nphenomenological form\nf(r) = \u0005N\nk=1fk(r) wherefk(r) = tanh((jr\u0000r0kj\u0000Rk)=wk): (6)\nThe function fattains the values \u00001 aroundNpinning sites positioned at r0k,\nfork= 1;2;:::;N . In the following we shall consider a two dimensional super-\nconductor where r= (x;y) and r0k= (x0k;y0k). This means that the pinning\nsites are circular with radius Rkand the transition from the superconducting\nstate to the normal state happens within an annulus of width wk. Assuming\na two dimensional superconductor means we strictly study an in\fnite prism,\nwhere the currents are \rowing in parallel with the xy-plane and the magnetic\n\feld is perpendicular to the xy-plane. However, the approach is valid for suf-\n\fciently thick \fnite size superconductors, where the geometric edge e\u000bects are\nnegligible. If the thickness is denoted by tthent>>\u0015 .\nOther choices for modelling pinning sites are available in the literature. In\nparticular we mention the local reduction of the mean free electron path at\npinning centres included in the Ginzburg-Landau model by Ge et al. [11]. Here\n4the mean free path enters as a factor on the momentum term in Eq. (2).\nThis approach is more based on \frst principles in the physical description than\nours. The above modelling strategy may also be used to investigate suppression\nof the order parameter. In particular we mention the experimental work by\nHaag et al. [6], where regular arrays of point defects have been inserted into\na superconductor by irradiation with He+ions. These defects acts as pinning\nsites.\nFigure 1: Numerical simulation of Eqs. (2) and (3) subject to the boundary conditions (5)\nshowingj j2. Dark red corresponds to j j2= 1 and dark blue corresponds to j j2= 0.\nThe initial conditions are  (r;0) = (1 +i)=p\n2 and A= (0;0). The parameter values are:\n\u0014= 4,\u001b= 1,Ba= 0:73. The positions of the pinning sites are: d1: r01= (\u00001;\u00003), d2:\nr02= (0;\u00002), d3: r03= (1;\u00001) and d4: r04= (2;0).\nNumerical simulations. In \fgure 1 we show snapshots of j j2from one simula-\ntion of the time dependent Ginzburg-Landau equations (2) and (3) subject to\n5the boundary conditions (5) from time t= 0 untilt= 750. We have chosen the\ninitial conditions  (r;0) = (1 +i)=p\n2 and A= (0;0). The external magnetic\n\feldBais turned on at time t= 0. This leads to a discontinuous mismatch\nbetween the initial vanishing magnetic \feld within the superconductor and the\nexternal applied magnetic \feld. The algorithm can handle this without prob-\nlems. Alternatively one could turn on the external magnetic \feld gradually\ngiving a more smooth transition.\nIn the region of interest we have inserted 4 pinning sites denoted d1, d2, d3\nand d4 at the respective positions r01= (\u00001;\u00003),r02= (0;\u00002),r03= (1;\u00001)\nandr04= (2;0). The pinning sites are modelled by fin Eq. (6) using Rk= 0:2\nandwk= 0:05 fork= 1;2;3;4. The external applied magnetic \feld Ba= 0:73 is\nchosen slightly smaller than the critical magnetic \feld for a superconductor with\nno pinning sites. This value leads to very complex dynamics of \ruxons entering\nthe superconductor resulting from mutual interactions and interactions with\nthe pinning sites as illustrated in Fig. 1. At time t= 12 we observe a \ruxon,\nf1, entering the superconductor, hopping from d1 to d2 in\ruenced by repulsive\nforces from the boundary and attractive forces from the pinning sites. At time\nt= 72 a second \ruxon, f2, has entered the superconductor and are attached to\nthe pinning site d1 and eventually pushing the \frst \ruxon f1 onto d3. As time\nprogress the \ruxon f2 deattaches d1 and moves into the superconductor and at\nthe same time a third \ruxon, f3, enters the superconductor at the right hand\nside moving toward d4, where it becomes trapped. A fourth \ruxon, f4, enters\nat the bottom boundary close to d1 and propagates into the superconductor\nand away from the pinning sites. Finally, a \ffth \ruxon, f5, enters close to d1\nand hops from d1 to d2, where it \fnally gets trapped. At t= 750 we have\nobtained a stationary state with two \ruxons in the bulk superconductor and\nthree \ruxons trapped on the pinning sites d2, d3 and d4. No \ruxon is attached\nto d1. In short the simulation results in Fig. 1 illustrate the intricate nonlinear\ndynamical behaviour of \ruxons hopping from pinning site to pinning site and\nat the same time experience mutual repulsive forces and repulsive forces from\nthe boundaries, controlled by the external applied magnetic \feld.\n63. Current carrying superconducting strips\nIn this section we study superconducting strips carrying currents along the\nstrip. The current is injected through metal contacts at two opposite boundaries\nof the superconductor, that is at x=\u0000Lx=2 andx=Lx=2, respectively, where\nLxis the length of the superconductor in the x-axis direction. At the side\nboundaries the superconducting current and the normal current are parallel to\nthe superconductor surface and therefore we use here the boundary conditions\n[1], [12]\nr\u0002A=Be=Ba+Bc;r \u0001n= 0 and A\u0001n= 0: (7)\nIn the above equations Beis the total external magnetic \feld composed of the\nsum of the applied magnetic \feld Baand the magnetic \feld Bcinduced from the\ntotal current J=Js+Jnin the superconducting strip. The induced magnetic\n\feld is given by\nBc=1\n4\u0019Z\n\nJ(r0)\u0002(r\u0000r0)\njr\u0000r0j3d\n: (8)\nAt the metal contacts we use the metal-superconductor boundary conditions\n[1], [12]\nr\u0002A=Be;  = 0 and\u0000\u001b@A\n@t\u0001n=Je\u0001n: (9)\nHerenis the outgoing normal vector to @\n and Jeis the external current density.\nIt has been shown in Ogren et al. [12] that the current induced magnetic \feld\nis well approximated by [13]\nBc=\u0006I\nLyyez; (10)\nwhereIis the total current \rowing from the metal lead into and through the\nsuperconducting strip. The width of the strip is Lyandezis the unit vector\nin thez-direction (here out of the plane). In the simulations we have used a\nuniform current density Jegiven by Je=I\nLyex, where exis the unit vector in\nthex-direction.\n70y\nxI I\n−5−55\n5 0(a) Anti vortexVortexBc\nBc\n05I Ia\nexy5\n0\n−5\n−5(b)=B  +B\n4 pinsc Be\n=B  +B c a BFigure 2: Numerical simulation of Eqs. (2) and (3) subject to the boundary conditions (7), (9)\nand (10). The initial conditions are  (r;0) = (1 +i)=p\n2 and A= (0;0). Parameters: \u0014= 4\nand\u001b= 4. (a) Simulations without pinning sites. Time t= 160 using Ba= 0 andI= 1:5.\n(b) Simulations with pinning sites. Time t= 280 using Ba= 0:5 andI= 0:5. The position\nof the four pinning sites are: r01= (\u00003;2),r02= (\u00001;2),r03= (1;2) and r04= (3;2).\nNumerical simulations. Figure 2(a) shows how vortex anti-vortex pairs are gen-\nerated at the top and bottom boundaries, propagating into the center of a\ncurrent carrying superconductor and eventually annihilate at the center. The\nexternally applied current, entering at the left hand side of the superconduc-\ntor, generates a magnetic \feld at the top boundary pointing out of the \fgure\nplane. At the bottom boundary the magnetic \feld points into the \fgure plane.\nNo external magnetic \feld Bais applied and the arrows in the \fgure show the\nstrength and direction of the super current within the superconductor. Genera-\ntion of vortex anti-vortex pairs has also been studied by Milo\u0015 sovi\u0013 c and Peeters\n[14] in a two dimensional superconductor structured with a lattice of magnetic\ndots. In this work di\u000berent complex patterns of vortex anti-vortex lattices have\nbeen found as the lattice constants of the magnetic dots are varied.\nIn Fig. 2(b) we have inserted four pinning sites in the superconductor placed\natr01= (\u00003;2),r02= (\u00001;2),r03= (1;2) and r04= (3;2). We also apply an\nexternal magnetic \feld Ba, which adds to the induced magnetic \feld from the\nexternal current \rowing through the superconductor from left to right. This\ngives rise to an asymmetry in the magnetic \feld between top and bottom of the\n8superconductor together with an asymmetry in the current \row as is evident\nfrom Fig. 2(b). Using the \feld strength Ba=0.5 and the current I= 0:5, two\n\ruxons enter from the top boundary and move toward the two center pinning\nsites, where they get trapped. The simulations demonstrate that \ruxon dynam-\nics can be controlled by applying an external magnetic \feld and by external\napplied currents.\n4. Conclusion\nWe have modi\fed the time dependent Ginzburg-Landau equations to model\nthe interaction between vortices and pinning sites in a planar two dimensional\nsuperconductor. The pinning sites are modelled by multiplying the quadratic\nterm in the Gibbs energy for the superconducting state by a suitable chosen\nfunction of the position, with a range from \u00001 at a pinning site to +1 away\nfrom the pinning sites. We found that pinning sites close to the boundary of the\nsuperconductor can lower the \frst critical magnetic \feld separating the Meiss-\nner state and the \rux penetration state. For magnetic \felds close to the \frst\ncritical \feld value we found complex nonlinear dynamical behaviour of the vor-\ntices interacting with the pinning sites, mutually and with the boundaries. The\nvortices can hop from pinning site to pinning site in\ruenced by repulsion from\nthe boundaries and repulsion from other vortices, which can push pinned vor-\ntices out of a given pinning site. For interaction energies and pinning energies\nof comparable magnitudes, the dynamics of vortices appears particular complex\nand intricate. Externally applied currents through the superconductor also in-\n\ruences the dynamics of vortices in superconductors with pinning sites. Hence,\nthe dynamics can to some extent be controlled by both external magnetic \felds\nand external currents. We speculate that further work could encompass deriva-\ntion of particle models of \ruxons in potentials modelling the pinning sites based\non collective coordinate approaches [15].\nAcknowledgement. We thank the EU Horizon 2020 (COST) program MP1201\nNanoscale Superconductivity: Novel Functionalities through Optimized Con-\n9\fnement of Condensate and Fields (NanoSC -COST) for \fnancial support.\nReferences\nReferences\n[1] D. Vodolazov, B.J. Baelus, F. Peeters, Dynamics of the superconducting\ncondensate in the presence of a magnetic \feld. Channelling of vortices in\nsuperconducting strips at high currents, Physica C 404 (2004) 400{404 doi:\n10.1016/j.physc.2003.10.027 .\n[2] P. Lipavsky, A. Elmurodov, P.-J. Lin, P. Matlock, G.R. Berdiyorov, E\u000bect\nof normal current corrections on the vortex dynamics in type-II supercon-\nductors, Physical Review B 86 (144516) (2012) 1{8.\n[3] F.-H. Lin, Q. Du, Ginzburg-Landau vortices: Dynamics, pinning and hys-\nteresis, SIAM J. Math. Anal. 28 (6) (1997) 1265{1293.\n[4] H. Nielsen, P. Olesen, Vortex-line models for dual strings, Nuclear Physics\nB 61 (1973) 45{61.\n[5] A.C. Scott, Encyclopedia of Nonlinear Science, Routledge, New York, 2005.\n[6] L.T. Haag, G. Zechner, W. Lang, M. Dosmailov, M.A. Bodea, J.D. Pedarnig,\nStrong vortex matching e\u000bects in YBCO \flms with periodic modulations of\nthe superconducting order parameter fabricated by masked ion irradiation,\nPhysica C 503 (2014) 75{81.\n[7] T.S. Alstr\u001cm, M.P. S\u001crensen, N.F. Pedersen, S. Madsen, Magnetic \rux lines\nin complex geometry type-II superconductors studied by the time dependent\nGinzburg-Landau equation, Acta Appl. Math. 115 (2011) 63{74. doi:10.\n1007/s10440-010-9580-8 .\n[8] L.P. Gorkov, G.M. Eliashburg, Generalization of the Ginzburg-Landau equa-\ntions for non-stationary problems in the case of alloys with paramagnetic\nimpurities. Sov. Phys. (JETP) 27 (1968) 328.\n10[9] COMSOL Multiphysics. [Cited January 6, 2016]. Available at the internet:\nhttps://www.comsol.com/.\n[10] W.B.J. Zimmerman, Multiphysics Modelling with Finite Element Methods,\nWorld Scienti\fc, Singapore, 2005.\n[11] J-Y. Ge, J. Gutierrez, V.N. Gladilin, J.T. Devreese, V.V. Moshchalkov,\nStrong vortex matching e\u000bects in YBCO \flms with periodic modulations of\nthe superconducting orderparameter fabricated by masked ion irradiation,\nNature Communications (2015) 1{8. DOI: 10.1038/ncomms7573.\n[12] M. Ogren, M.P. S\u001crensen and N.F. Pedersen, Self-consistent Ginzburg-\nLandau theory for transport currents in superconductors, Physica C 479\n(2012) 157{159. doi:10.1016/j.physc.2011.12.034.\n[13] M. Machida and H. Kaburaki, Direct simulation of the time-dependent\nGinzburg-Landau equation for type-II superconducting thin \flm: Vortex\ndynamics andV-Icharacteristics, Phys. Rev. Lett. 71 (19) (1993) 3206{3209.\ndoi:10.1103/PhysRevLett.71.3206.\n[14] M. Milo\u0014 sevi\u0013 c, F. Peeters, Vortex-antivortex lattices in superconducting\n\flms with magnetic pinning arrays, Journal of Low Temperature Physics\n139 (1/2) (2005) 257{272. doi:10.1007/s10909-005-3929-9 .\n[15] J.G. Caputo, N. Flytzanis and M.P. S\u001crensen, Ring laser con\fguration\nstudied by collective coordinates, Optical Society of America B 12 (1995)\n139{145.\n11"    },    {        "title": "2309.04243v1.Emergence_of_Chaos_in_Magnetic_Field_Driven_Skyrmions.pdf",        "content": "1 \n Emergence of Chaos in Magnetic -Field -Driven Skyrmions  \nGyuyoung Park and Sang -Koog Kima) \nNational Creative Research Initiative Center for Spin Dynamics and Spin -Wave Devices, Nanospinics  \nLaboratory, Research Institute of Advanced Materials, Department of Materials Science and Engineering, Seoul \nNational University, Seoul 151- 744, Republic of Korea  \n \nWe explore magnetic- field-driven chao s in magnetic skyrmion s. Oscillating magnetic field s \ninduce  nonlinear dynamics in skyrmions , arising from the coupling of the secondary gyrotropic \nmode  with a non-uniform , breathing -like mode.  Through micromagnetic simulations, we  \nobserve complex patterns of hypotrochoidal motion in the orbital trajectories of the skyrmions , \nwhich  are interpreted  using  bifurcation diagram s and local Lyapunov exponents. Our findings  \ndemonstrate that different nonlinear behavior s of skyrmions emerge  at distinct temporal stages , \ndepending on the nonlinear dynamic parameters. Investigating the abundant dynamic patterns \nof skyrmion s during  the emergence of chaos  not only enhances device reliability but also \nprovide s useful guidelines for establish ing chaos  computing based on skyrmion dynamics . \n \na) Author to whom all correspondence should be addressed; electronic mail: sangkoog@snu.ac.kr  \n  2 \n Introduction \nChaos presents a fundamental phenomenon in nonlinear dynamics, observed in various \nsystems, both classical and  quantum mechanical oscillators. Nonlinear dynamical systems can \nexhibit diverse  behaviors, including fixed points , periodic and  quasiperiodic , and chaotic \nmotions . This transition from the regular  nonlinear behavior to chaos  is a typical proc ess on \nthe route to chaos.  Importantly, these behavior s can manifest at different temporal stages  within \na single system [1]. For instance, a system may initially exhibit periodic  motion before \ntransitioning to  chaos , or it may start with quasiperiodic  motion and gradually become chaotic  \nover time.  \nIn the field of spintronics, topologically protected spin textures  like skyrmions have  \ngarnered  significant attention due to their potential applicability in  low-power -driven high-\ndensity data storage [2,3] , and ultrafast information processing , thanks to their topological \nstability at the nano- scales [4]. Therefore, delving into the nonlinearity wit hin skyrmion \ndynamics and any potential chaotic effects is crucial for ensuring reliable device utilization. \nWhile c haotic dynamics of other topological spin textures, such as current -driven magnetic \nvortice s[5,6]  and antiferromagnetic bimerons [7], have been studied, reports on chaotic \ndynamics  of skyrmions  are relatively rare, even though their nonlinear dynamics have been \nexamined in previous studies [8]. Further research is needed to fill this gap and explore chaoti c \ndynamics in magnetic skyrmions.  \nIn this letter, we begin by investigating  the field-driven nonlinear dynamics of \nmagnetic skyrmion s, which exhibit  complex hypotrochoidal trajectories. Subsequently, we \nmap the various regimes of the skyrmion’s nonlinear behavior over time, building  a \ncomprehensive nonlinear dynamic phase map. This map illustrates the transition from regular 3 \n to chaotic regime along temporal space versus the amplitude of the oscillating magnetic fields.  \nAdditionally, by exploring and harness ing the chaotic behavior of skyrmions, we may pave the \nway for novel applications in chaos -based computing and signal processing. Therefore, our \nfindings provide valuable insights for designing future skyrmion- based technologies with \nenhanced functionaliti es and reliability.  \n \nMethods  \nThe model employed in the present study involves  a Néel-type skyrmion formed in a \ncircular nano -disk with a radius ( Rdisk) of 30nm and a thickness of 0.6 nm , as shown in its \nground state in Fig. 1(a). To reduce the magnetostatic energy at the boundary of the disk, spins \nat the edge are tilted toward the center of the skyrmion. To numerically calculate the dynamic  \nmotion  of the skyrmion, we used  the MuMax3 code [9], which utilizes the Landau -Lifshitz -\nGilbert (LLG) equation : 𝜕𝜕𝑴𝑴/𝜕𝜕𝜕𝜕 =−𝛾𝛾𝑴𝑴×𝑯𝑯𝒆𝒆𝒆𝒆𝒆𝒆+(𝛼𝛼𝐺𝐺/𝑀𝑀𝑠𝑠)𝑴𝑴×𝜕𝜕𝑴𝑴/𝜕𝜕𝜕𝜕 , where 𝛾𝛾  is the \ngyromagnetic ratio, 𝛼𝛼𝐺𝐺 is the Gilbert damping constant, and 𝑯𝑯𝒆𝒆𝒆𝒆𝒆𝒆 is the effective field given \nas 𝑯𝑯𝒆𝒆𝒆𝒆𝒆𝒆=−(1𝜇𝜇0⁄)𝜕𝜕𝐸𝐸𝑡𝑡𝑡𝑡𝑡𝑡𝜕𝜕𝑴𝑴⁄  . The total energy , Etot, includes the magnetostatic, \nmagnetocrystalline anisotropy, exchange, and intrinsic Dzyaloshinskii -Moriya Interaction \n(DMI), and Zeeman energies. For the  material , we assume d Co interfaced with Pt and us ed the \nfollowing  parameters: saturation magnetization M s = 0.58× 106 A/m, exchange constant A ex = \n1.5× 1011 J/m, perpendicular anisotropic constant K u = 4×105 J/m3, Gilbert damping constant \n𝛼𝛼𝐺𝐺 = 0.01, and interfacial DMI constant D int = 3 mJ/m2. The cell size was set to 0.6× 0.6× 0.6 \nnm3. \nWith the aforementioned conditions, we intentionally formed a  Néel-type skyrmion with \na core polarization of - 1 (spin down)  and allowed it to relax to reach its ground state. In the 4 \n ground state, the skyrmion exhibits a  circular domain wall represented by the gray line and \ninset circle , indicating  a radius ( Rsky)[10] of about 12.5 nm. We applied a  linearly polarized \nharmonic field, H AC = 𝛼𝛼sin(2𝜋𝜋f+t), uniformly over the entire  disk, where  𝛼𝛼 = A/As[11] and As \n(= 2730 Oe) represents  the static field used to annihilat e the skyrmion. We gradually increased \n𝛼𝛼  from 0 to 0.30 in increments of 5×10−4 . By considering the skyrmion’ s inertial mass \nℳ[12] and using Thiele ’s equation of motion[13] , \n −ℳ𝑹𝑹̈+𝐺𝐺×𝑹𝑹̇−𝐾𝐾𝑹𝑹=0. (1) \nWe obtained t he angular frequencies of two gyrotropic modes as follows:  \n 𝑤𝑤±=−𝐺𝐺2ℳ⁄ ±�(𝐺𝐺2ℳ⁄ )2+(𝐾𝐾/ℳ). (2) \nThe higher (secondary) gyrotropic mode, denoted as f + (𝑓𝑓+=𝑤𝑤+/2𝜋𝜋≈ 21.37 GHz, cf.  𝑓𝑓−≈ \n2.19 GHz at zero external field) , represents a counter -clockwise rotation  in this core polarity \nof spin down. Here, 𝐺𝐺  is the gyrocoupling constant, 𝐾𝐾  is the spring constant, and 𝑹𝑹=\n(𝑋𝑋,𝑌𝑌)  represents the guiding center  with  𝑋𝑋=∫𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 /∫𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥   and 𝑌𝑌=∫𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 /\n∫𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 , where 𝑥𝑥=(1/4𝜋𝜋)𝒎𝒎∙�𝜕𝜕𝑥𝑥𝒎𝒎×𝜕𝜕𝑦𝑦𝒎𝒎� is the topological charge density [14]. \nUnlike the lower  or fundamental gyrotropic mode , w-, where the entire skyrmion moves \nas a rigid body, the  higher  or secondary  gyrotropic mode , w+, exhibits a unique behavior  (see \nsupplemental material) . In the w + mode, the core and peripheral spins gyrate in the same \ndirection, but they are out-of-phase when the skyrmion is excited [15]. This leads to  a coupling \nbetween the in -plane gyrotropic mode and a non- uniform out-of-plane breathing [16]-like mode. \nConsequently , during the application of a  driving force, the skyrmion undergoes a deformation \nthat breaks its rotational symmetry. When  the topological soliton experiences deformation, its  \nmoment of inertia  and spring constant  are altered,  giving rise to  the soliton ’s nonlinear \ndynamics [17,18] . This deformation -induced nonlinear dynamics  exclusively occur s when the 5 \n skyrmion is driven by the higher gyrotropic mode. In Fig. 1(b), we illustrated  the deformed \nskyrmion during its gyration . The Gray circle show s the distorted contour of the skyrmion’ s \ndomain wall , which assumes an  irregular ellipse -like shape  due to the broken rotational \nsymmetry of the breathing -like mode  being coupled with its gyration mode.  A profile of the \nindividual spins ’ orientations  across the center is shown below the skyrmion illustration . \nCompared to the ground state , the number of black spins inside the skyrmion’ s domain wall  \nincreas es from 5 to 7. Small black circles just above the black spins  denote the position of the \nguiding center at th at moment. The guiding center follows an orbital motion during the gyration, \nresembling  a hypotrochoid, which belongs to the  family of curves known as roulette s. \n \nResults  \nThe hypotrochoid  is traced by a point attached to a circle with  radius b, rolling inside a \nfixed circle with radius a, where the point is a distance  d from the interior circle ’s center . When \nthe higher gyrotropic mode is excited, its orbital motion represented by  the guiding center \ninevitably follows  one of the hypotrochoids [12,19] , which can be parametrized as follows : \n 𝑧𝑧(𝜕𝜕)=𝑟𝑟1𝑒𝑒𝑖𝑖𝑤𝑤1𝑡𝑡+𝑟𝑟2𝑒𝑒−𝑖𝑖𝑤𝑤2𝑡𝑡, (3) \nwhere the geometric parameters are related as follows:  𝑥𝑥=𝑟𝑟2, 𝑏𝑏=(𝑤𝑤1𝑤𝑤2)⁄𝑟𝑟1, and  𝑎𝑎=\n((𝑤𝑤1+𝑤𝑤2)𝑤𝑤2⁄ )𝑟𝑟1. The Thiele’ s equation (Eq. 1)  can be modified  with the linearly -polarized \noscillating magnetic field  as: \n −ℳ𝑹𝑹̈+𝐺𝐺×𝑹𝑹̇−𝐾𝐾𝑹𝑹+𝜇𝜇(𝒛𝒛�×𝑯𝑯)=0, (4) \nwith H = (0,  𝜇𝜇𝛼𝛼Hssin(w+t), 0), and the solution of Eq. ( 4) is, \n 𝑅𝑅(𝜕𝜕)=𝑅𝑅−𝑒𝑒𝑖𝑖𝑤𝑤−𝑡𝑡+(𝑅𝑅++ℎ\n2)𝑒𝑒𝑖𝑖𝑤𝑤+𝑡𝑡−ℎ\n2𝑒𝑒−𝑖𝑖𝑤𝑤+𝑡𝑡 (5) 6 \n with ℎ=𝜇𝜇𝛼𝛼𝐻𝐻𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2𝑤𝑤+𝜕𝜕(ℳ𝑤𝑤+2+𝐾𝐾) ⁄  . Eq. (5) takes a form similar to that of the  \nhypotrochoids , Eq. (3) . Note that the two gyrotropic modes 𝑤𝑤±  have opposite signs , \nindicat ing the opposite rotation sense , with 𝑅𝑅−=−𝑅𝑅+=(−𝑤𝑤−+𝑤𝑤+)ℎ𝑤𝑤+⁄. Therefore, the \nhypotrochoidal parameters for the skyrmion are given as 𝑟𝑟1=(−𝑤𝑤−+𝑤𝑤+)ℎ𝑤𝑤+⁄ and 𝑟𝑟2=\n(𝑅𝑅++ℎ\n2)=(2𝑤𝑤−−𝑤𝑤+)ℎ2𝑤𝑤+⁄ . So far, the equation of motion has been  linear.  However, it \nbecomes nonlinear as the eigenfrequencies , denoted by Eq. 2, vary during the motion. D ue to \nthe skyrmion ’s deformation  under  oscillation causes changes in  ℳ and 𝐾𝐾, where ℳ=ζ𝑟𝑟̅ \nand 𝐾𝐾=𝜉𝜉𝑟𝑟̅2  [19], with ζ  and 𝜉𝜉  being deformation -induced coefficients , and 𝑟𝑟 \nrepresenting  the local radial distance of the closed domain  wall. Furthermore, 𝑟𝑟  varies \nsignificantly for irregularly shaped skyrmions [20] . \n The shape of the hypotrochoid can be defined by two parameters, namely, the number \nof cusps , 𝜈𝜈=𝑎𝑎𝑏𝑏⁄=(𝑤𝑤1+𝑤𝑤2)𝑤𝑤1⁄, and the ratio of the distance to the smaller radius , 𝜀𝜀=\n𝑥𝑥𝑏𝑏⁄=𝑟𝑟2𝑤𝑤2𝑟𝑟1𝑤𝑤1⁄ . 𝜀𝜀 determines the type of the hypotrochoid. For example, in  either case of \n(𝑤𝑤1>𝑤𝑤2  and 𝜀𝜀<1 ) or (𝑤𝑤1<𝑤𝑤2  and 𝜀𝜀>1 ), the curve is a prolate hypotrochoid. \nConversely, if  the opposite  conditions  are met,  the curve is a curtate hypotrochoid. When 𝜀𝜀 = \n1, the curve is a hypocycloid. \n Figure 2(a) illustrates the representative shapes of the orbital trajectories for the  \nindicated values of ( 𝜈𝜈, 𝜀𝜀), which are also label ed by ① ~ ⑫ in Fig. 2( b) for four different \nvalues of 𝛼𝛼  (i.e.,  𝛼𝛼  = 0.0245, 0.0365, 0.126, and 0.1485) . These 𝛼𝛼  values fall within \ndistinct transition space s between different nonlinear behaviors . The parameter 𝜈𝜈  was \nestimated from the Fast Fourier Transform of the trajectories over time , while  𝜀𝜀 was derived \nfrom a geometrical analysis of R (X, Y) without any fitting. The number of cusps in each orbital \ngeometry  corresponds  well with each 𝜈𝜈  value. Meanwhile, when 𝜀𝜀  = 1, the hypotrochoid 7 \n becomes a hypocycloid , as seen in cases  ④ and ⑦. In this skyrmion system, 𝑤𝑤1 (= 𝑤𝑤−) is \nalways smaller than 𝑤𝑤2  (= 𝑤𝑤+ ). Therefore , trajectories with 𝜀𝜀>1  exhibit  prolate \nhypotrochoids , as observed  in cases ③, ⑥, and ⑨, while trajectories with 𝜀𝜀<1 display  \ncurtate hypotrochoids , as in cases  of ⑧, ⑩, ⑪, and ⑫. \nFigure 2( b) depict s the hypotrochoidal parameters of 𝜈𝜈  (height) and 𝜀𝜀  (color) as \nfunctions of evolution time ( 𝜏𝜏) and reduced field amplitude α   (0 ~ 0.157) for skyrmion motions \nexcited by oscillating magnetic field s. It is important to note that b eyond 𝛼𝛼=  0.157, the \nskyrmion  break s down within the observation time  from  0 to 534 𝜏𝜏, where 𝜏𝜏 represents  one \nperiod of the modulation frequency and  is equivalent to the inverse of  the frequency of the \nhigher gyrotropic mode , 1/f+. In the plots of  (𝜈𝜈 , 𝜀𝜀 ), two distinct regions  (black color ) are \nseparated  by the maximum 𝜀𝜀  values  (yellow color ) along given alpha values. When  the \ncondition of τ  and α below the yellow line, the skyrmion motions show fixed points or \nperiodic motions,  while they exhibit  quasiperiodic and chaotic motion on the condition above \nthe yellow line. Furthermore, there is a transition from prolate -type to curtate -type after the \nmaximum 𝜀𝜀 line. \n Next, we turn our attention to various nonlinear  regimes related to the nonlinear \nparameters.  Analyzing a single variable, such as the orbital radius 𝜌𝜌=|𝑹𝑹|/𝑅𝑅𝑠𝑠𝑠𝑠𝑦𝑦 , offers  a \nhighly effective method to classify different types of nonlinear behaviors depending on the \nnonlinear parameters. In our analysis, we identified  six critical points : 𝛼𝛼 = 0.0245 for the shift \nfrom a fixed point to periodic motion;  𝛼𝛼  = 0.0365 for the transition from periodic  to \nquasiperiodic motion;  𝛼𝛼  = 0.126 and 𝛼𝛼  = 0.1485 for the gradual transitions to complex \nquasiperiodicities ; 𝛼𝛼 = 0.152 for the transition from quasiperiodic motion to chaos ; and finally,  \nat 𝛼𝛼 = 0.1565, the skyrmion begins to break down during its motion. 8 \n For a more qualitative analysis of nonlinear behaviors, we plotted 𝜌𝜌 versus time for \n𝛼𝛼 = 0.0245, 0.0365, 0.126, 0.1485, and 0.152, as  illustrated in Fig. 3(a). When the skyrmion \nis driven by  the higher gyrotropic mode, 𝜌𝜌  exhibits  two steady -states , 𝜌𝜌± , at different \ntemporal stages: an initial unstable 𝜌𝜌+ state,  followed by a later stable  𝜌𝜌− state. In the 𝜌𝜌+ \nstate, the skyrmion is underdamped, while in the 𝜌𝜌− state, it becomes  overdamped due to the \nchanges in 𝐾𝐾 . For 𝛼𝛼  < 0.0245, the skyrm ion doesn’ t reach the 𝜌𝜌−  state, hindered by  an \nenergy barrier  that primarily originate s from the damping force . This regime is a fixed point \nwhere the guiding center asymptotically approaches the 𝜌𝜌+ state after  extensive gyration. As \n𝛼𝛼 surpasses 0.0245, 𝜌𝜌 increases , and its 𝜌𝜌𝑠𝑠 oscillates around the 𝜌𝜌− state. As 𝛼𝛼 continues \nto increa se, the transition time from 𝜌𝜌+ to 𝜌𝜌− decreas es. Specifically, at  𝛼𝛼 = 0.0365, the 𝜌𝜌− \nstate is achieved after a few hundred periods. In this regime  of  𝛼𝛼 < 0.0365, 𝜌𝜌 exhibits a \nperiodic behavior in its steady -states, wh ereas  it becomes  quasiperiodic (or loosely periodic) \nfor 𝛼𝛼 ≥ 0.0365. In this latter regime, some variations in the signal do not conform to regular \nperiodicities. For 𝛼𝛼 = 0.126 and 𝛼𝛼 = 0.1485, the quasi -periodicities gradually become much \nmore complex. Finally, at 𝛼𝛼 = 0.152, regular periodicities start to diminish, and chaos begins  \nto occur . The signals  shown at the top of Fig. 3(a) for 𝛼𝛼 = 0.126, 0.1485, and 0.152 allow us \nto better visualize the se periodicities. Unlike  the cases of  𝛼𝛼 = 0.126 and 0.1485, the case of  \n𝛼𝛼= 0.152 has no periodicities , indicating a transition  from  quasiperiodic to chaotic motion s. \nWhen  𝛼𝛼  ≥  0.152, a well -known property of chaos, the sensitive  dependence o n initial \nconditions  (SDI), arises . For these values of 𝛼𝛼 , signal s with minor differences in initial \nconditions  can result in substantial  variation s in the resulting signals.  \nFigure 3(b) displays the two steady  states, 𝜌𝜌±, versus  𝛼𝛼. The  separate 𝜌𝜌± values were \nidentified when  the variance of the <  𝜌𝜌> values within a moving window reached a minimum.  9 \n The different nonlinear regimes are labeled as fixed points (‘FP ’), periodic (‘P’) and \nquasiperiodic (‘QP’) motion, and c haos (‘C’). In the chaotic regime, the concept of SDI  was \nutilized. With SDI, even an extremely small variance in the initial conditions can lead to vastly \ndifferent outcomes. 𝜌𝜌±  values  obtained from four different initial conditions (see \nSupplemental Material s) were averaged . In the breakdown regime , indicated by black dotted \nboxes  for 𝛼𝛼  > 0.152, 𝜌𝜌±  exhibit  large var iations . This  distribution  signifies that even an \nextremely small difference in initial conditions, such as  ∆𝑥𝑥< 10-5, can significantly alter the \nlifetime of the skyrmion within this regime. Furthermore, the 𝜌𝜌− value generally decreases  in \nthe breakdown regime , indicative of  high instability in the dynamics of the  skyrmion.  As 𝛼𝛼 \nincreases beyond  𝛼𝛼 = 0.2365, unstable 𝜌𝜌+ states are no longer  observed.  \nA bifurcation diagram  can facilitate a comprehensive understanding of potential long -\nterm behaviors and highlight different periodicities that arise  with changes in  a key  bifurcation \nparameter . In this context, 𝜌𝜌𝑠𝑠 represents  the particular periodic  behavior, while α  acts as the \nbifurcation parameter. For example, Fig . 3(c) presents the bifurcation diagram of 𝜌𝜌𝑠𝑠 versus  α, \noffering a visua l guide to the field -driven skyrmion ’s quasiperiodic route to chaos. The diagram \noutlines  different stages , previously detailed  in Fig. 3(b) , and marks  each bifurcation point with \nblack -dashed vertical lines . As α increases, both 𝜌𝜌− and 𝜌𝜌+ bifurcate, but the  skyrmion does \nnot oscillate between  the 𝜌𝜌± states , back and forth. The existence of a breakdown regime and  \nfinite simulation time necessitate the inclusion of SDI in the chao s regime , by plotting the 𝜌𝜌𝑠𝑠 \nvalues at  other initial conditions. Unlike the ‘ FP’ regime, 𝜌𝜌𝑠𝑠 increases exponentially in the  ‘P’ \nregion. Through a series of bifurcations in the  ‘QP’ regime, denoted as  Ⅰ, II, and Ⅲ, the \nskyrmion enters the chaos regime , where 𝜌𝜌𝑠𝑠 can take on every possible value , as evidenced \nby the scattered data points . This result effectively describes the series of quasiperiodic routes  10 \n to chaos.  \nFinally, we evaluated the local Lyapunov exponent (LLE ), denoted as  𝜆𝜆(𝜕𝜕), a measure \nof sensitivity to initial conditions in a dynamical system, evaluated locally in the phase space \n(i.e., at a specific point or over a short period of time ). We constructed the 𝜆𝜆(𝜕𝜕) map from the \ntime trace of the speed, 𝑣𝑣, of the guiding cente r R (X, Y) as follow s [21] : \n 𝜆𝜆(𝜕𝜕)=𝛿𝛿(𝜕𝜕+∆𝜕𝜕)−𝛿𝛿(𝜕𝜕)\n∆𝜕𝜕 (6) \nand   \n 𝛿𝛿(𝜕𝜕)=1\n𝑁𝑁�𝑙𝑙𝑠𝑠��𝑉𝑉𝑖𝑖+𝑡𝑡−𝑉𝑉𝑗𝑗+𝑡𝑡�\n�𝑉𝑉𝑖𝑖−𝑉𝑉𝑗𝑗��𝑁𝑁\n𝑖𝑖=1 (7) \nwhere 𝑉𝑉𝚤𝚤��⃗=[𝑣𝑣𝑖𝑖,𝑣𝑣𝑖𝑖+𝛵𝛵]  is the reconstructed phase space with time delay 𝛵𝛵  and 𝑉𝑉𝚥𝚥��⃗  is the \nnearest neighbor  such that �𝑉𝑉𝚤𝚤��⃗−𝑉𝑉𝚥𝚥��⃗� is minimized . In the resultant LLE map, with respect to \n𝛼𝛼  and time (𝜏𝜏 ), as shown in Fig. 4, blue regions (where 𝜆𝜆  < 0) depict the exponential \nconvergence of trajectories in the phase space, representing stable ordered states . In contrast, \nred regions (where 𝜆𝜆 > 0) indicate  the exponential divergence of trajectories in the phase space, \nsignifying chaotic states  or ordered but unstable states . White regions  (where λ≈ 0), situated  \nbetween the red and blue regions , represent a transition space between order and chaos , known \nas the edge of chaos.  A black  region at the top of  the map  denotes the breakdown regime . \nThe p reviously classified regimes of nonlinear dynamics in  the bifurcation map match  \nwell with the LLE map at 𝜏𝜏 ≈ 0. However, the LLE map strongly suggest s that the nonlinear \nbehavior changes over time, as evidenced  in the map of hypotrochoidal parameters . The first \nwhite line  on the left  corresponds exactly to  the maximum 𝜀𝜀 line in Fig. 2. This line indicates \na transition from unstable to stable states  (for 𝛼𝛼 < 0.152)  or from ordered to chaotic states  (for 11 \n 𝛼𝛼  > 0.152) . For 0 < 𝛼𝛼  < 0.0365, the skyrmion  maintains  its ordered state throughout the \noscillation. For 0.0365 ≤ 𝛼𝛼 < 0.1, it begin s with a stable limit cycle or torus  and display s \nintermittent unstable ordered states due to repelling trajectories between 𝜌𝜌+ and 𝜌𝜌− states . \nFor 0.1 ≤ 𝛼𝛼 < 0.15, the skyrmion ’s dynamic states shift  from stable  state to unstable . For \n0.15 ≤ 𝛼𝛼 < 0.17, the skyrmion shows transient chaos in  the initial states. For 0.17 ≤ 𝛼𝛼, the \nskyrmion display s chaotic dynamics throughout the oscillation until it breaks down. \n \nConclusion  \nWe investigated the nonlinear dynamic behaviors  of a magnetic skyrmion  driven by \noscillating magnetic fields. The nonlinear mode arises  from the coupling  of gyrotropic modes  \nwith a non- uniform breathing -like mode . The deformation of the skyrmion altered key \nproperties such as  its moment of inertia and  spring constant , which in turn affected the \neigen frequency of the fundamental gyrotropic mode. This led to  changes in the characteristic \nparameters , including the cusp number and the type of the hypotrochoid, leading to the complex \nhypotrochoidal motions  of the skyrmion over time. By constructing a bifurcation map and a \nlocal Lyapunov exponent map, we were able to distinguish the entire range  of nonlinear \nbehavior s, spanning from ordered regimes  to chaos , across various field amplitudes and \ntemporal stages.  This in-depth understanding of the field- strength  and time -dependent chaotic \ndynamics of magnetic skyrmion s provides essential  insights  into different nonlinear dy namic  \nroutes to chaos. Furthermore, it can foster  the development of innovative computation schemes  \nthat leverage the initial condition -sensitive, deterministic chaos  dynamics . \n 12 \n Acknowledgment  \nThis research was supported by the Basic Science Research Program through the National \nResearch Foundation of Korea (NRF ), funded by the Ministry of Science , ICT, and Future \nPlanning  (Grant No. NRF -2021R1A2C2013543) . The Institute of Engineering Research at \nSeoul National University provide d additional research facilities for this work . \n  13 \n Figure captions  \nFig. 1. (a) Ground- state magnetization ( 𝒎𝒎=𝑴𝑴𝑀𝑀𝑆𝑆⁄) configuration of a Néel -type skyrmion \nconfined within a nano -disk. (b) A snapshot of the  magnetization  configuration of the same \nskyrmion at an arbitrary moment , excited by a sinusoidal magnetic field ( HAC) applied along \nthe y-axis. The domain wall of the skyrmion is represented by a gray circle shown in each spin \nconfiguration. Inset figures show spin profiles of the gray-shaded boxes. Black wide circles \nindicate  the position of the  guiding center at each state.  \n \nFig. 2. ( a) Orbital trajectories  of skyrmion motions calculated using the guiding center over a \ntime period of 𝜕𝜕𝑖𝑖 ~ 𝜕𝜕𝑖𝑖+1𝑓𝑓−⁄ at indicated values of  (𝜈𝜈, 𝜀𝜀) (see ① ~ ⑫), (b) Calculations \nof 𝜈𝜈 and 𝜀𝜀 on the plane of evolution time ( 𝜏𝜏) and reduced field amplitude  (𝛼𝛼). The 𝜀𝜀 value \nis represented by the color shown in the color bar on the right. White dotted lines correspond \nto 𝛼𝛼=0.0245 ,0.0365 ,0.126 ,and 0.1485 . \n \nFig. 3. (a) The o rbital radi us 𝜌𝜌 versus time calculated using of the guiding center for  selected \nfield amplitudes ( 𝛼𝛼=0.0245 ,0.0365 ,0.126 ,0.1485 ,and 0.152). (b) Two steady -states 𝜌𝜌±  \nas a function of 𝛼𝛼. (c) The bifurcation diagram of  𝜌𝜌𝑠𝑠 as a function of 𝛼𝛼. ‘F’, ‘P’, ‘QP ’, and \n‘C’ symbolize the regime of distinct nonlinear behaviors of fixed point, periodic, quasiperiodic, \nand chaotic motion.  \n Fig. 4. Calculation of the local Lyapunov exponent  (LLE ) as a functio n of time ( τ) and reduced \nfield amplitude , 𝛼𝛼. The color bar indicates the values of the LLE. 14 \n Figure 1 (one column)  \n \n(a) \n(b) 15 \n Figure 2 (one column)  \n \n  \n(a) \n③ (11,9) \n④ (11,1) \n ② (11,0.1)  \n ① (11,10) \n⑦ (15,1) \n⑧ (17,0.25)  \n ⑥ (13,9) \n ⑤ (12,10) \n⑪ (17,0.5)  \n⑫ (20,0.25)  \n ⑩ (15,0.75)  \n ⑨ (13,9) \n (𝜈𝜈, 𝜀𝜀) = \n-1 0 1\n10-3-1-0.500.5110-3\n-1 0 1\n10-3-10110-3\n-0.5 0 0.5-0.500.5\n-0.1 0 0.1-0.15-0.1-0.0500.050.10.15\n① \n② \n③ \n④ \n⑤ \n⑥ \n⑦ \n⑧ \n𝜀𝜀 \n⑨ \n⑩ \n⑪ \n⑫ \n(b) -2 -1 0 1 2-2-1012\n-0.2 -0.1 0 0.1 0.2-0.2-0.100.10.2\n-0.4 -0.2 0 0.2 0.4-0.4-0.200.20.4\n-5 0 5-6-4-20246\n-0.4 -0.2 0 0.2 0.4-0.4-0.200.20.4\n-5 0 5-6-4-20246\n-5 0 5-505\n-5 0 5-6-4-2024616 \n Figure 3 (one column)  \n  \n0 200 400 600\ntime  ()00.10.20.30.40.50.6\n𝛼𝛼 = 0.152  \n𝛼𝛼 = 0.1485  \n𝛼𝛼 = 0.126  \n𝛼𝛼 = 0.0365  \n𝛼𝛼 = 0.0245  380 390 400 410 420\ntime  ()0.60.620.640.66\nC \n𝜌𝜌− \n𝜌𝜌+ \nFP \n P+QP  \n𝜌𝜌𝑠𝑠 ���� \n(a) \n(b) \n(c) \nFP \nP \n QPⅠ \n QPⅡ \nQPⅢ \nC 17 \n Figure 4 (one column)  \n \n  \nbreakdown  \nchaos \norder 18 \n References  \n[1] M. 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Kim, Applied Physics Letters 91 (2007).  \n[12] I. Makhfudz, B. Krüger, and O. Tchernyshyov, Physical Review Letters 109, 217201 (2012).  \n[13] A. A. Thiele, Physical Review Letters 30, 230 (1973).  \n[14] N. Papanicolaou and T. N. Tomaras, Nuclear Physics B 360, 425 (1991).  \n[15] C. Moutafis, S. Komineas, and J. A. C. Bland, Physical Review B 79, 224429 (2009).  \n[16] S.-Z. Lin, C. D. Batista, and A. S axena, Physical Review B 89, 024415 (2014).  \n[17] Z. Chen, X. Zhang, Y. Zhou, and Q. Shao, Physical Review Applied 17, L011002 (2022).  \n[18] M.-W. Yoo, F. Mineo, and J. -V. Kim, Journal of Applied Physics 129 (2021).  \n[19] K.-W. Moon, B. S. Chun, W. Kim, Z. Q.  Qiu, and C. Hwang, Physical Review B 89 , 064413 \n(2014).  \n[20] D. R. Rodrigues, A. Abanov, J. Sinova, and K. Everschor -Sitte, Physical Review B 97, 134414 \n(2018).  \n[21] M. T. Rosenstein, J. J. Collins, and C. J. De Luca, Physica D: Nonlinear Phenomena 65 , 117 \n(1993).  \n Supplemental Materials  \nEmergence of Chaos in Magnetic -Field -Driven Skyrmion s \nGyuyoung Park and Sang- Koog Kima) \nNational Creative Research Initiative Center for Spin Dynamics and Spin -Wave Devices, Nanospinics  \nLaboratory, Research Institute of Advanced Materials, Department of Materials Science and Engineering, Seoul \nNational University, Seoul 151- 744, Republic of Korea  \na) Author to whom all correspondence should be addressed; electronic mail: sangkoog@snu.ac.kr  \n \nAppendix A: Linearity and nonlinearity of the in -plane  modes  \nFigure S1 . The FFT spectrum for the in -plane excitations of  the skyrmion driven by a sinc -\nfunction field along the y -axis. \n To obtain a spectrum of in- plane excitation modes for the magnetic field -driven \nskyrmion, we initially applied a sinc -function magnetic field given by  Hsinc(t) = H 0sinc(2𝜋𝜋fHt) \nover the entire nanodisk where H(t) = (0, H sinc(t), 0). Here, H0 is 10 Oe  and fH is 100 GHz . The \nmodes ’ spectrum, depicted in Fig. S1, was determined using a Fast -Fourier- Transform (FFT) \nof my components of the spins. Two gyrotropic modes appear at 2.19 GHz and 21.37 GHz, \nwhile other in -plane excitations, known as azimuthal modes, manifest in higher frequency \ndomains. \nTo assess the linearity of each distinct mode, we applied a linearly polarized harmonic \nfield corresponding to each mode ’s frequency along the y -axis. We then measured the 𝑋𝑋̈ of 0 20 40 60 80 100\n f (GHz)-10-50FFT Power (arb. unit)\nf+ \n f- Supplemental Materials  \nthe guiding center at the initial phase  of the field application w hile varying  the field amplitude , \n𝛼𝛼 , as shown in Fig. S2(a). Notably, 𝑋𝑋̈  only increases nonlinearly at  the higher gyrotropic \nmodes . See the supplemental video to check the raw data. Fig . S2(b) displays  the FFT results \nof the Y component of the guiding center  when the skyrmion is driven  by a linearly polarized \nharmonic field , aligned  to the higher gyrotropic mode ’s frequency , Asin(2𝜋𝜋sinf+t), with A being \n10 Oe,  along the y -axis. All six modes were simultaneously excited , even though the harmonic \nfrequency was f +[1]. In contrast , when driven by  the f- frequency, modes other than the lower \ngyrotropic mode weren ’t stimulated . Yet, this simultaneous excitation of different in -plane \nmodes doesn't account for the nonlinearity, as their coupling represents a straightforward linear \ncombination of the modes. Azimuthal modes also exhibited similar FFT spectra. Lastly,  Fig. \nS2(c) illustrates  the spatial distribution of the FFT phase for the  f+ frequency. Both the core and \nperipheral spins gyrate  in a counter -clockwise direction but are in antiphase , with  an azimuthal \nnumber of  0. \n \nFigure S 2. (a) Linearity and non- linearity of the in -plane excitation modes in relation to the \namplitude (𝛼𝛼) of the applied oscillating magnetic field. Each graph color corresponds to the \n- 0 \nf+ \n0.7 \n1.4 \n(a) \n (b) \n(c) \nFFT Power (arb. unit) \nfrequency \n f- \n𝛼𝛼 Supplemental Materials  \ncolors indicated in Fig. S1. (b) Abbreviated FFT spectrum of the Y component of the \nskyrmion ’s guiding center when driven by the  f+ mode. (c) FFT phase distribution of the f + \nfrequency . \nFigure S3. The FFT spectrum for the  mx, my, and m z components of the spins is shown when \nthe skyrmion is driven by an oscillating magnetic field at the frequency of the higher gyrotropic \nmode. The inset displays the spatial distributions of the FFT power for three distinct out -of-\nplane modes. White dashe d circles denote R sky. \n Figure S3 presents the mode spectra for both in- plane and out -of-plane directions when \nthe skyrmion is excited by a higher gyrotropic mode. At the frequency of 2f +, a uniform \nbreathing mode larger than R sky is excited. At frequencies lower than the f+ mode, two non -\nuniform out -of-plane modes couple to the f + mode. This is primarily because the two chiral \nedge spin wave modes possess different rotational senses [2]. The rotational symmetry of the \nFFT power distribution is disrupted, suggesting that these modes induce deformation in the skyrmion. A simple , non- uniform breathing mode could distort the skyrmion into an ellipse. \nHowever, this non-unif orm breathing -like mode is intertwined with the gyrotropic mode. As a \n0 20 40 60 80 100\n f (GHz)-5051015FFT Power (arb. unit)FFT(mx)\nFFT(my)\nFFT(mz)\n42.78 GHz  \n 10.59 GHz  \n 0.10 GHz  \n  \nminmax\nFFT Power (arb. unit)Supplemental Materials  \nresult, this combined mode contorts the skyrmion into a shape devoid of symmetries. The \namorphous form of the deformed skyrmion complicates analytical approaches. Supplemental Materials  \nAppendix B: Skyrmion breakdown  \n \nFigure S4. The  magnetization ( 𝒎𝒎=𝑴𝑴𝑀𝑀𝑆𝑆⁄) configuration of the  skyrmion breaks down. The \nclosed domain wall opens  at the bottom of the figure , as indicated by the gray line.  \nBreakdown – A skyrmion, when driven, can break down if the driving current density surpasses \na critical value, either due to overpowering the damping force or inducing significant \ndeformation in the skyrmion[3 ,4]. This breakdown signifies the skyrmion losing its topological \ncharge or its closed domain wall disappearing. Practically, establishing a numerical boundary for the topological charge— where it's still considered a skyrmion —can be challenging. \nConsequently, we define the skyrmion's breakdown as the point when its closed- loop domain \nwall opens, primarily due to the loss of its topological structure when approaching the disk boundary  (Fig.  S4).  \nStatic breakdown field - The static breakdown field ( A\ns) is characterized as a linear, static \nmagnetic field uniformly applied across the skyrmion, leading to its disintegration upon sustained application. In our system model, breakdown initiates when the field amplitude exceeds approximately 2730.4 Oe , as illustrated in Fig. S5(a). At this field intensity, the \nskyrmion's guiding center traces a chaot ic trajectory from the instant the field is applied up to \nits eventual breakdown. For the sake of clarity, we've approximated this value to 2730 Oe  as \nSupplemental Materials  \nthe representative static magnetic field. By comparison, with a precise amplitude of 2730 Oe , \nthe guiding  center begins with chaotic motion  and then transitions to stable trajectories with a \ngyration, as shown in Fig. S5(b). This gyration occurs at a frequency of 1.86 GHz, highlighting \nthe altered eigenfrequency due to the skyrmion's deformation. The equilibrium  state, marked \nby a gray dot, is located approximately at (1.31 nm, 1.14 nm). Interestingly, while the magnetic field was directed along the y -axis, the stabilization point is more skewed towards the x -axis. \nFigure S5 . The trajectories of the guiding center under the influence of a static magnetic field \napplied along the y -axis. The field amplitude is 2730.4 Oe in (a) and 2730 Oe in (b). The \nprogression of time is represented by the color index, and the gray dot indicates the equilibrium state.  -10 -5 0 5 10\nX (nm)-10-50510Y (nm)\n00.10.20.30.40.50.6\nt (ns)\n(a) \n(b) \n-10 -5 0 5 10\nX (nm)-10-50510Y (nm)\n0510152025\nt (ns)Supplemental Materials  \nAppendix C : Sensitive dependence of Initial conditions  \nInitial condition  <mz> q \n1 0.5971 7800 -0.9307 78116463741  \n2 0.5971 6004 -0.9307 84236318334  \n3 0.5971 5694  -0.9307 85326955563  \n4 0.5971 2756  -0.9307 95599997689  \n5 0.5971 1570 -0.9307 99777066074  \n \nTable S1. Possible numerical values of ground states achieved from the relaxation of the same \nmagnetic skyrmion are presented. Each state corresponds to a different initial condition. \nExtremely small differences between the values are denoted  with an underline. \n \nRegardless of the choice of differential equation solver, multiple numerical values for \nthe skyrmion's ground state can be obtained during micromagnetic simulations. Among these, only five states were selected, as detailed in Table 1. The <m\nz> = M z/Ms for every spin within \nthe skyrmion  exhibits differences up to the fifth decimal place. While these variations are too \nminimal to influence typical linear and nonlinear motions, they provide an excellent means to investigate SDI in the chaotic dynamics of the spin system. For instance, overlaying the time series of the X -signal from the five initial conditions, as depicted in Figure S6, offers a \nvisualization of SDI. Even though all other conditions remain constant, the five distinct skyrmion ground states commence their movements following identical trajectories. However , \ntheir coherence is lost around 350 τ. The eventual breakdown of the skyrmion, inferred from \nthe endpoint of each signal, occurs at markedly different times. \n Supplemental Materials  \nFigure S6 . X-signal for five different initial conditions, labeled IC1 through IC5. The amplitude \n(α) of the os cillating magnetic field is set to 0.16. \n \nFigure S 7. The breakdown time , 𝜏𝜏𝑏𝑏 , of the skyrmion for different initial conditions  on a \nlogarithmic scale, labeled IC1 through IC5, is plotted with respect to the field amplitude , 𝛼𝛼. \n \nThe SDI is particularly pronounced in the breakdown regime. The breakdown time , \n𝜏𝜏𝑏𝑏, for the five different initial conditions  have been  plotted against  varying field amplitude on \na logarithmic  scale, as depicted in Fig. S 7. A skyrmion ’s breakdown in the chaotic regime can \nbe interpreted as a manifestation of the self -criticality intrinsic to complex spin systems. It's \nIC1 \nIC2 \nIC3 \nIC4 \nIC5 0 100 200 300 400 500 600\ntime  ()-10-50510X (nm)\nIC1 \nIC2 \nIC3 \nIC4 \nIC5 Supplemental Materials  \nwidely accepted that self -criticality adheres to a power law, and in this instance, the breakdown \ntime conforms to this la w. Three distinct trend lines can be observed. In the first trend line, \nwhere  α  ranges from 0.157 to 0.233, and t spans from 3.5 to 6.5, the SDI effect is so \npronounced that the skyrmion's breakdown time fluctuates significantly based on its initial \nconditi on. In contrast, the second (with a longer breakdown time) and third (with a shorter \nbreakdown time) trends, where  α lies between 0.233 and 0.3, exhibit a reduced SDI effect, \nsuggesting that the chaos is, to some extent, mitigated.   Supplemental Materials  \nReferences  \n[1] M. Mruczkiewicz, P. Gruszecki, M. Krawczyk, and K. Y. Guslienko, Physical Review B 97, \n064418 (2018).  \n[2] I. Makhfudz, B. Krüger, and O. Tchernyshyov, Physical Review Letters 109, 217201 (2012).  \n[3] L. Liu, W. Chen, and Y. Zheng, Phys ical Review Applied 14, 024077 (2020).  \n[4] Z. Chen, X. Zhang, Y. Zhou, and Q. Shao, Physical Review Applied 17, L011002 (2022).  \n "    },    {        "title": "2306.13270v1.Spintronic_reservoir_computing_without_driving_current_or_magnetic_field.pdf",        "content": "Spintronic reservoir computing without driving\ncurrent or magnetic field\nTomohiro Taniguchi1,*, Amon Ogihara2, Yasuhiro Utsumi2, and Sumito Tsunegi1,3\n1National Institute of Advanced Industrial Science and Technology (AIST), Research Center for Emerging\nComputing Technologies, Tsukuba, Ibaraki 305-8568, Japan\n2Department of Physics Engineering, Faculty of Engineering, Mie University, Tsu, Mie, 514-8507, Japan\n3PRESTO, Japan Science and Technology Agency (JST), Saitama, Japan\n*tomohiro-taniguchi@aist.go.jp\nABSTRACT\nRecent studies have shown that nonlinear magnetization dynamics excited in nanostructured ferromagnets are applicable to\nbrain-inspired computing such as physical reservoir computing. The previous works have utilized the magnetization dynamics\ndriven by electric current and/or magnetic field. This work proposes a method to apply the magnetization dynamics driven by\nvoltage control of magnetic anisotropy to physical reservoir computing, which will be preferable from the viewpoint of low-power\nconsumption. The computational capabilities of benchmark tasks in single MTJ are evaluated by numerical simulation of the\nmagnetization dynamics and found to be comparable to those of echo-state networks with more than 10 nodes.\nRecent development of neuromorphic computing with spintronics devices1–4, such as pattern recognition and associative\nmemory, has provided a bridge between condensed matter physics, nonlinear science, and information science, and become\nof great interest from both fundamental and practical viewpoints. In particular, an application of nonlinear magnetization\ndynamics in ferromagnets to physical reservoir computing5–19is an exciting topic1, 20–30. Physical reservoir computing is a\nkind of recurrent neural network, which has recurrent interaction among large number of neurons in artificial neural network\nand, for example, recognizes a time sequence of the input data, such as human voice and movie, from the dynamical response\nin nonlinear physical systems19. In reservoir computing, only the weights between neurons and output are trained, whereas the\nweights among neurons are randomly given and fixed, and therefore, low calculation cost of training is expected. It has been\nshown that several kinds of physical systems, such as optical circuit10, soft matter12, quantum matter15, fluid18, and spintronics\ndevices, can be used as reservoir for information processing19.\nIn physical reservoir computing with spintronics devices, nonlinear magnetization dynamics has been excited in nanostruc-\ntured ferromagnets by applying electric current and/or magnetic field. For example, spin-transfer effect31, 32has been frequently\nused to excite an auto-oscillation of the magnetization in magnetic tunnel junctions (MTJs)1, 20–22, 24–26, 28–30, where the spin\nangular momentum from conducting electrons carrying electric current is transferred to ferromagnet and excites magnetization\ndynamics. It is, however, preferable to excite magnetization dynamics without driving current and magnetic field from the\nviewpoints of low-power consumption and simple implementation.\nIn this work, we propose that physical reservoir computing can be performed by magnetization dynamics induced by voltage\ncontrol of magnetic anisotropy in solid devices33–50. The voltage control of magnetic anisotropy is a fascinating technology\nas the low-power information writing scheme in magnetoresistive random access memory, instead of using spin-transfer\ntorque effect. An application of electric voltage to a metallic ferromagnet/insulator interface modifies electron states near the\ninterface34, 36, 37and/or induces magnetic moment46, and changes magnetic anisotropy. The magnetization in the ferromagnetic\nmetal changes its direction to minimize the magnetic anisotropy energy. Therefore, the voltage application can cause the\nrelaxation dynamics of the magnetization in the ferromagnet. In the practical application of nonvolatile random access memory,\nan external magnetic field is necessary to achieve a deterministic magnetization switching guaranteeing reliable writing40–43.\nOn the other hand, we notice that the magnetization switching, as well as magnetic field, is not a necessary condition in\nphysical reservoir computing. Accordingly, the voltage control of magnetic anisotropy can be used to realize physical reservoir\ncomputing by spintronics devices without driving current or magnetic field. Here, we perform numerical simulation of the\nLandau-Lifshitz-Gilbert (LLG) equation and find that the computational capabilities of benchmark tasks in single spintronics\ndevice are comparable to those of echo-state networks with more than 10 nodes.\n1arXiv:2306.13270v1  [cond-mat.mes-hall]  23 Jun 2023Figure 1. (a) Schematic illustration of an MTJ. The unit vector pointing to the magnetization direction in the ferromagnetic\nfree layer is m. The zaxis is perpendicular to the film plane. (b) An example of the time evolutions of mx(red), my(blue), and\nmz(black). (c) Trajectory of the relaxation dynamics on a sphere. In (b) and (c), the first order magnetic anisotropy field HK1is\nchanged from −0.1HK2to−0.9HK2by the voltage application. The red circle and blue triangle in (c) represent the initial and\nfinal states of the dynamics.\nModel\nLLG equation\nThe system under investigation is a cylinder-shaped MTJ schematically shown in Fig. 1(a), where the zaxis is perpendicular\nto the film plane. The MTJ consists of ferromagnetic free layer, MgO insulator, and ferromagnetic reference layer. The\nferromagnetic free layer has the perpendicular magnetic anisotropy, where the magnetic energy density is given by\nε=∑\ni=x,y,z2πM2Nim2\ni+K1\u0000\n1−m2\nz\u0001\n+K2\u0000\n1−m2\nz\u00012. (1)\nThe first term on the right-hand side in Eq. (1) represents the shape magnetic anisotropy energy density with the saturation\nmagnetization Mand the demagnetization coefficients Ni. Since we assume the cylinder shape, Nx=Ny. The unit vector\npointing in the magnetization direction of the free layer is denoted as m= (mx,my,mz). The second and third terms are the first\nand second order magnetic anisotropy energy densities with the coefficients K1andK2. Note that the energy density relates to\nthe magnetic field inside the free layer as\nH=\u0002\nHK1+HK2\u0000\n1−m2\nz\u0001\u0003\nmzez, (2)\nwhere HK1= (2K1/M)−4πM(Nz−Nx)andHK2=4K2/M; see also Methods. The magnetization in the reference layer points\nto the zdirection, and therefore, mzis experimentally measured through tunnel magnetoresistance effect.\nThe first order magnetic anisotropy energy coefficient K1consists of the bulk and interfacial contributions, KvandKi, and\nthe voltage-controlled magnetic anisotropy effect described as K1d=Kvd+Ki−ηE. The thickness of the ferromagnetic free\nlayer is d, whereas E=V/dIis the electric field with the voltage Vand the thickness of the insulator dI. In typical MTJs\nconsisting of CoFeB free layer and MgO insulator, Kidominates in K1, where Kiincreases with the increase of the composition\nof Fe51–53. It can reach on the order of 1.0 mJ/m2at maximum, which in terms of magnetic field, 2Ki/(Md), is typically on\nthe order of 1 T. Note that the magnitude of the shape magnetic anisotropy field −4πM(Nz−Nx)is also on the order of 1 T,\nwhere a typical value of the saturation magnetization in CoFeB, i.e., Mof about 1000 emu/c.c., is assumed. As a result of the\ncompetition between them, the ferromagnetic free layer in the absence of the voltage application can be either in-plane or\nperpendicular-to-plane magnetized51–53. The voltage control of magnetic anisotropy also modifies the magnetic anisotropy field\nHK1through the modification of the electron occupation states near the ferromagnetic interface34, 36, 37and/or the generation of\nthe magnetic dipole moment46. The coefficient of the voltage-controlled magnetic anisotropy effect, η, is recently achieved in\nthe experiment to be about 300 fJ/(Vm)45, 50, whereas the thickness of the insulator is about 2.5nm. A typical values of the\napplied voltage is about 0.5V at maximum48. Thus, the tunable range of the magnetic anisotropy by the voltage application in\nterms of the magnetic field, (2|η|V)/(Mdd I), is about 1.0 kOe, where we assume that M=1000 emu/c.c., d=1nm,dI=2.5\nnm, and |η|=250fJ/(V m). Note that the sign of the voltage-controlled magnetic anisotropy effect depends on that of the\nvoltage. Summarizing these contributions, HK1in the presence of the voltage can also be either positive or negative, depending\non the materials and their compositions, as well as the magnitude and sign of the applied voltage. For example, Ref.38uses\nan in-plane magnetized ferromagnet, i.e., HK1<0forV=0. The voltage control of magnetic anisotropy in Ref.38enhances\nthe perpendicular anisotropy K1and makes HK1positive at nonzero V. On the other hand, perpendicularly magnetized free\n2/13layers where HK1>0forV=0have been used in Ref.43. Contrary to HK1, the dependence of HK2∝K2on the applied voltage\nis still unclear, where Ref.47reports that HK2is approximately independent of the voltage while Ref.48observes the voltage\ndependence of HK2. Throughout this paper, for simplicity, we assume that only HK1depends on the voltage. As mentioned\nin the following, we performed numerical simulation by changing the value of HK1. It means that we do not specify the size\n(the thickness and cross-section area) of MTJ explicitly because HK1includes the information of the shape of MTJ through\nthe demagnetization coefficients Ni. It is, however, useful to mention that macrospin model has been proven to work well\nto describe the magnetization dynamics for MTJ whose typical size is 1-2nm in thickness and the diameter less than 200\nnm40, 42, 49.\nIn typical experiments on voltage control of magnetic anisotropy, a relatively thick (typically 1.5-2.5nm) MgO barrier is\nused as an insulator42, 43, 49, compared with MTJ manipulated by spin-transfer torque, where the thickness of the barrier is about\n1.0nm54. As a result, the resistance of MTJ used for experiments of voltage control of magnetic anisotropy, on the order of\n10-100kΩ, is two or three orders of magnitude larger than that used for spin-transfer torque experiments. On the other hand,\nthe maximum voltage used in both experiments is almost identical. Accordingly, current flowing in MTJ used for experiments\nof voltage control of magnetic anisotropy is two or three orders of magnitude smaller than that used for spin-transfer torque\nexperiments (see also Methods). In this sense, we mention that the driving force of magnetization dynamics is voltage control\nof magnetic anisotropy effect, although current cannot be completely zero in experiments. As mentioned in Methods, typical\nvalue of current Iflowing in MTJ is on the order of 1µA, while the current used in physical reservoir computing utilizing\nspin-transfer torque is on the order of 1mA29. On the other hand, the magnitude of the voltage Vapplied to MTJ is nearly the\nsame for both experiments on voltage control of magnetic anisotropy and spin-transfer effects. Accordingly, using the voltage\ncontrol of magnetic anisotropy effect could reduce energy consumption by three orders.\nThe magnetization in equilibrium points to the direction at which the energy density is minimized. For example, when HK1>\n(<)0andHK2=0, the energy is minimized when the magnetization is parallel (perpendicular) to the zaxis. Another example\nis studied in Ref.55, where, if HK1<0and|HK1|<HK2, the energy density εis minimized when mz=±p\n1−(|HK1|/HK2).\nWhen the voltage is applied to the MTJ and the minimum energy state is changed as a result, the magnetization relaxes to the\nstate. The relaxation dynamics is described by the LLG equation,\ndm\ndt=−γm×H+αm×dm\ndt, (3)\nwhere γandαare the gyromagnetic ratio and the Gilbert damping constant, respectively. Note that the macrospin model works\nwell to describe the magnetization dynamics driven by the voltage application40, 42, 44. The values of the parameters used in\nthe following are derived from typical experiments35, 38–44, 47, 48. The gyromagnetic ratio and the Gilbert damping constant are\nγ=1.764×107rad/(Oe s) and α=0.01. The second order magnetic anisotropy field HK2is 500 Oe.\nLet us show an example of the magnetization dynamics driven by the voltage control of magnetic anisotropy. We firstly\nsetHK1to be H(0)\nK1=−0.1HK2=−50Oe and solve the LLG equation with an arbitrary initial condition. The magnetization\nsaturates to a certain point where mzsaturates to mz→m(0)\nz≃0.95. We use this state as a new initial state and solve the\nLLG equation by changing HK1toH(1)\nK1=−0.9HK2=−450Oe. Then, the magnetization starts to move to a new stable state\nwhere mzsaturates to mz→m(1)\nz≃0.32. Figures 1(b) and 1(c) show time evolution of mand its spatial orbit from the initial\nstate of m(0)\nzto the final state m(1)\nz. We confirm that the initial and final states are those expected from the minimum energy\nstate mentioned above, i.e., m(0)\nz=q\n1−|H(0)\nK1|/HK2=√1−0.1≃0.95andm(1)\nz=q\n1−|H(1)\nK1|/HK2=√1−0.9≃0.32. We\nemphasize that mzmonotonically changes with respect to the change of HK1. Since the value of HK1can be manipulated by the\nvoltage application, the time evolution of mzcan be used to identify the value of the applied voltage. The estimation of the\ninput data, which is the sequence of the applied voltage in the present case, from the dynamical response of physical system is\nthe aim of physical reservoir computing. Therefore, the magnetization dynamics driven by the voltage control of magnetic\nanisotropy is applicable to physical reservoir computing. In the following, we evaluate its computational ability.\nResults\nMemory capacity\nThe ability in physical system for reservoir computing has been quantified by memory capacity15, 18, 20, 21, 25, 28, 30. The memory\ncapacity corresponds to the number of past data physical reservoir can store. For example, let us imagine injecting random\nbinary input b=0or1to reservoir, as done in experiments21, 25. The input data are often injected as pulses with the pulse\nwidth of tp, i.e., the value of bis constant during time tp. Therefore, it is convenient to add a suffix k=1,2,···tobasbkto\ndistinguish the order of the input data. We also introduce an integer D=0,1,2,···, called delay, characterizing the order of the\n3/13past input data. In this case,\nySTM\nk,D=bk−D, (4)\nare called target data of short-term memory (STM) task. We predict the value of the target data from the output of the reservoir\nand evaluate the reproducibility. The predicted data are called system output. The reproducibility is quantified by the correlation\ncoefficient between the target data and system output. Roughly speaking, if the reservoir can reproduce the past data up to D,\nthe STM capacity is defined as D. There is another kind of memory capacity, called parity-check (PC) capacity, where the\ntarget data are defined as\nyPC\nk,D=D\n∑\nℓ=0bk−D+ℓ(mod2 ). (5)\nAccording to their definitions, the STM and PC capacities quantify the number of the target data the reservoir can store, where\nthe target data are defined as linear and nonlinear transformations of the input data, respectively. A large memory capacity\nmeans that reservoir can store, recognize, and/or predict large data. See also Methods for the detail of the evaluation method of\nthese capacities.\nIn the present system, the random binary inputs are injected as voltage pulses, which change the first order magnetic\nanisotropy field HK1as\nHK1=H(0)\nK1(1−bk)+H(1)\nK1bk. (6)\nAccordingly, when the input is bk=0(1), the value of HK1isH(0)\nK1[H(1)\nK1]. In the following, we fix H(0)\nK1=−50Oe, i.e.,\nH(0)\nK1/HK2=−0.1, whereas H(1)\nK1varies in the range of −450≤H(1)\nK1≤ −100Oe, i.e., −0.9≤H(1)\nK1/HK2≤ −0.2. Figures 2(a)\nand 2(b) show the STM and PC capacities as a function of H(1)\nK1and the pulse width of the input data. The highest value of the\nSTM capacity, 3.29, is found at the conditions of H(1)\nK1=−430Oe and tp=69ns, as shown in Fig. 2(c). On the other hand, the\nhighest value of the PC capacity, 3 .40, is found at the conditions of H(1)\nK1=−445 Oe and tp=43 ns, as shown in Fig. 2(d).\nNARMA task\nAnother benchmark task to quantify the computational ability of physical system to reservoir computing is nonlinear autore-\ngressive moving average (NARMA) task12, 15, 18, 30, 56. NARMA task is a function-approximation task to reproduce a nonlinear\nfunction defined from input data by using output data in recurrent neural networks. The task is classified as NARMA Dwith\nD=2,5,10and so on, where Drepresents the delay included in the nonlinear function. In other words, the target data of\nNARMA Dtask consist of data defined until Dtimes before from the present data. For example, in NARMA2 task, the system\nis aimed to reproduce the target data,\nyNARMA2\nk+1 =0.4yNARMA2\nk +0.4yNARMA2\nk yNARMA2\nk−1 +0.1z3\nk+0.1, (7)\nfrom output data, where zk=0.2rkis defined from uniform random input data rkat a discrete time k; see Methods. The\ncomputational ability of NARMA task is evaluated from normalized mean-square error (NMSE) defined as\nNMSE =∑k\u0000\nyNARMA2\nk−vNARMA2\nk\u00012\n∑k\u0000\nyNARMA2\nk\u00012, (8)\nwhere vNARMA2\nkis the data reproduced from the output data (see also Methods). A low NMSE corresponds to high reproducibility\nof the target data. Figure 3(a) shows an example of the target data (red line), yNARMA2\nk, and the system output (blue dots). By\nevaluating the difference between the target data and the system output as such, the NMSE is obtained as shown in Fig. 3(b).\nThe NMSE is on the order of 10−6−10−5and is minimized to be 8.43×10−6attp=16ns and H(1)\nK1=−325Oe; see Fig. 3(c).\nDiscussion\nWe have developed theoretical analysis of the magnetization dynamics in nanostructured ferromagnetic multilayers driven by\nthe voltage control of magnetic anisotropy, and showed that the dynamics is applicable to physical reservoir computing through\nthe evaluations of the memory capacity and the NMSE of NARMA task. Neither electric current nor external magnetic field\nis introduced in the computation, contrary to the previous works focusing on the application to nonvolatile memory, because\nmagnetization switching is unnecessary. This fact will be preferable for reducing power consumption in reservoir computing.\n4/13Figure 2. Dependences of (a) STM (linear) and (b) PC (nonlinear) capacities on the pulse width and the first order magnetic\nanisotropy field. Their highest values are indicated by the red triangles in (c) and (d).\nFigure 3. (a) Examples of the target data (red line) and the system output (blue dots) of NARMA2 task, where tp=16ns and\nH(1)\nK1=−325 Oe. (b) Dependence of the NMSE of NARMA2 task on the pulse width and the first order magnetic anisotropy\nfield. The lowest value of the NMSE is indicated by the red triangle in (c).\n5/13Figures 2(a) and 2(b) show that the memory capacity increases with the difference between H(0)\nK1andH(1)\nK1increasing. This\nis because when the difference between H(0)\nK1andH(1)\nK1is large, the range of the dynamical response of mzalso becomes large,\nwhich makes it easy to identify the input data from the change of mz. Due to a similar reason, the memory capacity increases\nwith the increase of the pulse width. When the pulse width is relatively long, the change of mzduring a pulse injection becomes\nlarge, which again makes it easy to identify the input data. However, when the pulse width is sufficiently long, mzfinally\nsaturates to a stable state, and becomes approximately constant, as implied from Fig. 1(b). When mzbecomes constant, it\nbecomes impossible to estimate the past input from the present output. Therefore, the memory capacity does not increase\nmonotonically with the pulse width increasing. As written above, the STM and PC capacities are maximized at the pulse width\nof69and43ns, respectively. A similar trend is found in NARMA2 task, where low NMSEs are achieved in a relatively large\nH(1)\nK1region. Note that the memory capacity at the maximum was found to be about 3, which is comparable to the computational\nability of echo-state network with approximately 10 nodes20, 28. The value is also comparable or larger than that obtained by\nthe other single spintronics reservoirs without additional circuits20, 21, 29, driven by electric current and/or magnetic field. This\nmight be due to a matching between the relaxation time of the output signal and the pulse width. Another possible reason is a\nlarge change in the dynamical amplitude, compared with an oscillator system29. The NMSE of NARMA2 task, minimized to\nbe on the order of 10−6, is also comparable or lower than that found in soft robot12and echo-state network with nodes more\nthan 1018. These results indicate the potential applicability of an MTJ driven by the voltage-controlled magnetic anisotropy\neffect to physical reservoir computing.\nAn empirical rule shared among the research community is that the computational ability of physical reservoir computing is\nmaximized at the edge of chaos13, 30, 57. Simultaneously, an existence of chaos might lose the reproducibility of the computation\ndue to the sensitivity to initial states. Note that chaos is prohibited in the present system when random inputs are absent. This\nis because the magnetization dynamics are described by two variables, θ=cos−1mzandϕ=tan−1(my/mx), whereas the\nPoincaré-Bendixson theorem argues that chaos does not appear in a two dimensional system. When the random input are\ninjected to the MTJ, the system becomes nonautonomous due to the presence of time-dependent torque. In this case, the number\nof the dimension in the phase space becomes three, and the possibility to induce chaos becomes finite. For example, Ref.30\nreported the appearance of chaos in a spin-torque oscillator due to the injection of random input current. However, we should\nnotice that the presence of time-dependent input does not necessarily guarantee the presence of chaos. The identification of\nchaos is done by, for example, evaluating the Lyapunov exponent. The Lyapunov exponent quantifies the time evolution of an\ninfinitesimal difference given at the initial state. The positive Lyapunov exponent implies the presence of chaos. On the other\nhand, when the Lyapunov exponent is negative, the dynamics saturate to fixed points. When the Lyapunov exponent is zero, the\ndynamics is periodic. The dynamics with negative or zero Lyapunov exponent are classified as ordered dynamics. Since the\nLLG equation describes the relaxation dynamics to stable states, one might consider that the largest Lyapunov exponent of an\nMTJ in the absence of random inputs is negative. However, notice that the axial symmetry of the present system enables us to\nmove the magnetization rotating around the zaxis without energy injection. In fact, the energy density, as well as the equation\nof motion for mzdepends on mzonly, as explained in Methods; in other words, the equation of motion for θis independent of\nϕ. As a result, an infinitesimal difference given to the phase ϕis not shortened by the LLG equation. Therefore, the largest\nLyapunov exponent in the absence of the random input is zero. The fact that the equation of motion for θdepends on θonly\nalso implies the absence of homoclinic bifurcations, as well as chaos, even when the pulse data, independent of θandϕ, are\ninjected; in fact, the numerically evaluated Lyapunov exponent was zero, as explained in Methods. The absence of chaos\nindicates the reproducibility of the computation in the present reservoir.\nIn summary, we perform numerical experiments of the magnetization dynamics in an MTJ driven by the voltage control of\nmagnetic anisotropy. Injecting the voltage pulse to the MTJ, the magnetization changes its direction to minimize the magnetic\nanisotropy energy. The time evolution of the relaxation dynamics reflects the value of the input voltage, and therefore, can\nbe used to reproduce the time sequence of the input data. We evaluate the computing abilities, such as the memory capacity\nand the error in the reproducibility, of common benchmark task, and show that even a single MTJ can show high computing\nperformance comparable to echo-state network consisting of multiple nodes more than 10. Since neither electric current nor\nexternal magnetic field is necessary, the proposal here will be of interest for energy-saving computing technologies.\nMethods\nDefinition of magnetic field and relaxation time\nThe magnetic field Hrelates to the energy density εasH=−∂ε/∂(Mm), and therefore, is obtained from Eq. (1) as\nH=\n−4πMN xmx\n−4πMN ymy\n[(2K1/M)−4πMN z]mz+(4K2/M)\u0000\n1−m2\nz\u0001\nmz\n. (9)\n6/13Figure 4. Examples of the time evolutions of mx(red), my(blue), and mz(black) in the presence of spin-transfer torque, where\nthe current density is (a) 0 .06 MA/cm2and (b) 0 .6 MA/cm2.\nWe should note that the magnetization dynamics described by the LLG equation is unchanged by adding a term proportional to\nmtoHbecause the LLG equation conserves the magnitude of m. Adding a term as such corresponds to shifting the origin of\nthe energy density εby a constant. In the present case, we added a term 4πMN xmtoHand obtained Eq. (2), where we should\nremind that Nx=Nybecause we assume a cylinder shaped MTJ. The added term to Hshifts the origin of the energy density ε\nby the constant −2πM2Nxm2=−2πM2Nxand makes it depend on mzonly.\nThe LLG equation in the present system can be integrated as\nt=1+α2\nαγ(HK1+HK2)\u0014\nlog\u0012cosθf\ncosθi\u0013\n−HK1+HK2\nHK1log\u0012sinθf\nsinθi\u0013\n+HK2\n2HK1log\u0012HK1+HK2sin2θf\nHK1+HK2sin2θi\u0013\u0015\n, (10)\nwhere θiandθfare the initial and final values of θ=cos−1mz. Equation (10) provides the relaxation time from θ=θito\nθ=θf. Note that the relaxation time is scaled by αγHK1/(1+α2)andHK2/HK1, which can be manipulated by the voltage\ncontrol of magnetic anisotropy. We also note that Eq. (10) has logarithmic divergence due to asymptotic behavior in the\nrelaxation dynamics.\nRole of spin-transfer torque\nWe neglected spin-trasnfer torque in the main text because the current magnitude in typical MTJ used for voltage control of\nmagnetic anisotropy effect is usually small. For example, when using typical values47, 49for the voltage ( 0.4V), resistance\n(60kΩ), and cross-section being π×602nm2, the value of the current density is about 0.06 MA/cm2(6.7µA in terms of\ncurrent). Such a value is sufficiently small compared with that used in spin-transfer torque switching experiments54. To verify\nthe argument, we perform numerical simulations, where spin-transfer torque, −Hsm×(p×m), is added to the right-hand\nside of Eq. (3). We fix the values of HK2=500Oe and HK1=−0.1HK2=−50Oe. The unit vector palong the direction\nof the magnetization in the reference layer points to the positive zdirection. Spin polarization Pin the spin-transfer torque\nstrength, Hs=ℏP j/(2eMd), is assumed to be 0.5. Figure 4(a) shows time evolution of mfor the current density jof0.06\nMA/cm2. Although the magnetization slightly moves from the initial (stable) state due to spin-transfer torque, the change of the\nmagnetization direction is small compared with that shown in Fig. 1(b). Therefore, we do not consider that spin-transfer torque\nplays a major role in physical reservoir computing, although current cannot be completely zero in experiments.\nFor comprehensiveness, however, we also show the magnetization dynamics when the current density jis increased by\none order Figure 4(b) shows the dynamics for j=0.6MA/cm2, where the magnetization switching by spin-transfer torque\nis observed. We note that the current density is sufficiently small compared with that used in typical MTJs in nonvolatile\nmemory54. Nevertheless, the switching is observed because of a small value of the magnetic anisotropy field in the present\nsystem. We assume that HK2is finite and |HK1|<HK2to make a tilted state of the magnetization [ mz=±p\n1−(|HK1|/HK2)]\nstable due to the following reason. Remind that there are other stable states, such as mz=±1forHK1>0andmz=0for\nHK1<0, when HK2=0. Note that these states ( mz=±1ormz=0) are always local extrema of energy landscape. Accordingly,\nonce the magnetization saturates to these states, it cannot change the direction even if another input is injected. This conclusion\ncan be understood in a different way, where the relaxation time given by Eq. (10) shows a divergence when θi=0(mz= +1),\n7/13Figure 5. (a) An example of the time evolution of mz(black) in the presence of several binary pulses (red). The dotted lines\ndistinguish the input pulse. The pulse width and the first order magnetic anisotropy field are 69 ns and -430 Oe, respectively,\nwhere the STM capacity is maximized. (b) An example of mzin the presence of a random input. The dots in the inset shows the\ndefinition of the nodes uk,ifrom mzduring a part of an input pulse. The node number is Nnode=250. (c) Examples of the target\ndatay′\nn,D(red line) and the system output v′\nn,D(blue dots) of STM task with D=1. (d) Dependence of [Cor(D)]2on the delay\nDfor STM task. The node number is Nnode=250. The inset shows the dependence of the STM capacity on the node number.\nπ(mz=−1), orπ/2(mz=0) is substituted. On the other hand, for a finite HK2, the magnetization can move from the state\nmz=±p\n1−(|HK1|/HK2)when an input signal changes the value of HK1and makes the state no longer an extremum. We\nnote that the assumption |HK1|<HK2restricts the magnitude of the magnetic field. In fact, the magnitude of His small due to\na small value of HK2=500Oe found in experiments47, 48and the restriction of |HK1|<HK2. Since a critical current density\ndestabilizing the magnetization by spin-transfer effect is proportional to the magnitude of the magnetic field, a small Himplies\nthat a small current mentioned above might induce a large-amplitude magnetization dynamics.\nIn summary, the magnitude of the current density is sufficiently small, and the magnetization dynamics are mainly driven\nby voltage control of magnetic anisotropy effect. The condition to stabilize a tilted state, however, might make the magnitude\nof the magnetic field, as well as the critical current density of spin-transfer torque switching, small. Thus, even a small current\nmay cause nonnegligible dynamics. Simultaneously, however, it is practically difficult to increase the current magnitude by one\norder, and therefore, in the present study, we still consider that voltage control of magnetic anisotropy effect is the main driving\nforce of the magnetization dynamics.\nEvaluation method of memory capacity\nThe memory capacity corresponds to the number of data which can be reproduced from the output data, as mentioned in\nthe main text. The evaluation of the memory capacity consists of two processes. During the first process called training (or\nlearning), weights are determined to reproduce the target data from the output data. In the second process, the reproducibility of\nthe target data defined from other input data is evaluated.\nLet us first describe the training process. We inject the random binary input bk=0or1into MTJ as voltage pulse.\nThe number of the random input is N. The input bkis converted to the first order magnetic anisotropy field through the\n8/13voltage control of magnetic anisotropy, which is described by Eq. (6). We choose mzas output data, which can be measured\nexperimentally through magnetoresistance effect. Figure 5(a) shows an example of the time evolution of mzin the presence of\nseveral random binary inputs, where the values of the parameters are those at the maximum STM capacity conditions, i.e., the\npulse width and the first order magnetic anisotropy field are tp=69ns and H(1)\nK1=−430Oe. As can be seen, the injection of\nthe random input drives the dynamics of mz.\nThe dynamical response mz(t), during the presence of the kth input bk, is divided into nodes, where the number of nodes is\nNnode. We denote the i(=1,2,···,Nnode)th output with respect to the kth input as uk,i=mz(t0+(k−1)tp+i(tp/Nnode)), where\nt0is time for washout. The output uk,iis regarded as the status of the ith neuron at a discrete time k. Figure 5(b) shows an\nexample of the time evolution of mzwith respect to an input pulse, whereas the dots in the inset of the figure are the nodes uk,i\ndefined from mz. The method to define such virtual neurons is called time-multiplexing method15, 20, 21. We also introduce bias\nterm uk,Nnode+1=1. In the training process, we introduce weight wD,iand evaluate its value to minimize the error,\nN\n∑\nk=1 Nnode+1\n∑\ni=1wD,iuk,i−yk,D!2\n, (11)\nwhere, yk,Dare the target data defined by Eqs. (4) and (5). For simplicity, we omit the superscripts such as “STM” and “PC” in\nthe target data because the difference in the evaluation method of the STM and PC capacities is merely due to the definition of\nthe target data. In the following, we add superscripts or subscripts, such as “STM” and “PC”, when distinguishing quantities\nrelated to their capacities are necessary. The weight should be introduced for each target data. According to the above statement,\nwe denote the weight to evaluate the STM (PC) capacity as wSTM(PC)\nD,i , when necessary. Also, we note that the weights are\ndifferent for each delay D.\nOnce the weights are determined, we inject other random binary inputs b′\nkto the reservoir, where the number of the input\ndata is N′. Note that N′is not necessarily the same with N. Here, we use the prime symbol to distinguish the input data from\nthose used in training. Similarly, we denote the output and target data with respect to b′\nkasu′\nn,iandy′\nn,D, respectively, where\nn=1,2,···,N′. From the output data u′\nn,iand the weight wD,i, we define the system output v′\nn,Das\nv′\nn,D=Nnode+1\n∑\ni=1wD,iu′\nn,i. (12)\nFigure 5(c) shows an example of the comparison between the target data y′\nn,D(red line) and the system output v′\nn,D(blue dots)\nof STM task with D=1. It is shown that the system output well reproduces the target data. The reproducibility of the target\ndata is quantified from the correlation coefficient Cor (D)between y′\nn,Dandv′\nn,Ddefined as\nCor(D)≡∑N′\nn=1\u0010\ny′\nn,D−⟨y′\nn,D⟩\u0011\u0010\nv′\nn,D−⟨v′\nn,D⟩\u0011\nr\n∑N′\nn=1\u0010\ny′\nn,D−⟨y′\nn,D⟩\u00112\n∑N′\nn=1\u0010\nv′\nn,D−⟨v′\nn,D⟩\u00112, (13)\nwhere ⟨···⟩ represents the averaged value. Note that the correlation coefficients are defined for each delay D. We also note\nthat the correlation coefficients are defined for each kind of capacity, as in the case of the weights and target data. In general,\n[Cor(D)]2≤1, where [Cor(D)]2=1holds only when the system output completely reproduces the target data. Figure 5(d)\nshows an example of the dependence of [Cor(D)]2for STM task on the delay D. The results implies that the reservoir well\nreproduces the target data until D=3, whereas the reproducibility drastically decreases with the delay Dincreasing. The STM\nand PC capacities, CSTMandCPC, are defined as\nC=Dmax\n∑\nD=1[Cor(D)]2. (14)\nNote that the definition of the memory capacity obeys, for example, Refs.18, 20, 21, 25, where the memory capacity in Eq. (14)\nis defined by the correlation coefficients starting from D=1. In some papers such as Refs.15, 30, however, the square of the\ncorrelation coefficient at D=0 is added to the right-hand side of Eq. (14).\nIn the present study, we introduce Nnode=250nodes and use N=1000 andN′=1000 random binary pulses for training of\nthe weight and evaluation of the memory capacity, respectively. The number of nodes is chosen so that the value of the capacity\nsaturates with the number of nodes increasing; see the inset of Fig. 5(d). We also use 300 random binary pulses before the\ntraining and between training and evaluation for washout. The maximum delay Dmaxis20. Note that the value of each node\nshould be sampled within a few hundred picosecond: specifically, in the case of an example shown in Fig. 2(c), it is necessary\nto sample data within tp/Nnode=69ns/250≃276ps. We emphasize that it is experimentally possible to sample data within\nsuch a short time. For example, in Ref.21,tp=20 ns and Nnode=200 were used, where the sampling step is 100 ps.\n9/13NARMA task\nThe evaluation procedure of the NMSE in NARMA task is similar to that of the memory capacity. The binary input data, bk=0\nor1, in the evaluation of the memory capacity are replaced by uniform random number rkin(0,1). The variable zkin Eq. (7)\nis generally defined as zk=µ+σrk30, where the parameters µandσare determined to make zkbe in (0,0.2)15. As in the\ncase of the evaluation of the memory capacity, the evaluation of the NMSE consists of two procedures. The first procedure\nis the training, where the weight is determined to reproduce the target data from the output data uk,i. Secondly, we evaluate\nthe reproducibility of another set of the target data from the system output vNARMA2\nn defined from the weight and the output\ndata. Then, the NMSE can be evaluated. Note that some papers13, 27, 30define the NMSE in a slightly different way, where\n∑N′\nn=1\u0000\nyNARMA2\nn\u00012in the denominator of Eq. (8) is replaced by ∑N′\nn=1\u0000\nyNARMA2\nn −yNARMA2\u00012, where yNARMA2is the average of\nthe target data yNARMA2\nn . In this work, we use the definition given by Eq. (8), which is used, for example, in Refs.12, 15, 18.\nEvaluation of Lyapunov exponent\nWe evaluated the conditional Lyapunov exponent as follows58. The LLG equation was solved by the fourth-order Runge-Kutta\nmethod with time increment of ∆t=1ps. We added perturbations δθandδφwithε=p\nδθ2+δϕ2=10−5toθ(t)andϕ(t)\nat time t. Let us denote the perturbed θ(t)andϕ(t)asθ′(t)andϕ′(t), respectively. Solving the LLG equation from time t\ntot+∆t, the time evolution of the perturbation is obtained as ε′(t) =p\n[θ′(t+∆t)−θ(t+∆t)]2+[ϕ′(t+∆t)−ϕ(t+∆t)]2.\nA temporal Lyapunov exponent is obtained as λ(t) = ( 1/∆t)log[ε′(t)/ε]. Repeating the procedure, the temporal Lyapunov\nexponent is averaged as λ(N) = ( 1/N)∑N\ni=1λ(ti) = [ 1/(N∆t)]∑N\ni=1log{ε′[t0+(i−1)∆t]/ε}, where t0is time at which the\nfirst random input is injected, whereas Nis the number of averaging. The Lyapunov exponent is given by λ=limN→∞λ(N).\nIn the present study, we used the time range same as that used in the evaluations of the memory capacity and the NMSE and\nadded uniform random input. Hence, notice that N=Mtp/∆tdepends on the pulse width, where Mis the total number of\nthe random inputs including washout, training, and evaluation. We confirmed that λ(N)monotonically saturates to zero; at\nleast,|λ(N)|is one or two orders of magnitudes smaller than 1/tp. Thus, the expansion rate of the perturbation, 1/λ(N),\nis much slower than the injection rate of the input signal. Considering these facts, we concluded that the largest Lyapunov\nexponent can be regarded as zero, and therefore, chaos is absent. Note that the absence of chaos in the present system relates to\nthe facts that the free layer is axially symmetric and the applied voltage modifies the perpendicular anisotropy only. 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Appl. Phys. Express 8, 063007 (2015).\n56.Atiya, A. F. New Results on Recurrent Network Training: Unifying the Algorithms and Accelerating Convergence. IEEE\nTrans. Neural. Netw. 11, 697 (2000).\n57.Bertschinger, N. & Natschläger, T. Real-Time Computation at the Edge of Chaos in Recurrent Neural Networks. Neural.\nComput. 16, 1413 (2004).\n58.Taniguchi, T. Synchronization and chaos in spin torque oscillator with two free layers. AIP Adv. 10, 015112 (2020).\nAcknowledgements\nT.T. acknowledges Takayuki Nozaki, Tomohiro Nozaki, and Yoichi Shiota for their valuable discussions. This paper was based\non the results obtained from a project (Innovative AI Chips and Next-Generation Computing Technology Development/(2)\nDevelopment of next-generation computing technologies/Exploration of Neuromorphic Dynamics towards Future Symbiotic\nSociety) commissioned by NEDO. The work is also supported by JPS KAKENHI Grant Number 20H05655.\nAuthor contributions statement\nT.T. designed the project with help from S.T. A.O., Y .U. and T.T. developed the codes and performed the simulations. T.T.\nwrote the manuscript and prepared the figures. All authors contributed to discussing the results.\n12/13Competing interests\nThe authors declare no competing interests.\nData availability\nThe datasets used and/or analysed during the current study available from the corresponding author on reasonable request.\nAdditional information\nCorrespondence and requests for materials should be addressed to T.T.\n13/13"    },    {        "title": "2005.09467v2.Numerical_Analysis_of_the_Screening_Current_Induced_Magnetic_Field_in_the_HTS_Insert_Dipole_Magnet_Feather_M2_1_2.pdf",        "content": "Numerical Analysis of the Screening Current-Induced Magnetic Field\nin the HTS Insert Dipole Magnet Feather-M2.1-2\nL. Bortot1;2, M. Mentink1, C. Petrone1, J. Van Nugteren1, G. Kirby1, M. Pentella1;3, A.P. Verweij1, and S. Sch ops2\n1CERN, Espl. des Particules 1, 1211 Geneva, CH\n2Technische Universit at Darmstadt, Karolinenplatz 5, 64289 Darmstadt, DE\n3Department of Applied Science and Technology, Polytechnic of Turin, Turin, IT\nlorenzo.bortot@cern.ch\nAbstract\nScreening currents are \feld-induced dynamic phe-\nnomena which occur in superconducting materials,\nleading to persistent magnetization. Such currents\nare of importance in ReBCO tapes, where the large\nsize of the superconducting \flaments gives rise to\nstrong magnetization phenomena. In consequence, su-\nperconducting accelerator magnets based on ReBCO\ntapes might experience a relevant degradation of\nthe magnetic \feld quality in the magnet aperture,\neventually leading to particle beam instabilities. Thus,\npersistent magnetization phenomena need to be accur-\nately evaluated. In this paper, the 2D \fnite element\nmodel of the Feather-M2.1-2 magnet is presented.\nThe model is used to analyze the in\ruence of the\nscreening current-induced magnetic \feld on the \feld\nquality in the magnet aperture. The model relies on\na coupled \feld formulation for eddy current problems\nin time-domain. The formulation is introduced and\nveri\fed against theoretical references. Then, the\nnumerical model of the Feather-M2.1-2 magnet is\ndetailed, highlighting the key assumptions and sim-\npli\fcations. The numerical results are discussed and\nvalidated with available magnetic measurements. A\nsatisfactory agreement is found, showing the capability\nof the numerical tool in providing accurate analysis\nof the dynamic behavior of the Feather-M2.1-2 magnet.\nIndex Terms { High-temperature superconductors,\nscreening currents, magnetic \felds, magnetization,\n\fnite-element analysis, superconducting coils, acceler-\nator magnets.\n1 Introduction\nHigh Temperature Superconducting (HTS) materials\nare a promising technology for high-\feld magnets in\nparticle accelerators. In particular, superconducting\ntapes based on ReBCO compounds [1] have a critical\ntemperature of 93 K and an estimated upper critical\n\feld of 140 T [2]. These properties are about one order\nof magnitude higher than in traditional Low Temperat-\nure Superconducting (LTS) materials, such as Nb-Ti or\nNb3Sn [3]. Thus, HTS materials might be used in build-ing accelerator magnets with higher magnetic \felds\nand thermal margins [4]. A signi\fcant milestone in\nthis direction was recently achieved by the EuCARD-\n2 [5] and ARIES [6] projects, which led to the con-\nstruction of the HTS accelerator dipole insert-magnet\nFeather-M2.1-2 [7, 8]. This demonstrator magnet is de-\nsigned to operate inside the aperture of the Nb 3Sn\nFRESCA2 dipole magnet [9, 10, 11, 12, 13], producing\na peak \feld of 5 T at a nominal current of 10 kA, in a\nbackground \feld of 13 T. The magneto-thermal beha-\nvior of the Feather-M2.1-2 magnet was recently tested\nin a stand-alone con\fguration [14], and the in\ruence\nof the superconducting coil dynamics on the magnetic\ntransfer function was measured [15].\nOne of the key requirements for accelerator magnets\nis to produce high-quality magnetic \felds in their mag-\nnet aperture (see e.g. [16]), as \feld imperfections can\nlead to particle beam instabilities [17]. Therefore, the\ncurrent density distribution within the superconductor\nshould be as uniform as possible. Conversely, HTS\ntapes behave as wide and anisotropic mono-\flaments,\nresulting in the dynamic regime in screening currents\nwhich are persistent, as they \row in a superconduct-\ning material. These currents prevent a homogeneous\ncurrent density distribution by producing a persistent\nmagnetization in the tape, potentially degrading the\nmagnetic \feld quality [18, 19, 20, 21, 22, 23, 24]. At-\ntempts in reducing persistent magnetization phenom-\nena either by tape striation [25] or tape-\feld align-\nment [26] were not yet fully satisfactory.\nThe screening currents magnitude is determined by\nthe operational margin of the tape, i.e. the di\u000berence\nbetween the supply and the critical current, which lim-\nits the superconducting state. As a consequence, per-\nsistent magnetization phenomena are more severe at\nlow current. This poses a major challenge for high-\n\feld accelerator magnets, whose supply current typ-\nically varies over one order of magnitude during the\nenergy ramp. For this reason, persistent magnetization\nphenomena need to be carefully evaluated, and pos-\nsibly predicted by means of numerical models, as they\nmay limit the use of HTS technology in accelerator\nmagnets.\nIn this paper, we present the time domain analysis\nof the magnetic \feld quality in the Feather-M2.1-2\n1arXiv:2005.09467v2  [physics.acc-ph]  6 Jan 2021magnet. A dedicated 2D numerical model is developed\nusing the \fnite element method (FEM, e.g. [27]).\nThe model implements a coupled A-H\feld formula-\ntion [28, 29] for HTS materials [30, 31], extended to\nthe simulation of HTS magnets [32]. This is achieved\nby following a domain decomposition strategy, solv-\ning the \feld problem for the magnetic \feld strength\nHin the superconducting regions, and for the mag-\nnetic vector potential Ain the normal-conducting and\nnon-conducting regions. Thus, the coupled \feld formu-\nlation accounts for electrodynamic phenomena in the\nsuperconducting coil by solving an eddy current prob-\nlem in time-domain. The advantages of this approach\nare discussed in [31].\nThe model of the Feather-M2.1-2 magnet is used\nto quantify the contribution of the screening currents-\ninduced magnetic \feld to the magnetic \feld quality.\nMoreover, simulations provide the current density dis-\ntribution within each superconducting tape, which is\ncrucial for the determination and understanding of the\nJoule losses and the Lorentz forces in the coil. The\nnumerical results are compared with measurements\nof the magnetic \feld quality in the aperture of the\nFeather-M2.1-2 magnet. In this comparison, a high de-\ngree of consistency is found.\nThe paper is organized as follows. The mathematical\nmodel is discussed in Section 2 and veri\fed in Section 3\nby comparing simulations of single tapes with theoret-\nical references. In Section 4, the numerical model of\nthe Feather-M2.1-2 magnet is presented, highlighting\nassumptions and simpli\fcations. The validation of the\nmodel with measurements is reported in Section 5, dis-\ncussed in Section 6 and followed by the conclusions.\n2 Mathematical Model\nHTS materials exhibit a highly nonlinear electric \feld-\ncurrent density relation (e.g. [1]), which determines\nthe magnetic \feld penetration in the superconductors\nand, ultimately, the dynamics of the screening currents.\nThus, the mathematical model of such relation is cru-\ncial for the time-domain analysis of the Feather-M2.1-2\nmagnet.\nThe resistivity \u001ain HTS materials can be modeled\nby means of a phenomenological percolation-depinning\nlaw proposed in [33]. The law introduces a lower limit\nfor the current density, below which the magnetic \feld\nis frozen in the superconductor and \rux creep [3] can-\nnot occur. However, the current density values used in\npractical applications are typically much higher than\nthe lower limit considered in the percolation law. Thus,\na further simpli\fcation into a power law [34] is su\u000e-\ncient, as shown in [35], and is adopted in this work.\nThe resistivity reads\n\u001a(jJj) =Ec\nJc\u0012jJj\nJc\u0013n\u00001\n; (1)\nwhere Jis the current density, E cis the critical elec-\ntric \feld strength, set to 1 \u000210\u00004V m\u00001[36], and the\nmaterial- and \feld-dependent parameters J candnare\nFigure 1. Decomposition of the domain \n into the source and\nsource-free domains \n H= \n H;s[\nH;cand \n A= \n A;c[\nA;i.\nThe two domains are separated by the interface \u0000 HA, shown as\na dashed line. The regions \n H;sand \n H;crepresent the super-\nconducting and normal-conducting parts in the source domain.\nThe regions \n A;cand \n A;ishow the normal-conducting and non-\nconducting parts in the source-free domain. The electrical ports\nare marked as \u0000 Jand \u0000 E. This \fgure is based on Fig. 2 from [32].\nthe critical current density and the power-law index, re-\nspectively. For the limiting cases of n!0 andn!1 ,\nthe power law approximates the behavior of normal\nconducting materials and superconducting materials in\ncritical state [37, 38].\nAt low currents, i.e. jJj! 0, the resistivity in (1)\nvanishes. Thus, the \feld problem is formulated avoid-\ning the electrical conductivity \u001bin superconducting\ndomains [39, 31], and the electrical resistivity in non-\nconducting domains, such that the material properties\nremain \fnite. This is done by combining a domain de-\ncomposition strategy with a dedicated coupled \feld\nformulation (see [32]), brie\ry discussed in the next sec-\ntion.\n2.1 Formulation of the Field Problem\nThe computational domain \n representing the super-\nconducting magnet is illustrated in Fig. 1. The domain\nis decomposed into the source and source-free regions\n\nHand \n Aoriented with the unit vector n, such that\n\nH[\nA=\n. The source region \n Hcorresponding\nto the coil is given by the union of the superconduct-\ning and normal-conducting parts \n H;sand \n H;c. The\nsource-free region \n A, containing the remainder of the\nmagnet such as the iron yoke, the mechanical supports,\nand the air region, is given by the union of the normal-\nconducting and non-conducting parts \n A;cand \n A;i.\nA constant magnetic permeability \u0016is assumed in \n H,\nwhereas a nonlinear dependency from the magnetic\n\feldBis considered for the iron yoke in \n A, as\u0016(B).\nThe \feld problem is solved under magnetoquasi-\nstatic assumptions for the reduced magnetic vector\npotential A?[40] in \n A, and for the magnetic \feld\nstrength H[41, 42] in \n H, with suitable boundary\nconditions on the exterior boundary. The formulation\n2reads\nr\u0002\u0016\u00001r\u0002A?+\u001b@tA?= 0 in \n A;(2)\nr\u0002\u001ar\u0002H+@t\u0016H\u0000r\u0002 \u001fus= 0 in \n H;(3)Z\n\nH\u001f\u0001(r\u0002H) d\n = i s; (4)\nwhere \u001fis a voltage distribution function [43, 44], u s\nis the source voltage treated as an algebraic unknown,\nand i sis the source current which is imposed via a\nconstraint equation (i.e., a Lagrange multiplier). The\nsources are provided by means of the electrical ports \u0000 J\nand \u0000 E. The \felds A?andHare linked via continuity\nconditions at the interface of the domains \u0000 HA, repres-\nented in Fig. 1 by a dashed line. In particular, the con-\ntinuity of the normal component of the magnetic \feld\nBnand the current density Jn, and the tangential com-\nponent of the magnetic \feld strength Htand electric\n\feld strength Etare imposed, ensuring the consistency\nof the overall \feld solution.\n2.2 Magnetic Field Quality\nThe magnetic \feld quality in accelerator magnets is\nde\fned as the set of Fourier coe\u000ecients, known also\nas \feld harmonics or multipole coe\u000ecients. The coe\u000e-\ncients are derived from the solution of the \feld problem\nin the source-free magnet aperture, which is given by\nthe Laplace equation r2A= 0. In the two-dimensional\napproximation of accelerator magnets, the axial \feld\nvariations are neglected along the z-direction (the lon-\ngitudinal axis of the magnet). Thus, the \feld can be\nexpressed as (e.g. [45])\nAz(r;') =1X\nk=1rk(Aksink'+Bkcosk'):(5)\nwhere A zis the longitudinal component of the mag-\nnetic vector potential, AkandBkare the multipole\ncoe\u000ecients, and ( r;';z ) are spatial coordinates in a\ncylindrical reference system consistent with the magnet\naperture. The \feld components are obtained from (5)\nas\nBr(r;') =1X\nk=1krk\u00001(Akcosk'\u0000Bksink'); (6)\nB'(r;') =\u00001X\nk=1krk\u00001(Aksink'+Bkcosk'):(7)\nThe indexkrepresents solutions of the Laplace equa-\ntion which can be associated to \feld distributions gen-\nerated by ideal magnet geometries. As an example, k=\n1,2,3 correspond to the dipole, quadrupole, and sextu-\npole \feld distributions. Once the radial \feld compon-\nent (6) is known at a reference radius r= r0(either\nby measurements or simulations), the skew and nor-\nmal multipole coe\u000ecients A kand Bkare obtained for\nk= 1;2;3;:::, as\nAk(r0) =1\n\u0019Z2\u0019\n0Br(r0;') cosk' d'; (8)Bk(r0) =1\n\u0019Z2\u0019\n0Br(r0;') sink' d'; (9)\nwhere the radius r 0is usually chosen as 2/3 of the mag-\nnet aperture.\nThe coe\u000ecients are often combined in the complex\nnotation C k(r0) = Bk(r0) +iAk(r0) as the skew and\nnormal pairs are orthogonal to each other. Moreover,\nthe coe\u000ecients are typically normalized with respect\nto the main \feld component B K(r0), and denoted as\nbkand ak. Thus, the \feld quality is quanti\fed as a\nrelative error c k, fork= 1;2;3;:::, as\nck(r0) = bk(r0) +iak(r0) = 104Ck(r0)\nBK; (10)\nand given in 1\u000210\u00004units with respect to B Kat the\nreference radius r 0. It is worth mentioning that for\nreaching accelerator quality standards, the \feld mul-\ntipoles shall be limited within a few units [45].\nThe magnetic \feld quality can also be conveni-\nently given in terms of total harmonic distortion factor\nFTHD(r0), which is a scalar quantity de\fned as\nFTHD(r0) =vuut1X\nk=1;k6=Kb2\nk(r0) + a2\nk(r0); (11)\nwhere K refers to the index of the main \feld compon-\nent.\n3 Veri\fcation of the Mathemat-\nical Model\nThe FEM implementation of the formulation proposed\nin (2){(4) is used to simulate the dynamic behavior\nof a single HTS tape, considered as an in\fnitely thin\nshell [46]. For this simpli\fed case, analytical solutions\nfrom previous literature are used for the veri\fcation\nof the numerical results in Sections 3.1.1 and 3.1.2.\nSubsequently, a mesh sensitivity analysis is carried out\nfor a known \feld solution to assess the precision of the\ncode in calculating the multipole coe\u000ecients. The mesh\nsensitivity results are presented in Section 3.2.\n3.1 Single Tape Model\nThe 2D magnetoquasistatic model of the HTS tape is\nused for calculating the speci\fc Joule losses per cycle,\nin the sinusoidal regime. The tape is composed only\nby one superconducting layer whose speci\fcations are\ngiven in Table I. Two scenarios are considered, di\u000bering\nonly in the source quantity applied to the tape: 1) An\nexternal magnetic \feld at zero current, (Fig. 2, left),\nand 2) a supply current, in self \feld (Fig. 2, right).\nThe results are veri\fed against analytical solutions and\npresented in Sections 3.1.1 and 3.1.2.\nThe numerical model of the HTS tape is used over\na frequency range of several orders of magnitude. For\nthis reason, an adaptive mesh distribution is used in the\ntape. The mesh elements are denser at the tape edges,\n3Figure 2. Magnetic \feld, in T, for a single HTS tape with an\nn-value of 20, at 1 :25 sec. a) Tape in external sinusoidal \feld of\n10 mT and frequency of 1 Hz. b) Tape in self-\feld, driven with a\nsinusoidal current of 500 A and frequency of 1 Hz.\nTABLE I. Tape speci\fcations\nName Unit Value Description\n\u000ew [mm] 10 Tape width\n\u000et [mm] 1\u000210\u00006Tape thickness\nJc [kA mm\u00002] 100 Critical current density\nfollowing a geometrical distribution of ratio 25. This al-\nlows for resolving the highly-nonlinear current density\ndistribution in the tape. About 500 elements are used\nfor the simulations at low \feld and current, whereas\nabout 20 elements are used for saturated tapes, in\naccordance with the relaxation of the magnetic \feld\nwithin the tape. The maximum time step size is given\nas \u0001t max= (50f)\u00001, wherefis the frequency of the\nsource quantity in the model.\n3.1.1 Tape in Perpendicular External Field\nA single HTS tape with no supply current is exposed\nto a time-dependent, perpendicular external \feld. The\nlayout of the simulated scenario is shown in the box of\nFig. 3. The source term is given by a sinusoidal mag-\nnetic \feld B( t) = B psin(2\u0019ft), applied perpendicularly\nto the tape. The speci\fc loss per cycle w Jis calculated\nforn= 5, 20 and 40. The case of n=1, which corres-\nponds to the critical state model [37, 38], is calculated\nanalytically. The theory of in\fnitely thin \flms with \f-\nnite width and one dimensional current distribution in\na perpendicular \feld [47, 48, 49] gives w Jas\nwJ=dwJcBc\u00122\nbpln(cosh b p)\u0000tanh b p\u0013\n;(12)\nwhere B c=\u00160(dhJc)=\u0019is the critical magnetic \feld,\ndwand dhare the width and the thickness of the tape,\nand b p= B p=Bcis the normalized magnetic \feld.\nThe losses w Jare given in Figs. 3 and 4 as a function\nof B pandf. The losses converge to the theoretical solu-\ntion in [47] for increasing n-values, which is to be expec-\nted given that the critical state model corresponds to\nthe power-law equation (1) where nis set to in\fnite.\nFor low \feld values, the losses follow a quartic scal-\ning law with respect to the magnetic \feld amplitude.\nOnce the magnetic \feld reaches B c, it fully penetrates\nin the tape, and the screening current distribution is\nmaintained. The losses grow proportionally with the\namplitude of the applied magnetic \feld, as the model\nconsiders the critical current density to be constant10\u0000410\u0000310\u0000210\u0000110010110210\u00001010\u0000910\u0000810\u0000710\u0000610\u0000510\u0000410\u0000310\u0000210\u00001100101102\n/B3\n/B4/B1Bc\nBp[T]wJ\u0002\nJ=mm3=cycle\u0003n= 5\nn= 20\nn= 40\nn=1\nFigure 3. Speci\fc Joule losses per cycle in a single HTS tape.\nLosses are given as a function of the sinusoidal magnetic \feld\napplied perpendicularly to the tape width, with peak value B p\nand frequency 1 Hz. The numerical results are parametrized by\nn= 5, 20, 40, and compared with the analytical solution from\nliterature where n=1.\n10\u0000410\u0000310\u0000210\u0000110010110210310\u0000510\u0000410\u00003\nBp= 10 mT\n/f0\nf[Hz]wJ\u0002\nJ=mm3=cycle\u0003n= 5\nn= 20\nn= 40\nn=1\nFigure 4. Speci\fc Joule losses per cycle in a single HTS tape.\nLosses are given as a function of the sinusoidal magnetic \feld\napplied perpendicularly to the tape width, with peak value B p=\n10 mT and frequency f. The numerical results are parametrized\nbyn= 5, 20, 40, and compared with the analytical solution from\nliterature where n=1.\nand \feld-independent. For the sake of completeness,\nFig. 3 reports also a trend line for a cubic scaling law,\nwhich is found in models accounting for \fnite n-values\nand two dimensional current density distributions in\nthe tape [50, 51, 4].\nThe losses presented in Fig. 4 are calculated for a\npeak \feld of 10 mT. The \feld is chosen below the pen-\netration limit, such that the \feld-screening behavior\nof the tape is included in the simulation. For high n-\nvalues (see Fig. 4) the frequency dependency tends to\nvanish, in accordance with a hysteresis-like behavior,\nand w Jconverges to the theoretical solution in [47] for\nn=1.\n3.1.2 Tape in Self-Field\nA source current is imposed to a single HTS tape, in\nself-\feld. The layout of the simulated scenario is shown\nin the box of Fig. 5. The source term is given by a\nsinusoidal current I( t) = I psin(2\u0019ft), applied to the\ntape as source. The calculation of w Jis done forn= 5,\n410010110210310410\u00001110\u00001010\u0000910\u0000810\u0000710\u0000610\u0000510\u0000410\u0000310\u0000210\u00001100101\n/I4/In+1\nIc\nIp[A]wJ\u0002\nJ=mm3=cycle\u0003n= 5\nn= 20\nn= 40\nn=1\nFigure 5. Speci\fc Joule losses per cycle in a single HTS tape\nof critical current I c= 1kA. Losses are given as a function of\nthe sinusoidal supply current, with peak value I pand frequency\n1 Hz. The numerical results are parametrized by n= 5, 20, 40,\nand compared with the analytical solution from literature where\nn=1.\n10\u0000410\u0000310\u0000210\u0000110010110210310\u0000410\u0000310\u0000210\u00001\nIp= 0:5Icrit\n/f0\nf[Hz]wJ\u0002\nJ=mm3=cycle\u0003n= 5\nn= 20\nn= 40\nn=1\nFigure 6. Speci\fc Joule losses per cycle in a single HTS tape\nof critical current I c= 1kA. Losses are given as a function of\nthe sinusoidal supply current, with peak value I p= 0:5Icand\nfrequencyf. The numerical results are parametrized by n= 5,\n20, 40, and compared with the analytical solution from literature\nwheren=1.\n20 and 40, whereas the case of n=1is calculated\nanalytically. The theory of in\fnitely thin \flms with\n\fnite width and one dimensional current distribution\nin self-\feld [49, 52] gives w Jas\nwJ=\u00160\ndwdhI2\nci4\np\n6\u0019; (13)\nwhere I cis the critical current of the tape and i p=\nIp=Icis the normalized supply current. The losses w J\nare given in Figs. 5 and 6 as a function of I pandf. Con-\nsistent with (13), for high n-values the numerical res-\nults show a quartic dependence for currents up to the\ncritical current. Beyond this value, the current density\ndistributes homogeneously in the tape, and the losses\nare proportional to In+1, in accordance with the power-\nlaw behavior in (1).\nThe losses presented in Fig. 6 are calculated for a\nsub-critical current I p= 0:5 Ic. From Fig. 5 it is clear\nthat with increasing n-value, the simulation results\nconverge to the analytical dependence given in literat-\nure [49, 52], whereas Fig. 6 shows that with increasing10110210310410\u0000510\u0000410\u0000310\u0000210\u00001100101102\n1\n\u0001x\u0002\nm\u00001\u0003e\u0001x[\u0000]\n102103104105106\nno. of elements [\u0000]THD error\nno. of elements\nFigure 7. Left axis: relative error in the calculation of the total\nharmonic distortion index, as function of the reciprocal of the\nmaximum mesh size. Right axis: number of elements in the mesh.\nn-value, the frequency dependency vanishes as expec-\nted.\n3.2 Mesh Sensitivity\nIn numerical models, the multipole coe\u000ecients are ob-\ntained by applying the Fast Fourier Transform al-\ngorithm to the radial component of the magnetic \feld,\ncalculated along the reference circumference in the\nmagnet aperture (see Section 2.2). Care has to be\ntaken, as the \fnite resolution of the mesh in the spa-\ntial discretization introduces a numerical error which\na\u000bects the calculation of the multipole coe\u000ecients [53].\nFor this reason, a mesh sensitivity analysis is carried\nout for a reference model where a known analytical\n\feld solution is simulated and calculated at the refer-\nence circumference. The relative error e\u0001xis de\fned\nas\ne\u0001x=jFTHD\u0000F\u0001x\nTHDj\nFTHD; (14)\nwhere F THD and F\u0001x\nTHD are the total harmonic distor-\ntion factors in (11) for the analytical and calculated\n\feld solutions. The error is shown in Fig. 7 as a function\nof the reciprocal of the maximum element size \u0001x max.\nBased on this investigation, triangular elements with\n\u0001xmax= 1 mm are chosen for the mesh, yielding an\nestimated error of 3 \u000210\u00005.\n4 Numerical Model of the\nFeather-M2.1-2 Magnet\nThe FEM model of the Feather-M2.1-2 dipole magnet\nrefers to the magnet version M.1-2, which is wound\nusing a coated conductor produced by Sunam [54]. The\ngeometric and superconducting properties of the tape\nare reported in Table II. This particular tape limits\nthe magnet current to 5 kA and the peak \feld in the\naperture to 3 T. A simpli\fed rendering of the magnet\nis given in Fig. 8, where for the sake of clarity, only\nthe components relevant for the numerical analysis are\n5Figure 8. Simpli\fed rendering of the Feather-M2.1-2 magnet.\nThe coil is composed by two pairs of central and wing decks.\nThe cable is made of 15 tapes fully transposed with the Roebel\ntechnique. The cross-section of the cable is shown in the lower-\nright corner. The magnetic circuit is composed by four iron poles\nand a cylindrical iron yoke (half-shown).\nFigure 9. Magnetic \feld in T, at 5 kA and 4 :5 K, shown for the 2D\ncross-section of the Feather-M2.1-2 magnet. The peak magnetic\n\feld reached in the aperture is about 2 :5 T.\nshown. The coil is composed by two poles, each made\nof two windings named central and wing decks, and\nis designed to optimize the tape-\feld alignment [26].\nThe magnetic \feld is shaped in the magnet aperture\nby means of iron poles. The outer iron yoke intercepts\nthe stray \feld and allows for operating the magnet in\na stand-alone con\fguration. The central cross-section\nof the magnet is used as geometry input for the 2D\nFEM model. The magnetic \feld solution, in Tesla, is\nshown in Fig. 9 for a current of 5 kA at 4 :5 K. The key\nfeatures and the relevant simpli\fcations of the model\nare discussed in the remainder of this section.\n4.1 Coil Geometry\nThe model is implemented for a 2D transverse \feld\ncon\fguration, thus neglecting the magnetic e\u000bects of\nthe end-coils. Due to the presence of the layer jumps\nconnecting the lower and the upper windings in the\ncoil, the magnetic symmetry in the cross-section of the\nmagnet is not preserved. For this reason, the modelTABLE II. Feather-M2.1-2 tape speci\fcations\nParameter Unit Value Description\nProducer Sunam [54]\nTechnology IBAD [55, 56]\nSubstrate Hastelloy\nStabilizer Copper\n\u000et;sub [µm] 100 Substrate thickness\n\u000et;stab [µm] 40 Stabilizer thickness\n\u000et [µm] 150 Tape thickness\n\u000ew [mm] 5.5 Tape width\nIc;meas [A] 300 @ 77 K, self-\feld\nJc(B;T) [A mm\u00002] \ft Fit in [57]\nn [-] 4\u0014n\u001430 Power-law index\naccounts for a four-quadrants geometry, including the\nlayer jumps in the \frst and third quadrant. The layer\njump is visible in Fig. 9, as a cable slightly misaligned\nwith respect to the coil decks.\nHTS tapes feature a multi-material and multi-layer\nstructure. At the same time, the tape used in the coil\nhas a width-to-thickness ratio of about two orders of\nmagnitude. This justi\fes approximating the geometry\nof the tape with a line [46]. In this way, the discretiza-\ntion of the thickness of the superconductor is avoided.\nAt the same time, the physical properties of the ma-\nterials composing the tape are homogenized. Such sim-\npli\fcation is adopted to ensure an acceptable compu-\ntational time, as the 2D model accounts for 648 tapes\nover four quadrants.\n4.2 Current Sharing Approximation\nThe cable used in the coil is made of 15 tapes, which are\nfully transposed using the Roebel technique [58, 59].\nThe cross-section of the cable used in the numerical\nmodel is sketched in the box of Fig. 8, where each line\nrepresents a tape. Each tape is electrically connected in\na parallel con\fguration, allowing for the redistribution\nof the supply current. Moreover, the Roebel transposi-\ntion enforces the same electrical impedance for each of\nthe tapes composing the cable, providing an even cur-\nrent distribution. For this reason, the same fraction of\nthe supply current is imposed in the numerical model\nto each of the tapes, excluding current redistribution\nphenomena. Coupling currents [3] are also excluded,\nsince they represent a second-order e\u000bect with respect\nto persistent currents [4].\nWithin each tape, current sharing phenomena are\nmodeled by means of an equivalent surface resistivity,\nwhich homogenizes the superconducting and normal-\nconducting layers, as detailed in [32]. The surface res-\nistivity depends from the power law in (1), thus is af-\nfected by the n-value. From magnet measurements [14],\nann-value of 5 was experimentally found, outside the\nexpected range of 20-30 [60], and attributed to unbal-\nanced tape joints. However, note that for persistent\nmagnetization the local critical current density is the\nrelevant quantity and so the joint resistance is not the\nrelevant quantify for calculating the persistent magnet-\nization. Unfortunately, the tape was not characterized\nindividually and so the uncertainty of the supercon-\n6TABLE III. Parameters used for the J c\ft\nName Unit Value\ng0 - 0:03\ng1 - 0:25\ng2 - 0:06\ng3 - 0:06\nTc0 K 93\npc - 0:5\nqc - 2:5\nBi0c T 140\n\rc - 2:44\n\u000bcMA T\nmm2 1:86Name Unit Value\n\u0017 - 1:85\na - 0 :1\nn0 - 1\nn1 - 1:4\nn2 - 4:45\npab - 1\nqab - 5\nBi0ab T 250\n\rab - 1:63\n\u000babMA T\nmm2 68:3\n0 1 2 3 4 5 6 7 8024681012\nB [T]Ic[kA]10 K\n20 K\n30 K\n40 K\n50 K\n60 K\n70 K\n80 K\nBp\nintersection\nFigure 10. Calculation of the critical current in the\nFeather-M2.1-2 magnet as a function of the magnetic \feld. The\ncritical current is obtained for each temperature as the intersec-\ntion point (markers) of the magnetic characteristic of the magnet\n(dotted line), known also as the load line, with the critical cur-\nrent provided by the \ft (solid lines), assuming a perpendicular\nmagnetic \feld to the cable.\nducting properties of the tapes is signi\fcant and the\nn-value is not known. To overcome this issue, a para-\nmetric sweep is performed for 4 \u0014n\u001430, quantifying\nthe sensitivity of the model. The results are compared\nwith measurements in Section 5.\n4.3 Critical Current Density Fit\nThe critical current density J cin (1) a\u000bects the persist-\nent currents dynamics and, ultimately, the \feld quality\nin the magnet. In ReBCO tapes, J cshows an aniso-\ntropic, \feld- and temperature-dependent behavior, as\nJc(B;T;jB), where jBis the magnetic \feld angle with\nrespect to the direction perpendicular to the tape wide\nsurface.\nThe behavior of J cis included in the model by means\nof the numerical \ft provided in [57]. The \ft parameters,\nreported in Table III, are taken from [4], since no data\nwas available for the used Sunam tape. For this reason,\nthe \ft is scaled in order to provide a critical current\nfor the Feather-M2.1-2 coil which is consistent with\nmeasurements [14], as follows.\nThe magnetic characteristic of the magnet, known\nalso as the load line, is calculated numerically by means\nof magnetostatic simulations. With respect to Fig. 10,\nthe load line is given in terms of peak magnetic \feld\nBp;coilin the coil as a function of the supply current0 10 20 30 40 50 60 70 80 90024681012\nT [K]Ic[kA]measurement\njB= 0\u000e\njB= 45\u000e\njB= 75\u000e\nFigure 11. Calculated critical current of the cable as a function\nof temperature, parametrized by the magnetic \feld angle with\nrespect to the cable perpendicular direction. The markers show\nthe measured critical current in the Feather-M2.1-2 magnet.\n0 10 20 30 40 50 60 70 80 9000:250:50:7511:251:51:752\nT [K]fc[\u0000]\njB= 0\u000e\njB= 45\u000e\njB= 75\u000e\nFigure 12. Correction factor applied to the critical current \ft, as\na function of temperature, parametrized by the magnetic \feld\nangle with respect to the cable perpendicular direction.\n(dotted line). The critical current is then given for each\ntemperature as the intersection of the load line (mark-\ners) with the critical current provided by the \ft (solid\nlines). The magnetic \feld is assumed to be perpendic-\nular to the cable, as jB= 0\u000e. In Fig. 11, the calculated\ncritical current is compared with the measurements,\nand parametrized by the \feld angle. The assumption\nof \feld perpendicularity gives the best agreement with\nthe measured data. The \ftting factor is \fnally obtained\nas\nfc(T) =Ic;meas(T)\nJc(Bp;coil(T);T;jB)SHTS\f\f\f\f\njB=0\u000e; (15)\nwhere I c;meas is the critical current obtained from\nmeasurements, and S HTSis the superconducting cross-\nsection of the cable. The \ftting factor is shown as a\nfunction of temperature in Fig. 12, and parametrized\nby the \feld angle. The factor obtained for jB= 0\u000eis\nused in the model for scaling the critical current dens-\nity \ft.\n7\u00002\u00001:5\u00001\u00000:50 0:5 1 1:5 2\u00002\u00001:5\u00001\u00000:500:511:52\nH\u0002\nkA m\u00001\u0003B [T]\n\frst magnetization\nJiles-Atherton\nFigure 13. Nonlinear magnetic characteristics of the iron used in\nthe Feather-M2.1-2 model, represented with: 1) the \frst magnet-\nization curve, and 2) the hysteresis loop provided by the Jiles-\nAtherton model.\nTABLE IV. Parameters for the Jiles-Atherton hysteresis model\nName Unit Value Description\nMs [A m\u00001] 1:35\u0002106Saturation magnetization\na [A m\u00001] 90 Domain wall density\nk [A m\u00001] 40 Pinning loss\nc [{] 1 \u000210\u00006Magnetization reversibility\na [{] 50\u000210\u00006Inter-domain coupling\n4.4 Iron Hysteresis\nThe magnetic material used in the yoke of the Feather-\nM2 magnet is chosen to minimize the detrimental in\ru-\nence of the iron hysteresis on the magnetic \feld quality.\nNo material characterization data was available, how-\never the iron is similar to the one of the LHC main\ndipoles. For this reason, the numerical model considers\nthe same nonlinear B(H) curve used for the LHC [61].\nThe curve is shown as a dashed line in Fig. 13. In stand-\nalone operations, most of the outer iron yoke of the\nFeather-M2.1-2 magnet remains unsaturated up to the\nmaximum current of 5 kA. For this reason, the contri-\nbution of the iron hysteresis on the \feld quality cannot\n00:511:522:533:544:550510152025\nI [kA]j\u0001b1j[units]\n012\nj\u0001b3;5;7j[units]\u0001b1\n\u0001b3\n\u0001b5\n\u0001b7\nFigure 14. Amplitude of the magnetization loop of the \feld mul-\ntipoles, as a function of current. The magnitude of \u0001b1 is given\nby the left axis, whereas the magnitude of \u0001b3, \u0001b5 and \u0001b7\nis given by the right axis.be neglected a priori, and is estimated as follows.\nA coercive \feld H cof 40 A m\u00001is assumed for the\nmaterial used in the yoke, in accordance with the LHC\nspeci\fcations (H c\u001460 A m\u00001[45]). The iron hyster-\nesis is included by using the Jiles-Atherton model [62],\nwhose loop is determined by using the B(H) curve as\nreference, and shown in Fig. 13. The relevant para-\nmeters for the hysteresis model are obtained using the\nopen-source algorithm from [63] and are reported in\nTable IV.\nThe hysteresis contribution is calculated on a sim-\npli\fed Feather-M2.1-2 model, including only the iron\nas nonlinear e\u000bect. The multipole coe\u000ecients are ob-\ntained as function of the current for both the upper and\nthe lower hysteresis curve, then the two data sets are\ncompared. Their di\u000berence \u0001b provides the amplitude\nof the magnetization loop for each coe\u000ecient, giving an\nestimation for the e\u000bect of the iron hysteresis on the\n\feld quality.\nWith respect to Fig. 14, the hysteresis of the iron has\na minor in\ruence at low current on the main \feld com-\nponent b 1. A peak value of about 20 units is found and\nit rapidly decreases once the current is increased, since\nthe with of the hysteresis loop narrows. Concerning the\nhigher order multipoles b 3, b5and b 7, there is almost\nno in\ruence since the contribution is always less than\none unit. As a consequence, the contribution to the\n\feld from the interaction between the iron hysteresis\nand the screening currents in the coil can be reasonably\nassumed a second order e\u000bect, thus negligible.\nThe analysis shows a limited in\ruence from the iron\nhysteresis on the magnetic \feld quality, at the price of\nan increased computational cost. For this reason, the\niron hysteresis is excluded from the numerical model of\nthe Feather-M2.1-2 magnet.\n5 Comparison of Simulations\nwith Measurements\nThe numerical model of the Feather-M2.1-2 magnet is\nvalidated by comparing the simulation results of the\nmagnetic \feld quality in the magnet aperture with\navailable experimental observations. The comparison is\ndone for four scenarios, which di\u000ber in the peak current\nIp(i.e., peak magnetic \feld) and operational temperat-\nure T opof the magnet. The relevant parameters char-\nacterizing the scenarios are reported in Table V. It is\nworth noting that as T opis increased, I pis reduced ac-\ncordingly, such that the ratio between the peak current\nand the critical current of the cable is kept constant.\nIn accordance with measurements, T opis assumed as\nhomogeneous and constant in the numerical model, for\neach scenario.\nIn the following, the measurement and simulation\nsetups are discussed, and the comparison between ex-\nperimental and numerical results is presented. All the\nsimulations are carried out on a standard worksta-\ntion (IntelrCore i7-3770 CPU @ 3 :40 GHz, 32 GB\nof RAM, Windows-10rEnterprise 64-bit operating\n8TABLE V. Simulated scenarios: main parameters\nScenario T op[K] I p[kA] B p[T]\n1 4.5 5 2.5\n2 9 4.75 2.4\n3 25 3.75 2.0\n4 68 1.75 1.0\n\u00001 000 0 1 000 2 000 3 00000:511:52\nt [s]I [kA]pre-cycle\nstaircase\nup\ndown\nFigure 15. Example of a current pro\fle used in the simulations\n(scenario at 1 :75 kA and 68 K). The current follows a trapezoidal\npre-cycle, then a staircase pro\fle, up to the peak current and\nback. The markers at the current plateaus ( upanddown labels)\nrepresent the evaluation points for the magnetic \feld quality for\nboth the ascending and descending part of the staircase.\nsystem), using the proprietary FEM solver COMSOL\nMultiphysicsr[64].\n5.1 Measurement Setup\nRotating-coil magnetometers, also known as harmonic\ncoils, are electromagnetic transducers for measuring\nthe Bkand Ak\feld multipoles. The coil shaft is po-\nsitioned parallel to the magnetic axis of the magnet,\nand it is rotated in the magnet aperture. The change\nof \rux linkage \b induces, by integral Faraday\u0013s law\nUm=\u0000dt\b, a voltage signal U mwhich is measured\nat the terminals of the coil. By integrating in time\nthe voltage signal, the \rux linkage is obtained and\ngiven as a function of the series expansion of the ra-\ndial \feld [45]. Assuming a coil of negligible thickness,\nperfectly centered in the aperture of a magnet, and ro-\ntating with angular velocity w, then for an arbitrary\nangle'(t) =wt+'0the \rux linkage is given at time t\nas\n\b(t) =1X\nk=1fs[Ak(rc0) cosk'\u0000Bk(rc0) sink'];(16)\nfs(k) =2Nclcrc0\nk; (17)\nwhere the coil sensitivity factor f s(k) embeds the coil\ngeometric parameters, namely the number of turns N c,\nlongitudinal length l cand the mean radius r c0. Such\nparameters are calibrated in a dipole and quadrupole\nreference magnet.\nEncouraged by the results obtained from the \rux\nsensors presented in [15], a dedicated rotating-coil mag-\nnetometer was developed and employed to test theFeather-M2.1-2 magnet in the variable temperature\ncryostat at CERN. The constructed coil shaft is com-\nposed of a chain of \fve Printed-Circuit Boards (PCBs),\n(200 mm in length and 35 mm in width), that span the\nentire magnet length including the fringe-\feld areas.\nEvery PCB board contains three coils mounted radi-\nally, with an active surface of 0 :1817 m2.\nFor the magnetic-\feld harmonics, the measurement\nsensitivity is improved by connecting two coils in anti\nseries; for the dipole magnet measurement the external\ncoil minus the central coil. CERN proprietary digital\ncards [65] integrate the induced voltages in the coils ro-\ntating at a frequency of 2 Hz. In this paper, the meas-\nurement results are taken from the longitudinal center\nof the magnet (the central element of the rotating shaft\nof 200 mm in length), delivering a measurement preci-\nsion of a magnetic-\feld harmonic of \u00060:05 units.\n5.2 Simulation Setup\nTo match the experimental procedure, a current excit-\nation is applied as a source for the numerical model.\nWith respect to the example provided in Fig. 15, the\ncurrent follows \frstly a trapezoidal pre-cycle, then a\nstaircase pro\fle spanning from a minimum value of\n0:25 kA up to the peak current, and back. The aim\nof the pre-cyle is to remove the dependency of the\nsuperconducting coil on the \frst magnetization cycle.\nThe staircase signal is composed of steps of steepness\n10 A s\u00001, which increase the current by \u0001I = 250 A,\nand then keep it constant for \u0001t \rat= 120 s. For each\nmidpoint in the staircase plateaus, sowed in Fig. 15\nwith a marker, the magnetic \feld quality is calcu-\nlated and compared with measurements. The number\nof steps is adapted for each scenario, in order to reach\nthe prescribed peak current. The shape of the current\nexcitation and the evaluation points for the \feld quality\nare consistent with the ones used in the measurements.\nFollowing [15], the staircase current pro\fle is used\nto quantify the in\ruence of hysteresis phenomena oc-\ncurring within both the superconducting coil and the\niron yoke of the magnet, as follows. With respect to\nFig. 15, for each current step the \feld quality is meas-\nured and simulated twice, once during the ramp-up\nand then during the ramp-down, and the results are\ngrouped in pairs. Subsequently, the di\u000berence of the\n\feld multipoles is calculated for each pair of \feld qual-\nity evaluations. Thanks to this operation, the contribu-\ntions of the non-ideal geometry of the coil and the iron\nsaturation are canceled out, being the same for both\nevaluations in each pair, and the residual is attributed\nto the hysteresis phenomena. Since the iron hysteresis\nwas previously found to produce only a second-order\ne\u000bect on the \feld quality (see Section 4.4), the hys-\nteresis contribution is fully attributed to the persistent\nmagnetization of the superconducting coil.\n5.3 Results\nThe measured and simulated \feld multipole coe\u000ecients\nare given in Fig. 16. The markers represent the meas-\n90123450123B1[T]Top= 4:5 K\n0123450123Top= 9 K\n0123450123Top= 25 K\n0123450123Top= 68 K\n0123450100200300400500b3[units]\n0123450100200300400500\n0123450100200300400500\n0123450100200300400500\n012345020406080100b5[units]\n012345020406080100\n012345020406080100\n012345020406080100\n0123450246810\n[kA]b7[units]\n0123450246810\n[kA]0123450246810\n[kA]0123450246810\n[kA]\nup down n=sweep n=20jJjhomogeneous\nFigure 16. Magnetic \feld quality in the magnet aperture as a function of the current, using a current staircase pro\fle (see Fig. 15).\nMeasurements are given by markers, whereas the shaded area corresponds to the envelope of the numerical solutions, obtained with\nthe parametric sweep of the n-value as 4\u0014n\u001430. The solution for n= 20 is marked with a solid line. The dotted line is obtained\nby assuming a homogeneous current density distribution in the superconducting tapes. From left to right: results at 4.5, 9, 25, and\n68 K. From top to bottom: results for the B 1, b3, b5, b7multipole coe\u000ecients.\nurements which are split in the upanddown datasets,\naccordingly to the upward and downward part of the\ncurrent staircase (see Fig. 15). The shaded area gives\nthe envelope of the numerical solutions obtained by a\nparametric sweep of the n-value between 4 and 30. As\nan example, the simulation results for n= 20 are high-\nlighted with a solid line. The dashed line represents the\nideal case in which the the screening currents do not\nhave any in\ruence on the \feld quality. This is obtained\nby arti\fcially increasing the resistivity of the supercon-\nducting coil until a homogeneous current density dis-\ntribution is achieved in the time domain simulation.\nThe rows show, from top to bottom, the normal dipole\n\feld B 1and the multipoles b 3, b5and b 7, as a function\nof the source current. The columns separate the results\nby the operational temperature of the magnet, namely\n4.5, 9, 25, and 68 K or, in other words, the simulated\nscenario.\nThe \feld multipoles keep qualitatively the same be-\nhavior through the di\u000berent scenarios (see Fig. 16, row\nby row). Moreover, the b 3and b 5multipoles are re-\nduced as the the current is increased. The b 7coe\u000ecient\nis negligible with respect to the others. The scenario\nat 4:5 K shows the highest variation in the magnitudeof the multipoles. At low current, the b 3contribution\nincreases of about a factor 2, from 200 to 400 units,\nand the b 5multipole shows an increase of a factor 8,\nfrom 10 to 80 units. This might be explained as screen-\ning currents are higher at low temperature, due to the\nhigher critical current density of the tape.\nHysteresis phenomena in the Feather-M2.1-2 magnet\ncreate the magnetization loops which are present in the\nmeasured and simulated data sets. The loops are found\nto be at least one order of magnitude smaller than the\nabsolute value of the multipole coe\u000ecients. For this\nreason, the width of the loops is shown separately in\nFig. 17. The layout and the meaning of symbols is the\nsame as before for Fig. 16. The rows show from top\nto bottom the variation in units for the multipoles b 1,\nb3, b5and b 7, as a function of the supply current. The\ncolumns separate the results by the operational tem-\nperature of the magnet, namely 4.5, 9, 25, and 68 K.\nThe width of the magnetization loops due to persist-\nent currents does not exceed twenty units for b 1and\nb3, two units for b 5and one unit for b 7. The trend is\ngenerally monotone, showing the multipoles decreasing\nas the current increases, and vanishing as the current\nreaches its peak value. The b 1coe\u000ecient is an excep-\n10012345\u0000200204060b1[units]Top= 4:5 K\n012345\u0000200204060Top= 9 K\n012345\u0000200204060Top= 25 K\n012345\u0000200204060Top= 68 K\n012345\u00002002040b3[units]\n012345\u00002002040\n012345\u00002002040\n012345\u00002002040\n012345\u00004048b5[units]\n012345\u00004048\n012345\u00004048\n012345\u00004048\n012345\u00004\u00002024\n[kA]b7[units]\n012345\u00004\u00002024\n[kA]012345\u00004\u00002024\n[kA]012345\u00004\u00002024\n[kA]\nmeas. n=sweep n=20jJjhomogeneous\nFigure 17. Screening currents-induced magnetic \feld contribution to the magnetic \feld quality, in units, as a function of the current\nin the magnet. Measurements are given by markers, whereas the shaded area corresponds to the envelope of the numerical solutions,\nobtained with the parametric sweep of the n-value as 4\u0014n\u001430. The solution for n= 20 is marked with a solid line. The dotted\nline is obtained by assuming a homogeneous current density distribution in the superconducting tapes. From left to right: results\nat 4.5, 9, 25, and 68 K. From top to bottom: results for the b 1, b3, b5, b7multipole coe\u000ecients.\ntion, as it has a peak around 3 :5 kA, when the pole\nof the iron yoke saturates. In the ideal case where the\nscreening currents are neglected, the width of the mag-\nnetization loops is always zero.\n6 Discussion\nThe \feld quality in the Feather-M2.1-2 magnet shows\nb3and b 5coe\u000ecients which are much higher than\nthe few units typically required by accelerator quality\nstandards [45] (see Fig. 16). This might be explained\nby the in\ruence of the outer iron yoke which is not yet\noptimized for \feld quality purposes. The \feld error is\ngoverned by the b 3coe\u000ecient, whereas b 5is about one\norder of magnitude smaller, and b 7is negligible.\nThe magnet design is optimized to deliver the\nhighest \feld quality when operating in nominal con-\nditions. As a consequence, for an increasing supply\ncurrent (i.e., increasing main dipole \feld), the multi-\npole coe\u000ecients are decreasing. If the temperature is\nincreased, the peak supply current needs to be reduced\naccordingly, to cope with the temperature dependency\nof the cable critical current. The working point of themagnet then shifts from nominal conditions, and the\nb3and b 5multipole coe\u000ecients increase.\nReferring to Fig. 17, the contribution of the screen-\ning current-induced magnetic \feld to the \feld quality\nnever exceeds 20 units, thus it is one order of magnitude\nsmaller than the total \feld error (see Fig. 16). The\nnumerical analysis gave better agreement with meas-\nurements for high n-values (\u001520), whereas for small\nn-values (\u001410), the contribution of the persistent cur-\nrents is overestimated. The results seem to con\frm that\nthe low quality of the tape (measured n-value of 5) is\ndue to the tape joints, which do not play any role in\nthe dynamics of the persistent currents. The limited\ncontribution of the persistent magnetization might be\nexplained with the coil design, which is optimized to\nalign the tapes with the magnetic \feld lines [26], limit-\ning the \rux linked to the surface of the tapes, and thus\nmagnetization phenomena.\nBy increasing the operational temperature of the\nmagnet, the critical current density of the tape is re-\nduced, leading to a faster \feld di\u000busion in the tape,\nand consequently to a more homogeneous current dens-\nity distribution in the cable. This is shown in Fig. 18\nwhere the current density distribution normalized to\n11ramp-up, 1 kA ramp-down, 1 kA\n4:5 K\n9 K\n25 K\n68 K\nFigure 18. Current density distribution in the most inner turn\nof the upper deck, normalized with the critical current density\nat zero \feld and 4 :5 K J crit ;0= 138 kA mm\u00002. The distribution\nis given at 1 kA for both, the ramp-up and the ramp-down, for\ndi\u000berent temperatures.\nJc(4:5 K;0 T;0°) = 138 kA mm\u00002is given for the most\ninner turn of the upper deck. As the supply current\nis increased, the persistent currents tend to vanish in-\ndependently from the operational temperature. This\nmight be explained by the saturation of the tape due\nto the supply current.\nNumerical simulations are in agreement with meas-\nurements, consistently reproducing both the magnetic\noverall \feld quality and the persistent magnetization\ncontribution. Still, simulation results are a\u000bected by\nthe uncertainty on the superconducting properties of\nthe tapes used in the Feather-M2.1-2 magnet. Nev-\nertheless, the analysis is relevant as it clearly shows\nwhich properties are important for understanding the\n\feld-quality-behavior of HTS accelerator magnets. For\nthis reason, a more extensive tape characterization is\nrecommended for future magnets, thus reducing the\nuncertainty in the material properties and enhancing\nthe con\fdence and accuracy in dynamic \feld quality\nsimulations.\n7 Conclusions and Outlook\nThis paper presents the time-domain analysis of the\ndemonstrator magnet Feather-M2.1-2, an HTS insert\ndipole designed to provide an additional 5 T in the\nNb3Sn FRESCA2 background magnet, up to peak\n\felds of 18 T in the magnet aperture. The ana-\nlysis quanti\fes the in\ruence of the screening current-\ninduced magnetic \feld on the magnetic \feld quality in\nthe magnet aperture. Simulations reproduce the power-\ning cycle of the magnet for di\u000berent temperatures and\noperating currents by using a staircase-shaped current\npro\fle. The magnet is simulated in a stand-alone con-\n\fguration, such that numerical results are veri\fed with\navailable measurements.\nFor this case study, the \feld quality error due to\npersistent magnetization phenomena a\u000bects mostly the\nmain \feld component, and it is limited to 20 units.\nMoreover, the error is signi\fcantly reduced once the\nsupply current is increased to the operational value,\nsaturating the tape. The coupling of the scrrening cur-\nrents with the hysteresis of the iron is found to be neg-\nligible. Thus, the aligned-coil design might be a key-\nfeature for ensuring accelerator quality standards inthe magnetic \feld of future HTS accelerator magnets.\nThe numerical analysis is carried out under mag-\nnetoquasistatic assumptions, using time-domain sim-\nulations based on a coupled A-Hformulation imple-\nmented in a 2D FEM model. The formulation is veri-\n\fed against analytical solutions from previous literat-\nure, and the model is validated with available experi-\nmental data. The model requires only one scalar cor-\nrection parameter for the power law, compensating for\nthe uncertainty in the critical current density of the\ntape. Simulations quantify the in\ruence of the coil elec-\ntrodynamics on the magnetic \feld, achieving satisfact-\nory agreement with measurements. The computational\ntime is less than one hour for each simulation, on a\nstandard workstation. The accuracy of the model may\nbe increased by a better knowledge of both the critical\nsurface current of the tape used for the coil, and the\nmagnetization curve of the iron used for the yoke.\nThe model provides for each tape an accurate quan-\nti\fcation of the dynamic distribution of the persistent\ncurrents, which can be used not only for the magnetic\n\feld quality analysis, but also for the calculation of\nthe Joule losses and the dynamic forces in the coil. As\nscreening currents provide the principal contribution\nto dynamic losses in HTS tapes, such valuable insights\ncan be integrated for the future design of HTS mag-\nnets, e.g. within a numerical optimization work\row for\nquench protection studies.\n8 Acknowledgments\nThis work has been sponsored by the Wolfgang Gent-\nner Programme of the German Federal Ministry of\nEducation and Research (grant no. 05E15CHA), by\nthe `Excellence Initiative' of the German Federal and\nState Governments and by the Graduate School of\nComputational Engineering at Technische Universit at\nDarmstadt. Parts of the work have been funded by\nthe BMBF project \\Diagnose of high-intensity hadron\nbeams (DIAGNOSE)\", under grant no. 05P18RDRB1.\nThe authors would like to acknowledge the fruit-\nful collaboration between CERN and the Technische\nUniversit at Darmstadt, within the framework of the\nSTEAM collaboration project [66]. The authors would\nlike to thank S. 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[Accessed: May 01,\n2020].\n14"    },    {        "title": "1903.08395v2.Nonlinear_magnetization_dynamics_driven_by_strong_terahertz_fields.pdf",        "content": "Nonlinear magnetization dynamics driven by strong terahertz \felds\nMatthias Hudl,1Massimiliano d'Aquino,2Matteo Pancaldi,1See-Hun Yang,3Mahesh G. Samant,3Stuart\nS. P. Parkin,3, 4Hermann A. D urr,5Claudio Serpico,6Matthias C. Ho\u000bmann,7and Stefano Bonetti1, 8,\u0003\n1Department of Physics, Stockholm University, 106 91 Stockholm, Sweden\n2Department of Engineering, University of Naples \\Parthenope\", 80143 Naples, Italy\n3IBM Almaden Research Center, San Jose CA 95120, USA\n4Max-Planck Institut f ur Mikrostrukturphysik, 06120 Halle, Germany\n5Department of Physics and Astronomy, Uppsala University, 751 20 Uppsala, Sweden\n6DIETI, University of Naples Federico II, 80125 Naples, Italy\n7SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA\n8Department of Molecular Sciences and Nanosystems,\nCa' Foscari University of Venice, 30172 Venezia Mestre, Italy\n(Dated: September 26, 2019)\nWe present a comprehensive experimental and numerical study of magnetization dynamics in a\nthin metallic \flm triggered by single-cycle terahertz pulses of \u001820 MV/m electric \feld amplitude\nand\u00181 ps duration. The experimental dynamics is probed using the femtosecond magneto-optical\nKerr e\u000bect, and it is reproduced numerically using macrospin simulations. The magnetization\ndynamics can be decomposed in three distinct processes: a coherent precession of the magnetiza-\ntion around the terahertz magnetic \feld, an ultrafast demagnetization that suddenly changes the\nanisotropy of the \flm, and a uniform precession around the equilibrium e\u000bective \feld that is relaxed\non the nanosecond time scale, consistent with a Gilbert damping process. Macrospin simulations\nquantitatively reproduce the observed dynamics, and allow us to predict that novel nonlinear mag-\nnetization dynamics regimes can be attained with existing table-top terahertz sources.\nSince Faraday's original experiment [1] and until two\ndecades ago, the interaction between magnetism and\nlight has been mostly considered in a unidirectional way,\nin which changes to the magnetic properties of a mate-\nrial cause a modi\fcation in some macroscopic observable\nof the electromagnetic radiation, such as polarization\nstate or intensity. However, the pioneering experiment of\nBeaurepaire et al. [2], where femtosecond optical pulses\nwere shown to quench the magnetization of a thin-\flm\nferromagnet on the sub-picoseconds time scales, demon-\nstrated that intense laser \felds can conversely be used\nto control magnetic properties, and the \feld of ultrafast\nmagnetism was born. Large research e\u000borts are nowadays\ndevoted to the attempt of achieving full and deterministic\ncontrol of magnetism using ultrafast laser pulses [3{10], a\nfundamentally di\u000ecult problem that could greatly a\u000bect\nthe speed and e\u000eciency of data storage [11].\nRecently, it has been shown that not only femtosecond\nlaser pulses, but also intense single-cycle terahertz (THz)\npulses [12] can be used to manipulate the magnetic order\nat ultrafast time scales in di\u000berent classes of materials\n[13{18]. The main peculiarity of this type of radiation,\ncompared with more conventional femtosecond infrared\npulses, is that the interaction with the spins occurs not\nonly through the overall energy deposited by the radia-\ntion in the electronic system, but also through the Zee-\nman torque caused by the magnetic \feld component of\nthe intense THz pulse. This is a more direct and e\u000ecient\nway of controlling the magnetization, and to achieve the\nfastest possible reversal [19, 20]. However, an accurate\ndescription of the magnetization dynamics triggered by\nstrong THz pulses is still missing.In this Letter, we present a combined experimental and\nnumerical study of the magnetization dynamics triggered\nby linearly polarized single-cycle THz pulses with peak\nelectric (magnetic) \felds up to 20 MV/m (67 mT). We\ninvestigate not only the fast time scales that are com-\nparable to the THz pulse duration ( \u00181 ps), but also\nthe nanosecond regime, where ferromagnetic resonance\n(FMR) oscillations are observed. Moreover, we write\nan explicit form of the Landau-Lifshitz-Gilbert (LLG)\nequation [21, 22] suitable to analyze terahertz-driven dy-\nnamics, that we use to predict yet-unexplored nonlin-\near magnetization dynamics regimes uniquely achievable\nwith this type of excitation mechanism.\nExperimental details. Room temperature experi-\nmental data is obtained from a time-resolved pump-\nprobe method utilizing the magneto-optical Kerr e\u000bect\n(MOKE), Refs. [23, 24]. A sketch of the experimental\nsetup is presented in Fig. 1 (a). Strong THz radiation is\ngenerated via optical recti\fcation of 4 mJ, 800 nm, 100\nfs pulses from a 1 kHz regenerative ampli\fer in a lithium\nniobate (LiNbO 3) crystal, utilizing the tilted-pulse-front\nmethod [25]. In contrary to the indirect (thermal) cou-\npling present in visible- and near-infrared light-matter in-\nteraction, THz radiation can directly couple to the spin\nsystem via magnetic dipole interaction (Zeeman inter-\naction) [26]. In this respect, a fundamental aspect is\nthe orientation of the THz polarization, which is con-\ntrolled using a set of two wire grid polarizers, one vari-\nably oriented at \u000645\u000eand a second one \fxed to +90\u000e\n(or -90\u000e) with respect to the original polarization direc-\ntion of ETHz. As depicted in Fig. 1 (b), the magnetic\n\feld component of the THz pulse HTHzis \fxed alongarXiv:1903.08395v2  [cond-mat.mes-hall]  25 Sep 20192\n(a)\n(e) (f)(b)\n+HTHz\n-HExtOUTWPLNO OAPM\nEM+SP P\nMPump\nProbe±45° ±90°\n0 2.5 5.0 7.5 10−100−50050100MOKE signal (a.u.)\n0 250 500 750−100−50050100\n+HExt\n-HExt\n0 2.5 5.0 7.5 10\nTime (ps)−100−50050MOKE signal (a.u.)\n0 250 500 750\nTime (ps)−100−50050\n+HTHz\n-HTHz\n(a)\n(c)\n(e)(d)\n(f)(b)\n+HTHz\n-HExtOUTWPLNO OAPM\nEM+SP P\nMPump\nProbe±45° ±90°\nFIG. 1. (Color online) (a) Schematic drawing of the exper-\nimental setup: BD - Balanced detection using two photodi-\nodes and a lock-in ampli\fer, WP - Wollaston prism, EM + S\n- Electromagnet with out-of-plane \feld and sample, OAPM\n- O\u000b-axis parabolic mirror, P - Wire grid polarizer, LNO -\nLithium niobate, and M - Mirror. (b) Sample geometry: The\nelectric \feld component of the THz pulse at the sample po-\nsition is oriented parallel to the y-axis direction, ETHzky,\nand the magnetic \feld component parallel to the x-axis direc-\ntion,HTHzkx. A static magnetic \feld HExtis applied along\nthez-axis direction. (c-f) Experimental MOKE data showing\nthe in\ruence of reversing the external magnetic \feld ( \u0006HExt,\nHTHz= const.) on the THz-induced demagnetization (c) and\non the FMR oscillations (d). The in\ruence of reversing the\nTHz magnetic \feld pulse ( \u0006HTHz,HExt.= const.) on the\nTHz-induced demagnetization and on the FMR oscillations is\nshown in (e) and (f), respectively. (The shaded green area is\na guide to the eye.)\nthex-axis direction and is therefore \ripped by 180\u000eby\nrotating the \frst polarizer. An amorphous CoFeB sample\n(Al2O3(1.8nm)/Co 40Fe40B20(5nm)/Al 2O3(10nm)/Si\nsubstrate) is placed either in the gap of a \u0006200 mT elec-\ntromagnet or on top of a 0.45 T permanent magnet. In\nboth cases, the orientation of the externally applied \feld\nHExtis along the z-axis direction, i.e. out of plane with\nrespect to the sample surface. However, a small compo-\nnent of this external bias \feld lying in the sample plane\nparallel to the y-axis direction has to be taken into ac-count, due to a systematic (but reproducible) small mis-\nalignment in positioning the sample. The THz pump\nbeam, with a spot size \u001f\u00191 mm (FWHM), and the 800\nnm probe beam, with a spot size \u001f\u0019200\u0016m (FWHM),\noverlap spatially on the sample surface in the center of\nthe electromagnet gap. Being close to normal incidence,\nthe MOKE signal is proportional to the out-of-plane com-\nponent M zof the magnetization, i.e. polar MOKE geom-\netry [27]. The probe beam re\rected from the sample sur-\nface is then analyzed using a Wollaston prism and two\nbalanced photo-diodes, following an all-optical detection\nscheme [4].\nResults and discussion. The experimental data demon-\nstrating THz-induced demagnetization and the magnetic\n\feld response of the spin dynamics is shown in Fig. 1\n(c-f) when \u00160HExt= 185 mT. For short timescales on\nthe order of the THz pump pulse, \u001c\u00181 ps, the polar\nMOKE is sensitive to the coherent response of the mag-\nnetization, i.e. its precession around the THz magnetic\n\feld, as shown in Fig. 1 (c)+(e). Within \u001c\u0018100 fs af-\nter time zero (t 0\u00185 ps) a sudden demagnetization step\nof the order of 0.1-0.2% of the total magnetization vec-\ntor is observed. The demagnetization step is followed by\na 'fast' relaxation process, \u001c\u00181 ps, and subsequently\nby a 'slow' recovery of the magnetization on a longer\ntimescale,\u001c\u0018100 ps. During and after magnetization\nrecovery, a relaxation precession (corresponding to the\nFMR) is superimposed, see Fig. 1 (d)+(f). The e\u000bect of\nreversing the externally applied magnetic \feld HExton\nthe MOKE measurements is illustrated in Fig. 1 (c)+(d).\nThis data shows that all the di\u000berent processes just iden-\nti\fed (demagnetization, coherent magnetization response\nin the range t 0<t.t0+ 2 ps, and FMR response)\nare indeed magnetic, as they all reverse their sign upon\nreversal of the sign of the bias magnetic \feld. Fig. 1\n(e)+(f) instead depict the e\u000bect of reversing the THz \feld\npolarity, while keeping the externally applied \feld con-\nstant. This data illustrate the symmetry of the magnetic\nresponse with respect to the THz magnetic \feld. The\ndemagnetization and FMR response, whose amplitude\nscales quadratically with the THz magnetic \feld HTHz,\nremain unchanged upon reversal of the terahertz mag-\nnetic \feld, while the coherent magnetization response,\nlinear in HTHz, reverses its sign.\nThe dependence of the THz-induced FMR response on\nthe magnitude of the magnetic bias \feld ( HExt) is shown\nin Fig. 2 (a). Oscillation amplitude and resonance fre-\nquency are summarized in Fig. 2 (b). The oscillation\namplitude is obtained from \ftting a damped sinusoidal\nfunction to the data shown in Fig. 2 (a), and the reso-\nnance frequency follows from subsequent Fourier trans-\nformation of that \ftting function. The relationship be-\ntween magnetic resonance \feld Hrand FMR frequency\nfFMR can be described by the phenomenological Kittel3\n0 200 400 600 800\nTime (ps)-1000100200300400500600700800MOKE signal (a.u.)2.0 2.2 2.4 2.6\nFrequency (GHz)0204060FMR amp. (a.u.)\n0102030405060\nMagnetic field (mT)02468Frequency (GHz)(a) (b)\n(c)\nFIG. 2. (Color online) Experimental data showing the mag-\nnetic \feld dependence of the FMR. (a) MOKE signal for\ndi\u000berent out-of-plane \felds \u00160HExt- 20, 50, 75, 100, 125,\n150, 175 and 185 mT in ascending order. The solid line is a\ndamped sine function \ftted to the experimental data. Both\n\ft and data curve are displaced on the y axis by a constant\no\u000bset. (b) Fourier transformation of the data shown in (a)\nvs. FMR amplitude. (c) FMR frequency as a function of the\nin-plane component of HExt. The solid line corresponds to\nKittel equation, Eq. (1).\nequation [28]\nfFMR =\r\u00160\n2\u0019p\nHr(Hr+Me\u000b) (1)\nwith gyromagnetic ratio \rand e\u000bective magnetization\nMe\u000b=MS\u0000H?\nK, where H?\nKis the perpendicular\nanisotropy \feld. From room-temperature magnetization\nmeasurements (see the Supplemental Material), a satu-\nration magnetization \u00160MS= 1.84 T and an anisotropy\n\feld\u00160H?\nK= 0.76 T are obtained. The value of H rrep-\nresents the small in-plane component of HExt(i.e. along\nthey-axis direction), which can be inferred from the mea-\nsurements in Fig. 2 (a) in order to get good agreement\nwith the experimental data, as shown in Fig. 2 (c).\nNumerical modelling. A detailed picture of the THz-\ninduced demagnetization and FMR response for an exter-\nnal magnetic \feld generated by a 0.45 T permanent mag-\nnet and a THz pulse of E THz= 18 MV/m ( \u00160HTHz\u001860\nmT) is presented in Fig. 3. Experimental data has been\nrecorded both on a short timescale after arrival of the\npump pulse showing coherent precession and demagneti-\nzation, Fig. 3 (a), and on long timescales showing mag-\nnetization recovery and FMR at a frequency of about 7.2\nGHz, Fig. 3 (b). By knowing the FMR frequency, the\nactual applied magnetic \feld can be deduced from the\ndata shown in Fig. 2 (a) and Eq. (1), as described in the\nprevious paragraph. Following this approach, we \fnd a\nvalue of 450 mT for the out-of-plane component and 56\nmT for the in-plane component of HExt, corresponding\nto the right-most point in Fig. 2 (c).\nThis complete set of good-quality data has been used\n(b)\n012345678\nTime (ps)53.853.954.054.154.254.354.4Mz/M 0(%)\n0 200 400 600 800\nTime (ps)(a) (b)FIG. 3. (Color online) Comparison of experimental data\n(open markers) and macrospin simulations (solid lines). (a)\nTHz-induced demagnetization for an external applied \feld of\n\u00160HExt\u0018450 mT and a THz pulse of E THz= 18 MV/m,\n(\u00160HTHz\u001860 mT). The blue markers/lines correspond to\n+HTHz, and the orange markers/lines to \u0000HTHz. (b) Exper-\nimental data and macrospin simulations under similar condi-\ntions as (a) showing FMR on a longer timescale.\nfor validating all the assumptions considered in our\nmacrospin simulations, whose results are shown as solid\nlines in Fig. 3. Indeed, magnetization dynamics of a uni-\nform ferromagnetic material can be modeled and under-\nstood from macrospin simulations where the description\nof the magnetic state is simpli\fed by a single-domain ap-\nproximation. The phenomenological LLG equation [22]\ncan be used to obtain a \frst approximate description of\nthe magnetization continuum precession and relaxation.\nSince the LLG equation preserves the length of the mag-\nnetization vector ( jMj= const.) [29], it does not account\nfor laser-induced demagnetization and it ignores relevant\nphysical phenomena such as scattering e\u000bects [11]. A\npossibility to account for optically- or thermally-induced\ndemagnetization is the use of the Landau-Lifshitz-Bloch\n(LLB) equation [30], which has been shown to describe\nultrafast demagnetization processes [31]. However, the\nparameters involved in the LLB equation depend on\ntemperature, and a description of their temporal evo-\nlution due to the interaction with ultrafast pulses is\nneeded. Such a description is often provided using a two-\ntemperature model [8, 31], which then has to be coupled\nto the LLB equation. However, such a two-temperature\nmodel relies on several phenomenological material pa-\nrameters, and little insight on the physics is gained by\nthis approach, in particular at longer time scales. Since\nhere we are interested in these time scales, we simplify\nthe problem by just assuming a phenomenological LLG\nequation with non-constant magnetization magnitude, a\nphysical quantity which can be readily measured.\nOur version of the LLG equation reads (see the Sup-4\nplemental Material)\ndM\ndt=\u0000\r0\u0002\nM\u0002He+\u000b\nMS(M\u0002(M\u0002He))\u0003\n\u0000c(M;He)M;\n(2)\nwith\r0=\r=(1 +\u000b2), where the gyromagnetic ratio is\nj\r=2\u0019j \u001928.025 GHz/T, and M= Mm, where M( t)\nis the magnetization amplitude and m(t) the unit vec-\ntor. The e\u000bective magnetic \feld Heincludes the exter-\nnal \feld HExt, THz \feld HTHzand demagnetization \feld\nHD. The Gilbert damping \u000b= 0:01 was independently\nmeasured with conventional FMR spectroscopy, as shown\nin the Supplemental Material. In Eq. (2), the \frst and\nsecond terms on the right-hand side describe coherent\nspin precession and the macroscopic spin relaxation, re-\nspectively, whereas the third term describes the fast de-\nmagnetization along the mdirection controlled by the\nfunctionc(M;He). A non-constant magnetization mag-\nnitude is also needed for reproducing the observed FMR\noscillation, as discussed in the Supplemental Material.\nHence, we propose a semi-empirical approach to iden-\ntifyc(M;He) and, consequently, the time evolution of\nthe magnetization vector length M( t) from experimental\ndemagnetization data. For this purpose, the change in\nM(t) as a function of time is calculated from the cumu-\nlative integral of the incident THz pulse [16]:\n\u0001M(t)/e\u0000t=\u001cRZt\n\u00001HTHz(\u0018)2d\u0018; (3)\nwhere the exponential term describes the recovery of the\nmagnetization on a timescale of \u0018100 ps for CoFeB. The\nshape of the incident THz pulse (in the time domain)\nis proportional to ETHz(t), which can be measured via\nelectro-optical sampling in GaP [32]. We notice that the\nTHz magnetic \feld HTHz(t) inside the material, to be\nused in Eq. (3), can be derived from the measured free-\nspace ETHz(t) after taking into account the whole sample\nstack, e.g. by means of the transfer matrix method [33].\nEventually, the function c(M;He) is obtained from Eq.\n(3) after re-scaling it with experimentally obtained de-\nmagnetization values (see Ref. [16] for more details).\nAfter determining c(M;He), all the terms in Eq. (2)\nare known, and the equation can be used to compute\nthe dynamics of the magnetization orientation in terms\nof the spherical angles \u0012(t); \u001e(t). Indeed, Eq. (2) is nu-\nmerically solved in spherical coordinates (M ;\u0012;\u001e) (see\nthe Supplemental Material for details) using a custom-\nmade Python solver based on the 4th-order Runge-Kutta\nmethod, which provides a simple implementation and\ngood accuracy of the numerical solution [29].\nIt is now interesting to explore the expected response of\nthe magnetization to THz \felds with strength larger than\nthe ones considered in this work, but nowadays accessi-\nble with table-top sources. For simplicity and generality,\nonly the free-space value of the THz peak electric \feld is\nreported in the below discussion. In case of THz \felds\n0 250 500 750 1000\nTime (ps)-0.4-0.200.2∆M y,z(%)\n0 5 10 15 20\nFrequency (GHz)0246Amplitude (a.u.)×10−9\nMz\nMy\n0 250 500 750 1000\nTime (ps)-40-20020∆M y,z(%)\n0 5 10 15 20\nFrequency (GHz)0246Amplitude (a.u.)×10−7\nMz\nMy(a)\n(c)(b)\n(d)FIG. 4. (Color online) Comparison of macrospin simulation\ndata for moderate (20 MV/m) and high (200 MV/m) THz-\npeak-\feld stimulus. The external magnetic \feld is set to\n\u00160HExt= 450 mT, i.e. replicating the experimental condi-\ntions from Fig. 3. Panel (a) shows the relative change of the\nMy(blue line) and M z(light blue line) magnetization com-\nponent for a E THzpeak \feld of 20 MV/m and (b) shows the\ncorresponding Fourier spectrum. (c) and (d) show the simu-\nlation data and corresponding Fourier spectrum for an E THz\npeak \feld of 200 MV/m, respectively.\nsmaller than E THz\u0018100 MV=m, the demagnetization\nfollows the square of the amplitude of the THz \feld, see\nRef. [16]. Lacking detailed experimental data, it is rea-\nsonable to assume that THz-induced demagnetization for\nhigher THz peak \felds (E THz>100 MV=m) can be de-\nscribed by the positive section of an error function, allow-\ning for a quadratic behavior for small demagnetization\nand a saturation for large demagnetization approaching\n100% [17]. From our experimental data, we derive a func-\ntional description of the demagnetization as a function of\nthe THz \feld such as Demag = f(E) = erf( A\u0001E2), with\nTHz peak \feld E and \ftting parameter A\u00196:0\u000110\u00006\nm2V\u00002. (See the Supplemental Material for further de-\ntails.)\nWith this assumption, the macrospin simulation re-\nsults for THz \felds E THz= 20 MV=m and E THz= 200\nMV=m are presented in Fig. 4 (a-b) and Fig. 4 (c-d),\nrespectively. For E THz = 200 MV =m, a clear nonlin-\near response of the magnetization to the THz \feld is\nfound, illustrated by the second harmonic oscillation in\nthe Mycomponent of the magnetization. The simulated\nTHz-induced demagnetization for E THz= 200 MV=m is\non the order of \u0001M z\u001820%. In Fig. 4 (b)+(d), the\nFourier spectrum of the FMR oscillation for the M yand\nMzcomponents of the magnetization at THz pump peak\n\felds of 20 MV/m and 200 MV/m are depicted. The\nFourier data of the 200 MV/m simulation shown in Fig. 4\n(c) clearly shows a second harmonic peak at \u001814 GHz,\npresent for M ybut not for M z. A similar behavior was5\nobserved recently by performing FMR spectroscopy of\nthin \flms irradiated with femtosecond optical pulses in-\nducing either ultrafast demagnetization [34], by exciting\nacoustic waves [35], and by two-dimensional THz mag-\nnetic resonance spectroscopy of antiferromagnets [36]. In\nour case, the high-harmonic generation process is solely\ndriven by the large amplitude of the terahertz magnetic\n\feld that is completely o\u000b-resonant with the uniform pre-\ncession mode. This would allow for exploring purely mag-\nnetic dynamics in regimes that are not accessible with\nconventional FMR spectroscopic techniques, where high-\namplitude dynamics are prevented by the occurrence of\nso-called Suhl's instabilities, i.e. non-uniform excitations\ndegenerate in energy with the uniform mode. Such non-\nresonant, high THz magnetic \felds are within the capa-\nbilities of recently developed table-top THz sources [37],\nand can also be generated in the near-\feld using meta-\nmaterial structures as described by Refs. [38{40].\nIn summary, we investigated magnetization dynam-\nics induced by moderate THz electromagnetic \felds in\namorphous CoFeB, in particular the ferromagnetic reso-\nnance response as a function of applied bias and THz\nmagnetic \felds. We demonstrate that semi-empirical\nmacrospin simulations, i.e. solving the Landau-Lifshitz-\nGilbert equation with a non-constant magnitude of the\nmagnetization vector to incorporate THz-induced de-\nmagnetization e\u000bect, are able to describe all the details\nof the experimental results to a good accuracy. Exist-\ning models of terahertz spin dynamics and spin pumping\nwould need to be extended to include the evidence pre-\nsented here [41, 42]. Starting from simulations describ-\ning experimental data for THz-induced demagnetization,\nwe extrapolate that THz \felds one order of magnitude\nlarger drive the magnetization into a nonlinear regime.\nIndeed, macrospin simulations with THz \felds on the or-\nder E THz\u0018200 MV=m (\u00160HTHz\u0018670 mT) predict a sig-\nni\fcant demagnetization of \u0001M z\u001820%, and a marked\nnonlinear behavior, apparent from second harmonic gen-\neration of the uniform precessional mode. We anticipate\nthat our results will stimulate further theoretical and ex-\nperimental investigations of nonlinear spin dynamics in\nthe ultrafast regime.\nM.H. gratefully acknowledges support from the\nSwedish Research Council grant E0635001, and the\nMarie Sk lodowska Curie Actions, Cofund, Project INCA\n600398s. The work of M.d'A. was carried out within the\nProgram for the Support of Individual Research 2017\nby University of Naples Parthenope. M.C.H. is sup-\nported by the U.S. Department of Energy, O\u000ece of Sci-\nence, O\u000ece of Basic Energy Sciences, under Award No.\n2015-SLAC-100238-Funding. M.P. and S.B. acknowledge\nsupport from the European Research Council, Starting\nGrant 715452 \\MAGNETIC-SPEED-LIMIT\".\u0003stefano.bonetti@fysik.su.se\n[1] M. Faraday, Philosophical Transactions of the Royal So-\nciety of London 136, 1 (1846).\n[2] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y.\nBigot, Phys. Rev. Lett. 76, 4250 (1996).\n[3] B. Koopmans, M. Van Kampen, J. T. 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Lett. 118, 257202 (2017)."    },    {        "title": "1909.04205v1.Are_gravitating_magnetic_monopoles_stable_.pdf",        "content": "Are gravitating magnetic monopoles stable?\nBen Kain\nDepartment of Physics, College of the Holy Cross, Worcester, Massachusetts 01610, USA\nThe gravitating Julia-Zee dyon is a particle-like solution with both electric and magnetic charge.\nIt is found in the Einstein-Yang-Mills-Higgs system of SU(2) with a scalar \feld in the adjoint\nrepresentation coupled to gravity. Within the magnetic ansatz this system is reduced from describing\ndyons to describing the gravitating 't Hooft-Polyakov magnetic monopole. The stability of the\nwell-known static gravitating magnetic monopole solutions with respect to perturbations within\nthe magnetic ansatz|so-called magnetic perturbations|is well studied, but their stability with\nrespect to perturbations outside the magnetic ansatz|so-called sphaleronic perturbations|is not.\nI undertake a purely numerical study by adding sphaleronic perturbations to gravitating magnetic\nmonopole solutions and then dynamically evolving the system. For large perturbations I \fnd that\nthe system heads toward a dyon con\fguration, as expected. For su\u000eciently small perturbations,\nhowever, the system oscillates about the magnetic ansatz in a manner consistent with oscillations\nabout a stable equilibrium.\nI. INTRODUCTION\nSU(2), when spontaneously broken by a real\ntriplet scalar \feld, has as a classical solution the\nJulia-Zee dyon [1], a spherically symmetric particle-\nlike solution with both electric and magnetic charge.\nWithin the magnetic ansatz, a physical constraint\nwhich sets the electric charge of the U(1) subgroup\nto zero, the theory no longer describes dyons and\nhas as a classical solution the 't Hooft-Polyakov\nmonopole [2, 3], a spherically symmetric particle-like\nsolution with only magnetic charge. When coupled\nto gravity, the system has regular and black hole\nstatic dyon solutions [4, 5] and, within the magnetic\nansatz, regular and black hole static monopole solu-\ntions [6{10].\nThe magnetic ansatz, which plays a central role\nin this work, is self-consistent, in that an evolution\nthat begins within the magnetic ansatz stays within\nthe magnetic ansatz. As I explain below, it is imple-\nmented by setting a certain group of \felds to zero.\nThus, if an evolution begins with the relevant \felds\nset to zero, these \felds stay zero throughout the evo-\nlution.\nThe stability of the static gravitating magnetic\nmonopole solutions has been studied in some detail,\nbut only with respect to magnetic perturbations,\nwhich are perturbations within the magnetic ansatz,\nwhere there is little question that stability exists in\na large area of parameter space [11{16]. This means\nthat, in this area of parameter space, a dynamic evo-\nlution that begins with initial data within the mag-\nnetic ansatz will settle down to a static monopole\ncon\fguration and not, say, disperse all matter \felds\nto in\fnity [16].\nIn addition to magnetic perturbations, there\nare sphaleronic perturbations, which are perturba-\ntions to the magnetic ansatz itself. As far as I\nam aware, sphaleronic perturbations to the staticmonopole solutions have not yet been studied|\npresumably because it is very di\u000ecult to do so an-\nalytically (or semianalytically)|and, consequently,\nit is an open question whether gravitating mag-\nnetic monopoles are stable. To avoid the di\u000eculties\nin a (semi)analytical stability analysis, I undertake\na purely numerical study by adding a sphaleronic\nperturbation to gravitating monopole solutions and\nthen dynamically evolving the system. Performing\nthe necessary evolutions requires code that can dy-\nnamically evolve the full gravitating dyon system.\nAs far as I am aware, this is the \frst time the grav-\nitating dyon has been dynamically solved.\nFor relatively large perturbations, the system ap-\npears to relax toward a dyon con\fguration. As the\nsize of the perturbation is made smaller, the electric\ncharge in the system decreases and the end states\nof the evolutions move toward the monopole. For\nsu\u000eciently small perturbations, the electric charge\ndensity oscillates about zero in a manner suggestive\nof oscillations about a stable equilibrium. This in\nturn is suggestive of the gravitating monopole be-\ning stable with respect to sphaleronic perturbations\nand, hence, of the static gravitating monopole so-\nlutions being stable with respect to both magnetic\nand sphaleronic perturbations.\nThe dynamic evolution of systems related to the\ngravitating dyon system studied here has a rich his-\ntory. Choptuik et al. [17, 18] dynamically evolved\npureSU(2) (i.e. unbroken and without a scalar \feld)\nin their study of black hole critical phenomena. This\nwas further studied in the same system by a num-\nber of authors [19{21], as were tails and other topics\n[19, 22{26]. Millward and Hirschmann [27] studied\ncritical phenomena in SU(2) with a scalar \feld in\nthe fundamental representation. Sakai [28] was the\n\frst to dynamically evolve the gravitating monopole\nand was interested in what happens when the scalar\n\feld vacuum value is near its upper limit. I re-arXiv:1909.04205v1  [gr-qc]  10 Sep 2019cently evolved the monopole system in a study of\ntype III critical phenomena and stability with re-\nspect to magnetic perturbations [16] and in a study\nof type II critical phenomena [29]. Finally, Gund-\nlach, Baumgarte, and Hilditch made a related type\nII study in a system with a scalar \feld and an SU(2)\nYang-Mills \feld, but with only gravitational inter-\nactions [30]. With the important exception of the\nwork of Rinne et al. in [21, 24], all of these papers\nworked within the magnetic ansatz. Thus, there has\nbeen limited dynamical study of SU(2) outside the\nmagnetic ansatz.\nIn the next section, I present the equations that\ndescribe the time-dependent gravitating Julia-Zee\ndyon and discuss gauge choices, the magnetic ansatz,\nand boundary conditions. In Sec. III I discuss nu-\nmerics. In Sec. IV I study the stability of gravitat-\ning monopoles with respect to sphaleronic perturba-\ntions. I conclude in Sec. V.\nII. EQUATIONS, GAUGES, THE\nMAGNETIC ANSATZ, AND BOUNDARY\nCONDITIONS\nIn this section I give the equations which describe\nthe gravitating dyon system. I gave many (though\nnot all) of these equations in [16], to which I refer the\nreader for additional information. After presenting\nthe equations, I discuss gauge choices for the matter\nsector, the magnetic ansatz, and boundary condi-\ntions.\nA. Metric equations\nMy study of monopoles and dyons is restricted to\nspherical symmetry. The general spherically sym-\nmetric metric in the Arnowitt-Deser-Misner (ADM)\nformalism [31, 32] is\nds2=\u0000\u0000\n\u000b2\u0000a2\f2\u0001\ndt2+ 2a2\fdrdt +a2dr2\n+Br2\u0000\nd\u00122+ sin2\u0012d\u001e2\u0001\n; (1)\nwhere the metric functions \u000b,\f,a, andBare func-\ntions oftandronly and I use units such that c= 1\nthroughout. These four functions obey the Einstein\n\feld equations,\nG\u0016\u0017= 8\u0019GT\u0016\u0017; (2)\nwhereT\u0016\u0017is the energy-momentum tensor.\nThe code I use to dynamically evolve the system\nuses radial-polar spacetime gauge. This gauge has\nthe bene\ft of simplifying equations by setting B= 1\nand\f= 0.aand\u000b, the only metric functions to besolved for then, obey the constraint equations\na0\na= 4\u0019Gra2\u001a\u0000a2\u00001\n2r\n\u000b0\n\u000b= 4\u0019Gra2Sr\nr+a2\u00001\n2r;(3)\nwhich follow from the Einstein \feld equations. In\n(3), primes denote rderivatives and \u001aandSr\nrcome\nfrom the energy-momentum tensor and are given be-\nlow. Although the code I use only makes use of the\nmetric functions aand\u000b, in the following I give the\ngeneral form of equations for completeness.\nB. Matter equations\nThe matter content of the 't Hooft-Polyakov\nmonopole and the Julia-Zee dyon is an SU(2) Yang-\nMills-Higgs theory, with gauge \feld Aa\n\u0016and real\nscalar \feld \u001eain the adjoint representation, where\na= 1;2;3 is the gauge index (which can equiva-\nlently be placed up or down). For SU(2) the gener-\nators satisfy [ Ta;Tb] =i\u000fabcTc, where\u000fabcis the com-\npletely antisymmetric symbol with \u000f123= 1. In the\nadjoint representation I de\fne the components of the\ngenerator matrices as ( Ta)bc=\u0000i\u000fabcwith normal-\nization Tr(TaTb) = 2\u000eab, where Tr here and below\nindicates a trace over generator matrices. De\fning\n\u001e\u0011Ta\u001ea; A\u0016\u0011TaAa\n\u0016; F\u0016\u0017\u0011TaFa\n\u0016\u0017;(4)\nwhere a sum over repeated gauge indices is implied\nandFa\n\u0016\u0017is the \feld strength, the Yang-Mills-Higgs\nLagrangian is\nLYMH =\u00001\n2Tr [(D\u0016\u001e) (D\u0016\u001e)]\u0000V+LSU(2);(5)\nwhere\nLSU(2)=\u00001\n8g2Tr (F\u0016\u0017F\u0016\u0017); (6)\ngis the gauge coupling constant,\nD\u0016\u001e=r\u0016\u001e\u0000i[A\u0016;\u001e]\nF\u0016\u0017=r\u0016A\u0017\u0000r\u0017A\u0016\u0000i[A\u0016;A\u0017];(7)\nandVis the scalar potential, whose form I give be-\nlow.\nSpherical symmetry constrains the \felds. The\ngeneral spherically symmetric SU(2) gauge \feld\ntakes the form [33{35]\nAt=T3ut\nAr=T3ur\nA\u0012=T1w2+T2w1\nA\u001e=\u0000\n\u0000T1w1+T2w2+T3cot\u0012\u0001\nsin\u0012;(8)\n2whereut,ur,w1, andw2parametrize the gauge \feld\nand are functions of tandronly, and the real triplet\nscalar \feld takes the form\n\u001e='p\n2T3; (9)\nwhere'is a canonically normalized real scalar \feld\nand is a function of tandronly. The components\nof the spherically symmetric \feld strength can be\nfound, for example, in [16, 18].\nWitten showed that spherical symmetry breaks\nSU(2) down to U(1) [33]. This can be shown ex-\nplicitly by writing the pure SU(2) Lagrangian (6),\nwith gauge \feld (8), as a Lagrangian for a complex\nscalar \feld gauged under U(1):\nLSU(2)=\u00002\ng2Br2(D\u0016w)(D\u0016w)\u0003(10)\n\u00001\n2g2B2r4(1\u0000jwj2)2\u00001\n4g2f\u0016\u0017f\u0016\u0017;\nwherew=w1+iw2,\nD\u0016w=r\u0016w\u0000ia\u0016w\na\u0016= (ut;ur;0;0)\nf\u0016\u0017=r\u0016a\u0017\u0000r\u0017a\u0016:(11)\nI thus \fnd that wacts as the complex \\scalar\" \feld\ngauged under U(1), but with noncanonical kinetic\nterms and an atypical \\scalar\" potential. The SU(2)\nLagrangian (10) is clearly invariant under a U(1)\ngauge transformation,\nui!u0\ni=ui\u0000ri\u001c; w!w0=we\u0000i\u001c;(12)\nwherei=t;rand\u001cis the gauge parameter. Since\nthe spherically symmetric kinetic term for the actual\nscalar \feld is\n\u00001\n2Tr [(D\u0016\u001e)(D\u0016\u001e)] =\u0000@\u0016'@\u0016'\u00002\nBr2jwj2'2;\n(13)\nwe \fnd that with 'invariant under the U(1) trans-\nformation the complete theory has a U(1) symmetry.\nThis symmetry will be made use of when I \fx the\ngauge below.\nThe scalar potential for the monopole and dyon is\nV=\u0015\n4\u0000\n'2\u0000v2\u00012; (14)\nwhere\u0015is the self-coupling constant and vis the vac-\nuum value of '. This scalar potential spontaneously\nbreaksSU(2) down to U(1) giving rise to massive\nvector bosons and a massive scalar \feld with masses\nmV=gv; m S=p\n2\u0015v: (15)\nI gave the equations of motion which fol-\nlow from the Einstein-Yang-Mills-Higgs LagrangianLEYMH =p\u0000gLYMH in [16], which I repeat here.\nFor numerical purposes it is important to have the\nequations of motion in \frst-order form. I thus de\fne\n\b\u0011'0\u0005\u0011aB\n\u000b( _'\u0000\f\b)\nQ1\u0011w0\n1+urw2P1\u0011a\n\u000b\u0010\n_w1+utw2\u0000\fQ1\u0011\nQ2\u0011w0\n2\u0000urw1P2\u0011a\n\u000b\u0010\n_w2\u0000utw1\u0000\fQ2\u0011\nY\u0011Br2\n2\u000ba( _ur\u0000u0\nt): (16)\nI list the equations of motion grouped into families,\nusing a dot to denote tderivatives. First ', \b, and\n\u0005:\n_'=\u000b\naB\u0005 +\f\b\n_\b =@r\u0010\u000b\naB\u0005 +\f\b\u0011\n_\u0005 =1\nr2@r\u0012\u000bBr2\na\b +r2\f\u0005\u0013\n\u0000\u000baB@V\n@'\n\u00002\u000ba\nr2(w2\n1+w2\n2)'; (17)\nthenw1,Q1, andP1:\n_w1=\u000b\naP1\u0000utw2+\fQ1\n_Q1=@r\u0010\u000b\naP1+\fQ1\u0011\n\u0000utQ2+ur\u0010\u000b\naP2+\fQ2\u0011\n+w22\u000ba\nBr2Y\n_P1=@r\u0010\u000b\naQ1+\fP1\u0011\n\u0000P2(ut\u0000\fur) +\u000b\naurQ2\n+\u000ba\nBr2w1(1\u0000w2\n1\u0000w2\n2)\u0000g2\u000baw 1'2;(18)\nandw2,Q2, andP2:\n_w2=\u000b\naP2+utw1+\fQ2\n_Q2=@r\u0010\u000b\naP2+\fQ2\u0011\n+utQ1\u0000ur\u0010\u000b\naP1+\fQ1\u0011\n\u0000w12\u000ba\nBr2Y\n_P2=@r\u0010\u000b\naQ2+\fP2\u0011\n+P1(ut\u0000\fur)\u0000\u000b\naurQ1\n+\u000ba\nBr2w2(1\u0000w2\n1\u0000w2\n2)\u0000g2\u000baw 2'2;(19)\nand \fnally urandY:\n_ur=2\u000ba\nBr2Y+u0\nt (20)\n_Y=\u000b\na(w1Q2\u0000w2Q1) +\f(w1P2\u0000w2P1):\nNote that I do not have an evolution equation for ut,\nwhich I will handle when \fxing the gauge. There ex-\nists one \fnal equation, which is the Gauss constraint:\nY0=w1P2\u0000w2P1: (21)\n3We shall see below that Yis proportional to the total\nelectric charge inside a sphere of radius rand thus\nY0=@Y=@r is proportional to the radial electric\ncharge density.\nIn [16] I gave the energy-momentum tensor thatfollows from the Lagrangian in (5), including each of\nits nonvanishing components and a number of com-\nmonly used matter functions which follow from it.\nHere I repeat only the matter functions used in (3):\n\u001a=1\n2a2\u0012\n\b2+\u00052\nB2\u0013\n+(w2\n1+w2\n2)'2\nBr2+V+(1\u0000w2\n1\u0000w2\n2)2\n2g2B2r4+Q2\n1+Q2\n2+P2\n1+P2\n2\ng2a2Br2+2Y2\ng2B2r4\nSr\nr=1\n2a2\u0012\n\b2+\u00052\nB2\u0013\n\u0000(w2\n1+w2\n2)'2\nBr2\u0000V\u0000(1\u0000w2\n1\u0000w2\n2)2\n2g2B2r4+Q2\n1+Q2\n2+P2\n1+P2\n2\ng2a2Br2\u00002Y2\ng2B2r4:(22)\nC. Electric charge and mass\nThe electric charge is found with the help of the\nconserved electric current, j\u0016, which follows from the\ninhomogeneous Maxwell equation,\nr\u0016f\u0016\u0017=gj\u0017; (23)\nwhere the factor of gis included because of my con-\nvention for the U(1) gauge \feld in (10) and (11).\nThatr\u0016j\u0016= 0, and hence that j\u0016is conserved,\nfollows immediately from f\u0016\u0017being antisymmetric.\nThe components of the current work out to be\njt=2Y0\ng\u000baBr2; jr=\u00002_Y\ng\u000baBr2: (24)\nThe total charge enclosed in a sphere of radius ris\ngiven by [36]\nq(t;r) =Zp\r(\u0000n\u0016j\u0016)drd\u0012d\u001e =8\u0019\ngZr\n0Y0dr\n=8\u0019\ngY(t;r); (25)\nwheren\u0016= (\u0000\u000b;0;0;0) is the time-like unit vector\nnormal to the spatial slices, \u0000n\u0016j\u0016is the electric\ncharge density, and \r=a2B2r4sin2\u0012is the deter-\nminant of the spatial metric. I explain below that\nthe \fniteness of the energy density at the origin re-\nquiresY(t;0) = 0, which allows for the evaluation\nof the limits above. The total charge in the sys-\ntem,q1\u0011q(t;1), is a conserved quantity. As\npromised,Yis proportional to the total electric\ncharge inside a sphere of radius r. I note that qis\nrelated to the radial component of the electric \feld,\nEr=\u0000fr\u0016n\u0016=g=q=(4\u0019aBr2), where the factor\nofgin the de\fnition for Eragain follows from my\nconvention for the U(1) gauge \feld in (10) and (11).\nA convenient form for the mass function can be\nmotivated by looking at the static solution in thelargerlimit. De\fning for convenience the function\nMas\n1\ngrr=1\na2\u00111\u00002GM\nr; (26)\nfrom which GM= (r=2)(1\u00001=a2), I have\nM0= 4\u0019r2\u001a; (27)\nwhere I used the a0equation in (3). I explain be-\nlow that the outer boundary conditions at r=1\nare'=\u0006vandw1=w2= 0, for which, in the\nstatic limit, the energy density, \u001ain (22), reduces\nsigni\fcantly and\nM0=2\u0019\ng2r2+q2\n8\u0019r2: (28)\nIn the large rlimitq!q1and is constant, allowing\nthe above equation to be easily integrated and we\nhave\n1\ngrr=1\na2= 1\u00002GM\nr+4\u0019G(1=g)2\nr2+Gq2\n1=4\u0019\nr2;\n(29)\nwhereMis the ADM mass. This is the Reissner-\nNordstr om solution with unit magnetic charge (in\nunits ofg) and electric charge q1. This solution\nmotivates de\fning the mass function as\nm(t;r) =r\n2G\u0014\n1\u00001\na2(t;r)+4\u0019G\ng2r2+Gq2(t;r)\n4\u0019r2\u0015\n;\n(30)\nwhose asymptotic value gives the ADM mass M=\nm(t;1).\nD. Matter gauges\nThe matter sector obeys the gauge transforma-\ntion (12) and it will be useful to \fx this gauge. In\nthis subsection I comment on a few gauge choices.\n4One choice is temporal gauge, which \fxes ut= 0\nand immediately solves the problem that there is no\nevolution equation for ut. In some gauges, static\nsolutions|in which gauge-invariant \felds, such as\njwj=p\nw2\n1+w2\n2andY, are time independent|\nhave gauge-dependent \felds, such as ur,w1, andw2,\nwhich retain a time dependence. Temporal gauge is\nperhaps the easiest gauge in which to see that this\nis the case. Y, being proportional to the total elec-\ntric charge inside a sphere of radius r, must be time\nindependent and nonzero for a static dyon. A look\nat its de\fnition in (16) shows that with ut= 0, _ur\nmust be nonzero.\nAnother possibility is radial gauge, which \fxes\nur= 0 and the urevolution equation in (20) reduces\nto an ODE for ut. Rinne et al. used radial gauge in\ntheir dynamical study of pure SU(2) [21, 24]. Radial\ngauge is the best choice for \fnding static solutions\ndirectly, since, for static solutions, all \felds are time\nindependent and one can additionally \fx w2= 0.\nThe \fnal gauge I mention is Lorenz gauge, which\nintroduces an evolution equation for utthrough the\nLorenz gauge condition, r\u0016a\u0016= 0. As with tem-\nporal gauge, some gauge-dependent \felds in Lorenz\ngauge retain a time dependence for static solutions.\nI use Lorenz gauge in this work because, for the nu-\nmerical scheme I am using, I found Lorenz gauge to\nbe the most stable. Introducing the auxiliary \feld\n\n\u0011aB\n\u000b(ut\u0000\fur); (31)\nthe Lorenz gauge condition can be written as\nut=\u000b\naB\n +\fur;_\n =1\nr2@r\u0014\nr2\u0012\u000bB\naur+\f\n\u0013\u0015\n:\n(32)\nE. Magnetic ansatz\nThe magnetic ansatz is a physical constraint on\nthe theory (and not a gauge choice) which sets the\nelectric charge of the Abelian subgroup to zero and\nreduces the dyon to the monopole. I take its de\fni-\ntion to be\nY0(t;r) = 0; (33)\nsinceY0is proportional to the electric charge den-\nsity.1Once the magnetic ansatz is made, convenient\n1It is easy to see that if Y0(t; r) = 0, then physically it must\nalso be that Y(t; r) = 0, since Y(t; r) is proportional to the\ntotal electric charge inside a sphere of radius r.Y(t; r) = 0\nis a common way of expressing the magnetic ansatz.gauge choices (see, for example, [16, 18] for details)\nsetut=ur=w2= 0 and the only nonvanishing\nmatter \felds are 'andw1.\nThe magnetic ansatz is self-consistent, in that\nan evolution that begins with initial data within\nthe magnetic ansatz remains within the magnetic\nansatz. That this is so is a big reason why nearly all\ndynamical gravitational studies of SU(2) have been\ndone within the magnetic ansatz (the only excep-\ntions I am aware of are [21, 24]). To be speci\fc, an\nevolution with initial data that has Y0=ut=ur=\nw2= 0 everywhere, keeps Y0=ut=ur=w2= 0\neverywhere. An immediate consequence is that an\nevolution that begins with the gravitating monopole,\nstays with the gravitating monopole.\nIt is an open question whether the magnetic\nansatz is stable in the gravitating monopole system\nand hence whether gravitating monopoles are stable.\nI study this issue numerically in Sec. IV.\nF. Boundary conditions\nTo solve the system of equations I need bound-\nary conditions for many of the variables. Boundary\nconditions include both conditions at the boundary\nof space and the boundary of the computational do-\nmain. I list a number of boundary conditions in this\nsubsection and discuss the outer boundary of the\ncomputational domain in the next section.\nThe inner boundary condition for aisa(t;0) = 1,\nwhich is the \rat space value ahas when inside a\nspherically symmetric matter distribution and fol-\nlows from \fniteness of the top equation in (3). As\ncan be seen from the bottom equation in (3), any\nsolution for \u000bcan be scaled by a constant and still\nbe a solution. I set \u000b(t;r) = 1=a(t;r) at larger, a\nchoice motivated by the spacetime being asymptoti-\ncally Reissner-N ordstrom. I take the parity of aand\n\u000bto be even near the origin.\nSome boundary conditions for matter functions\nfollow from the energy density, \u001ain (22), being \fnite\nat the origin and r2\u001avanishing as r!1 so that the\ntotal integrated energy is \fnite. At the inner bound-\nary I have'=O(r),jwj2=w2\n1+w2\n2= 1+O(r2), and\nY=O(r2). Additional inner boundary conditions\ncan be found by solving the equations of motion af-\nter expanding them around the origin. I \fnd\nw1= cos\u0012w(t) +O(r2); w 2= sin\u0012w(t) +O(r2);\n(34)\nwhere I have introduced the angle \u0012was a\nparametrization of w2\n1+w2\n2= 1 +O(r2), and\nut=_\u0012w(t) +O(r2); ur=O(r): (35)\nI note in particular that the equation for utis the\nsolution to the Gauss constraint in (21). It is easy\n5to see that the equation for utmay also be written\nasut=\u0000_w1=w2+O(r2) andut= _w2=w1+O(r2).\nThese two forms are precisely what is needed for\nthe (P2\n1+P2\n2)=r2term in the energy density to be\n\fnite at the origin. At r=1I have'=\u0006vand\nw1=w2= 0. I take the parity of the matter \felds\nto be \b,w1,w2,P1,P2,ut, and \n are even and ',\n\u0005,Q1,Q2,ur, andYare odd near the origin.\nIII. NUMERICS\nIn this section I describe numerical aspects, in-\ncluding the code I use to dynamically evolve the\nsystem of equations listed in the previous section.\nAs mentioned there, I evolve the system in radial-\npolar spacetime gauge, which \fxes B= 1 and\f= 0.\nThe constraint equations in (3) determine the metric\nfunctionsaand\u000bon a given time slice and I solve\nthem using second-order Runge-Kutta. The evolu-\ntion equations in (17){(20) and the Lorenz gauge\ncondition equations in (32) determine the matter\n\felds', \b, \u0005,w1,Q1,P1,w2,Q2; P2,ut,ur,Y,\nand \n and I solve them using the method of lines and\nthird-order Runge-Kutta. I note in particular that\nI solve forYusing its evolution equation in (20) in-\nstead of the Gauss constraint in (21) because I found\nthis to be more stable. I use centered sixth-order\n\fnite di\u000berencing for spatial derivatives. In solv-\ning the evolution equations I include fourth-order\nKreiss-Oliger dissipation [31] to help with stability.\nInner boundary conditions at the origin are as given\nin Sec. II F.\nSince the outer boundary of the computational do-\nmain does not extend to r=1I need outer bound-\nary conditions for the matter \felds that allow them\nto exit the computational domain. I use standard\noutgoing wave conditions with 'modeled as a spher-\nical wave and w1andw2modeled as one-dimensional\nwaves, just as in [16]. Additionally, I model urand\n\n as spherical waves. utandYdo not need outer\nboundary conditions since utis given by an algebraic\nequation in Lorenz gauge and the evolution equation\nforYdoes not contain spatial derivatives and can be\nintegrated right up to the outer boundary.\nIn any numerical study it is best to use dimension-\nless variables. In the literature there exist two com-\nmon mass scales used for constructing dimensionless\nquantities: mPandv, wheremP= 1=p\nGis the\nPlanck mass and vis the vacuum value of the scalar\n\feld. As in [16], I use mPand de\fnemG\u0011mP=p\n4\u0019\n(where thep\n4\u0019is included for convenience) and the\ndimensionless quantities\n\u0016r\u0011(gmG)r;\u0016t\u0011(gmG)t;\n\u0016v\u0011v=mG;\u0016\u0015\u0011\u0015=g2; (36)\u0016'\u0011'=mG;\u0016ut\u0011ut=gmG;\u0016ur\u0011ur=gmG;\nalong with \u0016 m\u0011(gmG=m2\nP)mand\n\u0011\n=gmG. I\nnote thatw1,w2, andYare already dimensionless\nand \u0016v=mV=gmGand \u0016\u0015= (mS=p\n2mV)2, where\nmVandmSare the vector and scalar masses in\n(15). The results presented in the next section will\nbe the radial energy and radial electric charge densi-\nties. For future convenience, then, the dimensionless\nquantities in terms of the dimensionful quantities are\n\u0016r2\u0016\u001a= 4\u0019r2\u001a=m2\nP; Y0=@Y\n@\u0016r=1\n4p\u0019mP@q\n@r;\n(37)\nwhere \u0016\u001a\u0011\u001a=g2m4\nGis the dimensionless energy den-\nsity and a prime now denotes a derivative with re-\nspect to \u0016rinstead ofr.\nThe code I use is second-order accurate and I\nhave con\frmed second order convergence. In Lorenz\ngauge it is surprisingly stable. I have not found any\nindications of instability using a uniform computa-\ntional grid and a grid-point spacing of \u0001\u0016 r= 0:06,\nor even larger, and a time step of \u0001 \u0016t=\u0001\u0016r= 0:5, in-\ncluding for very long runs. Further, there is no dis-\ncernible di\u000berence between results using \u0001\u0016 r= 0:06\nand a smaller grid-point spacing. By using the rel-\natively large grid-point spacing \u0001\u0016 r= 0:06, I can\nalso use a large value for \u0016 rmax, the position of the\nouter boundary, and still have run times that are\nnot impractical. Any numerical scheme that allows\n\felds to exit the computational domain will have\n(arti\fcial) re\rections due to \felds not perfectly ex-\niting. By pushing the outer boundary far enough\nout, these re\rections will take so long to return that\nthey cannot in\ruence what happens near the origin.\nFor the results presented in the next section I use\n\u0001\u0016r= 0:06, \u0001 \u0016t=\u0001\u0016r= 0:5, \u0016rmax= 5000, and evolve\nthe system to \u0016t= 10 000.\nIV. SPHALERONIC STABILITY OF\nGRAVITATING MONOPOLES\nIn this section I study the stability of gravitat-\ning monopoles with respect to sphaleronic perturba-\ntions, i.e. perturbations to the magnetic ansatz. I do\nso by taking initial data within the magnetic ansatz,\nand thus initial data for a gravitating monopole, and\nadding to it a magnetic ansatz-breaking perturba-\ntion. I then dynamically evolve the system. My\nfocus will primarily be on the quantity Y0. This is\nbecauseY0is gauge invariant and Y0= 0 de\fnes the\nmagnetic ansatz.2\n2One can just as easily use Yinstead of Y0and obtain the\nsame results found below.\n6-0.050.000.050.10\nt=05153050110\n020 40 60 80100-0.0050.0000.0050.010\n110 300\nr6001000500010000FIG. 1. A time evolution of the radial energy density, \u0016 r2\u0016\u001a(purple), and Y0=@Y=@ \u0016r(blue), which is proportional to\nthe radial electric charge density, as a function of \u0016 rfor \u0016v= 0:2,\u0015= 0, initial data (38) with \u0016 s'= 10, \u0016r1= 2, and\n\u0016s1= 5, and sphaleronic perturbation (39) with \u0016 r2= 15, \u0016s2= 4, andf2= 1. Although only plotted out to \u0016 r= 100,\nthe outer boundary of the computational domain extends to \u0016 r= 5000. The perturbation is large and the evolution\nappears to head toward a dyon con\fguration and maintain a large nonzero value for Y0, which breaks the magnetic\nansatz. Starting in the bottom row, I change the vertical scale to better see the solutions. The value of \u0016tis given in\nthe corner of each frame.\nThe parameters of the system are \u0016 vand\u0016\u0015. In the\nfollowing I restrict attention to \u0016 v= 0:2 and \u0016\u0015= 0.\nFor these values, there exists a unique regular static\nmonopole solution [9].3This means that all initial\ndata with \u0016v= 0:2,\u0016\u0015= 0, and without a sphaleronic\nperturbation evolves to the same static monopole\nsolution (as long a black hole does not form) [16].\nFurther, it means that all sphaleronic perturbations\nare perturbing the same static monopole solution.\nI explained in Sec. II E that the magnetic ansatz\ncan be thought of as Y0= 0, along with ut=ur=\nw2= 0 (the latter set of conditions being gauge de-\npendent). Thus, in constructing initial data, I begin\nwithY0=ut=ur=w2= 0 and then add a nonzero\nvalue to one of these \felds. For reasons having to\ndo with constructing initial data, I only consider\nnonzero values for w2andur. I present results for a\nw2perturbation to generic magnetic initial data and\naur-perturbation to the static gravitating monopole\nsolution. I have studied evolutions for various initial\ndata and found the results that follow to be typical.\nFor thew2perturbation, I use for the magnetic\npart of the initial data [16]\n'(0;r) =vtanh (r=s')\n3I am referring to the fundamental solution, in which the\ngauge \feld w1only equals zero at r=1, and not to excited\nsolutions [9] which are expected to be unstable.w1(0;r) =1\n2(\n1 +\u0014\n1 +a1\u0012\n1 +b1r\ns1\u0013\ne\u00002(r=s1)2\u0015\n\u0002tanh\u0012r1\u0000r\ns1\u0013)\n; (38)\nalong with _ '(0;r) = _w1(0;r) = 0. The parameters\nr1ands1give the center and spread of the w1pulse\nand the parameters a1andb1are chosen such that\nthe boundary conditions for w1are satis\fed at the\norigin and are given by a1= coth(r1=s1)\u00001 andb1=\ncoth(r1=s1) + 1. The sphaleronic w2perturbation is\na Gaussian:\nw2(0;r) =f2(r=r2)2e\u0000(r\u0000r2)2=s2\n2; (39)\nalong with _ w2(0;r) = 0. The parameters r2ands2\ngive the center and spread of the perturbation and\nf2can be thought of as its strength.\nIn Fig. 1 I show a typical evolution for a large\nperturbation. The main purpose of this \fgure is to\ngive an impression of what an evolution looks like.\nI plot the radial energy density, \u0016 r2\u0016\u001a(purple), and\nY0=@Y=@ \u0016r(blue), which is proportional to the ra-\ndial electric charge density. One can thus see how\nenergy and electric charge distribute themselves over\nthe course of an evolution. Both the energy and\nelectric charge appear to maintain a localized con-\n\fguration at late times and thus the system appears\nto settle toward a gravitating dyon. The expecta-\ntion for a large perturbation is that the system is\npushed far from the monopole and stays far from\n70.0000.0050.010t=875587628768877587828789\n020 40 60 801000.0000.0050.0108795 8802\nr8809881688228829FIG. 2. A time evolution of the radial energy density, \u0016 r2\u0016\u001a(purple), and Y0=@Y=@ \u0016r(blue), which is proportional to\nthe radial electric charge density, as a function of \u0016 rwith the same initial data given in the caption of Fig. 1, except\nwith the sphaleronic perturbation strength f2= 0:02, making this a small perturbation. The value of \u0016tis given in\nthe corner of each frame. I do not show the beginning of the evolution because it looks similar to that shown in Fig.\n1. I show instead late times where we can see Y0oscillating.\nthe monopole. That the evolution in Fig. 1 appears\nto maintain a large nonzero electric charge density\nis consistent with this.\nI show a typical evolution when the perturbation\nis small in Fig. 2, where again the purple curve is\nthe radial energy density and the blue curve is Y0,\nwhich is proportional to the radial electric charge\ndensity. The beginning of the evolution is similar\nto Fig. 1 and is not shown. I focus instead on late\ntimes where we can see Y0oscillating. Physically,\nit would appear that shells of positive and negative\ncharge trade places as they oscillate closer and then\nfarther from the center of the system.\nAs mentioned above, in analyzing stability with\nrespect to sphaleronic perturbations I focus on Y0.\nI show an alternative view of the evolutions of Y0in\nFig. 3. The top curve in Fig. 3(a) is for the same\nevolution shown in Fig. 1 and Fig. 3(g) is for the\nsame evolution shown in Fig. 2. The plots in Fig. 3\nare all for the speci\fc value \u0016 r= 5:01 and the \frst\ntwo rows show a series of evolutions with decreasing\nperturbation strengths. One can see that as the size\nof the perturbation decreases, the oscillations of Y0\nmove toward being around zero.\nFigure 2 and the middle row of Fig. 3 are sug-\ngestive of oscillations about a stable equilibrium. If\nthis is the case, the stable equilibrium appears to be\nY0= 0, which is the magnetic ansatz. To further\nanalyze this possibility I take a closer look at the\nindividual oscillations. The bottom row of Fig. 3\nshows the same evolutions as the middle row except\nzoomed in so that individual oscillations can be seen.\nIf these oscillations do actually contain harmonic os-cillations about a stable equilibrium, we would ex-\npect a number of things about the period of the os-\ncillations. One would expect the period to be both\n\u0016tand \u0016rindependent as well as independent of the\nstrength of the perturbation (as long as the pertur-\nbation is su\u000eciently small). A precise determination\nof the periods of the oscillations can be made from\nthe Fourier transform, which I will label F(Y0). My\ninterest is in the most rapid oscillations and I show in\nFig. 4 the Fourier transform of the data given in the\n\frst two rows of Fig. 3. The Fourier transform pre-\nsented is for data with \u0016t>2000, so as to ignore ini-\ntial transient e\u000bects which precede the steady-state\noscillations. I \fnd two narrow spikes with periods\n\u0016\u001c1= 33:3\u00060:1 and \u0016\u001c2= 36:0\u00060:1. As the strength\nof the perturbation decreases, the locations of the\nspikes do not change and thus the periods of the os-\ncillations are independent of the strength of the per-\nturbation (as long as the perturbation is su\u000eciently\nsmall). Further, I have Fourier transformed the data\nat di\u000berent values of \u0016 rand for di\u000berent ranges of \u0016t\nand found the locations of the two spikes to be both \u0016t\nand \u0016rindependent. That there are two spikes whose\nperiods are very close is expected after seeing beats\nin Fig. 3.\nI now perturb the well-known static gravitating\nmonopole solutions. These solutions were \frst stud-\nied in [6{9], with a comprehensive analysis given in\n[9, 10]. I gave a limited review, using the same nota-\ntion used here, of constructing the solutions in [16].\nIn terms of matter \felds, the static monopole solu-\ntions have nonzero values for ', \b,w1, andQ1. After\nconstructing a static solution for the initial data, I\n80 5000 104-0.00250.00000.00250.00500.0075Y′(a)0 5000 104(b)0 5000 104(c)\n0 5000 104(d)\n0 5000 104(e)\n0 5000 104-0.00250.00000.00250.00500.0075Y′(f)\n0 5000 104(g)\n0 5000 104(h)\n0 5000 104(i)\n0 5000 104(j)\n8500 9000-0.00250.00000.00250.00500.0075\ntY′(k)8500 9000\nt(l)8500 9000\nt(m)8500 9000\nt(n)8500 9000\nt(o)FIG. 3. Each plot is a time evolution of Y0=@Y=@ \u0016r, which is proportional to the radial electric charge density, for\nthe speci\fc value \u0016 r= 5:01 and the same initial data given in the caption of Fig. 1, except with the perturbation\nstrengthf2equal to (a) (from top to bottom) 1, 0.5 (b) 0.1, (c) 0.08, (d) 0.05, (e) 0.04, (f) 0.03, (g) 0.02, (h) 0.01,\n(i) 0.008, and (j) 0.005. (a) is the same evolution shown in Fig. 1 and (g) is the same evolution shown in Fig. 2. The\nbottom row is the same as the middle row except zoomed in so that individual oscillations can be seen.\n|ℱ(Y′)|(a)(b)(c)(d)(e)\n20 30 40 50\nτ|ℱ(Y′)|(f)20 30 40 50\nτ(g)20 30 40 50\nτ(h)20 30 40 50\nτ(i)20 30 40 50\nτ(j)\nFIG. 4. Each plot gives the Fourier transform of the data (for \u0016t>2000) shown in the corresponding plot in the \frst\ntwo rows of Fig. 3. (The vertical scale is arbitrary, but consistent across the plots.) Once the perturbation size is\nsu\u000eciently small, there are always two narrow spikes with periods \u0016 \u001c1= 33:3\u00060:1 and \u0016\u001c2= 36:0\u00060:1, which are\nindependent of the perturbation size.\n90 5000 104-0.004-0.0020.000\ntY′(a)\n0 5000 104\nt(b)\n0 5000 104\nt(c)\n0 5000 104\nt(d)\n0 5000 104\nt(e)\n8500 9000-0.004-0.0020.000\ntY′(f)8500 9000\nt(g)8500 9000\nt(h)8500 9000\nt(i)8500 9000\nt(j)\n20 30 40 50\nτ|ℱ(Y′)|(k)20 30 40 50\nτ(l)20 30 40 50\nτ(m)20 30 40 50\nτ(n)20 30 40 50\nτ(o)FIG. 5. Each plot in the top row is a time evolution of Y0=@Y=@ \u0016r, which is proportional to the radial electric charge\ndensity, for the speci\fc value \u0016 r= 5:01. The initial data for the evolutions is the regular static monopole solution\nwith \u0016v= 0:2,\u0015= 0, and sphaleronic perturbation (40) with \u0016 rr= 25, \u0016sr= 10, and perturbation strengths \u0016frequal\nto (a) (from bottom to top) 1, 0.5, 0.3, (b) 0.2, (c) 0.1, (d) 0.08, and (e) 0.05. The middle row is the same as the\ntop row except zoomed in so that individual oscillations can be seen. The bottom row is the Fourier transform of\nthe top row for \u0016t>2000. (The vertical scale for jF(Y0)jis arbitrary, but consistent across the plots.) As in Fig. 4,\nonce the perturbation size is su\u000eciently small there are always two narrow spikes with periods \u0016 \u001c1= 33:3\u00060:1 and\n\u0016\u001c2= 36:0\u00060:1, which are independent of the perturbation size.\nadd aur-sphaleronic perturbation, which is again a\nGaussian:\nur(0;r) =fr(r=rr)e\u0000(r\u0000rr)2=s2\nr; (40)\nalong with _ ur(0;r) = 0. The parameters rrandsr\ngive the center and spread of the perturbation and\nfrcan be thought of as its strength. I note that this\nperturbation gives a nonzero value for Q2, as can be\nseen from (16).\nI show the evolution of Y0for a series of per-\nturbations of decreasing size in Fig. 5, again for\n\u0016r= 5:01. In the top row we see that as the pertur-\nbation strength decreases the oscillations of Y0move\ntoward being around zero. The middle row of Fig. 5\npresents the same evolutions as the top row except\nzoomed in so that individual oscillations can be seen.\nThe bottom row shows the Fourier transform of the\ntop row for \u0016t >2000. I \fnd spikes in the Fourier\ntransform at the exact same locations that we did in\nFig. 4: \u0016\u001c1= 33:3\u00060:1 and \u0016\u001c2= 36:0\u00060:1. I haveFourier transformed this data at di\u000berent values of \u0016 r\nand for di\u000berent ranges of \u0016tand found the locations\nof the two spikes to be both \u0016tand \u0016rindependent.\nThe results in this section are evidence for\nthe stability of gravitating magnetic monopoles\nwith respect to sphaleronic perturbations. As the\nsphaleronic perturbation decreases in size, the sys-\ntem is found to oscillate about the magnetic ansatz.\nFor su\u000eciently small perturbations the periods of\nthe oscillations are both \u0016tand \u0016rindependent and\nindependent of the strength of the perturbation.\nIndeed, the periods are independent of the initial\ndata entirely. Two distinct and narrow spikes in\nthe Fourier transform were found with periods \u0016 \u001c1=\n33:3\u00060:1 and \u0016\u001c2= 36:0\u00060:1.\nIn this section I displayed results only for \u0016 v= 0:2\nand\u0016\u0015= 0. I have looked at other values of \u0016 v(but\nkept \u0016\u0015= 0) and found that the oscillation periods\ndepend on \u0016 v, but are otherwise independent of ini-\ntial data (for su\u000eciently small perturbations). In-\n10dications are that there exists a branch of static\nmonopole solutions (parametrized by \u0016 v) that are sta-\nble with respect to sphaleronic perturbations. Map-\nping this out is beyond the scope of this work, but\nit would be interesting to look at this more closely.\nV. CONCLUSION\nSU(2) with a scalar \feld in the adjoint representa-\ntion coupled to gravity has as a classical solution the\ngravitating Julia-Zee dyon [1, 4, 5]. Within the mag-\nnetic ansatz this system no longer contains dyons\nand instead has as a classical solution the gravitat-\ning 't Hooft-Polyakov magnetic monopole [2, 3, 6{\n10]. I developed second-order code to dynamically\nsolve the full gravitating dyon system. As far as I\nam aware, this is the \fst time the gravitating dyon\nhas been dynamically solved and is one of the only\ntimes (aside from the important papers of Rinne et\nal. [21, 24]) that SU(2) has been dynamically solved\noutside the magnetic ansatz.\nIn pureSU(2) (i.e. unbroken and without the\nscalar \feld), regular static gravitational solutions are\nknown as Bartnik-McKinnon solutions [34], which\nare well-known to be unstable with respect to both\nmagnetic [37] and sphaleronic [38{40] perturbations.\nThe stability of regular static gravitating monopoles\nwith respect to magnetic perturbations was studied\nby Hollmann [15], who found that they are always\nstable for values of \u0016 vnot too large. Other studies\ncorroborated this [13, 14, 16], leaving little question\nas to their stability with respect to magnetic per-\nturbations. As far as I am aware, the stability of\ngravitating monopoles with respect to sphaleronic\nperturbations has not been studied and thus it is\nan open question whether gravitating monopoles are\nstable. That it has not been studied is presum-\nably because it is very challenging to do so. Indeed,\n[12{15] indicated that a standard harmonic stabil-\nity analysis of magnetic perturbations (let alonesphaleronic perturbations) is very di\u000ecult to per-\nform (semi)analytically.\nI chose to avoid these di\u000eculties by making a\npurely numerical study of sphaleronic stability. I did\nthis by adding a sphaleronic perturbation to both\ngeneric magnetic initial data and the static grav-\nitating monopole solutions [6{10] and then evolv-\ning the system. For large perturbations the system\nheads away from the gravitating monopole and to-\nward a dyon con\fguration. As the perturbation de-\ncreases in size, the system begins oscillating about\nthe magnetic ansatz. I found that the periods of\nthe oscillations are independent of time, position,\nand initial data (as long as the perturbation is suf-\n\fciently small), exactly what one would expect for\noscillations about a stable equilibrium. I thus found\nnumerical evidence for gravitating monopoles being\nstable with respect to sphaleronic perturbations.\nA numerical stability analysis can rarely replace\nan analytical one and the results presented here\ndo not prove that gravitating monopoles are stable.\nNevertheless, the results are, as far as I am aware,\nthe \frst piece of evidence discovered for the possi-\nble stability of gravitating monopoles with respect\nto sphaleronic perturbations.\nIn this work I did not consider black holes.\nThere exist static gravitating black hole monopole\nsolutions [7, 9, 10] and it is an important ques-\ntion whether they too are stable with respect to\nsphaleronic perturbations. There are a few reasons\nwhy I did not consider them. Some of the reasons are\nnumerical: the code I use is less stable when a black\nhole forms and it looks to be very di\u000ecult to con-\nstruct initial data that is a perturbation of a static\nblack hole monopole solution. It is not di\u000ecult to\nconstruct initial data that is a sphaleronic perturba-\ntion of generic regular magnetic data which forms a\nblack hole during the evolution. However, the static\nblack hole monopole solutions are not unique (there\nexists a continuum of solutions with the same \u0016 vand\n\u0016\u0015[9]) and it is therefore not clear which solution is\nbeing perturbed in a given evolution.\n[1] B. Julia and A. Zee, Poles with Both Magnetic\nand Electric Charges in Nonabelian Gauge Theory,\nPhys. Rev. D 11, 2227 (1975).\n[2] G. 't Hooft, Magnetic Monopoles in Uni\fed Gauge\nTheories, Nucl. Phys. B 79, 276 (1974).\n[3] A. M. Polyakov, Particle Spectrum in the Quantum\nField Theory, JETP Lett. 20, 194 (1974), [Pisma\nZh. Eksp. Teor. Fiz. 20, 430 (1974)].\n[4] Y. Brihaye, B. Hartmann, and J. Kunz, Gravitating\ndyons and dyonic black holes, Phys. Lett. B 441, 77\n(1998), arXiv:hep-th/9807169 [hep-th].[5] Y. Brihaye, B. Hartmann, J. Kunz, and N. 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L.\nMarsa, New critical behavior in Einstein-Yang-Mills\ncollapse, Phys. Rev. D 60, 124011 (1999), arXiv:gr-\nqc/9903081 [gr-qc].\n[19] P. Bizon, A. Rostworowski, and A. Zenginoglu,\nSaddle-point dynamics of a Yang-Mills \feld on\nthe exterior Schwarzschild spacetime, Class. Quant.\nGrav. 27, 175003 (2010), arXiv:1005.1708 [gr-qc].\n[20] O. Rinne, Formation and decay of Einstein-Yang-\nMills black holes, Phys. Rev. D 90, 124084 (2014),\narXiv:1409.6173 [gr-qc].\n[21] M. Maliborski and O. Rinne, Critical phenomena\nin the general spherically symmetric Einstein-Yang-\nMills system, (2017), arXiv:1712.04458 [gr-qc].\n[22] A. Zenginoglu, A Hyperboloidal study of tail decay\nrates for scalar and Yang-Mills \felds, Class. Quant.\nGrav. 25, 175013 (2008), arXiv:0803.2018 [gr-qc].\n[23] M. Purrer and P. C. Aichelburg, Tails for the\nEinstein-Yang-Mills system, Class. Quant. Grav.\n26, 035004 (2009), arXiv:0810.2648 [gr-qc].\n[24] O. Rinne and V. Moncrief, Hyperboloidal Einstein-\nmatter evolution and tails for scalar and Yang-\nMills \felds, Class. Quant. Grav. 30, 095009 (2013),arXiv:1301.6174 [gr-qc].\n[25] P. Bizon and P. Mach, Global dynamics of a Yang-\nMills \feld on an asymptotically hyperbolic space,\nTrans. Am. Math. Soc. 369, 2029 (2017), [Erra-\ntum: Trans. Am. Math. Soc.369,no.4,3013(2017)],\narXiv:1410.4317 [math.AP].\n[26] P. Bizon and M. Kahl, A YangMills \feld on the ex-\ntremal ReissnerNordstrm black hole, Class. Quant.\nGrav. 33, 175013 (2016), arXiv:1603.04795 [gr-qc].\n[27] R. S. Millward and E. W. Hirschmann, Critical be-\nhavior of gravitating sphalerons, Phys. Rev. D 68,\n024017 (2003), arXiv:gr-qc/0212015 [gr-qc].\n[28] N. Sakai, Dynamics of gravitating magnetic\nmonopoles, Phys. Rev. D 54, 1548 (1996), arXiv:gr-\nqc/9512045 [gr-qc].\n[29] B. Kain, Type II critical behavior of gravitating\nmagnetic monopoles, Phys. Rev. D. 99, 104017\n(2019), arXiv:1905.04355 [gr-qc].\n[30] C. Gundlach, T. Baumgarte, and D. Hilditch, Crit-\nical phenomena in gravitational collapse with two\ncompeting massless matter \felds, arXiv:1908.05971\n[gr-qc] (2019).\n[31] M. Alcubierre, Introduction to 3+1 numerical rela-\ntivity (Oxford, Oxford, UK, 2008).\n[32] T. W. Baumgarte and S. L. Shapiro, Numerical rela-\ntivity: Solving Einstein's equations on the computer\n(Cambridge, Cambridge, UK, 2010).\n[33] E. Witten, Some Exact Multi - Instanton Solutions\nof Classical Yang-Mills Theory, Phys. Rev. Lett. 38,\n121 (1977).\n[34] R. Bartnik and J. McKinnon, Particle - Like Solu-\ntions of the Einstein Yang-Mills Equations, Phys.\nRev. Lett. 61, 141 (1988).\n[35] M. S. Volkov and D. V. Gal'tsov, Gravitating\nnonAbelian solitons and black holes with Yang-\nMills \felds, Phys. Rept. 319, 1 (1999), arXiv:hep-\nth/9810070 [hep-th].\n[36] R. J. W. Petryk, Maxwell-Klein-Gordon \feld\nin black hole spacetimes , Ph.D. thesis, British\nColumbia U. (2006).\n[37] N. Straumann and Z.-H. Zhou, Instability of the\nBartnik-mckinnon Solution of the Einstein Yang-\nMills Equations, Phys. Lett. B 237, 353 (1990).\n[38] M. S. Volkov and D. V. Galtsov, Odd parity neg-\native modes of Einstein Yang-Mills black holes\nand sphalerons, Phys. Lett. B 341, 279 (1995),\narXiv:hep-th/9409041 [hep-th].\n[39] G. V. Lavrelashvili and D. Maison, A Remark\non the instability of the Bartnik-McKinnon solu-\ntions, Phys. Lett. B 343, 214 (1995), arXiv:hep-\nth/9409185 [hep-th].\n[40] M. S. Volkov, O. Brodbeck, G. V. Lavrelashvili,\nand N. Straumann, The Number of sphaleron insta-\nbilities of the Bartnik-McKinnon solitons and non-\nAbelian black holes, Phys. Lett. B 349, 438 (1995),\narXiv:hep-th/9502045 [hep-th].\n12"    },    {        "title": "1505.07566v1.Dynamics_of_an_ensemble_of_clumps_embedded_in_a_magnetized_ADAF.pdf",        "content": "arXiv:1505.07566v1  [astro-ph.GA]  28 May 2015Keywords: accretion— accretiondisc— blackholephysics—magnetohyd rodynamicResearchin AstronomyandAstrophysics manuscriptno.\n(LATEX: sample2.tex; printedonJuly7,2021; 4:30)\nDynamicsofanensembleofclumpsembeddedinamagnetizedAD AF\nFazeleh Khajenabi1, MinaRahmani2,\n1DepartmentofPhysics,GolestanUniversity,Gorgan49138- 15739,Iran; f.khajenabi@gu.ac.ir\n2DepartmentofPhysics,DamghanUniversity,Damghan,Iran\nAbstract Weinvestigateeffectsofaglobalmagneticfieldonthedynam icsofanensembleof\nclumps within a magnetized advection-dominated accretion flow by neglecting interactions\nbetween the clumps and then solving the collisionless Boltz man equation. In the strong-\ncouplinglimit,inwhichtheaveragedradialandtherotatio nalvelocitiesoftheclumpsfollow\nthe ADAF dynamics, the averaged radial velocity square of th e clumps is calculated analyt-\nically for different magnetic field configurations. The valu e of the averaged radial velocity\nsquare of the clumps increases with increasing the strength of the radial or vertical compo-\nnentsofthemagneticfield.Butapurelytoroidalmagneticfie ldgeometryleadstoareduction\nof the value of the averaged radial velocity square of the clu mps at the inner parts with in-\ncreasingthestrengthofthiscomponent.Moreover,dynamic softheclumpsstronglydepends\nontheamountoftheadvectedenergysothatthe valueoftheav eragedradialvelocitysquare\nof the clumps increasesin the presence of a global magnetic fi eld as the flow becomesmore\nadvective.\n1 INTRODUCTION\nAccretion processes have been extensively studied during l ast decades and several types of the accretion\nmodels have been proposed to explain certain observational features of some of the astrophysical objects.\nMost of the models assume that the accretion process occurs a s one-component gaseous fluid. But there\nare strong observational and theoretical arguments which i mply at least some of accreting systems are\nclumpy so that they consist of cool clumps embedded in a much h otter and more tenuous gaseous fluid.\nFor example, observational evidences show that broad-line region of active galactic nuclei (AGN) has a\nclumpystructure( Rees1987 ;Krolik&Begelman1988 ;Nenkovaetal.2002 ;Risalitietal.2011 ;Torricelli-\nCiamponiet al. 2014 ). Thebroademission linesin thespectrumofAGNsare attrib utedtoan assemblyof\ncloudswhicharemovingthroughahotintercloudmedium.Clo uds’basicpropertiesareestimatedaccording\ntothephotoionisationmodels.Thesemodelspredictthatcl ouds’typicalsizeis 1012±1cmandtheirnumber\ndensityis 1010±1cm−3(e.g.,Rees1987 ;Krauseetal.2012 ).OrbitalmotionofBLRcloudsisarichsource\nof information for estimating the mass of the central black h ole (e.g.,Netzer & Marziani 2010 ). One can\nneglect collisions between the clumps and investigate the o rbit of an individual clump in the presence of\nthe central gravitational force and possible radiation fiel d like a two-body classical problem (e.g., Netzer\n& Marziani 2010 ;Krause et al. 2011 ;Krause et al. 2012 ;Plewa et al. 2013 ;Khajenabi 2015 ). Although2 F.Khajenabi & M.Rahmani\nthere are theoreticalconcernsaboutthe stability of the cl umps,it is generallybelievedthat magneticfields\nprovidea confinementmechanism(e.g., Rees1987 ).\nAnotherapproachtostudydynamicsoftheclumpsembeddedin ahotmediumisbasedonanalyzingthe\ncollisionless Boltzmann equation as has been done by Wang et al. 2012 (hereafter WCL). They described\nthe gaseous ambient medium using the classical similarity s olutions of Advection-Dominated Accretion\nFlows (ADAFs) presented by Narayan & Yi (1994) for the non-magnetized systems. Although collisions\nbetween the clumps have been neglected for simplicity, thei r interactions with the surrounding gaseous\nmediumwas includedthrougha dragforce as a functionof the r elative velocityof the clumpsand the gas.\nIn the strong-coupling limit, it was shown that the root of av eraged radial velocity square of the clumps\nis much larger than radial velocity of the gas flow. The analys is has been extended to the magnetizedcase\nbyKhajenabi et al. (2014) where the authors considered a purely toroidal magnetic fie ld geometry for the\ngaseous component. They found that when magnetic pressure i s less than the gas pressure, the averaged\nradialvelocityofclumpsdecreasesattheinnerregionsoft hesystemwhereasitincreasesattheouterparts,\nthoughthisenhancementisnotverysignificantunlessthesy stem becomesmagneticallydominated.\nIn this work, we extend the analysis by Khajenabi et al. (2014) to include all three components of the\nmagnetic field in the gaseous component. Since properties of the gas flow is significantly modified in the\npresence of a global magnetic field ( Zhang & Dai 2008 ), the drag force varies depending on the strength\nof the magnetic field and the assumed magnetic geometry which will eventually lead to a considerable\nmodificationinthevelocitydispersionofclumps.Inthenex tsection,wepresentourbasicassumptionsand\nthe equations. In section 3, a parameter study for the dynami cs of the clumps are presented. We conclude\nwitha summaryoftheresultsin section4.\n2 GENERALFORMULATION\nOur analysis for describing an ensemble of clumps is based on WCL approach which implements the\ncollisionless Boltzman equation in the cylindrical coordi nates(r,φ,z)including the components of the\ndrag force. If we assume the distribution function of clumps is represented by F, then Boltzman equation\niswrittenas\n∂F\n∂t+˙R∂F\n∂r+˙φ∂F\n∂φ+ ˙z∂F\n∂z+ ˙vr∂F\n∂vr+ ˙vφ∂F\n∂vφ+ ˙vz∂F\n∂vz+F(˙vr\n∂vr+˙vφ\n∂vφ+˙vz\n∂vz) = 0,(1)\nwhere˙r=vr,˙φ=vφ/rand˙z=vz. Thecentral objectwith mass Mis at the originandthe gravitational\npotential becomes Φ =GM/(r2+z2)1/2. The componentsof the drag force are Fr=fr(vr−Vr)2and\nFφ=fφ(vφ−Vφ)2wherefrandfφareconstantsoforderunity.Here,theradialandtherotati onalvelocities\nof the ADAF where clumpsare movingwithin it are denotedby VrandVφ. Thus,dynamicalpropertiesof\nthe background medium affect dynamics of the clumps through these components of the velocity. Thus,\nBoltzmanequationbecomes(WCL)\n∂F\n∂t+vr∂F\n∂r+vz∂F\n∂z+/parenleftBigg\nv2\nφ\nr−∂Φ\n∂r+Fr/parenrightBigg\n∂F\n∂vr+/parenleftBig\nFφ−vrvφ\nr/parenrightBig∂F\n∂vφ−∂Φ\n∂z∂F\n∂vz+2F[Fφ(vφ−Vφ)+Fr(vr−Vr)] = 0.\n(2)Dynamics of clumps inADAFs 3\nIt is very unlikely to solve this equation analytically in a g eneral case unless we apply further simplifying\nassumptions.We assumethattheclumpsarestronglycoupled withthebackgroundgaseousmediumwhich\nimpliesthe meanradial andthe rotationalvelocitiesofthe clumpsare equal to theradial andthe rotational\nvelocityofADAF.Underthesesimplifyingassumptions,iti spossibletoobtaintherootmeanradialvelocity\nsquareofthe clumps /an}bracketle{tv2\nr/an}bracketri}ht1/2analytically(WCL):\n/an}bracketle{tv2\nr/an}bracketri}ht=c2{1\n2[α2c2\n1ΓrΛ3\n2(Γr,r)+(1−c2\n2)Λ5\n2(Γr,r)−αc1c2\n2ΓφΛ7\n2(Γr,r)]+V2\nout\nc2r1\n2\noute−Γrrout}×r1\n2eΓrr,(3)\nwhereΓr=frRschandΓφ=fφRschare the coefficients of the drag force. Function Λqis introduced\nby WCL as Λq=/integraltextrout\nrx−qexp(−ΓRx)dx. The outer boundary condition is at r=rout, so that/an}bracketle{tv2\nr/an}bracketri}ht=\nV2\nout. Moreover, properties of the ADAF are described using a set o f radially self-similar solutions where\nc1andc2are coefficients of the radial and the rotational velocities of the ADAF. In WCL, the standard\nnonmagnetizedADAF solutions ( Narayan & Yi 1994 ) have been used for their analysis. Then, Khajenabi\net al.(2014) extended the analysis by including purely toroidal compon ent of the magnetic field using\nsimilarity solutions of Akizuki & Fukue (2006). But we extend previous studies by considering all three\ncomponentsofthemagneticfieldusingsimilaritysolutions ofZhang&Dai (2008).Magnetizedself-similar\nsolutionsof Zhang& Dai (2008)are writtenas\nvr(r) =−c1α/radicalbigg\nGM\nr, (4)\nvφ(r) =c2/radicalbigg\nGM\nr, (5)\nc2\ns(r) =c3GM\nr, (6)\nc2\nr,φ,z(r) =B2\nr,φ,z\n4πρ= 2βr,φ,zc3GM\nr, (7)\nwhere the coefficients βr,βφ,βzmeasure the ratio of the magnetic pressure in three directio ns to the gas\npressure,i.e. βr,φ,z=Pmag,r,φ,z/Pgas. The coefficients c1,c2andc3are obtainedusing a set of algebraic\nequations( Zhang& Dai2008 ):\n−1\n2c2\n1α2=c2\n2−1−[(s−1)+βz(s−1)+βφ(s+1)]c3, (8)\n−1\n2c1c2α=−3\n2α(s+1)c2c3+c3(s+1)/radicalbig\nβrβφ, (9)\nc2\n2=4\n9f(1\nγ−1+s−1)c1, (10)\nHere,γandsare adiabatic index of the gas and mass loss parameter, respe ctively. Also, fmeasures the\ndegreetowhichtheflowisadvectiondominated.Now,wecansu bstitutetheabovemagnetizedself-similar\nsolutionsintotheequation( 3)tostudydynamicsoftheclumpsinthepresenceofa globalma gneticfield.4 F.Khajenabi & M.Rahmani\n3 ANALYSIS\nWenowstudytherootmeanradialvelocitysquareoftheclump s/an}bracketle{tv2\nr/an}bracketri}ht1/2asafunctionoftheradialdistance\nfordifferentvaluesoftheinputparametersusingthemaine quation(3).InallFigures,weassumecoefficient\nof the viscosity is α= 0.1and the adiabatic index is γ= 1.4. Moreover,the mass of the central object is\nfixedatonesolarmass. Thecoefficientsofthedragforceare Γr= 5×10−2andΓφ= 2.8×10−3.\nFigure 1 shows the root mean radial velocity square of the clu mps as a function of the radial distance\nnormalized by Rschfor different values of βrwhereas the other magnetic parameters are fixed as βz=\nβφ= 1. ThisFigure indicatesthat the valueof /an}bracketle{tv2\nr/an}bracketri}ht1/2increaseswith βr, althoughits variationis not very\nsignificant.\nIn Figure 2, we assume that the radial component of the magnetic field doe s not exist and the toroidal\ncomponent is fixed, i.e. βr= 0andβφ= 1. We can then vary the parameter βzto study its effect on the\nradial dynamicsof the clumps.Again,we see that the valueof /an}bracketle{tv2\nr/an}bracketri}ht1/2increasesasthe verticalcomponent\nofthemagneticfieldbecomesstronger,thoughitsvariation withβzislesssignificantat largevaluesof βz.\nMoreover,at theinnerpartsofthe systemclumpsareradiall ymovingfaster astheparameter βzincreases.\nDependenceof the root mean radial velocity square of the clu mps on the variations of the the toroidal\ncomponent of the magnetic field is more complicated as it has a lready been explored by ( Khajenabi et al.\n2014) for a purely toroidal configuration. In Figure 3, we assume that βr= 0andβz= 1, but different\nvaluesof βφare considered.In comparisonto the previousstudy ( Khajenabiet al. 2014 ),here,the vertical\ncomponentofthemagneticfieldisconsideredtoo.Thevalueo fthe rootmeanradialvelocitysquareofthe\nclumpsdecreasesattheinnerpartsofthesystemwith βφwhereasthevalueof /an}bracketle{tv2\nr/an}bracketri}ht1/2increasesattheouter\npartsofthesystem.\nSince the radial and the rotational velocities of the gas com ponent strongly depend on the amount of\nthe advectedenergy,thenobviouslythe clumpsexperienced ifferentvaluesof the dragforce dependingon\nthe advection parameter f. We explore dependence of /an}bracketle{tv2\nr/an}bracketri}ht1/2on the parameter ffor different magnetic\nfieldconfigurationsinFigure 4.Forpurelyradialortoroidalmagneticfieldgeometries,th evalueof /an}bracketle{tv2\nr/an}bracketri}ht1/2\nstronglyincreaseswiththeamountoftheadvectedenergy.B ut forpurelyverticalmagneticfield,thistrend\nchangestoareductionofthemeanvelocitysquareoftheclum pswithincreasingtheparameter βz.However,\nthis reductionis not verysignificant.Thus, one may conclud ethat the value of /an}bracketle{tv2\nr/an}bracketri}ht1/2generallyincreases\nas the flow becomes more advective even in the presence of all t hree components of the magnetic field\n(Figure5).\nIn all of the figures, most of the curves cannot extend to the ve ry inner parts. It is actually because of\nthelimitationof similaritysolutions.We describedthe ga seouscomponentusingself-similar solutionsand\nas we knowsimilarity solutionsare validonlyat the regions far fromthe boundaries.Inotherwords, these\nsimilaritysolutionsforgascomponentare notvalidat thev eryinnerparts.But at theintermediateregions,\nsimilaritysolutionsrepresentdynamicsofthegasflowwith averygoodaccuracy.Sincedynamicsofclumps\ninourmodelisdeterminedmainlyduetointeractionofclump swiththegasandsimilaritysolutionsforthe\ngasarenotvalidat theinnerboundary,wenotinvestigatefa teoftheclumpsattheinnerpartsbasedonour\nsolutions.Dynamics of clumps inADAFs 5\n/s51/s120/s49/s48/s50\n/s54/s120/s49/s48/s50\n/s57/s120/s49/s48/s50/s49/s48/s55/s49/s48/s56\n/s114\n/s114\n/s114\n/s32\n/s114/s122\n/s115/s61/s45/s48/s46/s53/s44 /s102/s61/s48/s46/s57/s48\n/s32/s32/s60/s32/s118/s32/s50 /s114\n/s32/s62/s49/s47/s50\n/s32/s40/s32/s99/s32/s109/s32/s32/s115/s32/s45/s49\n/s41\n/s82/s97/s100/s105/s117/s115/s32 /s40/s82\n/s83/s99/s104/s41\nFig.1Rootoftheaveragedradialvelocitysquare /an}bracketle{tv2\nr/an}bracketri}htofclumpsversustheradialdistancefora\ncentralobjectwithonesolarmass.Differentvaluesforthe parameter βrareconsideredandeach\ncurve is labeled by the corresponding value of this paramete r. The rest of the input parameters\nareγ= 1.4,s=−0.5,f= 0.9,βz= 1andβφ= 1.\n4 CONCLUSIONS\nWestudieddynamicsofanensembleofcoldclumpsembeddedin ahotmagnetizedaccretionflow.Although\nmagnetic field has a vital role in stability and confinement of cold clouds, their role on the orbital motion\nof these clumps has not been studied. In our work, properties of the gas component is modified in the\npresence of a global magnetic field and so, the drag force on ea ch clump changesaccordingly.Comparing\nto the previous study by Khajenabi et al. (2014) who assumed the toroidal component of the magnetic\nfield is dominant, we showed that both the radial and the verti cal components of the magnetic field also\nlead to some changes in the averaged radial velocity square o f clumps. The value of /an}bracketle{tv2\nr/an}bracketri}ht1/2increases\nwith increasing the strength of the radial and the vertical c omponents of the magnetic field. Moreover,\nvelocity dispersion of clumps increases as the flow becomes m ore advective when all components of the\nmagneticfieldareconsidered.Althoughresultsofouranaly sisarenotdirectlyapplicabletotherealsystems\nbecauseoflimitationsofthissimplifiedmodel,thepresent studyclearlydemonstratestheimportanceofthe\nmagneticfieldinthedynamicsofclumpswhichcannotbenegle cted.Theresultsofthepaperareobtained\nwithintheconditionsofstrongcouplingandsimplification ofthemagneticfield. Itisalso possibletorelax\nthesesimplifyingassumptionsbutthenitwouldbeveryunli kelytoobtainanalyticalsolutionswhichisour\ngoalinthepresentstudy.6 F.Khajenabi & M.Rahmani\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51/s49/s48/s55/s49/s48/s56/s49/s48/s57/s122/s32/s122\n/s122\n/s122\n/s122/s114\n/s115/s61/s45/s48/s46/s53 /s32/s32 /s102/s61/s48/s46/s57/s48 /s32/s32\n/s32/s32/s60/s32/s118/s32/s50 /s114\n/s32/s62/s49/s47/s50\n/s32/s40/s32/s99/s32/s109/s32/s32/s115/s32/s45/s49\n/s41\n/s82/s97/s100/s105/s117/s97/s115/s32 /s40/s82\n/s83/s99/s104/s41\nFig.2SameasFigure1,butfordifferentvaluesof βz.Here,wehave βr= 0andβφ= 1.\nAs the clumps move toward the central black hole, they will gr adually accumulate at the inner parts\nbecauseofthetidaldisruptionoftheblackhole’sgravitat ionalfield.Infact,tidaldisruptiondeterminesthe\ninner edge of the clumpy disc. In the presence of a global magn etic field, we find that on the average the\nclumpsareradiallymovingfasterincomparisontoasimilar configurationbutwithoutmagneticfield.WCL\ncalculated the capture rate of clumps and found that it is dir ectly proportional to the ratio of /an}bracketle{tv2\nr/an}bracketri}ht1/2/Vr.\nThus,presenceofaglobalmagneticfieldincreasesthecaptu rerateofclumps,butthelevelofenhancement\ndependson the detailed input parametersas we exploredin th is study.In other words, capturingclumpsis\nfasterwhenmagneticfieldsareconsidered.\nReferences\nAkizuki,C., &Fukue,J. 2006,PASJ, 58,469 3\nKhajenabi,F. 2015,MNRAS, 446,1848 1\nKhajenabi,F., Rahmani,M.,&Abbassi,S. 2014,MNRAS, 439,2 4682,3,4,5\nKrause,M., Burkert,A.,&Schartmann,M. 2011,MNRAS, 411,5 501\nKrause,M., Schartmann,M.,&Burkert,A. 2012,MNRAS, 425,3 1721\nKrolik,J. H.,& Begelman,M.C. 1988,ApJ, 329,702 1\nNarayan,R.,& Yi, I.1994,ApJ, 428,L13 2,3\nNenkova,M., Ivezic,Z.,&Elitzur,M.2002,ApJ,570,L9 1\nNetzer,H.,&Marziani,P.2010,ApJ,724,318 1\nPlewa, P.M., Schartmann,M.,&& Burkert,A.2013,MNRAS, 431 ,L1271Dynamics of clumps inADAFs 7\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51/s49/s48/s55/s49/s48/s56/s49/s48/s57/s49/s48/s49/s48/s114 /s122\n/s115/s61/s45/s48/s46/s53/s44/s32/s102/s61/s48/s46/s57/s48/s32\n/s32/s32/s60/s32/s118/s32/s50 /s114\n/s32/s62/s49/s47/s50\n/s32/s40/s32/s99/s32/s109/s32/s32/s115/s32/s45/s49\n/s41\n/s82/s97/s100/s105/s117/s115/s32 /s40/s82\n/s83/s99/s104/s41\nFig.3SameasFigure1,butfordifferentvaluesof βφ. Here,wehave βr= 0andβz= 1.\nRees, M.J. 1987,MNRAS,228,47P 1,2\nRisaliti, G., Nardini,E.,Salvati,M.,et al.2011,MNRAS,4 10,1027 1\nTorricelli-Ciamponi,G.,Pietrini,P.,Risaliti, G., &Sal vati,M.2014,MNRAS, 442,2116 1\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51/s49/s48/s56/s49/s48/s57\n/s48/s46/s51/s48/s46/s53/s32/s102/s61/s48/s46/s56\n/s115/s61/s45/s48/s46/s53\n/s114\n/s122/s61/s50\n/s32/s32/s60/s32/s118/s32/s50 /s114\n/s32/s62/s49/s47/s50\n/s32/s40/s32/s99/s32/s109/s32/s32/s115/s32/s45/s49\n/s41\n/s82/s97/s100/s105/s117/s115/s32 /s40/s82\n/s83/s99/s104/s41\nFig.4Root of the averaged radial velocity square /an}bracketle{tv2\nr/an}bracketri}ht1/2of clumps versus the radial distance\nfor a central object with one solar mass Here, we explore depe ndence of the root of the aver-\naged radial velocity square on the amount of the advected ene rgy for different magnetic field\nconfigurations.8 F.Khajenabi & M.Rahmani\n/s51/s120/s49/s48/s50\n/s54/s120/s49/s48/s50\n/s57/s120/s49/s48/s50/s49/s48/s54/s49/s48/s55/s49/s48/s56\n/s115/s61/s45/s48/s46/s53\n/s114\n/s122/s61/s50/s48/s46/s51/s48/s46/s52 /s48/s46/s53/s48/s46/s54/s48/s46/s56/s102/s61/s48/s46/s57\n/s32/s60/s32/s118/s32/s50 /s114\n/s32/s62/s49/s47/s50\n/s32/s40/s32/s99/s32/s109/s32/s32/s115/s32/s45/s49\n/s41\n/s82/s97/s100/s105/s117/s115/s32 /s40/s82\n/s83/s99/s104/s41\nFig.5SameasFigure4,butall threecomponentsofthemagneticfiel d areconsidered.\nWang,J.-M.,Cheng,C., && Li,Y.-R.2012,ApJ,748,147,13 2\nZhang,D.,&Dai, Z.G.2008,MNRAS, 388,1409 2,3"    },    {        "title": "1905.03352v2.Flare_Energy_Release_at_the_Magnetic_Field_Polarity_Inversion_Line_During_M1_2_Solar_Flare_of_2015_March_15__II__Investigation_of_Photospheric_Electric_Current_and_Magnetic_Field_Variations_Using_HMI_135_second_Vector_Magnetograms.pdf",        "content": "Flare Energy Release at the Magnetic Field Polarity Inversion\nLine During M1.2 Solar Flare of 2015 March 15. II. Investigation\nof Photospheric Electric Current and Magnetic Field Variations\nUsing HMI 135-second Vector Magnetograms\nI.N. Sharykin1;2, I.V. Zimovets1, I.I. Myshyakov3\nSpace Research Institute of Russian Academy of Sciences (IKI), Moscow, Russia\nABSTRACT\nThis work is a continuation of Paper I (Sharykin et al. 2018) devoted to\nanalysis of nonthermal electron dynamics and plasma heating in the con\fned\nM1.2 class solar \rare SOL2015-03-15T22:43 revealing energy release in the highly\nsheared interacting magnetic loops in the low corona, above the polarity inver-\nsion line (PIL). The scope of the present work is to make the \frst extensive\nquantitative analysis of the photospheric magnetic \feld and photospheric verti-\ncal electric current (PVEC) dynamics in the con\fned \rare region near the PIL\nusing new vector magnetograms obtained with the Helioseismic and Magnetic\nImager (HMI) onboard the Solar Dynamics Observatory (SDO) with high tem-\nporal resolution of 135 s. Data analysis revealed sharp changes of the magnetic\nstructure and PVEC associated with the \rare onset near the PIL. It was found\nthat the strongest plasma heating and electron acceleration were associated with\nthe largest increase of the magnetic reconnection rate, total PVEC and e\u000bective\nPVEC density in the \rare ribbons. Observations and non-linear force-free \feld\n(NLFFF) extrapolations showed that the magnetic \feld structure around the PIL\nis consistent with the tether-cutting magnetic reconnection (TCMR) geometry.\nWe gave qualitative interpretation of the observed dynamics of the \rare ribbons,\nmagnetic \feld and PVEC, and electron acceleration, within the TCMR scenario.\nSubject headings: Sun: chromosphere; Sun: corona; Sun: \rares; Sun: magnetic\n\felds; Sun: photosphere\n1Space Research Institute (IKI) of the Russian Academy of Sciences\n2Moscow Institute of Physics and Technology\n3Institute of Solar-Terrestrial Physics (ISTP) of the Russian Academy of Sciences, Siberian BrancharXiv:1905.03352v2  [astro-ph.SR]  20 Apr 2020{ 2 {\n1. INTRODUCTION\nInitiation of solar \rares is connected with magnetic \feld dynamics in active regions of\nthe Sun. Free magnetic energy contained in active regions in the form of electrical currents\nis enough to explain any small and large, con\fned and eruptive solar \rares (Emslie et al.\n2012; Aschwanden et al. 2014). One of the key observational objects determining magnetic\n\feld topology of active regions is the photospheric polarity inversion line (PIL). It has been\nknown since the observations of Severnyi (1958) that solar \rares appear near the PIL of the\nline-of-sight (LOS) magnetic \feld (e.g. Rust 1972; Schrijver 2009; Sadykov & Kosovichev\n2017). The recent statistical study by Schrijver (2016) illustrated that X-class \rares are\nassociated with strong-\feld, high-gradient polarity inversion lines (SHIL) created during the\nemergence of magnetic \rux. Many recent works (e.g. Chifor et al. 2007; Zimovets et al.\n2009; Bamba et al. 2017; Wang et al. 2017) also reported pre-\rare activity around the PIL\nin the form of brightenings seen in di\u000berent ranges of the electromagnetic spectrum. Thus,\nobservations of the PIL regions are crucial for understanding the processes of accumulation\nand release of energy in solar \rares.\nStrong changes of photospheric magnetic \feld near the PIL during \rares were reported\nin many works (e.g. Wang et al. 1994; Kosovichev & Zharkova 2001; Sudol & Harvey 2005;\nPetrie 2013; Fainshtein et al. 2016; Wang et al. 2018). In particular, according to Sun et al.\n(2012); Petrie (2013) magnetic \feld near the PIL became stronger and more horizontal, and\nthe magnetic shear increased during \rares. It was also found that the horizontal gradient\nof radial magnetic \feld increases during a \rare (e.g. Sharykin et al. 2017). The nonlinear\nforce-free \feld (NLFFF) extrapolations have shown that magnetic \feld lines around the PIL\nbecame shorter during \rares (e.g. Sun et al. 2012; Sharykin & Kosovichev 2015). Probably,\nsuch behavior of magnetic loops can be connected with the process of magnetic reconnection\nabove the photospheric PIL during \rares.\nAccording to the main modern paradigm, the primary energy release of solar \rares,\naccompanied by the transformation of free magnetic energy into the kinetic energy of heated\nplasma and accelerated particles and radiation, occurs in the corona as a result of magnetic\nreconnection above the photospheric PIL (Priest & Forbes 2002; Aschwanden 2004; Shibata\n& Magara 2011). In a common situation, where there is one main PIL in an active region, a\n\rare occurs in a bipolar magnetic con\fguration. Usually this bipolar structure is an arcade\nof sheared magnetic loops or a twisted magnetic \rux rope embedded in the arcade along\nthe PIL. Conventionally, the innermost part of the magnetic arcade is called the core \feld,\nand the outer layers are called the envelope \felds (Moore & Roumeliotis 1992; Moore et al.\n2001). The core \feld lines are highly sheared, such they can almost be parallel to the PIL.\nThe core \feld lines can constitute a magnetic \rux-rope before a \rare, or form it as a result{ 3 {\nof the magnetic reconnection during a \rare (e.g Gibson et al. 2004, 2006; Wang et al. 2015;\nCheng et al. 2017). The reconnection can also result in the increase of magnetic \rux and\ntwist of a pre-existing \rux-rope.\nThe \rare and eruption can be triggered in various ways in the bipolar magnetic con\fg-\nuration (e.g. Priest 2014). In particular, the widely discussed possibilities include (but not\nlimited by) various instabilities of a magnetic \rux-rope (e.g. Hood & Priest 1979; Gibson\net al. 2004; T or ok & Kliem 2005; Kliem & T or ok 2006), or the so-called tether-cutting mag-\nnetic reconnection (TCMR) of the core \feld leading to formation of an unstable \rux-rope\n(Moore & Roumeliotis 1992; Moore et al. 2001). The TCMR concept considers the situation\nwhen a group of highly sheared \feld lines interact and reconnect above the PIL, forming two\nnew groups of \feld lines: one group of lower-lying, shorter and less sheared loops than initial\n\feld lines, and another group of higher \feld lines forming (or contributing to) a magnetic\n\rux-rope. In such a scenario, a \rare may develop in two consecutive or partially overlap-\nping phases. The \frst phase is related to the TCMR or the so-called \\zipper-reconnection\"\n(Priest & Longcope 2017; Threlfall et al. 2018), and is associated with the apparent system-\natic motion of H \u000b, ultraviolet (UV), soft X-ray (SXR; .10 keV) and hard X-ray (HXR;\n&10 keV) emission sources along the PIL, and the elongation of \rare ribbons (e.g. Vorpahl\n1976; Krucker et al. 2003; Bogachev et al. 2005; Grigis & Benz 2005; Qiu 2009; Yang et al.\n2009; Zimovets & Struminsky 2009; Kuznetsov et al. 2016; Qiu et al. 2017). This \frst phase\nusually coincides with the \rare impulsive phase, often accompanied by successive bursts\nor pulsations of energetic electromagnetic radiation. The second phase is related to the\nformation of a three-dimensional quasi-vertical current sheet beneath an erupting \rux-rope\nand magnetic reconnection in this current sheet. This phase is predominantly accompanied\nby expansion and separation of \rare ribbons away from the PIL, and is well-explained by\nthe \\standard\" (CSHKP) two-dimensional \rare model and its three-dimensional extensions\n(Carmichael 1964; Sturrock 1966; Hirayama 1974; Kopp & Pneuman 1976; Shibata & Magara\n2011; Janvier 2017).\nOne may suggest that depending on the initial con\fguration of the magnetic \feld and\nthe development of the process of formation and eruption of a \rux-rope, the two phases\nunder consideration can be expressed di\u000berently. For example, in the case of the presence of\na highly twisted unstable \rux-rope before the onset of eruption and its quasi-homogeneous\n(symmetric) eruption along the PIL, the \\zipper-reconnection\" phase may be very quick\nor absent, and the motion along the PIL is not pronounced. In the event when a \rux-\nrope eruption cannot fully develop, i.g., due to a strongly suppressing overlying magnetic\n\feld, the second phase of \rare ribbon expansion may be mild or absent, and one observes\na con\fned \rare (e.g. Thalmann et al. 2015; Amari et al. 2018). It should be noted here\nthat the observed speeds of movement of the \rare footpoint sources are used to estimate{ 4 {\nthe reconnection electric \feld ( E) and dimensionless reconnection rate in the coronal energy\nrelease sources, i.e. the Alfven Mach number MA=vin=vA, wherevinandvAare the in\row\nand Alfven speeds respectively (e.g. Forbes & Lin 2000; Qiu et al. 2002; Isobe et al. 2002; Asai\net al. 2004; Isobe et al. 2005; Miklenic et al. 2007; Yang et al. 2011; Hinterreiter et al. 2018).\nThe values found in this way are in the ranges of E\u00181\u0000100 V cm\u00001andMA\u00180:001\u00000:1.\nSince the free energy in the active regions is contained in the form of electric currents\n\rowing along magnetic \feld lines or having the form of current sheets, currents play an\nimportant role in the processes of \rare energy release (Melrose 2017; Fleishman & Pevtsov\n2018; Schmieder & Aulanier 2018). However, routine measurements of electric currents in\nthe corona are not available yet. Measurements of the photospheric magnetic \feld vector in\nthe active regions producing solar \rares reveal sites of enhanced vertical electric currents in\nthe vicinity of the PIL (e.g. Moreton & Severny 1968; Leka et al. 1993; van Driel-Gesztelyi\net al. 1994; Janvier et al. 2014; Sharykin & Kosovichev 2015; Janvier et al. 2016; Wang et al.\n2018). Using vector magnetograms with high spatial resolution (up to 100 km) from the New\nSolar Telescope (NST) of the Big Bear Solar Observatory (BBSO) Wang et al. (2017) have\nshown that pre-\rare activity in the form of optical brightenings near the PIL is associated\nwith multiple inversions of the radial magnetic \feld and regions of enhanced photospheric\nvertical electric currents (PVEC).\nOne can suspect that electrons accelerated in the sheared magnetic structures, possibly\nassociated with magnetic \rux-ropes and/or the TCMR, may correspond to the regions of\nenhanced PVEC, as they trace twisted magnetic \feld lines stretched along the PIL. Many\nobservations (e.g. Romanov & Tsap 1990; Abramenko et al. 1991; Can\feld et al. 1992; Leka\net al. 1993; D\u0013 emoulin et al. 1996) have shown that \rare H \u000bor HXR emission sources do\nnot exactly correspond to the most intensive PVEC. In particular, it was demonstrated by\nLi et al. (1997) that the sites of accelerated electrons precipitation to the chromosphere,\nfound from the observations of the Hard X-ray Telescope (HXT) onboard Yohkoh, avoid the\nsites of the highest PVEC density and occur adjacent to these current channels. The recent\nstudies by Musset et al. (2015); Sharykin et al. (2015) using vector magnetograms from the\nHelioseismic and Magnetic Imager (HMI; Scherrer et al. (2012)) onboard the Solar Dynamics\nObservatory (SDO; Pesnell et al. (2012)) have shown that only a part of the HXR emission\nsources are located in the vicinity of the enhanced PVEC near the PIL. This indicates\nthe absence of a direct spatial connection between the enhanced vertical electric currents\nand beams of accelerated particles. Musset et al. (2015) have also demonstrated co-spatial\nappearance of a new HXR source and a region of new enhanced PVEC during the same period\nof the powerful eruptive X2.2 \rare on 15 February 2011. Moreover, it was shown recently\nthat the photospheric vertical electric currents integrated over the \rare regions near the\nPIL has tendency to increase abruptly and stepwise during some \rares (Janvier et al. 2014;{ 5 {\nSharykin & Kosovichev 2015; Janvier et al. 2016; Wang et al. 2018). These observations are\nnot consistent with the electric circuit type models (Alfv\u0013 en & Carlqvist 1967; Colgate 1978;\nZaitsev & Stepanov 2015; Zaitsev et al. 2016), where nonthermal electrons are assumed to be\nlocalized in magnetic loops with the strongest electric current, and the latter must dissipate,\nat least in part, during \rares. Rather they support the concept that the processes of \rare\nenergy release and acceleration of particles occur in the coronal reconnection regions above\nthe photospheric PIL. In this case, the presence of regions of enhanced PVEC indicates the\npresence of free magnetic energy above the PIL, a part of which can be released during a\n\rare. The detected increase of vertical currents and the appearance of new sites of enhanced\ncurrents on the photosphere during \rares can be interpreted as the generation of new currents\nin the coronal sources and their closure (at least partial) to or through the photosphere.\nThe major fraction of recent (since 2010) works dedicated to studying magnetic \feld\nand PVEC during \rares were made using HMI/SDO vector magnetograms (Centeno et al.\n2014; Hoeksema et al. 2014). Angular resolution of standard HMI vector magnetograms\nis about 100when temporal resolution is 720 s. Comparing with typical duration of the\n\rare impulsive phase ( \u00191\u000010 min), such time cadence is insu\u000ecient to resolve changes\nof magnetic \feld during \rares. In this situation, one is mainly able to compare pre-\rare\nand post-\rare states of magnetic \feld. However, it is known that electron acceleration and\nplasma heating develops on shorter than 12 min time-scales (e.g. Aschwanden 2004). Thus,\nto understand development of the \rare energy release processes one need to investigate\ndynamics of the \rare emission sources, photospheric magnetic \feld and PVEC with a time\nresolution signi\fcantly less than 12 min.\nNew high-cadence 135-s vector magnetograms from HMI, which recently became in\nopen access, have a large potential for this kind of research. Some long-duration ( \u001510 min)\n\rare impulsive phases can be roughly investigated using these magnetograms. In the case\nof the shorter impulsive phases (with duration less than 2-3 min), it can be used to obtain\nmore reliable change rates of magnetic \rux and PVEC, because standard magnetograms give\nunderestimated results due to averaging over the longer time interval of 12 min.\nStandard (12-minute) HMI vector magnetograms are the result of summation of 135-\nsecond vector magnetograms. The time step of 135 s is an instrumental time needed for\nmeasuring the full set of Stokes pro\fles for calculation of all magnetic \feld components.\nSummation of 135-second magnetograms is made for increasing the signal-to-noise ratio.\nSun et al. (2017) demonstrated that 135-second magnetograms are more noisy than the\nstandard ones. However, for the magnetic \feld values larger than 300 G the uncertainty\nfor all magnetic \feld components is small. Sun et al. (2017) described the high-cadence\nobservations of magnetic \feld changes during the powerful X2.2 \rare of 2011 February 15,{ 6 {\nwhich were mostly pronounced for the horizontal magnetic component. Kleint et al. (2018)\npresented the non-linear force-free \feld (NLFFF) modeling of the X1 \rare region of 2014\nMarch 29 based on 135-second magnetograms. They found that the magnetic \feld changes\non the photosphere and in the chromosphere are surprisingly di\u000berent, and are unlikely to\nbe reproduced by a force-free model. These works demonstrated that the new HMI data\nproduct can be successfully used to study the \rare processes in the active regions with strong\nmagnetic \felds. However, this data product has not been widely used yet.\nThe present work is a continuation of our previous work (hereinafter, Paper I, Sharykin\net al. 2018). Paper I was devoted to detailed quantitative multiwavelength analysis of non-\nthermal electron dynamics and plasma heating in the system of highly sheared magnetic\nloops interacting with each other above the PIL during the \frst sub-\rare of the con\fned\nM1.2 class solar \rare on 2015 March 15 (SOL2015-03-15T22:43). This \rare was selected\nbecause there were emission sources in di\u000berent ranges of electromagnetic spectrum located\nin the PIL region with strong photospheric horizontal magnetic \feld and PVEC. We ana-\nlyzed X-ray, microwave, ultraviolet and optical observational data. To investigate magnetic\nstructure in the \rare region we used a NLFFF extrapolation. To model the microwave\ngyrosynchrotron emission of nonthermal electrons with the power-law energy distribution\n(low-energy and high-energy cuto\u000b are 20 keV and 1 MeV, respectively) we used the GX\nSimulator tool (Nita et al. 2015). It is worth noting that we ignored other two subsequent\nsub\rares, as they were characterized by less energetic electrons that were in the focus of the\nPaper I. Moreover, the third sub\rare was mostly thermal \rare without nonthermal electrons,\nor electrons producing HXR emission were under background RHESSI level.\nIt was found that the observed structure and dynamics of the \rare emission sources\nand magnetic \feld is consistent with the TCMR scenario. We found that appearance of\nnonthermal electrons with the hardest spectrum and super-hot plasma ( T\u001940 MK) was si-\nmultaneous. By simulating the gyrosynchrotron radiation and comparing with observations,\nit was inferred that accelerated electrons were localized in a thin magnetic channel with a\nwidth of around 0.5 Mm elongated along the PIL within the twisted low-lying ( \u00193 Mm above\nthe photosphere) magnetic structure with high average magnetic \feld of about 1200 Gauss.\nIn other words we observed some kind of \flamentation of the \rare energy release. The ac-\ncelerated electron density in this energy release region above the PIL was about 109cm\u00003\n(the peak energy rate was about 2 \u00021028erg/s) that is much less than the density of the\nthermal super-hot plasma.\nDespite the detailed analysis of the multiwavelength observations of the \rare emission\nsources and its modeling done in Paper I, there was no investigation of dynamics of pho-\ntospheric magnetic \feld and PVEC near the PIL. There was also no discussion of electron{ 7 {\nacceleration and plasma heating as a consequence of magnetic \feld and PVEC changes\naround the PIL. It was just mentioned that the observed \rare emission sources were located\nclose to the PIL, and the HXR sources were almost co-spatial with the regions of strong\nPVEC found from the standard (12-minute) HMI vector magnetograms.\nThe aim of the present work is to perform an extensive quantitative analysis of mag-\nnetic \feld and PVEC dynamics in the SOL2015-03-15T22:43 event using new 135-second\nHMI/SDO vector magnetograms to study magnetic reconnection near the PIL accompanied\nby e\u000ecient plasma heating and electron acceleration. The main tasks are as follows:\n1. to compare positions of the \rare X-ray sources and \rare ribbons with the photospheric\nmagnetic \feld components and PVEC;\n2. to compare variations of the photospheric magnetic \rux and PVEC in the entire PIL\nregion and in the developing \rare ribbons with variations of the \rare electromagnetic\nemissions;\n3. to estimate the reconnection rate and electric \feld in the reconnecting current sheet\n(and some of its physical parameters) to evaluate potential e\u000eciency of the \rare energy\nrelease;\nFinally, we will discuss the data analysis results in the frame of the TCMR scenario, which\nagrees well with the observations of this con\fned \rare.\nThe paper is organized as follows. Section 2 describes some general observations of\nthe \rare including vector magnetograms, X-ray and optical images. Analysis of HMI 135-\nsecond vector magnetograms and NLFFF extrapolations based on these magnetograms are\ndescribed in Sections 3 and 4, respectively. In Section 5 we present time pro\fles of di\u000berent\nparameters inside the \rare ribbons calculated from 135-second HMI vector magnetograms.\nThe discussion and conclusions of the results obtained are presented in the two \fnal sections.\n2. OBSERVATIONS\n2.1. TIME PROFILES OF FLARE EMISSION\nFigure 1 shows temporal evolution of emission lightcurves of the \rare studied. Three\ncolumns in \fgure correspond to three sub\rares. The top panel demonstrates comparison of\nthe radio \ruxes detected by the Nobeyama Radio Polarimeter (NoRP) with the SXR \rux in\n0.5-4 and 1-8 \u0017A bands detected by the X-ray Sensor onboard the Geostationary Operational{ 8 {\nEnvironmental Satellite (GOES). The NoRP time pro\fles are shown for \fve frequencies: 2,\n3.75, 9.4, 17, and 35 GHz. The bottom panels show the time pro\fles of the Ramaty High-\nEnergy Solar Spectroscopic Imager (RHESSI; (Lin et al. 2002)) 4-second count rate in the\nenergy band of 25{50 keV (red histogram) and of the time derivatives of the GOES X-ray\nlightcurves. We do not present RHESSI data in c3 panel as the signal was on background\nlevel. It can be seen from Figure 1 that the three successive sub-\rares had approximate\nduration of the corresponding microwave bursts of about 2{5 minutes. We also should\nmention that the observed three bursts are considered as parts of the large \rare process, as\nit is shown (see next sections) that all emission sources and changes of the magnetic \feld\nwas in the same region near the PIL. Moreover, UV maps (Sharykin et al. 2018) revealed\ncontinual transition of the energy release from the decay phase of the previous sub-\rare to\nthe impulsive phase of the next sub-\rare. In other words, the subsequent sub-\rares are\ndeveloped from the initial magnetic con\fgurations prepared by the previous sub-\rare energy\nrelease.\nThe detected HXR and microwave emissions were generated by accelerated electrons.\nIntensities of these emissions were maximal during the \frst sub-\rare, which was investigated\nin details in Paper I. The spectral analysis (performed in Paper I) of the X-ray emission\ndetected in the \frst sub-\rare has revealed that the spectrum consists of the two main com-\nponents: thermal ( .20 keV) and nonthermal ( &20 keV). By thermal and nonthermal we\nmean the spectral components, which can be \ftted by the Maxwellian and power-law distri-\nbutions, respectively. In particular, the detailed modeling of the microwave gyrosynchrotron\nemission was performed in Paper I, and we remind that the dominating emission at 35 GHz\nduring the \frst sub-\rare is a result of the gyrosynchrotron radiation of nonthermal electrons\nlocalized in the low-lying magnetic loops stretched along the PIL with the average magnetic\n\feld\u00191200 G. Figure 1 also reveals that the HXR and microwave emission peaks correspond\nto peaks in the time derivative of the GOES lightcurves (known as the Neupert e\u000bect; Ne-\nupert 1968). Further, in this paper, we will use the time derivative of the GOES lightcurve\nin 1-8 \u0017A band to compare qualitatively the time evolution of the \rare energy release process\nwith the dynamics of the photospheric magnetic \felds and PVEC.\nTo sum up, the selected \rare reveals non-stationary energy release process with the three\nstages characterized by di\u000berent e\u000eciency of plasma heating and acceleration of electrons.\nPaper I was concentrated on the study of the \frst sub-\rare, as it reveals the most intense\nHXR and microwave emission sources in the vicinity of the PIL that allowed to investigate\naccelerated electrons and plasma heating at the PIL in details. However, in this work we will\ninvestigate dynamics of the magnetic \feld and PVEC for the entire \rare including all three\nsub-\rares. We know from the HXR emissions that the \frst sub\rare was more impulsive and\nenergetic (HXR emission with the hardest spectrum) than the second sub\rare which was{ 9 {\nmore energetic than the third sub\rare. Further we will compare magnetic \feld dynamics in\nthe \rare energy release sites with the onsets of the sub\rares and try to answer the question\nwhat determines the energy release e\u000eciency (acceleration and heating rate).\n2.2. COMPARISON OF X-RAY EMISSION SOURCES WITH MAGNETIC\nFIELD AND PVEC\nThe X-ray maps (Figures 2 and 3) were reconstructed with the CLEAN algorithm\n(Hurford et al. 2002) using the data of the RHESSI detectors 1, 3, 5, 7, and 8 for three time\nintervals during the \frst sub-\rare (panels a-c) and two time intervals for the second sub-\rare\n(panels d, e). We do not present maps for the third sub-\rare due to the low RHESSI count\nrate above 25 keV that leads to too noisy images. Figure 2 demonstrates the X-ray sources\nin two energy bands of 6{12 and 25{50 keV shown by the contours at 50, 70 and 90 % of the\nmaximum of each map. The \frst band corresponds mostly to the thermal emission, while\nthe second one is mostly produced by accelerated (nothermal) electrons precipitated into the\nchromosphere. The X-ray maps are plotted as the contours of constant levels and overlayed\nonto the HMI magnetic \feld maps. For this comparison we used the standard HMI vector\nmagnetogram with temporal resolution of 720 s, as they are less noisy than 135-s ones. Vector\nmagnetic \feld components were calculated from the reprojected onto the heliographic grid\nHMI vector magnetogram in the cartesian heliocentric coordinates. Vertical and horizontal\nmagnetic \feld maps are shown in the top and bottom panels of Figure 2, respectively. The\nPIL has the S-shaped form that is typical for active regions with strong electric currents and\ncontaining sigmoids.\nAs visible in Figure 2, thermal and nonthermal HXR emission sources were generated\nclose to the PIL (see top panels), where the horizontal magnetic \feld dominates (see bottom\npanels) and reaches values of 1500 Gs. In the beginning of the impulsive phase (panel a)\nthere were three major non-thermal HXR sources with di\u000berent intensities, and the thermal\nX-ray source at this time had an elongated (along the PIL) shape. The thermal X-ray\nemission was generated from the thin channel with the approximate length of \u00194000, \flled\nwith hot plasma. From the Paper I we also know that this elongated X-ray emission source\nis also seen in the \\hot\" 94 and 131 \u0017A AIA channels (which are sensitive to emission of\n\raring plasmas with temperatures log10T\u00196:8\u00007:0) and is associated with the strongly\nsheared magnetic loops. Thus, the observed EUV and X-ray emissions were generated, most\nprobably, from the same \rare loops. The X-ray contour maps in the subsequent two time\nintervals (panels b, c) show the rather compact thermal X-ray source located between the\ntwo strongest non-thermal HXR sources. In these cases we observe the highly-sheared (about{ 10 {\n80 degree) magnetic structure. During the second sub-\rare (panels d, e), we observed two\nmajor non-thermal HXR sources on opposite sides of the PIL. However, the single elongated\nthermal source was not located exactly at the PIL, as in the case of the \frst sub-\rare. It was\nslightly (\u0019200) shifted west. This could be a result of the two e\u000bects: (1) the expansion of\nthe \rare energy release region leading to the apparently upward displacement of the loop-top\nthermal X-ray source, and (2) the projection e\u000bect (since the \rare was not exactly in the\ncenter of the solar disk).\nTo reconstruct vertical PVEC density ( jz) maps shown in the top panels of Figure 3\nwe applied Ampere's law to the horizontal magnetic \feld components in the HMI vector\nmagnetograms (e.g. Guo et al. 2013; Musset et al. 2015). The two ribbon-like regions of\nthe highest PVEC density are located near the PIL and close to the two strongest non-\nthermal HXR sources (the intense PVEC regions are covered by 50% HXR contours) for\nboth considered sub-\rares. However, similar to the previous works, there are no exact spa-\ntial correspondence, i.e. the most intense HXR pixels are not co-spatial with the pixels of\nthe highest PVEC density, at least for some time intervals considered. The best correspon-\ndence between the brightest HXR pixel and the nearest HMI pixel with the locally maximal\nPVEC density was at the initial time (panel a). The southern strongest HXR source almost\ncoincided with the strongest PVEC region (the displacement was less than 100). But the\ndisplacement of the northern strongest HXR source from the HMI pixel with the locally\nmaximal PVEC value was about 300. For the panel b, the displacement distance is about 300\nfor the southern HXR source and 500for the northern one. The largest displacement of the\nHXR source from the region of the locally maximal PVEC density was at the time of panel c\nfor the southern source, when the distance was about 700and strongest PVEC region was out\nfrom the 50% HXR contour. At later times, it is di\u000ecult to estimate the distances between\nthe local maxima of PVEC density and non-thermal HXR sources due to their relatively low\nbrightness and increased noise in the synthesized images.\nIn the bottom panels of Figure 3 we show the absolute value of the vertical magnetic\n\feld gradientrhBz=p\n(@Bz=@x)2+ (@Bz=@y)2in the local plane of the solar surface. The\nstrongest values of rhBz(\u00191\u00002 kG/Mm) are found along the PIL, and also concentrated\nin the vicinity of the two strongest non-thermal HXR sources. The best coincidence between\nthe HXR sources and the regions of the strongest rhBzwas achieved in the beginning of the\n\rare impulsive phase (panel a). The best coincidence between the non-thermal HXR sources\n(brightest pixels) and maximal rhBzwas for the southern sources in panels a,d and for the\nnorthern sources in panels a,b (with displacements .100). In other cases, the displacement\nvalues varied in the approximate range of 200\u0000700that is similar for the PVEC maps.\nTo resolve \fne spatial structure of the \rare energy release in the lower solar atmosphere{ 11 {\nand to compare it with the PVEC density maps we used Ca II ( \u0015=3968.5 \u0017A, this emission is\nformed in the lower chromosphere) images from the Solar Optical Telescope (SOT: Tsuneta\net al. (2008)) onboard the space solar observatory Hinode (Kosugi et al. 2007). In Figure 4\nwe present comparison between two jzmaps, deduced from two successive 720-second HMI\nvector magnetograms, and two optical cumulative images from SOT. The cumulative image\nis a sum of all available SOT images within the corresponding 720-second integration time\ninterval of the corresponding HMI magnetogram. Such images present information about\nspatial distribution of total photospheric response during the \rare impulsive phase and allow\nto compare roughly the high-cadence Hinode data with the HMI maps with the low temporal\nresolution. The most intense optical emission was generated in the regions of enhanced PVEC\ndensity, and the most intense emission sources had the best coincidence with the strongest\nPVEC during the \frst 12-min time interval (left panel), but were out of the places with the\nhighestjzvalues during the subsequent 12-min interval (right panel).\nTo sum up, in general, the \rare non-thermal HXR and optical emission sources were\nin the vicinity (\u0019200\u0000700) to the regions of enhanced PVEC and at some time intervals,\nespecially in the begin of the \rare impulsive phase, the localized regions of the most intense\nemissions were very close (within \u0019100) to the strongest PVEC.\n3. CHANGES OF PHOTOSPHERIC MAGNETIC FIELD AND PVEC\nAROUND THE PIL FROM HMI VECTOR 135-SECOND\nMAGNETOGRAMS\nIn the previous section we compared the X-ray and optical images with the magnetic\n\feld maps deduced from the HMI vector magnetograms with the time cadence of 720 s. This\ntemporal resolution is insu\u000ecient to resolve magnetic \feld dynamics during the sub-\rares,\nwhose impulsive phase has typical duration of \u00195 minutes. In this section, we will describe\nthe magnetic \feld dynamics during the \rare studied using the high-cadence HMI vector\nmagnetograms with the temporal resolution of 135 s.\nThe HMI 135-seconds vector magnetograms were previously described by Sun et al.\n(2017). It was shown that the maximum of magnetic \feld distribution is achieved at \u0019\n130 Gs. It is typical for the quiet Sun, as the polarization degree is low, and the most probable\nvalue is due to the noise. This noise mostly a\u000bects the determination of the horizontal\nmagnetic \feld component. We reconstructed distribution of magnetic \felds in the quiet Sun\noutside the active region where the \rare was triggered and found the most probable value is\n110 Gs. This value was taken as the upper boundary for the magnetic strength error. Due\nto the summation, the magnetic \rux error is very small for the \rare regions composed of{ 12 {\nmany pixels and negligible compared to the \rux values.\nWe should also mention that the observed dynamics of the magnetic \felds and PVEC\nduring the \rare studied is not distorted by artifacts connected with the wrong measurements\nof the Fe I line. Such incorrect HMI data could be connected with sharp energy release in the\nlower solar atmosphere leading to distortion of the Stokes pro\fles of the measured line. The\nargument that it is not our case is the absence of fast and high amplitude variations from\npixel to pixel on the magnetograms analyzed. Such variations are quite typical for strong\nwhite-light solar \rares. For example, in the work of Sun et al. (2017) all 4 Stokes pro\fles\nwere compared for the \rare ribbon and quite Sun regions. There were strong distortions in\nthe \rare ribbon that explains magnetic \feld artifacts. We also checked the Stokes pro\fles\nfor the selected points in the vicinity of the PIL, where HXR, UV and optical emissions\nwere generated. We found that the Fe I line did not reveal strong distortions, and, thus, we\nassume that the magnetic \feld is measured correctly. Therefore, PVEC is also calculated\ncorrectly. We also emphasize here that the \rare studied was quite a weak M1.2 class \rare.\nThe time sequence of the magnetic \feld component maps is shown in Figure 5. There are\nmaps for the magnetic \feld absolute value (bottom panels), horizontal (middle panels) and\nvertical (top panels) components. These maps reveal that there were no signi\fcant changes\nin the vertical magnetic \feld component, while the horizontal magnetic \feld near the PIL\nwas signi\fcantly intensi\fed during the \rare. Figures 6(a, b) demonstrate the time pro\fles of\nmagnetic \ruxes Fz=jPBij\nzjSpixfor the negative and positive vertical components, where\nijsuperscript means summation through all pixels inside a particular region and Spixis the\nsingle pixel area. These \ruxes were calculated in the area (shown by the black thick contour\nin Figure 8(a)) around the PIL with the high PVEC density. Both positive and negative\nmagnetic \ruxes do not reveal signi\fcant changes around the \rare onset. The negative and\npositive \ruxes have opposite trends: decreasing and increasing, respectively.\nTo compare real magnetic \rux Fz(integrated over the region-of-interest, ROI, for ~B\u0001~ n,\nwhere~ nis the normal vector to the photosphere) with dynamics of the other magnetic \feld\ncomponents through all analyzed area, we introduce the nominal magnetic \ruxes for the\nhorizontal component ( Bh) and for the absolute value of the magnetic \feld ( jBj), same as\nfor the vertical magnetic component. Figures 6(c1{c3) show temporal dynamics of magnetic\n\ruxPBij\nhSpixfor the horizontal magnetic \feld component. These three panels present the\ntime pro\fles of the \ruxes calculated by summing pixels with magnetic \feld values higher\nthan three thresholds: of 0, 1, and 2 kG. The relative magnetic \rux change in Figure 6(c1)\n(Fh\nmax\u0000Fh\nmin)=Fh\nmin\u00190:4\u00021021=2:1\u00021021\u00190:19, whereFh\nminandFh\nmaxare the magnetic\n\ruxes before the \rare onset (the minimal value) and after the \rare (the maximal value),\nrespectively. The relative \rux change in Figure 6(c2) is about 0.58 with Fh\nmax\u00191:9\u00021021Mx{ 13 {\nwhich is 76 % from the total horizontal magnetic \rux. In other words, the largest fraction\nof the horizontal magnetic \rux is contained in the strong magnetic \feld with Bh>1000 G.\nThe strongest ( Bh>2000 G) horizontal magnetic \feld appeared around the major peak\n(the third sub-\rare) of the \rare X-ray \rux, approximately 40 minutes after the \rare onset\n(Figure 6(c3)).\nThe time pro\fles for the nominal total magnetic \ruxesPjBijjSpix(jBj=p\nB2\nh+B2\nz)\nare plotted in Figures 6(d1{d3). The \ruxes shown in Figures 6(d1{d3) were calculated, as\nbefore, by summing pixels with the magnetic \feld absolute values higher than three thresh-\nolds: of 0, 1, and 2 kG. The change of the total magnetic \rux is also associated with the \rare\nonset. According to Figure 6(d1), the maximal total magnetic \rux Fmax\u00193:2\u00021021Mx\nand the relative change is \u00190:12. Comparing panels (c) and (d) we can conclude that the\nlargest fraction of the total magnetic \rux is contained in the horizontal magnetic \feld. For\nexample, comparing panels (c1) and (d1) we have Fh\nmax=Fmax\u00190:78, or comparing panels\n(c2) and (d2) Fh\nmax=Fmax\u00190:59. We also found that more than half (53 %) of the total\nmagnetic \rux (d1) is in the strong ( >1000 G) horizontal magnetic \feld (c2).\nThe spatial distributions of PVEC density and horizontal gradient of vertical magnetic\n\feld around the PIL are shown in Figure 7 for \fve time instants. Visual analysis of these\n\fgures does not show obvious changes. To demonstrate changes in the PVEC system quan-\ntitatively we calculated the average characteristics (see Figure 8) for the pixels with the\nenhanced PVEC density in the total PIL region, marked by the black thick contour in panel\n(a). Three panels (from 1 to 3) in each raw (c{e) show cases considering the pixels with\nPVEC density above three thresholds with values of 1 \u001b(jz), 3\u001b(jz) and 5\u001b(jz). The black\nand red colors in each panel of Figure 8(c{e) correspond to the positive and negative PVEC,\nrespectively.\nTo estimate the background (noise) level \u001b(jz) for the PVEC density we calculated\nthejzdistribution in the non-\raring region marked by the black rectangle in the lower\nright corner in Figure 8(a). This distribution is shown in Figure 8(b) by the histogram,\nwhere the solid line is a Gaussian \ft and the two dashed vertical lines mark the 1 \u001b(jz)\nlevel. The calculated sigma is about 11 mA/m2, which is comparable with the estimation\n\u001bj=cp\n6\u001bB=(16\u0019\u0001x)\u001914\u00064 mA/m2for\u001bB\u0019110\u000630 G and pixel size \u0001 x= 0:500. Here\ncis the speed of light in vacuum. To estimate errors in determination of the total PVEC\n(c1{c3) in the ROI, the area of the ROI (d1{d3) with the enhanced PVEC density, and\nthe e\u000bective PVEC density (or, by the other words, the PVEC density averaged over the\nROI; e1{e3) we used the Monte-Carlo simulation technique. In the selected ROIs we added\nGaussian noise with \u001bj= 11 mA/m2to calculate Izmap and deduced all needed parameters\nin 100 runs. Then, by calculation of the standard deviations, we found all needed sigmas{ 14 {\nfor each time interval, and show the errors in Figure 8(c1-e3) by the vertical bars of three\nsigmas length.\nThe temporal dynamics of the total PVEC Izin the PIL region is shown in Figure 8(c1{\nc3).Izwas calculated as the sum of PVEC density values for pixels inside the ROI and\nmultiplied by the pixel area. One can see that Izhad a jump during the \rare onset (the \frst\nsub-\rare) and continued increasing | until around the major \rare SXR emission peak for\nIz<0 and even further for Iz>0. Considering the case jjzj>1\u001b(jz) shown in Figure 8(c1),\nthe relative value of Izchange is estimated as [ max(Iz)\u0000min(Iz)]=min (Iz)\u00190:26 for both\npolarities. The higher amplitudes were achieved in the cases jjzj>3\u001b(jz) in Figure 8(c2)\nandjjzj>5\u001b(jz) in Figure 8(c3): 0.41 and 0.22 (for jz<0), 0.69 and 0.73 (for jz>0),\nrespectively.\nTemporal dynamics of the total area of the regions with the enhanced PVEC density\nis shown in Figure 8(d). The area was calculated as a sum of all pixels area above the\ncorresponding thresholds of 1 \u001b(jz), 3\u001b(jz), and 5\u001b(jz). The changes of the total area similar\ntoIz, i.e. the increasing after the \rare onset. Figures 8(d2) and (d3) reveal the largest\njumps of the area for the positive PVEC density: from 26 to 48 and from 12 to 20 Mm2,\nrespectively. Thus, the total PVEC in the ROI increased simultaneously with increasing\nphotospheric cross-sectional area of the electric current carrying magnetic structure.\nFigures 8(e) present the temporal pro\fles of the estimated average (or e\u000bective) PVEC\ndensity< jz>in the PIL region. It is calculated as the ratio of the total PVEC Iz(Fig-\nures 8(c)) to the area (Figures 8(d)). The changes of <jz>have di\u000berent trend comparing\nwith the total PVEC and area of the regions with enhanced PVEC density. There is a peak\nof< jz>after the \rare onset. This increase of < jz>is especially pronounced for jz<0\nshown in Figure 8(e2) and Figure 8(e3), where jmax(< jz>)j\u001970 and\u0019105 mA/m2,\nrespectively. After this peak, < jz>gradually decreased with some weak \ructuations.\nOne can see this tendency in each panel of Figures 8(e) for both current signs, except for\njjzj>1\u001b(jz) (forjz>0) in Figure 8(e1) that could be due to contribution of the background\ninto<jz>estimation.\nTo sum up this section, the high-cadence 135-second HMI magnetograms helped to\nreveal the intensi\fcation of the horizontal magnetic \feld component around the PIL, where\nwe observed the \rare emission sources. The vertical magnetic \feld component did not show\n\rare-related dynamics. The total PVEC Izin the \rare region around the PIL gradually\nincreased during the entire \rare, while the estimated averaged PVEC density showed non-\nmonotonic dynamics with the peak during the \frst sub-\rare and decreasing trend after that.{ 15 {\n4. 3D MAGNETIC STRUCTURE OF THE FLARE REGION\nThe HMI vector magnetograms allow to investigate magnetic \feld dynamics only on the\nphotospheric level. To study temporal dynamics of the 3D coronal magnetic \feld structure\nwe used the NLFFF extrapolation applied to the time sequence of the HMI 135-second vector\nmagnetograms used as the boundary condition. The magnetic \feld extrapolation was made\nusing the implementation of the optimization algorithm (Wheatland et al. 2000) developed\nby Rudenko & Myshyakov (2009). The same procedure and extrapolation parameters are\nused as in Paper I (see Paper I for details). It should be noted here that the \rare studied\nwas a con\fned event without eruption of a \flament and magnetic \rux-rope. Consequently,\nthe \rare was not accompanied by destruction of the magnetic structure of the active region.\nThis partly justi\fes the use of a force-free approach to describe the magnetic structure and\nits dynamics in a given event.\nThe NLFFF extrapolation results are shown in Figures 9(a) and (b), where a set of\nselected magnetic \feld lines (violet) is plotted. These \feld lines are started from the points\nin the PIL region, where the strong PVEC density (shown by the blue-red base maps) are\nconcentrated. Figures 9(a) and (b) correspond to the very begin and the major peak (i.e.\nthe GOES X-ray \rux maximum) of the entire \rare, respectively. At the beginning of the\n\rare, the highly sheared intersecting magnetic \feld lines were involved into the initial energy\nrelease process above the PIL. This magnetic con\fguration is favorable for the TCMR, and\nthe compact twisted magnetic \feld lines observed at the \rare maximum along the PIL in\nthe form of a magnetic \rux rope is a result of the magnetic restructuring due to the TCMR\nprocess (see also Paper I). It can be seen that, in general, magnetic \feld lines around the PIL\nbecame more pressed to the solar surface during the \rare. (More detailed dynamics of the\nmagnetic \feld lines in the PIL region is shown in the movie available in the supplementary\nmaterials to the paper on the journal website.)\nFigures 9(c) and (d) present the 3D structure of coronal electric current density in the\nPIL region for the two time instants (the same as in Figures 9(a) and (b), respectively). It\nis shown by the white semitransparent surface corresponding to the coronal electric current\ndensity of constant level 55 mA/m2(this value is arbitrary selected and is about 5 \u001bjz), which\nis calculated using the Ampere's law applied to the NLFFF extrapolation results. One can\nsee that the \rare energy release lead to expansion of the current-carrying region elongated\nalong the PIL. The pre-\rare state was characterized by the thinner channel of strong electric\ncurrent density. Then this channel became thicker across the PIL. Here we need to note that\nthe NLFFF approximation gives only electric currents \rowing along magnetic \feld lines.\n(Temporal evolution of the current surface is also presented in the movie which can be found\nin the supplementary materials to the paper on the journal website.){ 16 {\nThe NLFFF modelling reveals the \rare-related magnetic \feld restructuring around the\nPIL. Interaction of the crossed sheared magnetic loops at the PIL lead to the formation\nof region at the PIL with the enhanced horizontal magnetic \feld component when vertical\ncomponent was quasi stable. It is resulted from formation of the \rux-rope-like magnetic\nstructure along the PIL and the sheared magnetic arcade and expansion of the elongated\nelectric current channel in the corona. It also corresponds to formation of the region where\nwe observed the enhancement of the total PVEC (described in the previous section).\n5. CHANGES OF MAGNETIC FIELD AND PVEC IN THE UV FLARE\nRIBBONS\nFlare ribbons are associated with footpoints of magnetic \feld lines directly connected\nwith the magnetic reconnection site and their observation is important for diagnostic of the\n\rare energy release process. Here we concentrate on a detailed analysis of variations of the\nmagnetic \feld and PVEC density inside the fare ribbons in the PIL region using 135-second\nmagnetograms.\nTo study the dynamics of the \rare ribbons we used AIA UV 1700 \u0017A images. This UV\nemission is generated in the chromosphere. Temporal resolution of this data product is 24\nseconds and the pixel size is \u00190:600. We decided to use these images instead of SOT Ca II\nimages with better spatial resolution mainly due to the limited \feld-of-view (FOV) of the last\none. SOT did not observe some distant \rare emission sources, while AIA observes the whole\nsolar disk. Moreover, SOT has only 1-minute temporal resolution when \rare observational\nregime stops. In our case, such resolution was before the \rare onset and after the second\nsub-\rare.\nIn each image, the area of the \rare ribbons is calculated as a sum of pixels with intensity\nvalues higher than the arbitrary selected threshold of 3800 DNs. Figure 10 presents temporal\nsequence of binary maps (the black-white background images) showing the \rare ribbon\npositions deduced from AIA UV 1700 \u0017A images. Each pixel in these maps can have value of\n0 (black) or 1 (white). Value 1 means that a given pixel is inside the \rare ribbons. Positions\nof the \rare ribbons are compared with the regions of strong PVEC density shown by the red\nand blue contours corresponding to the negative and positive PVEC density, respectively.\nIn general, the \rare ribbons appeared around the PIL in the regions of enhanced PVEC\ndensity. However, parts of the ribbons did not coincide with the strong PVECs in some\ntime intervals, in particular, the southern part of the ribbons ( x\u0019\u000024000;y\u001947600) at\n22:46:31{22:50:07 UT and 22:54:55{22:58:31 UT.{ 17 {\nIn Figure 11 the temporal pro\fles of the \rare ribbon area (a1-c1) and the total UV\nintensity (a2-c2) are shown by the black histograms and compared with the GOES 1{8 \u0017A\nlightcurve (cyan) and its time derivative (blue). Three columns of the \fgure correspond to\nthree sub\rares. The fastest UV intensity growth and the maximal value was during the \frst\nsub-\rare with the largest area change of \u00191:3\u00021018cm2/s. The maximal total area of the\nribbons is about 1 :4\u00021018cm2.\nThe sequence of the UV images reveals the sharp enhancement of the ribbon area during\nthe \frst frames (Figure 11) of the \frst sub-\rare. Dynamics of the ribbons can be splitted\ninto two phases. The \frst one (before around 22:49 UT) is not very evident as we do not\nhave su\u000ecient temporal and spatial resolution. However, comparing the \frst and the second\nimages (two left top images in Figure 10) one can see the fast expansion (or elongation) of\nthe ribbons along the PIL from the initial compact brightenings. The second stage (mainly\nafter around 22:49 UT) is characterized by the gradual separation of the ribbons out from\nthe PIL.\nTo investigate details of the evolution of the ribbons during the \frst sub-\rare we present\nFigure 12 with six (non-binary) UV images compared with PVEC density contour maps.\nInitially (panel a) we observe the small weak emission sources in the regions of strong PVEC\ndensity around the PIL. Then (in panel b), we see the large scale ribbons with non-uniform\nbrightness distribution from \u0019\u000024500up to\u0019\u000017500along the south-north direction. This\nexpansion occurred during the 24-second time interval of the temporal resolution for AIA\nUV 1700 \u0017A images. One can also see (panels (c){(f)) that later the length of the \rare ribbons\nalmost did not change, while the ribbon emission intensity varied with time.\nThe ribbon separation during the \frst sub\rare (the distance between the ribbons in a\ndirection perpendicular to the PIL) was about 1 Mm, that means that the emission sources\nwere very close to the PIL and comparable with the ribbons width. Considering the whole\nevent (all three sub-\rares), the averaged ribbon velocity ( Vrib) in the perpendicular direction\nto the PIL is\u0019500per 10 minutes or V?\u00196:25 km/s. That's why, during the \frst sub\rare the\nperpendicular displacement of the ribbons was very small <100. To estimate expansion rate\nV?during the very beginning of the impulsive phase (the upper limit) of the \frst sub-\rare\nlet's assume fast expansion of the \rare ribbon up to its width from the initial brightenings.\nIn this case, the approximate velocity is \u0019300during 24 seconds (the time cadence of AIA\n1700 \u0017A images), or V?\u0019100 km/s. Actual velocity during 24 seconds can be higher or lower\nthan this value within range of \u001940\u0000160 km/s calculated considering \u001bt= 12 seconds (half\nduration of AIA time resolution) and \u001bpix= 0:300(half size of AIA pixel).\nLet's estimate the parallel velocity Vjjof the ribbon elongation during the \frst sub-\rare.\nIt cannot be measured directly, because of its fast elongation time relatively to the AIA{ 18 {\n1700 A cadence (24 s). However, we can estimate it by the following way. Considering ribbons\nwidthV?tmuch smaller then its length Vjjt, the total area S(t) of both \rare ribbons can be\napproximated as S(t) = 2VjjV?t2and the expansion rate is dS=dt = 4VjjV?t. Considering the\n\frst sub-\rare, one can estimate the elongation velocity as Vjj= \u0001S=(4V?\u0001t2)\u0019560 km/s\nfor the beginning of the impulsive phase taking \u0001 t= 24 seconds, \u0001 S\u00191:3\u00021018cm2and\nV?\u0019100 km/s (see estimation above). As we considered the maximal possible value of V?,\nthe obtained Vjjis the lower limit. If we will take smaller \u0001 tthan velocity will be larger up\nto the Alfven velocity VA\u00196900 km/s (upper limit) considering B= 1000 Gauss and ion\ndensity ofni= 1011cm\u00003(see Paper I). For vjj=vAthe ribbons elongation developed on\nthe time scale \u0001 t\u00190:5 s that is much less than actual temporal resolution of AIA data.\nTo sum up, Vjj\u001dV?during the beginning of the impulsive phase. Than the \rare ribbon\nelongation was \fnished and we observed their slow motion out from the PIL. This picture is\nquite similar to the two step reconnection process discussed in (e.g. Qiu et al. 2010; Priest\n& Longcope 2017).\nFurther we will investigate dynamics of the magnetic \feld and PVEC inside the \rare\nribbons for all three sub-\rares. To calculate the magnetic \feld parameters inside the \rare\nribbons at the time of the selected AIA UV 24-second frame we used linear interpolation\nbetween corresponding pixels of two neighboring HMI 135-second magnetograms (the UV\nframe is between these two magnetograms). Time derivatives of the magnetic (vertical\ncomponent) \rux inside the \rare ribbons is shown in Figure13(a1-c1) by the black (negative)\nand red (positive) histograms. Such time derivative is usually referred as the magnetic\nreconnection rate. The largest reconnection rate of \u00197\u00021018Mx/s was achieved in the\nbeginning of the \frst sub-\rare. This reconnection rate is almost two and six times larger than\nfor the second and third sub-\rares, respectively. Thus, dynamics of reconnection rate nicely\ncorrelates to heating rate of the whole \rare. It is also worth noting that two enhancements of\nthe reconnection rate during the second sub-\rare are in accordance with two major heating\nbursts deduced from the time derivative of the GOES 1{8 \u0017A lightcurve.\nFigures 13(a2-c2) and (a3-c3) present variations of the magnetic \feld components inside\nthe \rare ribbons. Panels (a2-c2) demonstrates the magnetic \rux (orange) around the PIL,\nwhich is calculated using the vertical magnetic \feld component. This panel also demonstrates\nthe nominal magnetic \ruxes deduced for the absolute value of the magnetic \feld (black)\nand for the horizontal magnetic \feld component (red). These nominal magnetic \ruxes are\nintroduced to make comparison between the magnetic \feld components over the total area\nof the \rare ribbons. The time pro\fles of the magnetic \feld components averaged over the\n\rare ribbon area are plotted in panels (a3-c3) with the same meaning of colors as in panels\n(a2-c2). Each sub-\rare was characterized by approximately the same enhancement of the\nnominal total magnetic \rux with the magnitude of \u00196:5\u00021019Mx. The \frst and second{ 19 {\nsub-\rares were characterized by the averaged absolute magnetic \feld value of \u00191100 G,\nwhen the third one had slightly less value of \u0019800 G. However, the main result is that the\nmagnetic \feld inside the \rare ribbons associated with the \frst sub-\rare were more horizontal\nthan in the case of the subsequent sub-\rares. At the time of the maximal j~Bjduring the\n\frst sub-\rare the ratio <Bh>=<B z>\u00191:62 and the \rux ratio was about 1.66. For the\nsecond and third sub-\rares these ratios had the following values: 1.3 and 1.2, and 1.2 and\n1.1, respectively.\nFigure 14 shows the following time pro\fles: (a1-c1) the total PVEC Iz(t) inside the\n\rare ribbons, calculated as a sum of pixel values above 35 mA/m2\u00193\u001b(jz) and multiplied\nby the pixel area; (a2-c2) the e\u000bective (or averaged) PVEC density hjz(t)iinside the \rare\nribbons estimated as the ratio of the total PVEC Iz(t) (panel (a)) to its area S(t), whose\ntime pro\fle is shown in panel (a3-c3). The time pro\fles are shown for the positive (red) and\nnegative (black) PVEC density signs. The total PVEC in the ribbons during the third sub-\n\rare was small comparing with the \frst and second sub-\rares having Iz\u00194\u00005\u00021011A. The\nmaximal e\u000bective PVEC density was about 60 mA/m2for the \frst and second bursts. The\n\rare ribbons were partially covered by the regions with strong PVEC density. From panels\n(a3-c3), the maximal area of the regions with strong ( >35 mA/m2) PVEC density was \u0019\n1017cm2, which is about 10 % from the total area of the \rare ribbons. From Figure 14 we can\nmake an important conclusion that the temporal variations of plasma heating and electron\nacceleration e\u000eciency (inferred from the time derivative of the GOES 1{8 \u0017A lightcurve and\nthe Neupert e\u000bect) is consistent, in general, with the variations of the e\u000bective PVEC density\ninside the \rare ribbons. Although there are some discrepancies, in particular, the peaks of\nthe positivehjz(t)iof the \frst and second sub-\rares are a bit (2{3 min) longer than the\ncorresponding peaks of the time derivative of the GOES 1{8 \u0017A lightcurve.\nThe analysis done in this section allowed us to make detailed comparison of the \rare\nenergy release with the changes of the magnetic \feld and PVEC in the \rare ribbons, con-\nnected with the energy release site. We found good time consistency of the \rare energy\nrelease e\u000eciency (approximated by the time derivative of the GOES 1{8 \u0017A lightcurve) with\nthe magnetic reconnection rate and the total PVEC inside the ribbons.\n6. DISCUSSION\n6.1. MAIN RESULTS\nUsing the new HMI vector magnetograms with the high time cadence of 135 s, we inves-\ntigated dynamics of the photospheric magnetic \feld and PVEC in the PIL region during the{ 20 {\nthree successive sub-\rares of the con\fned M1.2 class solar \rare on 2015 March 15 (SOL2015-\n03-15T22:43), previously studied in Paper I (Sharykin et al. 2018). The following main data\nanalysis results were obtained:\n\u000fThe \rare X-ray sources, as well as the optical and UV \rare ribbons, were located,\nin general, in the regions of the strong horizontal photospheric magnetic \feld, PVEC\ndensity and the horizontal gradient of the vertical magnetic \feld component on the\nphotosphere around the PIL.\n\u000fThe photospheric magnetic \feld was mostly horizontal near the PIL region. The\nmagnetic \feld in the \rare region became more horizontal during the \rare. The large\nfraction (more than half) of the magnetic \rux was concentrated in the strong ( B >\n1000 G) magnetic \feld. There were no fast \rare-associated changes of the vertical\nmagnetic \feld component.\n\u000fThe total PVEC near the PIL region increased simultaneously with the increasing pho-\ntospheric cross-sectional area of the current-carrying magnetic structure. The e\u000bective\nPVEC density \frst increased impulsively with the \rare onset and then decreased grad-\nually with the \rare development. Expansion of the regions with the strong PVEC is\nin qualitative accordance with the motion of the \rare ribbons from the PIL.\n\u000fThe \rare ribbons penetrated into the regions with more vertical magnetic \feld in course\nof the \rare, while they were initially characterized by the dominant horizontal magnetic\ncomponent. The expansion of the \rare ribbons was characterized by enhancement\nof the total PVEC and e\u000bective PVEC density inside them, which had tendency to\ncoincide temporally with the lightcurves of microwave emission and the time derivative\nof the SXR lightcurve.\n\u000fThe NLFFF extrapolation based on the HMI 135-sec magnetograms showed that (a)\nthe pre-\rare state was characterized by the highly sheared magnetic structure intersect-\ning above the PIL favorable for the three-dimensional (3D) TCMR; (b) the magnetic\n\feld lines started from the regions of strong PVEC became shorter and formed the\nlow-lying magnetic \rux-rope-like structure embedded in the sheared magnetic arcade\nalong the PIL at the end of the \rare.\nIn the following subsections we will discuss these observational phenomena jointly with\nthe results from Paper I in the context of the TCMR above the photospheric PIL. We will\ntry to give qualitative interpretation of the phenomena observed during the \rare and make\nthe order of magnitude estimations of some important physical parameters related to the\nenergy release processes and electron acceleration.{ 21 {\n6.2. DYNAMICS OF MAGNETIC FIELD AND ELECTRIC CURRENTS IN\nTHE FLARE REGION\nHere we will discuss dynamics of the photospheric magnetic \feld components and PVEC\n(see the second and third items in the list of the main results in Section 6.1) in the vicinity\nof the PIL where the \rare energy release developed in the conditions of 3D restructuring of\nthe magnetic \feld lines within the found sheared magnetic structure.\nThe found increase of the horizontal magnetic component near the PIL during the \rare\ncan be explained in the frame of the TCMR scenario. The sheared magnetic loops interact\nand reconnect above the PIL, which results in formation of a small magnetic arcade mov-\ning/collapsing towards the photosphere. That is why the growth of the horizontal magnetic\ncomponent near the PIL is observed during the \rare.\nThe larger and higher magnetic \feld lines are also formed due to this reconnection\nprocess. These \feld lines move upwards and interact with the overlying magnetic \feld above\nthe primary energy release site. Subsequent episodes of magnetic reconnection are triggered\nin this case. We observe it, \frstly, as the initial fast elongation of the \rare ribbons along\nthe PIL, and after that as the small displacement of the \rare ribbons out from the PIL and\nappearance of the expanding hot sheared magnetic arcade (Figure 16(b)). The velocity of this\nexpansion is rather small that is in agreement with the \\slipping\" reconnection in the con\fned\n\rares (Hinterreiter et al. 2018). In other words the expansion is a result of involvement of\nnew magnetic loops lying above the already reconnected ones. The reconnection happens\nbetween higher and higher, more distant magnetic loops from the PIL, which are less sheared\nand in a more potential state (Priest & Longcope 2017; Qiu et al. 2017). If a full eruption\ndoes not occur, then this process stops at some point (at some height). Determining the\nreason for stopping this process and a lack of developed eruption in the \rare studied is\nbeyond the scope of this work (see, e.g., Amari et al. 2018).\nLet's now discuss the observed dynamics of the PVEC in the \rare studied (see the\nthird and forth items in the list of the main results in Section 6.1). First of all, we need\nto mention that the observed dynamics is not related to the projection e\u000bect. The thing\nis that the observed enhancement of the total PVEC in the \rare region may be connected\nwith the magnetic \feld verticalization near the PIL. However, the magnetic \feld near the\nPIL become more horizontal during the \rare and, thus, the increase of PVEC cannot be\nthe result of magnetic \feld inclination change at the foot of the \rare loops. Our working\nhypothesis is that the detected changes of the PVEC density could be connected with the\nelectric current generation/ampli\fcation in the current sheet(s) above the PIL in course\nof the 3D TCMR. Indeed, in contrast to the 2D \\standard\" model, in the 3D magnetic\ncon\fguration, the current \rowing in the coronal current sheet(s) must be closed somewhere.{ 22 {\nWe assume that this may take place, at least partially, under the photosphere. Thus, the\nappearance/ampli\fcation of the current in the corona can be accompanied, possibly with\nsome delays, by the appearance/ampli\fcation of vertical currents on the photosphere. This\nis ideologically similar to the results obtained by Janvier et al. (2014, 2016) for two powerful\neruptive X-class \rares on 2011 February 15 and 2011 September 6 (see also Schmieder &\nAulanier 2018, for discussions).\nAs we discussed above, the magnetic reconnection in the \rare studied is a non-stationary\nprocess involving di\u000berent magnetic loops of various spatial scales. To analyze dynamics of\nelectric current in the current sheet qualitatively, \frstly, let's consider a single reconnection\nepisode and try to understand the situation when we have a pulse of electric current density\nand gradual increase of the total electric current. For simplicity, we will consider a standard\nrectangular di\u000busion region with height land width \u000e(Fig. 15b). Magnetic \feld at the\nboundaries of this region has three components: the guide ( Bjj; along the PIL), perpendicular\n(B?=gBin; i.e. vertical), and transverse Bx(i.e. across the sheet) components. Here gis\na geometric factor connected with the magnetic shear across the current sheet. It is di\u000ecult\nto estimate gfrom observations, and we just assume that it is less than unity. To estimate\nthe total electric current through the current sheet we use the Ampere's law in the integral\nform:\nJjj=Z\nSCjjjdS=c\n4\u0019I\nC~B\u0001~ds=c\n2\u0019B?l\u0014\n1 +Bx\nB?\u000e\nl\u0015\n\u0019c\n2\u0019B?l (1)\nHereSC=\u000elis a cross-sectional area of the current sheet and Cmeans the contour marking\nthe boundary of this area. This approximate formula for Jjjis derived assuming Bx\u000e=B?l\u001c1\nthat is quite natural considering magnetic \rux conservation vinB?=vABx. Thus,jjj/\nB?=\u000eandJjj/B?l. The time derivatives of the electric current density and the total\nelectric current are jjjt/B?t=\u000e\u0000B?\u000et=\u000e2andJjjt/lB?t+B?lt, respectively, where\nsubscripttmeans time derivative. Thus, to achieve jjjt>0 andJjjt>0 one has to guarantee\nsimultaneous ful\fllment of the following conditions: B?t=B?> \u000et=\u000eandB?t=B?>\u0000lt=l.\nIn other words, relative growth rate of the magnetic \feld near the di\u000busion region boundary\nshould be larger than change of the current sheet width and the change of the height with\nopposite sign. The obvious way to ful\fll these inequalities is to assume the current sheet\nthinning, elongation and enhancement of B?value at the reconnection region boundary.\nSuch dynamics of a current sheet was simulated for eruptive solar \rares and discussed by\nJanvier et al. (2016). We suggest that the similar behavior for the 3D magnetic reconnection\nwith a guide \feld in the closed magnetic con\fguration like in the frame of the 3D TCMR\nscenario considered here is also possible. To achieve B?t>0 we can suggest the induction\nequationB?t= [vinBjj]x>0. It means that the in\row across the current sheet leads to\nenhancement of the perpendicular magnetic \feld.{ 23 {\nAfter the pulse of < jz>, we observed the situation when the total electric current\nand current density show di\u000berent trends: jjjt<0 andJjjt>0 (see Figure 8(e1{e3)). It can\ndescribed by the following inequalities: B?t=B?< \u000et=\u000eandB?t=B?>\u0000lt=l. We suggest\nthat the most reliable and simple scenario explaining these inequalities is the magnetic\nannihilation process without magnetic advection described by the simple equation for the\ndi\u000busion region: B?t=\u0016B?xx, where\u0016is the magnetic di\u000busivity and xxmeans double x\nderivative. Considering B?t\u00190 at the boundary as there is no advection, we will obtain\n\u000et/p\n\u0016=t> 0 andjjjt<0.\nAbove we discussed the observed dynamics of jjjandJjjin the frame of a single re-\nconnection episode. But enhancement of Izon the photosphere and the area with strong\nPVEC density could be a result of successive involvement of new magnetic \feld lines, located\nhigher than previously reconnected ones, into the reconnection process. In other words, the\n\rare process may consist of a large set of reconnection episodes, probably in di\u000berent loca-\ntions in the corona above the PIL (e.g. Zimovets et al. 2018). The moving \rare ribbons\ntrace magnetic \rux associated with magnetic \feld lines passing through the reconnection\nregion(s). Analysis of high cadence HMI vector magnetograms revealed expansion of the\nregions with strong PVEC density associated with the moving \rare ribbons seen in the UV\nimages. It con\frms the idea that the total PVEC in the \rare region has tendency to grow\ndue to successive involvement of new magnetic \feld lines in the reconnection process in the\ncorona. Good time matching between the \rare energy release e\u000eciency (observed as the\ntime derivative of the SXR emission lightcurve) and electron acceleration (observed as the\nHXR and microwave light curves) from one side, and the \rare ribbon area, total PVEC\nand average PVEC density through the ribbons during three subsequent sub-\rares (see Fig-\nure 14) from another side, con\frms additionally that the \rare energy release is associated\nwith the involvement of new current-carrying magnetic elements and local ampli\fcation of\nPVEC near the PIL.\nHere it is appropriate to note a certain analogy of the observed phenomena with the\nprocesses in the Earth's magnetosphere during substorms. It is well known that substorms\nare the result of magnetic reconnection in the near-Earth magnetotail current sheet (see for\nreview, e.g., Baker et al. 1996; Angelopoulos et al. 2008). This phenomenon is associated\nwith plasma heating and acceleration, generation of earthward and tailward plasma \rows\naccompanied by the magnetic \feld enhancement, so called dipolarization. These plasma\n\rows distort magnetic \feld lines that leads to generation of magnetic shear and the \feld-\naligned currents as a consequence (see e.g. Kepko et al. 2015, and references therein). Recent\nobservations in the magnetosphere and numerical simulations con\frm the close link between\nplasma out\rows from the near-Earth magnetotail reconnection region and generation of the\n\feld-aligned currents (e.g. Artemyev et al. 2018). These results indicate \\that the dominant{ 24 {\nrole of the near-Earth magnetotail reconnection in the \feld-aligned current generation is\nlikely responsible for their transient nature\". The situation could be ideologically similar in\nthe \rare investigated in the present work. The active region around the PIL already contained\nsigni\fcant quasi-steady electric currents before the \rare (and after it), probably generated\nas a result of long-lasting horizontal movements of the foot of the magnetic loops. The\ntransient ampli\fcations of the vertical currents on the photosphere during the energy release\nin the sub-\rares could be associated with plasma out\rows and magnetic \feld enhancement as\nthe result of magnetic reconnection episodes in the current sheet(s) above the PIL. Further\nresearch should show how justi\fed such an analogy is.\n6.3. MAGNETIC RECONNECTION IN THE PIL REGION\nIn Figure 15 we present the proposed geometry of the magnetic reconnection region in\nthe frame of the 3D TCMR scenario (panel (a)). The geometric sizes and magnetic structure\nof the reconnection site are shown in panel (a1-c1). The basic parameter of the magnetic\nreconnection is the reconnection rate d\u001e=dt , de\fned as a time derivative of magnetic \rux\nthrough the \rare ribbons (e.g. Forbes & Lin 2000). This \rux determines magnetic in\row into\na current sheet where magnetic reconnection develops. The reconnection rate is proportional\nto the electric \feld Ein the reconnecting current sheet: d\u001e=dt\u0018cEL, wherecis the speed\nof light,Lis the length scale of the current sheet (e.g. Forbes & Lin 2000). The maximal\nvalued\u001e=dt\u00197\u00021018Mx/s and, thus, E\u00182=Lstatvolt/cm inside the current sheet\nwithLin Mm. Using the results of Paper I we take L\u00181 Mm, which is the width of\nthe hot channel at the PIL seen in the \\hot\" EUV bands (AIA/SDO 94 and 131 \u0017A; see\nFigure 16). The magnetic reconnection rate can be aslo calculated as d\u001e=dt\u0018vingBinL,\nwherevinis a velocity of plasma \rowing into the current sheet with the length scale L\n(Figure 15b). It means that the parallel electric \feld along the sheet is connected with gBin\nat the current sheet boundaries. It is di\u000ecult to estimate the current sheet length L. The\npossible assumption is L\u0018l, wherelis the current sheet height in the vertical direction out\nfrom the photosphere and it is about 1 Mm (i.e. a cross section size of the \rare loops; see\nPaper I). However, it seems that due to the high shear, Lshould be a few times larger than\nl. For further estimations let's take in mind that gL\u00181, forLdescribed in Mm. But, not\nto loss generality, this factor will be written in the following expressions.\nWe found that vin\u0019700=(gL[Mm]) km/s, and the dimensionless reconnection rate (i.e.\nthe Alfven Mach number in the in\row region) MA=vin=vA\u00190:1=(gL[Mm]) for the Alfven\nvelocityvA= 6900 km/s considering B= 1000 G and ion density of 1011cm\u00003. The\nestimated value of MAis close to the upper limit of the magnetic reconnection rate found{ 25 {\nin other works (e.g. Yokoyama et al. 2001; Isobe et al. 2005; Lin et al. 2005; Narukage &\nShibata 2006; Takasao et al. 2012; Su et al. 2013; Nishizuka et al. 2015; Cheng et al. 2018).\nThe characteristic in\row velocities in that works were in the range of 10 \u0000100 km/s that\nis signi\fcantly smaller relative to our estimation of vin\u0019700=(gL[Mm]) km/s. From our\npoint of view, it is unlikely that the driver of the \rare energy release in this non-eruptive\n\rare was the large scale displacement of magnetic loops with this velocity. Possibly, such\nvelocity could arise locally within the current sheet due to motion of the separate magnetic\nelements across the current sheet. For example, formation of magnetic islands (their three-\ndimensional analog), their relative motion and acceleration up to the Alfven velocity within\nthe current layer will lead to subsequent acceleration of magnetized plasma incoming to the\nspace between magnetic islands (e.g. plasmoid induced magnetic reconnection, Shibata &\nTanuma 2001).\nLet's estimate the current sheet width \u000e. It is reasonable to use the continuity equation\nfor the steady state (reconnection rate maxima when d2\u001e=dt2\u00180):\nvinninl=vAn0\u000e (2)\nThus,\u000e=l(nin=n0)MAis determined by the plasma compression ratio nin=n0and\ndimensionless reconnection rate MA. From our estimations MA\u00190:1=(gL). To estimate the\ncompression ratio we assume an incompressible limit as the simplest case. Another option is\nto assume strong compression from the initial (background) coronal density nin\u0018109cm\u00003\nto the \rare super-hot plasma density n0\u00191011cm\u00003(see Paper I). As a result, we obtain\n\u000e\u0018(10\u00003\u000010\u00001)=(gL) Mm.\nAnother important characteristics of the magnetic reconnection process is the ratio\n\u0015mfp=L, where\u0015mfpis the plasma collisional mean free path and Lis the characteristic length\nscale (in our case we assume it to be of the current sheet size). The estimation gives \u0015mfp\u0019\n0:5 Mm for the super-hot plasma temperature T= 40 MK and density n\u00191011cm\u00003. Thus,\nwe have collisionless conditions taking the current sheet width \u000e\u0018(10\u00003\u000010\u00001)=(gL) Mm\nand height l\u00180:5 Mm. However, for electrons propagating along the current we consider a\nweakly collisional regime ( \u0015mfp=L=\u0015mfpg=l&1). It is likely that the magnetic reconnection\nprocess was collisionless due to the high plasma temperature.\nLet's also estimate the energy release rate in the current sheet with the height of l=\n1 Mm (the width of the \\hot\" EUV channel, see Paper I for details) and incoming magnetic\n\feldBin= 1000 G using the formula (e.g. Aschwanden 2004):\ndE\ndt= 2gB2\nin\n4\u0019vinLl=gBinl\n2\u0019\u0001d\u001e\ndt\u00181029g; (3){ 26 {\nergs/s.\nTo estimate the in\row energy we neglected the kinetic and thermal energy as the mag-\nnetic energy is dominant. During the impulsive phase of the \frst sub-\rare of duration\n\u0001t\u0019100 s the total magnetic energy release is ( dE=dt )\u0001t\u00181031gLergs that is one order of\nmagnitude larger than the nonthermal particle energy ()1030ergs) and a few times smaller\nthan the change of the free magnetic energy \u00192:9\u00004:6\u00021031ergs bearing in mind that\ngL\u00181 (see Paper I). Thus, the estimated magnetic reconnection energy release rate does\nnot contradict to the found \rare energetics deduced from di\u000berent emissions. The di\u000berence\nbetween total magnetic free energy and dissipated magnetic energy in the current sheet can\nbe connected with those fact that we did not take into account subsequent reconnection\nepisodes (the second and the third sub\rares).\nTo sum up, despite the fact that the magnetic reconnection during the studied \rare has\nquite high rate (up to d\u001e=dt\u00197\u00021018Mx/s), its dimensionless value ( MA\u00190:1=(gL))\nis comparable with the upper limit found in other works, where the current sheets located\nhigher in the corona in the eruptive events considered.\n6.4. CURRENT SHEET STABILITY AND FILAMENTATION\nThe fast in\row velocity ( vin\u0019700=(gL) km/s) leading to the large reconnection rate\nis probably not resulted from the fast macroscopic motions, because we did not found them\nin the \rare region. It is likely to be connected with some local processes around the current\nsheet. Thus, to trigger the fast magnetic reconnection one need to assume the current sheet\nachieving unstable conditions somehow. This type of magnetic reconnection is usually called\nthe spontaneous reconnection (e.g. Priest, & Forbes 2000). In the case of the spontaneous\nreconnection, it is necessary that the current sheet is formed and accumulated su\u000ecient\namount of energy to be released during a \rare.\nIn Figure 16(b) we present the time-distance plot for the observational slit (the white\nhorizontal line in panel (a) with the AIA 94 \u0017A image for one time instant \u001920 min before the\n\rare onset) crossing the bright EUV channel at the PIL. From Figure 16(b) we conclude that\nthe energy release (heating) in the bright \\hot\" source elongated along the PIL was observed\nlong before the \rare impulsive phase onset (at least before 45 min in Figure 16(b)). A\nsimilar picture was previously observed before many \rares (Cheng et al. 2017, and references\ntherein). Sudden enhancements of the channel brightness were detected episodically in the\npre-impulsive \rare phase. The \rare initiation was preceded by the gradual increase of the\nEUV luminosity for \u001910 min. It seems that the energy release site was already prepared for{ 27 {\nthe \rare onset, and once it reached some special conditions for an instability the \rare started.\nThe current sheet could exist in the quasi-stationary state and the magnetic reconnection\nwas slow, then the reconnection rate started to grow suddenly, which led to the beginning\nof the \rare and its development. This is consistent with the discussion of the TCMR \rare\nscenario by Moore & Roumeliotis (1992).\nIt is di\u000ecult to \fnd a reason for the spontaneous equilibrium loss of the current sheet\nbecause of the limited capabilities of the available observational data. However, one of the\npossible ways to trigger transition from the slow to fast regime of reconnection is to assume\nthe tearing instability leading to formation of current \flamentation inside the current sheet.\nThe reason to consider this scenario is the fact that the accelerated electrons and hot plasma\nwere localized in the thin magnetic \flament (described in Paper I). We assume that such\n\flament is a bundle of magnetic \rux tubes formed due to the tearing instability inside a non-\nneutralized current sheet. It was shown (e.g. Kliem 1994) that magnetic islands (\flaments\nin 3D) in the current sheet lead to e\u000ecient trapping of electrons inside the \flaments.\nAs we know from the theory (e.g. see Priest 2014), the characteristic time \u001ctearof\nthe tearing instability is the shortest for large scale perturbations with k\u000e\u00181 (kis the\nwavenumber of perturbation). This time can be estimated asp\u001cd\u001cA(e.g. subsection 10.2.1\nin Aschwanden 2004), where \u001cAis the Alfven time and \u001cd=\u000e2=\u0016is the di\u000busion time across\na current sheet with the width \u000eand magnetic di\u000busivity \u0016=c2=(4\u0019\u001b), where\u001bis the\nelectrical conductivity. Considering \u000e\u0018(10\u00003\u000010\u00001)=(gL) Mm (estimated above), one can\ndeduce\u001beff= (\u001ctearc)2vA=(\u000e34\u0019)\u0018(2:5\u00021011\u00002:5\u00021017)(gL)3, when the classic electrical\nconductivity is \u001bSp\u00194:8\u00021017forT= 40 MK. Thus, even the classical conductivity could\nexplain formation of magnetic islands in the more narrow current sheet. But in the case of a\nthicker current sheet one should consider the suppressed (anomalous) electrical conductivity\nwhich can be \fve orders less than the classical one. The electric conductivity reduction can\narise due to the presence of turbulence.\nAnother reason to suggest the presence of anomalous transport connected with turbu-\nlence is the appearance of the super-hot plasma (for details see Paper I). Con\fnement of\nthe super-hot plasma on a time scale of the impulsive phase could be due to anomalously\nslow heat losses from the heated region. Let's estimate characteristic cooling time via heat\nconduction as \u001ccond\u00194nkBL2=(kT5=2)\u00190:13 s forL= 5 Mm (half length of 10 Mm mag-\nnetic loop) and T= 40 MK. Here kBis the Boltzmann constant and kis the thermal Spitzer\nconductivity coe\u000ecient. If we consider saturation limit of heat conduction, when electrons\nwith their thermal velocity transfer energy, we \fnd \u001ccond\u00181:5VTeL\u00190:2 s, whereVTeis the\nvelocity of thermal electrons. The values obtained are too small comparing with the dura-\ntion of the impulsive phase \u0018100 s. To resolve this discrepancy, we assume the suppressed{ 28 {\nheat conduction coe\u000ecient by four orders. Some discussion of such possibility, related to the\ngeneration of turbulence by beams of accelerated particles, can be found in (e.g. Sharykin\net al. 2015). More detailed discussion of the possible physical reasons for the suppressed\nheat conduction is out of the scope of the present work.\n6.5. ACCELERATION OF ELECTRONS\nAs we know from Paper I, electrons were accelerated up to the kinetic energy K0\u0018\n0:1 MeV in the \frst sub-\rare of the \rare studied. A small fraction of electrons could be\naccelerated to higher energies ( \u00181 MeV), as evidenced by the presence of detectable non-\nthermal gyrosynchrotron microwave radiation, but could not be registered in the HXR range\nbecause of the background. Thus, the lower limit for the maximal kinetic energy of non-\nthermal electrons is considered here as K0\u00180:1 MeV. From tracing of the \rare ribbons\nwe found the magnetic reconnection rate related to the electric \feld in the current sheet.\nFirstly, we notice that the reconnection rate estimated for three sub-\rares is in qualitative\naccordance with the observed heating rate (the time derivative of the SXR light curve) and\nmicrowave light curves related to the gyrosynchrotron radiation of accelerated electrons. In\nother words, temporal dynamics of electron acceleration and heating rate correlates roughly\nwith the electric \feld strength (the reconnection rate) dynamics in the current sheet.\nBelow we will discuss a possible acceleration process during the \frst sub-\rare when the\npopulation of non-thermal electrons was the most energetic among three sub-\rares. In the\nsecond and third sub-\rares, the situation could, in general, repeat the situation in the \frst\nsub-\rare, but with lower intensity.\nThe maximum value of the electric \feld in the current sheet was estimated above in order\nof magnitude as E\u00181=L[Mm]\u00181 statvolt/cm (or \u00183\u0002102V/cm) forL= 1 Mm. This\nelectric \feld highly exceeds the Dreicer \feld ED\u001910\u000011ln \u0003neT\u00001\ne\u001810\u00006statvolt/cm (or\n\u001810\u00004V/cm), where ln \u0003 \u001920 is the Coulomb logarithm, ne\u00191011cm\u00003andTe\u001940 MK\nis the electron plasma density and temperature in the acceleration site, respectively (see\nPaper I). Considering the acceleration length scale Lacc\u001910 Mm, corresponding to the\nlength of the accelerated electron capture region obtained in Paper I (i.e. the region where\nthe accelerated electrons were trapped and emitted the gyrosynchrotron radiation detected),\none can deduce the maximal accelerated electron energy W1=eELacc\u0019300 GeV, which\nwas actually not observed ( W1\u001dK0). This indicates that, most probably, the electron\nacceleration length scale was several orders of magnitude less than the electron capture length\nscale. Considering the results of Litvinenko (1996), one can deduce the maximal energy\ngained asW2=eEjj\u000eBjj=B?. Here the current sheet is assumed to be non-neutralized with{ 29 {\nthe longitudinal Bjjand transverse B?magnetic components. For \u000e\u00186\u0002104\u00006\u0002106cm\n(see above) and Bjj=B?\u001810 (that is reasonable for the \rare studied, see Paper I), we obtain\nW2\u00190:2\u000020 GeV.\nThe estimated energies W1andW2of non-thermal electrons are much higher than the\nobserved ones, i.e. W1\u001dW2\u001dK0. It means that if the mechanism of the super-Dreicer\nelectric DC \feld acceleration is valid, than: (1) the in\row plasma velocity should be 103\u0000105\ntimes lower (i.e. v0\nin\u00183:5\u0000350 m/s), or (2) the reconnection current sheet width ( \u000erec) should\nbe at least\u0018103times smaller than the minimum value of the current sheet width \u000emin\u0019\n6\u0002104cm estimated above from the continuity equation, i.e. \u000erec\u001810\u0000100 cm. The \frst\npossibility seems to be unrealistic, as the in\row velocity was estimated from the reconnection\nrate, which is consistent in order of magnitude with the previous estimations. The second\npossibility is more likely. Our estimation of \u000eis based on the continuity equation (2), where\nall parameters ( nin,n0,vin,vA,l) can not be measured precisely. Probably, the greatest\nuncertainty is in estimating the height of the reconnecting current sheet l. We took an\nestimate of l\u00181 Mm from the EUV observation of the bright structure elongated over the\nPIL. In fact, the height of the current sheet could be much lower, and the apparent thickness\nof this EUV structure could be determined by the expansion (out\row) of the heated plasma\nleaving the reconnection region. It is possible that the actual height scale of the reconnecting\ncurrent sheet may be several orders of magnitude lower than our estimation of l, and then\nthe reconnecting current sheet width could be \u000erec\u001810\u0000100 cm. Here we can also add\nthat this\u000erecshould be considered as a characteristic spatial scale of electron acceleration to\nthe observed energies. This means that the current sheet itself could be much thicker (with\n\u000e\u00186\u0002104\u00006\u0002106cm), but it had \flamentation with a characteristic scale \u000erec, which\ncould impede e\u000bective acceleration to higher energies.\nLet's estimate the electron acceleration time: \u0001 tacc=p\n(Bjj=B?)(2\u000erecme)=(eEjj)\u0018\n10\u00008\u000010\u00007s, wheremeandeis the electron mass and charge, respectively (see Aschwanden\n2004). The resulting estimate shows that electrons are gaining energy very quickly, much\nfaster than the observed time scales. From this point of view, this acceleration process\ndoes not contradict the available observations. Electrons can gain energy quickly, and then\nthey can precipitate to the chromosphere or can be trapped in magnetic loops for quite\na long time, emitting HXR and microwave radiations. Each observed burst of HXR and\nmicrowave emission with time scale of several seconds (or tens of seconds) could consist of\nmillions of \\elementary\" bursts (e.g. Kaufmann 1985; Emslie & Henoux 1995), each of which\nis associated with acceleration in the current \flament of scale \u000erec.\nLet's also compare the rate of energy gain due to acceleration by the super-Dreicer\n\feld (dWSD=dt) and the loss rate due to gyrosynchrotron radiation ( dWgs=dt):dWSD=dt\u0018{ 30 {\n107Ejj[statvolt=cm ]p\n1\u0000(mec2=W)2MeV/s and dWgs=dt\u0018 \u0000 10\u00009(B[G])2(W=mec2)2MeV/s\n(e.g. Longair 1981). Consequently, dWSD=dt>jdWgs=dtjfor electron energy W.500 GeV\nunder the estimated Ejj\u00191 statvolt/cm and B\u00191000 G. This means that for the kinetic\nenergies of accelerated electrons ( K.1 MeV) observed in the \rare studied the energy\ngain due to acceleration in the super-Dreicer \feld estimated far surpasses the energy loss\ndue to gyrosynchrotron radiation, and the latter can be neglected. The characteristic time\nof energy loss due to gyrosynchrotron radiation (after leaving the acceleration region) is\n\u0001tgs\u0019109\u0002(mec2)K0=[B2\n[G](K0+mec2)]\u001980\u0000330 s forK0\u00180:1\u00001 MeV and B\u00191000 G,\nwhich is comparable with the duration of the sub-\rares observed.\nTo sum up the above estimations, there is no problem to produce electrons with energies\nsu\u000ecient to generate HXR and microwave emissions by the estimated super-Dreicer electric\n\feld. The problem is that such \\ideal\" accelerator seems to be too e\u000ecient and could accel-\nerate electrons to much higher energies than the observed ones. Possibly, an e\u000eciency of a\nreal accelerator may be much di\u000berent due to presence of electric current \flamentation (frag-\nmentation). In particular, it was shown on the base of 3D kinetic simulation that electrons\ncan e\u000bectively gain energy by a Fermi-like mechanism due to re\rection from contracting \feld\nlines during reconnection in a \flamenting current sheet with a guide magnetic \feld (Dahlin\net al. 2015). A similar situation, but in a much larger physical volume than the simulated\none, could well be realized in the \rare studied. Another possibility of plasma heating and\nelectron acceleration is related to collapsing magnetic loops (traps), which could have been\nformed above the PIL as a result of the TCMR (see discussion above). In this process, the\nkinetic energy gain by a particle is roughly proportional to the square of the ratio of lengths\nof stretched and unstretched trap (Somov & Bogachev 2003; Borissov et al. 2016). It is\nunlikely that this ratio exceeded \u00192\u00003 in the con\fned \rare studied. Thus, we suppose\nthat this mechanism, although it could serve as a source of some additional plasma heating\nand electron acceleration, did not play a crucial role.\nFrom Paper I we know that the ratio of nonthermal electrons density to thermal super-\nhot electrons density was \u00190:01. Large-scale electric \feld acts equally on all electrons,\nwhereas we see that only a small part of them was accelerated. Runaway electrons can\nquickly setup a strong charge separation that screens the electric \feld. This process can\nbe also accompanied by excitation of waves/turbulence by beams of runaway electrons and\nsubsequent interaction with them, preventing further e\u000bective acceleration (e.g. Boris et al.\n1970; Holman 1985). On the other hand, it is known that stochastic acceleration, based\non wave-particle interaction, is selective to electrons which are in resonance with waves.\nThus, only a fraction of electrons from thermal or initially pre-accelerated populations has\na chance to be accelerated further. In the above subsection we discussed necessity to intro-\nduce \flamentation/turbulence to explain tearing instability and the presence of super-hot{ 31 {\nplasma. Possibly, this turbulence could accelerate electrons in the frame of the stochastic\nacceleration models. It could explain the ratio of accelerated electrons to thermal particles\nand less energies than the estimated ones in the frame of the super-Dreicer DC electric \feld\nacceleration. Unfortunately, it is di\u000ecult to prove the presence of turbulence in the \rare\nstudied on the base of the available observational data. Finally, it should be mentioned that\nother acceleration mechanisms could be also considered (e.g. Aschwanden 2004; Zharkova\net al. 2011, and references therein) that is, however, out of the scope of this work.\n6.6. SUMMARY AND CONCLUSIONS\nThis work and Paper I present detailed investigation of the \rare energy release in con-\nditions far from those ones assumed in the \\standard\" 2D model of eruptive solar \rares.\nOur interest to the selected con\fned \rare (SOL2015-03-15T22:43), composed of three sub-\nsequent sub-\rares, was due to its initial energy release development very low in the corona\n(H\u00193 Mm, possibly even in the chromosphere), in the region with strong vertical PVEC\nnear the the PIL, where we found interaction of the highly stressed sheared magnetic loops.\nIn other words one can say that the selected \rare is nice natural experiment showing energy\nrelease of the pure three dimensional magnetic reconnection in the con\fned region (without\neruption).\nIn Paper I we studied \rare in the context of plasma heating and dynamics of nonther-\nmal electrons using multiwavelength observations. In this work we focused on analysis of\nhigh-cadence 135-second HMI vector magnetograms that allowed us to investigate dynamics\nof the photospheric magnetic \feld and PVEC on \rare time scale. We compared maps of\nphotospheric magnetic \feld components and PVEC with the \rare images in di\u000berent wave-\nlength ranges (optical, UV, EUV, SXR and HXR). Locations of the \rare emission sources, as\nwell as the retrieved dynamics of the photospheric magnetic \feld and extrapolated coronal\nmagnetic \feld are nicely explained by the TCMR-based \rare scenario.\nAnalysis of 135-second vector magnetograms revealed that the total PVEC in the \rare\nregion shows sharp increase during the \rare that con\frms the observations obtained with\n720-second cadence in the previous works. However, we found that the temporal pro\fle of the\ne\u000bective PVEC density (averaged over the \rare region around the PIL) has maximum during\nthe \frst sub\rare and subsequent gradual decrease. We think that, we found manifestations\nof electric current reorganization and dissipation connected with the \rare energy release\nprocess.\nFrom our point of view, the most important result of this paper is the deduced dynam-{ 32 {\nics of the magnetic \feld and PVEC inside the \rare ribbons. High-cadence magnetograms\nallowed to investigate non-stationary dynamics of the UV ribbons relative to the structure\nof magnetic \feld and PVEC in the \rare region. For three consecutive sub-\rares, we found\nrough matching of the plasma heating and electron acceleration e\u000eciency with the magnetic\nreconnection rate, total PVEC and PVEC density in the \rare ribbons. We argued that this\nobservation can be also qualitatively interpreted within the TCMR scenario.\nMagnetic reconnection (TCMR) was developped in the non-neutralized current sheet\n(formed between two crossed magnetic structures at the PIL) with dominating guide \feld\n\u00181000 Gauss, existed long before the \rare onset. Considering the maximal value of magnetic\nreconnection rate 7 \u00021018Mx=s we found its dimensionless rate MA\u00190:1 and estimated\nplasma in\row (into reconnecting current sheet) velocity as vin\u0019700 km/s. Reconnection\nrate is comparable with the upper limit found in other works, where eruptive events were\nconsidered. However in\row velocity is larger comparing with the eruptive \rares. We think\nthat such fast in\row is not connected with external driver (gradual displacement of magnetic\nloops due to photspheric motions, or eruption), but is a result of spontaneus equilibrium\nloss. Estimations also reveal that magnetic reconnection was collisionless and accompanied\nby formation of thin channels (possibly 3D magnetic islands, see Paper I) where accelerated\nelectrons and hot plasma were localized.\nWe would like to emphasize that we hope to continue studies of similar events to extract\nmore detailed quantitative information about \rare energy release processes around the PIL,\nwhere the three dimensional magnetic reconnection develops. In particular it is still not clear\nthe reason for observed dynamics of the electric currents. To explain it one need data-driven\nMHD simulations to interprete observations. We also should focus on comparison between\nmagnetic \feld dynamics and acceleration process with plasma heating. However, to achieve\nprogress in such comparison we should have vector magnitograms with time cadence of one\norder better than the avalaible ones (135 seconds). Without new generation instruments we\ncan only investigate magnetic reconnection at the PIL indirectly using di\u000berent assumptions.\nIn conclusion, it is also worth noting that today we have a lot of models of the eruptive\n\rares, but we have a lack of models explaining three dimensional magnetic reconnection (in\nparticular, TCMR) in the conditions similar to the \rare studied in this paper. Observational\nresults in our two papers can be used as a input data and expected output for future solar\n\rare models describing three dimensional magnetic restructuring at the PIL. We think that\nit is important to develop the new self-consisting models of \rare energy release reproducing\n3D magnetic reconnection in the PIL with strong magnetic \feld, spatial \flamentation of\nenergy release, formation of high energy density populations of nonthermal electrons and\nappearance of the super-hot plasma.{ 33 {\nWe are grateful to the teams of HMI/SDO, AIA/SDO, RHESSI, SOT/Hinode, NoRP\nand GOES for the available data used. We thank Drs A.V. Artemyev and D.Y. Kolotkov for\nfruitful discussions. We also appreciate to the anonymous reviewer for a number of useful\ncomments, which helped to improve the paper. This work is supported by the Russian\nScience Foundation under grant No. 17-72-20134.\nREFERENCES\nAbramenko, V. I., Gopasiuk, S. I., & Ogir', M. B. 1991, Sol. Phys., 134, 287\nAlfv\u0013 en, H., & Carlqvist, P. 1967, Sol. 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M. 2018, Journal of Atmospheric and Solar-Terrestrial\nPhysics, 174, 17\nThis preprint was prepared with the AAS L ATEX macros v5.2.{ 40 {\n110100\n10-910-810-710-6\n110100\n10-610100\n10-610-5\n0 2 4 6 8012345\n050100150200\n10 12 14 16 1801234\n020406080\n25 30 35 40 45 50 55-1.0- 5 0.00.51.0\ndI /dt and dI /dt, 10 Watts m s1-8 0.5-4-8 -2 -1I and I , Watts m1-8 0.5-4-2\ncountrate, counts detector-1Radio flux, sfuNoRP data:\n2 GHz\n3.75 GHz\n9.4 GHz\n17 GHz\n35 GHzGOES 1-8 A\nGOES 0.5-4 A\nRHESSI 25-50 keVGOES\ntime derivative:\n1-8 A\n0.5-4 Aa2)a1) b1)\nb2) c2)c1)\n10 20 60\ntime, minutes since 15-mar-2015   22:40:31 UT1 subflarest2 subflarend3 subflarerd\nno RHESSI data\n(night time)\nFig. 1.| Time pro\fles of the M1.2 solar \rare on 15 March 2015 for three sub\rares (columns\na-c). a1-c1) GOES 1-8 \u0017A lightcurve (thick black line; arbitrary units) and NoRP Stokes I\ntime pro\fles at 2, 3.75, 9.4, 17, and 35 GHz (colors are indicated in the plot). a2-c2) Time\nderivatives of the GOES lightcurves in both bands (black and grey) and RHESSI lightcurve\nin the energy band of 25-50 keV in arbitrary units (red histogram). In panel c2 RHESSI\ndata are not plotted due to night time.{ 41 {\n470 480 490 500 510\nX, arcsec\n470 480 490 500 510\nX, arcsec\n-240-220-200-180\nY, arcsec\n470 480 490 500 510\nX, arcsec-240-220-200-180\nY, arcsec\n-230-220-210-200-190-180\n470 480 490 500 510\nX, arcsec-230-220-210-200-190-180\n-1.5-1.0-0.50.00.51.01.5\nBz, kG\n0.60.81.01.21.41.6\nBh, kG\n470 480 490 500 510\nX, arcsecHMI, ( 1 - 53:57) 22: 4 :57 UT\nRHESSI, ( 5 08-45:38) 22: 4 : UTHMI, ( 1 - 53:57) 22: 4 :57 UT\nRHESSI, ( 5 40-46:08) 22: 4 : UTHMI, ( 1 - 53:57) 22: 4 :57 UT\nRHESSI, ( 6 18-47:00) 22: 4 : UTHMI, ( 1 - 53:57) 22: 4 :57 UT\n22: 4 : UTHMI, ( 1 - 53:57) 22: 4 :57 UT HMI, ( 1 - 53:57) 22: 4 :57 UT\nRHESSI, 22:(53:28 - 55:32) UTHMI, ( 1 - 53:57) 22: 4 :57 UT\nRHESSI, 22:(51:52 - 53:12) UTHMI, ( 1 - 53:57) 22: 4 :57 UT\na b c d e\nFig. 2.| Location of the X-ray sources detected by RHESSI in the 6 \u000012 (red) and 25\u000050 keV\n(light blue) energy bands in three time intervals (from left to right) during the \frst sub\rare\nof the 15 March 2015 \rare studied. Contours correspond to 50, 70, and 90 % levels of\nthe maximal X-ray intensity for each map. The corresponding time intervals are indicated\nabove the top panels. Top and bottom panels present HMI vector magnetograms showing\ndistributions of the vertical and horizontal magnetic \feld components on the photosphere,\nrespectively. The photospheric PIL is shown by the white curves. Note that the \feld-of-view\nsize is centered and adjusted relative to the RHESSI contours and slightly di\u000bers for the left\nand other four panels.{ 42 {\n470 480 490 500 510\nX, arcsec\n470 480 490 500 510\nX, arcsecHMI, ( 1 - 53:57) 22: 4 :57 UT\nRHESSI, 22:(51:52 - 53:12) UTHMI, ( 1 - 53:57) 22: 4 :57 UTHMI, ( 1 - 53:57) 22: 4 :57 UT\nRHESSI, ( 5 08-45:38) 22: 4 : UTHMI, ( 1 - 53:57) 22: 4 :57 UT\nRHESSI, ( 5 40-46:08) 22: 4 : UTHMI, ( 1 - 53:57) 22: 4 :57 UT\nRHESSI, ( 6 18-47:00) 22: 4 : UT\n-240-220-200-180\nY, arcsec\n470 480 490 500 510\nX, arcsec-240-220-200-180\nY, arcsec\n-230-220-210-200-190-180\n470 480 490 500 510\nX, arcsec-230-220-210-200-190-180\n-0.10-0.050.000.050.10\njz, A/m2\n0.40.60.81.01.21.4\ngradhBz, kG/Mm\n470 480 490 500 510\nX, arcsecHMI, ( 1 - 53:57) 22: 4 :57 UT HMI, ( 1 - 53:57) 22: 4 :57 UT\nRHESSI, 22:(53:28 - 55:32) UT\nab c d e\nFig. 3.| Comparison between the RHESSI X-ray contour maps in the energy bands of 6-12\n(red) and 25-50 keV (light blue) for three di\u000berent time intervals (indicated above the top\npanels) and the maps showing distribution of the vertical electric currents (top panels) and\nhorizontal gradient of vertical magnetic \feld component (bottom panels) deduced from the\n720-second HMI vector magnetogram. The photospheric PIL is shown by the black (top) and\nwhite (bottom) curves. Contours correspond to 50, 70, and 90 % levels of the maximal X-ray\nintensity for each map. Note that the \feld-of-view size is centered and adjusted relative to\nthe RHESSI contours and slightly di\u000bers for the left and other four panels.{ 43 {\n2015-03-15T22: : 41 57 UT + 12 min\n470 480 490 500 510\nX,arcsec-220-210-200-190-180-170\nY, arcsec\n2015-03-15T22:5 : 3 57 UT + 12 min\n470 480 490 500 510\nX,arcsec\nFig. 4.| Comparison between Ca II SOT/Hinode cumulative images (background) and\ncontour maps of the PVEC and the PIL (black curve) deduced from the HMI vector mag-\nnetograms. The left and right panels show two subsequent time intervals. The red and\nblue contours correspond to the negative and positive PVEC density (15, 45, 75, 105, and\n135 mA/m2), respectively. The SOT cumulative image is a result of summing Ca II images\ntaken in time interval of 12 min corresponding to the HMI vector magnetograms.{ 44 {\n-13.0-12.5-12.0-11.5\nY [degrees]\n29.6 29.8 30.0 30.2 30.4 30.6 30.8\nX [degrees]-13.0-12.5-12.0-11.5\nY [degrees]\n-13.0-12.5-12.0-11.5\nY [degrees]\n29.6 29.8 30.0 30.2 30.4 30.6 30.8\nX [degrees]\n29.6 29.8 30.0 30.2 30.4 30.6 30.8\nX [degrees]\n29.6 29.8 30.0 30.2 30.4 30.6 30.8\nX [degrees]\n-2000-1000010002000\nBz, Gauss\n500100015002000\n|B|, Gauss\n29.6 29.8 30.0 30.2 30.4 30.6 30.8\nX [degrees]\n50010001500\nBhoriz, Gauss2015.03.15 23:59:57 UT 2015.03.15 23:19:27 UT 2015.03.15 22:54:42 UT 2015.03.15 22:47:57 UT 2015.03.15 22:41:12 UT\n10 Mm 10 Mm 10 Mm 10 Mm 10 Mm\n10 Mm 10 Mm 10 Mm 10 Mm 10 Mm10 Mm 10 Mm 10 Mm 10 Mm 10 Mm\nFig. 5.| Time sequence (time is growing from left to right) of magnetic \feld component\nmaps, deduced from HMI 135-second vector magnetograms and reprojected onto the heli-\nographic grid. The time is indicated above the top panels. The top, middle and bottom\npanels correspond to the vertical, horizontal magnetic component and absolute value of the\nmagnetic \feld, respectively. The photospheric PIL is shown by the black (top) and cyan\n(middle, bottom) curves. The black and white contours in the left column show the region-\nof-interest (similar to the black contour in Figure 8(a) where we calculated magnetic \ruxes\nshown in Figure 6.{ 45 {\n2.2•10212.3•10212.4•10212.5•1021\n1.2•10211.4•10211.6•10211.8•10212.0•1021\n0 50 100 150 200 25002•10194•10196•10198•10191•10202.8•10212.9•10213.0•10213.1•10213.2•1021\n2.5•10212.6•10212.7•10212.8•10212.9•1021\n0 50 100 150 200 25001•10202•10203•10204•10205•10200 50 100 150 200 2505.0•10205.5•10206.0•10206.5•1020\n0 50 100 150 200 2509.50•10201.00•10211.05•10211.10•10211.15•10211.20•1021\n|B|S, |B|>2000 G|B|S, |B|>1000 G|B|S, |B|>0 G\nB S, B >2000 Gh hB S, B >1000 Gh hB S, B >0 Gh h|B |S, B <0z z|B |S, B >0z z\nMagneticflux, Mx\nMagneticflux, Mx\nMagneticflux, MxMagneticflux, Mx\ntime, min (from 2015.03.15 20:59:57 UT)a) b)\nc1)\nc2)\nc3)d1)\nd2)\nd3)time, min (from 2015.03.15 20:59:57 UT)\ntime, min (from 2015.03.15 20:59:57 UT) time, min (from 2015.03.15 20:59:57 UT)\nFig. 6.| Time pro\fles of magnetic \ruxes in the PIL region around the strongest PVEC. To\ncompare with the real magnetic \rux BzS, we introduced the nominal magnetic \rux for the\nhorizontal component and for the absolute value of the magnetic \feld. The analyzed region\nis shown by the black contour in Fig. 8(a). Panels show the following information: (a) and\n(b) magnetic \ruxes for the vertical magnetic \feld components with positive and negative\nsign, respectively. (c1{c2) Nominal magnetic \ruxes for the horizontal component summing\npixels with Bh>0, 1000, and 2000 G, respectively. (d1{d2) Nominal magnetic \ruxes the\nabsolute value summing pixels with jBj>0, 1000, and 2000 G, respectively. The blue curve\nin panels (a{d) is the GOES X-ray lightcurve (1-8 \u0017A) in arbitrary units.{ 46 {\n2015.03.15 22:41:12 UT\n-13.0-12.5-12.0-11.5\nY [degrees]\n29.6 29.8 30.0 30.2 30.4 30.6 30.8\nX [degrees]-13.0-12.5-12.0-11.5\nY [degrees]\n2015.03.15 22:47:57 UT\n29.6 29.8 30.0 30.2 30.4 30.6 30.8\nX [degrees]\n2015.03.15 22:54:42 UT\n29.6 29.8 30.0 30.2 30.4 30.6 30.8\nX [degrees]\n2015.03.15 23:19:27 UT\n29.6 29.8 30.0 30.2 30.4 30.6 30.8\nX [degrees]\n-0.020.000.02\njz, A/m22015.03.15 23:59:57 UT\n2004006008001000\n(gradB)horiz, Gauss/Mm\n29.6 29.8 30.0 30.2 30.4 30.6 30.8\nX [degrees]10 Mm10 Mm 10 Mm 10 Mm 10 Mm 10 Mm\n10 Mm 10 Mm 10 Mm 10 Mm\nFig. 7.| Temporal sequence (corresponding time is indicated above the top pannels) of\nmaps showing spatial distributions of PVEC (top panels) and horizontal gradient of the\nvertical magnetic \feld component (bottom panels). These maps are deduced from the HMI\n135-sec vector magnetograms and reprojected onto the heliographic grid. The photospheric\nPIL is shown by the black (top) and white (bottom) curves. The orange contours mark\nPVEC density levels of 39, 62, 85, and 115 mA/m2for bothjzsigns.{ 47 {\n-0.020.000.02jz, A/m22015.03.15 21:17:57 UT\n26 28 30 32 34 36\nX [degrees]-16-14-12-10-8\nY [degrees]a)\n>1 sigma\n2.42.62.83.03.2\nI, A 1012>3 sigma\n1.41.61.82.02.22.4>5 sigma\n0 8 .1.01.21.41.6\n95100105110115120\nS, Mm2\n25303540455055\n10152025\n0 50 100 150 200\ntime, min (\"0\": 2015.03.15 20:59:57 UT)2426283032\naveraged jz, A/m m2\n0 50 100 150 200\ntime, min (\"0\": 2015.03.15 20:59:57 UT)45505560\n0 50 100 150 200\ntime, min (\"0\": 2015.03.15 20:59:57 UT)60708090100sigma = 0.011 A/m2\n-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06\njz, A/m20200400600800\nnamber of pixels\nGOES 0.1-0.8 nmj >0zj <0zb)\nc1\nd1\ne1 e2 e3d2 d3c2 c3\nFig. 8.| Temporal dynamics of the total PVEC Izin the \rare region (marked by the thick\nblack contour in the panel (a) is shown in the panels (c1){(c3). Temporal dynamics of the\ntotal area (S) of the regions with the enhanced PVEC is shown in the panels (d1){(d3). The\npanels (e1){(e3) present temporal pro\fles of the estimated averaged (or e\u000bective) PVEC\ndensity (the ratio of the total PVEC Izto the total area S). The measurement errors are\nshown by the vertical bars. Three panels (from left to right) in each raw (c{e) show cases\nconsidering the PVEC density above the threshold values 1 \u001b(jz), 3\u001b(jz), and 5\u001b(jz), respec-\ntively. The thin black rectangle in the right-bottom corner of the panel (a) corresponds to\nthe non-\raring (background) region where we calculate the PVEC distribution to determine\nthe noise level \u001b(jz). This distribution is shown in the panel (b) by the histogram, where\nthe solid line is a Gaussian \ft. The blue curve in panels (c{e) is the GOES X-ray lightcurve\n(1-8\u0017A) in arbitrary units.{ 48 {\na)2015.03.15 22:41:12 UT\nb)2015.03.15 23:14:57 UT\nc)\nd)\nFig. 9.| Magnetic \feld lines in the PIL region calculated from the NLFFF magnetic \feld\nextrapolations using the HMI 135-sec vector magnetograms as the boundary condition for\ntwo di\u000berent times just before the \rare impulsive phase (a) and during main \rare X-ray\nemission peak (b). The regions of strong electric currents \rowing along the magnetic \feld\nlines for the same times are shown by the white semi-transparent surfaces with the constant\nlevel of electric current density of 55 mA/m2in (c) and (d), respectively. The base maps\nshow the distributions of vertical electric current density on the photosphere by blue-red\n(negative-positive) color palette. The thick black curves mark the photospheric PIL.{ 49 {\n2015-03-15□22:45:19.21 UT\n-240-220-200-180\n2015-03-15□22:46:31.21 UT\n 2015-03-15□22:47:43.21 UT\n 2015-03-15□22:48:55.21 UT\n2015-03-15□22:50:07.21 UT\n-240-220-200-180\n2015-03-15□22:51:19.24 UT\n 2015-03-15□22:52:31.21 UT\n 2015-03-15□22:53:43.21 UT\n2015-03-15□22:54:55.21 UT\n470 480 490 500 510 520 530-240-220-200-180\n2015-03-15□22:56:07.21 UT\n470 480 490 500 510 520 530\n2015-03-15□22:57:19.24 UT\n470 480 490 500 510 520 530\n2015-03-15□22:58:31.21 UT\n470 480 490 500 510 520 530\nX, arcsec X, arcsec X, arcsec X, arcsecY, arcsecY, arcsecY, arcsec\nFig. 10.| Temporal sequence of binary maps (black-white background images) showing\nthe \rare ribbon positions deduced from the AIA/SDO UV 1700 \u0017A images. Corresponding\ntimes are shown above the panels. Pixels in these maps have only two values: 0 or 1.\nValue 1 means that this pixel belongs to the \rare ribbons. The ribbons were extracted\nby thresholding images with the threshold value of 3800 DNs. The red and blue contours\n(23, 39 and 54 mA/m2) correspond to the negative and positive PVEC, respectively. The\nphotospheric PIL is marked by the cyan curves.{ 50 {\n0 2 4 6 8 102468101214\nS, cm1017 2\n10 12 14 16 18 202468101214\n25 30 35 40 45 50 55 6046810121402468\nIntensity, 10 DNs6\n012345\n012345\na1) b1) c1)\nc2) b2) a2)1 subflarest2 subflarend3 subflarerd\nGOES 1-8 A\nGOES 1-8 Atime derivativelightcurve\ntime, minutes since 15-mar-2015   22:40:31 UTintensity\nAIA 1700 A\nribbons area\nAIA 1700 A\nFig. 11.| Time pro\fles are plotted for three sub\rares (columns a-c). The temporal dynamics\nof the total UV intensity of the \rare ribbons is shown in panel (a1-c1) by the black histogram.\nTemporal dynamics of the \rare ribbon area (black histogram) deduced from the AIA UV\n1700 \u0017A images (a2-c2). The area is calculated as a sum of pixels with the intensity values\nhigher than the threshold of 3800 DNs (the background images in Fig. 10). The cyan and\nblack lines mark the GOES 1-8 \u0017A lightcurve and its time derivative, respectively.{ 51 {\n2015-03-15 22:45:19 UT\n-240-220-200-1802015-03-15 22:45:43 UT 2015-03-15 22:46:07 UT\n2015-03-15 22:46:31 UT\n470 480 490 500 510-240-220-200-1802015-03-15 22:46:55 UT\n470 480 490 500 5102015-03-15 22:47:19 UT\n470 480 490 500 510\nX, arcseconds X, arcseconds X, arcsecondsY, arcseconds Y, arcseconds2015-03-15 22:45:19UT\na) b) c)\nd) e) f)\nFig. 12.| Temporal sequence of AIA 1700 \u0017A images (white-black background maps) showing\npositions of the \rare ribbons during the \frst sub-\rare of the 15 March 2015 \rare studied.\nThe corresponding time is shown above the panels. The red and blue contours (23, 39 and\n54 mA/m2) correspond to the negative and positive PVEC, respectively. The photospheric\nPIL is marked by the cyan curves.{ 52 {\n-6-4-202468\nMagnetic flux change, Mx/s1018\n-20246\n-2-1012345\n0.20.40.60.81 0 .1 2 .\n<|B|>, <Bz>, <Bh>, G k\n0.20.40.60.81 0 .1 2 .\n0.20.40.60.81 0 .\n0 2 4 6 8 1002468101214\n|B| , B S S, B S, 10 Mxh z20\n10 12 14 16 18 20051015\n25 30 35 40 45 50 55 600510151 subflarest2 subflarend3 subflarerd\nGOES 1-8 A\nGOES 1-8 Atime derivativelightcurve a1) b1) c1)\na2)\na3)b2)\nb3)c2)\nc3)\ntime, minutes since 15-mar-2015   22:40:31 UTd(B S)/dt for B <0z z\nd(B S)/dt for B >0z z\n|B| , S , BhSB Sz\n< > |B| , ,<Bh><B >z\nFig. 13.| Time pro\fles are plotted for three sub\rares (columns a-c). (a1-c1) Time deriva-\ntives of the (vertical) magnetic \rux inside the \rare ribbons deduced by the thresholding\nAIA UV 1700 \u0017A images (Fig. 10). The negative and positive \ruxes are marked by the black\nand red colors, respectively. (a2-c2) Temporal dynamics of the magnetic \ruxes, which are\ncalculated for the magnetic \feld absolute value (black), vertical (orange) and horizonal (red)\ncomponents. (a3-c3) Time pro\fles of the average magnetic \feld values in the \rare ribbons.\nColors mark di\u000berent magnetic \feld components, similar to (a2-c2). The cyan and blue lines\nmark the GOES 1-8 \u0017A lightcurve and its time derivative, respectively.{ 53 {\n0123456\nIz, A 1011\n0123456\n00.51.01.52.02.5\n20406080100\njz, A/m m2\n0 2 4 6 8 1000.20.40.60.81.01.21.4\nS, cm1017 2\n10 12 14 16 18 2000.20.40.60.81.01.2\n25 30 35 40 45 50 55 6000.10.20.30.420406080100\n20406080100GOES 1-8 A\nGOES 1-8 Atime derivativelightcurvea1)\na2)\na3)b1) c1)\nb2) c2)\nb3) c3)1 subflarest2 subflarend3 subflare\ntime, minutes from 15-mar-2015   22:40:31 UTI <0z\nI >0z\nj <0z\nj >0z\nS(j <0)z\nS(j >0)z\nFig. 14.| Time pro\fles are plotted for three sub\rares (columns a-c). (a1-c1) Time pro\fles of\nthe total PVEC inside the \rare ribbons. (a2-c2) Time pro\fles of the average PVEC density\ninside the \rare ribbons. (a3-c3) Time pro\fles of the total area characterizing the considered\nregions with strong PVEC within the \rare ribbons. Estimates of the measurement errors\nare shown by the vertical bars. The red and black colors correspond to positive and negative\nelectric currents, respectively. The cyan and blue lines mark the GOES 1-8 \u0017A lightcurve and\nits time derivative, respectively.{ 54 {\nPIL\nBxB┴\nB||J||vinvA\nin owflout owfl\nδLlFootpoint\nemission sources\nGeometry of magnetic reconnection regionMagnetic arcade over the\nprimary magnetic reconnection region\nMagnetic loops interacting the PIL aboveBin\nTether-cutting magnetic\nreconnection at the PILa)\nb)\nFig. 15.| This scheme presents geometry of the magnetic reconnection region discussed.\nPanel (a) describes the general magnetic \feld topology in the frame of the TCMR scenario.\nPanel (b) shows the magnetic \feld structure and geometry of the magnetic reconnection\nregion.{ 55 {\n1.01.52.02.53.03.5\nlog10(I [DNs])\n0 20 40 60\ntime, min (from 2015-03-15T22:00:02.57Z)0510152025\nX, Mm (along observational slit)\nAIA 94 UT Å 2015-03-15 22:22:26.57\n460 480 500 520 540\nX, arcsec-240-220-200-180\nY, arcseca)\nb)\nFig. 16.| (a) The UV AIA 94 \u0017A image of the active region made around 20 min before the\n\rare impulsive phase onset, showing the presence of the bright elongated source above the\nPIL. The white horizontal line marks the slit used to make the time-distance plot, shown\non (b), to trace the dynamics of the \rare energy release around the PIL. To show the time-\ndistance plot with more contrast we overplotted it with the black contours. Both color plots\non (a) and (b) are made on a logarithmic scale of intensities. Red curve in the right-bottom\npart of panel (b) marks the time derivative of the GOES 1-8 \u0017A lightcurve to show the \rare\nenergy release rate. Note that time is on a linear scale on the horizontal axis.{ 56 {\nSingle TCMR\nreconnection episodeFormation of a small magnetic arcade.\nExpansion of the region with\nthe strong horizontal magnetic componentlarge scale magnetic loop\nmoving upwards\na) b)MR site\nlarge scale magnetic loop moves upwards\nand interacts with overlying magnetic arcadePILOverlying magnetic arcade stops magnetic loops  moving upwards\nRibbons move out from the PIL\ntracing reconnected magnetic loopsFormation of the large-scale\nsheared magnetic arcade\nc)Single reconnection episode above initial MR site.\nIt leads to formation of the sheared magnetic arcadeMR siteExpansion of the are ribbons fl\nalong the PIL due to TCMR+ - +\n-\n-+PIL\nPIL\nFig. 17.| This scheme shows our explanations of the \rare energy release process. Panel\n(a) demonstrates the \rare onset as TCMR. Then upward moving magnetic \feld lines inter-\nact with the overlaying magnetic arcade (panel (b)) that leads to the secondary magnetic\nreconnection (panel (c)) and cause the \rare ribbons to move from the PIL."    },    {        "title": "2101.10628v2.Independent_Control_and_Path_Planning_of_Microswimmers_with_a_Uniform_Magnetic_Field.pdf",        "content": "Independent Control and Path Planning of Microswimmers with\na Uniform Magnetic Field\nLucas Amoudruz Petros Koumoutsakos*\nL. Amoudruz, Prof. P. Koumoutsakos\nComputational Science and Engineering Laboratory, ETH Z¨ urich, CH-8092, Switzerland.\nEmail: petros@seas.harvard.edu\nL. Amoudruz, Prof. P. Koumoutsakos\nJohn A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA.\nKeywords: micro-swimmers, reinforcement learning, magnetically driven\nArtificial bacteria flagella (ABFs) are magnetic helical micro-swimmers that can be remotely controlled via a uniform, rotating mag-\nnetic field. Previous studies have used the heterogeneous response of microswimmers to external magnetic fields for achieving inde-\npendent control. Here we introduce analytical and reinforcement learning control strategies for path planning to a target by multiple\nswimmers using a uniform magnetic field. The comparison of the two algorithms shows the superiority of reinforcement learning in\nachieving minimal travel time to a target. The results demonstrate, for the first time, the effective independent navigation of realistic\nmicro-swimmers with a uniform magnetic field in a viscous flow field.\n1 Introduction\nThe magnetic control of micro-swimming devices [1, 2, 3, 4, 5] through micro-manipulation [6, 7], tar-\ngeted drug delivery [8, 9] or convection-enhanced transport [10], has created new frontiers for bio-medicine.\nA particularly promising technology involves corkscrew-shaped magnetic micro-swimmers (artificial bac-\nterial flagella (ABFs)) that propel themselves when subjected to a rotating magnetic field [11]. Rotat-\ning magnetic fields can form propulsive gradients and they are arguably preferable to alternatives, such\nas electric fields, for in-vivo conditions [12, 13]. However, the independent, yet coordinated, control of\nindividual ABFs is challenging as it requires balancing between the magnetic forces and the hydrody-\nnamic interactions between the swimmers while the employed magnetic fields are practically uniform\nover lengths of few micrometers. We note that independent navigation of mm-sized micro-swimmers has\nbeen shown in [14] through experiments and simulations, while in [15] a reinforcement learning (RL) al-\ngorithm was applied to adjust the velocity of an idealized swimmer in simulations with one way coupling\nwith a complex flow field. Control of swimmers using two-way coupling and RL have been demonstrated\nwith linked-spheres at low Reynolds numbers [16] and for artificial fish in macro-scales [17]. Similarly,\ngenetic algorithms have been used to navigate micro-swimmers towards high concentrations of chemicals\n[18].\nThe problem of heterogeneous micro-robots navigation via a uniform input has been studied in two di-\nmensions on surfaces [19] and in a fluid at rest [20]. The steering of two micro-propellers along two dis-\ntinct paths in 3 dimensions has been accomplished with the help of magnetic fields gradients [21]. These\nadvances exploited the heterogeneous response of micro-swimmers to a uniform input to achieve inde-\npendent trajectories along a prescribed path. These control methods are based on short horizon objec-\ntives (stay on the prescribed path) and do not provide the trajectory that minimizes the travel time to a\ntarget position, particularly in the presence of a background flow. In addition, strong background flows\nrestrict the set of feasible paths for given micro-swimmers. To the best of our knowledge the steering of\nmultiple micron-sized swimmers towards a target in a minimal time under a background flow and a uni-\nform magnetic field has not been reported before.\nIn this work, we present two methods to independently guide two micro-ABFs towards a single target in\nthe presence of a uniform magnetic field. The two methods rely on simulations of swimming ABFs using\nan ordinary differential equation (ODE) model. The model is calibrated with the method of dissipative\nparticle dynamics (DPD) [22, 23], taking into account the particular geometry of the swimmers and their\ninteractions with the viscous fluid. We first present a semi-analytical solution for the simple yet instruc-\ntive setup of multiple, geometrically distinct ABFs in free space, with zero background flow. This result\n1arXiv:2101.10628v2  [physics.flu-dyn]  4 Jan 2022enables understanding of the design constraints for the ABFs necessary for independent control and how\ntheir geometric characteristics relate to their travel time. We then employ RL to control multiple ABFs\ntrajectories in a broad range of flow conditions including a non-zero background flow.\n2 Artificial bacterial flagella\nThe ABFs are modeled as microscopic rigid bodies of length lwith position xand orientation q(rep-\nresented by a quaternion), immersed in a viscous fluid and subjected to a rotating, uniform, magnetic\nfield. We estimate that the magnetic and hydrodynamic interactions between ABFs are orders of mag-\nnitude smaller than those due to the magnetic field for dilute systems (see supplementary material) and\nwe ignore inertial effects due to their low Reynolds number (Re ≈10−3). Following this approximation,\nthe system is fully described by the position and orientation of the ABFs.\nAdditionally, the linear and angular velocities of the ABF, VbandΩb, are directly linked to the external\nforce and torque, FbandTb, via the mobility matrix [24],\n/bracketleftbigg\nVb\nΩb/bracketrightbigg\n=/bracketleftbigg\n∆Z\nZTΓ/bracketrightbigg/bracketleftbigg\nFb\nTb/bracketrightbigg\n, (1)\nwhere the superscriptbindicates that the quantity is expressed in the ABF frame of reference, for which\n∆,Zand Γ are diagonal. The matrices Γ and Zrepresent the application of torque to changing the an-\ngular and linear velocity, respectively. The ABFs are propelled by torque applied through a magnetic\nfield and we assume that it can swim only in the direction of its main axis so that Zhas only one non-\nzero entry ( Z11). The coefficients in the mobility matrix are often estimated by empirical formulas for\nlow Reynolds number flows [25]. Here we estimate the components of the mobility matrix for the specific\nABF by conducting flow simulations using DPD [22, 23], which we validate against experimental data of\n[8] (see supplementary material). We remark that the shape (pitch, diameter, length, thickness) of the\nABF influence the elements of these matrices and the present approach allows to account for these ge-\nometries.\nThe ABF with a magnetic moment mis subjected to a uniform magnetic field Band hence experiences\na torque\nT=m×B. (2)\nNo other external force is applied to the ABF, hence F=0. Combining eq. (1) with the kinematic equa-\ntions for a rigid body gives the following system of ODEs:\n˙x=V, (3a)\n˙q=1\n2q⊗ˆΩ, (3b)\nVb=ZTb, (3c)\nΩb= ΓTb, (3d)\nwhere⊗denotes the quaternion product, and ˆΩthe pure quaternion formed by the vector Ω. The trans-\nformations between the laboratory frame of reference and that of the ABF are given by:\nTb=R(q)T, (4a)\nm=R(q⋆)mb, (4b)\nV=R(q⋆)Vb, (4c)\nΩ=R(q⋆)Ωb, (4d)\nwhereq⋆is the conjugate of qandR(q) is the rotation matrix that corresponds to the rotation by a quater-\nnionq[26]. The system of differential equations (2) and (3) is advanced in time with a fourth order Runge-\nKutta integrator.\n20.00 0.25 0.50 0.75 1.00\nω/ωc,10.0000.0020.0040.0060.008V/lωc,1\nFigure 1: Left: Dimensionless time averaged forward velocity of two ABFs, differing in shape and magnetic moment,\nagainst the field rotation frequency (in units of the step-out frequency of the first swimmer, ωc,1). Right: The ABFs ge-\nometries. The arrows represent the magnetic moment of the ABFs.\nWe note that when simulating multiple non-interacting ABFs in free space, we use the above ODE sys-\ntem for each swimmer with the common magnetic field but different mobility coefficients and magnetic\nmoments.\n3 Forward velocity\nABFs were designed to swim under a rotating, uniform magnetic field [27, 11]. We first study this sce-\nnario by applying the field B(t) =B(0,cosωt,sinωt) to ABFs initially aligned with the xaxis of the\nlaboratory frame. Note that in the later sections, the magnetic field is able to rotate in any direction\nso that the swimmers can navigate in three dimensions. We consider two ABFs with the same length\nbut different pitch and magnetic moments, as shown in fig. 1. In both cases, the magnetic moment is\nperpendicular to the helical axis of the ABF. Under these conditions, by symmetry of the problem, the\nswimmers swim along the xaxis. The difference in pitch results in different coefficients of the mobility\nmatrix and along with the different magnetic moments results in distinct propulsion velocities for the\ntwo ABFs. For each ABF velocity we distinguish a linear and a non-linear variation with respect to the\nfrequency of the magnetic field. First, the ABF rotates at the same frequency as the magnetic field and\nits forward velocity increases linearly with the frequency of the magnetic field, consistent with the low\nReynolds approximation [28, 8, 29, 3]. In the non-linear regime, the magnetic torque is no longer able to\nsustain the same frequency of rotation as the magnetic field. The onset of non-linearity depends on the\ngeometry and magnetic moment of the ABF as well as the imposed magnetic field. Indeed, the magni-\ntude of the magnetic torque is bounded while that of the hydrodynamic torque increases linearly with\nthe ABF angular velocity Ω. The torque imbalance at high rotation frequencies causes the ABF to slip,\nresulting in an alternating forward and backward motion (see supplementary material). Increasing the\nfrequency further increases the effective slip and accordingly decreases the forward velocity. The two\nregimes are distinguished by the step-out frequency ωccorresponding to the maximum forward velocity\nof the ABF.\nThe differences in propulsion velocities for the ABFs can be exploited to control independently their tra-\njectories. The slope V/ω in the linear regime depends only on the shape of the ABF. The step-out fre-\nquency depends on both the shape and the magnetic moment (it can also be changed by varying the\nsurface wetability of the ABF [30]). These two properties can be chosen such that the forward velocities\nof two ABFs react differently to the magnetic rotation frequency (fig. 1). By changing ω, it is then pos-\nsible to control the relative velocities of the two ABFs: one is faster that the other in one regime while\nthe opposite occurs in an other regime. This simple observation constitutes the key idea for independent\ncontrol of several ABFs even with a uniform magnetic field. We remark that, while this potential has\nbeen previously identified [28, 30, 31, 32, 33], the control of similar systems have been performed in the\nsimple case of free space, non interacting propellers and no background flow [21, 20]. To the best of our\n3knowledge, this is the first time that such independent controlled navigation of multiple micro-swimmers\nis materialised in three dimensions with a complex background flow. In the following sections we propose\ntwo methods to tackle the problem of steering ABFs towards a target in a minimal amount of time.\n4 Independent control I: semi-analytical solution\nIn the absence of an external flow field, we derive a semi-analytical strategy for the navigation of NABFs\ntowards a particular target. Each ABF has a distinct magnetic moment and without any loss of general-\nity, we set the target position of all swimmers to the origin and define the initial position of the ithABF\nasx(i). We assume that the time required by one ABF to align with the rotation direction of the field is\nmuch smaller than |x(i)|/v, wherevis the typical forward velocity of the ABF. The proposed strategy\nconsists in gathering all ABFs along one direction nkat a time, such that x(i)·nk= 0,i= 1,2,...,N af-\nter phasek. We choose a sequence of orthogonal directions, nk·nk/prime=δkk/prime. The choice of the orientations\nofnkis not restricted to the basis vectors of the laboratory frame and is described at the end of this sec-\ntion. In three dimensions, the strategy consists of three phases, k= 1,2,3, until all ABFs have reached\ntheir target: they first gather on a plane, then on a line and finally to the target.\nAll ABFs are gathered along a given direction nkby exploiting the different forward responses of the\nABFs when we alternate the frequency of rotation of the magnetic field. More specifically, for NABFs,\nthe field rotates in the direction nkfortjtime units at frequency ωc,j,j= 1,2,...,N , whereωc,jis the\nstep-out frequency of the jthswimmer. We define the velocity matrix with elements Uij=Vi(ωc,j), de-\nnoting the velocity of swimmer iwhen the field rotates with the step out frequency of swimmer j. We\ncan relate the above quantities to the (signed) distances djcovered by the ABFs as\ndi=N/summationdisplay\nj=1sjtjUij,\nwheresj∈{− 1,1}determines if the field rotates clockwise/counterclockwise. Equivalently, the vector\nform of the above is d=Uβ, whereβj=tjsj. Settingdi=x(i)·nk, we can invert this linear system of\nequations for each phase kand obtain the times spent at each step-out frequency β=U−1Xnk, where\nwe have set Xij=x(i)\nj. We emphasize that this result holds only if the velocity matrix is invertible, re-\nstricting the design of the ABFs to achieve independent control. The total time spent at phase kis then\ngiven by\nT(nk) =N/summationdisplay\ni=1ti=N/summationdisplay\ni=1|βi|=/vextenddouble/vextenddoubleU−1Xnk/vextenddouble/vextenddouble\n1.\nThe yet unknown directions nk,k= 1,2,3, are chosen to minimize the total travel time. The directions\nare parameterized as nk=R(φ,θ,ψ )ek,k= 1,2,3, whereR(φ,θ,ψ ) is the rotation matrix given by\nthe three Euler angles φ,θandψ. Note that this choice of handedness of the three directions does not\ninfluence the final result. The optimal angles satisfy\nφ⋆,θ⋆,ψ⋆= arg min\nφ,θ,ψ3/summationdisplay\nk=1T(R(φ,θ,ψ )ek).\nWe solve the above minimization problem numerically with derandomised evolution strategy with covari-\nance matrix adaptation (CMA-ES) [34] (see supplementary material for the configuration of the opti-\nmizer).\n5 Independent control II: Reinforcement Learning\nWe now employ a RL approach to solve the problem introduced in section 4. Each of the NABFs is\ninitially placed at a random position xi∼N (x0\ni,σ),i= 1,2,...,N . The RL agent controls the mag-\n40204060r/lAnalytic RL\n0 2500 5000 7500 10000 12500 15000\ntωc,10.00.51.0ω/ωc,1\n0 2000 4000 6000 8000 10000 12000\ntωc,1Figure 2: Distance to target of the two controlled ABFs (in units of body length l) against dimensionless time\n(\n and\n ) in free space, zero background flow, and corresponding magnetic field rotation frequency (\n ),\nwhere ωc,1is the step-out frequency of the first swimmer.\nnetic field frequency of rotation and direction, and has the goal of bringing all ABFs within a small dis-\ntance (here two body lengths, d= 2l) from the target origin. This small distance is justified by the as-\nsumption of non-interacting ABFs. The agent sets the direction and magnitude of the magnetic field fre-\nquency every fixed time interval. An episode is terminated if either of the two conditions occur: (a) all\nABFs reached the target within a small distance d, or (b) the simulation time exceeds a maximum time\nTmax. The positions xiand orientations qiof the ABFs describe the state sof the environment in the RL\nframework. The action performed by the agent every ∆ ttime encodes the magnetic field rotation fre-\nquency and orientation for the next time interval. The reward of the system is designed so that all ABFs\nreach the target and the travel time is minimized. Additionally, a shaping reward term [35] is added to\nimprove the learning process. The training is performed using VRACER, the off-policy actor critic RL\nmethod described in [36]. More details on the method can be found in the supplementary material.\n6 Reaching the targets\nIn this section, we demonstrate the effectiveness of the two methods introduced in sections 4 and 5. We\nfirst consider 2 ABFs in free space with zero background flow. Figure 2 shows the distance of the ABFs\nto their target over time, and the corresponding magnetic field rotation frequency for both methods. In\nboth cases, the ABFs successfully reach their target. Interestingly, the rotation frequencies chosen by the\nRL agent correspond to the step-out frequencies of the ABFs. Indeed, these frequencies allow the fastest\nabsolute velocity difference between the ABFs, so it is consistent that they are part of the fastest solu-\ntion found by the RL method. Furthermore, the RL trained swimmer was about 25% faster than the\nsemi-analytical swimmer. We remark that the RL solution amounts to first blocking the forward motion\nof one swimmer while the other continues swimming (see fig. 3). The blocked swimmer is first reoriented\nsuch that its magnetic moment is orthogonal to the plane of the magnetic field rotation, thus the result-\ning magnetic torque applied to this swimmer is zero. On the other hand, the method presented in sec-\ntion 4 makes both ABFs swim at all time, even if one of them must go further from its target. In such\nsituations, the “blocking” method found by RL is advantageous over the other method.\nWe now employ the RL method in the case of 2 ABFs swimming in a background flow with non zero ve-\nlocity. The assumptions required for deriving the semi-analytical approach are violated and therefore we\ndo not use this approach in this case.\n5exeyezFigure 3: Trajectories of the ABFs from their initial positions (ABF representations) to the target area (sphere) obtained\nwith the two methods in three dimensions: semi-analytical (dotted lines) and RL (solid lines). The arrows show the suc-\ncessive axes of rotation of the magnetic field. The size of the ABFs has been scaled up by a factor of 7, for visualization\npurpose.\nIn the presence of a background flow u∞, eqs. (4c) and (4d) become\nV=R(q⋆)Vb+u∞(x),\nΩ=R(q⋆)Ωb+1\n2∇×u∞(x) +λ2−1\nλ2+ 1p×(E(x)p),\nwhere we approximated the rotation component by the effect of the flow on an axisymmetric ellipsoid of\naspect ratio λ(Jeffery orbits). Here E(x) =/parenleftbig\n∇u∞(x) +∇uT\n∞(x)/parenrightbig\n/2 is the deformation rate tensor of\nthe background flow evaluated at the swimmer’s position and p=R(q⋆)exis the orientation of the ellip-\nsoid. We used λ= 2 in the subsequent simulations. The background flow is set to the initial conditions\nof the Taylor-Green vortex,\nu∞(r) =\nAcosaxsinbysincz\nBsinaxcosbysincz\nCsinaxsinbycoscz\n, (5)\nwithA=B=C/2 =V1(ωc,1) anda=b=−c= 2π/50l. With these parameters, the maximum velocity\nof the background flow is larger than the maximum swimming speed of the ABFs.\nThe distances between the swimmers and the target over time are shown for the RL method on fig. 4.\nDespite the background flow perturbation, the RL method successfully navigates the ABFs to their tar-\nget. The magnetic action space exhibits a similar behavior as in the free space case: the rotation fre-\nquency of the magnetic field oscillates between the step out frequencies of both swimmers and never ex-\nceeds the highest of these frequencies, where the swimming performance would degrade considerably.\nThe trajectories of the ABFs seem to make use of the velocity field to achieve a lower travel time: fig. 4\nshows that the trajectories tend to be parallel to the velocity field. The RL method not only found a so-\nlution, but also made use of its environment to reduce the travel time.\n7 Robustness of the RL policy\nThe robustness of the RL method is tested against two external perturbations, unseen during the train-\ning phase. In both cases, the robustness of the method is measured in terms of success rate (expected\nratio between the number of successful trajectories and the number of attempts).\nFirst, a flow perturbation δu(r) =εu∞(r/p) is added to the background flow described in the previous\nsection, where εcontrols the strength of the perturbation and pcontrols the wave length of the pertur-\nbation with respect to the original one. Figure 5 shows the success rate of the RL approach against the\n60204060r/l\n0 2000 4000 6000 8000 10000\ntωc,10.000.250.500.751.00ω/ωc,1\nFigure 4: Left: Distance to target of the two controlled ABFs (in units of body length l) against dimensionless time\n(\n and\n ) in free space with the background flow described by eq. (5) and corresponding magnetic field ro-\ntation frequency (\n ), where ωc,1is the step-out frequency of the first swimmer. Right: Trajectories of the ABFs\nwith non-zero flow obtained with the RL method. The arrows represent the velocity field and the colors represent the mag-\nnitude of the vorticity field. The flow field is only shown for a distance less than 4 lfrom the trajectories, where lis the\nlength of the swimmers.\n0.0 0.2 0.4 0.6 0.8 1.0\nε0.00.20.40.60.81.0success rate\n10−1100101102103\nkBT/kBT00.00.20.40.60.81.0success rate\nFigure 5: Left: Success rate of the RL method to guide swimmers to their target against the flow perturbation strength ε,\nfor different wave numbers, p= 1/2 (\n ),p= 1 (\n ),p= 2 (\n ). Right: Success rate of the RL method to\nsteer swimmers to their target against thermal fluctuation kBT/kBT0.\nperturbation strength εfor different p. For large wave lengths ( p= 2), the RL agent is able to success-\nfully steer the ABFs to their target in more than 90% of the cases when the perturbation strengths of\nless than 20% of the original flow. In contrast, the success rate degrades more sharply for smaller wave\nlengths (p= 1/2), suggesting that the method is less robust for short wave length perturbations. The\nRL policy seems more robust to perturbations with the same wavelengths as the original flow ( p= 1) for\nlarge perturbation strengths: the success rate is above 30% even for large perturbations.\nAt small length scales, micro-swimmers are subjected to thermal fluctuations. We investigate the ro-\nbustness of the RL policy (trained with the background flow, eq. (5)) on swimmers subjected to ther-\nmal noise and background flow (eq. (5)). The thermal fluctuations are modeled as an additive stochastic\nterm to the linear and angular velocities of each swimmer, following the Einstein relation with the mo-\nbility tensor given by eq. (1). Defining the generalized undisturbed velocity ¯V= (V,Ω), the resulting\n7stochastic generalized velocities satisfy\nV=¯V+δV,\n/angbracketleftδVi/angbracketright= 0, i= 1,2,..., 6,\n/angbracketleftδVi,δVj/angbracketright=kBTMij, i,j = 1,2,..., 6,\nwhereMis the mobility tensor and kBTis the temperature of the system, in energy units. The above\nproperty is achieved by adding a scaled Gaussian random noise with zero mean to the velocities at every\ntime step of the simulation.\nThe success rate of the policy is shown in fig. 5 for various temperatures kBT, in units of the room tem-\nperaturekBT0. As expected, a large thermal noise causes the policy to fail at its task. Nevertheless, this\nfailure only occurs at relatively high temperatures: the success rate falls below 50% for kBT > 25kBT0,\nwhich is well above the normal operating conditions of ABFs. With temperatures below 2 kBT0, the RL\nmethod sustains a success rate above 99%. We remark that this robustness is achieved successfully even\nwith a policy trained with kBT= 0.\n8 Conclusion\nWe have presented two methods to guide multiple ABFs individually towards targets with a uniform\nmagnetic field. The semi-analytical method allows to understand the basic mechanisms that allow in-\ndependent control and we derive the necessary condition for the independent control of multiple ABFs:\ntheir velocity matrix must be invertible, a condition that can be accommodated by suitably choosing the\ngeometries of the swimmers. This result may help to optimize the shapes of ABFs to further reduce the\ntravel time.\nThe RL approach can control multiple ABFs in quiescent flow as well as in the presence of a complex\nbackground flow. Additionally, this approach is resilient to small flow perturbations and to thermal noise.\nWhen the background flow vanishes, the RL method recovers a very similar behavior as the first method:\nthe rotation frequency is alternating between the step-out frequencies of the swimmers. Furthermore,\nthe RL method could reach lower travel time than the first method by blocking one swimmer while the\nother is swimming. Steering an increased number of swimmers requires longer travel times according to\nthe first method. We thus expect that applying the RL approach to more than two swimmers requires\nlonger training times and might become prohibitively expensive as the number of swimmers increases.\nPossible solutions to this problem might include pre-training the RL agent with the policy found by the\nsemi-analytical method.\nThe current work focused on the simplified case of non-interacting swimmers. In practice, the ABFs may\ninteract hydrodynamically and magnetically with each other, encounter obstacles, evolve in confined ge-\nometries or experience time varying flows. Nevertheless, we expect the RL method to be a good candi-\ndate to overcome these variants, in the same way as it naturally handled the addition of a background\nflow.\nAcknowledgments\nWe acknowledge insightful discussions with Guido Novati (ETHZ) and his technical support for the us-\nage of smarties . We acknowledge support by the European Research Council (ERC Advanced Grant\n341117).\nConflicts of Interest\nThe authors declare no financial or commercial conflicts of interest.\nReferences\n[1] Q. Cao, X. Han, L. Li, Lab Chip 2014 ,142762.\n8REFERENCES\n[2] L. Yang, L. Zhang, Annual Review of Control, Robotics, and Autonomous Systems 2021 ,4, 1 null.\n[3] P. Tierno, R. Golestanian, I. Pagonabarraga, F. Sagu´ es, The Journal of Physical Chemistry B\n2008 ,112, 51 16525.\n[4] Y. Liu, D. Ge, J. Cong, H.-G. 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Tawakol, A. Klingner, N. El Gohary, B. Mizaikoff, M. Sitti,\nIEEE Robotics and Automation Letters 2018 ,3, 3 1703.\n[34] N. Hansen, S. D. M¨ uller, P. Koumoutsakos, Evolutionary computation 2003 ,11, 1 1.\n[35] A. Y. Ng, D. Harada, S. Russell, In ICML , volume 99. 1999 278–287.\n[36] G. Novati, P. Koumoutsakos, In Proceedings of the 36thInternational Conference on Machine Learn-\ning.2019 .\n10Supplementary Material: Independent Control and Path Plan-\nning of Microswimmers with a Uniform Magnetic Field\nLucas Amoudruz Petros Koumoutsakos*\nL. Amoudruz, Prof. P. Koumoutsakos\nComputational Science and Engineering Laboratory, ETH Z¨ urich, CH-8092, Switzerland.\nEmail: petros@seas.harvard.edu\nL. Amoudruz, Prof. P. Koumoutsakos\nJohn A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA.\n1 Neglecting the interactions between swimmers\nThe artificial bacterial flagella (ABFs) experience hydrodynamic drag forces and torques. Additionally,\nthey are subjected to a magnetic torque from the external, imposed, magnetic field. Furthermore, when\nmultiple swimmers compose the system, hydrodynamic and magnetic interactions occur. Here we esti-\nmate the magnitude of these interaction forces by considering a simplified representation of the ABFs\nimmersed in a fluid of viscosity η= 10−3Pa s, subjected to a rotating magnetic field of magnitude B=\n1 mT. To have a rough estimate of the hydrodynamic torque, we represent the ABFs as ellipsoidal bod-\nies of size 2 a×2b×2c= 10 µm×2µm×2µm. The magnetic fields rotates with angular velocity\nω= 10·2πs−1which is also the step-out frequency ωcof the ABFs. These choices correspond to simi-\nlar conditions as those found in experimental work for ABFs [1].\nConsidering the equivalent sphere of the ABF, with a radius R= (abc)1/3, the hydrodynamic torque\nmagnitude can be expressed as [2]\nTH= 8πηR3ω. (1)\nAt the step out frequency, the magnetic torque reaches its maximum magnitude,\nTM=mB, (2)\nwheremis the magnetic moment of the sphere. Equating eqs. (1) and (2) gives an estimate for the mag-\nnetic moment\nm=8πηR3ω\nB, (3)\nhencem≈8×10−15Am2.\nThe magnetic field generated by a dipole of moment mis given by\nBD(r) =µ0\n4πr3(3(m·ˆr)ˆr−m), (4)\nwhere ˆr=r/randr=|r|.\nTherefore, the magnetic torque exerted by one sphere of moment m2to another of moment m1(sepa-\nrated by a distance r) is given by\nTDD(r) =m1×BD(r) =µ0\n4πr3(3(m1·ˆr)(m2׈r)−m1×m2). (5)\nConsidering a small distance between the spheres r= 10a(five body lengths between the centers) and\nreplacing with the numerical values, we obtain TDD/TM≈10−5, hence we neglect the magnetic interac-\ntions between swimmers. Similarly, the magnetic force caused by the gradient of the magnetic field gen-\nerated by the dipole is negligible compared to that of the hydrodynamic drag.\nWe now estimate the hydrodynamic interactions between 2 rotating ABFs by computing the fluid veloc-\nity caused by the rotation of one at the other’s location. The magnitude of the velocity caused by the\nequivalent sphere of radius Rrotating with angular velocity ωis given by\nvint(r) =ωR3/r2. (6)\n1arXiv:2101.10628v2  [physics.flu-dyn]  4 Jan 20220 25 50 75 100\ntωc0.00.10.20.3x/lFigure 1: Displacement (in body lengths l) of an ABF against dimensionless time ( wcis the step out frequency) subjected\nto a constant rotating magnetic field, with frequency ω= 0.6ωc(\n ),ω=ωc(\n ) andω= 1.4ωc(\n ).\nThis rough estimate, at a distance r= 10a(5 body lengths), gives a flow velocity of about vint≈0.12µm s−1.\nABFs are able to swim at velocities of around one body length per second, that is v≈10µm s−1. Hence\nthe velocity due to the interactions amounts to about 1 .2% of the swimming velocity when the swim-\nmers are close to each other. This velocity decreases further when the swimmers are separated by larger\ndistances (0 .03% when they are separated by 10 body lengths), which is the most common configuration\nin the current problem. We therefore neglect the hydrodynamic interactions between ABFs in our simu-\nlations.\n2 Forward slip\nAs explained in the main text, the velocity of an ABF subjected to a uniform magnetic field rotating at\nconstant frequency first increases linearly (its rotation follows that of the magnetic field, hence its veloc-\nity is constant). After the frequency of rotation of the field reaches the critical value ωc, the hydrody-\nnamic forces that would be needed to follow the magnetic field frequency are too high, causing the ABF\nto “slip”. This can be seen on fig. 1. Below ωc, the velocity is constant, hence the displacement of the\nABF increases linearly with time. For higher magnetic field frequency, the ABF slips and hence perform\na back and forth motion. The time averaged velocity hence decreases.\n3 Quaternions\nQuaternions are useful mathematical tools to represent rotations in 3 dimensions [3]. Unlike the rotation\nmatrices, which have 9 components in 3 dimensions, quaternions store only 4 components, which make\nthem also desirable on a computational point of view. A quaternion qcan be represented as\nq=qw+qxi+qyj+qzk, (7)\nwhereqw,qx,qy,qz∈Rare real numbers and i2=j2=k2=ijk=−1. A common notation is to\ndecompose the quaternion into a scalar part and a vector part,\nq= (qw,q), (8)\nwhere q= (qx,qy,qz). A quaternion with zero scalar part is called a pure vector quaternion, and a quater-\nnion with the null vector is called a pure scalar quaternion. The conjugate of qis defined as q⋆= (qw,−q).\n2Furthermore, the norm of a quaternion is\n|q|=/radicalBig\nq2\nw+q2\nx+q2\ny+q2\nz. (9)\nThe product between two quaternions, noted ⊗, can be derived from the properties of i,j,k . In vector\nnotations, it is given by\nq⊗p= (qwpw−q·p,qwp+pwq+q×p), (10)\nwhere×denotes the vector product. Rotations can be represented by unit quaternions |q|= 1. The ro-\ntation of angle φaround the unit vector uis represented by the quaternion\nq=/parenleftbigg\ncosφ\n2,sinφ\n2u/parenrightbigg\n. (11)\nThe rotation of a vector xbyqis given in terms of quaternion product,\ny=q⋆⊗x⊗q, (12)\nwherex= (0,x) is the pure vector quaternion of xandyis the result of the rotation on x. The same\noperation can be written in matrix form,\ny=R(q)x, (13)\nwhereR(q) =E(q)GT(q) with\nE(q) =\n−qxqw−qzqy\n−qyqzqw−qx\n−qz−qyqxqw\n, (14)\nG(q) =\n−qxqwqz−qy\n−qy−qzqwqx\n−qzqy−qxqw\n. (15)\n4 Calibration of the propulsion matrix from dissipative particle dynamics\nThe mobility matrix presented in the main text links the external forces and torques applied to a rigid\nbody with its instantaneous linear and angular velocities, relatively to the flow:\n/bracketleftbigg\nVb\nΩb/bracketrightbigg\n=/bracketleftbigg\n∆Z\nZTΓ/bracketrightbigg/bracketleftbigg\nFb\nTb/bracketrightbigg\n. (16)\nThe elements of this mobility matrix depend on the shape of the body. In this work, we estimated the\nmobility matrix of ABFs of different pitch but with the same length and diameter. The body was im-\nmersed in a fluid at rest and subjected to rotations and translations with constant speed, and the hydro-\ndynamic torques and forces were collected. The flow solver, Mirheo [4], employs the dissipative particle\ndynamics (DPD) method [5], that we briefly summarize here for completeness. The flow is discretized\nintonparticles of mass, positions and velocities mi,riandvi, respectively, with i= 1,2...,n . We con-\nsider particles with the same mass, mi=m. They evolve in time according to Newton’s law of motion,\ndri\ndt=vi, (17a)\ndvi\ndt=1\nmifi, (17b)\n3where fiis the total force acting on the ithparticle,\nfi=/summationdisplay\nj/negationslash=ifC\nij+fR\nij+fR\nij, (18)\nfC\nij=aw(rij)eij, (19)\nfD\nij=−γwD(rij) (eij·vij)eij, (20)\nfR\nij=σξijwR(rij)eij, (21)\nwhere rij=ri−rj,rij=|rij|,eij=rij/rijandvij=vi−vj. The coefficients a,γandσare the\nconservative, dissipative and random force coefficients, respectively. ξijis a random variable with zero\nmean and unit variance satisfying\n/angbracketleftξij(t)/angbracketright= 0,\n/angbracketleftξij(t)ξkl(t/prime)/angbracketright= (δikδjl+δilδjk)δ(t−t/prime).\nThe kernel w(r) = max (1−r/rc,0) vanishes after the cutoff radius rc. The dissipative and random ker-\nnels are linked through the fluctuation-dissipation theorem\nσ2= 2γkBT, (22a)\nwD=w2\nR, (22b)\nwherekBTis the temperature of the fluid in energy units. We set wD=wswiths= 1/4.\nThe ABF is modeled as a rigid object consisting of frozen DPD particles that move as a rigid body and\ninteract with the surrounding solvent particles. The frozen particles are particles chosen from a separate\nsimulation of a DPD solvent at rest. Once equilibrated, the particles that are inside the volume of the\nABF are selected. Furthermore, to enforce no-slip and no-penetrability boundary conditions, the solvent\nparticles are bounced back from the surface of the ABF.\nThe ABFs were first oriented along their principal axes and we assumed that the matrices ∆, Zand Γ\nhad non zero elements only on their diagonal. For each ABF, we perform 6 simulations: in each simula-\ntion, only one of the 6 components of the linear and angular velocities, VandΩ, is non zero. The time\naveraged forces and torques were collected for each of these simulations, FiandTi, wherei= 1,2,..., 6\nis the simulation index. The coefficients of the mobility matrix are then estimated as\n∆ii=Vi\ni/Fi\ni, (23)\nΓii= Ωi+3\ni/Ti+3\ni, (24)\nfori= 1,2,3. The coefficient Z11, on the other hand, is computed from the simulation of a freely trans-\nlating ABF with an imposed rotation with constant angular velocity Ω along its swimming direction in a\nDPD fluid. After reporting the averaged swimming velocity V, we obtain\nZ11=V\nΩΓ11. (25)\nAll simulations are performed with a particle number density of nd= 10r−3\nc. The length of the ABF\nis set to 27rc, resulting in more than 3000 ABF frozen particles in each case. The simulation domain is\nperiodic and has dimensions L= 270rc(ten times the length of the swimmer), which was high enough to\navoid disturbances from the periodic images.\nWe report the coefficients of the propulsion matrix in dimensionless form ∆⋆= ∆ηl,Z⋆=Zηl2and Γ⋆=\nΓηl3, whereηis the dynamic viscosity of the fluid and lis the length of the ABF. The DPD simulations\nare run with a Reynolds number Re = 0 .2 and a Mach number Ma = 0 .05 in a periodic domain of size\n10l×10l×10lwith the ABFs shown on fig. 2. The nondimensional propulsion coefficients are reported\nin table 1.\n4Figure 2: The geometries of the two ABF for which the propulsion matrix has been calibrated. In arbitrary units, the left\none has a pich of 2 and the right one has a pitch of 5. The arrow represents the magnetic moment mof the ABFs.\nPitch ∆⋆\n11 ∆⋆\n22 ∆⋆\n33Z⋆\n11 Γ⋆\n11 Γ⋆\n22 Γ⋆\n33\n2 0.36 0.32 0.32 0.096 11.9 1.7 1.7\n5 0.41 0.37 0.37 0.39 18.2 2.42 2.42\nTable 1: Non dimensional coefficients of the propulsion matrix obtained from DPD simulations with the two geometries\nshown on fig. 2.\nWe choose the magnetic moment such that the maximum velocity of the swimmer equals one body length\nper second, which is a typical value found in experiments for a magnetic field of the order of 1 mT. This\nconstraint is expressed as Vmax=Z11mB, hence we set\nm=Vmax\nZ11B. (26)\n5 Validation of the DPD method\nThe simulation of a single swimmer in 3 dimensions, in free space, is modeled similarly as in the previ-\nous section with the DPD method. The swimmer has a magnetic moment perpendicular to its principal\ncomponent and is immersed in a uniform, rotating magnetic field B(t) =B(0,cosωt,sinωt). The simula-\ntion employs the same resolution as described in the previous section. The swimmer’s geometry is repro-\nduced from [6] and the magnetic moment (not reported in the experiments) has been tuned to match the\nexperimental data (m = 1 .0×10−14N m T−1), with a magnetic field magnitude of B= 3 mT. The mean\nswimming velocity along the xaxis is reported for different angular velocities ωon fig. 3 and agree well\nwith the experimental data.\n0 5 10\nf[Hz]05v[µm/s]\nFigure 3: Swimming velocity of the swimmer against the angular frequency of the magnetic field. DPD simulations (trian-\ngles) and experimental data from [6] (crosses).\n6 Settings of the CMA-ES minimization\nThe minimization of the travel time with respect to the travel directions nkis performed with the gra-\ndient free method derandomised evolution strategy with covariance matrix adaptation (CMA-ES). The\npopulation size is set to µ= 16, the initial standard deviation is σ=π/2 and the minimization is\n5stopped when the standard deviation reaches δ= 10−5. The first generation is centered at ( φ,θ,ψ ) =\n(0,0,0).\n7 Reinforcement Learning\nThe reinforcement learning (RL) method consists in an agent controlling the magnetic field through an\naction A= (aω,ax,ay,az). This action sets the frequency and direction of the rotating magnetic field\nfor ∆tsimulation time. The angular velocity is set to ω=aωand the direction is set to the unit vec-\ntoru= (ax,ay,az)//radicalbiga2\nx+a2\ny+a2\nz. Every ∆ttime, the environment returns its state sand reward rto\nthe agent. The state is composed of all positions and quaternions of the ABFs. Due to the absence of in-\nertia, this state describes the full system. The reward has two components, designed to guide the ABFs\ntowards the target in a minimal time:\nrt=gt−ct. (27)\nThe first term is designed from reward shaping [7]: it increases if the ABFs come closer to the target,\ngt=φt−1−φt,\nφt=Kφ/radicaltp/radicalvertex/radicalvertex/radicalbtN/summationdisplay\ni=1/bardblxi(t)/bardbl2,(28)\nso that the cumulative reward (we use a discount factor γ= 1), until the final simulation time T, de-\npends only on the initial conditions if the ABFs reached their goal. The constant Kφin eq. (28) is set\nsuch that the sum of the shaping part of the reward over one successful episodes is of the order of 1. There-\nfore, this part of the cumulative reward does not affect the resulting policy [7]. The second term in eq. (27)\nis designed to penalize long travel times, ct= ∆t/Tmax, Furthermore, an additional positive term is\nadded to the reward if all ABFs reached the target within the given distance dsuccess . We set this con-\nstant toKsuccess = 1. The parameters used for the two-swimmers setup are listed in table 2.\nParameter Value\n∆t 128.5/ωc,1\ndsuccess 2l\nTmax 139l/V\nKsuccess 1\nKφ 1/72.3l\nx0\n1 (50l,20l,−5l)\nx0\n2 (−10l,30l,20l)\nσ 3l\nTable 2: Parameters used in the RL setup.\nWe used the off-policy actor-critic algorithm V-RACER , implemented in the software smarties [8, 9], with\nthe default parameters. A single deep neural network (with 2 hidden layers of 128 units each) maps the\nstate space to a continuous approximation of the policy, the state value function and the action value\nfunction. More precisely, the policy is parameterized as a gaussian whose means and variances are out-\nput of the neural network. The neural network weights are updated through policy-gradient. More de-\ntails on the algorithm and implementation can be found in [8].\nThe number of episodes needed to converge for the RL method is of the order of 2 .5×104, as shown\non fig. 4. The travel time is first equal to the maximum allowed one (when the RL agent is not able to\nguide the ABFs towards the target). After about 5000 episodes, the ABFs reach the target and the travel\ntime decreases with the number of training episodes. The travel time converges to a smaller value than\nthat of the semi analytical solution.\n60 20000 40000 60000 80000 100000\nEpisode0100200300Tl/V\n0 20000 40000 60000 80000 100000\nEpisode0100200300Tl/VFigure 4: Dimensionless travel time Tl/V against number of episodes during training obtained with the RL method\n(\n ). The left figure corresponds to the case with 2 swimmers and no background flow. The time obtained with\nthe semi-analytical method is shown for comparison (\n ). The right figure corresponds to the case with the back-\nground flow. Vis the maximum forward velocity of the ABFs and ltheir length. The line corresponds to the median over\n200 episodes, and the shaded area corresponds to the 0.05 and 0.95 quantiles over the same number of episodes.\n0204060r/lAnalytic RL\n0 2500 5000 7500 10000 12500 15000 17500\ntωc,10.00.51.0ω/ωc,1\n0 2500 5000 7500 10000 12500 15000\ntωc,1\nFigure 5: Distance to target of three controlled ABFs (in units of body length l) against dimensionless time (\n ,\nand\n ) in free space, zero background flow, and corresponding magnetic field rotation frequency (\n ),\nwhereωc,1is the step-out frequency of the first swimmer.\n8 Three swimmers solution with the RL approach\nThe results obtained on two swimmers in the manuscript are here extended to three swimmers in the\ncase of zero background flow. Figure 5 shows the distance of the swimmers to the target. Similarly to\nthe two-swimmers case, the RL method seems to alternate between the step-out frequencies of the three\nABFs.\nReferences\n[1] L. Zhang, K. E. Peyer, B. J. Nelson, Lab on a Chip 2010 ,10, 17 2203.\n[2] S. Kim, S. J. Karrila, Microhydrodynamics: principles and selected applications , Courier Corporation,\n2013 .\n[3] B. Graf, arXiv preprint arXiv:0811.2889 2008 .\n[4] D. Alexeev, L. Amoudruz, S. Litvinov, P. Koumoutsakos, Comput. Phys. Commun. 2020 , 107298.\n7REFERENCES\n[5] P. Espanol, P. Warren, EPL (Europhysics Letters) 1995 ,30, 4 191.\n[6] R. Mhanna, F. Qiu, L. Zhang, Y. Ding, K. Sugihara, M. Zenobi-Wong, B. J. Nelson, Small 2014 ,10,\n10 1953.\n[7] A. Y. Ng, D. Harada, S. Russell, In ICML , volume 99. 1999 278–287.\n[8] G. Novati, P. Koumoutsakos, In Proceedings of the 36thInternational Conference on Machine Learn-\ning.2019 .\n[9] G. Novati, L. Mahadevan, P. Koumoutsakos, Physical Review Fluids 2019 ,4.\n8"    },    {        "title": "1802.01460v1.Diamagnetism_of_2D_Fermions_in_the_Strong_Nonhomogeneous_Static_Magnetic_Field______bf_B___B__0__0__1_cosh__2____frac_x_x__0_____δ________gas_magnetization______and_gas_compressibility.pdf",        "content": "arXiv:1802.01460v1  [cond-mat.mes-hall]  11 Oct 2017DIAMAGNETISM OF 2D-FERMIONS IN THE STRONG NONHOMOGENEOUS\nSTATIC MAGNETIC FIELD B =B(0,0,1/cosh2(x−x0\nδ)): gas magnetization, static\nmagnetic susceptibility, chemical potential and gas compr essibility.\nM. Hud´ ak\nStierova 23, SK - 040 23 Kosice, Slovak Republik\nand\nO. Hud´ ak\nDepartment of Aviation Technical Studies, Faculty of Aerodynamic s, Technical University Kosice,\nRampova 7, SK - 040 01 Kosice, Slovak Republik\nWe study diamagnetism of a gas of fermions moving in a nonhomo geneous magnetic field\nB=B(0,0,1/cosh2(x−x0\nδ)).The gas magnetization, the static magnetic susceptibility , the chemi-\ncal potential and the gas compressibility are discussed and compared with the uniform field case.\nGeneral need to study dynamics of electrons in different type s of magnetic fields follows from a\nlarge number of experimental situations in which its unders tanding enables physicists to obtain new\ninformation.\nPACS Numbers:\n71.45Electron gas - electron states (condensed matter) ,\n75.20Diamagnetism ,\nRecently, using results of our exact description of the spinless fer mion motion in a nonhomogeneous magnetic field\nB=B(0,0,1/cosh2(x−x0\nδ)), we studied ground state energy properties of a gas of spinless fermions moving in this\nfield. For densities lower than some critical value, ν < νc(B,δ),the corresponding total energy is lower than that of\nthe uniform field state, [1]. However, physical properties of a gas o f spinless two-dimensional (2d) -fermions moving\nin this field were not studied in [1]. It is the aim of this paper to fill in this ga p.\nEnergy of a gas of free spinless fermions moving in a plane in a uniform fi eld perpendicular to this plane is larger\nor equal to its total energy ET(0,ν) = 2πtNν2in the zero field:\n∆Eh(n)≡ET(B,ν)−ET(0,ν) = 2πtN(νn+1−ν)(ν−νn), (1)\nwhereνn+1≥ν > νn,νn≡n.Φ\nΦ0,n= 0,1,...;Φ0is a unit of the magnetic flux, Φ ≡Ba2,t≡¯h2\n2ma2;νis the number\ndensity of the gas, ν≡Nf\nN,Nfis the total number of fermions, Na2is the area of the square with the side length\nL=a√\nN,to which the motion is bounded, m is the fermion mass. Here ais a characteristic length of the system, its\nvalue is of the order of a lattice constat value. The energy degener acy occurs in (1) only if a number of Landau levels\nis completely filled (e.i. if for some n νn=ν).A nonhomogeneity of the field introduced by a local field intensity\ndecrease leads to competition of two tendencies: a decrease of th e single fermion energy level due to decreased value\nof the field intensity and a decrease of the every energy level dege neracy due to larger spacing between centers of\nneighboring orbits within the region of smaller fields. Spectrum of 2d B loch electrons in a periodic magnetic field\nwas studied in [2], where the magnetic unit cell is assumed to be commen surate with the lattice unit cell. Their work\nis extension of previous studies of free electrons in periodic magnet ic field [3] to the lattice case. In our work we\nconcentrate our attention on the nonperiodic modulated field.\nLet us consider motion of a spinless fermion gas bounded to the squa re LxL in a nonhomogeneous static magnetic\nfield perpendicular to this plane. We neglect the lattice periodic poten tial influence on the gas energy spectrum\nin this paper. We consider in more details the limit in which nonhomogeneit y disappears and a uniform field ap-\npears. In difference to [2] and [3] we do not consider a periodic magne tic field. Recently, [4], an exact description\nof motion of the quantum spinless fermion in a nonhomogeneous magn etic field described by the vector potential\nA= (0,Bδtanh (x−x0\nδ),0) was found. Let us now describe those results from [4] which are relevant for our further\ncalculations in this paper. In the case of motion of a quantum spinless fermion in a nonhomogeneous magnetic field\ndescribed by the vector potential A= (0,Bδtanh (x−x0\nδ),0) the energy spectrum of the motion in the x-direction is\nsplitted, see in [4], into a discrete and a continuous parts for genera l values of the field B and of the nonhomogeneity2\nparameter δ.We take x0= 0 in the following, thus the field has its maximum intensity at x= 0.Let us consider the\nlimit of strong fields ( F≡2πΦ′\nΦ>>1/2,where Φ′≡Bδ2).In this limit it is sufficient to take into account the lowest\nenergy levels of the spectrum. The eigenvalues of the energy corr esponding to this part of the spectrum are given by,\nsee in [4] :\nEn(p) =p2\ny\n2m(1−F2\n((1\n4+F2)1\n2−((1/2)+n))2)+\n(¯h2\n2mδ2)(F2−((1\n4+F2)1\n2−(1\n2+n))2,\nwheren= 0,1,...[nmax],here [n] denotes an integer part of a real number n, pyis the y-momentum. Let us define\nP≡|py|δ\n¯h. The number nmaxis defined by:\nnmax= (1\n4+F2)1\n2−(1/2)−(|P|F)1\n2,\nfor given values of P and F.\nForF2>>1\n4and for small quantum numbers n, n= 0,1,2,...,the energy En(py) expanded into series of 1/F\npowers takes the form :\nEn(py)≈¯hω(n+1\n2)−(¯h2\n2mδ2)((n+1\n2)2+1\n4)−p2\ny\nmF(n+1\n2)+\n(¯h2\n8mFδ2)(n+1\n2)+O(1\nF3).\nwhereω≡Bc\nemis the cyclotron frequency. The energy levels become degenerate d in the limit of strong but modulated\nfields if the energy expansion above is restricted to the first two te rms, which are of the F1andF0orders respectively.\nThe largest value of the third term in this expansion is negligible with res pect to the second term\nmax(p2\ny\nmF(n+1\n2))<<(¯h2\n2mFδ2)(n+1\n2).\nif we take into account that there exists a natural cut-off for pymomenta, max(|py|) =π¯h\na,due to the underlying\ncrystal and if we assume that the field intensity B satisfies the inequ ality:\nΦ\nΦ0>>8π2,\nwhere Φ ≡B.a2.The degeneracy of the n-th level appears due to the lost of the en ergy dependence on pymomentum.\nThese levels are, [4], modified Landau levels with energies in the form\nEn= ¯hω(n+1\n2)−¯h2\n2mδ2[(n+1\n2)2+1\n4]+O(1/F), (2)\nwhere\nn= 0,1,... << n m;nm≈F.\nNote that\n¯h2\n2mδ2= 4t(L\n2δ)2/N.\nEvery energy level Enis degenerated within considered approximation, its degeneracy Dnis found to be:\nDn=DLtanh(L\n2δ)\nL\n2δ. (3)3\nHere the characteristic length L and the nonhomogeneity paramet erδsatisfy the inequality:\ntanh(L/2δ)<(1−2\nF(n+1\n2)).\nThe Landau level degeneracy is DL≡Bea2\nhcNas in the case of the uniform field. The form of the degeneracy\nDngiven above holds for all orders of F. The large F expansion in (2) limits its validity to the region of system\nparameters given by the inequality below (3) which follows from the us ual, [5], boundary conditions: periodicity in\nthe y-direction perpendicular to the x-axis and limits on the position o f the orbit center in the x-direction to the\nregion<−L/2,+L/2> .The orbit center x-coordinate xcis given, [4], by\ntanh(xc/δ) = (−pyδ\n¯h)/F.\nThe ground state energy ET(B,δ,ν) for spinless fermion gas with density νin the limit of strong but nonhomo-\ngeneous fields specified by B,δgiven as difference between the nonhomogeneous field state and th e zero field state\nenergy is found to be:\n∆Enh(n)≡ET(B,δ,ν)−ET(0,ν) = 2πtN[(ν−νntanh(L\n2δ)\nL\n2δ)(νn+1tanh(L\n2δ)\nL\n2δ−ν)+ (4)\n(1−tanh(L\n2δ)\nL\n2δ)(ν(2νn+ν1)−tanh(L\n2δ)\nL\n2δνn+1νn)]−\n−ta2\nδ2N[ν(n2+n+1\n2)−νn(2n2\n3+n+1\n3)].\nThe total energy difference (4) is found assuming that there are n levels 0,1,...,n−1 filled and that the n-th level\nis filled partially. The gas density νin (4) is limited by the following inequalities:\nνn+1tanh(L\n2δ)\nL\n2δ≥ν > νntanh(L\n2δ)\nL\n2δ, (5)\nνn≡nΦ/Φ0.\nThe uniform field result (1) follows from (4) and (5) in the limit δ−→ ∞keeping values of all the other system\nparameters constant.\nLet us now calculate some of physical characteristics of our gas of spinless fermions in the considered nonhomoge-\nneous magnetic field at zero temperature. Then let us compare the se characteristics ( magnetization, static magnetic\nsusceptibility, chemical potential and compressibility ) with the corr espondingcharacteristicsin the uniform field case.\nThe comparison enable us to describe influence of nonhomogeneity o f the magnetic field in the large intensity and\nnonhomogeneity limit on a gas of fermions.\nThe magnetization is found to be given by\nmz(B,δ)≡1\nN∂ET(B,δ,ν)\n∂B= (a2\nΦ0).[2πt(ν(2n+1)−tanh(L\n2δ)\nL\n2δ(n+1)2nν1)]+ (6)\nta2\nδ2n(2n2\n3+n+1\n3)].\nWe see from (6) that when the nonhomogeneity parameter δincreases its value to infinity we obtain the uniform\nfield result. When this parameter is finite then the magnetization incr eases its value ( its absolute value decreases).\nThe static magnetic susceptibility χzz(B,δ) is found to be given by:\nχzz(B,δ)≡∂mz\n∂B= (a2\nΦ0)2.[−4πtn.(n+1)tanh(L\n2δ)\nL\n2δ] (7)4\nWe see that the susceptibility becomes less diamagnetic than in the un iform field case due to presence of the field\nnonhomogeneity where the field intensity is decreased, and thus als o where the density of states is lower.\nThe chemical potential difference between the nonhomogeneous fi eld state and the zero field state ∆ µis given by\n∆µ=1\nN.∂∆Enh(n)\n∂ν= 2πt[(ν1(2n+1)−2ν]−ta2\nδ2[n2+n+1\n2]. (8)\nThe single fermion energy shift down due to presence of nonhomoge neity decreases the chemical potential and is\ndescribed in (8) by the second term. The first term has the same fo rm as in the uniform field case. Note, however,\nthat the density of particles, ν, is in the nonhomogeneous field case bounded from above and below b y limiting values\nwhich are dependent on the nonhomogeneity of the field, see (5). I n the uniform field case there exists symmetry\nwhen filling a given energy level ( ν→ν−\nn+1) and when emptiing the same level ( ν→ν+\nn):\nµ(ν→ν+\nn) =−µ(ν→ν−\nn+1).\nThis symmetry is lost whenever the nonhomogeneity parameter is fin ite.\nThe difference of the inverse compressibility ∆1\nκof the gas in the nonhomogeneous field state and the zero field\nstate is given by\n∆1\nκ≡ −∂µ\n∂ν= 4πt (9)\nFrom (9) it follows that the inverse compressibility is not affected by t he presence of the nonhomogeneity. Thus\ndecrease of the chemical potential with increase of the gas densit y is the same in the uniform field case and in the\nnonhomogeneous field case.\nResults of this paper describe some of the physical properties of 2 d-fermion gas moving in our specific type of the\nnonhomogeneous field. We expect that qualitatively these results d escribe modification of the uniform field case due\nto presence of a static nonhomogeneity of general type. Genera l need to study dynamics of electrons in different\ntypes of magnetic fields follows from a large number of experimental situations in which its understanding enables\nphysicists to obtain new information. In the solid state physics thes e situations occur studying such effects as: the Hall\neffect; magnetoresistivity; anomalous skin effects in a magnetic field ; cyclotron resonances in metals; magnetoplasma\nwaves in metals and semiconductors; quantum oscillatory effects like : the de Haas - van Alfen effect, oscillations\nof the entropy, of the volume, of the specific heat, of the thermo power and of other thermodynamic characteristics,\nthe Shubnikov-de Haas effect, oscillations of the surface impedanc e, of the Hall coefficients, of the sound absorption\ncoefficients and of the sound velocity, magnetoacoustic oscillations (Pippard’s geometric resonance oscillations), giant\nquantum oscillations of absorption in metals, quasiclassical size effec ts (radiofrequency size effects, cutt off effects of\ncyclotron resonance and of quantum oscillations, oscillations on cut ted off orbits, the Sondheimer effect) and quantum\nsize effects (mesoscopic phenomena).\n[1] M, Hud´ ak and O. Hud´ ak , ”Transition of the Uniform Stati stical Field Anyon State to the Nonuniform One at Low Particl e\nDensities”, arXiv preprint arXiv:1704.01640\n[2] A. Barelli, J. Bellisard and R. Rammal, J. Phys. France 51(1990) 2167-2185\n[3] B.A. Dubrovin and S.P. Novikov, Sov. Phys. JETP 52(1980) 511\nB.A. Dubrovin and S.P. Novikov, Sov. Math. Dokl. 22(1980) 240\nS.P. Novikov Sov. Math. Dokl. 23(1981) 298\n[4] O. Hud´ ak, Dynamics of a quantum particle in a static modu lated magnetic field B=(0, 0, B/cosh ((x x 0)/)), Zeitschrift\nfuer Physik B Condensed Matter 88(1992) 239-246\n[5] R. E. Peierls, Quantum Theory of Solids, Oxford, Clarend on Press, (1955)"    },    {        "title": "0911.2377v1.Groebli_solution_for_three_magnetic_vortices.pdf",        "content": "arXiv:0911.2377v1  [cond-mat.mes-hall]  12 Nov 2009Gr¨ obli solution for three magnetic vortices\nStavros Komineas1and Nikos Papanicolaou2,3\n1Department of Applied Mathematics, University of Crete, 71 409 Heraklion, Crete, Greece\n2Department of Physics, University of Crete, 71003 Heraklio n, Crete, Greece\n3Institute for Theoretical and Computational Physics, Univ ersity of Crete, Heraklion , Greece\nThe dynamics of Npoint vortices in a fluid is described by the Helmholtz-Kirch hoff (HK) equa-\ntions which lead to a completely integrable Hamiltonian sys tem forN= 2 or 3 but chaotic dynamics\nforN >3. Here we consider a generalization of the HK equations to de scribe the dynamics of mag-\nnetic vortices within a collective-coordinate approximat ion. In particular, we analyze in detail the\ndynamics of a system of three magnetic vortices by a suitable generalization of the solution for three\npoint vortices in an ordinary fluid obtained by Gr¨ obli more t han a century ago. The significance of\nour results for the dynamics of ferromagnetic elements is br iefly discussed.\nI. INTRODUCTION\nFerromagnetic media are described by the density of magnetic mome nt or magnetization\nm=m(r,t) which is a vector field of constant length that satisfies the highly no nlinear Landau-\nLifshitz (LL) equation. Magnetic bubbles and vortices are special s olutions of the LL equation\ncharacterizedby a topological invariant sometimes called the Pontr yaginindex or skyrmion num-\nber. Upon suitable normalization, the skyrmion number is integer for magnetic bubbles occuring\nin magnetic films with perpendicular easy-axis anisotropy [1] and half in teger for magnetic vor-\ntices that may be realized in the case of magnetic films with easy-plane anistotropy [2].\nMost of the early studies were carried out on magnetic films of infinite extent where vortices\nand bubbles occur as excited states above a uniform ground state . More recently it has been\nrealized that the ground state of a disc-shaped magnetic element, with a diameter of a few\nhundred nanometers, is itself a vortex configuration. Hence the v ortex is a nontrivial magnetic\nstate that can be spontaneously created on magnetic elements of finite extent. Such a possibility\nis of obvious significance for device applications and explains the curr ent interest on this subject\n[3, 4, 5].\nThedynamicsofmagneticvorticesisgreatlyaffectedbytheunderly ingtopologicalstructure[6,\n7]. This is described by a topologicalvorticity which givesthe skyrmion number upon integration\nover the plane of the magnetic film. As a result, the essential featu res of magnetic vortex\ndynamics are similar to those displayed by ordinary vortices in fluid dyn amics. A single vortex\nor antivortex is spontaneously pinned and thus cannot move freely within the ferromagnetic\nmedium. However, vortex motion relative to the medium is possible in th e presence of other\nvortices, or externally applied magnetic field gradients, and displays characteristics analogous\nto those of two-dimensional (2D) motion of electric charges in a unif orm magnetic field. In\nparticular, two like vortices orbit around each other while a vortex- antivortex pair undergoes\nKelvin motion. A recent review of two-vortex magnetic systems tog ether with some progress in\nthe study of three-vortex dynamics may be found in Refs. [8, 9].\nIn the present paper we study the dynamics of a system consisting ofNinteracting magnetic\nvortices within a collective-coordinate approximation which evades t he complexities of the com-\nplete LL equation but retains important dynamical features due to the underlying topology. The\nrelevant equations are a generalization of the HK equations encoun tered in the study of Npoint\nvortices in ordinary fluids [10, 11]. The latter equations were solved e xplicitly by Helmholtz\nhimself for the two-vortex ( N= 2) system, while Kirchhoff concluded that the three-vortex\n(N= 3) system is completely integrable. But a complete solution for N= 3 was given in the\n1877 dissertation of Gr¨ obli [12] recently reviewed by Aref et al [13 ] (see also [14]). For N >3,\nthe system is not completely integrable and thus leads to chaotic dyn amics.2\nIn Section II, we state the appropriate modifications of the HK equ ations to account for N\ninteracting magnetic vortices, derive the corresponding conserv ation laws, and provide a first\ndemonstration with an explicit solution for two-vortex systems. Th e generalization of Gr¨ obli’s\nsolution to three magnetic vortices is given in Section III and is illustra ted in detail in Section\nIV for important special cases such as scattering of a vortex-an tivortex pair off a target vortex\ninitially at rest, three-vortex collapse, bounded three-vortex mo tion, etc. Our main conclusions\nare summarized in Section V.\nII. GENERALIZATION OF THE HELMHOLTZ-KIRCHHOFF EQUATIONS\nWe consider a system of Nmagnetic vortices labeled by α= 1,2,...N. Each vortex is\ncharacterized by a pair of indices ( κα,λα) whereκαis the vortex number and λαthe polarity,\nwhich take the values κα=±1 andλα=±1 in any combination. We also define the skyrmion\nnumber\nsα=καλα (1)\nwhich also takes the distinct values sα=±1 (in any combination for varying α= 1,2,...N)\nand differs from the standard definition of the Pontryagin index by a n overall normalization\n(Nα=−1\n2sα) [8, 9].\nNow, inacollective-coordinateapproximation[15, 16, 17], the posit ionofeachvortexis labeled\nby a vector rα= (xα,yα) in the 2D plane and the energy functional is given by\nE=−/summationdisplay\nα<βκακβln|rα−rβ| (2)\ninsuitableunits. Notethattheenergyisafunctionofthevortexnu mberκαbutnotthepolarities\nλα. The equations of motion are then written in Hamiltonian form as\nsαdxα\ndt=∂E\n∂yα, s αdyα\ndt=−∂E\n∂xα;α= 1,2,...N, (3)\nor, more explicitly, as\nλαdxα\ndt=−/summationdisplay\nβ/negationslash=ακβyα−yβ\n|rα−rβ|2, λ αdyα\ndt=/summationdisplay\nβ/negationslash=ακβxα−xβ\n|rα−rβ|2. (4)\nThese differ from the HK equations for Npoint vortices in an ordinary fluid by the presence of\nthe polarities λα(or the skyrmion numbers sα) in addition to the vortex numbers κα. Equations\n(4) reduce to the HK equations in the special limit where all polarities a re taken to be equal\n(λ1=λ2=...=λN). Also note that the vortex number καmay assume any continuous value\nfor a fluid vortex but takes the quantized values ±1 for a magnetic vortex.\nIn addition to the conserved energy of Eq. (2), the Hamiltonian sys tem (3) or (4) possesses\nconservation laws which originate in translational and rotational inv ariance; i.e., the analog of\nlinear momentum P= (Px,Py) with\nPx=−/summationdisplay\nαsαyα, P y=/summationdisplay\nαsαxα, (5)\nand angular momentum\nL=1\n2/summationdisplay\nαsα(x2\nα+y2\nα) (6)3\nboth defined in convenient units. Actually, the physical significance of these conservation laws\ndepends crucially on whether or not the total skyrmion number\nS=/summationdisplay\nαsα=/summationdisplay\nακαλα (7)\nvanishes. The linear momentum (5) assumes its customary meaning o nly when S= 0. For\nS/negationslash= 0, one may instead define the conserved guiding center R= (Rx,Ry) with\nRx=1\nS/summationdisplay\nαsαxα, R y=1\nS/summationdisplay\nαsαyα, (8)\nwhich is a measure of position rather than linear momentum. The inher ent transmutation of\nmomentum into position occursalsowithin the complete LL equationan d hasbeen the subject of\nseveral investigations [6, 7, 9]. To complete the discussion of cons ervation laws we note that the\nlinear momentum (5) and the angular momentum (6) may be combined t o yield the derivative\nconserved quantity\n1\n2/summationdisplay\nαβsαsβ(rα−rβ)2(9)\nwhich depends only on the relative distances |rα−rβ|and will play an important role in the\nfollowing.\nThis section is concluded with a simple application of the general forma lism to the case of\ntwo-vortex systems ( N= 2). Equations (4) then read\nλ1dx1\ndt=−κ2y1−y2\nd2, λ 2dx2\ndt=−κ1y2−y1\nd2,\nλ1dy1\ndt=κ2x1−x2\nd2, λ 2dy2\ndt=κ1x2−x1\nd2, (10)\nwhered2≡(x2−x1)2+(y2−y1)2is the squared relative distance between the two vortices which\nis conserved by virtue of the conservation of the energy Eof Eq. (2) restricted to N= 2 for any\nchoice of vortex numbers and polarities. But the details of the motio n depend on the specific\nchoice of κ’s andλ’s as demonstrated below with two representative examples.\nFirst, we consider a vortex-antivortex pair with equal polarities:\n(κ1,λ1) = (1,1),(κ2,λ2) = (−1,1), (11)\nhence opposite skyrmion numbers s1= 1, s2=−1 and total skyrmion number S=s1+s2= 0.\nA straightforward solution of Eqs. (10) then leads to Kelvin motion w here the vortex and the\nantivortex move in formation at constant relative distance d, along parallel trajectories that are\nperpendicular to the line connecting the vortex and the antivortex , with speed\nV=1\nd(12)\nand linear momentum (5) that is parallel to the velocity.\nIn contrast, a vortex-antivortex pair with opposite polarities:\n(κ1,λ1) = (−1,1),(κ2,λ2) = (1,−1), (13)\nhences1=s2=−1 and nonvanishing total skyrmion number S=s1+s2=−2, undergoes\nrotational motion at constant diameter d, around a fixed guiding center (8) which may be taken\nto be the origin of the coordinate system ( x1+x2= 0 =y1+y2), with angular frequency\nω=2\nd2. (14)4\nAdetailedstudyofKelvinandrotationalmotionwascarriedoutwithin thecompleteLLequation\nin Refs. [18] and [19], respectively, the results of which were recent ly reviewed in Ref. [8]. In\nparticular, their results confirm the validity of Eqs. (12) and (14) f or sufficiently large relative\ndistance d, as expected for the collective-coordinateapproximationadopte d in the present paper.\nFinally we note the the motion of a two-vortex system is Kelvin-like whe n the total skyrmion\nnumber vanishes ( S=s1+s2=κ1λ1+κ2λ2= 0) and rotational otherwise ( S=s1+s2=\nκ1λ1+κ2λ2=±2) irrespectively of the detailed configuration of the vortex numbe rsκ1,κ2and\npolarities λ1,λ2.\nIII. THREE MAGNETIC VORTICES\nConsider a system of three interacting magnetic vortices with vort ex numbers and polarities\ngiven by the three pairs of indices ( κ1,λ1),(κ2,λ2),(κ3,λ3). The three vortices form a triangle\nwhose vertices are located at r1= (x1,y1),r2= (x2,y2),r3= (x3,y3) and move about the 2D\nplane according to Eqs. (4) written here explicitly as\nλ1dx1\ndt=−κ2y1−y2\nC2\n3+κ3y3−y1\nC2\n2, λ 1dy1\ndt=κ2x1−x2\nC2\n3−κ3x3−x1\nC2\n2,\nλ2dx2\ndt=−κ3y2−y3\nC2\n1+κ1y1−y2\nC2\n3, λ 2dy2\ndt=κ3x2−x3\nC2\n1−κ1x1−x2\nC2\n3,(15)\nλ3dx3\ndt=−κ1y3−y1\nC2\n2+κ2y2−y3\nC2\n1, λ 3dy3\ndt=κ1x3−x1\nC2\n2−κ2x2−x3\nC2\n1,\nwhere\nC1=|r2−r3|, C2=|r3−r1|, C3=|r1−r2| (16)\nare the lengths of the triangle sides.\nOur task is to solve the initial-value problem defined by Eqs. (15) with in itial conditions\nfurnished by the t= 0 configuration of the three vortices. Needless to say, a numeric al solution\nis straightforward and has indeed been used to confirm our main ana lytical results obtained in\nthe following by a generalization of Gr¨ obli’s original solution for ordina ry fluid vortices [12].\nThe key observation is that the scalar quantities C1,C2andC3satisfy a closed system of\nequations of their own, which may be employed to determine the time e volution of the shape\nand size of the vortex triangle, irrespectively of its relative orienta tion and motion in the plane.\nSpecifically, as a consequence of Eqs. (15),\nd\ndt(C2\n1) = 4κ1A/parenleftbigg1\nλ3C2\n2−1\nλ2C2\n3/parenrightbigg\nd\ndt(C2\n2) = 4κ2A/parenleftbigg1\nλ1C2\n3−1\nλ3C2\n1/parenrightbigg\n(17)\nd\ndt(C2\n3) = 4κ3A/parenleftbigg1\nλ2C2\n1−1\nλ1C2\n2/parenrightbigg\n.\nHereA=ν|A|whereν= 1 or−1 when the vortices (1 ,2,3) appear in the counterclockwise or\nclockwise sense in the plane, and |A|is the area of the vortex triangle which may be expressed\nentirely in terms of C1,C2andC3:\n|A|= [C(C−C1)(C−C2)(C−C3)]1/2, C≡1\n2(C1+C2+C3). (18)5\nEqs. (17) constistute a reduced dynamical system that is complet ely integrable because it pos-\nsesses two conservation laws; namely\nB=C1/κ1\n1C1/κ2\n2C1/κ3\n3,Γ =s2s3C2\n1+s3s1C2\n2+s1s2C2\n3\ns1+s2+s3, (19)\nwhere the conservation of Bfollows from the conservation of the energy of Eq. (2), and Γ from\nthe conservation of the quantity defined in Eq. (9), both restrict ed to the three-vortex system\nand normalized in a manner suitable for subsequent analysis. The con servation of Band Γ may\nbe confirmed directly from Eqs. (17). Thus, having determined Band Γ from the initial ( t= 0)\nvalues of C1,C2andC3, one may use Eqs. (19) to eliminate two of these variables, say, C2and\nC3in favor of C1and thereby reduce Eqs. (17) to a single equation for C1that may be integrated\nby a simple quadrature.\nIn order to complete the solution we must also determine the relative orientation and motion\nof the vortex triangle in the plane and thus solve for the actual vor tex trajectories. The latter\nmay be conveniently described in polar coordinates ( rα,θα), α= 1,2,3. Using these coordinates\nHamilton’s equations (3) read\nsαrαdrα\ndt=∂E\n∂θαsαrαdθα\ndt=−∂E\n∂rα. (20)\nWe write explicitly the three equations for the polar angles:\nλ1r1dθ1\ndt=κ2r1−r2cos(θ1−θ2)\nC2\n3+κ3r1−r3cos(θ3−θ1)\nC2\n2,\nλ2r2dθ2\ndt=κ3r2−r3cos(θ2−θ3)\nC2\n1+κ1r2−r1cos(θ1−θ2)\nC2\n3, (21)\nλ3r3dθ3\ndt=κ1r3−r1cos(θ3−θ1)\nC2\n2+κ2r3−r2cos(θ2−θ3)\nC2\n1.\nA crucial step for simplifying the latter equations is to set the guiding center (8) at the origin,\ni.e.,Rx= 0 =Ry. In polar coordinates this constraint reads:\ns1r1cosθ1+s2r2cosθ2+s3r3cosθ3= 0\ns1r1sinθ1+s2r2sinθ2+s3r3sinθ3= 0. (22)\nSimple combinations of Eqs. (22) yield\n2s2s3r2r3cos(θ2−θ3) =s2\n1r2\n1−s2\n2r2\n2−s2\n3r2\n3\n2s3s1r3r1cos(θ3−θ1) =−s2\n1r2\n1+s2\n2r2\n2−s2\n3r2\n3 (23)\n2s1s2r1r2cos(θ1−θ2) =−s2\n1r2\n1−s2\n2r2\n2+s2\n3r2\n3.\nFurther, we write the sides of the triangle using polar coordinates:\nC2\n1=r2\n2+r2\n3−2r2r3cos(θ2−θ3)\nC2\n2=r2\n3+r2\n1−2r3r1cos(θ3−θ1) (24)\nC2\n3=r2\n1+r2\n2−2r1r2cos(θ1−θ2),\nand we can now use Eqs. (23) and (24) to obtain explicit relations bet ween the radial coordinates\nrαand the sides of the triangle Cα:\ns2s3C2\n1= (s2+s3)Γ−s1(s1+s2+s3)r2\n1\ns3s1C2\n2= (s3+s1)Γ−s2(s1+s2+s3)r2\n2 (25)\ns1s2C2\n3= (s1+s2)Γ−s3(s1+s2+s3)r2\n3,6\nwhere Γ is the conserved quantity already defined in Eq. (19).\nWe are now ready to write Eqs. (21) in a form where the radial coord inatesrαare eliminated\nand only the variables Cαandθαappear. We use Eqs. (24) to eliminate the cosines and Eqs. (25)\nto express the radial coordinates in terms of the Cα’s in Eqs. (21), to obtain\n2λ1λ2λ3(s1+s2+s3)r2\n1C2\n2C2\n3dθ1\ndt=λ3C2\n2[s2s3(−C2\n1+C2\n2+C2\n3)+2s2\n2C2\n3]\n+λ2C2\n3[s2s3(−C2\n1+C2\n2+C2\n3)+2s2\n3C2\n2]\n2λ1λ2λ3(s1+s2+s3)r2\n2C2\n3C2\n1dθ2\ndt=λ1C2\n3[s3s1(C2\n1−C2\n2+C2\n3)+2s2\n3C2\n1] (26)\n+λ3C2\n1[s3s1(C2\n1−C2\n2+C2\n3)+2s2\n1C2\n3]\n2λ1λ2λ3(s1+s2+s3)r2\n3C2\n1C2\n2dθ3\ndt=λ2C2\n1[s1s2(C2\n1+C2\n2−C2\n3)+2s2\n1C2\n2]\n+λ1C2\n2[s1s2(C2\n1+C2\n2−C2\n3)+2s2\n2C2\n1],\nwhere it is understood that the rα’s are given through Eqs. (25) in terms of the Cα’s. Explicit\napplications of the preceding general results are given in Section IV .\nIV. AN INTERESTING THREE-VORTEX SYSTEM\nFor an explicit demonstration we consider a three-vortex system w ith vortex numbers and\npolarities given by\n(κ1,λ1) = (1,1),(κ2,λ2) = (−1,1),(κ3,λ3) = (1,−1) (27)\nand corresponding skyrmion numbers s1= 1, s2=−1 =s3. The pair (1 ,2) is the vortex\n-antivortex pair of Eq. (11) which would undergo Kelvin motion in the a bsence of vortex 3. In\ncontrast, the pair (2 ,3) would undergo rotational motion in the absence of vortex 1, as d iscussed\ninSec.IIfollowingEq.(13). Thusthemotionofthethreevorticesis expectedtobeacombination\nof Kelvin and rotational motion. Since the total skyrmion number S=s1+s2+s3=−1 is\ndifferent from zero, the guiding center for the three-vortex sys tem defined from Eq. (8), i.e.,\nRx=−x1+x2+x3, R y=−y1+y2+y3, (28)\nis conserved and thus remains fixed during the motion.\nWe now return to the reduced system (17) applied for the specific c hoice (27):\nd\ndt(C2\n1) =−4A/parenleftbigg1\nC2\n2+1\nC2\n3/parenrightbigg\nd\ndt(C2\n2) =−4A/parenleftbigg1\nC2\n3+1\nC2\n1/parenrightbigg\n(29)\nd\ndt(C2\n3) = 4A/parenleftbigg1\nC2\n1−1\nC2\n2/parenrightbigg\n,\nwhile the conservation laws defined from Eq. (19) are given by\nB=C1C3\nC2,Γ =−C2\n1+C2\n2+C2\n3. (30)7\nOne may then express C2andC3in terms of C1as\nC2\n2=C2\n1+Γ\nC2\n1+B2C2\n1, C2\n3=C2\n1+Γ\nC2\n1+B2B2(31)\nand the area of the triangle defined earlier in Eq. (18) as\n4A=ν/radicalBig\n4C2\n2C2\n3−Γ2=ν\nC2\n1+B2/radicalBig\n4B2C2\n1(C2\n1+Γ)2−Γ2(C2\n1+B2)2. (32)\nThen the system of Eqs. (29) reduces to the single equation\nd\ndt(C2\n1) =−ν\nC2\n1+Γ/parenleftbigg1\nC2\n1+1\nB2/parenrightbigg/radicalBig\n4B2C2\n1(C2\n1+Γ)2−Γ2(C2\n1+B2)2, (33)\nwhich may readily be integrated to yield C1=C1(t), as well as C2=C2(t) andC3=C3(t) from\nEq. (31), for any specific values of the conserved quantities Band Γ calculated from the initial\nthree-vortex configuration.\nIn order to complete the solution we must also determine the relative orientation and motion\nof the vortex triangle in the plane and thus solve for the actual vor tex trajectories. We will\nfollow the derivation given in Section III and quote only important res ults that will allow us to\nprovide a detailed illustration of the three-vortex motion for the sp ecial case of vortex numbers\nand polarities given by Eq. (27).\nIt is convenient to specify a reference frame such that its origin co incides with the conserved\nguiding center of Eq. (28):\n−x1+x2+x3= 0,−y1+y2+y3= 0. (34)\nIn this frame the radial distances r1,r2andr3are related to the triangle sides C1,C2andC3by\nEq. (25) applied for s1= 1,s2=−1 =s3:\nr2\n1= 2(C2\n2+C2\n3)−C2\n1, r2\n2=C2\n2, r2\n3=C2\n3, (35)\nwhile the angular variables satisfy Eqs. (26) now reduced to\ndθ1\ndt=Γ\n2(C2\n1+2Γ)/parenleftbigg1\nC2\n2−1\nC2\n3/parenrightbigg\ndθ2\ndt=−Γ\n2C2\n2/parenleftbigg1\nC2\n1+1\nC2\n3/parenrightbigg\n+1\nC2\n1(36)\ndθ3\ndt=Γ\n2C2\n3/parenleftbigg1\nC2\n2−1\nC2\n1/parenrightbigg\n+1\nC2\n1.\nTo summarize, for the specific choice of vortex numbers and polarit ies given by Eq. (27), the\nradialvariables r1,r2andr3areexpressedintermsof C1,C2andC3fromEqs.(35)andultimately\nin terms of C1alone using the conservation laws as in Eqs. (31); while the time evolut ion ofC1\nis determined from Eq. (33) by a straightforward integration. Simila rly, we may use Eqs. (31)\nto express the right-hand sides of Eq. (36) in terms of C1and thus the time evolution of the\nangular variables θ1,θ2andθ3is also reduced to simple quadratures.\nWithout loss of generality, the initial three-vortex configuration is shown in Fig. 1. In par-\nticular,bis the initial length of the base (1 ,2) of the triangle and hthe corresponding height.\nThe impact parameter ais defined as the distance of vortex 2 from the y-axis and is taken to be\nnegative when vortex 2 is located to the left of the y-axis and positiv e otherwise. Also note that8\n2 h\n2a 1O (guiding center)b3\nC\nC =b1\n3C\nFIG. 1: Initial configuration of three magnetic vortices (1, 2,3) with vortex numbers and polarities given\nby Eq. (27). The origin of the coordinate frame is taken to coi ncide with the conserved guiding center\nof the three-vortex system. The impact parameter ais taken to be negative when vortex 2 lies to the\nleft of the y-axis and positive otherwise.\nvortex 3 is initially located on the negative x-axis at a distance bfrom the origin, thus enforcing\nEqs. (34). Hence, the initial values of the triangle sides are C2\n1=h2+(a+b)2, C2\n2=h2+a2,\nandC2\n3=b2, and the conserved quantities of Eq. (30) are given by\nB2=h2+(a+b)2\nh2+a2b2,Γ =−2ab, (37)\nwhere Γ is positive for left impact ( a <0) and negative for right impact ( a >0). This completes\nthe description of the general procedure which is explicitly implement ed in the continuation of\nthis section using examples of increasing complexity.\nA. Special case\nFirst we employ the general formalism developed in the present pape r to recover a special\nsolution already given in the Appendix of our earlier paper [9]. The initial configuration is\ntaken to be an isosceles triangle defined by choosing the impact para metera=−b/2 in Fig. 1.\nThe vortex-antivortex pair (1 ,2) would then move upward with initial velocity V≈1/b(Kelvin\nmotion) and eventually collide against the target vortex 3. After co llision the vortex-antivortex\npair moves off to infinity at some scattering angle while the target vor tex comes to rest at a new\nlocation. Our aim in the following is to calculate the scattering angle as w ell as the final position\nof the target vortex.\nNow, inserting a=−b/2 in Eq. (37), we find that the values of the conserved quantities ar e\ngiven by\nB2=b2= Γ (38)\nand are independent of the initial triangle height h. Then Eqs. (31) read\nC1(t) =C2(t), C 3(t) =b (39)\nwhich establish that the triangle remains isosceles ( C1=C2) and the length of its base remains\nconstant ( C3=b) at all times t >0. ButC1itself undergoes a nontrivialtime evolution governed9\nby Eq. (33):\nd\ndt(C2\n1) =−νb/parenleftbigg1\nC2\n1+1\nb2/parenrightbigg/radicalBig\n4C2\n1−1. (40)\nA more convenient parametrization is obtained by\nC1=/radicalbigg\nH2+1\n4b2=C2, C 3=b, (41)\nwhereH=H(t) is the instantaneous height of the triangle. Eq. (40) then reads\ndH\ndt=−ν\nbH2+5\n4b2\nH2+1\n4b2(42)\nand must be solved with initial condition H(t= 0) =h.\nDuring the initial stages of the evolution the three vortices (1 ,2,3) appear in the plane in a\ncounterclockwise sense and thus ν= 1. Eq. (42) is then integrated to yield\nH\nζb−4\n5arctan/parenleftbiggH\nζb/parenrightbigg\n=t0−t\nζb2,0≤t≤t0 (43)\nζ≡√\n5\n2, t0≡ζb2/bracketleftbiggh\nζb−−4\n5arctan/parenleftbiggh\nζb/parenrightbigg/bracketrightbigg\n.\nAstapproaches t0, heightHapproaches zero and the triangle reduces to a straight line element\n(1,3,2) att=t0with vortex 3 located at the center of the line element (1 ,2). Also recalling\nthat the length of the line element (1 ,2) remains constant (equal to b) we conclude that t0is\nthe instance of the closest encounter of the three vortices. For t > t0, vortex 3 emerges from the\nother side and thus Eq. (42) must be integrated with ν=−1 and initial condition H(t=t0) = 0:\nH\nζb−4\n5arctan/parenleftbiggH\nζb/parenrightbigg\n=t−t0\nζb2, t > t 0. (44)\nIn the far future ( t→ ∞)H→t/bwhich is consistent with a vortex-antivortexpair (1,2) moving\naway from the target vortex 3 with asymptotic speed V= 1/b, in agreement with Eq. (12)\nThe preceding results already suggest a process in which the vorte x-antivortex pair (1,2)\ninitially approaches the target vortex 3 but eventually moves off to in finity at a scattering angle\nthat remains to be calculated. One must also ascertain the fate of t he target vortex 3 well after\ncollision. Both of these questions are answered below by applying Eqs . (35) and (36) to the\nspecial case treated in this subsection.\nFirst, we note that Eqs. (35) suggest that the three vortices (1 ,2,3) together with the origin of\nthe coordinatesystem O(i.e., the guiding centerofthe three-vortexsystem) forma paralle logram\n12O3 at all times during the collision. In particular, the third equation (35 ) establishes that\nr3=C3=bis time independent, or, vortex 3 moves on a circle whose center coin cides with\nthe guiding center and its radius is a constant bequal to the constant length of the base of the\ntriangle. Furthermore, the polar angle θ3changes during collision by the same amount as the\nbase of the triangle (1,2) rotates in the 2D plane. Therefore, the t otal change ∆ θ3is equal to\nthe scattering angle of the vortex-antivortex pair (1,2) and also s pecifies the final location of the\ntarget vortex 3.\nIn order to actually calculate ∆ θ3in the present special case, we apply Eq. (36) with C1=C2:\ndθ3\ndt=1\nC2\n1=1\nH2+1\n4b2, (45)10\n0 0.2 0.4 0.6 0.8 1\nt/t00246C1, C2, C3 (in units of b)C1C2\nC3\nFIG. 2: Time evolution of C1,C2andC3for an initial three-vortex triangle given by Fig. 1 with a=\n0,b= 1,h= 5. Note that all three lengths simultaneously collapse to z ero at a finite time t0calculated\nfrom Eq. (53).\nwhere we may insert H=H(t) from Eqs. (43) and (44) to obtain θ3=θ3(t) by a further\nintegration. In fact, a more direct calculation of the total scatte ring angle ∆ θ3is obtained by\ncombining Eqs. (42) and (45) to write\ndθ3\ndH=−b\nν1\nH2+5\n4b2, (46)\nand\n∆θ3=b/integraldisplayh\n0dH\nH2+5\n4b2+b/integraldisplay∞\n0dH\nH2+5\n4b2, (47)\nwhere the two terms may be interpreted as the scattering angles b efore and after collision and\ncorrespond to the two branches of the solution H=H(t) given by Eqs. (43) and (44). In\nparticular, for pure scattering where the vortex-antivortex pa ir originates very far from the\ntarget vortex ( h→ ∞):\n∆θ3= 2b/integraldisplay∞\n0dH\nH2+5\n4b2=2π√\n5, (48)\nwhich reproduces the special result given in the Appendix of Ref. [9 ]. A detailed illustration of\nthe actual vortex trajectories may be found in Fig. A1 of the abov e reference (using a different\nconvention for the origin of the coordinate system) and in Fig. 5 of t he present paper (where the\norigin of the coordinate system coincides with the guiding center).11\nB. Three-vortex collapse\nAt first sight, the special example treated in the preceding subsec tion would seem to cor-\nrespond to a head-on collision of a vortex-antivortex pair off a targ et vortex. However, such\nan interpretation seems to be unjustified in view of the peculiar scat tering angle calculated in\nEq. (48). Indeed, an example that is closer to head-on collision is obt ained by choosing vanishing\nimpact parameter ( a= 0) in the initial vortex configuration shown in Fig. 1.\nThe initial vortex triangle is then orthogonal, with initial values of the length of its sides\nC1(t= 0) =√\nh2+b2, C2(t= 0) = handC3(t= 0) = b. The two conserved quantities of\nEq. (30) now read\nB=C1C3\nC2=b\nh/radicalbig\nh2+b2,Γ =−C2\n1+C2\n2+C2\n3= 0. (49)\nA notable consequenceofthe conservationofΓ = 0 is that the vort extriangleremains orthogonal\nat all times, even though the length of its sides C1,C2, andC3, as well as its relative orientation\nin the 2D plane, undergo a nontrivial time evolution.\nIn particular, the evolution of C1is governed by Eq. (33) applied with Γ = 0 and Bgiven by\nEq. (49):\ndC1\ndt=−C2\n1+B2\nBC2\n1, (50)\nwhileC2, andC3obtained from Eq. (31) are now given by\nC2=C2\n1/radicalbig\nC2\n1+B2, C3=BC1/radicalbig\nC2\n1+B2. (51)\nEq. (50) may easily be integrated to yield\nC1\nB−arctan/parenleftbiggC1\nB/parenrightbigg\n=t0−t\nB2, (52)\nwhere the integration constant t0is calculated from the initial condition C1(t= 0) =√\nh2+b2,\nor\nt0=b2\nh2(h2+b2)/bracketleftbiggh\nb−arctan/parenleftbiggh\nb/parenrightbigg/bracketrightbigg\n. (53)\nThe physical significance of t0becomes apparent by applying Eq. (52) in the limit t→t0:\nC1(t→t0)∼[3B(t0−t)]1/3, (54)\nand Eqs. (51) in the same limit to yield\nC2(t→t0)∼1\nB[3B(t0−t)]2/3, C3(t→t0)∼[3B(t0−t)]1/3. (55)\nTherefore, t0calculated explicitly from Eq. (53) is the instance at which all three C1,C2and\nC3vanish simultaneously, or, the vortex triangle collapses to a point. Also taking into account\nEq. (35), we conclude that the vortex triangle collapses to the guid ing center ( r1=r2=r3= 0)\nof the three-vortex system in a finite time interval t0. The exact time dependence of C1,C2and\nC3calculated from Eqs. (52) and (51) over the entire time interval 0 ≤t≤t0is depicted in\nFig. 2.12\n-6-4 -2 0 2 4 6\nx/b-6-4-20246y/b\n123\nFIG. 3: Trajectories of the three vortices 1,2 and 3 emanatin g form an initial three-vortex configuration\ngiven by Fig. 1 with a= 0,b= 1,h= 5. Note that the three trajectories merge at the origin of th e\ncoordinate frame which is taken to coincide with the guiding center of the three-vortex system.\nComplete information about the vortex trajectories may be obtain ed by further utilizing the\nresults of Eqs. (35) and (36) or by a direct numerical integration o f Eqs. (15). The actual\ntrajectories are illustrated in Fig. 3 which explicitly demonstrates th e predicted three-vortex\ncollapse. A notable feature of Fig. 3 is that vortex 1 moves on a stra ight line that connects its\noriginal position with the guiding center of the three-vortex syste m.\nAlthough this special case of vanishing impact parameter ( a= 0) is clearly singular from the\npoint of view of scattering theory, it is still possible to calculate the s cattering angle ∆ θ3before\ncollapse. First, we apply the third of Eqs. (36) with Γ = 0:\ndθ3\ndt=1\nC2\n1(56)\nand combine this result with Eq. (50) to obtain\ndθ3=−BdC1\nB2+C2\n1. (57)\nThe scattering angle before collapse is then obtained by integrating Eq. (57) between the values\nC1(t= 0) =√\nh2+b2andC1(t= 0) = 0:\n∆θ3|before=/integraldisplay√\nh2+b2\n0BdC1\nB2+C2\n1= arctan/parenleftbiggh\nb/parenrightbigg\n. (58)\nIn the limit where the vortex-antivortex pair originates at a large dis tanceh→ ∞from the\ntarget vortex, we find ∆ θ3|before=π/2 and, by convention, ∆ θ3|total= 2(π/2) =π. This result\nagrees with the interpretation of this special case as a head-on co llision whose singular nature\nwill also become apparent in the discussion of the following subsection .13\nC. Scattering in the general case\nOur aim in this subsection is to determine the scattering angle as a fun ction of impact pa-\nrameter a. The general strategy is suggested by the special examples trea ted in the preceding\ntwo subsections. Thus we return to the initial configuration depict ed in Fig. 1, now applied for\narbitraty a, which yields values for the conserved quantities Band Γ as given in Eq. (37). In\nparticular, we shall be interested in the limit h→ ∞:\nB2=b2,Γ =−2ab, (59)\nwhich corresponds to pure (asymptotic) scattering where the vo rtex-antivortex pair is initially\nlocated very far from the target vortex and also diverges to infinit y after collision.\nIn order to calculate the scattering angle we return to Eq. (36) an d express its right-hand side\nentirely in terms of C1using Eq. (31) to eliminate C2andC3. We then combine the resulting\nequation with Eq. (33) to write\ndθ3\ndC2\n1=−1\nνΓ\n2B2−Γ\nC2\n1+Γ+C2\n1+Γ\nC2\n1+B2B2\n/radicalbig\n4B2C2\n1(C2\n1+Γ)2−Γ2(C2\n1+B2)2. (60)\nSimple inspection of this equation suggests using scaled variables uandγfrom\nu≡C2\n1\nB2, γ≡Γ\nB2=−2a\nb(61)\nto obtain\ndθ3\ndu=−1\nνγ(1−γ)\n2(u+γ)+u+γ\nu+1/radicalbig\n4u(u+γ)2−γ2(u+1)2. (62)\nBased on this equation the total scattering angle is calculated as a f unction of the parameter γ\nalone by a procedure similar to that employed in the special example tr eated in subsection A:\n∆θ3= 2/integraldisplay∞\nu0γ(1−γ)\n2(u+γ)+u+γ\nu+1/radicalbig\n4u(u+γ)2−γ2(u+1)2du (63)\nwhere the factoroftwoand the upper limit ofintegrationaredictat ed bythe fact that the vortex-\nantivortex pair is located very far from the target vortex well bef ore and after the collision. The\nlower limit u0is a specific value of u=C2\n1/B2for which the area of the vortex triangle vanishes.\nTherefore, u0must be chosen among the three roots of the cubic equation\n4u(u+γ)2−γ2(u+1)2= 0. (64)\nIt is thus important to examine the behavior of the roots as functio ns of the parameter γ.\nForγ <0 there exists a real and positive root u1and two complex roots u2andu3such that\nu2=u∗\n3. In this region the lower limit in the integral of Eq. (63) must be chose n asu0=u1.\nNow,γ= 0 is an exceptional point in that all three roots vanish ( u1=u2=u3= 0) and thus\nthe lower limit must be chosen as u0= 0. Applying Eq. (63) for γ= 0 and u0= 0 yields\n∆θ3=/integraldisplay∞\n0du\n(u+1)√u=π (65)\nin agreement with the conclusion of Subsection B. Indeed, γ= 0 correspondsto vanishing impact\nparameter ( a= 0) which is the special case discussed in Subsection B (three-vort ex collapse).14\nNext we consider the region 0 < γ <1 where the cubic equation again possesses a real and\npositive root u1and two complex roots u2=u∗\n3. Thus the lower limit must again be chosen\nasu0=u1. Now, γ= 1 is another exceptional point in that all three roots become real\n(u1=1\n4,u2=−1 =u3) and the lower limit must be chosen to coincide with the positive root\nu0=u1=1\n4. Applying Eq. (63) for γ= 1 and u0=1\n4yields\n∆θ3= 2/integraldisplay∞\n1\n4du\n(u+1)√4u−1=2π√\n5, (66)\nwhich coincides with the result of Eq. (48). This is not surprising beca useγ=−2a/b= 1 leads\nto an impact parameter a=−b/2 which is indeed the special case discussed in Subsection A.\nForγ >1, the cubic equation again possesses a real and positive root u1and two complex\nrootsu2=u∗\n3, thus we must choose u0=u1. This generic picture continues to hold until γ\nreaches the critical value\nγc=27\n2(67)\nwhere all three roots become real ( u1= 0.5625,u2= 9 =u3) and remain real and positive for\nγ > γc. In this range of γ, the three roots may be ordered as u1< u2< u3and the lower limit\nof integration in Eq. (63) must now be chosen as u0=u3.\n-10 -5 0 5\nimpact parameter  a/b-1-0.500.51scattering angle (in units of π)\nFIG. 4: Scattering angle during collision of a vortex-antiv ortex (Kelvin) pair (1,2) with a target vortex\n3, as a function of impact parameter a. Note the crossover behavior at the critical values a/b= 0 and\na/b=−27/4 discussed in the text. The results depicted in this figure sh ould be read modulo 2 π.\nIn all cases the denominator in Eq. (63) does not possess singularit ies in the domain of inte-\ngration [u0,∞) except for an integrable square root singularity at the lower end. It can also be\nshown that the numerator in Eq. (63) does not exhibit singularities in the integration domain.\nThus we have provided a complete prescription for the calculation of the scattering angle, which\nentails locating the appropriate root u0=u0(γ) of the cubic equation and an elementary numer-\nical integration of the integral in Eq. (63) for any value of γ=−2a/b. The calculated scattering\nangle as a function of impact parameter ameasured in units of bis depicted in Fig. 4.\nThus the scattering angle displays critical (crossover) behavior a t two characteristic values of\nthe impact parameter; namely, a/b= 0, which corresponds to the three-vortex collapse discussed15\nin subsection B, and a/b=−27/4, which corresponds to the critical coupling γ=−2a/b= 27/2\nof Eq. (67).\n-10 -5 0 5 10\nx/b-10-50510y/b\n123a\nb_-2=-10 -5 0 5 10\nx/b-10-50510y/b\n123_a\nb=-426__\n-10 -5 0 5 10\nx/b-10-50510y/b\n123a\nb_=-1\n2_\n-10 -5 0 5 10\nx/b-10-50510y/b\n123a\nb_ 1\n2=_-10 -5 0 5 10\nx/b-10-50510y/b\n123a\nb=_ 28\n4-__\n-10 -5 0 5 10\nx/b-10-50510y/b\n123a\nb=-_4\nFIG. 5: Vortex trajectories for six characteristic values o f the impact parameter a. In all cases a vortex-\nantivortex (Kelvin) pair (1,2) collides with a target vorte x 3 and is deflected by a scattering angle in\nagreement with the results of Fig. 4. The target vortex 3 is at rest well before collision and also comes\nto rest well after collision, as expected for an isolated vor tex which is always spontaneously pinned.\nIn Fig. 5 we display the actual vortex trajectories calculated nume rically from Eqs. (15) for a\nset of six characteristic values of a/b. The change of pattern becomes clear as one crosses the\ntwo critical values a/b= 0 and a/b=−27/4. The nature of the critical parameter γcof Eq. (67)\nis further illuminated in the following subsection.16\nD. Bounded three-vortex motion\nThe analysis of the cubic equation (64) revealed the importance of t he critical parameter\nγc= 27/2 of Eq. (67) above which all three roots are real (and positive). W e may then order\nthe roots as u1< u2< u3and note that the argument under the square root in, say, Eq. (6 2) is\npositive when either u > u3oru1< u < u 2. The former case,\nγ > γc=27\n2, u > u 3, (68)\ncorresponds to the scattering solution discussed in subsection C w hereu=C2\n1/B2approaches\ninfinity well before or after the collision, while it reaches its minimum valu eu3at some instance\nduring collision. In contrast, the case\nγ > γc=27\n2, u 1< u < u 2, (69)\nmay lead to a motion where uoscillates between the values u1andu2. Also taking into account\nEq. (31) we conclude that all three C1,C2andC3would then remain bounded. The aim of this\nsubsection is to explicitly construct a special class of solutions that realize Eqs. (69) and thus\nlead to bounded (quasiperiodic) three-vortex motion. In order to achieve the two conditions\ndisplayed in Eqs. (69) we consider an initial configuration where the t hree vortices (1,2,3) lie on\na straight line in the order indicated and thus the relative distances a re such that C1+C3=C2.\nSpecifically, we choose\nC1= (2+ǫ)b, C2= (3+ǫ)b, C3=b (70)\nand initially place the three vortices along the negative x-axis at point s such that\nx1=−C2−C3, x2=−C2, x3=−C3,\ny1=y2=y3= 0. (71)\nThen the origin of the coordinate system coincides with the guiding ce nter defined by Eqs. (28).\nOur aim is to calculate the vortex trajectories emanating from the in itial configuration defined\nby Eqs. (70) and (71).\nWe first consider the two conserved quantities calculated from suc h an initial configuration,\nB2=C2\n1C2\n3\nC2\n2=/parenleftbigg2+ǫ\n3+ǫ/parenrightbigg2\nb2,Γ =−C2\n1+C2\n2+C2\n3= 2(3+ǫ)b2, (72)\nand thus\nγ=Γ\nB2=2(3+ǫ)3\n(2+ǫ)2(73)\nis now the important dimensionless quantity that controls the behav ior of the roots of the cu-\nbic equation. Note that ǫ= 0 leads to the critical parameter γc= 27/2 of Eq. (67) while,\ninterestingly, γ > γcfor both positive and negative values of the parameter ǫ.\nA numerical solution of the initial value problem reveals an interesting picture. For ǫ >0, we\nencouter a situation of the type described by Eq. (68) which corre sponds to a scattering solution\nwherethevortex-antivortexpair(1,2)movesofftoinfinityleavingb ehindvortex3whichrelocates\nto a new final position. However, ǫ <0 leads to a situation of the type anticipated by Eq. (69).\nThe resulting bounded (quasiperiodic) three-vortex motion is illustr ated in Fig. 6 for ǫ=−1/4.17\n-6 -3 0 3 6\nx/b-6-3036y/b123\nFIG. 6: Trajectories of three vortices in bounded (quasiper iodic) motion emanating from an initial\nconfiguration specified by Eqs. (70) and (71) with ǫ=−1\n4.\nAlthough the present calculation is performed in an infinite magnetic fi lm, this type of bounded\nmotion may be relevant for the dynamics in finite, disc-shaped ferro magnetic elements.\nIn the special limit ǫ→0 the three vortices lie on a straight line for all times at constant\nrelative distances:\nC1= 2b, C2= 3b, C3=b. (74)\nSuch a linear configuration rotates rigidly around a fixed guiding cent er with constant angular\nfrequency\nω=dθ1\ndt=dθ2\ndt=dθ3\ndt=−1\n6b2, (75)\na relation obtained by applying Eqs. (36) to the present degenerat e case.\nV. CONCLUSIONS\nWe have studied the equations of motion for magnetic vortices which are approximated as\npoint vortices. These are a generalization of the Helmholtz-Kirchho ff equations [10, 11] derived\nfor ordinary fluid vortices. They differ from them in that magnetic vo rtices have an additional\ncharacteristic, their polarity, which enters in the equations. The m ain body of the paper is\ndevoted to the case of three magnetic vortices which constitute a n integrable system. For the\nintegration of this system we follow the steps of the seminal paper o f Gr¨ obli [12] modified to\ninclude the vortex polarity.\nMagnetic vortices are commonly observed in magnetic elements with t ypical dimensions of the\norder of a few micrometers or hundreds of nanometers. A single vo rtex spontaneously created\nat the center of an element can be driven by a magnetic field or an elec trical current to produce\nexcitations in its vicinity. It has been observed in various numerical s imulations that such18\nexcitations often have the form of a lump of magnetization opposite to that of the original\nvortex. In some cases the lumps are amplified to become vortex-an tivortex pairs with clearly\ndistinct constituent vortices. Such a process was apparently obs erved in experiments where\nthe original vortex was driven by a magnetic field [3, 4] or an electrica l current [5]. The fact\nthat the created vortex-antivortex pair has opposite polarity to the original vortex should most\nprobably be attributed to the effect of the magnetostatic (dipole- dipole) interaction which is the\nonly magnetic interaction that primarily favors domains of opposite m agnetization. From these\nexamples it follows that the three-vortex configuration, given in Eq . (27), on which we have\nmainly focused in this paper, is a particularly interesting one for the p urposes of understanding\nvortex dynamics in magnetic elements.\nIn view of the above comments it is interesting to compare our result s with simulations of\nthe Landau-Lifshitz equation. We have used the Landau-Lifshitz e quation with an exchange\ninteraction and an on-site easy-plane anisotropy so that vortices are the relevant excitations of\nthe system. Some results have been published in Refs. [8, 9]. For the case of vortex-antivortex\npairs where the vortices are well-separated the simulations show ve ry good agreement with the\nanalytical solutions for the vortex trajectories. It is worth discu ssing separately the special\ncase of three-vortex collapse of Subsection IVB which would appea r to be specific to the point\nvortex system. We have simulated the case of zero angular moment um within the Landau-\nLifshitz equation and have observed vortex trajectories very sim ilar to those shown in Fig. 3.\nThe original vortex and the antivortex are annihilated when the thr ee-vortex system collapses\nto the origin and an isolated vortex 1 is the final outcome of the simula tion. The results of the\nsimulation are closer to the analytical solutions when the vortex-an tivortex pair is large, i.e., the\nvortex and the antivortex are initially well separated.\nAknowledgements\nWe are grateful to Mahir Hussein for stimulating discussions that re focused our interest on\nthis subject.\n[1] A. P. Malozemoff and J. C. Slonczewski, “Magnetic Domain Walls in Bubble Materials” (Academic\nPress, New York, 1979).\n[2] D. L. Huber, Phys. Rev. B 26, 3758 (1982).\n[3] A. Neudert, J. McCord, R. Sch¨ afer, and L. Schultz, J. App l. Phys.97, 10E701 (2005).\n[4] B. V. Waeyenberge, A. Puzic, H. Stoll, K. W. Chou, T. Tylis zczak, R. Hertel, M. F¨ ahnle, H. Br¨ uckl,\nK. Rott, G. Reiss, I. Neudecker, D. Weiss, C. H. Back, and G. Sc h¨ utz, Nature(London) 444, 461\n(2006).\n[5] K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A. Thiaville, and T. Ono, Nature\nMaterials 6, 269 (2007).\n[6] N. Papanicolaou and T. N. Tomaras, Nucl. Phys. B 360, 425 (1991).\n[7] S. Komineas and N. Papanicolaou, Physica D 99, 81 (1996).\n[8] S. Komineas and N. Papanicolaou, “Dynamics of vortex-antivortex pairs in ferromagnets” in E.\nKamenetskii (ed.), “Electromagnetic, Magnetostatic, and exchange-interaction vortices in confined\nmagnetic structures”, (Transworld Research Networks, 200 8); arXiv:0712.3684v1\n[9] S. Komineas and N. Papanicolaou, New J. Phys. 10, 043021 (2008).\n[10] H. Helmholtz, J. Reine Angew. Math. 55, 22 (1858) [P. G. Tait, Philos. Mag. 33, 485–510 (1867)].\n[11] Kirchhoff, “Vorlesungen ¨ uber mathematische Physik. Mechanik” (Teubner, Leipzig, 1876).\n[12] W. Gr¨ obli, “Spezielle Probleme ber die Bewegung geradliniger paralle ler Wirbelf¨ aden” (Z¨ urcher and\nFurrer, Z¨ urich, 1877). Reprinted in Vierteljahrsschr. Na tforsch. Ges. Zur. 22, 37–81 (1877); ibid.\n22, 129–165 (1877).19\n[13] H. Aref, N. Rott, and H. Thomann, Annu. Rev. Fluid Mech. 24, 1 (1992).\n[14] H. Aref, Phys. Fluids 22, 393 (1979).\n[15] A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n[16] V. L. Pokrovskii and G. V. Uimin, JETP Lett. 41, 128 (1985).\n[17] A. Kovalev, S. Komineas, and F. G. Mertens, Eur. Phys. J. B65, 89 (2002). q\n[18] N. Papanicolaou and P. N. Spathis, Nonlinearity 12, 285 (1999).\n[19] S. Komineas, Phys. Rev. Lett. 99, 117202 (2007)."    },    {        "title": "2007.07325v1.Inverse_energy_transfer_in_decaying__three_dimensional__nonhelical_magnetic_turbulence_due_to_magnetic_reconnection.pdf",        "content": "MNRAS 000, 1–14 (2020) Preprint 16 July 2020 Compiled using MNRAS L ATEX style file v3.0\nInverse energy transfer in decaying, three dimensional, nonhelical\nmagnetic turbulence due to magnetic reconnection\nPallavi Bhat1;2?, Muni Zhou2and Nuno F. Loureiro2\n1Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, UK\n2Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\n16 July 2020\nABSTRACT\nIt has been recently shown numerically that there exists an inverse transfer of magnetic en-\nergy in decaying, nonhelical, magnetically dominated, magnetohydrodynamic turbulence in\n3-dimensions (3D). We suggest that magnetic reconnection is the underlying physical mecha-\nnism responsible for this inverse transfer. In the two-dimensional (2D) case, the inverse trans-\nfer is easily inferred to be due to smaller magnetic islands merging to form larger ones via\nreconnection. We find that the scaling behaviour is similar between the 2D and the 3D cases,\ni.e., the magnetic energy evolves as t\u00001, and the magnetic power spectrum follows a slope of\nk\u00002. We show that on normalizing time by the magnetic reconnection timescale, the evolution\ncurves of the magnetic field in systems with different Lundquist numbers collapse onto one\nanother. Furthermore, transfer function plots show signatures of magnetic reconnection driv-\ning the inverse transfer. We also discuss the conserved quantities in the system and show that\nthe behaviour of these quantities is similar between the 2D and 3D simulations, thus making\nthe case that the dynamics in 3D could be approximately explained by what we understand\nin 2D. Lastly, we also conduct simulations where the magnetic field is subdominant to the\nflow. Here, too, we find an inverse transfer of magnetic energy in 3D. In these simulations, the\nmagnetic energy evolves as t\u00001:4and, interestingly, a dynamo effect is observed.\nKey words: (magnetohydrodynamics) MHD–turbulence–magnetic reconnection\n1 INTRODUCTION\nTurbulent processes are of fundamental importance to a wide range\nof systems, from quantum fluids to astrophysical plasmas (Skrbek\n& Sreenivasan 2012; Biskamp 2003). In a typical turbulent sys-\ntem, energy injected at a certain scale direct cascades down to\nsmaller and smaller scales until it is dissipated by microphysical\nprocesses. On the other hand, an inverse cascade, or inverse trans-\nfer, involves energy being transferred from smaller to larger scales.\nThis can occur in both forced or freely decaying turbulent systems\n(e.g., Davidson 2004). The best known inverse cascading system\nis two-dimensional (2D) hydrodynamic turbulence, where energy\ninverse cascades, while enstrophy direct cascades (Batchelor 1969;\nKraichnan 1967). Indeed, the 2D hydrodynamic inverse cascade is\nwidely considered one of the most important results in turbulence\n(Frisch 1995; Falkovich & Sreenivasan 2006) since Kolmogorov’s\n1941 work. Both energy and enstrophy are inviscid invariants in\n2D hydrodynamics. Here, the existence of more than one ideally\nconserved quadratic quantity in the system can lead to an inverse\ncascade (Nazarenko 2011). The 3D system mimics 2D-like inverse\ntransfer when there is anisotropy due to strong rotation or the pres-\nence of a strong magnetic field (Yakhot & Pelz 1987; Baggaley\n?P.Bhat@leeds.ac.uket al. 2014; Pouquet et al. 2019). Biferale et al. (2012) demon-\nstrated that even in the case of 3D isotropic and homogeneous hy-\ndrodynamic turbulence, there can be an inverse cascade when parity\n(mirror symmetry) of the flow is broken.\nSimilarly, in 3D magnetohydrodynamics (MHD), it is well\nknown that even in isotropic and homogenous decaying turbu-\nlence, inverse cascade occurs due to the presence of non-zero net\nmagnetic helicity which breaks the parity in the system (Pouquet\net al. 1976; Christensson et al. 2001). Magnetic helicity is a well-\nconserved quantity in the limit of large magnetic Reynolds number\n(Rm). Thus it is possible to have an inverse transfer in decaying\nturbulence in 3D MHD as long as it has helical magnetic fields\n(Christensson et al. 2001). However, recent simulations (Branden-\nburg et al. 2015; Zrake 2014; Berera & Linkmann 2014; Reppin &\nBanerjee 2017; Zhou et al. 2020) have shown that there exists an in-\nverse transfer of magnetic energy in 3D MHD decaying turbulence,\neven in the absence of magnetic helicity.\nIn this paper we investigate the underlying cause of such a\n3D nonhelical inverse transfer. We find that there are similarities\nbetween the 2D and 3D cases. The 2D inverse transfer has been\npreviously well-studied and the ideal conserved quantities have\nbeen identified such as the total energy and vector-potential squared\n(Fyfe & Montgomery 1976; Pouquet 1978; Biskamp & Welter\n1989). However, earlier 2D studies used Kolmogorov-type argu-\nc\r2020 The AuthorsarXiv:2007.07325v1  [astro-ph.CO]  14 Jul 20202\nments to obtain scaling solutions for the decaying field (Biskamp\n2003). These arguments do not shine light upon the underlying\nphysical processes responsible for the inverse transfer. In recent\nwork by Zhou et al. (2019), a simple model based on merging mag-\nnetic islands provides a physical picture for the inverse transfer in\nthe 2D system, and finds that the relevant timescale is that dictated\nby magnetic reconnection, which underlies such mergers. Here, we\npropose that magnetic reconnection is responsible for the 3D non-\nhelical inverse transfer as well. Using direct numerical simulations,\nwe study 3D, nonhelical, decaying MHD turbulence and build con-\nnections to the 2D case. We present evidence of similarities be-\ntween 2D and 3D systems, and suggest magnetically dominated\n3D systems display a 2D-like behavior.\nWe believe these findings are pertinent to several cosmo-\nlogical and astrophysical contexts. This reconnection-based un-\nderstanding of the nonhelical inverse transfer, if true, affects the\ntimescales of magnetic field evolution in the early universe (Baner-\njee & Jedamzik 2004; Sethi & Subramanian 2005; Subramanian\n2016). Occurrence of reconnection in magnetically dominated de-\ncaying turbulence can be relevant to the understanding of high en-\nergy phenomena such as gamma-ray bursts and Crab nebula flares\n(Asano & Terasawa 2015; Zrake 2016; Blandford et al. 2017). Fur-\nthermore, such decaying turbulence has been studied in the context\nof star-formation in molecular clouds (Mac Low et al. 1998; Gao\net al. 2015), and is relevant to the seeding of magnetic fields in pro-\ntogalaxies from supernovae ejecta (Beck et al. 2013), and also in\nthe case of galaxy-clusters after a merger event (Subramanian et al.\n2006; Sur 2019). Zhou et al. (2020) present a discussion on the\nsignificance of inverse transfer in obtaining seed magnetic fields\nrequired for galactic dynamos (see their Appendix A).\n2 NUMERICAL SETUP\n2.1 The model\nWe use the P ENCIL CODE1to simulate decaying MHD turbulence\nin both 2D and 3D. We solve the MHD equations given by\nDln\u001a\nDt=\u0000r\u0001u; (1)\nDu\nDt=\u0000c2\nsrln\u001a+J\u0002B\n\u001a+Fvisc\n\u001a; (2)\n@A\n@t=u\u0002B\u0000\u0011\u00160J; (3)\non a Cartesian N2orN3grid, with periodic boundary conditions,\nwhereNis the number of grid points in any given direction. The\noperatorD=Dt =@=@t +u\u0001ris the advective derivative, with u\nthe fluid velocity field. We solve the uncurled version of the in-\nduction equation, in terms of the vector potential, A, related to\nthe magnetic field by B=r\u0002A. We adopt the Weyl gauge\n\b = 0 , where \bdenotes the scalar potential. The current den-\nsity is J=r\u0002B=\u00160, with\u00160, the vacuum permeability. The\nviscous force is Fvisc=r\u00012\u0017\u001aS, where\u0017is the kinematic\nviscosity, and Sis the traceless rate of strain tensor with compo-\nnents Sij=1\n2(ui;j+uj;i)\u00001\n3\u000eijr\u0001u(commas denote par-\ntial derivatives). Finally, \u0011is the magnetic diffusivity. In the 2D\n1DOI:10.5281/zenodo.2315093, github.com/pencil-codeTable 1. A summary of all runs and their respective parameters (in dimen-\nsionless units). urms0andBrms0are the initial root-mean-squared val-\nues of the flow and the magnetic field respectively. In all runs the initial\nkp\u001925.\nRun Resolution \u0011\u0002104urms0Brms0S\nA2D 204820.5 0.0 0.2 1000\nB2D 102421.0 0.0 0.2 500\nC2D 102422.0 0.0 0.2 250\nF2D 102421.0 0.2 0.02 50\nA3D 102430.5 0.0 0.2 1000\nB3D 102431.0 0.0 0.2 500\nC3D 51232.0 0.0 0.2 250\nD3D 51234.0 0.0 0.2 125\nE3D 51238.0 0.0 0.2 50\nF3D 102430.25 0.2 0.02 200\nruns, we solve a 2D version of equations (1 - 3) obtained by set-\nting@z= 0 and eliminating vector components in the zdirection.\nOther than compressibility effects (which are minor in our simula-\ntions), this 2D version of the equations is identical to the 2D ver-\nsion of the reduced-MHD equations Kadomtsev & Pogutse (1974);\nStrauss (1976); Schekochihin et al. (2009). The code uses dimen-\nsionless quantities by measuring length in units of the domain size\nL, speed in units of the isothermal sound speed cs, density in units\nof the initial value \u001a0and magnetic field in units of (\u00160\u001a0c2\ns)1=2.\nWe chooseL= 2\u0019, andcs=\u001a0=\u00160= 1.\n2.2 Initial conditions and parameters\nThe initial magnetic field is generated in the wavenumber space\nwith a certain spectrum and random phases, similar to the method\nin Brandenburg et al. (2015). The magnetic power spectrum is k4\n(Brandenburg et al. 2015) for k < k 0, and is exponentially cutoff\nbeyondk0. Such a spectrum is obtained from the vector potential\nin Fourier space, ^Aj(k), whose three components jare given by,\n^Aj(k) =A0(k2=k2\n0)n=4\u00001=2exp (\u0000k2=k2\n0) exp (i\u001e(k)) (4)\nwhere exponent n= 2,\u001e(k)are random phases and A0is the\namplitude.\nWe define the Lundquist number in our simulations as S=\nVA(2\u0019=kp)=\u0011, whereVAis the Alfv ´en velocity and kpis the\nwavenumber at which the magnetic power spectrum peaks. In our\nmain runs, the initial Alfv ´en velocity is VA= 0:2(which implies\nthat compressibility effects are weak and can be ignored in the anal-\nysis of the dynamics) and the initial velocity field is zero, analysed\nin sections 3.1-3.3. We have also carried out runs with non-zero ini-\ntial velocity field; these are reported in section 3.4. In all runs, ini-\ntialkp\u001925. We have run simulations across a range of Lundquist\nnumbers as allowed by the resolution limit of 20482in 2D and\n10243in 3D. The magnetic Prandtl number in all our simulations\nis1. A list of all runs reported in this paper is shown in Table 1.\n3 RESULTS\n3.1 Decaying turbulent magnetic fields in 2D\nWe first present our study of 2D simulations of decaying MHD tur-\nbulence (simulations A2D, B2D and C2D in Table 1). In the top\nMNRAS 000, 1–14 (2020)3\nFigure 1. Top panel: Evolution of magnetic energy (solid black) and ki-\nnetic energy (dashed blue) in a 2D simulation, A2D, with S= 1000 and\nresolution of 20482. Bottom panel: Compensated magnetic power spectra\nk2M(k;t)are plotted at regular intervals of \u0001t= 10 with a thick final\ncurve att= 70 .\npanel of Fig. 1, the evolution of magnetic energy is shown in a log-\nlog plot. It decays in time as a power law, with an exponent close\nto\u00001at late times. This result matches with that obtained in the\n2D simulations performed by Zhou et al. (2019), which focused\non an initial condition consisting of an ordered array of current\nfilaments (or, equivalently, magnetic islands) with alternating po-\nlarities. Upon introducing small perturbations into that system, the\ncurrent filaments move out of the initial (unstable) equilibrium. The\nsubsequent evolution of the system is then primarily dictated by the\ncoalescence, via magnetic reconnection, of filaments with equal po-\nlarity. Mergers of island pairs lead to larger islands, albeit at the cost\nof magnetic energy dissipation. Successive mergers lead to progres-\nsively larger structures, resulting in an inverse transfer of magnetic\nenergy. This occurs hierarchically in a self-similar manner, giving\nrise to power-law-in-time behaviour. Similarly, even in the case of a\nrandom initial condition such as employed here, we observe an in-\nverse transfer as the system evolves in time. This is quite evident in\nthe time progression of the magnetic power spectrum, shown in the\nbottom panel of Fig. 1. The initial spectrum (random field peaked at\nkp\u001825) is seen to shift from large wavenumbers to smaller ones,\ndepicting an inverse transfer. The spectra are compensated by k2to\nreveal a range where they flatten; the same power law is observed\nby (Zhou et al. 2019), who attribute it to the dominance of sharp\ncurrent sheets. (i.e., a Burgers’ spectrum (Burgers 1948)).\nAs in Zhou et al. (2019), the growth of magnetic energy at\nlarge scales that we find in our 2D simulations is due to magnetic\nreconnection. This can be seen explicitly and clearly from a se-\nquence (a movie) of time evolving contour plots of Az(see sup-\nplementary material), or from the corresponding stills at specific\nmoments of time shown in Fig. 2. Current sheets — sharply local-\nized enhancements of current density in Fig. 3 — are seen to form\nat the interface of any pair of interacting islands, leading to their re-\nconnection and resulting in larger islands. The magnetic islands can\nbe seen to grow progressively larger in time due to island mergers.A complementary way to understand inverse transfer in this\n2D system is to consider the conserved quantities in the system. For\nthe ideal 2D MHD equations (in the absence of dissipation), these\nare the total energy, EM+EK=hB2i=2 +\u001ahu2i=2(given weak\ncompressibility), and vector-potential squared, P=hA2i, (where\nhirepresents integral over the domain) (e.g., Biskamp 2003). In the\nfollowing we show that in our non-ideal system where the kinetic\nenergy is subdominant, by considering the evolution of EMandP,\nwe can deduce that decaying 2D MHD turbulence displays inverse\ntransfer of energy. The evolution equations for magnetic energy and\nvector-potential squared in a closed domain (given periodic bound-\naries in the DNS) in the non-ideal case are given by,\nZZ\ndS@\n@t\u0012B2\n2\u0013\n=\u0000ZZ\ndSu\u0001(J\u0002B) +\u0011J2;(5)\nZZ\ndSD\nDt\u0012A2\nz\n2\u0013\n=\u0000ZZ\ndS\u0011B2: (6)\nIn the 2D limit that we consider here, only one component of the\nvector potential is needed, i.e. B=r\u0002Az^z, where the ^zis the\nunit vector orthogonal to the 2D plane. Similarly, only one compo-\nnent of the current density survives, Jz=@xBy\u0000@yBx.\nFrom Eqs. (5) and (6), it is possible to deduce the following\nimplications for a freely evolving turbulent system. While the evo-\nlution of vector potential squared is governed by only a decay term\non the RHS of Eq. (6), the equation for magnetic energy, Eq. (5),\nalso consists of a source term given by u\u0001(J\u0002B). Depending on\nthe sign of this term, either the energy is being transferred from the\nmagnetic field to the velocity field, or vice versa . Now, in these sim-\nulations, the velocity field is initially zero, and it is entirely driven\nby the magnetic field. We assume that the back-reaction from the\ngenerated flow on the field is negligible: this is a reasonable as-\nsumption if the kinetic energy is subdominant, as is indeed the case\nin our system (see the top panel of Fig. 1). Thus, as the system is\nallowed to evolve freely, the magnetic field loses its energy to ei-\nther the velocity field or to resistive decay. Given that the system is\nturbulent, the magnetic field is expected to decay even as \u0011!0be-\ncause the field can develop small enough scales (current sheets). As\na result,\u0011hJ2ican remain finite in that limit. However, as \u0011!0,\nthe term on the right hand side of Eq. (6), \u0011hB2i(wherehB2iis\nessentially independent of resistivity) will go to zero, thereby ren-\ndering volume-integrated vector-potential squared, P, to be nearly\ninvariant. In short, in the limit of \u0011!0(or, equivalently, in the\nlimit of very large RmorS), the vector potential squared is better\nconserved than the magnetic energy, EM=hB2i=2.\nWe can now use this conservation property to argue why such\na freely decaying turbulent system can exhibit inverse transfer. In\nthe Fourier domain we have ^B=ik\u0002^A^z, where k=kx^x+ky^y.\nIt follows that\nj^Bj2=k2j^Aj2: (7)\nNow, let us use the expressions for EMandPin the Fourier\ndomain,\nEM=1\n2Z\nj^Bj2d2k; (8)\nP=Z\nj^Aj2d2k; (9)\nand consider that most of the magnetic energy is concentrated in\na single scale in the system; we shall call it the correlation scale,\nkcorr. It then follows from Eq. (7) that\nkcorr\u0018p\nEM=P: (10)\nMNRAS 000, 1–14 (2020)4\nFigure 2. Evolution of the vector potential ( Az) in the 2D simulation A2D, with S= 1000 . The times plotted are t= 1,t= 15 andt= 45 , from left to\nright.\n−3.0 −2.8 −2.6 −2.4 −2.2 −2.0 −1.8 −1.6−3.0−2.8−2.6−2.4−2.2−2.0−1.8−1.6\n−36.8−27.6−18.4−9.20.09.218.427.636.8\n−3.0 −2.8 −2.6 −2.4 −2.2 −2.0 −1.8 −1.6−3.0−2.8−2.6−2.4−2.2−2.0−1.8−1.6\n−17.6−13.2−8.8−4.40.04.48.813.217.6\n−3.0 −2.8 −2.6 −2.4 −2.2 −2.0 −1.8 −1.6−3.0−2.8−2.6−2.4−2.2−2.0−1.8−1.6\n−3.36−2.52−1.68−0.840.000.841.682.523.36\nFigure 3. Evolution of the current density ( Jz) in1=8of the domain from the 2D simulation A2D, at times t= 1,t= 15 andt= 45 , from left to right. The\noverlaid lines are contours of Az.\nSince this is an unforced stochastic system, the magnetic en-\nergy,EM, will decay. Given that Pis better conserved than EM,\nPremains nearly constant as EMdecreases; thus, the wavenumber\nkcorris expected to decrease. This implies a shift of the correla-\ntion scale in the system to larger and larger scales — the spectral\nsignature of an inverse transfer. Indeed, if we substitute the scaling\nEM/t\u00001into Eq. (10), and consider Pto be constant in time, we\nobtainkcorr/t\u00001=2. This is consistent with what we find from\nour simulations when we trace kcorras a function of time. Both of\nthese scalings are predicted by the reconnection-based hierarchi-\ncal model of Zhou et al. (2019), and are verified by the RMHD\nnumerical simulations carried out in that paper (note that the hier-\narchical model itself is based on mass and magnetic flux ( Az) being\nconserved during island mergers through reconnection). Thus, we\nconclude that the implications from 2D conservation properties are\nconsistent with the physical picture of island mergers via reconnec-\ntion in 2D; together, they provide a solid explanation for the inverse\ntransfer of magnetic energy in the 2D system.\n3.2 Decaying turbulent nonhelical magnetic fields in 3D\nNext, we turn to 3D simulations (runs A3D, B3D, C3D, D3D and\nE3D in Table 1). The 3D run resolutions go up to 10243grid points,\nand all have an initial condition similar to the 2D case of randommagnetic fields with power peaked at small scales, as specified in\nEq. (4).\nAs in the 2D case, we again observe a power-law-in-time mag-\nnetic energy decay with exponent \u00001, as shown in the top panel of\nFig. 4 (at later times, the decay of magnetic energy steepens, pos-\nsibly due to diffusion beginning to dominate the system. Branden-\nburg et al. (2015), who use a similar setup, do not report such a\ntransition, possibly because the higher resolution that they employ\n(23043) reduces diffusive effects in their simulation).\nFrom the bottom panel of Fig. 4, a flat range indicating k\u00002\nslope in the magnetic spectrum can be observed (Brandenburg et al.\n2015; Zrake 2014), however with limited range given the resolu-\ntion. These scalings are intriguingly similar to the ones seen already\nin the 2D case, thus triggering the following questions:\n(i) To what extent are the 3D simulations similar to the 2D ones?\n(ii) Can we conclude that, even in 3D, ”structure mergers” via\nreconnection are responsible for this inverse transfer of magnetic\nenergy?\nThis section and the next are concerned with answering these ques-\ntions.\nFirstly, we see in Fig. 5 qualitative similarities with the 2D\nruns; namely, the evolution of the magnetic field structures (from\na slice out of the 3D domain) resembles the behaviour of the mag-\nnetic islands seen in the contour plots from the 2D system (Fig. 2).\nMNRAS 000, 1–14 (2020)5\nFigure 4. Top panel: evolution of magnetic energy (solid black) and kinetic\nenergy (dashed blue) in the 3D simulation A3D, with S= 1000 and res-\nolution of 10243. The bottom panel shows compensated magnetic power\nspectrak2M(k;t)for the same run, plotted at regular intervals of \u0001t= 5,\nwith a thick final curve at t= 50 .\nWe also show the evolution of the x-component of the field, Bx,\nin Fig. 6. It is clearly seen that the field structures grow in scale.\nHowever, here in the 3D case, the structures are more elongated\nand are not as symmetric as in the 2D case. Nonetheless, they do\nnot exhibit any specific directionality overall. In other words, while\nlocally each field structure does seem to prefer a certain direction\n(given the elongation), these preferences are randomly distributed\nover the domain. Thus, there is no development of a large-scale\nstructure which can bias the system in a certain randomly cho-\nsen direction, as is routinely seen, for example, in helical dynamos\n(Brandenburg & Subramanian 2005).\nFrom comparisons with the 2D results, there is a suggestion\nthat perhaps, even in the 3D system, a reconnection-based mech-\nanism might be responsible for the growth of the structures over\ntime (Zhou et al. (2020) have explored the suggestion in this work,\nalso in the context of reduced-MHD, and found it to correctly\ndescribe their numerical results). To further support this sugges-\ntion, we show in Fig. 7 the absolute value of the current density,\njJj=p\nJ2x+J2y+J2z). The wispiness of the current density\nstructures corroborates the existence of current sheets where recon-\nnection can take place.\nAlready at this point it is possible to argue for why there\nare similarities between the 2D and the 3D results. Given that\nthe system is magnetically dominated, we think that strong local\nanisotropy spontaneously arises. This is reflected in the previously\nmentioned elongation of the magnetic structures in Figs. 5 and 6.\nThis local anisotropy, then, could lead to 2D-like behaviour. While\nthe Sweet-Parker scaling of magnetic reconnection rate does not\nchange from 2D to 3D, this local anisotropy entails the possibil-\nity of the existence of local guide-fields, if required to render the\n3D reconnections with 2D-like behaviour. This would explain why\nwe see results in 3D which are similar to that in 2D (such as the\nmagnetic energy scaling of t\u00001and the spectral scaling of k\u00002).\nNext, we look at the conservation properties in both the 2Dand 3D systems. First, we show in the top panel of Fig. 8 the evo-\nlution of the rate of change of the 2D MHD ideal invariants P\n(black) andEM(red-dashed) (given that kinetic energy is subdom-\ninant here), \u0015A=d(lnP)=dtand\u0015B=d(lnEM)=dt, respec-\ntively, calculated from run A2D. As expected, \u0015Ais much smaller\nthan\u0015B, thus demonstrating Pto be better conserved than EM, as\nwe have argued earlier. In the bottom panel of Fig. 8, we show the\nevolution of\u0015Aand\u0015Bfrom the 3D simulation A3D, and again we\nfind the former to be much smaller than the latter. While theoreti-\ncallyPis strictly an ideal invariant only in 2D, these results suggest\nthat it is possible to make a case for its approximate conservation\nin 3D as well.\nConsider, therefore, the evolution of Pin 3D,\nZ\ndVD\nDt\u0012A2\n2\u0013\n=Z\ndVu\u0001(A\u0001rA)\u0000\u0011B2: (11)\nThis equation differs from the 2D case only by the term u\u0001\n(A\u0001rA)on the RHS. Here, again, we appeal to the fact that flow\nis subdominant to the field in order to assume that backreaction\nof the flow on the field is negligible. Such subdominance can be\nseen in Fig. 9: in the vicinity of the peak wavenumber, the am-\nplitude of the kinetic power spectra are lower than the magnetic\npower spectra by about an order of magnitude, in both 2D and\n3D cases. Furthermore, the source term u\u0001(A\u0001rA)in question\nfrom Eq. (11) can be compared to the analogous source term in\nthe equation for the magnetic energy Eq. (5), u\u0001(B\u0001rB)(Note\nthat Eq. (5) is valid in 3D also). This term arises on expanding\nu\u0001(J\u0002B) =u\u0001(\u0000r(B2=2) +B\u0001rB). Assuming that\njBj\u0018kcorrjAj(in a scenario where most of the power is in a sin-\ngle scale, represented by the wavenumber kcorr\u001d1), then these\nsources differ by a factor of k2\ncorr, with the term u\u0001(A\u0001rA)be-\ning smaller of the two. Thus, again, we conclude that in the limit of\n\u0011!0,Pdecays much slower than EM. Consequently, it follows\nfrom Eq. (10) that there can be an inverse transfer in 3D as well, as\nseen in the 3D simulations.\nSince we are dealing with quantities based on vector poten-\ntial, a fair concern is with regard to the gauge dependence. As men-\ntioned earlier, our model equations adopt the Weyl gauge ( \b = 0 ).\nTo check for possible gauge-related effects in the results, we per-\nformed a simulation using instead the Lorenz gauge, with the same\nparameters and initial conditions as those employed in our main\nruns with the Weyl Gauge. In the Lorenz gauge (or the pseudo-\nLorenz gauge), we have @t\b =\u0000c2\nsr\u0001A(Brandenburg & K ¨apyl¨a\n2007), where csis the speed of sound instead of the speed of light.\nWe overplot the result in the bottom panel of Fig. 8 (dotted blue\nline). It can be seen that the results from the Lorenz gauge are in-\ndistinguishable from those with the Weyl gauge. This is consistent\nwith the expectation of better conservation of Pthan ofEMto hold\nup in any gauge within a closed domain, as the sink terms in the\nequations for PandEMremain the same.\nWhile these arguments based on ideal conserved quantities are\nuseful to provide plausibility to the notion that the understanding of\n3D nonhelical inverse transfer lies in its 2D like behaviour, we still\ndo not have more substantial evidence for reconnection being the\ndriving factor for the inverse transfer. To gain a better understand-\ning of the system, we study the timescale governing its dynamical\nevolution. In doing so, we continue to probe the similarities be-\ntween the 2D and 3D cases.\nThe power law governing the evolution of the magnetic field in\nthe 2D system is expected to be Brms=B0(t=\u001crec)\u00001=2as shown\nby Zhou et al. (2019), where \u001crecis the reconnection time scale,\ngiven by\u001crec=\f\u00001\nrec(2\u0019=kcorr0)=VA0, with\frecthe normalized\nMNRAS 000, 1–14 (2020)6\nFigure 5. Contour plots of the z-component of the vector potential ( Az), in an arbitrary 2D slice (in the x\u0000yplane) from the 3D simulation C3D. The times\nplotted aret= 2,t= 15 andt= 50 , from left to right.\nFigure 6. Evolution of a component of the magnetic field, Bx, shown on the 3D domain from the 3D simulation, C3D, at times t= 2,t= 15 andt= 50\nfrom left to right.\n−3 −2 −1 0 1 2 3−3−2−10123\n0.03.67.210.814.418.021.625.228.832.4\n−3 −2 −1 0 1 2 3−3−2−10123\n0.0000.3450.6901.0351.3801.7252.0702.4152.7603.105\n−3 −2 −1 0 1 2 3−3−2−10123\n0.0000.0920.1840.2760.3680.4600.5520.6440.7360.828\nFigure 7. Evolution of the absolute value of the current density ( jJj) in 1/4 of the domain of an arbitrary 2D slice from the 3D simulation, A3D, shown in\ncontour plots at times t= 3,t= 21 andt= 60 , from left to right.\nreconnection rate, kcorr0 is the wavenumber associated with the ini-\ntial correlation scale and VA0is the initial Alfv ´en velocity. Here\nwe use the Sweet-Parker scaling for the reconnection rate Sweet\n(1958); Parker (1957), \frec=S\u00001=2, which is appropriate for\nvalues ofSlower than the value critical to trigger the plasmoidinstability (Loureiro et al. 2007; Samtaney et al. 2009). Note that\nas the simulation proceeds, the correlation scale, (2\u0019=kcorr)(we\ntakekcorr=kp) increases and the Alfv ´en velocity,VA, decreases,\nand thus the Lundquist number, S=VA(2\u0019=kcorr)=\u0011is expected\nto remain constant (Zhou et al. 2019). For two different runs with\nMNRAS 000, 1–14 (2020)7\nFigure 8. The rate of change of vector potential squared, \u0015A(dashed red),\nand magnetic energy, \u0015B(solid black), is shown for 2D and 3D simula-\ntions, A2D and A3D (where S= 1000 ) in the top and bottom figures,\nrespectively. In each figure, the upper panel is a log-linear plot, whereas the\nlower panel is a linear-linear one. In the bottom figure, an additional curve\nfrom a 3D simulation employing the Lorenz gauge is shown in dotted blue.\ndifferent Lundquist numbers S1andS2, at any given time t, the\nratio of the magnetic field strengths is then predicted to scale as\nBrms1=Brms2= (S1=S2)1=4.\nIn Figs. 10 and 11, we compare Brmsevolution curves from\n2D and 3D runs, respectively, with different values of S, which\nvary by a factor of 2 from one run to another. In the bottom panels\nof Figs. 10 and 11, we normalize the time axis by the reconnection\ntimescale\u001crec(note that the normalization \u001crecis computed for\nthe initialkcorrand not varied with time; this is because kcorris\na discrete quantity and thus its variation does not lead to a secular\nevolution of the time axis t=\u001crec). On applying this normalization,\nthere is a notable tendency for curves from different simulations to\ncollapse on top of each other. The collapse of the curves is better\nin the 2D case than the 3D case; but, even in the 3D case, for runs\nwith increasing values of S, the gap between the successive curves\ndecreases. The curves from runs with the highest resolution and\nLundquist numbers, S= 500 , shown in dash-dotted green, and\nS= 1000 , shown in dotted black, very nearly collapse on top of\neach other. These results suggest that the reconnection timescale\ndictates the dynamical evolution of both the 2D and the 3D systems.\nA point to be noted is that when the time axes are normalized by\nthe resistive timescale instead, the curves do not collapse together.\nThis result of curves collapsing together on normalization of\ntime by\u001crecstrongly supports the possibility of magnetic recon-\nFigure 9. Magnetic and kinetic power spectra plotted at times, t= 2 and\n12from the 2D and 3D runs, A2D and A3D, in upper and lower panels\nrespectively.\nFigure 10. Time evolution of Brmsfrom 2D runs with varying values of S.\nIn the lower panel, the time axis has been normalized by the reconnection\ntimescale\u001crecpertaining to each value of S.\nMNRAS 000, 1–14 (2020)8\nFigure 11. Time evolution of Brmsfrom 3D runs with varying values of S.\nIn the lower panel, the time axis has been normalized by the reconnection\ntimescale\u001crec.\nnection being the key mechanism responsible for this 3D nonheli-\ncal inverse transfer.\n3.3 Energy transfer functions\nThe previous sections have provided both qualitative and quantita-\ntive information in support of the notion that magnetic reconnection\nis the physical mechanism underlying the inverse transfer that we\nobserve in both the 2D and 3D simulations. Additional arguments\nconsistent with this conclusion arise from the analysis of the energy\ntransfer functions, as we discuss in this section.\nWe calculate spectral transfer functions involving transfer be-\ntween different scales in the magnetic energy, given by Tbb, be-\ntween magnetic and kinetic energies, given by Tub, and between\ndifferent scales in the kinetic energy, given by Tuu. For the calcu-\nlation of the transfer functions, we follow the formalism discussed\nby Grete et al. (2017). The transfer function Txy(Q;K )denotes the\ntransfer of energy from shell Qto shellK, with the subscript refer-\ning to the energy reservoir, ufor kinetic energy and bfor magnetic\nenergy. In other words, Txy(Q;K )>0denotes a transfer from the\nreservoirxtoy, andTxy(Q;K )<0denotes transfer from ytox.\nThese functions are antisymmetric when x=y,i.e.,TuuandTbb.The transfer functions are given by\nTbb(Q;K ) =\u0000Z\nBK\u0001(u\u0001r)BQ\n+1\n2BK\u0001BQ(r\u0001u)dx; (12)\nTub(Q;K ) =Z\nBK\u0001r\u0001\u0012Bp\u001a\nwQ\u0013\n\u0000BK\u0001Br\u0001\u0012wQ\n2p\u001a\u0013\ndx; (13)\nTuu(Q;K ) =\u0000Z\nwK\u0001(u\u0001r)wQ\n+1\n2wK\u0001wQ(r\u0001u)dx; (14)\nwhere\ndenotes tensor product, w=p\u001au, and the shell-filtered\nquantities in real space are given by \u001eK(x) =R\nK^\u001e(k)eik\u0001xdk.\nWe intend to look for signatures of magnetic reconnection in\nthe transfer function plots calculated from our simulations. En-\nergetically, MHD reconnection involves energy transfer from the\nmagnetic to the velocity fields, manifested by the Alfv ´enic outflows\nalong the length of the current sheet that it generates. There is also,\nin addition, Ohmic dissipation in the current sheet.\nIn previous sections, we have mentioned that the merging of\nmagnetic islands facilitated by reconnection results in inverse trans-\nfer in a 2D system; these mergers take place in hierarchical fashion,\nwhere each generation of mergers produces islands of larger sizes,\nwhich then merge to produce larger islands, and so on (Zhou et al.\n2019). We conjecture that the 3D system evolves in a similar way,\nwith reconnection merging current filaments, and resulting in an in-\nverse cascade of magnetic energy. If this conjecture is true, then we\nexpect to observe, at any given point in time, significant transfer of\nmagnetic to kinetic energy at a scale corresponding to the dominant\nisland size at that time (the current sheet length scales as the size of\nthe islands).\nWe suppose that the 3D system evolves in a similar way, with\nreconnection merging current filaments, and resulting in an inverse\ncascade of magnetic energy. Given this theoretical understanding,\nwe have the following expectations for the transfer function plots:\n(i) In theTbbplot, the scales at which the merging of islands\n(or current filaments) predominantly takes place (corresponding to\nkcorr) should exhibit inverse transfer, while rest of the (smaller)\nscales should decay or direct transfer to further smaller scales.\n(ii) In theTubplot, the transfer from magnetic to kinetic energy\nshould stand out at scales comparable to those at which the inverse\ntransfer (i.e., reconnection) is dominant.\n(iii) In theTuuplot, there should be a similarity with the Tbb\nplot as the flows accompanying the fields will behave similarly.\nNote that the expectations for the behaviour of transfer func-\ntions for a system where magnetic reconnection drives the inverse\ntransfer are quite specific, as opposed to a case where a generic\nturbulence-related process drives the inverse transfer. For exam-\nple, in a generic turbulence-related process, we do not expect the\ntransfer from the magnetic to the velocity fields to be concentrated\naround certain scales, but to be spread out over a wide range of\nscales.\nIn the upper panel of Fig. 12, the Tbbplot from a 2D simu-\nlation shows both inverse and direct transfer of energy for certain\nranges of scales. Notice that the reflection of the patterns around the\ndiagonal is due to antisymmetry. Next, observe that on the lower\nside of the diagonal, there is a change in the dominant color of red\nMNRAS 000, 1–14 (2020)9\nFigure 12. Top, middle and bottom panels show the transfer functions Tbb,\nTubandTuu, respectively, from the 2D simulation A2D. At this point of\ntime,t= 10 in the simulation, kp\u00189.\nin lower wavenumbers to the dominant color of blue in the higher\nwavenumbers. This means that there is inverse transfer of energy\nfromQ= 10 toK= 6–9indicated by the red color, and for\nQ > 10forward transfer is dominant, as indicated by the blue\ncolor.\nIn the middle panel of Fig. 12, the Tubplot shows that en-\nergy transfer from the magnetic field to the velocity field is from\nK= 11\u000014toQ\u00186, as indicated by the blue patch. Since the\nblue color refers to negative values, it implies the direction of the\nFigure 13. Top, middle and bottom panels show the transfer functions Tbb,\nTubandTuu, respectively, from the 3D simulation C3D. At this point of\ntime,t= 20 in the simulation, kp\u00189.\ntransfer to be from KtoQand thus from the magnetic to the ki-\nnetic energy reservoirs. This confirms that the transfer is localized\nto a certain set of scales as expected for a phenomenon (reconnec-\ntion) dependent process as opposed to a generic turbulence driven\nprocess.\nThe bottom panel of Fig. 12 shows the Tuuplot. Below the\ndiagonal line, the darkest red spot at Q= 10 and the surrounding\nsmall red patch is indicative of minor inverse transfer of energy.\nThis is mainly due to the reasoning that as the magnetic structures\nMNRAS 000, 1–14 (2020)10\nFigure 14. Top panel: Evolution of magnetic energy (solid black) and ki-\nnetic energy (dashed blue) in a 2D simulation, F2D, with non-zero initial\nvelocity. Bottom panel: Magnetic and kinetic power spectra from the same\nsimulation, plotted at regular intervals of \u0001t= 5, with a thick final curve\natt= 45 .\nmerge, the underlying flow structures also acquire a larger size.\nIn that sense, the features in Tuuplot mimic the Tbbplot. Also,\ngiven that the flow is energetically subdominant to the field, the\nTuutransfers are expected to be small.\nSimilarly, we show transfer function plots for the 3D case in\nFig. 13. In the plot of Tbb, we find that the pattern changes trend\naroundQ\u001810. The scales larger than the wavenumber Q'10\nexhibit inverse transfer (these are the scales where reconnection\nwould be taking place), while Q>10show forward transfer, as\nexpected. In the plot for Tub, the transfer from magnetic to kinetic\nreservoirs is localized around Q\u001812andK\u001810, as expected\nfrom a reconnection-dependent process dominantly happening at\nthese scales. Again, as in 2D case, the Tuuplot here in 3D shows\nsimilarity to the Tbbplot, with a minor inverse transfer of energy\nfrom around Q= 7. Note that the energy transfers are mostly local\nand thus the patterns seen in all the plots are mostly concentrated\naround the diagonal in both 2D and 3D cases.\nThe 2D and the 3D transfer function plots tell a similar story\n— with greater clarity in the 3D case, we think, because turbu-\nlence in that limit is unconstrained. The behaviour of the transfer\nfunctions is what is expected for a magnetic-reconnection-driven\ninverse cascade.\n3.4 The case when the initial velocity field is non-zero\nIn all our simulations up until this point, the velocity field was ini-\ntialized to be zero. The flows that arose in these simulations were\ngenerated by the magnetic field, and were shown to be subdomi-\nnant to it. Magnetic reconnection is typically accompanied by the\nconversion of magnetic to kinetic energy. These generated flows,\nthus, are largely Alfv ´enic in nature. And such flows, where uand\nBare mostly parallel, lead to negligible induction.\nHowever if the velocity field is non-zero (and the system is not\nmagnetically dominated) to begin with, it can lead to a non-trivial\nFigure 15. Top panel: Evolution of magnetic energy (solid black) and ki-\nnetic energy (dashed blue) in a 3D simulation, F3D, with non-zero initial\nvelocity. Bottom panel: Magnetic and kinetic power spectra M(k;t)plot-\nted at regular intervals of \u0001t= 5, with a thick final curve at t= 50 .\nstretching term ( B\u0001ru), resulting in conversion of kinetic to mag-\nnetic energy. Then the simple arguments for showing \u0015A\u001c\u0015B\nwill not hold true anymore. This invites the question that if we con-\nsider a non-zero initial velocity field, will we observe energy decay\nof a different nature, one without an inverse transfer? To clarify this\nquestion, we have also performed simulations where the initial ve-\nlocity field is not only finite, but dominant, which we discuss in this\nsection.\nWe first examine the 2D case (run F2D). We initialize the flow\nfield in a manner similar to the magnetic field, as specified in sec-\ntion 2.2. While the slope of the magnetic power spectrum is set to\nk4, the kinetic spectrum is set to k2(chosen because this is the\nslope that develops in the runs when the initial velocity field is\nzero). Also, urmsis initialized to be larger than Brmsby a factor\nof 10. In Fig. 14, we show the evolution curves of the magnetic and\nkinetic energies, and also their spectra. It is seen that there is no\ninverse transfer in energy (there is a minor growth at the k= 1\nwhich we will address below), and also the temporal scaling of the\nmagnetic energy evolution curve is much steeper than the \u0018t\u00001\nevolution found in the case of zero initial velocity (Fig. 1).\nNext, we show in Fig. 15 the evolution curves of the mag-\nnetic and kinetic energies, and their spectra, for the 3D case (run\nF3D). Here, surprisingly, we do find an inverse transfer. However\nthe magnetic energy (and the kinetic energy) does not evolve as\n\u0018t\u00001but as\u0018t\u00001:4. This numerical scaling of \u0018t\u00001:4is close to\nthe decay law of\u0018t\u000010=7, as governed by the Loitsyanky invariant\n(Davidson 2000) (obtained in the case of hydrodynamic turbulence\nbut not unreasonable to consider here, given the dominance of the\nkinetic energy).\nIt is not at once obvious why there is a continued inverse trans-\nfer behaviour also when the initial kinetic energy is non-zero in the\n3D case. To understand this, we have to consider that there exists a\ncrucial difference between 2D and 3D cases with respect to dynamo\naction. It is well-known from anti-dynamo theorems (Moffatt 1978;\nZeldovich 1957) that there can be no sustained dynamo action in\nMNRAS 000, 1–14 (2020)11\nFigure 16. Evolution of the vector potential ( Az) from the 2D simulation, F2D, with non-zero initial velocity shown in contour plots at times t= 2,t= 10\nandt= 40 from left to right.\nFigure 17. Evolution of a component of the vector potential ( Az) in an arbitrary 2D slice (in x\u0000yplane) from the 3D domain of the 3D simulation, F3D,\nwith non-zero initial velocity shown in contour plots at times t= 5,t= 20 andt= 60 from left to right.\n2D. A random velocity field can give rise to anomalous diffusion.\nIn the absence of any sustained dynamo action, such an anomalous\ndiffusion can lead to rapid decay of the field in 2D. In Fig. 16, it\ncan be seen that the system indeed looks turbulent. The stretching\nof the fields by turbulence can grow the fields in a certain direc-\ntion while thinning them out in the perpendicular direction. Thus,\neven though the structures seem to grow in size over time, they are\nextremely thin and drawn out.\nIn 3D, besides an anomalous diffusion, these same underlying\nrandom motions can also lead to a dynamo, which can mitigate the\neffect of the anomalous diffusion. The presence of dynamo in our\n3D simulations with initial flow can be seen from the top panel of\nFig. 15, where the Brmsactually increases slightly before it decays.\nThe dynamo effect could explain the difference in the nature of de-\ncay of magnetic fields in 2D and 3D, when fields are subdominant\nto random flows.\nIn Fig. 17, we find that on a 2D plane from within the 3D\ndomain, the magnetic fields structures are not as drawn out as in\nthe 2D case. They, in fact, retain a more definitive form similar\nto the earlier cases, as in Figs. 2 and 5. It is not clear if magnetic\nreconnection has a role to play in the inverse transfer seen in the 3D\ncase. To investigate this further we now study the transfer function\nplots obtained for the 3D case.\nFig. 18 presents the energy transfer function plots for the run\nH3D. Even though the spectra in Fig. 15 show the signature of in-verse transfer, a corresponding distinctive signature in Tbbis lack-\ning. The red spots below the diagonal (or equivalently, the blue\nspots above the diagonal), which indicate inverse transfer, are very\nfew. Here, direct or forward transfer dominates the plot. Also the\nTubplot is dominated by red color, indicating that the transfers are\nfrom kinetic to magnetic energy, supporting a scenario of dynamo\naction. Similarly, the Tuuplot mostly shows forward transfers as\none would expect for a fairly turbulent flow. Thus, overall, the trans-\nfer function plots in this case of non-zero initial velocity, fail to\nuncover any signatures of reconnection-based inverse transfer.\nNonetheless, an interesting feature can be observed in the Tub\nplot. While most of the energy transfers are from low wavenum-\nbers in the kinetic energy reservoir to the high wavenumbers in the\nmagnetic energy reservoir, there is also energy transfer to smaller\nwavenumbers. For example, the wavenumber Q= 10 contributes\nsignificant energy to K= 7\u00009. This could be the tail of the small-\nscale dynamo at low wavenumbers (Haugen et al. 2004; Bhat &\nSubramanian 2013). Then the question which arises is why is there\nan inverse transfer in decaying turbulence with dynamo effects. In\nsuch a system, the eddies which are supercritical to carry out the\ndynamo action would pertain to the peak in the kinetic spectrum. It\ncan, then, be seen from the Fig. 15, that due to selective decay, this\npeak shifts to the lower wavenumbers. As the peak in the kinetic\nspectrum shifts, it could also shift the scales at which the magnetic\nenergy grows, thus leading to an effect that resembles the inverse\nMNRAS 000, 1–14 (2020)12\nFigure 18. Top, middle and bottom panels show the transfer functions Tbb,\nTubandTuu, respectively, from the 3D simulation F3D, with non-zero\ninitial velocity. At this point of time, t= 30 in the simulation, kp\u00189.\ntransfer. A similar effect of flow (which exhibits inverse transfer)\ndragging the field could be the reason for the growth of magnetic\nenergy atk= 1 as seen in Fig. 14 in the 2D simulation, F2D. A\nmore detailed investigation of the case of non-zero initial veloc-\nity field in decaying nonhelical MHD turbulence is left to a future\npaper.4 DISCUSSION AND CONCLUSIONS\nWe have investigated the inverse transfer of magnetic energy in the\ndecay of nonhelical MHD turbulence in 2D and 3D simulations.\nWe find that the scaling of magnetic energy with time ( \u0018t\u00001)\nand that of power spectrum with wavenumber ( \u0018k\u00002) is similar\nbetween both 2D and 3D cases (when the initial velocity field is\nzero). This is suggestive of similar mechanisms being responsible,\nin both cases, for the inverse transfer. In the 2D case, Zhou et al.\n(2019) have shown that island mergers via magnetic reconnection\nare key to understanding formation of larger and larger structures\nthat lead to inverse transfer. We find that our simulation results sup-\nport the idea that magnetic reconnection is responsible for the in-\nverse transfer in 3D nonhelical turbulent systems as well.\nOur investigations have yielded two main results via the study\nof conserved quantities, timescales, and length scales (via transfer\nfunction plots). In 2D MHD, the ideal invariants include energy and\nvector-potential squared. We have provided analytical arguments\nto show that in a turbulent system, for large Lundquist numbers,\nvector-potential squared Pis better conserved than magnetic en-\nergyEM(the dominant component of enegry in our system) and\nhow, for a decaying system, this can lead to inverse energy transfer.\nWe have calculated the rate of change of the two ideal invariants\nfrom the 2D simulation and shown that indeed Pis better con-\nserved thanEM. Further, we found that this was the case even in\nthe 3D simulations, indicating that the dynamics in 3D have 2D-\nlike tendencies. This is our first main result.\nOur second main result is that this inverse transfer, both in 2D\nand in 3D, is due to magnetic reconnection. Indeed, on normaliz-\ning the time axis by the magnetic reconnection timescale, we find\nthe evolution curves of the magnetic energry from runs with vary-\ning values of Lundquist numbers collapse on top of each other in\nboth 2D and 3D (the collapse being better with larger values of\nS). Additionally, the transfer function plots show clear signatures\nof magnetic reconnection driving the inverse transfer. We find from\ntheTbbplots that only those scales either at or above the peak corre-\nlation scale, at any given time, exhibit inverse transfer as expected\nfrom a physical picture of island (or filament) mergers being domi-\nnant at a certain scale. The more clinching evidence arises from the\nTubplots, where it is seen that a set of scales compatible with our\nunderstanding of the reconnection process in this system stand out\nin the transfer of magnetic to kinetic energy.\nFrom these results, an emergent characteristic of the magnet-\nically dominated 3D system is its tendency to align with the be-\nhaviour observed in 2D. The overarching question is then what el-\nement in 3D renders it with 2D-like behaviour? We think the an-\nswer lies in the fact that the system is magnetically dominated. The\nfield can provide anisotropy at small-scales i.e. the current sheets\ncan have local guide fields. Magnetic reconnection in 3D, when\npresided by guide field, leads to familiar 2D results (Onofri et al.\n2004). Indeed, in another recent study of inverse energy transfer us-\ning the reduced-MHD model (which assumes a strong background\nmagnetic field), Zhou et al. (2020) find mergers between magnetic\nflux tubes driving inverse transfer.\nReturning to the result of k\u00002slope in the magnetic power\nspectrum, it has been pointed out that this corresponds to the theo-\nretical expectation for weak turbulence (Brandenburg et al. 2015).\nHowever Zhou et al. (2019) find in their 2D simulations that it cor-\nresponds to the presence of thin current sheets. In accordance with\nour findings of 2D-like behaviour in 3D, this explanation of thin\ncurrent sheets for k\u00002slope may carry over to 3D as well. Zhou\net al. (2020) report a k\u00001:5slope in their reduced-MHD simula-\nMNRAS 000, 1–14 (2020)13\ntions but unlike the case in the simulations here, they also find ki-\nnetic energy is not subdominant to the magnetic energy.\nTo ascertain whether by making magnetic field subdominant,\nthe inverse transfer in energy ceases to appear, we performed simu-\nlations where the initial velocity was set to a large non-zero value.\nIn the 2D simulation, the system becomes turbulent leading to\nmuch faster decay of energy, likely due to anomalous diffusion\nand there is no significant inverse transfer. In contrast, in the 3D\ncase, the energy decay follows a t\u00001:4scaling, and we do observe a\ndefinitive signature of inverse transfer in the evolution of the mag-\nnetic spectrum. Furthermore, the evolution of magnetic energy re-\nveals a dynamo effect which possibly counters the anomalous dif-\nfusion, leading to a decay rate that is slower than the one seen in the\n2D case. On studying the transfer function plots for the 3D simu-\nlation, we find that the signature for inverse transfer is surprisingly\nabsent inTbbplot. However, the Tubreveals that there is transfer\nof energy from the kinetic energy reservoir to magnetic energy to\nboth small and large scales, where the larger portion goes to the\nsmall scales. This kinetic energy transfer to larger magnetic scales\nis a possible signature of the tail of small-scale dynamo action at\nsmall wavenumbers. This tail can possibly shift further to lower\nwavenumbers as the peak in kinetic spectrum shifts due to selec-\ntive decay, leading to an inverse transfer type effect (as seen in the\nevolving magnetic spectra).\nWe have mentioned several astrophysical and cosmological\napplications to which our results might be relevant in the introduc-\ntion section. In all of the applications mentioned, the astrophysi-\ncal systems under consideration consist of highly conducting, large\nLundquist number (or magnetic Reynolds number) plasmas. The\nrange of Lundquist numbers explored in this paper is limited by the\nresolution and thus our simulations are in a regime where Sweet-\nParker model for magnetic reconnection is valid. However at higher\nvalues ofS, the nature of reconnection changes with the onset of\nthe plasmoid instability (Loureiro et al. 2007). Recent research has\nrevealed that the plasmoid instability renders the magnetic recon-\nnection rate independent of SforS&104, with a reconnection\nrate of\u00180.01VA(Bhattacharjee et al. 2009; Uzdensky et al. 2010;\nLoureiro & Uzdensky 2016). This would be the timescale to be\nconsidered in the astrophysical systems which can be described\nwith the MHD framework. If, instead, the environment under con-\nsideration is weakly collisional, the adequate reconnection rate to\nconsider would be faster, on the order of 0:1VA(e.g., Cassak et al.\n2017)\nA previous study of this problem had shown that the inverse\ntransfer is weak or altogether absent upon increasing the magnetic\nPrandtl number Pr M(Reppin & Banerjee 2017). This is consistent\nwith the understanding that magnetic reconnection at higher Pr M\nbecomes increasingly inefficient (Park et al. 1984). However, it is\nnot clear if at both higher Sand Pr Mthis trend will continue, as the\nensuing plasmoid instability could potentially change it (Loureiro\net al. 2013).\nIn conclusion, we provide a physical understanding to the puz-\nzling and unexpected 3D nonhelical inverse transfer via analysis\nof direct numerical simulations of magnetically dominated, decay-\ning MHD turbulence. We argue that magnetic reconnection is the\nphysical mechanism responsible for the emergence of progressively\nlarger structures. Further, we show that the behavior in the 3D sys-\ntem is intriguingly similar to that in 2D, possibly because of local\nanisotropy in this system. These results could have important con-\nsequences for a wider range of astrophysical applications.ACKNOWLEDGMENTS\nWe thank K. Subramanian for useful feedback on the paper. PB and\nNFL acknowledge support from the NSF-DOE Partnership in Basic\nPlasma Science and Engineering Award No. DE-SC0016215. MZ\nand NFL acknowledge support from the NSF CAREER Award No.\n1654168. This project was completed using funding from the Eu-\nropean Research Council (ERC) under the European Unions Hori-\nzon 2020 research and innovation programme (grant agreement no.\nD5S-DLV-786780). 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Conventional magnets \nare generally composed of rigid materials with shapes and structures unchangeable which may face \nchallenges sometimes to answer the ab ove questions. Here, from an alternative other than rigid \nmagnet, we disclosed an unconventional way to generate endogenous magnetism and then construct \nmagnetic monopole through tuning liquid metal machine. Through theoretical interpretation and \nconceptua l experiments, we illustrated that when gallium base liquid metal in solution rotates under  \nactuation of an external electric field, it forms an endogenous magnetic field inside which well \nexplains the phenomenon that two such discrete metal droplets could  easily fuse together, indicating  \ntheir reciprocal attraction via N and S poles. Further, we conceived that the self -driving liquid me tal \nmotor was also an endogenous magnet owning the electromagnetic homology. When liquid metal \nin solution swallowed alumi num inside, it formed a spin motor and dynamically variable charge \ndistribution which produced an endogenous magnetic field. This finding explains the phenomena \nthat there often happened reflection collision and attraction fusion between running liquid met al \nmotors which were just caused by the dynamic adjustment of their N and S polarities. Finally, we \nconceived that such endogenous magnet could lead to magnetic monopole and four technical routes \nwere suggested as: 1. Matching the interior flow field of li quid metal machines; 2. Superposition \nbetween external electric effect and magnetic field; 3. Composite construction between magnetic \nparticles and liquid metal motor; 4. Chemical ways such as via galvanic cell reaction. Overall, the \nfluidic endogenous mag net and the promising magnetic monopole it enabled may lead to \nunconventional magnetoelectric devices and applications in the near future.  \n \nKeywords: Fluidic Magnet; Magneti c Monopole; Liquid Metal; Endogenous Magnetism; Droplet \nMachine; Self -fueled Motor  \n  \n \n 2 \n 1 Introduction  \nAlthough the magnetic field is invisible and intangible, it is a special substance that exists \nobjectively and can be found from cells to the periphery of s tars. So far, researchers have conducted \nin-depth studies on various existing known magnetic fields[1, 2], such as coils, earth, galaxies, etc., \nas shown in Figure 1a -c. The magnets exert forces and moments on each other through the \nrespective magnetic fie lds. With the aid of the magnetic field as a medium, the object can transmit \nthe magnetic force without contact. The study of magnetic fields is of great significance to scientific \nand technological progress, such as electronic cooling[3], microfluidic chi p[4], graphene orientation \ncontrolling[5], medical imaging and tumor treatment[6-8], visual probe of amino acid[9], life science \nmagnetic protein[10] and magnetotactic bacteria[11, 12] and so on.  \n \n \nFigure 1.  Several typical types of magnetic fields: (a) Inside the atom; (b) The energized \ncoil; (c) The earth.  \n \nAccording to modern physics, the magnetic field produced by moving electric charges ( Figure \n1a) is fundamentally determined by moving electrons or movin g protons. Atom is the smallest basic \nunit that makes up a substance, and the magnetism of a substance is the collection of the magnetism \nof all the atoms it contains. Inside the atom, the nucleus has spin motion, and the electrons outsid e \nthe nucleus are rotating while moving around the nucleus. The above motion will generate tiny \ncircular currents. According to Maxwell ’s electromagnetic theory, electricity and magnetism are \ninduced by each other. These tiny circular currents will induce a corresponding ma gnetic field. \nTherefore, the magnetic field generated by current or point charge is the macroscopic manifestation \nof the magnetic field generated by a large number of moving electrons or protons.  \nTraditionally, from microscopic magnetic nanoparticles to ma croscopic natural magnets, as well \nas artificially manufactured excitation coils ( Figure 1b ), permanent magnets, etc., are composed of \n3 \n rigid materials, and their N and S poles are located at the fixed ends of the magnet. Generally, their \nshape and structur e cannot be changed on the eigen -scale, which makes the applications somewhat \nlimited. In order to improve the adaptability of the magnet to different situations, people have \ninvented the magnetic fluid functional material, which is a colloidal solution fo rmed by surface \nactive agent -encapsulated nanomagnetic particles dispersed in the base fluid[13], and can exhibit \nspecial magnetic and optical properties. Although the fluidity has been enhanced, the part that \nexhibits magnetism has not deviated from the e ssence of the rigid magnet. Recently, as a base fluid \nmaterial for loading magnetic nanoparticles, liquid metal has been gradually developed as multi -\nfunctional materials due to its outstanding electrical and thermal conductivity and ductility. Its \nexcelle nt deformability allows droplets to adapt to paths of different sizes. In particular, the chemical \nproperties of room temperature liquid eutectic alloys such as GaIn and GaInSn are relatively stable \nunder normal conditions. Their low toxicity and good oper ability allow researchers to adjust the \nalloy ratio to achieve various melting points and characteristics, so as to be competent for tasks that \nrigid materials cannot do . \nPrevious studies on the magnetic properties of liquid metal have mostly involved addi ng iron \nparticles[14], or chemically coating a layer of nickel on the surface of the droplet[15], and then usin g \nmagnets to manipulate the composite material. The orientation of the magnet will cause a change in \nthe motion state of the droplet. Even when t he position of the droplet is unknown, the magnet can \nattract the liquid metal to achieve rapid control. However, the story will not just end there. Here,  we \nconceived that on a microscopic level, the spherical liquid metal droplet in fact formed a weak \nendogenous magnetic field when excited by its own circular current or an external current during \nthe rotation process. This rapidly  rotating liquid metal droplet not only owns the magnetic properties  \nof rigid materials,  but also flows like water. It belong s to a new kind of magnetic matter, or termed \nas fluidic magnet.  \nFurther, we conceived that the discloser of the fluidic magnet still offers another important clue  \nto answer the classical issue of magnetic monopole, one of the most intriguing scientific myst eries \nin nature. When people try to tackle the fundamental problem of whether the magnetic monopole \nexists or not, they tend to explore  clues in the rigid materials or magnets which however may not \nalways be rational in reality. The magnetic poles of such type of material are usually in a fixed \nposition. The magnetic field lines are emitted from the N pole outside the magnet and then return to  \nthe S pole, and inside the magnet, the S pole points to the N pole, forming countless closed circuit s. \nThis fixed p attern of magnetic field distribution limits the routes to verify the actual existence o f \nmagnetic monopoles. So far, the magnetic monopoles only appear as quasi -particles in condensed \nmatter, such as the flipping excitation of spin ice[16-18], and the sim ilar structure produced by the \nvortex of super -cold rubidium atoms in Bose -Einstein condensate (BEC)[19]. However, we realized \nthat as a variable fluidic conductor, the liquid metal machine generates a constantly changing \nendogenous magnetic field in the r andom spin motion. When an external electric or magnetic field \nis superimposed, or compounding with magnetic particles, or just artificially modifying the \nmagnetic field through internal chemical reactions, even much diverse magnetic field configurations \nor behaviors will be generated. This is fundamentally different from the conventional rigid magnetic  \nmaterial with fixed -poles. This article  is dedicated to present a new conceptual fluidic magnet and \nprovide s everal  possible technical route s towards finding  the magnetic monopoles.  \n \n 4 \n 2 Experimental Evidences of Fluidic Endogenous Magnetism from Liquid Metal Machine  \n2.1 Basic Properties of Conductive Fluidic M etal \n  At the macro level, the flexibility or rigidity of material has an important impact on i ts application. \nIn recent years, with the development of material science, various unique effects of soft matter have \nbeen gradually discovered and utilized. Particularly, room temperature liquid metal owns many \nfavorable properties, including large surfac e tension, ideal flexibility, high conductivity, low toxicity, \netc. From Table 1[20], it can be seen that the typical liquid metals generally own very similar fluidity \nwith water while their thermal conductivities are dozens of times higher than the later.  Especially, \nthe excellent electric conductivity of such matters molds them into intrinsically conductive fluid. \nBasically, electrically conductive solution such as aqueous NaOH, NaCl, etc. may also display \nsimilar behavior with liquid metal. But consideri ng the huge conductivity of liquid metal, such fluid \nwill particularly play dominate role in developing liquid magnet  and is therefore the core of current \nanalysis .  \n \nTable  1 Typical physical properties of Ga -based  liquid  metal  with those of other liquids[20] \nComposition  Ga Ga75.5In24.5 Ga67In20.5Sn12.5 Ga61In25Sn13Zn1 Hg Water \nMelting Point (oC) 29.8 15.5 10.5 7.6 −38.8 0 \nBoiling Point (oC) 2204  2000  >1300  >900  883 100 \nDensity (kg/m3) 6080  6280  6360  6500  13530  1000 \nElectrical  conductivity (/Ω/m)  3.7× 106 3.4× 106 3.1× 106 2.8× 106 1.0× 106 - \nThermal  conductivity  (W/m/K)  29.4 26 16.5 - 8.34 0.6 \nViscosity ( m2 s) 3.24× 10-7 2.7× 10-7 2.98× 10-7 7.11× 10-8 13.5× 10-7 1.0× 10-6 \nSurface tension (N/m)  0.7 0.624  0.533  0.5 0.5 0.072 \nSound speed (m/s)  2860  2740  2730  2700  1450  1497 \nWater compatibility  Insoluble  Insoluble  Insoluble  Insoluble  Insoluble  - \n \nAs gradually realized by recent studies, the excellent characteristics of liquid metal make it \navailable in the fields of fluidics ( Figure 2a ), printed circuits (Figure 2b ), flexible sensors, \ntransformable machine ( Figure 2c ), self -driving motors ( Figure 2d ), and the base carrier fluid for \nmagnetic particles ( Figure 2e ), etc. The present research focuses on the electromagnetic effects of \nsuch fluidic conductive ma terial. Particularly, the shape and motion of liquid metal have great \nvariability and controllability, the magnetic field generated by it will have rather rich possibilit ies, \nwhich appears more competent in some occasions. In this sense, we can come up to a generalized \nmatter state, which can be termed as electromagnefluid, indicat ing the material that simultaneously \nconsisted of electronics, magnet and fluid inside together.  \n 5 \n  \nFigure 2.  Typical material features of liquid metal which may serve to make end ogenous magnetic \nfields: (a) Fluidic behavior; (b) Conductive electronic ink[21]; (c) Morphologically transformable \nmachine[22]; (d) Self -fueled motor[23]; (e) Loadable by magnetic particles[24]. \n \n2.2 Magnet Induced from Liquid Metal Energized Coil  \n  Liquid metal has good flexibility and conductivity, and can replace traditional rigid coils, making  \nthe moving parts of the electromagnetic actuator more flexible, thereby improving the ability to dea l \nwith complex situations. At present, there are two ma in methods to make liquid metal \nelectromagnetic coils. One is to use optical mask technology to lithography microchannels on the \nPDMS substrate, and then inject liquid metal into it[25]. In this way, the liquid metal is still in fl uidic \nform ( Figure 3a ). \nAnother method is to use a mask with a specific shape to cover the PDMS substrate, and \nuniformly print the liquid metal on the PDMS film through a liquid metal spray gun ( Figure 3b )[26]. \nCompared to the first method, this direct printing method of liquid me tal has a shorter production \ncycle and simple r operation. In this case, the liquid metal exists on the PDMS in a semi -solid form, \nthat is, the part in contact with air is oxidized to solid, but the liquid metal inside still has th e \npossibility of flowing b ehaviors.  \nIn the magnetic driving device, the magnet was placed above or below the liquid metal \nelectromagnetic coil ( Figure 3c ), the magnetic field lines passed through the electromagnetic coil. \nWhen an alternating current was applied, the Lorentz force F was generated in the coil:  \n𝐅=∫𝐼𝑑𝑙×𝑩                                 (1) \nwhere, dU as indicated in Figure 3d  is the potential difference between the two ends of the micro \nsegment dl. Macroscopically, the Lorentz force generated by the magnetic field component parallel \nto the coil was embodied as driving the coil to move closer or away from the magnet.  \nFor the first injection method, since the flow channel is in a sealed state and the operating \ntemperature is higher than the freezing point of the liqui d metal, the alloy in the flow channel is \nalways liquid. Under the action of an external electric field, in addition to inducing a magnetic field, \nthe micro segment would also generate an endogenous magnetic field under an alternating current. \nThis segment  had weak attractive or repulsive interaction with the placed magnet and other micro \n6 \n segments. However, this force was too weak, only the device action dominated by the Lorentz force \nof the coil could be observed.  \n \n \nFigure 3.  Schematic diagrams of liquid metal coil: (a) Method of injection; (b) Method of printing; \n(c) Magnetic field around the coil; (d) Partial enlarged view of the charge movement and rotation \nof the liquid metal in the coil.  \n \n2.3 Endogenous Magnet Generated b y Electrically Controlled Liquid Metal Machine  \n  Applying an electric field to the liquid metal can induce its trans formation, movement or \nrotation, and its motion state depends on the direction and strength of the applied electric field, the \ncontact with the electrode, and the surrounding solution environment, respectively[22]. To illustrate \nthe basic principle to generate e ndogenous magnet by electrically controlled liquid metal machine , \nwe designed a cylindrical channel with a smooth surface, and put a spherical liquid metal droplet in  \nthe electrolyte -filled solution  (Figure 4 ). After applying an external field, the liquid metal \nimmediately responded, rotating and moving towards the positiv e electrode. The original spherical \ndroplet was stretched, taking the advancing direction of the droplet as the head, it was observed th at \nthe tail had a slight deformation. When the applied electric field was large enough, the droplets we re \ndragged instan taneously ( Figure 4a -d), and even separated into two small droplets, as shown in \nFigure 4d .   \nDue to the difference between the physical properties of the liquid metal and the electrolyte \nsolution, the electric field would be stepped at the two -phase conta ct interface, thereby generating  \nelectrical stress. At the same time, GaIn liquid metal could react with NaOH solution to generate  \n[Ga(OH) 4]-, [Ga(OH) 4]- would be specifically adsorbed to the Ga surface, that is, in the Helmholtz \nlayer, and the surface of Ga was negatively charged due to the abundance of reactive electrons, and \nthe diffusion layer was positively charged, forming an electric double layer (EDL). According to \nLippmann ’s equation, there was a connection between surface tension and potential dif ference[23]: \nγ=𝛾0−1\n2𝑐𝑉2                                (2) \nwhere, γ is the surface tension, c is the capacitance per unit area of EDL, V is the potential difference \n7 \n of EDL, and γ0 is the maximum surface tension when V=0. \n \n \nFigure 4.  The liquid metal droplet continues to move to the electrode under the electric field: (a) \nThe droplet was a regular spherical shape; (b) The droplet moved to the positive electrode with \nslight deformation; (c) The droplet had obvious head and tail; (d) Dr oplet dispersed.  \n \n  EDL makes the charge distribution of two droplets similar. Using an electrode to drag a droplet \nand approach another in the solution, when the distance between the two was small enough, they \nwould quickly merge. This internal rotation c aused by an external electric field was similar to the \nspontaneous rotation of the droplet after swallowing aluminum, and we will explain this in detail in  \nsection 2.4.  \nIn the system composed of liquid metal and water, it could be found that the influence of liquid \nmetal on the surrounding water is rather evident under the action of an external electric field. The  \nsystem layout is shown in Figure 5a , placing a spherical liquid metal droplet in water and extending \na pair of electrodes into the surrounding so lution. When the current was applied, the liquid metal \nsphere started to rotate, and two eddies appeared in the surrounding solution, and they kept spinnin g \nwith the rotation of the liquid metal sphere. Therefore, the external electric field could induce t he \nliquid metal to rotate, causing the real -time change of the electronic charge distribution on the \nsurface of the droplet. This moving charge would cause the droplet to generate an endogenous \nmagnetic field , as shown in Figure 5b .  \nBecause the flow direction of the droplet  surface was along the electric field gradient, when \napplying an alternating current, the direction of the electric field changed periodically, and the f low \nstate of the liquid metal would also change accordingly, a s shown in  Figure 5c . Before and after the  \nexternal electric field was applied, the electric field on the liquid metal -electrolyte interface wo uld \nchange ( Figure 5d ). As the alternating frequency increased, the vortex current formed inside could \nbe stronge r, and the magnetic field generated inside will be strengthened at this time. It should be  \nnoted that such endogenous magnetic field was a manifestation of electricity and magnetism on a \nmicroscopic level, and its intensity was much smaller than the magnet ic field induced by the change \nof the external electric field.  \n \n8 \n  \nFigure 5. The rotation and transformation of liquid metal in the solution under an external electric \nfield: (a) Experimental case of rotating liquid metal droplet under direct current[22]; (b) Schematic \nfor rotating liquid metal droplet under direct current; (c) Resonant oscillation phenomenon of liqui d \nmetal droplet under matching alternating field; (d) Electric field on the liquid metal -electrolyte \ninterface before and after applying extern al electric field[27]. \n \n2.4 Endogenous Magnetic Field Generated from Self -Driving Liquid Metal Motor \nAccording to our former discovery, small liquid metal droplets owns intriguing self -driving \ncapability after fueled with aluminum[28], and can automaticall y converge or diverge[29], which is a \nvery unconventional feature that traditional rigid machines do not own. We placed a spherical liquid  \nmetal droplet in the NaOH solution and added some aluminum foil, as shown in Figure 6a . \nAccording to Rebinder's effec t, after a period of time, the aluminum foil was completely infiltrated  \nand corroded by the liquid metal, the oxide layer on the aluminum surface was destroyed, causing \nAl to activate, triggering a redox reaction, and the electrons from the aluminum interi or \npreferentially deoxidated the oxidized gallium near the Al. The charge distribution of the EDL had \nchanged, resulting in a potential gradient and asymmetric surface tension on the surface of the \ndroplet[30]. According to the Young -Laplace equation, the pressure difference p between the solution \nand the liquid metal droplet can be expressed as:  \np=γ∙2\n𝑟                                     (3) \nwhere, 1/r is the curvature of the droplet surface[31]. \nThe liquid metal sphere moved randomly in the solution and was accompanied by its own rapid \nrotation, as shown in Figure 6b . And their lifetime lasted for more than 1 hour without any other \nexternal energy[23]. According to our former measurement, this sel f-fueled liquid metal motor \n9 \n generated electrical voltage and current between its surface and the surrounding electrolyte ( Figure \n6c). It is this dynamically variable electrical field leads to the generation of an endogenous \nmagnetism inside the liquid meta l motor.  \n \n \nFigure 6. The rotation and transformation of liquid metal in the solution after swallowing aluminum: \n(a) Experimental case of rotating liquid metal droplet after swallowing aluminum[32]; (b) Schematic \nfor rotating liquid metal droplet after swallowing aluminum and generated magnetic field; (c) \nMeasured average voltage and electrical current on self -fueled liquid metal and electrolyte[23]. \n \nIn addition, there was a galvanic reaction composed of liquid metal, aluminum foil and electrolyte  \nsolut ion, which accelerated the speed of electrochemical reaction. After being corroded by GaIn \nalloy, aluminum aggregated or dispersed on the surface, then it was uniformly distributed in the \nliquid metal in the form of small particles, and hydrogen was genera ted by the electrochemical \nreaction (Equation 4). The average voltage and current between the liquid metal and the electrolyte \ncould be measured by A vometer[23], as shown in Figure 6c . The gas released from the interface also \npushed the liquid metal forwar d, forming a self -driving motor. When the size of the droplet was \nlarger, the specific surface area would reduce and the electrochemical reaction sites were limited, \nresulting in less gas generated, and the driving force of the gas does not have an obvious  effect on \nthe large droplet. Therefore, the self -driving of liquid metal in a large volume mainly depended on \nthe tension gradient, and the effect of the latter was negligible.  \n2Al +2NaOH +2𝐻2O=2NaAl 𝑂2+3𝐻2↑                     (4) \nNext, we explore the mergence of rotating droplets ( Figure 7 ). In the solution, two liquid metal \ndroplets rotated inside after swallowing aluminum, changing the flow line of the surrounding \nfluid[32], and when the distance between the two is small enough, they will quickly mer ge into one. \nIt could be seen from section 2.3 that the surface charge distribution of the droplets changed in the \nelectrolyte and formed an electric double layer. This structural similarity promoted the mergence of \nthe droplets. Here, we proposed that the  rotation inside the droplet not only enabled itself and the \nsurrounding flow field to maintain a state of motion, but also had the ability to excite an endogenous \nmagnetic field. During the rotation process, through adaptive adjustment, it induced a magne tic field \nthat attracted each other. The two droplets were brought closer and merged into one, as shown in \n10 \n Figure 7a -d. Over the process, various endogenous magnetic fields were induced inside the liquid \nmetal ( Figure 7e -h). This mechanism allows discrete droplet machines to be quickly assembled, \nwhich has profound significance for s elf-assembly machine with internal trigger motion . \n \n \nFigure 7.  Droplets mergence  and induced flow and magnetic field: (a -d) Experimental picture of \nthe mergence process of two droplets; (e -h) Spin trajectory of the liquid metal droplets and the \nsurrounding flow lines.  \n \n2.5 Liquid Metal Spin Superimposed on an External Magnetic Field  \nAdding  aluminum in the liquid metal can form tiny motors in the NaOH solution, but the \nmovement i s random and lacks of a specific direction and speed[32]. If this kind of random movement \ncan be controlled, the liquid metal self -driving motor can be used in more occasions, such as \ntrans formable intelligent robots, precise drug delivery, detectors/sensors, etc. We placed a \npermanent magnet under the petri dish to introduce a magnetic field to adjust the random movement \nof the motor. The GaIn 10 liquid metal motor had a significant group effect at the boundary of the \npermanent magnet. Over  the time, the motor group  gathered near the boundary of the magnet, and \nthen rebounded after a short stay. Tan et al.[33] suggested  that this peculiar phenomenon was related \nto the magnetic flux density of the bottom magnet, and the magnetic intensity was z ero at the \nboundary of the magnet. At first, these droplet motors were attracted by the magnet, which leaded \nto aggregation.  The higher magnetic intensity on the side of the magnet prevented the motor from \npassing, while the peak of the magnetic field on t he side away from the magnet was lower. The \ndroplet motor selectively tended to the low peak position, thus limiting the motor's range of motion . \nBased on our former experimental video [33], we tracked the motion path of a single droplet motor, \nand carried  out a more in -depth interpretation on the interaction mechanism between the liquid metal  \nand the permanent magnet. Here, we proposed a conjecture that the spin of liquid metal generated \na weak magnetic field on a microscopic level. Since the motor was sma ll enough and in a solution \nenvironment, the spin drive of the motor could more clearly show the characteristics related to the \nmagnet. The magnetic field generated by the spin is non -directional, and the droplet exhibited the \nsame or opposite magnetic pol es as the magnet at the boundary of the magnetic field, resulting in \nthe effect of being repelled or attracted.  \nWe have studied the effect of the external magnetic field on the moving liquid metal droplet in \nthe above phenomenon. From the present analysis,  it is known that the rotating liquid metal droplet \n11 \n will generate a ma gnetic field, and there will be negatively charged free electrons flowing through \nit. The magnetic field intensity of a permanent magnet around it is H and the magnetic induction \nintensi ty is B. Then the liquid metal sphere will be subjected to the Lorentz force of permanent \nmagnet induction intensity on the charged sphere, and the force of magnetic field intensity on the \nsphere's own magnetic field at the same time. Firstly, the Lorentz force on the sphere is analyzed. \nAssuming that the moving speed of the liquid metal sphere is v and the overall charge is Q at a \ncertain time, the liquid metal sphere can be regarded as a large charged particle. Then the force F1 \nat this time can be obtained from the Lorentz force formula in classical electromagnetics:  \n𝑭𝟏=𝑄𝒗×𝑩                                   (5) \nThe stress of the liquid metal sphere in the magnetic field intensity at the same time is analyzed  \nby using the e quivalent magnetic charge theory  which can be available in classical text books . \nAssuming that the magnetization of the liquid metal sphere is Mp and the magnetization in the \nsolution is Ms, then the equivalent surface magnetic charge density on the surfac e of the liquid metal  \nsphere κ is: \n𝜅=𝜇0(𝑴𝒑−𝑴𝒔)⋅𝒏                             (6) \nwhere, μ0 is the vacuum permeability and n is the normal vector of the surface of the liquid metal \nsphere.  \nBy integrating the surface S of the sphere, it can be obtained that the magnet product force of the \nsphere under the magnetic field intensity is:  \n𝑭𝟐=∯𝜅𝑯𝑑𝑠𝑆=𝜇0∯(𝑴𝒑−𝑴𝑺)⋅𝒏𝑆𝑯𝑑𝑠                    (7) \nTherefore, the total force of the applied magnetic field on the liquid metal sphere in this system \nis: \n𝐹𝑚=𝐹1+𝐹2                                  (8) \nFor a small -volume liquid metal motor, as shown in Figure 8(a -d), the magnetic field strength \nwas the smallest near the magnetic field boundary, and the magnetic force on the droplet motor was \nweak. Therefore, the motor oscillated in a small range nearby. The force was converted into the \nmomentum of the drop, making th e momentum continuously accumulated. When the motor moved \naway from the magnetic field, the magnetic pole facing the magnet was the same as the magnet, \nwhich produced a repulsive force. When it reached a certain value, it could break away from the \nrestrain t of the magnetic field boundary, and it appeared to be bounced off by the magnet on the \nmacroscopic view. Due to the increase of the magnetic field along the diameter of the magnet, the \ndroplet far away from the magnet would be in a larger magnetic flux d ensity space, the magnetic \nforce would become more  obvious, and the momentum would further increase. Although the \nmagnetic poles generated by the endogenous  magnetic field m ight be attracted by the permanent \nmagnet at the next moment, the attractive force could not pull back the large momentum droplet \naway from the magnet at this time, so  the droplet eventually moved away from the magnet.  \nHowever, large -volume motors were not as sensitive to weak magnetic fields at the boundary of \nmagnets as small -volume mo tors, as shown in  Figure 8(e -h). When the large -volume motor \nhappened to approach the permanent magnet with the same pole, it would be hindered by the \nexternal magnetic field. In the process of approaching the center of the magnetic field, the motor ’s \npotential energy continued to accumulate and the momentum gradually decreased. Under the \ncontinuous accumulation of reverse acceleration, the liquid metal motor began to move away from \nthe magnet  (Figure 8i ), and the accumulated potential energy turn back to i ts own momentum. When 12 \n the speed reached a certain level , it could break away from the restraint of the magnetic field \nboundary.  \nBased on the above discussion, we suggested that when the motor had the same or opposite \nmagnetism as the bottom permanent magne t, it was bounced or attracted ( Figure 8j ), resulting in \nthe macroscopic effect of the motor group rotating, oscillating, gathering and bouncing off the \nboundary of the magnet, a nd this effect was affected by the volume of the motor.  \n \n \nFigure 8.  The motions of the liquid metal motor under the action of magnetism: (a -d) The position \nof the small liquid metal motor; (e -h) The position of the large liquid metal motor;  (Data from[33]) \n(i) Magnetic flux density along the diameter of the magnet; (j) Schematic diagram of magnetic \ninteraction between liquid m etal motor and magnet.  \n \n2.6 Magnetic Field Generated from Liquid Metal under Chemical Environment  \nLiquid metal reacts with other metals to form a galvanic cell, which is also a form of causing sel f-\nrotation. Since liquid metal is easily oxidized in the air and forms an oxide layer on the surface, \nwhich is not conducive to the observation and analysis of the surface and internal movement. \nUsually, we put the liquid metal in an acid or alkaline soluti on for operation to remove the surface  \noxide film, and on this basis, build a galvanic battery with other metals.  \nPrevious studies have revealed the Marangoni effect caused by gallium -copper galvanic corrosion \ncouple[34], that is, the phenomenon in which t he tension gradient between the liquid interface moves \nthe mass. Here, liquid metal was used as the anode, Ga was oxidized to Ga3+, and copper was \nadopted as the anode. The cathode had a corrosion potential between (Ga/Ga3+) and (H/H+), so H+ \nwas reduced t o H 2 on the cathode (Equation 9, 10). Therefore, the essence of the liquid metal \nMarangoni phenomenon was the galvanic corrosion.  \n13 \n 𝐺𝑎→𝐺𝑎3++3𝑒−                              (9) \n𝐻++2𝑒−→𝐻2                              (10) \nGa reacted with HCl solution to produce gas at the liquid interface. By tracking the movement \ntrajectory of the bubble, the flow state of the liquid metal could be obtained, as shown in Figure 9a , \nthe liquid metal on the left was in contact with the copper  sheet, and the flow lines of liquid gall ium \nwas marked in yellow. Here, the liquid metal participated in the reaction as a part of the galvanic \nbattery, and the internal rotation could make the charge continue to flow through it ( Figure 9b, c ), \nwhich also  had the conditions to excite the endogenous magnetic field.  \n \n \nFigure 9.  Gallium -copper galvanic corrosion: (a) Superficial solution streamline [34]; (b) The x -z \nplan view of galvanic corrosion system in the channel; (c) The y -z plan view of the liquid ga llium \nand HCl solution in the channel . \n \nIn addition to the above -mentioned characteristics, liquid metal also has unique reversibility, th at \nis, via synthetically chemical -electrical mechanism (SCHEME) to control its structure[35]. As shown \nin Figure 10a , two platinum electrodes were inserted into the liquid metal and electrolyte. \nAccording to our former research, when direct current was applied, an oxide layer was formed on \nthe gallium surface at the anode, and the surface tension decreased. The originally  spherical drople t \nappeared to be spread out ( Figure 10b ), and the surface area increased, even reaching five times the  \noriginal size. When the applied electric field was removed, the gallium oxide on the surface \nchemically reacted with NaOH solution, and the surface tension of the droplet increased and \nreturned to the spherical state, as shown in Figure 10c, d . This reversible SCHEME  was affected \nby many factors, such as current intensity, electrode spacing, liquid metal volume, electrolyte \nconcentration a nd so on.  \nIn this system that combines chemical dissolution and electrochemical oxidation, liquid metal \nwas both a conductor and a reactant. In the process of continuous expansion and contraction, the \nchange of surface tension and the redistribution of cha rge could be achieved by controlling the \nexternal electric field. In the process, the internal oscillations were synergistic with the movement \nof electric charges, and the electric charges of this kind of motions were also related to the dynamic \n14 \n magnetic f ield. Therefore, the internal chemical mechanism of liquid metal also had the possibility \nof generating an endogenous magnetic field.  \n \n \nFigure 10.  Reversible shape and charge redistribution mechanism of liquid metal [35]: (a) \nExperimental schematic diagra m and equivalent circuit after applying DC power ; (b) Experimental \nschematic diagram and equivalent circuit without external power supply ; (c) The effect of surface \noxide layer on droplet morphology ; (d) The change process of the droplet shape when the ext ernal \npower supply was removed after 5s.  \n \n3. Theoretical Interpretation  \n3.1 Generation of Endogenous Magnetic Field from Liquid Metal Machines  \nWhen applied an external electric field or reacts with other metals to form a galvanic cell, the \nliquid metal act s as a conductor, there is current passing through its surface, the inside of the dr oplet \nrotates at the same time. From the above experiments, we have observed that liquid metal can \ngenerate endogenous magnetic fields through rotation. Next, based on some  basic principles \navailable in classical electromagnetics textbooks, we will conduct theoretical analysis and formula \nderivation of this peculiar phenomenon, and try to make a more in -depth explanation. For the \nphysical properties of the liquid metal spher ical droplet, its density ρ, electrical conductivity σ, and \nmagnetic permeability μ0 are all constants. According to Maxwell's electromagnetic equation, the \nelectromagnetic equation describing the space around the spherical droplet of liquid metal can be \nobtained as:  \n∇×𝑯=𝜕𝑫\n𝜕𝑡+𝑱𝒆                              (11) \n∇×𝑬=−𝜕𝑩\n𝜕𝑡                               (12) \n∇⋅𝑩=0                                 (13) \n∇⋅𝑫=𝜌𝑒                                (14) \n15 \n And its physical property equations are:  \n𝑩=𝜇0𝑯                                 (15) \n𝑱=𝜎𝑬                                  (16) \n𝑫=ε𝑬                                  (17) \nwhere,  H is the magnetic field strength, the unit is A·m-1. E is the electric field strength, the unit is \nV·m-1. B is the magnetic induction, the unit is T. J is the current density, the unit is A·m-2; D is the \nelectric displacement, the unit is C·m-1; ε is the dielectric constant.  \n  Firstly, we analyze the liquid metal droplet in rotating state after swallowing aluminum foil, as \nshown in Figure 11a, b . The liquid metal droplet, aluminum foil and electrol yte solution together \nconstitute a short -circuit galvanic cell system, with aluminum foil as the cathode and liquid metal \nas the anode. For the rotating liquid metal droplet, its internal current will be composed of two parts , \nthe first part is the galvani c current I1 produced by the electrochemical reaction of galvanic cell, a nd \nthe second part is the rotating current I2 produced by the charge moving in the rotating process. The \ncurrent density of the liquid metal droplet is also composed of these two parts, named  J1 and J2 \nrespectively, and the synthetic magnetic field of liquid metal droplet is generated under these two  \ndifferent currents.  \nWe define the angular velocity of the liquid metal droplet as ω, and suppose that when there is \ncurrent flowing inside, the direction of the current is parallel to the rotation axis of the liquid metal \ndroplet in dynamic equilibrium. T he radius of the sphere of the liquid metal droplet is ‘a’, the cen ter \nof the sphere is the origin, and the axis of rotation is the polar axis to establish a spherical coo rdinate \nsystem, r is the distance from any point to the center of the sphere. Assumin g that Q1 is the total \ncharge passing through the maximum cross section (radius=a) of the liquid metal droplet within \ntime t, the current passing through the maximum cross section can be equivalent to:  \n𝑰𝟏=𝑸𝟏\n𝒕                                  (18) \nDuring time t, the current density flows on any circular cross section perpendicular to the current \ndirection on the sphere is as follows : \n𝑱𝟏(𝒓)=𝐼1\n𝑠1                                (19) \n𝑠1=𝜋(𝑟𝑠𝑖𝑛𝜃)2                            (20) \nwhere, s1 is the area of  circular  cross section, and there is 0<r ≤a. \n  For the second part, the liquid metal droplet will also produce current  I2 in the process of rotat ing. \nAssuming that in the process of constant rotation, the charge amount of the liquid metal droplet is \nQ2, then the volume charge density is:  \n𝑞=𝑄2\n4\n3𝜋𝑎3                                (21) \n  The corresponding current density is:  \n𝑱𝟐(𝒓)=𝑄2\n4\n3𝜋𝑅3⋅𝝎×𝒓=3𝑄2\n4𝜋𝑅3𝝎×𝒓                   (22) \n  Under steady conditions, using the formula of the vector potential generated by the current in the \nclassical theory of electromagnetics, the vector potential of any point  A (r 1, θ, φ) outside the liqui d \nmetal droplet can be obtained as:  \n𝑨𝟏(𝒓𝟏)=𝜇0\n4𝜋∫𝑱𝟏(𝒓)+𝑱𝟐(𝒓)\n|𝒓𝟏−𝒓|𝑉𝑑𝑉                      (23) 16 \n where,  |r1-r| is the distance between point r1 (r1, θ1, φ1) and point r (r, θ, φ). \n  According to the basic equation under steady electric fields, the galvanic current and the rotating \ncurrent  flowing through the liquid metal droplets can be obtained together at any point A outside \nthe body. The resulting expression of the total magnetic induction is:  \n𝑩𝟏(𝒓𝟏)=𝛻×𝑨𝟏(𝒓𝟏)                              (24) \nNext, we analyze the liquid metal sphere rotating under a constant external electric field, as shown  \nin Figure 11c, d . This is similar to the previous analysis. The only difference is that the liquid me tal \nsphere droplet does not have the current generated by the galvanic reaction, but has an external \nconduction current under the external electric field. When an electric  field is applied, the liquid \nmetal droplet in the electrolyte solution will not only conduct current, but also form a rotating \ncurrent. In this case, it is also assumed that I3 is the conduction current flowing through the larges t \ncross -section (r=a) of t he sphere, then the area current density flows on the circular cross -section \nperpendicular to the current direction is:  \n𝑱𝟑(𝒓)=𝐼3\n𝑠2                                  (25) \nSimilarly, it is assumed that liquid metal droplets will also generate a rotating cu rrent I4. During \nthe constant rotating process, it always has a charge of Q4, so the charge density is:  \n𝑞=𝑄4\n4\n3𝜋𝑎3                                  (26) \nIts current density is:  \n𝑱𝟒(𝒓)=𝑄4\n4\n3𝜋𝑅3⋅𝝎×𝒓=3𝑄4\n4𝜋𝑅3𝝎×𝒓                        (27) \nSimilar to the above analysis, the vector potential of any point B (r 2, θ2, φ2) outside the body of \nthe liquid metal droplet can be obtained by using the same current to generate the vector potential \nformula as:  \n𝑨𝟐(𝒓𝟐)=𝜇0\n4𝜋∫ 𝑱𝟑(𝒓)+𝑱𝟒(𝒓)\n|𝒓𝟐−𝒓|𝑉𝑑𝑉                         (28) \nwhere,  |r2-r| is the distance between point r2 (r2, θ2, φ2) and point r (r, θ, φ). \n  Then the liquid metal droplet under external electric field is integrated by the conduction current  \nand the rotating current, and the total magnetic induction intensity generated by any point B outside \nthe body is expressed as:   \n𝑩𝟐(𝒓𝟐)=𝛻×𝑨𝟐(𝒓𝟐)                            (29) \nThrough the analysis and calculation of the above two motion phenomena of the liquid metal  \nsphere droplet, it can be seen that when the liquid metal droplet is rotating in the solution and  there \nis current passing through it , the sphere itself will also generate a rotating circular current due  to the \nrotating motion. The superimposition of these two currents will generate a magnetic field  together. \nAt the same time, due to the instability of the rotation of the liquid metal s pherical droplet  on the \nplane, its rotational direction will suddenly change within a period of time, resulting in a  change in  \nthe direction of the magnetic field.  \nWhat needs to be pointed out here is that the spherical liquid metal droplets that rotate d by \nswallowing aluminum foil generate a specific magnetic field, which is essentially different from the  \nmagnetic fluid widely reported in the past. This kind of liquid metal is still in the  form of full fluid. \nIt can adjust the rotation speed, volume and cu rrent in the rotation process, and  can also possiblly \nbe applied to various research applications that require flexible magnetic fluid by virtue of  the 17 \n characteristics of full fluid.  \n \nFigure 11.  Schematic diagram of the movement mechanism and equivalent circuit of surface current \nin NaOH solution: (a) and (b) liquid metal sphere swallowing aluminum; (c) and (d) Under external \nelectric field.  \n \n3.2 Technical  Routes to Realize  Magnetic Monopole from  Liquid Metal Machines  \nMost of the liquid metal magnetoelectric devices and magnetic drive structures reported in the \npast used an external electric field to change the overall magnetic field of the structure, thereby \ncausing the device to act. For example , a layer of spiral GaIn alloy was printed on PDMS to obtain \na flexible electromagnetic driver[26]. When alternating current was applied to both ends of the coil, \nthe coil would generate a magnetic field, which was attracted and repelled by the magnets on both \nsides according to the alternating frequency of the current, which could simulate jellyfish swimming , \nfish tail swinging and other actions  as made in the present authors ’ lab. There were also studies \nusing the electromagnetic interaction between a mag net and a Galinstan liquid metal coil to generate  \nsound waves and made a retractable dynamic acoustic device[36]. \nHowever, the spin motion of liquid metal can excite the endogenous magnetic field in essence. \nBecause the two magnetic poles of a traditional rigid materials  are relatively fixed, it is difficult to \nfind a matching substance when searching for magnetic monopoles, and it is even more difficult to \nfind a case through experimental observation. Considering the aforementioned four basic ways of \nexcit ing the magnetic field with liquid metal, we proposed several possible methods of artificially \nsynthesizing a liquid metal sphere similar to a magnetic monopolar state.  \nRoute 1: Matching the interior flow field of liquid metal droplet machines.  As shown in  \nFigure 12a , while the liquid metal sphere rotates itself after swallowing the aluminum foil, there \n18 \n may be vortex -shaped small spheres inside which move opposite to the outer fluid. This unique \nnested structure makes it possible to artificially synthesize magnetic monopoles. When the outer \nsphere rotates around the axis, the inner four small spheres perform their respective circular motions \nin opposite directions, and then through micro -scale manipulation, such as regulating the position \nand speed of the ro tation, the magnetic effect of the N pole or S pole generated in the outer area and \nthe magnetic poles generated by the four small spheres in the interior are mutually offset, so that the \nentire liquid metal sphere exhibits only a single magnetic pole.  \nRoute 2: Superposition between external electric effect and magnetic field.  Secondly, an \nexternal field superposition method can be adopted. As shown in Figure 12b , a tiny permanent \nmagnet is inserted into the bottom end of the liquid metal sphere. When the liquid metal sphere is \nspinning, the permanent magnet is fixed, so that the magnetic field of the permanent magnet and the \nmagnetic field generated by the spin of the liquid metal sphere are superimposed . Since the \npermanent magnet is at one end of the  liquid metal sphere, the endogenous magnetic field excited \nby the spin interacts with the magnetic field of the permanent magnet. When the two cancel out at \na certain moment, the remaining magnetic field is located at the end far away from the permanent \nmagnet. At this time, the liquid metal sphere will exhibit such un -cancelled magnetism.  \nRoute 3: Composite construction between magnetic particles and liquid metal motor.  In \naddition, adding appropriate magnetic nanoparticles into the liquid metal sphere is  also a possible \nmethod, as shown in Figure 12c . Since the ferromagnetic particles are very tiny, when they are \nuniformly mixed with the liquid metal spheres, the spinning liquid metal spheres will also generate \nmany tiny magnetic fields. The endogenous ma gnetic field can also interact with the ferromagnetic \nparticles added. Proper control of the number, shape and position of magnetic nanoparticles can also  \nmake the synthetic magnetic field of the liquid metal sphere exhibit a certain unipolar characterist ic. \nRoute 4: Chemical ways such via galvanic cell reaction between liquid metal motor and \nsubstrate or foreign substances.  Finally, chemical reaction can generally  be a possible option , as \nshown in Figure 12d . As mentioned above, when the copper contacts th e surface of the liquid metal \nto form an electrode pair, a galvanic battery system that can undergo electrochemical reactions is \nformed in an alkaline solution. When the current flow through the liquid metal, the droplet spins \nunder the influence of the su rface tension gradient. In such an environment where two different \nmetals are in contact with each other, it is allowed to simultaneously manipulate another metal \nsubstrate (like copper) and a spinning liquid metal sphere, so that the sphere produces a syn thetic \nmagnetic field with an apparent single pole characteristic.  \nOverall, i n the above four different methods of constructing the single magnetic pole, the control  \nof the spin vortex inside the liquid metal is a certain challenge due to the changeable ro tation stat e \nof the droplet, and the other three methods have certain feasibility. Therefore, it is possible to \nsynthesize a magnetic substance that acts as a source or sink of magnetic field lines by using the \nspin characteristics of the liquid metal sphe re and through artificial control, which is expected to  \nprovide supporting evidence for the actual existence of magnetic monopoles  in the coming time . \n 19 \n  \nFigure 12.  Four methods of forming a liquid metal sphere with a combined magnetic monopole \nfield: (a) Internal vortex interaction; (b) Combined magnetic field with external magnetic field; (c)  \nAdding magnetic particles; (d) Chemical reaction pathway.  \n \n4 Discussion  \n4.1 Revisit  of Magnetic Monopole Theory  \nUnder active vacuum conditions, Maxwell's equations do not satisfy the invariance of \nelectromagnetic duality, that is, electrons are regarded as the source and sink of electric field li nes, \nimplying that particles with basic magnetic charges are related to  the emission of magnetic field \nlines. Therefore, Dirac proposed the magnetic monopole theory in 1931 to resolve this problem[37]. \nA magnetic monopole is a magnetic substance with only a single magnetic pole of N or S pole in \ntheoretical physics, and its m agnetic line of induction is similar to the electric field lines of a point \ncharge, as shown in Figure 13a . Dirac used a mathematical formula to guess its existence by \nanalyzing the phase uncertainty of the wave function of a quantum system. It should be p ointed out \nthat if the magnetic monopole really exists, the electric charge and magnetic charge must be \nquantized in quantum mechanics. In modern physics, charge quantization has become an important \ntheory, and this theory has also promoted the process of researchers looking for magnetic monopoles.  \nIn order to find magnetic monopoles, researchers have conducted a lot of explorations for decades.  \nA well -known theory related to magnetic monopoles is the Grand Unified Theory (GUTs), in which \nstrong interaction  forces, weak interaction forces and electromagnetic forces can be unified into a \nnormative interaction. Unlike elementary particles, a magnetic monopole is a regional energy called \nsolitary wave, the size and mass of a magnetic monopole can be estimated t hrough unified field \ntheory , and the quality of a magnetic monopole is approximately about 1016 GeV[38, 39]. At the same \ntime, the internal space freedom of the gauge theory makes the magnetic monopole have a certain \ntopological structure and provides a gu arantee for stability. Although GUTs is dedicated to unifying  \nthe interaction forces between microscopic particles, it has not been finally verified, nor can it s olve \nproblems such as the \"excess\" of magnetic monopoles[40]. In addition, String theory, M th eory, etc. \nalso conjectured the existence of magnetic monopoles.  \n20 \n  \n4.2 Extension of Magnetic Monopole Concepts  \nExcept for the classical way, s ome physicist s turned to  strictly mathematical approaches to find \nour possible clues towards magnetic monopole . For example, Yang et al.[41] introduc ed the \noverlapping area of the sphere to construct a dual coordinate system to eliminate singular chords, \nand proved that any curl of the two vector potentials could reasonably give the magnetic field of \nmonopole, and  gave a magnetic monopole solution in the overall description of the gauge field, that \nis, Wu -Yang magnetic monopole. Abrikosov[42] proposed that the quantum state on the sphere of \ngraphene was related to monopole harmonics. Under certain operators, keepin g the magnetic \nmonopole at the center of the sphere unchanged, rotating the sphere would produce a part related to \nspin, and this part was caused by a magnetic monopole.   \nOn the basis of rich theories, some researchers have tried to explore the existence o f magnetic \nmonopoles through  experimental means. For example, by mixing tiny magnetic spins[43], as shown \nin Figure 13b , or artificially constructing a magnetic monopole at the end of a nano -magnetic \nneedle[44]. Fang et al.[45] found that there was an anom alous Hall effect in ferromagnetic crystals, \nand only the hypothesis of the existence of magnetic monopoles could explain this phenomenon. \nThis result might serve as indirect evidence for the existence of magnetic monopoles in the \nmomentum space of the cry stal. \nBecause spin ice exhibits unique magnetic field characteristics, some  studies have found \nanalogues of magnetic monopoles in strange spin ice[16-18]. This is due to the dipolar magnetic \nexcitation caused by s  spin flipping that produces free defects d istributed in the crystal lattice \n(Figure 13c ), thus exhibiting the properties of a magnetic monopole[46], which has magnetic charges \nand electric dipoles[47], but it cannot be separated from the material[18]. Dusad et al.[48] used a \nsuperconducting quantu m interference device (SQUID) to detect the quantization jump of the \nmagnetic flux of the Dy 2Ti2O7 crystal, and found that the magnetic noise generated by the magnetic \nmonopole could be heard by humans after amplification. This is the more intuitive observation of \nmagnetic monopoles in spin ice so far . \nIn the field of condensed matter physics, theoreti cally, the spin structure of BEC could be \nmanipulated with point -like topological defects through an external magnetic field[49]. On this basis,  \nRay et al.[50] used cold atoms to generate Raman transitions from lasers of different frequencies \nunder a magne tic field gradient to simulate the effect of the magnetic field. Since the magnetic spi n \narrangement of rubidium  atoms was controllable, the magnetic spins were arranged in the form of \nlarge vortices, and the middle part would produce the effect of synthes izing magnetic monopoles. \nAt this time, the spin direction of the atom could point to a certain point in space at the same tim e, \nwhich also proved the basic quantum characteristics of the magnetic monopole, as shown in Figure \n13d. 21 \n  \nFigure 13.  Hypothesis a nd simulation of magnetic monopole: (a) Conceptual diagram of magnetic \nmonopoles; (b) Sketch of a magnetic configuration describing the merging of two skyrmions[43]; (c) \nSchematic diagram of spin -ice excited state; (d) Magnetic monopole model composed of c old atom \ncondensed state[50]. \n \n4.3 Experimental Challenges and New Fluidic Magnet Opportunity  \nAlthough abundant evidence for the existence of analogues of magnetic monopoles has been \nfound, the direct observation of Dirac monopoles in the medium of a quant um field has not yet \nachieved a breakthrough. For decades, with the support of GUTs, String theory, M -theory and other \ntheories, researchers have even searched for relevant magnetic monopole trajectories in particle \naccelerators and lunar geotechnical samp ling. But researchers have not yet observed its actual \nexistence by experimental means.  However, liquid metal combines the properties of electrical \nconductors, magnets, and fluids, which endorsed it rather profound possibilities. When an electric \nor magnetic field is applied, or a galvanic cell is formed with other metals, the inter ior flow field \nwill keep moving, which is closely related to the movement of the electric charge and the excitation  \nof the endogenous magnetic field. In this sense, the ric h operability of liquid metal machines offer \nquite a few experimental approaches for the discovery of magnetic monopoles.  \nThe magnetic field described by Maxwell's electromagnetic equations is a passive field, that is, \neach closed magnetic line of inductio n does not have a starting point and an end point. If the \nmagnetic monopole exists, the magnetic field will become an active field. Maxwell ’s equations need \nto undergo appropriate coordinate transformation and conditional restrictions, which does not \nviola te its correctness. The revised equations are as follows,  \n∇×𝑬=−𝜕𝑩\n𝜕𝑡−𝑱𝑚                             (30) \n∇⋅𝑩=ρ𝑚                                 (31) \nCompared with Maxwell's equations, which originally did not consider the existence of magnetic \nmonopoles, the two differential equations related to the magnetic field have changed. Since the \nmovement of the magnetic monopole may also excite the electric fi eld, the magnetic current density \nvector Jm is introduced into the Equation (12) to obtain the Equation (3 0). At the same time, the \nmagnetic monopole can excite the magnetic field by itself, and the divergence of the magnetic field \nis no longer zero  (Equation 3 1). The equations will show a more symmetrical electromagnetic field \n22 \n excitation form. Under the new category of liquid metal magnetic monopole, many theoretical and \nexperimental works are worthy of pursuing in the coming time.  \n \n4.4 Endogenous Magnet from Rigid, Soft to Fluidic Matters  \nFrom the initial discovery of natural magnetic fields, the manufacture of artificial magnets, to t he \nconclusion of electromagnetic related theories, people have experienced a long exploration in this \nprocess. Based on th e above systematic interpretation, we outlined Figure 14  as follows which \ncomparatively illustrate s the evolution diagram of various forms of magnets that people have \nmanufactured , i.e. From rigid matter, to soft material until not fluidic magnetism .  \nFigu re 14a  depicts magnets of various shapes that are widely known and commonly used in daily \nlife. Many important electromagnetic field theories are based on that. Figure 14b refers to a soft \nmagnetic substance, which can be attached to a specific substrate. This magnetic material has been \napplied to various electronic devices, such as sensors and flexible circuits. Figure 14c  is a composi te \nof magnetic nanoparticles and fluid proposed in recent years. This fluid has no magnetic attraction \nin static state, and  only exhibits magnetism when an external magnetic field is applied.  It has a wid e \nrange of applications in the fields of medical equipment, magnetic fluid beneficiation, but is still  not \nan ideal all -magnetic fluid . Stepping further, t he rotating liquid m etal machine or motor as proposed \nin this work belong s to a new kind of magnetic fluid, or electromagnefluid, as shown in Figure 14d . \nIn this category, t here is no need to add additional magnetic particles, which is a complete magneti c \nfluid in the true se nse. The discovery of room temperature liquid metal full magnetic fluid will bri ng \nnew vision s and directions for research in diverse  fields, which has profound significance for the \nacademic society  to further understand magnets and fully use them . \n \n \nFigure 14.  Different forms of magnets: (a) Rigid magnet; (b) Soft magnet; (c) Semiliquid magnet; \n(d) Liquid magnet.  \n \nIn nature, the north and south poles always exist at the same time. Cut a magnet into two pieces, \nand each piece will form its own north an d south poles, which makes it difficult to achieve the \nexistence of one pole alone. Compared with traditional rigid magnets, the advantage of liquid metal \nlies in its fluidity, which makes it easier to adjust to different shapes and the restriction on part icle \n23 \n movement is reduced. While the surface charges are moving fast, the inside of the fluid is constantly \nrotating, making the surrounding magnetic field in a state of constant change. If the single charge \non the surface of the liquid metal can be manipul ated, even controlling the positive and negative  \nfeatures  of the charge in a certain position, then the direction of a single magnetic particle can be \nreversed, it is possible to construct a liquid metal magnetic monopole.  \nFurthermore, if the direct experi mental observation of the magnetic monopole can be achieved, \nthe charge quantification can be better explained. Electrons are particles with no internal structure. \nThey can be reshaped to carry orbital angular momentum (OAM). By coupling the center of mass  \nof the wave function with the interior, the total angular momentum information can be obtained \nfrom the vortex beam, and then it can be manipulated artificially[51]. Similarly, if the magnetic \nmoment of the liquid metal micromotor can be measured to obtai n momentum information, it may \nbe manipulated and improved microscopically, and it will even be possible to develop  technology \nbased on moving magnetic charges to break through the limitations of current charge engineering  \nand create  magnetic materials tha t exhibit rich er behavior.  \n \n5 Conclusion  \nLiquid metal can interact with sound, light, heat, electricity and magnetism etc. The present study \ndisclosed that as a fluidic conductor, liquid metal machine opens a generalized way to generate \ntransformable endog enous magnet through tuning its interior rotational configurations. This \nmechanism is rather difficult to implement through rigid  materials  other wise. It may change \npeople ’s basic understanding of the classical magnetism science. A most  noteworthy tool enabled  \nfrom this finding still lies in that such fluidic magnet suggest ed a new promising way to realize \nmagnetic monopole  in reality . And the fundamental routes can be based on either self -fueled liquid \nmetal m otor or the electrically controlled spin mot ion of liquid metal machine inside the electrolyte. \nMeanwhile, it should also be pointed out that, such liquid metal endogenous magnetism is \nfundamentally different from the conventional  magnetic fluids which was in fact caused by the \nmagnetic particles  rather than the fluid  itself . This raised important fundamental and practical issues \nwaiting to be addressed  in the coming time. It also opens  perspective for making new conceptual \nliquid machines which may find some emerging  uses in future basic physics, information technology, \nsmart devices , and advanced functional materials. 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Since magnetic \nanisotropy energies of nanomagnets are comparable to the therm al energy scale, temperature can \nhave a profound effect on the dynamics of a nanomagnet driven  by spin transfer torque. Here we \nreport the observation of unusual types of microwave-frequ ency nonlinear magnetization dynamics \nco-excited by alternating spin transfer torque and therm al fluctuations. In these dynamics, \ntemperature amplifies the amplitude of GHz-range prec ession of magnetization and enables \nexcitation of highly nonlinear dynamical states of magnetization  by weak alternating spin transfer \ntorque. We explain these thermally activated dynamics in ter ms of non-adiabatic stochastic \nresonance of magnetization driven by spin transfer torque. Th is type of magnetic stochastic \nresonance may find use in sensitive nanometer-scale microwave signal d etectors. X. Cheng et al.  2 December 18, 2009 \n Spin transfer torque (STT) from spin-polarized current 1,2,3,4 applied to a ferromagnet can give rise \nto several types of magnetization dynamics such as auto-oscill ations of magnetization excited by direct \ncurrent 5,6,7,8,9,10,11 , magnetization reversal driven by STT 12,13,14,15,16,17  and current-induced magnetic domain \nwall motion 18,19,20 . Practical applications of STT in non-volatile memory, microwa ve oscillators and \ndetectors are under development 21 , and non-equilibrium states of magnetization induced by STT are of \nfundamental interest in nonlinear science 22 , 23 , 24 , 25 . Alternating STT is employed in spin torque \nferromagnetic resonance (st-FMR)26,27,28  measurements of spin waves in magnetic nanostructures 26,29 . In \nthis paper, we describe unusual nonlinear GHz-range magnetizat ion dynamics that are co-excited by ac \nSTT and temperature. The pronounced thermally activated character  sets these dynamics apart from the \ntypes of excitations seen in st-FMR and conventional FMR expe riments. Temperature-dependent studies \nreveal that these dynamics arise from non-adiabatic stochastic resonance (NSR) of magnetization excited \nby ac STT. The amplitude of the observed NSR exceeds that of typi cal st-FMR-driven excitations by \nnearly two orders of magnitude for the same level of ac STT drive.  \n     We study magnetization dynamics of the permalloy (Py=Ni 81 Fe 19 ) free layer in Pt(30 nm)/ Cu(20 \nnm)/ Py(3 nm)/ Cu(6 nm)/ Co(3 nm)/ Ir 20 Mn 80 (8 nm)/ Cu(80 nm) spin valve (SV) nanopillars. All \nmagnetic layers are patterned into 120×60 nm 2 elliptical pillars with the long axis parallel to the exc hange \nbias field from Ir 20 Mn 80 . Electrical current, I, is applied between the top and the grounded bottom leads as \nshown in Fig. 1a. We report data from a single sample; nine other  samples show similar behavior. For all \nmeasurements, a magnetic field, Hr\n, is applied at 10 ° from the sample normal, with the in-plane field \ncomponent in the exchange bias direction. Fig. 1b shows resistance, R, of the SV as a function of H at \ntemperatures of 295 K and 120 K. The in-plane component of Hr\nswitches the magnetization of the \nuniaxial Py nanomagnet, Mr\n, between the high-resistance, nearly antiparallel (HR), and the low-\nresistance, nearly parallel (LR), states of the SV. The L R↔HR transition is continuous at 295 K and \nhysteretic at 120 K, indicating that the free layer is superpa ramagnetic at 295 K. Figures 1c and 1d X. Cheng et al.  3 December 18, 2009 \n illustrate the LR ↔HR transition induced by STT 1, ( )Co dc MM MIrrrr× ×⋅ ~τ , from direct current, Idc . Fig. 1e \nshows differential resistance, dV/dI,  versus H at 120 K measured for several Idc . This figure illustrates that \nthe free layer hysteresis loop width decreases with increasi ng |Idc | and becomes zero at Idc  = -2.6 mA, \nindicating that the free layer is superparamagnetic for Idc  ≤ -2.6 mA. Figure 1f shows telegraph noise of \nthe free layer between the HR and LR states at T = 120 K, H = 2.2 kOe and Idc  = -2.7 mA, which \ndemonstrates that the free layer indeed becomes superparamagnetic  at this current. This current-induced \nsuperparamagnetism results from renormalization of the amplitude  of thermal fluctuations of Mr\nby \nSTT 14,30 . For Idc  < 0, STT pushes Mr\n away from the LR state, thereby enhancing the amplitude of thermal \nfluctuations in this state, and increases the rate of thermally  activated transitions out of the LR state. In \ncontrast, the amplitude of thermal fluctuations in the HR state  is suppressed for Idc  < 0 because STT \npushes magnetization towards the HR state and thereby decreases the rate of thermally activated \ntransitions out of the HR state. As a result, the Kramers transition rates between the HR and LR states \nbecome functions of current 30 : \n( )\n\n\n\n\n\n\n\n\n−− ≈→TkH E\nIIf K\nBHR \nb\nHR LR HR \n00 1 exp     (1a), \n( )\n\n\n\n\n\n\n\n\n+− ≈→TkH E\nIIf K\nBLR \nb\nLR HR LR \n00 1 exp                                                  (1b), \nwhere ()H ELR HR \nb)( are the zero-current barriers for transition out of the HR(LR)  states, kB is the \nBoltzmann constant, ≈0f 10 9-10 10  Hz and )(\n0LR HR I = const are the critical currents 30 .   \n We measure magnetization dynamics driven by ac STT using the setup in Fig. 2a26 . In these \nmeasurements, ac STT from a square-wave modulated microwave c urrent of rms amplitude Iac  is applied \nto the SV in addition to Idc . The ac STT excites oscillations of Mr\n, which give rise to oscillations of R via \ngiant magnetoresistance,  R = R0+δRac sin( ω t+ϕ), where ϕ is the phase shift between the resistance and \ncurrent oscillations. As a result, a rectified voltage ( )ϕ δ cos \n21\nac ac RI  is generated by the SV. At Idc ≠ 0, X. Cheng et al.  4 December 18, 2009 \n an additional voltage, Idc⋅δRdc, is induced in response to Iac , where δRdc is the difference between the time-\naverage SV resistance in the presence and in the absence o f Iac 31 . The resulting rms voltage measured by \nthe lock-in amplifier is ( )dc dc ac ac dc RI RI V δπϕ δπ2cos 1+ = , where the first term is dominant for small-\namplitude oscillations of Mr\n while the second term dominates for large-amplitude oscillati ons of Mr\n, \nprovided | Idc |>>I ac 31 . In our measurements, we sweep the frequency of Iac , f, and record Vdc (f). When f \ncoincides with a spin wave eigenfrequency of the Py nanomagnet, fn, the eigenmode is resonantly excited, \nand a peak or a trough in Vdc (f) is observed at f = fn. These peaks and troughs give the st-FMR spectrum of \nspin wave eigenmodes of the nanomagnet 26,27 .  \n Figure 2c-e shows Vdc (f) at T = 120 K and H = 2.2 kOe for several values of Idc : far from (-1.5 mA, \n-3.5 mA) and at (-2.6 mA to -2.8 mA) the current-driven  LR ↔HR transition. The st-FMR spectra, Vdc (f) , \nfar from the LR ↔HR transition shown in the insets of Fig. 2c-d exhibit typical spin wa ve \nresonances 26,27,32,33 . However, for Idc  at the LR ↔HR transition, we observe qualitatively different Vdc (f). \nAt Idc  = -2.6 mA (Fig. 2c), Vdc (f) develops a negative low-frequency tail LF Vmax ≡|Vdc (0) | while at Idc  = -2.8 \nmA (Fig. 2d), Vdc (f) exhibits a positive maximum, HF Vmax , at f = 2.2 GHz.  Both LF Vmax  and HF Vmax are two \norders of magnitude greater than the amplitudes of the high-fre quency resonances, Vmax ≡ \nmax {| Vdc (f) |} f>1GHz , in the LR and HR states, indicating that large-amplitude dynami cs are excited by Iac  at \nthe LR ↔HR transition. Fig. 2e shows the response curve at Idc  = -2.7 mA in the crossover regime \nbetween the large-amplitude high- and low-frequency resonances. Fig. 2f displays Vdc (t) at Idc  = -2.8 mA \nand  f = 2.2 GHz for Iac  = 0 mA and Iac  = 0.4 mA. We also make spectral measurements of microwave \nsignal emission by the sample 5,6  at Idc  = -2.8 mA and Iac = 0 mA, and do not observe a detectable signal in \nthe 0.1-10 GHz band. These measurements show that the large-amplitude  oscillations of Mr\nare only \nexcited in the presence of Iac. \n Figure 3 further illustrates the difference of the  dynamic response at the LR ↔HR transition from \nthat in the HR and LR states. Fig. 3a-b shows the full width at hal f maximum and the amplitude, Vmax , of X. Cheng et al.  5 December 18, 2009 \n the high frequency ( f > 1 GHz) spectral peak in Vdc (f)  versus Idc . For | Idc | < 2.4 mA, the linewidth of the \npeak decreases while Vmax  increases with increasing | Idc |. This behavior is due to renormalization of the \neffective damping by dc STT 26,34 . However, for -3.1 mA < Idc  < -2.6 mA, both the amplitude and the line \nwidth increase dramatically, signaling a transition to a ne w dynamic regime. Fig. 3c-d shows the \ndependence of Vmax  on Iac for several values of Idc . For Idc  far from the LR ↔HR transition, Vmax (Iac ) is \nquadratic. In this small-amplitude regime, δRdc≈031  and ac ac RI V δπ1\nmax ≈  ~2\nac I. In contrast, at the \nLR↔HR transition, Vmax (Iac ) crosses over from quadratic to linear behavior and eventually s aturates at a \nvalue close to \nπ22 RIdc ∆, where ∆R = 35 m Ω is the resistance difference between the HR and LR states  at \nIdc  = -2.8 mA. This type of response indicates that Iac  induces a transition from the HR state with \nresistance R0+∆R to a state with time-average resistance of ≈ R0+∆R/2. The large-amplitude high- and \nlow-frequency responses of the types shown in Fig. 2c-d are observed in a strip in the ( Idc ,H ) plane, which \ncoincides with the region of the LR ↔HR transitions as illustrated in Fig. 3e. \n The origin of the large-amplitude dynamics at the LR ↔HR transition is revealed by the \ntemperature dependence of Vdc (f). Fig. 3f shows the maximum amplitudes of the high, HF Vmax ~≡ \nmax{ Vdc (f,H)} f, H, and low, LF Vmin ~≡ min{ Vdc (0,H)} H, frequency resonances versus T. Below a threshold \ntemperature, Tth (I dc ), HF Vmax ~ and LF Vmin ~are small and Vdc (f) shows small-amplitude eigenmode resonances. \nAbove Tth (I dc ), HF Vmax ~ and LF Vmin ~rapidly rise to their maximum values at T = ()dc LF HF \nSR I T)( and the response \ncurves become similar to those in Fig. 2c-d. For T > TSR (I dc ), HF Vmax ~ and LF Vmin ~slowly decrease. This type \nof temperature dependence of the amplitude of motion is a salient feat ure of stochastic resonance (SR)36 .  \nSR is an effect of noise-induced amplification of the response o f a nonlinear system to a weak \nperiodic drive 35,36,37 . To describe SR, we consider the magnetic energy 38  of the Py nanomagnet (Fig. 4): \n  ( )dHHMMNM Errrrtr\n+⋅−⋅⋅ =π2 ,     (2)  X. Cheng et al.  6 December 18, 2009 \n where Nt\n={N x, N y, N z} is the diagonal demagnetization tensor of the free layer 38 , Hr\n is the external field, \ndHr\n~Co Mr is the stray field from the Co layer and Co Mr is the magnetization of Co. Fig. 4b-d shows that \nE(θ,φ) is an asymmetric double-well potential with the wells corresponding to the HR and LR states.  \nAccording to Eq. (1), the Kramers transition rates between t he HR and LR states depend on STT: \nKHR →LR  decreases and KLR→HR increases with increasing | Idc |. Therefore, equal dwell times in the HR and \nLR states ( KHR →LR = K LR→HR ≡ KE) can be achieved via tuning of Idc  even though the double-well potential \nis asymmetric. Analysis of Eq. (1) and previous experiments 30  show that KHR →LR =K LR→HR is observed on a \nline H=H E(I dc ) in the (I dc ,H) plane, and KE(I dc , H E(I dc )) on this line exponentially increases with | Idc |. \nAlternating current, Iac , applied in addition to Idc periodically modulates KHR →LR  and KLR→HR, and, due to \nthe exponential sensitivity of the transition rates to current, small Iac  can induce periodic transitions \nbetween the HR and LR states at the frequency of Iac . In this case, the LR ↔HR transitions are random at \nIac = 0 and nearly periodic at Iac ≠0. The periodic LR ↔HR transitions induced by Iac  cease if T becomes \ntoo low ( T < T th (I dc )) so that KHR →LR  or KLR→HR becomes small compared to f. This temperature-induced \namplification of the amplitude of motion of Mr\nat T > T th (I dc ) under the action of weak ac drive is SR 39 .  \n Low-frequency (adiabatic) SR driven by ac STT can explain the  large low-frequency tail of Vdc (f)  \nin Fig. 2c. Indeed, such a tail is observed when the system is near the LR end of the LR ↔HR transition \n(Fig. 2b) where KHR →LR  >> KLR→HR at Iac=0. According to Eq. (1), KLR→HR is much more sensitive to small \nvariations of current than KHR →LR  because LR \nbHR \nb E E << . Therefore, in the presence of Iac , KHR →LR  is nearly \ntime-independent while KLR→HR oscillates with the frequency of the ac drive. This implies t hat at large \nenough Iac ,  KHR →LR  >> KLR→HR(t) for a fraction of the Iac period, while for another fraction of the period \nKHR →LR  << KLR→HR(t) . If T is high enough so that KHR →LR  >> f and max{ KLR→HR(t) }t >> f, then the \nLR→HR →LR transition takes place in almost every cycle of Iac. This results in large-amplitude \nresistance oscillations at low f and gives rise to the large low-frequency tail in Vdc (f) . The negative sign of \nVdc (0) shows that the oscillations of Mr\nare in phase with the ac STT oscillations (more negative curr ent X. Cheng et al.  7 December 18, 2009 \n favors the HR state), as expected for adiabatic SR.  Figure 3 f confirms the SR nature of the effect as the \nresonance turns on only at T > Tth , and quickly reaches the maximum amplitude at the SR tempera ture, \nTSR . The turn-on of the SR is sharp in T due to the exponential dependence of the transitions rates on 1/ T. \nThe slow decay of Vdc (0) for T > T SR  is due to partial thermal randomization of the LR ↔HR transitions 36 . \nThe adiabatic SR is observed only for Idc  at the LR end of the LR ↔HR transition because the \nsystem returns to the LR state with current-sensitive KLR→HR in almost every period of Iac. In contrast, for \nIdc  = -2.8 mA at the HR end of the LR ↔HR transition, a low-frequency Iac  does not induce the HR →LR \ntransition because KHR →LR  is weakly sensitive to Iac, and the system remains in the HR state. For this Idc , \nonly a signature of small-amplitude intra-well resonance in the  HR state is expected in Vdc (f) . However, \nFig. 2d shows that this is not the case. Although the low frequency tail in Vdc (f)  disappears at Idc  = -2.8 \nmA, a peak due to unexpected high-frequency ( f = 2.2 GHz) large-amplitude dynamics is observed. \nThis surprising high-frequency dynamics can be explained if Iac  excites large-amplitude \noscillations of magnetization with a ~180 ° phase shift with respect to the ac STT drive. Fig. 4e illust rates \nhow such a phase shift can result in a non-zero time-average compone nt of STT perpendicular to the \nsample plane and thereby stabilize large-amplitude precession o f Mr\n on an out-of-plane trajectory 40 . Since \nthe angle between Mr\nand Co Mris not zero for H > 0, there is a non-zero component of STT perpend icular \nto the sample plane, ( )zCo z MM Mrrr\n× ×~τ . For Idc  < 0, τz > 0 near the LR state and τz < 0 near the HR state. \nTherefore, if Mr\n oscillates on a large-amplitude trajectory passing near both t he HR and LR states, the \ntime-average τz due to Idc  is close to zero. In contrast, Iac at the frequency of the oscillations of Mr\n on the \nlarge-amplitude trajectory generates positive time-averag e τz if the phase shift between the oscillations of \nMr\n and ac STT is ~ 180 °. Therefore, for motion of Mr\n phase locked to the ac STT with a ~ 180 ° phase \nshift, the STT from Iac  always pushes Mr\n in the direction perpendicular to the sample plane (increases  \nMz), towards higher energy precessional trajectories. Figure 4c- d shows that for large enough Mz, Mr\ncan \nprecess on large-amplitude trajectories encircling both the L R and the HR energy minima. The frequency X. Cheng et al.  8 December 18, 2009 \n of precession on such high-energy trajectories can be lower th an the frequency of the trajectories \nencircling only the HR or the LR energy minima 5. Such a large-amplitude dynamic state (D) stabilized by \nIac is consistent with the data in Fig. 2d showing a resonance with the amplitude c orresponding to peak-to-\npeak resistance oscillations similar to ∆R and a frequency below the eigenmode frequencies in the LR and \nHR states. The width of the resonance peak in the D state is determined by the bandwidth of phase \nlocking of the oscillations of Mr\n to Iac  rather than by damping. This explains the large line width of  the \nresonance in Fig. 2d, and the dependence of the line width on Idc  in Fig. 3a. This also explains the increase \nof the line width with Iac shown in the inset of Fig. 3c because larger Iac  increases the phase locking \nbandwidth 25 . The saturation of Vmax  with Iac in Fig. 3c is due to the saturation of the amplitude of \nresistance oscillations in the D state at a value close to ∆R (Vmax  ≈ \nπ22 RIdc ∆). \nSurprisingly, the amplitude of the D-state resonance in Vdc (f) has almost the same temperature \ndependence as the adiabatic SR low frequency tail, Vdc (0), (Fig. 3f). Therefore, the D state is thermally \nactivated and the observed D-state resonance belongs to the class of SR phenomena. Since the frequency \nof the D state is greater than the Kramers transition rate s between the HR and LR states, the D-state \nresonance is a high-frequency or non-adiabatic stochastic resona nce (NSR) 41,42,43,44 . We now discuss the \norigin of the NSR effect in our SV system and explain why NSR is  only seen near the HR end of the \nLR↔HR transition. To understand NSR, we consider thermally activa ted transitions among the HR, LR \nand D states. The time-average magnetic energies as well as the  widths of the energy distributions in these \nstates depend on STT as illustrated in Fig. 4b. Consideration of the available traj ectories of Mr\n in the HR, \nLR and D states gives the relation between the time-average e nergies in these states: 〈ED〉 > 〈EHR 〉 > 〈ELR 〉. \nAt the HR end of the LR ↔HR transition, where the NSR is observed, K LR →HR >>K HR →LR , and thus if \nMr\nfalls from the D state to the low-energy LR state, it is rapidly returned to the higher-energy HR state \nas illustrated in Fig. 4f. Since 〈EHR 〉, is close to 〈ED〉, small STT from Iac  pushing Mr\n towards the D state \nis sufficient to supply energy 〈ED〉-〈EHR 〉 and thereby induce the HR →D transition in the half the period of X. Cheng et al.  9 December 18, 2009 \n Iac  for which τz(t) >0. STT from  Iac  constantly supplies energy to the D state but not to the HR and L R \nstates, for which ac STT adds energy for half a period of Iac  and removes energy for the other half. Due to \nthe energy supplied by Iac  to the D state, this state becomes the most stable of the three non-equilibrium \nstates (HR, LR and D) at sufficiently large Iac  and T resulting in K HR →D>> K D→HR  and K HR →D>> K D→LR . \nAs a result, the system spends most of its time in the D sta te and a large-amplitude high-frequency peak \nappears in Vdc (f)  (Fig. 2d). Energy considerations also explain why NSR is not se en at the LR end of the \nLR ↔HR transition. In the LR state, the energy gap to the D stat e is large 〈ED〉-〈ELR 〉 > 〈ED〉-〈EHR 〉 and the \nenergy supplied by ac STT is insufficient to induce the LR →D transition. A detailed quantitative \nunderstanding of NSR requires a Fokker-Planck description of the transiti on rates 23,45 . \nThe large rectified voltage due to NSR can be employed for mi crowave signal detection 46 . Indeed, \nby using high-magnetoresistance MgO magnetic tunnel junctions (M TJ) 47,48 , a rectified voltage of greater \nthan 0.1 V in response to microwave currents of ~ 100 µA may be expected at NSR. The expected \nsensitivity of such an MTJ-based detector, ~20 mV/ µW, is greater than that of Schottky diode detectors.  \nIn conclusion, we observe adiabatic and non-adiabatic SR of magnetiz ation co-excited by ac STT \nand temperature in nanoscale spin valves. Our work demonstrates that combined dc  and ac STT applied to \na nanomagnet stabilize unusual dynamic states of magnetization f ar from equilibrium. The rates of \nthermally activated transitions from these states are sensi tive to the amplitude and phase of the applied \nSTT, and thus nanomagnets driven by ac and dc STT 24,25  provide a convenient playground for studies of \nthermodynamic processes far from equilibrium. The amplitude of mag netization oscillations in the non-\nadiabatic SR regime is nearly two orders of magnitude gre ater than that in the FMR regime for the same \nlevel of the ac STT drive. The non-adiabatic SR dynamics giv e rise to large rectified voltages generated \nby spin valves in response to applied microwave currents, and th us non-adiabatic SR can be utilized in \nsensitive microwave signal detectors of nanoscale dimensions.  \n \n X. Cheng et al.  10 December 18, 2009 \n Acknowledgements \nThis work was supported by the NSF (grants DMR-0748810 and ECCS-0701458) and by the \nNanoelectronics Research Initiative through the Western Instit ute of Nanoelectronics. We thank D. Ralph \nand R. Buhrman, in whose labs the samples were made, for helpful dis cussions and assistance with the \nsample preparation. We acknowledge the Cornell Nanofabrication Fa cility/NNIN and the Cornell Center \nfor Nanoscale Systems, both supported by the NSF, which facilities were used f or the sample fabrication. \nAuthor contributions \nX.C. collected and analyzed data and wrote the paper; I.K. made the s amples and wrote the paper. \nAll authors contributed to the data collection and the preparation of the  manuscript. X. Cheng et al.  11 December 18, 2009 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1| Spin valve magneto-resistance and current-induced switching.  a , Schematic view of the \nspin valve with approximate directions of magnetization of the pin ned, Co Mr\n, and the free, Mr\n, layers in \nthe high resistance (HR) state of the spin valve. External magnetic field, Hr\n, is applied at 10 ° to the \nsample plane normal. b, Resistance R as a function of H at two temperatures: 295 K and 120 K. R as a \nfunction of direct current, Idc , at 295 K ( c) and 120 K ( d). e, Differential resistance, dV/dI,  as a function of \nH at 120 K measured for several values of Idc . f, Telegraph noise of R between the high (HR) and low (LR) \nresistance states at T = 120 K, H = 2.2 kOe and Idc  = -2.7 mA.  \n \n IrMn Co Cu Py \nMmH 10 o\nMco M\n-2 0 2 424.2 24.4 24.6 \n-2 0 2 420.2 20.4 20.6 \n0 1 2 320.2 20.4 20.6 20.8 \n0 5 10 20.27 20.28 20.29 20.30 20.31 20.32 (a) \n0.0 0.5 1.0 1.5 2.0 2.5 3.0 24.28 24.30 24.32 24.34 \n120 K 295 K \nField (kOe) R (ΩΩΩΩ)\n20.20 20.22 20.24 20.26 20.28 \n (b) \n \n \noffset 0.1 ΩΩ ΩΩ2.0 kOe \n1.8 kOe \n1.7 kOe \n1.5 kOe \n1.0 kOe (c) 295 K R (ΩΩΩΩ)\nDirect current (mA)  \n 2.2 kOe \n2.0 kOe \n1.5 kOe \n1.0 kOe \n0.75 kOe offset 0.1 ΩΩ ΩΩ(d) 120 K R(ΩΩΩΩ)\nDirect current (mA)  \n \noffset 0.1 ΩΩ ΩΩ(e) 120 K \n-2.6 mA \n-2.4 mA \n-2.0 mA \n-1.0 mA dV/dI (ΩΩΩΩ)\nField (kOe) 0 mA  \n (f) 120 K   H = 2.2 kOe \n                 I dc = - 2.7 mA \n              R (ΩΩΩΩ)\nTime (ms) X. Cheng et al.  12 December 18, 2009 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2| Measurements of magnetization dynamics driven by ac spin transfer torque.  a , Spin \ntorque FMR (st-FMR) measurement setup. b, R versus Idc  at T = 120 K, H = 2.2 kOe. Symbols mark the \ndirect current values for which st-FMR data are shown in c-e. Measured response curves, Vdc (f) , at the \nLR↔HR transition:  c, Idc  = -2.6 mA, d, Idc  = -2.8 mA,  e, Idc  = -2.7 mA. Insets: st-FMR spectra far from \nthe LR ↔HR transition ( c,  Idc  = -1.5 mA) and ( d , Idc  = -3.5 mA). f, Vdc (t)  at Idc  = -2.8 mA, for Iac = 0 (open \ncircles) and   Iac = 0.4 mA at f = 2.2 GHz (solid squares). The data in c-e are measured at T = 120 K, H = \n2.2 kOe, Iac = 0.4 mA. \n \n \n \n \n Microwave \nsignal \ngenerator \nLock-in Amplifier DC \ncurrent \nsource rf signal \nIdc \nVdc Reference H         10 o\n-3 -2 -1 020.24 20.26 20.28 20.30 20.32 \n0 1 2 3 4 5-15 -10 -5 0\n0 1 2 3 4 50510 15 20 \n0 1 2 3 4 5-10 -5 05\n0.0 0.5 1.0 0510 15 20 25  - 2.8 mA \n -  2.7 mA \nLR (a) \n R (ΩΩΩΩ)\nDirect current (mA)  - 2.6 mA (b) H= 2.2 kOe \nHR \n-0.10 -0.05 0.00 \n \n   \n5.5 4.5 3.5  I dc = - 1.5 mA  \n Idc  = - 2.6 mA (c) Vdc  (µµµµV)\nFrequency (GHz) 2 3 4-0.3 -0.2 -0.1 0.0 \n \n   Idc = - 2.8 mA \nIdc = - 3.5 mA  \n (d)  Vdc  (µµµµV)\nFrequency (GHz)  \n (e) Idc  = - 2.7 mA Vdc (µµµµV)\nFrequency (GHz)  \nIdc = - 2.8 mA \n Vdc  (µµµµV)\nTime (s)  Iac = 0.4 mA \n I ac  = 0 mA (f) X. Cheng et al.  13 December 18, 2009 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3| Dependence of ac-STT-driven magnetization dynamic s on current and temperature. a,  \nFull width at half maximum and b, the amplitude, Vmax , of the high-frequency peak in Vdc (f)  response \ncurves as functions of Idc . Grey bands in (a) and (b) mark the crossover regime between t he large-\namplitude high- and low-frequency peaks. In this crossover regime, br oadband response curves Vdc (f)  \nsuch as that shown in Fig 2(e) are observed. Inset in b: blow-up of the low-current region of Vmax (Idc ). c, \nThe dependence of  Vmax  on ac drive current, Iac , for three values of Idc : far from (-1.5 mA, -3.5 mA) and at \n(-2.8 mA) the LR ↔HR transition. Inset: Vdc (f)  response curves in the regime of large Iac  for Idc =-2.8 mA . \nd, Quadratic dependence of Vmax  on Iac  for Idc  far from the LR ↔HR transition. Lines are quadratic fits to \nthe data. e, Phase diagram of the system in the ( H, Idc ) plane. Grey bands mark regions of the LR ↔HR \nresistance transitions at fixed field and temperature. Solid  squares and open circles mark the direct current \nvalues at which maximum rectified signal due to adiabatic and non-adiabatic SR are observed at fixed \nfield and temperature. f, Temperature dependence of the amplitudes of the high frequency HF Vmax ~and low \nfrequency LF Vmin ~st-FMR signals at two values of Idc  : -2.2 mA (open symbols) and -2.8 mA (solid symbols). -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 0.0 0.5 1.0 \n-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 0510 15 20 \n0.0 0.2 0.4 0.6 0.8 0510 15 20 \n-3 -2 -1 01.6 1.8 2.0 2.2 2.4 \n0 100 200 300 -20 -10 010 20 0.0 0.2 0.4 0.6 0.8 0.0 0.3 0.6 0.9 Broadband Response \n(f) (e) (d) (c) (b)   FWHM (GHz) \nDirect current (mA) (a) \nBroadband Response \n-2.0 -1.5 -1.0 0.00 0.15 0.30 \nA\n  Vmax  (µµµµV)\nDirect current (mA) \nIdc =\n0 1 2 3 40510 15 20 \n  Vdc  (µµµµV)\nFrequency (GHz)  0.8  mA \n 0.6  mA \n 0.4  mA Iac  = \n  \n - 2.8 mA \n - 1.5 mA \n - 3.5 mA Vmax  (µµµµV)\nAlternating current (mA) Idc =\n295 K 200 K \n  \n Non-adiabatic SR \n Adiabatic SR \n     HR     LR transition Field (kOe) \nDirect current (mA) 120 K \n - 2.8 mA  VLF \nmin  - 2.2 mA VLF \nmin  - 2.8 mA  VHF \nmax  - 2.2 mA VHF \nmax Voltage ( µµµµV)\nTemperature (K)  - 1.5 mA \n - 3.5 mA \n  Vmax  (µµµµV)\nAlternating current (mA) \n~~~~X. Cheng et al.  14 December 18, 2009 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 4| Stochastic resonance energy diagrams.  a, Spherical coordinate system in which the energy \nof the Py nanomagnet,  E( θ,ϕ), is described by Eq. (2). b, E( θ,ϕ) as a function of ϕ for θ = 3 π/8, the \napproximate equilibrium polar angle of the Py magnetization in t he HR and LR states. This energy is a \ndouble-well potential with the minima corresponding to the HR and LR states. Schematic energy \ndistributions of magnetization in the HR, LR and the dynamic st ate D at non-zero Idc  and Iac  are shown. c,  \n3D sketch of the energy surface E( θ,ϕ); thick solid lines schematically show magnetization traject ories \nwith time-average energies, 〈ELR 〉, 〈EHR 〉, 〈ED〉 in the LR, HR and D states. d, Contour plot of E( θ,ϕ); \ndashed lines schematically show 〈ELR 〉, 〈EHR 〉, 〈ED〉. e, Alternating STT, τac , always pushes magnetization \nout of the sample plane towards high energy trajectories of the D state if the phase of the magnetization \noscillations is ~ 180 ° with respect to the ac STT drive. This τac  stabilizes the large-amplitude dynamic \nstate D. f,  Sketch of the Kramers transition rates among the HR, LR a nd D states at the non-adiabatic \nstochastic resonance. 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"    },    {        "title": "1105.0280v1.Magnetization_Oscillation_of_a_Spinor_Condensate_Induced_by_Magnetic_Field_Gradient.pdf",        "content": "arXiv:1105.0280v1  [cond-mat.quant-gas]  2 May 2011Magnetization Oscillation of a Spinor Condensate Induced b y Magnetic Field Gradient\nJie Zhang,1Baoguo Yang,1and Yunbo Zhang1,∗\n1Institute of Theoretical Physics, Shanxi University, Taiy uan 030006, People’s Republic of China\nWe studythespin mixingdynamicsof ultracold spin-1atoms i n aweak non-uniformmagnetic field\nwith field gradient G, which can flip the spin from +1 to −1 so that the magnetization m=ρ+−ρ−\nis not any more a constant. The dynamics of mF= 0 Zeeman component ρ0, as well as the\nsystem magnetization m, are illustrated for both ferromagnetic and polar interact ion cases in the\nmean-field theory. We find that the dynamics of system magneti zation can be tuned between the\nJosephson-like oscillation similar tothe case of double we ll, and the interesting self-trapping regimes,\ni.e. the spin mixing dynamics sustains a spontaneous magnet ization. Meanwhile the dynamics of\nρ0may be sufficiently suppressed for initially imbalanced numb er distribution in the case of polar\ninteraction. A ”beat-frequency” oscillation of the magnet ization emerges in the case of balanced\ninitial distribution for polar interaction, which vanishe s for ferromagnetic interaction.\nPACS numbers: 03.75.Mn, 67.85.Fg, 67.85.De\nI. INTRODUCTION\nSince the successful realization of23Na condensate in\nthe optical trap [1], with the spin degrees of freedom\nliberated, the coherent spin-mixing dynamics inside a\nspin-1 BEC has been studied intensively [2–8]. The spin\nmixing interaction allows for exchanging atoms among\nspin components but conserving the total angular mo-\nmentum. Two atoms in Zeeman state |0/an}b∇acket∇i}htcan coher-\nently scatter into the states |1/an}b∇acket∇i}htand|−1/an}b∇acket∇i}ht,and vice versa:\n2|0/an}b∇acket∇i}ht⇋|1/an}b∇acket∇i}ht+|−1/an}b∇acket∇i}ht. As one of the most active topics\nin quantum gases, such spin-exchang dynamics was first\nstudied by Law et. al. [2] and has been observed in\nthe way of population oscillations of the Zeeman states\ninside87Rb condensates [5], where atoms interact ferro-\nmagnetically. A temporal modulation of spin exchange\ninteraction, which is tunable with optical Feshbach reso-\nnance, was recently proposed to localize the spin mixing\ndynamics in87Rb condensate [8].\nThe properties of a three-component ( F= 1) spinor\ncondensate are first studied by Ho [9] and Ohmi [10].\nFor a spin-1 system, atom-atom interaction takes the\nformV(r) =δ(r)(c0+c2F1·F2), where ris the dis-\ntance vector between two atoms, and c0,c2denote spin-\nindependent and spin-exchange interaction respectively.\nManypredictionsareverifiedexperimentally[11,12], and\nthe most fundamental property concerns the existence of\ntwo different phases determined by c2: the so-called po-\nlar (c2>0) and ferromagnetic ( c2<0) states, corre-\nsponding to the F= 1 state of23Na and87Rb atomic\ncondensates respectively. The fragmented condensate in\na uniform magnetic field can be turned into a single con-\ndensate state by a field gradient [13, 14]. More exotic\nground state phases both in the mean field level and the\nfully quantum many body theory have been extended\nto condensates with higher spins [15–17] and recently to\nspinor mixtures [18–20].\n∗Electronic address: ybzhang@sxu.edu.cnIn this paper we study the dependence of the spin dy-\nnamics on a small magnetic field gradient, which prac-\ntically provides a process to flip the spin between +1\nand−1 states thus turns a fragmented condensate into\na coherent one. We adopt the mean-field approximation,\nin which a spinor condensate is described by a multi-\ncomponentvectorfield. It hasprovidedsurprisinglygood\ndescriptions for most properties of the spinor condensate\nas evidenced by the experimental verification of many\npredictions [4]. As the spin flipping term is considered in\nthe condensate, the system magnetization exhibits obvi-\nous macroscopic oscillation similar to the Josephson os-\ncillation of a scalar condensate in a double well [21] and\nwefindthat the dynamicsofspin-0component ρ0maybe\ngreatly suppressed in the case of polar interaction. This\nprovides us an intriguing tool to manipulate the atomic\npopulation in spinor condensate.\nII. THE EFFECTIVE HAMILTONIAN FOR THE\nSYSTEM\nThe many-body Hamiltonian of Nspin-1 atoms of\nmassMin a uniform magnetic field reads\nH=N/summationdisplay\nk=1(P2\nk\n2M+Vtrap+γB0·Fk)+/summationdisplay\nk<lVint(rk−rl).(1)\nThe atoms are loaded into an optical trap VtrapandVint\ndenotes the collisional interaction between atoms. Pk\nandFkare the momentum operators and spin-1 opera-\ntors of the k-th atom, respectively. We consider a field\ngradientGas has already been applied in the MIT ex-\nperiment [11], and to be more specific we just replace B0\nin (1) with\nB(r) =B0rB=B0[ˆ z+G(xˆ x−zˆ z)].(2)\nWe choose the local field direction rBas the spin quan-\ntization axis [13] by performing a unitary transforma-\ntionU=N/producttext\nk=1e−i\n/planckover2pi1n(rk)·Fkon the Hamiltonian (1) where2\nn=ˆ z×rB=Gxˆ y. The contact interaction Vintis spin\nconserving, hence it is invariant under the transforma-\ntion. On the other hand, an additional Berry phase as-\nsociated with the local change of basis emerges in the\nmomentum term. Take the single particle hamiltonian as\nan example, h=P2\n2M+Vtrap+γB·F, andU=e−i\n/planckover2pi1GxFy,\nwe find that\nU†P2U= (Px−A)2+P2\ny+P2\nz\n=P2−2GFyPx+G2F2\ny (3)\nwhere the operator A=GFy. The Zeeman energy term\nis transformed into [22]\nU†B·FU=B0Fx[Gxcos(Gx)−(1−Gz)sin(Gx)]\n+B0Fz[Gxsin(Gx)+(1−Gz)cos(Gx)].\n(4)\nFor small field gradient G, we approximate the trigono-\nmetric functions up to order of O(G2) in accordancewith\nEq. (3), cos( Gx)≃1 +G2x2/2,sin(Gx)≃Gx, which\nleads us to\nU†B·FU=B0[Fz+CG+DG2+O(G3)] (5)\nwhere the operator C=−zFz,D=xzFx+x2Fz/2.\nThe many-body Hamiltonian is finally transformed\ninto\nHeff=N/summationdisplay\nk=1{P2\nk\n2M−G\nMFk\nyPk\nx+G2\n2M(Fk\ny)2\n+Vtrap+γB0Fk\nz+γB0GCk+γB0G2Dk}\n+/summationdisplay\nk<lVint(rk−rl) (6)\nThe atomic interaction takes the form [9, 10]\nVint(r) = (c0+c2F1·F2)δ(r)\nandc0= 4π/planckover2pi12(a0+ 2a2)/3M, c2= 4π/planckover2pi12(a2−a0)/3M.\nThe representation of the Hamiltonian in the second\nquantized form reads\nˆHeff=/integraldisplay\ndr{ˆΨ†\ni[P2\n2Mδij−G\nMPx(Fy)ij+G2\n2M(F2\ny)ij\n+Vtrap+p0(Fz)ij+p0GCij+p0G2Dij]ˆΨj\n+c0\n2ˆΨ†\niˆΨ†\njˆΨjˆΨi\n+c2\n2ˆΨ†\niˆΨ†\ni′Fij·Fi′j′ˆΨj′ˆΨj} (7)\nwhere repeated indices are to be summed over and\nˆΨ(r)(ˆΨ†(r)) is the field operator that annihilates (cre-\nates)anatominthe i-thhyperfinestateswith i= 1,0,−1\nat location r, andp0=γB0withγthe gyromagnetic ra-\ntio of the bosonic atoms.The dynamics of the condensate components reveal a\nrichcouplingbetweenthespin andspatialdegreesoffree-\ndom resulting in a variety of interesting phenomena, in-\ncluding spin mixing, spin domain formationand spin tex-\ntures [4]. The internal and external dynamics are both\nvery sensitive to the external magnetic fields and field\ngradients and they can be decoupled under certain con-\nditions, in particular, when the available spin dependent\ninteraction is insufficient to create spatial spin structures\nin the condensates. This occurs when the spin healing\nlength is larger than the size of the condensate, which al-\nlowsus to focus on the coherent spin mixing oscillationof\nthe spin populations. Practicallywechooseaproperfield\ngradientGto induce an energy in the same magnitude of\nspin interactionterm c2. The gradientwill flip the spin of\natoms in the condensate but keep the three components\nstill miscible and free of spin texture ( G <2cm−1as in\nthe MIT experiment [11]). As a result we can still safely\nadopt the single spatial mode approximation (SMA) in\nthe following [2–4, 7, 23].\nWe take\nˆΨ†\ni=√\nNφ(r)ˆa†\ni (8)\nwithφ(r) is defined by the Gross-Pitaevskii equation\nthrough the spin-independent part ˆH0\n/parenleftbigg\n−/planckover2pi12∇2\n2M+Vtrap+c0N|φ|2/parenrightbigg\nφ=ˆH0φ=µφ(9)\nwhereNis the total number of the atoms and µis the\nmean field energy or the chemical potential. The ˆ a†\ni(ˆai)\nis the spin component operator that annihilates (cre-\nates) an atom with spin i(i= 1,0,−1).Substitute the\nansatz (8) into the Hamiltonian (7) and neglect the spin-\nindependent part, our model then reads\nHeff=−ǫ(ˆa†\n1ˆa−1+ˆa†\n−1ˆa1)+ǫˆa†\n0ˆa0+c2ˆF2−pˆFz(10)\nwith\nˆF2=ˆF2\nz+(ˆF+ˆF−+ˆF−ˆF+)/2\nˆF±=√\n2(ˆa†\n±1ˆa0+ˆa†\n0ˆa∓1)\nˆFz= ˆa†\n1ˆa1−ˆa†\n−1ˆa−1. (11)\nThe parameter ǫ=G2/4Mcharacterizes the spin-\nflipping process induced by the field gradient, and\nthe spin interaction parameter is scaled as c′\n2=\n(c2/2)/integraltext\ndr|Φ(r)|4and we keep the original notation c2\nfor ease of representation. The term Cijvanishes due to\nthe fact that in ground state φis a symmetric function\n(/integraltext\ndrφ∗xφ= 0 and/integraltext\ndrφ∗zφ= 0,etc.), so does the term\nPx(Fy)ijbecause\n/integraldisplay\ndrφ∗Pxφ=M\ni/planckover2pi1/integraldisplay\ndrφ∗[x,H0]φ= 0.\nThe termDijamounts to a shift of the linear Zeeman\nenergyp=p0+ ˜pwith ˜p=p0G2/integraltext\ndrφ∗x2φ.3\nTheǫterm in Hamiltonian (10) denotes the process\nthat flips the spin from 1 to −1 or vise versa, and it\nplays the same role as the hopping term of scalarconden-\nsates in a double well. We want to emphasize that this\nterm in the form of ˆ a†\ni(F2\ny)ijˆajinduces the oscillation of\nthez-component magnetization mand goes against the\nquadratic Zeeman effect term ˆ a†\ni(F2\nz)ijˆajin Refs. [4, 7],\nwhich instead adheres to the constancy of the magneti-\nzationmand keep the system in the polar phase [11]. In\na uniform field, the z-component magnetization mis a\nconstant, and the system can be described by two canon-\nical conjugate variables: the population on spin-0 com-\nponentρ0and the relative phase θ=θ1+θ−1−2θ0.The\nsystem can be finally reduced to a nonrigid pendulum\nmodel [4]. Our model system conserves only the total\nnumber of atom and another relative phase θ′=θ1−θ−1\narises. We will mainly study the dynamics of the pop-\nulation on spin-0 component ρ0(t) =N0(t)/N, and the\nz-component magnetization m(t) = (N1(t)−N−1(t))/N.\nThe populations on ±components are related to ρ0and\nmthroughρ1+ρ−1+ρ0= 1 andm=ρ1−ρ−1.\nIII. THE SEMICLASSICAL MODEL\nInstead of considering a Hilbert space of the Fock\nstates|N1,N0,N−1/an}b∇acket∇i}ht, where the population dynamic can\nbe describe as\nρ0=/an}b∇acketle{tψi|eiˆHt//planckover2pi1ˆa†\n0ˆa0e−iˆHt//planckover2pi1|ψi/an}b∇acket∇i}ht/N\nwith|ψi/an}b∇acket∇i}htan initial state which can be taken as one of the\nbasis in the Hilbert space, here we consider the conden-\nsate to be in a coherent state, associated with a macro-\nscopic wave function with both magnitude and phase. In\nthe study of spin mixing dynamics [3, 7, 8], it is custom-\nary to replace the operator ˆ a†\ni(ˆai) bycnumbersa∗\ni(ai),\nand the coherent state is analogous to a classical field\nof complex amplitude with a definite phase in each spin\ncomponent\n|Φ/an}b∇acket∇i}ht=|a1,a0,a−1/an}b∇acket∇i}ht=/vextendsingle/vextendsingle/vextendsingle/radicalbig\nN1eiθ1,/radicalbig\nN0eiθ0,/radicalbig\nN−1eiθ−1/angbracketrightBig\nSemiclassical equations of motion can be derived from\nHamiltonian (10) as\ni/planckover2pi1˙a1= 2c2(˜Fza1+˜F−a0/√\n2)−ǫa−1−pa1\ni/planckover2pi1˙a0=√\n2c2(˜F+a1+˜F−a−1)+ǫa0 (12)\ni/planckover2pi1˙a−1= 2c2(−˜Fza−1+˜F+a0/√\n2)−ǫa+1+pa−1\nwhere the quantities ˜F±,˜Fzare c number counterparts\nof the operators ˆF±andˆFzrespectively. The neglected\nspin-independent part of the Hamiltonian (7) give each\nof the three equations in Eq. (12) a constant energy\nshiftµthat can be trivially eliminated by changing ai\ntoaie−iµt//planckover2pi1. For simplicity we rescale the time as t→\n|c2|t//planckover2pi1. The three coupled Gross-Pitaevskii equations/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s40/s97/s41/s32\n/s40/s98/s41\n/s32/s32\n/s40/s99/s41\n/s32/s32\n/s40/s100/s41/s32 /s32\n/s40/s101/s41/s40/s102/s41/s109/s109\n/s109\n/s116/s105/s109/s101\n/s32\nFIG. 1: (Color online) The dependence of the dynamics of\nρ0(t) (red solid) and m(t) (blue dashed) on the parameters\nofǫat fixed values of p= 0,c2= 1 (in units of |c2|), and\nǫ= 0(a),1.45(b),1.49,1.50(c),1.51 (d), 1.55 (e)and2 .25 (f).\nFig.1(c) shows the critical transition parameters of ρ0(t),m(t)\nwithǫ=1.50 (black dashed dot line). Time is in units of\n|c2|t//planckover2pi1.\n(12) for the interacting condensate amplitudes a∗\ni(ai)\ndescribe the dynamics in terms of the inter-component\nphase difference and population imbalance.\nIV. RESULTS AND DISCUSSION\nFirst, we consider an initial distribution with imbal-\nance between |+1/an}b∇acket∇i}htand|−1/an}b∇acket∇i}htcomponents, i.e. |Φ(0)/an}b∇acket∇i}ht=/vextendsingle/vextendsingle/vextendsingle0,/radicalbig\nN/2eiθ0,/radicalbig\nN/2eiθ−1/angbracketrightBig\nandillustratethedynamicsof\npopulation ρ0(t) and magnetization m(t) for bothc2>0\nandc2<0.In order to give prominence to the effect of\ntheǫterm, we consider p= 0 first, and set the phases\nθ0=θ−1= 0.This initial imbalance N1(0)−N−1(0) =\n−N/2 provides a ”Junction voltage” and the magnetiza-\ntion oscillation was induced by ǫ. In Fig. 1, we show\nthe solutions of equations (12) for c2>0 and illustrative\nparameters ǫ= 0,1.45,1.49,1.50,1.51,1.55 and 2.25, in\nthe unit of |c2|, respectively. We find that at the very4\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s40/s97/s41/s32\n/s32/s32\n/s40/s98/s41\n/s32/s32/s32\n/s40/s99/s41\n/s32/s32\n/s40/s100/s41\n/s32\n/s32/s32\n/s40/s101/s41 /s40/s102/s41/s109/s109/s109\n/s116/s105/s109/s101\n/s32\nFIG. 2: (Color online) The dependence of the dynamics of\nρ0(t) (red solid) and m(t) (blue dashed) on the parameters\nofǫat fixed values of p= 0,c2=−1 (in units of |c2|), and\nǫ= 0 (a),0.45 (b),0.49,0.50 (c),0.51 (d),0.85 (e), and 1 .50\n(f). Fig.2 (c) shows the critical transition parameters of m(t)\nwithǫ=0.50 (black dashed dot line). Time is in units of\n|c2|t//planckover2pi1.\nbeginning when ǫis small the magnetization oscillates\nwith small amplitude around an equilibrium above the\ninitial value of m(0) =−0.5 as in Fig.(1b), which is\nanalogous to the ”macroscopic quantum self-trapping”\neffect in double well system [21]. Meanwhile the dy-\nnamics ofρ0(t) experiences a crossover from sinusoidal\nto non-sinusoidal oscillation, with the population ρ0(t)\naveraged over time changing from less than the initial\nvalueρ0(0) = 0.5 to larger than it. As ǫincreases, there\nis a critical transition for ǫ= 1.50,black dashed line in\nFig.(1c), then the oscillation extends to the range be-\ntween−0.5 and 0.5. Accompanied by the arising of the\n”Josephson tunneling” [21] of the magnetization, the dy-\nnamics ofρ0has been sufficiently suppressed in Fig.(1f).\nThe coherent scattering of the internal Zeeman compo-\nnents 2|0/an}b∇acket∇i}ht⇋|1/an}b∇acket∇i}ht+|−1/an}b∇acket∇i}htwas suppressed by the ǫterm\nwith the process |1/an}b∇acket∇i}ht⇋|−1/an}b∇acket∇i}ht.The critical behavior de-\npends onǫ, as can be easily found from the energy con-\nservation and the extreme point for the minimization\nof the energy. Considering an arbitrary wave function/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s32/s32/s32\n/s40/s97/s41\n/s40/s98/s41\n/s32/s32\n/s32/s32\n/s40/s99/s41\n/s116/s105/s109/s101/s109/s109\n/s32 /s32/s32/s40/s100/s41\nFIG. 3: (Color online) The dependence of the dynamics of\nρ0(t) (red solid) and m(t) (blue dashed) on the parameters\nofpat fixed values of ǫ= 1.55,c2= 1 (in units of |c2|), and\np= 0 (a),0.2 (b),0.4 (c),0.6 (d). Time is in units of |c2|t//planckover2pi1.\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s40/s97/s41/s32\n/s40/s98/s41/s32\n/s32\n/s40/s99/s41/s40/s100/s41/s109/s109\n/s116/s105/s109/s101\n/s32\nFIG. 4: (Color online) The dependence of the dynamics of\nρ0(t) (red solid) and m(t) (blue dashed) on the parameters of\npat fixed values of ǫ= 0.55,c2=−1 (in units of |c2|), and\np= 0 (a),0.2 (b),0.4 (c),0.6 (d). Time is in units of |c2|t//planckover2pi1.\n|Φ/an}b∇acket∇i}ht=/vextendsingle/vextendsinglexeiθ1,yeiθ0,zeiθ−1/angbracketrightbig\n,the relative average energy\nof the system when p= 0 can be described as\nE=c2[2y2(x2+z2+2xzcosθ)+(x2−z2)2]\n−2ǫxzcosθ′+ǫy2(13)\nwithθ=θ++θ−1−2θ0,andθ′=θ+−θ−1.Forc2>0,the\ncritical point favors that y=/radicalbig\n2/3,x=z=/radicalbig\n1/6, θ=\nπ,θ′=π.Accordingto the energyconservationcondition\nEc=Einitial,we can get the critical value ǫc/c2= 1.50.5\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s109/s109/s109\n/s40/s98/s41/s109\n/s116/s105/s109/s101\n/s32/s32/s40/s97/s41\n/s32/s32/s32\n/s40/s100/s41\n/s32/s32/s40/s99/s41\n/s32/s32\nFIG. 5: (Color online) The dependence of the dy-\nnamics of ρ0(t) (red solid) and m(t) (blue dashed)\non the phases of the initial condition |Φ(0)/angbracketright=/vextendsingle/vextendsingle/vextendsingle/radicalbig\nN/3eiθ1,/radicalbig\nN/3eiθ0,/radicalbig\nN/3eiθ−1/angbracketrightBig\nwith parameters of\nc2= 1, ǫ= 0.55, p= 0.6 andθ1=θ0=θ−1= 0 for (a),\nc2= 1, ǫ= 0.55, p= 0.6 andθ1=θ−1=π/2, θ0=πfor (b),\nc2=−1, ǫ= 0.55, p= 0.6 andθ1=θ−1=θ0= 0 for (c),\nc2=−1, ǫ= 0.55, p= 0.6 andθ1=θ−1=π/2, θ0=πfor\n(d). Time is in units of |c2|t//planckover2pi1.\nForc2<0,we illustrate the dynamics for different pa-\nrameters in Fig.(2) with ǫ= 0,0.45,0.49,0.50,0.51,0.85\nand 1.50,in the unit of |c2|respectively. We find that\nthe oscillation of m(t) here is almost the same as in the\npolar interaction case c2>0. On the other hand, ρ0(t)\nis not suppressed at the beginning (Fig.(2b)), instead, it\nwas enhanced. When ǫ>0.5,and the oscillation of ρ0is\nstill active until ǫreaches 1 or even large value. This fer-\nromagneticfeature is quite different from the c2>0 case.\nThelattercaseshowssome”repulsive”effectbetweenthe\nZeeman components. The critical value ǫc/(−c2) = 0.5 is\nderived analytically through the equation (13) with the\nextreme point y= 0,x=z=/radicalbig\n1/2, θ=θ′= 0.\nNext, we consider a balanced initial distribution with\n|Φ(0)/an}b∇acket∇i}ht=/vextendsingle/vextendsingle/vextendsingle/radicalbig\nN/3eiθ1,/radicalbig\nN/3eiθ0,/radicalbig\nN/3eiθ−1/angbracketrightBig\nandθ1=\nθ−1=θ0= 0, where the ”Junction voltage” between\nthe|1/an}b∇acket∇i}htand|−1/an}b∇acket∇i}htcomponents vanishes. We consider theeffect of the parameter p,while we choose a fixed value\nofǫ= 1.55 (in the unit of |c2|) forc2>0 andǫ=0.55 for\nc2<0 as shown in Fig.(3) and Fig.(4).\nWe find that the pterm actsasaswitch forthe dynam-\nics of the magnetization m(t),and the cases for c2>0\nandc2<0 are quite different. For c2>0 in Fig.(3),\nthe fast oscillation of m(t) is modulated by a beat fre-\nquency, and,whentheamplitudeoftheenvelopefunction\nreaches its maximum the population of ρ0is completely\nsuppressed to zero. For the c2<0 case, the oscillation\nofm(t) is also induced by p, but the magnetization is\nalways positive. The dynamics of ρ0shows completely\nanharmonic behavior with the amplitude first enhanced\nthen reduced as pincreases, and no beat frequency mod-\nulation of the m(t) occurs.\nHowever, specific features of quantum nature cannot\nbeaddressedsatisfactorilywithinamean-fieldtreatment.\nIn an early experiment [5] on an F= 187Rb condensate,\natoms all prepared initially in the state |0,N,0/an}b∇acket∇i}htare ob-\nserved to exhibit a damped oscillation accompanied by\nlarge fluctuations during the spin-mixing evolution. The\ndynamics of a polar initial state |0,N,0/an}b∇acket∇i}htis trivial, i.e.\nthepopulation ρ0remainsaconstant,within amean-field\ntreatment, but many-body quantum dynamics shows in-\nteresting damped oscillation [2]. The presence of a field\ngradient will enlarge the Hilbert space in quantum treat-\nment due to the failure of the conservation of magneti-\nzationmand the related calculation on this feature will\nbe published elsewhere.\nFinally, let’s consider the effect of the phase difference\nbetweenthethreecomponents. Fig.(5)showsthedynam-\nics in the presence of an initial phase difference for both\nc2>0 andc2<0 cases. We find that the influence of\nthe phase difference for both cases are obvious. The beat\nfrequency oscillation of the magnetization remains in the\ncase ofc2>0 with a shift of the envelop center, but for\nthec2<0 the phase difference changes the amplitude of\nm(t) which extends down to the negative part of the axis\nand the dynamics of ρ0is also altered drastically.\nV. CONCLUSION\nThe dynamics of spin-1 BEC in a nonuniform mag-\nnetic field is studied with the emergence of an additional\nspin-flipping term induced by the field gradient, which\nhas an effect to reverse the spin from +1 to −1 and\nvise versa. Due to this spin flipping process the system\nmagnetization m(t) is not a constant any more, instead,\nit shows characteristic oscillation identical to that of a\nscalar BEC in double well. Meanwhile, the dynamics of\nρ0was greatly altered. We present the dynamics of ρ0(t)\nandm(t) under different initial conditions and the effect\nofphase difference is alsoshown. We find that the results\nfor the polar ( c2>0) and ferromagnetic ( c2<0) cases\nare quite different. The small magnetic field gradient is\nchosen properly to give rise to the flipping of the spin\nbetween +1 and −1 but still keep the three components6\nmiscible. These results highlight the possibility to ma-\nnipulate the coherent dynamics of the spinor condensate\nwith a field gradient, which is accessible to the current\nexperimental techniques.\nThis work is supported by the NSF of China un-der Grant No. 11074153, the National Basic Research\nProgram of China (973 Program) under Grant No.\n2011CB921601, the NSF of Shanxi Province, Shanxi\nScholarship Council of China, and the Program for New\nCentury Excellent Talents in University (NCET).\n[1] D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur,\nS. Inouye, H.-J. Miesner, J. Stenger, and W. Ketterle,\nPhys. Rev. Lett. 80, 2027 (1998).\n[2] C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. Lett.\n81, 5257 (1998).\n[3] H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N.\nP. Bigelow, Phys. Rev. A 60, 1463 (1999); H. Pu, S.\nRaghavan, and N. P. Bigelow, ibid.61, 023602 (2000).\n[4] M.-S. Chang, Q. S. Qin, W. X. Zhang, L. You, and M. S.\nChapman, Nat. Phys. 1, 111 (2005); W.X. Zhang, D. 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Mueller, T.-L. Ho, M. Ueda, and G. Baym, Phys.\nRev. A74, 033612 (2006).\n[15] C. V. Ciobanu, S.-K. Yip, and T.-L. Ho, Phys. Rev. A\n61, 033607 (2000).\n[16] M. Koashi and M. Ueda, Phys. Rev. Lett. 84, 1066\n(2000); M. UedaandM. Koashi, Phys.Rev.A 65, 063602\n(2002).\n[17] R. B. Diener and T.-L. Ho, Phys. Rev. Lett. 96, 190405\n(2006); L. Santos and T. Pfau, ibid96, 190404 (2006); H.\nM¨ akel¨ a and K.-A. Suominen, Phys. Rev. A 75, 033610\n(2007).\n[18] Z. F. Xu, Y. Zhang, and L. You, Phys. Rev.A 79, 023613\n(2009).\n[19] Z. F. Xu, J. Zhang, Y. Zhang, and L. You, Phys. Rev. A\n81, 033603 (2010).\n[20] J. Zhang, Z. F. Xu, L. You and Y. Zhang, Phys. Rev. A\n82, 013625 (2010).\n[21] A. Smerzi, S. Fantoni, S. Giovanazzi and S.-R. Shenoy,\nPhys. Rev. Lett. 79, 4950 (1997).\n[22] T.-L. Ho and V. B. Shenoy, Phys. Rev. Lett. 77, 2595\n(1996).\n[23] S. Yi, ¨O. E. M¨ ustecaplioglu, C. P. Sun, and L. You, Phys.\nRev. A66, 011601(R) (2002)."    },    {        "title": "1010.4366v4.Nonequilibrium_phase_transition_in_the_kinetic_Ising_model_driven_by_propagating_magnetic_field_wave.pdf",        "content": "arXiv:1010.4366v4  [cond-mat.stat-mech]  6 Aug 2011Nonequilibrium phase transition in the kinetic Ising\nmodel driven by propagating magnetic field wave\nMuktish Acharyya\nDepartment of Physics, Presidency University,\n86/1 College Street, Calcutta-700073, India.\nE-mail: muktish.acharyya@gmail.com\nThe two dimensional ferromagnetic Ising model in the presence of a propagating magnetic\nfield wave (with well defined frequency and wavelength) is studied by Mone Carlo simu-\nlation.This study differs from all of the earlier studies done so far, where the oscillating\nmagnetic field was considered to be uniform in space. The time average magnetisation\nover a full cycle (the time period) of the propagating magnetic field a cts as the dynamic\norder parameter. The dynamical phase transition is observed. Th e temperature varia-\ntion of the dynamic order parameter, the mean square deviation of the dynamic order\nparameter, the dynamic specific heat and the derivative of the dyn amic order parameter\nare studied. The mean square deviation of the dynamic order param eter, dynamic spe-\ncific heat show sharp maxima near the transition point. The derivativ e of dynamic order\nparameter shows sharp minimum near the transition point. The tran sition temperature\nis found to depend also on the speed of propagation of the magnetic field wave.\nKeywords: Ising model, Monte Carlo simulation, Dynamic transition, Propagating wave\nPACS Nos. 05.50.+q, 75.78.-n, 75.10.Hk, 42.15.Dp\n11 Introduction:\nThe nonequilibrium response of Ising ferromagnet in the presence o f time varying mag-\nnetic field is widely studied [1]. Among all these dynamical responses (e .g., hysteretic\nresponse, dynamic phase transition, stochastic resonance etc.) , the nonequilibrium dy-\nnamical phase transition is an important phenomenon[1] and became an interesting field\nof research recently. These dynamic phase transition has severa l similarities with that\nobserved in the case of equilibrium thermodynamic phase transition. The effort in study-\ning the invariance of time scale (i.e., critical slowing down) [2], the diverg ence of specific\nheat [2], divergence of critical fluctuations in energy [3], divergence of length scale near\nthe transition point [4], the order of the transition [5] established th e dynamic transition\nas an interesting nonequilibrium phase transition. This dynamic trans ition is very closely\nrelated to the hysteretic loss[6] and the stochastic resonance[7]. Experimentally the exis-\ntence of dynamic transition was found [8] in Co film on Cu surface (at r oom temperature)\nby surface magneto optic Kerr effect. Recently, the evidence of d ynamic phase transition\nwas found expreimentally [9], in [Co(4 ˚A)Pt(7˚A)]3multilayer system with strong perpen-\ndicular anisotropy by applying a time varying (sawtooth type) out-o f-plane magnetic field\nin the presence of small additional constant magnetic field. In this s tudy, the dynamic\nphase boundary was drawn and found similar to that obtained from t he simulation in\nkinetic Ising model with analogous condition.\nThe dynamic phase transition is also observed[10, 11, 12, 13] in othe r ferromagnetic\nmodels. It is studied[14] in the Ginzburg-Landau model of anisotrop ic XY ferromagnet\nand different types of chaotic behaviour is observed. Recently, th e multiple dynamic tran-\nsitions is observed[15, 16] in anisotropic Heisenberg model. These st udies are reviewed[17]\nrecently.\nHowever, all these studies, done so far for the dynamic phase tra nsition are made\nwith time varying magnetic field which was uniform over the space. No a ttempt has\nbeen made to study the dynamic phase transition with magnetic field d epending on both\nspace and time. In this article, the dynamic phase transition is studie d, by Monte Carlo\nsimulation[18], in Ising ferromagnet in the presence of a propagating magnetic field wave.\nThis article is organised as follows: the next section is devoted to des cribe the model\nand the Monte Carlo simulation method. The simulation results are rep orted in section\n-3 and the article ends with a summary in section -4.\n2 Model and Simulation:\nThe Hamiltonian, of an Ising model (with ferromagnetic nearest neig hbor interaction)\ndefined in two dimensions (square lattice) in the presence of a propa gating magnetic field\nwave, can be represented as\nH=−J/summationdisplay\n<ij>sisj−/summationdisplay\nih(/vector r,t)si. (1)\n2Here,si(=±1) is the Ising spin variable, J(>0) is the ferromagnetic interaction strength\nandh(/vector r,t)isthevalueofthepropagatingmagneticfieldwave atanytime tandatposition\n/vector r. Here, the propagating magnetic field ( h(/vector r,t)) wave is represented as\nh(/vector r,t) =h0cos(ωt−Ky) (2)\nwhereh0is the amplitude and ω(= 2πf) is the angular frequency of the oscillating field\nandK(= 2π/λ) is the wave vector. Here, fis the frequency and λis the wavelength\n(measured in the unit of lattice spacing) of the propagating magnet ic field wave. The\nwavelength, considered here, is smaller than and commensurate wit h the lattice size ( L).\nHere, the direction of propagation of the magnetic field wave ( h(y,t)) is taken along\ntheydirection only. It may be noted here, that all earlier studies of the dynamica l phase\ntransitions are done with oscillating (in time) but uniform (over the space) magnetic field.\nThe boundary condition is taken periodic in all directions. This complet es the description\nof the model.\nIn the simulation, the system is cooled gradually from a high temperat ure. Randomly\nselected 50% up ( si= +1) spins, is taken as the initial configuration. Physically, this\ncorresponds to the high temperature configuration of spins. In t he cooling process, the\nlast spin configuration corresponding to a particular temperature was used as the initial\nconfiguration of next lower temperature. At any finite temperatu reT, the dynamics of\nthis system has been studied here by Monte Carlo simulation using Met ropolis single\nspin-flip rate [18]. The transition rate is specified as\nW(si→ −si) = Min[1 ,exp(−∆H/kBT)] (3)\nwhere ∆His the change in energy due to spin flip ( si→ −si) andkBis the Boltzmann\nconstant. Any lattice site is chosen randomly and the spin variable ( si) is updated accord-\ning to the Metropolis spin flip probability. L2such updates constitute the unit (Monte\nCarlo step per spin or MCSS) of time here. The instantaneous bulk ma gnetisation (per\nsite),m(t) = (1/L2)/summationtext\nisihas been calculated. The time averaged (over the complete\ncycle of the propagating magnetic field wave) magnetisation,\nQ=1\nτ/contintegraldisplay\nm(t)dt, (4)\ndefines the dynamic order parameter[1]. The frequency is f= 0.01 (kept fixed throughout\nthe study). So, one complete cycle of the propagating field takes 1 00 MCSS (time period\nτ=1\nf= 100 MCSS). A time series of magnetisation m(t) has been generated up to 2 ×105\nMCSS. This time series contains 2 ×103(sinceτ= 100 MCSS) number of cycles of the\noscillating field. Here, first 103numbers of such transient values are discarded to get\nthe stable values of the dynamical quantities. The dynamic order pa rameterQhas been\ncalculated over 103values. It is checked (for a few data) that these number of sample s\n(Ns) is sufficient to get the stable values of the dynamical quantities. So , the statistics\n(distributionof Q)isbasedon Ns= 103differentvaluesof Q. Tohavetheconfidence(with\nthesenumberofsamples), themeansquaredeviation(i.e., <(δQ)2>=< Q2>−< Q >2)\n3ofQis also calculated and studied as a function of temperature. It may b e noted here,\nthat values of the dynamic order parameter (at lower temperatur es) become both positive\nand negative with equal probability. Here, only the positive values of Qare shown. The\nstatistical error (∆) in calculating Qmay be defined as the square root of <(δQ)2>. The\nmaximum error (∆ max) occurs near the transition point and this reasonably indicates the\ncritical fluctuations.\nThe time average dynamic energy is defined as\nE=1\nτ/contintegraldisplay\nHdt. (5)\nThe dynamic specific heat ( C=dE\ndT) is also calculated. The temperature variations of all\nthese (above mentioned) quantities are studied.\nHere, the temperature Tis measures in the unit of J/kB, the field amplitude h0and\nenergyEare measured in the unit of J.\n3 Results:\nTo investigate the nature of the spatio-temporal variations of fie ldh(y,t) and the local\n’strip magnetisation’, m(y,t) (=/integraltexts(x,y)\nLdx, wheres(x,y) =±1 is the spin variable at\nposition (x,y)), are studied as a function of coordinate y(along the direction of propa-\ngation of field wave) for different times ( t). Fig-1, shows such plots. From the figure,\nthe propagating nature of the field wave and the ’strip magnetisatio n’ is clear. Here,\nit may be noted that, for a particular instant of time, the magnetic fi eld and the ’strip\nmagnetisation’ differ by a phase. It is observed that, this phase diff erence depends on\ntemperature of the system, wavelength and the frequency of th e propagating magnetic\nfield wave. The systematic study of this dependence requires lot of computational effort\nand time.\nThe temperature variation of the dynamic order parameter is stud ied. This is shown\nin Fig-2. For the fixed values of the amplitude, frequency and the wa velength of the\npropagating magnetic field wave, it is observed that below a certain t emperature the\ndynamic ordering develops ( Q/negationslash= 0) and vanishes ( Q= 0) above it. Keeping the values\nof frequency ( f) and the wavelength ( λ), of the propagating magnetic field wave, fixed,\nif the amplitude ( h0) of the field increases the dynamic phase transition occurs at lower\ntemperature. For comparison, a similar studies are done for nonpr opagating (sinusoidally\noscillating in time but uniform over the space) magnetic field with same f requency and\namplitude. This clearly indicates that the dynamic transition occurs a t different higher\ntemperatures than observed in the case of a propagating field.\nThese dynamic transition temperatures can be estimated by study ing the temperature\nvariations of the mean square fluctuations ( < δQ2>) of the dynamic order parameter\nQ. These results are shown in Fig-3. Here, the < δQ2>shows very sharp maximum,\nindicating the dynamic transition temperature. From this one can es timate the maximum\nerror (∆ max) involved in statistical calculation for the dynamic order parameter Q. For\n4propagating magnetic field wave of f= 0.01 andλ= 25, the dynamic phase transitions\n(indicated by the maxima of < δQ2>) occur at T= 1.50 andT= 1.88 for the field\namplitudes h0= 0.5 andh0= 0.3 respectively. Here also, for comparison, the similar\nstudies are done in the case of nonpropagating magnetic field. Here , forf= 0.01 the\ndynamic phase transitions occur at T= 1.68 andT= 1.94 forh0= 0.5 andh0= 0.3\nrespectively.\nThe derivative (dQ\ndT) of the dynamic order parameter Qis calculated by central differ-\nence formula[19]\ndQ\ndT=Q(T+∆T)−Q(T−∆T)\n2∆T. (6)\nIn the simulation, the system was being cooled from a high temperatu re (random spin\nconfiguration)toacertaintemperatureslowlyinthestep∆ T= 0.02. Itmaybenotedhere\nthat the error in calculating thederivative numerically by this centra l difference formula is\nO((∆T)2)[19]. So, the error involved is of the order of 0.0004. The temperat ure variation\nof the derivative of the dynamic order parameter is studied and the results are shown\nin Fig-4. Here, the derivative shows very sharp minimum, indicating th e dynamic phase\ntransition temperature. For propagating magnetic field wave of f= 0.01 andλ= 25, the\ndynamic phase transitions (indicated by very sharp minima ofdQ\ndT) are observed to occur\natT= 1.50 andT= 1.88 for the field amplitudes h0= 0.5 andh0= 0.3 respectively. For\na comparison, the similar studies are done in the case of nonpropaga ting magnetic field.\nHere, for f= 0.01 the dynamic phase transitions occur at T= 1.68 andT= 1.94 for\nh0= 0.5 andh0= 0.3 respectively.\nThedynamic specific heat( C)iscalculatedfromthederivative(dE\ndT)ofdynamicenergy\n(E). Here also, the derivative is calculated by using central difference formula (described\nabove). The results are shown in Fig-5. The specific heat becomes m aximum near the\ndynamic transition point indicating the dynamic transition independen tly. Here, the term\nindependently means the following: Here, the dynamic phase transition is studied an d the\ntransition temperature is estimated from two types of quantities. One is dynamic order\nparameter Qand its derivatives (dQ\ndT), moments ( < δQ2>) etc. These depend directly\nonQ. Another quantity is dynamic specific heat ( C=dE\ndT), which is not directly related\ntoQ. For propagating magnetic field wave of f= 0.01 andλ= 25, the dynamic phase\ntransitions (indicated by the maxima of C=dE\ndT) occur at T= 1.50 andT= 1.88 for the\nfield amplitudes h0= 0.5 andh0= 0.3 respectively. Here also, for comparison, the similar\nstudies are done in the case of nonpropagating magnetic field. For f= 0.01 the dynamic\nphase transitions occur at T= 1.68 andT= 1.94 forh0= 0.5 andh0= 0.3 respectively.\nThe dependence of the dynamic phase transition, on the speed of p ropagation of the\npropagatingmagneticfield, isstudiedbriefly. Here, for f= 0.01,h0= 0.5thetemperature\nvariations of the dynamic order parameters for λ= 25 and λ= 50 are studied. The\nresults are shown in Fig-6. It is observed that the dynamic transitio n occurs at higher\ntemperature for higher speed ( v=fλ) of propagation of the propagating magnetic field.\nThe dynamical transition temperature Tcis measured here for a system of linear size\nL= 100. The systematic finite size analysis is not yet done. However, f ew results are\n5checked for smaller (say L= 50) system sizes. No appreciable change in Tcwas observed.\n4 Summary:\nThedynamical responseoftwodimensional Ising ferromagnetinpr esence ofapropagating\nmagneticfieldwave isstudied byMonte Carlosimulation. Adynamical ph asetransitionis\nobserved. Thisdynamical phasetransitionisobserved fromthest udies ofthetemperature\nvariationsofthe dynamic order parameter, thederivative ofthe d ynamic order parameter,\nthe mean square deviation of the dynamic order parameter and the dynamic specific heat.\nAll these studies indicate the dynamic phase transition and the tran sition temperatures\nare estimated.\nFor comparison the dynamic transition is also studied for a nonpropa gating (sinu-\nsoidally oscillating in time but uniform over space) magnetic field. It is ob served that the\ndynamic transition temperatures are different from that observe d in the case of propagat-\ning magnetic field wave. It is observed, from figures 3, 4 and 5 that t he propagating field\nwave causes the dynamical phase transition at lower temperature than that obtained from\na non-propagating field of same amplitude. One may argue that since the propagating\nmagnetic field makes, the strip magnetisation, a wave-like structur e, the value of Qwill\nbe less than that for a non-propagating field of same amplitude and f requency at the same\ntemperature. This would govern the transition to take place at lowe r temperature.\nHere, the dependence of the transition temperature on the spee d of propagation of the\npropagating magnetic field wave is studied briefly and it is observed th at the transition\ntakes place at higher temperature for the higher value of the spee d of propagation. One\nmay try to understand this fact in the following way: the increasing w avelength (or speed\nfor a fixed frequency) simply makes the field more nearly homogeneo us, approaching\nthe infinite wavelength spatially homogeneous limit. It does appear th at the transition\nfor propagating field (with h0= 0.5) has shifted from T= 1.50 to that obtained for\napproximately spatially homogeneous case, i.e., T= 1.68.\nThe present observations, based on the Monte Carlo simulation, ar e reported here\nbriefly. The dynamical phase boundary for propagating magnetic fi eld wave is yet to be\nsketched and the dependence of the phase boundary on the freq uency and wavelength\nof the propagating wave has to be determined. The finite size analys is and the detailed\nstudy of the behaviour of phase difference between propagating m agnetic field wave and\n’strip magnetisation’ have to be done. It requires lot of computatio nal efforts and will be\nreported later. The nonequilibrium dynamic phase transition in Ising f erromagnet, in the\npresence of propagating magnetic field wave, will become challenging in near future.\nAcknowledgements: The library facilities provided by Calcutta University is gratefully\nacknowledged.\n6References:\n1. B. K. Chakrabarti and M. Acharyya, Rev. Mod. Phys. ,71847 (1999)\n2. M. Acharyya, Phys. Rev. E ,562407 (1997)\n3. M. Acharyya, Phys. Rev. E ,561234 (1997)\n4. S. W. Sides, P. A. Rikvold, M. A. Novotny, Phys. Rev. Lett. ,81834 (1998)\n5. G. Korniss, P. A. Rikvold and M. A. Novotny, Phys. Rev. E ,66056127 (2002)\n6. M. Acharyya, Phys. Rev. E ,58179 (1998)\n7. M. Acharyya, Phys. Rev. E ,61218 (1999)\n8. Q. Jiang, H. N. Yang and G. C. Wang, Phys. Rev. B ,5214911 (1995).\n9. D. T. Robb, Y. H. Xu, O. Hellwig, J. McCord, A. Berger, M. A. Novo tny, P. A.\nRikvold, Phy. Rev. B ,78134422 (2008)\n10. M. Keskin, O. Canko and U. Temizer, Phys. Rev. E ,72036125 (2005)\n11. O. Canko et al, Physica A ,38824 (2009)\n12. U. 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Scarborough, Numerical Mathematical Analysis , Oxford and\nIBH (1930)\n7-1-0.500.51\n102030405060708090100(a)❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜\n✉\n✉✉✉✉ ✉✉ ✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉ ✉✉✉\n✉✉✉✉ ✉✉ ✉✉ ✉ ✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉\n✉\n✉✉\n✉✉✉✉✉ ✉ ✉✉✉ ✉✉✉✉✉✉✉✉ ✉✉✉✉ ✉✉✉✉✉✉ ✉✉✉ ✉ ✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉\n-1-0.500.51\n102030405060708090100(b)\n❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜ ✉✉✉✉✉✉✉✉✉✉✉ ✉ ✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉\n✉✉\n✉ ✉✉✉✉ ✉✉ ✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉\n✉\n✉✉✉ ✉✉✉✉ ✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉\n-1-0.500.51\n102030405060708090100(c)\n❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜✉✉✉✉✉✉ ✉✉✉ ✉✉✉\n✉✉✉✉✉✉✉✉ ✉✉✉ ✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉\n✉✉✉✉✉✉✉ ✉ ✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉\n✉\n✉✉ ✉✉ ✉✉ ✉✉✉ ✉✉ ✉ ✉✉✉✉✉ ✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉ ✉ ✉ ✉✉✉\n-1-0.500.51\n102030405060708090100\ny(d)\n❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜\n✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉ ✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉\n✉✉ ✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉\n✉✉\n✉✉✉✉ ✉ ✉✉✉✉ ✉ ✉✉✉✉✉✉✉✉✉✉✉ ✉ ✉ ✉✉✉✉✉✉ ✉\nFig-1.The spatio-temporal variations of propagating magnetic field wave (h(y,t)) and\n’strip magnetisation’ ( m(y,t)) forh0= 0.5,T= 1.50 andλ= 25. The magnetic field and\nmagnetisation are represented by open circles and bullets respect ively. Continuous lines\njoining the data points act as guide to the eye. The plots for differen t times (t) are shown\nas follows: (a) t= 100001 MCSS, (b) t= 100025 MCSS, (c) t= 100050 MCSS and (d)\nt= 100075MCSS.\n800.20.40.60.81\n1 1.5 2 2.5 3Q\nT❞❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞❞ ❞ ❞ ❞ ❞ ❞❞ ❞ ❞ ❞❞❞ ❞ ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ ❞ ❞\n✸ ✸ ✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸✸✸✸✸✸ ✸ ✸ ✸ ✸ ✸✸ ✸✸ ✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸ ✸✸✸ ✸ ✸\n✷✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷✷ ✷ ✷ ✷ ✷ ✷ ✷✷✷✷ ✷ ✷ ✷ ✷✷ ✷✷ ✷ ✷✷✷ ✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷\n✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉✉✉✉ ✉ ✉ ✉ ✉ ✉ ✉✉ ✉ ✉ ✉ ✉✉ ✉ ✉ ✉✉ ✉ ✉ ✉ ✉✉ ✉✉ ✉ ✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉ ✉✉ ✉✉\nFig.2.The temperature ( T) variations of dynamic order parameter Qfor different types\nof fields. (o) for propagating wave field with h0= 0.5,λ= 25 (and ∆ max= 0.216), (•) for\npropagating wave field with h0= 0.3,λ= 25 (and ∆ max= 0.197) (⋄) for non-propagating\nfield with h0= 0.5 (and ∆ max= 0.167), (✷) for non-propagating field with h0= 0.3 and\n(∆max= 0.200). Here, the frequency f= 0.01 for both type of fields. Continuous lines\njust join the data points.\n900.010.020.030.040.05\n1 1.5 2 2.5 3< δQ2>\nT❞❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞❞❞❞❞❞❞❞❞❞❞❞\n❞\n❞❞❞❞❞❞❞ ❞❞ ❞ ❞ ❞ ❞ ❞ ❞ ❞ ✸ ✸ ✸ ✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸\n✸\n✸\n✸✸✸✸✸✸✸ ✸✸✸✸ ✸ ✸✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✸ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷✷ ✷ ✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷\n✷✷\n✷ ✷✷✷✷✷✷✷✷ ✷ ✷ ✷✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉✉ ✉ ✉✉ ✉✉ ✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉\n✉\n✉✉✉✉✉✉✉✉ ✉ ✉✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉\nFig.3.The temperature ( T) variations of mean square deviation ( < δQ2>) of the\ndynamic order parameter ( Q) for different types of fields. (o) for propagating wave field\nwithh0= 0.5 andλ= 25, (•) for propagating wave field with h0= 0.3 andλ= 25, (⋄) for\nnon-propagating field with h0= 0.5, (✷) for non-propagating field with h0= 0.3. Here,\nthe frequency f= 0.01 for both type of fields. Continuous lines just join the data points .\n10-12-10-8-6-4-202\n1 1.5 2 2.5 3dQ\ndT\nT❞❞❞ ❞❞ ❞ ❞ ❞❞ ❞❞❞ ❞❞❞❞❞❞❞❞ ❞❞❞❞❞❞❞❞❞❞❞❞\n❞\n❞\n❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ ❞ ❞ ✸✸ ✸ ✸✸ ✸ ✸✸ ✸ ✸✸ ✸ ✸✸✸ ✸✸ ✸✸✸ ✸✸✸ ✸✸✸✸✸✸✸✸✸✸✸✸✸✸\n✸\n✸\n✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸ ✸✸ ✸✸ ✸ ✸ ✷✷ ✷ ✷✷ ✷✷ ✷ ✷ ✷✷ ✷ ✷✷✷✷✷✷✷ ✷✷ ✷ ✷✷✷ ✷✷✷✷ ✷✷ ✷ ✷✷✷ ✷✷✷✷✷✷✷✷✷✷✷✷\n✷✷\n✷\n✷✷\n✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷ ✷✷ ✷✷✉ ✉✉ ✉ ✉✉✉✉ ✉✉✉✉ ✉ ✉✉ ✉✉ ✉✉ ✉ ✉✉✉ ✉✉✉✉✉✉ ✉✉✉ ✉✉✉✉✉✉✉✉ ✉✉✉✉✉ ✉✉✉ ✉✉✉\n✉\n✉\n✉\n✉\n✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉ ✉✉ ✉✉ ✉ ✉ ✉✉ ✉ ✉ ✉ ✉\nFig.4.The temperature ( T) variations of the derivative (dQ\ndT) of the dynamic order pa-\nrameter ( Q) for different types of fields. (o) for propagating wave field with h0= 0.5 and\nλ= 25, (•) for propagating wave field with h0= 0.3 andλ= 25, (⋄) for non-propagating\nfield with h0= 0.5, (✷) for non-propagating field with h0= 0.3. Here, the frequency\nf= 0.01 for both type of fields. Continuous lines just join the data points . Here, the\nerror involved in calculating each data point is of the order of 0.0004.\n1100.511.522.5\n1 1.5 2 2.5 3C=dE\ndT\nT❞❞❞❞❞❞❞\n❞\n❞\n❞\n❞❞❞❞❞❞❞❞❞❞❞❞✸✸✸✸✸✸✸✸✸✸✸\n✸\n✸\n✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷\n✷\n✷✷\n✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉\n✉\n✉\n✉\n✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉\nFig.5.The temperature ( T) variations of the derivative (dE\ndT) of the dynamic energy i.e.,\ndynamic specific heat, for different types of fields. (o) for propag ating wave field with\nh0= 0.5 andλ= 25, (•) for propagating wave field with h0= 0.3 andλ= 25, (⋄) for\nnon-propagating field with h0= 0.5, (✷) for non-propagating field with h0= 0.3. Here,\nthe frequency f= 0.01 for both type of fields. Continuous lines just join the data points .\nHere, the error involved in calculating each data point is of the order of 0.0004.\n1200.20.40.60.81\n11.21.41.61.822.22.4Q\nT❡ ❡ ❡ ❡ ❡❡ ❡ ❡ ❡❡ ❡❡ ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ ❡❡❡ ❡ ❡❡ ❡ ❡❡ ❡ ❡ ❡ ❡\n✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷ ✷✷ ✷ ✷ ✷ ✷ ✷✷ ✷ ✷ ✷✷✷ ✷ ✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷ ✷ ✷\n△△△ △ △ △ △ △ △ △ △△△△△△ △ △ △ △ △△ △△ △△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△△ △△△ △ △\nFig.6.The temperature ( T) variation of dynamic order parameter ( Q) for propagating\nfields for two different velocities ( v=fλ). (✷) represents f= 0.01,λ= 25,h0= 0.5\n(and ∆ max= 0.216) and (o) represents f= 0.01,λ= 50,h0= 0.5 (and ∆ max= 0.188).\nFor comparison, QversusTis also plotted (represented by ( △)) for a non-propagating\noscillating magnetic field with f= 0.01,h0= 0.5 (and ∆ max= 0.167).\n13"    },    {        "title": "2207.06711v1.Attosecond_magnetization_dynamics_in_non_magnetic_materials_driven_by_intense_femtosecond_lasers.pdf",        "content": "1 \n Attosecond magnetization dynamics in non -magnetic materials driven by \nintense femtosecond lasers  \n \n \nOfer Neufeld1,*, Nicolas Tancogne -Dejean1, Umberto De Giovannini1,2, Hannes Hübener1, and Angel Rubio1,3,* \n1Max Planck Institute for the Structure and Dynamics of Matter and Center for Free -Electron Laser Science, \nHamburg, Germany, 22761.  \n2Università degli Studi di Palermo, Dipartimento di Fisica e Chimica —Emilio Segrè, Palermo I -90123, Italy.  \n3Center for Computational Quantum Physics (CCQ), The Flatiron Institute, New York, NY, USA, 10010.  \n*Corresponding author E -mails: ofer.neufeld@gmail.com , angel.rubio@mpsd.mpg.de  \n \nIrradiating solids with ultrashort laser pulses  is known to  initiate femtosecond  timescale  \nmagnetization  dynamics. However , sub-femtosecond spin dynamics have not yet been observed \nor predicted. Here, we explore ultrafast light -driven spin dynamics in a highly  non-resonant  \nstrong -field regime. Through state -of-the-art ab-initio  calculations, we predict that a non -\nmagnetic mater ial can be transiently transformed into a magnetic one via dynamical extremely \nnonlinear spin -flipping processes, which occur on attosecond timescales and are mediated by a \ncombination of multi -photon and spin -orbit interactions . These  are non -perturbative  non-resonant  \nanalogues to the inverse Faraday effect  that build up from cycle -to-cycle as electrons gain angular \nmomentum . Remarkably, we show  that even for linearly polarized driving , where one does not \nintuitively expect any magneti c response , the magnetization transient ly oscillat es as the system \ninteracts with light . This oscillating response  is enabled by transverse anomalous light -driven \ncurrents  in the solid, and typically occurs on timescales of ~500 attoseconds . We further \ndemonstrate  that the speed of magnetization can be controlled by tuning  the laser wavelength and \nintensity . An experimental set-up capable of measuring these dynamics through pump -probe \ntransient absorption spectroscopy  is outlined and simulated . Our results pave the way for new \nregimes of ultrafast manipulation of magnetism . \n \nI. INTROUCTION  \nMagnetism is one of the most fundamental physical phenomena in nature. It arises from internal spin degrees \nof freedom of quantum particles  [1,2] , and can create complex  spin textures such as skyrmions  [3–6] or other \nmagnetic and topological order ed phases  [7–9]. Over the past two d ecades immense efforts have been  \ndevoted to wards  the study of ultrafast magnetism, i.e. the  manipulation of materials’ magnetic structures on \nfemtosecond timescales with ultrashort laser pulses  [10–15]. However, despite many years of research \nnumerous open questions remain regarding the mechanis ms and pathways that control ultrafast magnetization \ndynamics  [10–19]. Part of the complexity arises because spin dynamics are often entangled with other \nprocesses and interactions (because spin is carried by charged particles that also interact with each other and \nwith other particles), and can also evolve over several orders of magnitudes of timescales ranging from \nfemtoseconds to nanoseconds. For instance, only recently the mechanism through which angular momentum \ntransfers from the spin order to the lattice during demagnetization was uncovered  [20]. \nIn a typical femto -magnetism experiment, an intense laser pulse  is irradiated  onto a magnetic material \nsuch as a ferromagnet, which initiates demagnetization dynamics, spin transfer  dynamics , or spin \nswitching  [21–30]. The ultrafast dynamics are subsequently tracked through time -resolved pump -probe \nspectroscopy  [31–34]. Recently it was shown that magnetization can even be  transiently  induced in non -\nmagnetic materials with resonant circularly polarized light  through  the inverse Faraday effect  [35], which is \na perturbative nonlinear optical effect  [36,37] , or through slowe r spin -phonon couplings  [38]. However, t o \nour knowledge  all experiments and theoretical works thus far have never observed or predicted the following: \n(i) An effect whereby a non -magnetic material is illuminated  by an intense non-resonant laser pulse t hat \ninitiates a few femtosecond  turn-on of magnetization . (ii) Magnetization dynamics driven in non -magnetic \nmaterials by linearly polarized light. (iii) Magnetization dynamics that occurs on sub -femtosecond timescales.  \nAll of t hese effects could pave the way to new regimes in ultrafast and non -equilibrium magnetism, e.g. \nallowing extremely fast manipulation of magnetic orders even in materials that have a  non-magnetic  ground -\nstate.  2 \n In parallel to advancements in femto -magnetism , strong -field physics in solids has developed  as a novel \napproach for controlling electron motion on sub -femtosecond  timescales  [39–45]. Strong -field interactions \nin solids enabled tailoring valley pseudo -spin occupations  [46,47] , controlling material topological \nproperties  [46], steering Dirac electrons  [48] and more  [49–52]. This regime provide s an ideal setting for \nexploring possibilities of attosecond magnetism, because it gives  natural access to attosecond electron motion  \n(whereby electrons  act as spin carrier s). To capitalize this, a strong spin -orbit interaction could  allow \nconverting electr onic angular momenta into magneti sm (because light does not directly couple to spin degrees \nof freedom). Extremely nonlinear light -matter interactions such as high harmonic generation (HHG)  have \nbeen explored in  some  material  systems  with strong spin -orbit  interactions  (e.g. in BiSbTeSe 2 [53], \nBi2Te3 [54], Bi2Se3 [55], Ca2RuO 4 [56], Na 3Bi [57]), but the induced magnetization  was not investigated . \nHere we report on f emto -magnetic phenomena that are driven in non -magnetic materials by intense \nultrashort laser pulses in the strong -field and highly nonlinear regime of light -matter interactions . We \ndemonstrate with state -of-the-art time -dependent spin density functional t heory calculations that strong \nmagnetization of ~0.1 µB (where µB is a Bohr magneton) can be turned -on extremely fast when driven by non-\nresonant circular ly polarized  light, within ~16 femtoseconds. This transient magnetic state is expected to live \nfor several tens of femtoseconds before it is destroyed by scattering and dephasing. We thoroughly analyze \nthis effect and show that the magnetization arises  from highly nonlinear mult i-photon processes, which \ntogether with spin -orbit interactions, allow for attosecond spin polarization  to build up over time. We also \nstudy systems irradiated by linearly polarized pulses, whereby one intuitively does not expect a magneti c \nresponse  (because there is no angular momentum in the driving pulses). Remarkably, we show that even \nlinearly polarized pulses , when sufficiently intense,  can induce magnetization dynamics, owing to an \ninterplay of electronic currents driven along the laser polarization axis  and transverse anomalous currents \nthat arise in some material systems (e.g . from  a nonzero Berry curvature  or other structural asymmet ry). \nTogether , these currents give rise to a sub -cycle electronic orbital angular momentum that is converted to \ntransient attosecond  magnetism. The spin expectation values can flip sign from a maximum of +0.01 µB to a \nminimum of -0.01µB in just ~411 attoseconds.  Strikingly,  the speed of magnetism can be tuned by changing \nthe laser parameters . We outline and simulate a circular dichroism attosecond transient absorption \nspectroscopy set -up that is capable of measuring these  unique phenomen a.  \nII. METHODOLOGY  \nWe begin by outlining our methodological approach. To model light -induced magnetization dynamics, we \nemploy ab-initio  calculations based on time -dependent spin density functional theory (TD SDFT) in the \nKohn -Sham (KS) formulation  [58]. The system’s gr ound -state is directly obtained within spin-polarized  \nDFT, and is then propagated in real -time with the following equations of motion (we use atomic units unless \nstated otherwise):  \n \n𝑖𝜕𝑡|𝜓𝑛,𝐤𝐾𝑆(𝑡)⟩=(1\n2(−𝑖𝛁+𝐀(𝑡)\n𝑐)2\n𝜎0+𝑣𝐾𝑆(𝑡))|𝜓𝑛,𝐤𝐾𝑆(𝑡)⟩ (1) \nwhere |𝜓𝑛,𝐤𝐾𝑆(𝑡)⟩ is the KS -Bloch state at k-point k and band index n, which is a Pauli  spinor:  \n \n|𝜓𝑛,𝐤𝐾𝑆(𝑡)⟩=[|𝜑𝑛,𝐤,↑𝐾𝑆(𝑡)⟩\n|𝜑𝑛,𝐤,↓𝐾𝑆(𝑡)⟩] (2) \nwith |𝜑𝑛,𝐤,𝛼𝐾𝑆(𝑡)⟩ the spin -up/spin-down part of the KS states  with spin index 𝛼. 𝜎0 in Eq. (1) is a 2 ×2 identity \nmatrix, and 𝐀(𝑡) is the vector potential of the impinging laser pulse within the dipole approximation such \nthat −𝜕𝑡𝐀(𝑡)=𝑐𝐄(𝑡), and c is the speed of light (in atomic units c≈137.036). 𝑣𝐾𝑆(𝑡) is the time -dependent \nKS potential given by:  3 \n  𝑣𝐾𝑆(𝑡)=∫𝑑3𝑟′𝑛(𝐫′,𝑡)\n|𝐫−𝐫′|𝜎0+ 𝑣𝑋𝐶[𝜌(𝐫,𝑡)]+𝑉𝑖𝑜𝑛  (3) \nwhere the first term in Eq. (3)  is the classical Hartree term – an electrostatic mean -field interaction between \nelectrons , where 𝑛(𝐫,𝑡)=∑ 𝑤𝐤|⟨𝐫|𝜑𝑛,𝐤,𝛼𝐾𝑆(𝑡)⟩|2\n𝑛,𝐤,𝛼  is the time -dependent electron density , with 𝑤𝐤 the k-point \nweights and the sum running over occupied bands . The s econd term in brackets, 𝑣𝑋𝐶, is the exchange -\ncorrelation (XC) potential that in the local spin density approximation is a functional of the spin density \nmatrix:  \n 𝜌(𝐫,𝑡)=1\n2𝑛(𝐫,𝑡)σ0+1\n2𝐦(𝐫,𝑡)∙𝛔 (4) \nwhere 𝐦(𝐫,𝑡) is the time -dependent magnetization vector:  \n 𝐦(𝐫,𝑡)=∑ 𝑤𝐤⟨𝜓𝑛,𝐤𝐾𝑆(𝑡)|𝐫⟩ 𝛔 ⟨𝐫 |𝜓𝑛,𝐤𝐾𝑆(𝑡)⟩\n𝑛,𝐤 (5) \n𝑉𝑖𝑜𝑛 in Eq. (3) represents the interactions of electrons with the lattice ions and core electrons. To reduce \nnumerical cost s we employ the frozen core approximation, and the bare Coulomb interaction between \nelectrons and ions is replaced with a fully -relativistic nonlocal norm -conserving pseudopotential  [59]. This \nterm also includes the full ab-initio  description of relativistic corrections to the Hamiltonian, including the \nmass term and the Darwin term. Most importantly, it incorporates a spin -orbit coupling term that is \nproportional to 𝐋∙𝐒, where 𝐋=(𝐿𝑥,𝐿𝑦,𝐿𝑧) is the angular momentum operator vector, and 𝐒=1\n2𝛔=\n1\n2(𝜎𝑥,𝜎𝑦,𝜎𝑧) is spin operator vector, with 𝜎𝑖 the 𝑖’th Pauli matrix. It is noteworthy that 𝑣𝑘𝑠 is non -diagonal \nin spin space due to the spin -orbit coupling term . \nThe interactions of electrons with the laser are described in the velocity gauge, where we employ the \nfollowing vector potential:  \n 𝐀(𝑡)=𝑓(𝑡)𝑐𝐸0\n𝜔sin(𝜔𝑡)𝐞̂ (6) \nwhere 𝑓(𝑡) is an envelope function (see  the Appendix  for details), 𝐸0 is the field amplitude, ω is the carrier \nfrequency, and 𝐞̂ is a unit vector that is generally elliptically polarized. Note that we neglect ion motion  and \nassume the frozen nuclei approximation ( i.e. omitting phononic excitations). This is expected to be a very \ngood approximation in attosecond to femtosecond timescales, especially for heavy atoms. The KS equations \nof motion are solved in a real -space grid representation with Octopus code  [60–62]. From the time -\npropagated KS states we calculate time -dependent observables of interest, including the total electronic \ncurrent, 𝐉(𝑡)=1\nΩ∫𝑑3𝑟 𝐣(𝐫,𝑡)Ω, where Ω is unit cell volume and 𝐣(𝐫,𝑡) is the microscopic time -dependent \ncurrent density:  \n \n𝐣(𝐫,𝑡)=∑ [𝜑𝑛,𝐤,𝛼𝐾𝑆∗(𝐫,𝑡)(1\n2(−𝑖𝛁+𝐀(𝑡)\n𝑐)+[𝑉𝑖𝑜𝑛,𝐫])𝜑𝑛,𝐤,𝛼𝐾𝑆(𝐫,𝑡)\n+𝑐.𝑐.]+𝐣𝑚(𝐫,𝑡)\n𝑛,𝐤,𝛼 (7) \n, where  𝐣𝑚(𝐫,𝑡) is the magnetization current density (which after spatial integration vanishes and does not \ncontribute to 𝐉(𝑡)). 𝐉(𝑡) is also used to obtain the HHG spectra , 𝐈(Ω)= |∫𝑑𝑡𝜕𝑡𝐉(𝑡)𝑒−𝑖𝜔𝑡|2. The spin \nexpectation values are calculated as 〈𝐒(𝑡)〉=⟨𝜓𝑛,𝐤𝐾𝑆(𝑡)|𝐒|𝜓𝑛,𝐤𝐾𝑆(𝑡)⟩, and are used to track the spin dynamics \nin the system. All other technical details about the numerical procedures are delegated to the Appendix . \nThis numerical approac h is employed in exemplary benchmark material s for exploring light -driven \nmagnetization dynamics. The main example used throughout the text is the two -dimensional (2D) topological \ninsulator, bismuthumane (BiH)  [63]. BiH is comprised of a monolayer of bismuth atoms arranged in a \nhoneycomb lattice (see illustration in Fig. 1(a)). The bismuth atoms are capped by hydrogen atoms that are 4 \n covalently bonded to the bismuth pz orbitals in a staggered configuration that preserves inversion symmetry, \nbut breaks some of the mirror planes of the honeycomb lattice. The electronic structure of BiH is strongly \naffected by spin -orbit coupling (SOC) – without SOC it exhibits Dirac cones in the K and K’ high symmetry \npoints, but SOC opens a large topological gap with nonzero Berry curvature throughout the Brillouin zone \n(see Fig. 1(b))  [63]. Each band is spin degenerate , and the ground -state is non -magnetic . BiH is an ideal \ncandidate for exploring light -driven magnetic phenomena bec ause the bismuth ions induce a relatively large \nSOC, and the system is an insulator which allows strong non -resonant nonlinear responses (the direct gap is \n1.35 eV  within the local spin density approximation ). As we will show below, all of our results are \nindependent of the topological insulator character of BiH.  \n \nFIG. 1. Ultrafast turn -on of magnetism in monolayer BiH.  (a) Illustration of the hexagonal BiH lattice  and schematic illustration of \nthe ultrafast turn -on of magnetism – an intense femtosecond laser pulse is irradiated onto the material, exciting electronic currents \nthat through spin -orbit interactions induce magnetization and spin flipping . (b) Band structure of BiH with and w/o SOC (red and \nblue bands indicat e occupied states  and unoccupied states, respectively) . In the SOC case, each band is spin degenerate. (c) Calculated  \nspin expectation value , <Sz(t)>, driven by circularly polarized pulses for several driving intensities  (for a wavelength of 3000nm) . \nThe x-component of the driving field is illustrated in arbitrary units to convey the different timescales in the dynamics.  \nIII. SUB -CYCLE  TURN -ON OF MAGNETIZATION  \nWe calculat e the electronic response of BiH to intense circularly -polarized laser pulses (polarized in the \nmonolayer xy plane) with a carrier wavelength of 3000nm, and intensities  in the ranges of 1011-1012 W/cm2 \n(see illustration in Fig. 1( a)). The corresponding carrier photon energy  of 0.41 eV  is well below the band gap, \nguaranteeing that the dominant light -matter response is non -resonant (at least four photons are required to \nexcite an electron from the valence to the conduction band). Thes e conditions result in HHG with harmonics \nup to  the ~30th order corresponding to photon  energies of ~12eV being emitted (see Appendix B). The \ncircularly polarized drive imparts angular momenta onto the electron ic system  through the light -matter \ncoupling term, and a combination of intraband acceleration and interband recollisions lead to the HHG \nemission  [43,64] . It is noteworthy that due to a 6 -fold improper -rotational symmetry in BiH, only harmonic \norders of 6𝑛±1 are emitted for integer 𝑛, which follows from dynamical symmetry selection rules  [65]. As \nwe will later show, similar selection rules can be derived for the total electronic excitation and the spin \nexpectation values, which play a significant role in the magnetization dynamics.  \nThe interesting question that  we now explore is whether the laser -driven electronic angular momenta  \ncan be converted to a net magnetization. While such conversion was recently shown in resonant \n           \n     \n              \n        a)\n b)  c)  b)  nergy  e ) ydrogen\n ismuth5 \n conditions  [35], it is not clear if it can be  obtained non -resonant ly. Moreover, it is unknown  whether  this can \nbe achieved in a system with  fully spin-degenerate bands  (precluding  spin-selectivity via optical transitions \nbetween  spin-split bands ). Figure 1( c) shows  the time dependent expectation value of spin along the z-axis, \n〈𝑆𝑧(𝑡)〉, for several laser intensities . The BiH s ystem is initially in a non -magnetic ground -state with \n〈𝐒(𝑡=0)〉=0, but shows an onset of magnetization about 12 femtoseconds after it starts interacting with \nthe laser. The characteristics of the magnetic response can be describ ed by  two main features : (i) sub -cycle \nfast oscillations of the spins, and (ii), a slower build -up of the magnetization that occurs over several laser \ncycles. By calculating the total occupation of electrons in just the up or down part of the spinor s we ve rify \nthat the induced magnetization  indeed results from spin -flipping processes (see Appendix B). In other words, \nduring the interaction with the laser spin-down electrons are flipped into a  spin-up state . By reversing the \nhelicity of the driv ing laser , one obtains the opposite picture with a conversion of up to down spins . We note \nthat these transitions cannot occur from a resonant optical transition because the photon energies are well \nbelow the gap .  \nThe very fast oscillations observed in Fig. 1( c) indicate that there are attosecond magnetization \ndynamics  involved . For circular ly polarized  driving these attosecond spin -flipping processes  accumulate  over \nthe laser cycle  on a timescale of about ten femtoseconds , yielding a net magnetization . Notably, the response \nstrongly depends on the driving power. Figure 1( c) shows that for stronger driving a larger net magnetization \ncan be obtained  (as high as ~0.1 𝜇𝐵). This result hints to the active mechanism at play: stronger drivi ng \nincreases the angular momentum of the excited electrons, which increases the contribution of the SOC term.  \nThis is also supported by calculations that show  that the induced magnetic response  diminishes with the \ndriving ellipticity (see Fig. 2(a) ). The process is thus analogous to the inverse Faraday effect, but in non -\nresonant and non -perturbative conditions.  Figure 2(b) presents the scaling of the magnetic response with \nintensity  – for weaker driving it follows a quartic dependence with the field ampl itude, indicating a 4 -photon \nresponse  (the minimal number  of photons  required  to excite an electron from the valence to the conduction \nband  in these conditions) , but above ~ 4×1011 W/cm2 this dependence breaks down  and behaves non -\nperturbatively . Interestin gly, we note that the induced magnetization saturates for ellipticities between 0.2 -\n0.4, i.e. the magnetic response stops increasing with ellipticity for that parameter range  (Fig. 2(a) ). This \nbehavior  differs from that of the inverse Faraday effect and reflects  the extreme nonlinearity  of the \nmagneti zation . \nThe onset time for the magnetization also strongly depends on the driving power and shows a non -\nperturbative nonlinear dependence (see Appendix B). Overall, this suggests  that the spin -flipping processes \nare directly driven by the electronic excitations to the conduction band  (because these are initiated by \ntunneling) , which subsequently undergo additional laser -induced acceleration in the bands that lead to spin -\nflipping. We further  support  this picture  with a k-space and band -resolved projection of the magnetization \ndensity, which validates  that the regions around the K and K’ points  minimal band gap positions) are the \ndominant contributors to the induced magneti zation , and that the first valence and conduction bands have the \nlargest contribution  (see Appendix ). We highlight that t hese observed dynamics differ from previously \nreported results in the perturbative resonant regime  – the non -resonant driving here does not directly excite a \nspin-selective optical transition , and spin-spilt bands do not play any role.   \nNext we analyze the elec tronic excitations that allow for the se spin flipping processes. Fig ures 2(c,d) \nplot the spectral components of the net magnetization of the system, i.e. the Fourier transform of 〈𝑆𝑧(𝑡)〉, vs. \nthe driving intensity and wavelength. The main observation from Fig. 2(c) and ( d) is that the spin excitation \nis inherently connected to the laser drive – remarkably, its spectral components are comprised of harmonics \nof the laser carrier frequency, and only 6𝑛 harmonics (for integer 𝑛) are allowed in the circular ly polarized  \ndriving case. The Appendix  presents results for elliptical driving which also support this conclusion, but \nwhere  2𝑛 (even ) harmonics are allowed. This result is important for several reasons: (i) it directly proves that \nthe laser -driven electron dynamics are converted to magnetization ( via SOC), because otherwise the spi n 6 \n would not evolve in temporal harmonics of the laser. (ii) It allows to tune the temporal behavior of the spin -\nflipping by changing the driving parameters . (iii) It establishes the main microscopic mechanism for the \nultrafast spin -flipping processes, which involves attosecond sub -cycle excitations of electrons from the \nvalence to the conduction band. The excited electrons are subsequently driven by the laser field that imparts \nangular momentum, which  is converted by the SOC term to spin flipping torques. In the Appendix  we show \nthat the total number of electrons excited to the conduction band over time evolves with the same exact  \nsymmetries as the spectral components of the magnetization (e.g. 6 -fold in the circular ly polarized  drivi ng \ncase), and also derive  the origin of this effect. Moreover, we confirm  that the onset time for the magnetization \ndynamics is inherently connected to the electronic excitation. Thus, the two processes of strong -field \ntunneling between the bands , and spin  flipping , are interlinked.  \nNotably, this mechanism follows the first steps in the HHG process in solids (i.e. tunneling of electrons \nto the conduction band, and subsequent acceleration in the bands)  [43,64] , but where spin and SOC play the \nadditional role in driving a magnetic response . Overall, the onset of ultrafast magnetization is enabled by t he \nfact that in the strong -field regime , electrons acquire relatively large momenta (i.e. large 〈𝐋〉), and that this \nhappens repeatedly every laser cycle. The fact that electrons are driven in an extremely nonlinear manner that \nis inherently sensitive to att osecond sub -cycle timescales allows possibilities of atto -magnetism.  Indeed, Fig. \n2 shows that the  electron’s  spin can oscillate with frequencies as high as the 42nd harmonic of the laser, which \npromises incredible potential for atto -magnetism  (that corres pond s to ~100 attosecond magnetization \ndynamics, provided  that the lower orders of the response can be suppressed ). In the Appendix  we validate \nthat the induced magnetism is not driven by  electronic correlations, which tend to slightly reduce the \nmagnitude of the phenomena (as expected due to enhanced scattering). We note that since  our simulations do \nnot include sufficient  dephasing and  relaxation  channels ( because of the use of semi -local approximations to \nthe XC functional, and because we do not incorporate electron -phonon couplings in the simulation ), the \nelectronic and spin excitations do not fully decay in our calculations . In reali stic experimental conditions , we \nexpect that these ma gnetic states will live for several tens of femtoseconds before decoher ing. 7 \n  \nFig. 2. Induced magnetic response for elliptical  driving  in BiH  for changing laser parameters . (a) Maximal induced magnetization vs. \nthe driving laser ellipticity, for driving power of 7×1011 W/cm2 and wavelength of 3000nm , where the elliptical major axis is \ntransverse to the Bi -Bi bonds.  (b) Same as (a) but for circularly polarized driving vs. laser power.  (c) Spectral components of < Sz(t)> \nvs. driving wavelength (calculated for a n intensity  of 5×1011 W/cm2, presented in log scale ). Dashed black lines indicate 6 n harmonics \nof the driving laser carrier frequency (for integer n), showing that the spin’s temporal evolution is driven by the laser , and that its \nsymmetries are connected to the light -driven electronic excitations . (d) Same as ( c) but for changing laser intensity  (for a  driving  \nwavelength of 3000nm) . Higher intensities  and longer wavelengths  are shown to lead to higher frequency components, indicating a \nfaster magnetic response.  \nIV. ATTOSECOND MAGNETISM  \nWe now  further analyze the very fast oscillations of magnetization  seen in Fig. 1( c). For circularly polarized \ndriving, they are overlayed on top of the dominant slow response that builds up the net magnetization from \ncycle to cycle.  That is, the dominant spin response is a zeroth order multi pole.  Consequently, it is very \ndifficult to measure these fast oscillations  experimentally, or to utilize them for applications. Still, the high \nenergy spectral components in 〈𝐒(𝑡)〉 appear be quite dominant compared to the weaker perturbative \nresponse , as lon g as there is a way to suppress the zeroth  order slow contribution  (see for instance Fig. 2(d)). \nIn order to try and extract this response we now explore linearly -polarized  driving , where the in -plane \npolarization axis is given by  the angle  𝜃, which is the offset angle from the x-axis ( that is transverse to the \nBi-Bi bonds) . Importantly, i n this case the cycle -averaged total angular momentum of the laser -matter system \nremains  zero.  In this respect , one expects no net magnetization to evolve, s uch that the zeroth  order multipole \nof 〈𝐒(𝑡)〉 should vanish.  At the same time, intuition would dictate  that no magnetization dynamics should  \noccur at all , since even if one considers timescales shorter than a laser cycle  the driving field  is linearly -\npolarized  and does not contain angular momenta . Nevertheless, Fig. 3(a) presents the temporal evolution of \n〈𝑆𝑧(𝑡)〉 for several driving intensities , which shows a strong magnetic response that rapidly oscillates in time \n(on attosecond timescales).  What is the origin of this  transient  magnetism?  \nFigure 3(b) presents the spectral components of the spin evolution vs. the in -plane laser polarization \nangle, which enables us to pinpoint the source of the effect. Clearly, the response follows fundamental  \nsymmetries of the material system – whenever  the laser is polarized along a plane of  BiH that exhibits a \nmirror or a 2 -fold rotational symmetry the magnetic response fully vanishes. At this stage we recall that for \nthese driving conditions, electrons are accelerated in the bands, which generates high harmonics and \nnonlinear currents in the system. The nonzero Berry curvature in BiH drives a transverse anomalous electr ic \n a)\n c)  d) b)8 \n current  [44,64,66–68]. However, from fundamental symmetries these transverse currents must vanish along \nthe same high symmetry axes  [65]. Figure 3(c) presents the HHG spectral components that are polarized only \ntransversely to the main driving axis vs. the in -plane driving angle, which verifies this result. Notably, the \nemitted HHG radiation polarized transverse to the driving axis, and the spectral components of the magnetic \nresponse, are extremely similar (compare Figs. 3(b) to (c)). We thus conclude that a transverse current \ncomponent is essential for generat ing the anomalous magnetic response. In accordance with the mechanism \ndescribed above , this result is clear – the electronic system mus t acquire a nonzero angular momentum, 〈𝐋〉, \nsince only then the SOC te rm can initiate  a net  magnetism  (given that the system starts out in a non -magnetic \nstate) . Such angular momenta  is only obtained if electron s are driven in at least two transverse axes. Thus, \nthe anomalous currents and Berry curvature in this case act as a key ingredient for the  magnet ic response , \nsince they initialize transverse electron motion that mimics the effects of circular ly polarized  driving . Indeed, \nif the SOC term is turned off the magnetic response completely vanishes (not shown).  \nWe stress that  after the laser matter interaction has ended , the system is  left in a non -magnetic state as \nexpected  (the magnetization along each half cycle cancels out) , remov ing the zeroth  order spin response . \nThus, linearly polarized driving enables the higher order  spin dynamics to be uncovered , and even dominate \nover the slower perturbative responses . Arguably, one of t he most exciting consequence s is that the magnetic \nresponse oscillate s very rapidly during the interaction with the laser. For instance, in an extrem e case we \nobserve the magnetization flipping from a local maximum of ~0.01 𝜇𝐵 to a local minimum of ~-0.01 𝜇𝐵 within \njust 411 attoseconds  (see Fig. 3(a)) .  \n \nFIG. 3. Attosecond light -driven m agnetic response in BiH. (a) Exemplary calculations of < Sz(t)> driven by intense linearly -polarized \npulses for several driving intensities  (for a driving wavelength of 3000nm, and a polarization angle of θ=100), showing attosecond \ntimescale magnetization dynamics . The driving field is illustrated in arbitrary units to convey the different timescales in the dynamics. \n(b) Spectral components of < Sz(t)> vs. driving angle . The magnetic response identically vanishes along high symmetry axes  (indic ated \nby white  dashed lines ). Plot calculated for a driving intensity  of 2×1011 W/cm2 and a wavelength of 3000nm. (c) The HHG spectra \npolarized transverse ly to the driving laser axis, vs. the driving laser polarization  axis in the monolayer plane . The transverse \nanomalous currents  identically vanishe  for the same driving conditions as the magnetic response, indicating that the two are \nconnected. In (b) and (c) the spectral power is presented in log scale.  \nV. OTHER MATERIALS  \nAt this point one may wonder about the generality of our results, since we studied light -driven dynamics in \na 2D topological insulator. Thus, a legitimate question is whether the obtained ultrafast magnetization is a \ntopological feature, and if it is also applicable  in 3D bulk solids. To address these questions  we explore the \nultrafast magnetic response in MoTe 2, which is a transition metal dichalcogenide , and bulk Na 3Bi [69]. \nSpecifically, we consider the 2H bulk phase and the H monolayer phases of MoTe 2, which are non -topological \nand have  non-magnetic  ground -states . Na3Bi on the other hand has a topological nature , and was recently \nshown to be a Dirac semimetal with bulk Dirac cones appearing at finite momenta  [69,70] . We consider it \nbecause it is a peculiar  example for a material system with strong SOC , a non -magnetic ground -state, and \ninherently vanishing  Berry curvature, but which still permits  generation of transverse  anomalous currents due \nto its hexagonal structure.  \nFigure 4(a) presents an exemplary  temporal evolution of 〈𝑆𝑧(𝑡)〉 driven by an intense linearly -polarized \npulse in a monolayer of H-MoTe 2. Figure 4(b) presents the spectral components of 〈𝑆𝑧(𝑡)〉 for linear driving  \n a)\n  b)  c)9 \n vs. the in -plane driving angle. Clearly, similar ultrafast  magnetic response s are obtained; albeit, they are \nslightly weaker because MoTe 2 exhibits  weaker SOC. Thus, a main conclusion is that the attosecond -based \nmagnetism is driven in non -magnetic materials independent ly on if the band s have nonzero  Chern numbers  \nor not . In general, very similar nonlinear behavior is obtained in MoTe 2 for all drivin g conditions (see \nAppendix C). Figure 4(c) presents the corresponding HHG spectra polarized transverse to the laser driving . \nVery good agreement between the two is obtained just as was seen in BiH (regardless of the different space \ngroup of MoTe 2 and BiH),  confirming the generality of the results  outlined above. In the Appendix  we also \npresent results for the bulk phase, which shows similar responses.  \nSimilarly, i n the Appendix we show that for linearly polarized driving there is a strong magnetic \nresponse in Na 3Bi. This is despite the fact that it has uniformly vanishing  Berry curvature  [69], and results \nfrom the hexagonal lattice  that permits nonzero transverse currents to first order of the driving field  (whereas \nBerry curvature is typically associated with a second -order nonlinear response) , as long as it is not driven \nalong high symmetry axes. Indeed, along mirror planes there are no transverse currents in Na 3Bi, and \nconsequently, also no light-induced magnetization dynamics.  Thus, our results indicate  a general mechanism \nthat enables atto second magnetism in otherwise non -magnetic systems, including 3D bulk solids and 2D \nsystems. The main ingredients for this response are: (i) highly non -resonant strong -field driving, (ii) strong \nspin-orbit coupling, and (iii) generation of  anomalous transverse currents  to the  driving which are permitted \nby crystal symmetries, and which effectively yield a strong oscillating orbital angular momentum . \n \nFIG. 4. Ultrafast turn -on of magnetization and attosecond magnetization dynamics  in monolayer H-MoTe 2 (topologically trivial) . (a) \nExemplary calculation of < Sz(t)> driven by a linearly -polarized pulse with an intensity  of 2×1011 W/cm2, a wavelength of 3000nm , \nand polarized at θ=200. The driving field is illustrated in arbitrary units to convey the different timescales in the dynamics. (b) Spectral \ncomponents of < Sz(t)> vs. driving angle in the same conditions as (a) . (c) The HHG spectra polarized transverse ly to the driving laser \naxis, vs. the driving laser polarization axis in the monolayer plane . In (b) and (c) the spectral power is presented in log scale. Dashed \nwhite lines indicate high symmetry axes.  \nVI. PUMP -PROBE CIRCULAR DICHROISM  \nLastly, we present a  potential  experimental set -up capable of  measuring these attosecond  magnetic responses. \nThis set -up is based on a pump -probe geometry, where the pump is an intense femtosecond pulse that excites \nmagneti zation  dynamics (just as in Eq. (6)), and the probe is an attos econd extreme ultraviolet ( XUV ) pulse \nthat is circularly (or elliptically) polarized. By measuring the  time-resolved  circular dichroism (CD) in the \nabsorption (or transmission) spectra, one can detect attosecond magneti sm (for numerical details see the \nmethods section).  Notably, since the ground -states for all of the material systems we examined are non -\nmagnetic, the CD is zero if the system is not pumped. Thus, any nonzero signal immediately indicates the \npresence of magnetization, making the detection sch eme potentially  simpler  and background -free.  \nFigure 5(a) presents an exemplary spectr um for BiH driven by a  circularly -polarized pulse – the strong \npump field induces changes to the imaginary part of the material’s dielectric function . When this driven state \nof matter is probed with left circularly polarized (LCP) or right circularly polarized (RCP)  pulses, the re is a \nnoticeable  deviation in the response. The subtraction defines the CD , which can be further normalized to the \nellipticity of the probe pulse in a given spectral region (because the probe pulse has a finite duration it is not \nperfectly circular ). Depending on the conditions, t he ellipticity -normalized  CD can reach  up to  50% changes  \n b)  c)  a)10 \n of the dielectric function in equilibrium, which should be well within experimental detectability . Figure 5(b) \npresents the attosecond -resolved CD for the linearly -driven c ase. A strong CD signal oscillate s rapidly in \naccordance with the induced magnetization  (peak to minima changes occur on a timescale of ~500 \nattoseconds) , illustrating that the  attosecond  magnetization dynamics should be experimentally accessible.  \n \nFIG. 5. Pump -probe time-resolved CD absorption  spectroscopy in BiH.  (a) Imaginary part of the dielectric function for the  driven \nsystem  (pumped with a circularly polarized pulse with a wavelength of 3000nm and intensity 7.5×1011 W/cm2), probed with either \nleft (LCP) or right (RCP) helical probes.  The driven system is  temporally  probed one cycle before  the end of the pump laser field.  \nThe CD curve represents the subtraction between the RCP and LCP curves  and is shifted down for clarity. Note that due to its finite \nduration , the probe pulse has a non -uniform ellipticity in this frequency region, which is not taken into account in (a). (b)  Attosecond -\nresolved CD in the linear driving case (for a wavelength of 3000nm and intensity 3×1012 W/cm2, driven at an angle of θ=200). The \nx-component of the driving electric field and the induced magnetization is plotted out of scale for reference, where positive/negative \nparts of the induced magnetization are highlighted by blue/red colors in correspondence with the CD.  The CD in (b) is normalized \nby the ellipticity of the probe pulse for each frequency region.  \nVII. SUMMARY AND OUTLOOK  \nTo summ arize, we investigated light -matter interactions between intense non-resonant femtosecond laser \npulses and solids with strong spin -orbit coupling. Through state -of-the-art ab-initio  calculations, we \ndemonstrated that the attosecond timescale excitation and acceleration of electrons in the bands  (which \ninvolves highly nonlinear multi -photon processes)  is converted to magneti sm by the SOC term. With \ncircular ly polarized  driving, this i nduces a net magnetization that typically  turns on within ~16 fem toseconds . \nConsequently, we establish a new regime of femto -magnetism where paramagnetic  materials can be \ntransiently transformed into magnetic states with non -resonant driving . Remarkably, e ven in linearly -\npolarized driving conditions there are significant magnetization dynamics during the interaction with the laser \npulse, which are enabled by light -driven anomalous currents  in materials that permit a transverse optical \nresponse . These dynamics evolve  intrinsically on very ultrafast timescales (much faster that previously \ndescribed) of ~500 attoseconds , as they result from the extreme nonlinear response of electrons to the driving \nlightwave itself, on a sub -cycle level . We studie d the connection between the symmetries of the laser -matter \nsystem and the induced nonlinear magnetic response , showing  that the speed of the magnetization dynamics \ncan be tuned with the  laser parameters . To our knowledge , these are the fastest known magnetization \ndynamics  in solids , which are enabled by the extreme nonlinearity in the strong -field regime , and the unique \nlinearly polarized drive  that effectively remove s slower terms  in the magnetic response . They should pav e \nthe way towards attosecond control of magnetism , and motivate utilizing strong -field physics as an avenue \nfor nonlinear spintronics with enhanced degrees of control over higher order spin and spin -photon \ninteractions . Lastly, we showed that these phenome na should be experimentally detectable with pump -probe \nattosecond transient absorption experiments, utilizing circular dichroism  [34,71–73]. \nIt is worth discussing some possible applications and extensions of our results. First, while we employed \nhere simple quasi -monochromatic laser pulses, our results are more general. In that respect, utilizing more \ncomplex waveforms such as bi -chromatic fields should enable enhan ced control over magnetism. This simply \nfollows from the enhanced control over electron dynamics that such fields offer  [46,74,75] . This possibility \n a)\nCD   ) b)11 \n is especially exciting because it could lead to even further increase of the speed of the magnetization \ndynamics, and to  controlling  ultrafast magnetization  by tuning the laser phases  (i.e. a form  of coherent \ncontrol) . Second, by us ing few -cycle pulses we expect  that one could induce a net magnetization even with \nlinearly -polarized pulses, since then anomalous contributions from sequential half -cycles would not cancel \nout. This would establish a  magnetic analogue  to transient injection currents that have recently been \nmeasured  [41,50,51] . Lastly, since  the magnetization is driven in the same conditions that allow for high \nharmonic generation, it could enable high harmonic spectroscopy for probing magnetism, which has not been \npossible before  [57]. Looking forward, we expect our results to motivate more experimental and theoretical \nwork in the field . \nACKNOWLDENGEMENTS  \nThis wo rk was supported by the Cluster of Excellence Advanced Imaging of Matter (AIM), Grupos \nConsolidados (IT1249 -19), SF 925, “Light induced dynamics and control of correlated quantum systems”  \nand has received funding from the European Union's Horizon 2020 research and innovation programme under \nthe Marie Skłodowska -Curie grant agreement No 860553 . The Flatiron Institute is a division of the Simons \nFoundation. O.N. gratefully acknowledges the ge nerous support of a Schmidt Science Fellowship . \nAPPENDIX A: TECHNICAL DETAILS   \nWe report here on technical details for calculations presented in the main text. We start with details of the \nground state DFT calculations that were used for obtaining the initial KS states. All DFT calculations were \nperformed using Octopus code  [60–62]. The KS equations were discretized on a Cartesian grid with the shape \nof the primitive lattice cell s. Atomic geometries and lattice parameters were taken from ref.  [63] for BiH  \n(a=b=5.53Å , and a Bi -H distance of 1.82 Å), taken as  a=b=3.55Å for H -MoTe 2, as a=b=3.56Å, c=15.35Å for \n2H-MoTe 2, and as a=b=5.448Å, c=9.655Å for Na3Bi [69]. In all  cases the space -group  symmetric primitive \nunit cell was employed  (with the hexagonal  lattice vectors residing in the xy plane) . For monolayer systems, \nthe z-axis (transverse to the monolayer) was described using non -periodic boundaries with a length of 60 \nBohr. The KS equations were solved to self -consistency with a tolerance <10-7 Hartre e, and the grid spacing \nused was 0.39 Bohr for BiH, and 0.3 6 Bohr for MoTe 2 and Na 3Bi. We employed a Γ -centered 24×24×1 k-\ngrid for BiH, of 30×30×1 k-grid for H -MoTe 2, 24×24×8 k-grid for 2H -MoTe 2, and 28 ×28×15 k-grid for  \nNa3Bi. Deep core states were replaced by Hartwigsen -Goedecker -Hutter  (HGH ) norm -conserving \npseudopotentials  [59]. \nFor the time -propagation of the main equations of motion we employed a time -step of 4.83 attoseconds. \nThe pro pagator was represented by a Lanczos expansion  and k-point  symmetries were not assumed . In the \ntime-dependent calculations of the monolayer systems, we employed absorbing boundaries through complex \nabsorbing potentials (CAPs) along the aperiodic z-axis with a width of 15 Bohr  [76] and a magnitude of 1 \na.u. We calculated the total electronic excitation induced in the system in a time -resolved manner ( 𝑛𝑒𝑥(𝑡)) \nby projecting the KS-Bloch states onto the ground state system:  \n 𝑛𝑒𝑥(𝑡)=𝑁𝑒− ∑ ∑ 𝑤𝐤|⟨𝜓𝑛′,𝐤𝐾𝑆(𝑡=0)|𝜓𝑛,𝐤𝐾𝑆(𝑡)⟩|\n𝐤∈𝐵𝑍2\n𝑛,𝑛′∈𝑉𝐵 (8) \nwhere 𝑁𝑒 is the total number of active electrons in the unit cell , the projections are performed onto the valence \nbands of the ground state system (i.e. 𝑛′∈𝑉𝐵), and the summation is performed in the entire first Brillouin \nzone (BZ). 𝑛𝑒𝑥(𝑡) gives a measure f or the number of excited electrons during the light -driven dynamics.   \nThe envelope function of the employed laser pulse, f(t) from Eq. (1), was taken to be of the following \n‘super -sine’ form  [77]: \n \n𝑓(𝑡)=(𝑠𝑖𝑛(𝜋𝑡\n𝑇𝑝))(|𝜋(𝑡\n𝑇𝑝−1\n2)|\n𝑤)\n (9) 12 \n where w=0.75, Tp is the duration of the laser pulse which was taken to be Tp=5T (~29.3 f emtoseconds  full-\nwidth -half-max (FWHM) for 3000nm light), where T is a single cycle of the fundamental carrier frequency. \nThis form is roughly equivalent to a super -gaussian pulse, but where the field starts and ends exactly at zero \namplitude, which is  numerically  more convenient.  \nCalculations of transient absorption spectroscopy employed the real-time propagation approach \ndetailed in ref.  [78]. We e mployed short XUV pulses as probes that were comprised of a set of 8 stepwise \njumps in the vector potential (each giving a Dirac Delta function peak in the time -domain electric field) , \nwhich had a rotating polarization direction. Each step was polarized at 45 degrees , and was separated by 62.9 \nattoseconds , with respect to its previous , resembling an optical centrifuge  [79]. The total temporal duration \nof the probe pulse is thus 503.1 attoseconds , and each peak had  an intensity of 1010 W/cm2 (which gives an \nintensity of ~108 W/cm2 in the frequency region of  interest of  1-5eV) . To change the helicity of the probe \npulses the direction of rotation of the centrifuge was rotated , and for normalization purposes the ellipticity of \nthe probe was calculated using stokes parameters  [80] in the frequency region of interest . This configuration \nallows calculating the CD in a wide frequency  range with attosecond temporal resolution wh ile avoiding \nperforming may separate calculations with changing carrier wavelength (because the step -like nature of the \nprobe pulse has an infinite frequency content).  \nBy calculating the light -driven current in the system we extracted the optical conducti vity via: \n \n𝜎𝑖𝑖(𝜔)=𝐽̃𝑝𝑟𝑜𝑏𝑒 ,𝑖(𝜔)\n𝐸̃𝑝𝑟𝑜𝑏𝑒 ,𝑖(𝜔) (10) \nwhere 𝐽̃𝑝𝑟𝑜𝑏𝑒 ,𝑖(𝜔) is the Fourier transform of the total current in the system that is induced solely by the probe \npulse. That is, in the time -domain 𝐉𝐩𝐫𝐨𝐛𝐞 (𝑡) is defined as the subtraction of the total current that is calculated \nwith the presence of the probe pulse, a nd the current that is calculated without a probe pulse that is driven \nsolely  by the pump. Here 𝐄̃𝐩𝐫𝐨𝐛𝐞  is the Fourier transform of the electric field vector of the XUV prob e pulse, \nand 𝑖,𝑗 are Cartesian indices. From the optical conductivity we ext racted the dielectric function : \n 𝜀𝑖𝑖(𝜔)=1+4𝜋\n𝜔𝜎𝑖𝑖(𝜔) (11) \nand the average dielectric function  𝜀(𝜔)=(𝜀𝑥𝑥(𝜔)+𝜀𝑦𝑦(𝜔))/2. The CD was calculated between the \nimaginar y part of the dielectric functions  for a right -circular probe and a left -circular probe:  \n 𝐶𝐷(𝜔)=Im{𝜀+(𝜔)−𝜀−(𝜔)} (12) \nwhere +/- refers to left or right circularly polarized probes. Eq. (12) was evaluated for different  pump -probe \ndelay s by changing the onset time of the probe pulse , where for the temporally -resolved plot in Fig. 5(b) we \nused steps of 500 attoseconds  in the pump -probe delay  grid and results were interpolated by splines on a \ndenser  grid. We also filtered the induced probe current, 𝐉𝐩𝐫𝐨𝐛𝐞 (𝑡), with an exponential mask in the time \ndomain to avoid numerical issues with the finite time propagation, and filtered 𝜀𝑖𝑖(𝜔) with an exponential \nmask below the band gap to remove issues of division by zero (because the probe pulse  has zero spectral \ncomponents at 𝜔=0).  \nAPPENDIX B: ADDITIONAL RESULTS IN BiH  \nWe present here additional  complimentary  results to those presented in the main text. Figure 6 presents the \nHHG spectra emitted from BiH in the same driving  conditions as those that induce the transient magnetization \ndiscussed in the main text. Fig ure 6(a) presents the HHG spectra for circular ly polarized  driving  while tuning  \nthe laser wavelength, showing that only 6 n±1 harmonics (for intege r n) are emitted due to dynamical \nsymmetry considerations  [65]. Figure 6(b) presents the emitted spectra for elliptical driving, where the \nellipti cal major axis is rotated within the monolayer plane. As seen, odd -only harmonics are emitted due to \nsimilar symmetry considerations  [65]. Figure 6(c) presents the emitted HHG spectra vs. the driving ellipticity  \n(changing from circular to linear ) where the main  elliptical axis is along the x-axis ( transverse to  the Bi -Bi \nbonds ). In this case we can track the harmonic emission as the system transitions from the odd -only inversion 13 \n symmetry  to the 6 -fold rotational symmetry obtained for circular ly polarized  driving.  The same symmetries \nthat constrain the HHG spect ral components were shown in the main text to constrain the temporal evolution \nof the electronic and spin excitations, clarifying that the atto -magnetic response is directly driven by the laser.  \n \nFIG. 6. HHG spectra and selection rules in BiH.  (a) HHG  spectra for circularly polarized driving vs. driving wavelength , calculated \nfor 5×1012 W/cm2. Dashed black lines indicate 6 n±1 harmonics (for integer n). (b) HHG spectra for elliptical driving with an ellipticity \nof 0.2, vs. driving angle of the elliptical major axis in the monolayer plane (for a driving wavelength of 3000nm and intensity  of \n2×1011 W/cm2). Dashed black lines indicate odd harmonics. (c) HHG spectra for elliptical driving vs. the driving ellipticity, where \nthe elliptical major axis is set at θ=0, for similar laser parameters as (b).  All spectra are presented in log scale.  \nNext, we further explore the femto -magnetic response in BiH. Figure 7(a) presents the difference in \ntotal occupations of spin -up and spin-down electrons from the ground state as the  system evolves in time \nwhen driven by intense circular pulses (which is compensated  for the small average ionization  in both \nchannels) , Δ𝑛𝛼. As is clearly shown, as the system builds up a magnetic response, the occupations of spin -\ndown states  is converted to up spins, or vice versa. In fact, the occupation of spin -up states  vs. time is  formally \nequivalent to the calculation of 〈𝑆𝑧(𝑡)〉. Thus, this result supports the mechanism responsible for this \nbehavior , which involves a SOC -driven flipping of spins.  Figure 7(b) presents a similar result but for \ncircular ly polarized  driving with an opposite helicity, showing that there is perfect spin -helicity symmetry – \nexchanging  light’s helicity flips the  magnetic response.  \n \nFIG. 7. Ultrafast magnetization  dynamics in BiH . (a) Time -dependent occupation s of spin-up and spin-down channels for circular ly \npolarized  driving (with a wavelength of 3000nm and intensity  of 1012 W/cm2). (b) same as (a) but for opposite driving light helicity. \nIn all plots the x-component of the driving laser field is illustrated in arbitrary units for clarity.  \nWe now present additional results that explore  the nonlinear  nature  of the induced magnetism . Figure \n8(a) presents the magnetization onset time (defined as the time for which the induced magnetization temporal \nderivative, 𝜕𝑡〈𝑆𝑧(𝑡)〉, surpasses 10-6 𝜇𝐵 per atomic unit of time ) vs. the driving intensity  for the circularly \npolarized driving case. The onset time behaves highly nonlinearly with the pump power, which  is additional  \ncorroborat ion for the nonlinear nature of the effe ct. Figure 8(b) presents the induced net magnetization in the \nsame conditions as in Fig. 1( c) in the main text, with the induced total electronic excitation , 𝑛𝑒𝑥. The plot \nconnects the onset time of the magnetization with the light-induced excitations to the conduction band, as the \ntwo curves  have a similar onset behavior . Moreover, the  rapid attosecond dynamics become most prominent \nwhen the conduction band is highly popula ted. Similar results are obtained for other driving conditions.  \n a)  b)  c)\n b)  a)14 \n  \nFIG. 8. Magnetization onset times and highly  nonlinear nature  of the  magnetization  dynamics in BiH . (a) Magnetization  onset times \nvs. driv ing intensity  for circularly polarized driving with a wavelength of 3000nm . (b) < Sz(t)> for circularly polarized driving with a \ndriving wavelength of 3000nm and intensity  of 5×1011 W/cm2, and total electronic excitation  in the same scale . \nWe next explore the connected symmetries of the electronic excitation and the spin dynamics, which \nfurther support the conclusions  presented in the main text . Figure 9(a) presents the spectral components of \nthe electronic excitation, i.e. the Fourier trans form of 𝑛𝑒𝑥(𝑡) to the frequency -domain. This analysis allows \ntracking the temporal dynamics of the electronic excitation and seeing if it complies to similar symmetries as \nthe induced magnetization. As seen in Fig. 9(a), for circular ly polarized  driving 𝑛𝑒𝑥(𝑡) is comprised of only \n6𝑛 harmonics of the driving laser (for integer 𝑛). This is a fundamental constraint that arises from dynamical \nsymmetr ies [65], and complements the symmetries of the emitted HHG radiation (follow ing 6𝑛±1 selec tion \nrules , see Fig. 6). Essentially, Eq. (8) evaluates the sum of projections of the time -dependent wave function s \nonto the ground -state wave  function s. However, because the dynamics of the wave  functions uphold the \nimproper -rotational dynamical symmetry in the light -matter system (which complies to 𝑆6𝐻(𝑡+𝑇\n6)𝑆6†=\n𝐻(𝑡), where 𝑆6 is a 6 -fold improper rotation in the monolayer plane ), this means that the occupations uphold \n𝑛𝑒𝑥(𝑡)≈𝑛𝑒𝑥(𝑡+𝑇/6), where 𝑇 is the laser period , which leads to the 6 𝑛 harmonic structure.  This equation \nis correct up to a constant shift at zero frequency that has to do with non -periodic tunneling dynamics , and \nsymmetry breaking due to the short duration of the laser pulse (which broadens  the 6 𝑛 harmonic peaks) . The \ninherent reason that the selection rules for 𝑛𝑒𝑥(𝑡) are different than for the HHG emission , is that 𝑛𝑒𝑥(𝑡) is \ncalculated with a  parity -even projection operator, whereas the dipole operator that evaluates the HHG \nemission is parity -odd. Nonetheless, th e two selection rules have similar origins.  A completely identical \nselection rule is obtained for the expectation value of the total magnetization, 〈𝑆𝑧(𝑡)〉 (e.g. as seen in Fig. 2 \nin the main text , and in Fig. 9(b) ), also allowing only 6 𝑛 harmonics . It means that the two quantities  of 𝑛𝑒𝑥(𝑡) \nand 〈𝑆𝑧(𝑡)〉 are inherently connected, because the induced magnetization is physically driven by the \nexcitations to the conduction band.  \nFigure 9(c) further presents the spectral components of 𝑛𝑒𝑥(𝑡) for a system driven by an elliptically -\npolarized pulse . In this case, the light -matter system follows a dynamical inversion symmetry \n(i𝐻(𝑡+𝑇\n2)i†=𝐻(𝑡), where i is an inversion operator ), which means that the projections follow 𝑛𝑒𝑥(𝑡)≈\n𝑛𝑒𝑥(𝑡+𝑇/2), leading to even -only harmonics. This complements  the odd -only HHG emission  (see Fig. 6) \nand has an identical origin. Figure 9(d) presents the spectral components of 〈𝑆𝑧(𝑡)〉 for the same conditions , \nalso showing even -only harmonics  further support ing the generality of the mechanism presented in the main \ntext. We obtained similar results that connect the symmetries of the light -driven currents, the electronic \nexcitation, and the induced magnetization, in all explored conditions .  \n b)  a)15 \n  \nFIG. 9. Symmetries of the electronic  excitation and their connection to magnetization  dynamics  in BiH . (a) Spectral components for \nnex(t) plotted in  arbitrary units in  log scale for circularly polarized driving  (with wavelength of 3000nm and intensity  of 5×1011 \nW/cm2). The x-axis is given in units of the laser frequency, and even harmonics are indicated with grey lines. Only 6 n harmonics are \nobserved with circularly polarized driving (for integer n). (b) Same as  (a) but for < Sz(t)> in similar conditions, showing similar \nsymmetry -based selection rules . (c) Same as (a) but for elliptical driving with an ellipticity of 0.5 , a major  elliptical  axis angle of \nθ=300, and driving intensity  of 2×1011 W/cm2. Only even harmonics of the drive are observed . Same as (c), but for < Sz(t)> in similar \nconditions, showing similar symmetry -based selection rules.  \nWe now present the band - and k-space -resolved  induced magnetization. The band -resolved \ncontributions to the magnetization were calculated by projecting the time -dependent wave  functions onto the \nground -state wave functions , just as was done for 𝑛𝑒𝑥(𝑡). However, in this case the projections were not \nsummed in k-space and over the bands,  and instead, the k-resolved projections for pairs of spin -degenerate \nbands were summed together after weighting the occupations by the expectation value 〈𝑆𝑧〉 at that particular \nband and k-point . This gives a measure for the diffe rent contributions of each band and k-point to the induced \nmagnetization , and for instance at t=0 this analysis leads to identically zero magnetization in all bands and \nk-points (because the initial state is non -magnetic) . Figure s 10(a-d) presents these re sults for a circular ly \npolarized  driving case after the laser pulse ends for the first and second valence and conduction bands (in the \nnotation where the ‘first’ bands includes two spin -degenerate bands, and so on). As is seen, the dominant \ncontribution to the magnetization arises from re gions near the minimal band gap  K and K’ points). Moreover, \nthe first valence and conduction  band contribute stronger  magnetic response s than other  bands. This trend \ncontinues in higher bands (not presented). These results thus support that the first step  in the mechanism \nbehind the induced magnetization is electronic excitation to the conduction band, as discussed in the main \ntext. In contrast  to these results , Figs. 10(e,f) present the complementary k- and band -resolved magnetization \nafter the interactio n with a linear driving pulse  for the first valence and conduction bands. Here, each region \naround K and K’ contributes to both positive and negative magnetization, and the magnetization at K and K’ \nis inverted, such that the net magnetism  overages out to zero, as expected for linear driving .  \n c) b)  a)\n d)16 \n  \nFIG. 10. k- and band -resolved projections  of the  light-induced  net magnetization after the pulse. (a -d) Magnetization after  circularly \npolarized driving with a wavelength of 3000nm and intensity  of 7.5×1011 W/cm2. (a) First valence band, (b) first conduction band, \n(c) second valence band, (d) second conduction band . (e,f) Same as (a ,b), respectively, but for linear driving with an intensity of \n2×1011 W/cm2 and a polarization axis at θ=200. \nWe also tested  the role of correlations in the ultrafast induced magnetization dynamics. We have thus \nfar employed TD SDFT in the local spin density approximation for the XC functional , which allows the e-e \ninteractions to evolve dynamically in time . However, it is helpful to explore the role of correlations by \nemploying the independent particle approximation (IPA), which is equivalent to freezing the time -evolution \nof the XC potential, and the time -evolutio n of the Hartree term. Within this approach the time propagation of \nall the KS -Bloch states is fully independent of each other, and dynamical e-e interactions are not included in \nthe simulation. Figure 11 presents the spin expectation value in circular ly polarized  driving, comparing the \nIPA to the full TD SDFT calculation. As is clearly seen, the magnetization dynamics are not driven by \ncorrelations  as a very similar response is obtained within the IPA .  \n \nFIG. 11. <Sz(t)> in BiH  calculated  with full TD SDFT compared to IPA for circularly polarized driving, with a wavelength of 3000nm \nand a intensity  of 2×1011 W/cm2. \n 6 4 20246 6 4 20246\n 0.02 0.0100.010.02\n 6 4 20246 6 4 20246\n 0.02 0.0100.010.02  e  f \n )  ) )\n 6 4 20246 6 4 20246\n 0.06 0.04 0.0200.020.040.06\n 6 4 20246 6 4 20246\n 0.06 0.04 0.0200.020.040.06 c  d  )\n 6 4 20246 6 4 20246\n 0.06 0.04 0.0200.020.040.06\n 6 4 20246 6 4 20246\n 0.06 0.04 0.0200.020.040.06 a  b  )\n17 \n APPENDIX C: ADDITIONAL RESULTS IN MoTe 2 \nWe present here additional complimentary results to those presented in the main text for the MoTe 2 material \nsystem, both in monolayer and bulk phases.  For the monolayer, Fig. 1 2(a) presents the spin expectation value \ndriven by a circular pump, showing a femtosecond turn -on of magnetism  that is  in correspondence with the \nresults obtained in the main text. The electronic excitation match es in onset time with the onset of \nmagnetization ( see Fig. 1 2(a)). Figure 1 2(b) presents the spectral components of 𝑛𝑒𝑥(𝑡) in the same driving \nconditions, showing that in this ca se only 3 𝑛 components (for integer 𝑛) are allowed  (because a monolayer \nof H-MoTe 2 is 3-fold symmetric instead of 6 -fold) . Similarly, Fig. 1 2(c) presents the spectral components of \n〈𝑆𝑧(𝑡)〉 that also support only 3 𝑛 harmonics. These results arise from  the different symmetr ies of MoTe 2 \ncompared to BiH. It demonstrates the generality of our analysis.  \n \nFIG. 12. Additional results for light -induced magnetization dynamics in monolayer H-MoTe 2. (a) < Sz(t)> and nex(t) for circularly \npolarized driving with a wavelength of 3000nm and intensity  of 2×1011 W/cm2. (b) Spectral components of nex(t) plotted in log scale \nfor the same driving conditions as in (a) . The x-axis is given in units of the laser frequency, and every 3rd harmonic is  indicated with \ngrey lines. Only 3n harmonics are observed with circularly polarized driving (for integer n). (c) Same as ( b) but for < Sz(t)> in similar \nconditions, showing similar sy mmetry -based selection rules.  (b) and (c) are plotted in arbitrary units in log scale.  \nFigure 1 3 presents results for the 2H bulk phase of MoTe 2 for circular and linear driving . For the circular \ncase (Fig. 1 3(a,b))  we obtain a femtosecond turn -on of the magnetic response which is comprised of 6𝑛 \nharmonics of the pump laser , owing to the inherent symmetries of the bulk 2H phase  driven by circular light  \n(the bulk phase exhibits  a 6-fold rotation coupled to a glide symmetry along the c-axis) . For the linear case \nwe obtain a ttosecond magnetization dynamics, in similar spirit to the results in the main text, which comprise \nof even -only harmonics of the laser (see Fig. 1 3(c,d)). Thus, the results in the bulk phase support the results  \nobtained  in the monolayer systems , and demonstrate that the effect is valid in 3D bulk systems.   \n a)  b)  c)18 \n  \nFIG. 13. Femtosecond turn -on of magnetization and attosecond magnetization dynamics in bulk 3D 2H-MoTe2. (a)  <Sz(t)> for \ncircularly polarized driving with a wavelength of 3000nm and intensity  of 2×1011 W/cm2. The x-component of the driving field is \nillustrated in arbitrary units to  convey the different timescales in the dynamics. (b) Spectral components of < Sz(t)> plotted in log scale \nfor the same driving conditions as in (a). The x-axis is given in units of the laser frequency, and every 3rd harmonic is indicated with \ngrey lines. Only 6n harmonics are observed with circularly polarized driving (for integer n). (c) Same as (a) but for linearly polarized \ndriving with θ=100. (d) same as in (b) but for the driving conditions in (c). Every 2nd harmonic is illustrated by grey lines, and only \neven harmonics are observed. (b) and (d) are plotted in arbitrary units in log scale.  \nAPPENDIX D: ADDITIONAL RESULTS IN Na 3Bi \nWe present additional complimentary results to those presented in the main text for Na 3Bi, the Dirac \nsemimetal. Figure 1 4 shows the induced magnetization dynamics driven by intense linearly -polarized light \nin the hexagonal planes for two different driving angles (either along a high -symmetry axis, or not). As clearly \nshown, when Na 3Bi is not driven along high symmetry axis it permits a strong oscillating magnetization. We \nverified that this is a result of nonzero transverse currents that permit transient  orbital angular momentum \n(not presented). On the other hand, along high symmetry a xes the transverse currents are not permitted, \nleading to an identically vanishing magnetism. Thus, these results support the generality of our conclusions, \neven in materials with vanishing Berry curvature.  \n \nFIG. 14. Attosecond magnetization dynamics in the Dirac semimetal Na 3Bi for linearly polarized driving with a wavelength of \n2100nm and intensity of 7×1011 W/cm2. 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Phys. 22, 351 (1954).  \n "    },    {        "title": "1007.1543v1.Polarization_and_magnetization_dynamics_of_a_field_driven_multiferroic_structure.pdf",        "content": "arXiv:1007.1543v1  [cond-mat.mtrl-sci]  9 Jul 2010Polarization and magnetization dynamics of a\nfield-driven multiferroic structure\nAlexander Sukhov1, Chenglong Jia1, Paul P. Horley2and Jamal\nBerakdar1\n1Institut f¨ ur Physik, Martin-Luther-Universit¨ at Halle-Wittenbe rg, 06120 Halle,\nGermany\n2Centro de Investigati´ on en Materiales Avanzados, S.C. (CIMAV), Chihuahua /\nMonterrey, 31109 Chihuahua, Mexico\nAbstract. We consider a multiferroic chain with a linear magnetoelectric coupling\ninduced by the electrostatic screening at the ferroelectric/ferr omagnet interface. We\nstudy theoretically the dynamic ferroelectric and magnetic respon se to external\nmagnetic and electric fields by utilizing an approach based on coupled L andau-\nKhalatnikov and finite-temperature Landau-Lifshitz-Gilbert equa tions. Additionally,\nwe compare with Monte Carlo calculations. It is demonstrated that f or material\nparameters corresponding to BaTiO 3/Fe the polarization and the magnetization are\ncontrollable by external magnetic and electric fields respectively.Polarization and magnetization dynamics of a field-driven m ultiferroic structure 2\n1. Introduction\nMagnetic nanostructures are intensively researched[1] due to t heir versatile use in\ntechnology. Multiferroics , i.e. systems that exhibit a coupled ferroelectric and\nferromagnetic order, have received much attention recently[2, 3, 4]. A variety\nof potential applications of multiferroics rely on a possible control o f magnetism\n(electric polarization) with electric (magnetic) fields due to the magn etoelectric\ncoupling. Indeed, recent experiments have demonstrated the ex istence of both\neffects[5, 6, 7]. Furthermore, a number of interesting phenomena associated with\nthe coupled polarization/magnetization dynamics at interfaces and in bulk have\nbeen reported such as the electrically controlled exchange bias[8], the electrically\ncontrolled magnetocrystalline anisotropy[9] and the influence of ele ctric field on the\nspin-dependent transport[10]. Theoretically, recent Monte-Car lo calculations for a\nthree-dimensional spinel lattice were performed to study the mag netic-field induced\npolarization rotation[11, 12].\nIn this work we investigate theoretically and numerically the field-driv en dynamics of\nthe electric polarization and the magnetization of a ferroelectric/f erromagnetic system\nthat shows a magnetoelectric coupling at the interface. For this pu rpose we consider a\ntwo-phase multiferroic chain consisting of50 polarizationsites and5 0 localizedmagnetic\nmoments, as sketched in Fig.1. The ferromagnetic (FM) part of the chain is a normal\nmetal (e.g., Fe), whereas the ferroelectric (FE) part is BaTiO 3(Fig. 1). Recently\nthis system has been shown to exhibit a magnetoelectric coupling[13, 14] and has\nbeen realized experimentally[15]. The multiferroic coupling arises as a r esult of an\naccumulationofspin-polarizedelectronsorholesattheFE-insulato r/FM-metalinterface\nwhen the FE is polarized[16]. At the metal/insulator interface, the sc reening of the\npolarization charge alters the FE polarization orientation resulting in a linear change\nof the surface magnetization. The resulting magnetization in the FM structure decays\nexponentially away from the interface. Taking the FM material as an ideal metal with\nthe screening length of around [17] 1 ˚A, the exchange interaction between the additional\nsurface magnetization and the FM part is thus limited to only the first site. Switching\nto dimensionless units, we introduce the reduced polarization ( pj(t) =Pj(t)/PS) and\nmagnetic moment ( Si(t) =µi(t)/µS) vectors, where PSis the spontaneous polarization\nof (bulk) BaTiO 3andµSis the magnetic moment at saturation of (bulk) Fe.\n2. Theoretical formalism\nThe total energy of the FE/FM-system in a very general one-dime nsional case consists\nof three parts\nFΣ=FFE+FFM+Ec. (1)Polarization and magnetization dynamics of a field-driven m ultiferroic structure 3\nThe ferroelectric energy contribution reads\nFFE=a3PSNFE−1/summationdisplay\nj=0/parenleftBigαFEPS\n2p2\nj+βFEP3\nS\n4p4\nj+\nκFEjPS\n2(pj+1−pj)2−pj·E(t)/parenrightBig\n, (2)\nwhereas the ferromagnetic energy part is\nFFM=NFM−1/summationdisplay\ni=0/parenleftBig\n−JiSi·Si+1−Di(Sz\ni)2−µSSi·B(t)/parenrightBig\n. (3)\nE(t) andB(t) are respectively external electric and magnetic fields.\nVarious pinning effects that may emerge in both the FE and the FM par ts due to\nimperfections and expressed by κFEjandJi,Direspectively, are not considered here,\ni.e.κFEj≡κFE,Ji≡J,Di≡D.\nWe consider that the linear FE/FM coupling appears as a result of the exchange\ninteraction ofthemagnetizationinduced bythe screening chargea t theinterface andthe\nlocal magnetization in the ferromagnet (for details we refer to [18]) and can be written\nas\nEc=λPSµSp0·S0. (4)\nThe meaning of the quantities appearing in these equations is explaine d in Table 1.\nBased on the parameters obtained from ab-initio calculations for BaTiO 3/Fe-\ninterface[13, 19], we estimate the coupling constant as λ=J a2\nFMαS/(εFEε0µ0µ2\nS)≈\n2·10−6s/F, where the surface ME-coupling constant is αS= 2·10−10Gcm2/V. We find\nthisvalueistoolowtoobtainasizableME-response fortheone-dimen sional multiferroic\ninterface. In what follows, we vary λand explore the dependence of the multiferroic\ndynamics on it.\nThe polarization dynamics is governed by the Landau-Khalatnikov (L Kh) equation[20,\n21], i.e.\nγνPSdpj\ndt=HFE\nj=−1\na3PSδFΣ\nδpj, (5)\nwhereγνis the viscosity constant (Table 1) and HFE\njstands for the total external and\ninternal fields acting on the local polarization. The magnetization dy namics obeys the\nLandau-Lifshitz-Gilbert[22] (LLG) equation of motion\ndSi\ndt=−γ\n1+α2FM/bracketleftbig\nSi×HFM\ni(t)/bracketrightbig\n−γαFM\n1+α2FM/bracketleftbig\nSi×/bracketleftbig\nSi×HFM\ni(t)/bracketrightbig/bracketrightbig\n,(6)\nwhereγis a gyromagnetic ratio (Table 1) and αFMis the Gilbert damping\nparameter. The total effective field acting on Siis defined as a sum of\ndeterministic and stochastic parts HFM\ni(t) =−1\nµSδFΣ\nδSi+ζi(t).The characteristics of\nthe additive white noise associated with the thermal energy kBTare[23]/an}b∇acketle{tζik(t)/an}b∇acket∇i}ht= 0\nand/an}b∇acketle{tζik(t)ζml(t+∆t)/an}b∇acket∇i}ht=2αFMkBT\nµSγδimδklδ(∆t).Hereiandmindex the correspondingPolarization and magnetization dynamics of a field-driven m ultiferroic structure 4\nsites in the FM-material. kandlare the Cartesian components of ζand ∆tis the time\ninterval. The coupled equations of motion (5) and (6) are solved num erically in reduced\nunits, re-normalizing the energy (1) over doubled anisotropy stre ngthD. Thus, the\ndimensionless time in both equations is τ=ωAt=γBAt=γ2Dt/µSand the reduced\neffective fields are hFM\ni(τ) =HFM\ni(τ)/BA,hFE\nj=HFE\nj/(γγνPSBA).\nTo endorse our results we conducted furthermore kinetic Monte C arlo (MC) simulations\nfor the description of the dynamics of a FE/FM chain subjected to e xternal magnetic\nand electric fields. The electric dipoles and the magnetic moments are understood as\nthree-dimensional classical unit vectors, which are randomly upda ted with the standard\nMetropolis algorithm[24]. The period of the external field is chosen to be 600 MC steps\nper site.\nIn the following the system is described via the reduced total polariz ationpΣ(t)\npΣ(t) =1\nNFENFE−1/summationdisplay\nj=0pj(t), (7)\nand the reduced net magnetization SΣ(t)\nSΣ(t) =1\nNFMNFM−1/summationdisplay\ni=0Si(t). (8)\n3. Results of numerical simulations\nTo demonstrate the response of the FE/FM-chain to external fie lds using the LKh and\ntheLLGequationsweusedadampingparameter αFM= 0.5,whichissignificantlyhigher\nthan the experimental value[25] ( α(Fe)=0.002), in order to achieve a faster relaxation\nof the net magnetization for both B- and E-drivings. It was assure d in our calculations\nthat the FE-subsystem is far from its phase transition temperatu re atT= 10 K and the\nFM-subsystem remainsnon-superparamagneticonthewholetimes caleofconsideration.\nFig. 2 shows the hysteresis loops in the presence of a harmonic exte rnal magnetic field\nBz(t) =B0zcosωt. The amplitude of the field B0zis chosen to be comparable with\nthe exchange interaction energy Jas well as the coupling energy Ec. The period\nof the external magnetic field is chosen to exceed the field-free pr ecessional period\n(2π/ω≈5Tprec) of the LLG. Irrespective of the temperature and the calculation al\nmethod thisfield is capable of switching themagnetization of theFM-c hain (Fig. 2b, d).\nThe FE-polarization indirectly driven by the external magnetic field is not completely\nswitched according to both methods (Fig. 2a, c). The role of therm al fluctuations on\nthe FM part only (cf. eq. (5)) is exposed by the p(B)-behavior sho wn in Fig. 2a,\nwhereas the MC method accounts for temperature effects also on the polarization (Fig.\n2c). As a result, the p(B)-hysteresis (Fig. 2c) shows a clear temperature dependence,\nwhich becomes especially pronounced for the one-dimensional chain in which thermal\nfluctuations degrade the polarization/magnetization ordering mor e intensively than for\nthe case of a two-dimensional system.\nThe hysteresis loops for the external electric field of the form Ez(t) =E0zcosωtarePolarization and magnetization dynamics of a field-driven m ultiferroic structure 5\npresented in Fig. 3. The energy of the applied electric field is compara ble with the\ncoupling energy. As a result, the total polarization can be complete ly switched (Fig.\n3a, c). As inferred from Fig. 3b, d the net magnetization is not fully s witched. Only\nseveral first spin sites follow the electric field due to the coupling at t he interface.\n4. Discussion\nIn real experiments several additional effects may affect the pola rization and the\nmagnetization dynamics. Below we estimate how strong these effect s might be for\nthe multiferroic chain.\n4.1. Effect of depolarizing fields\nIn the general case the field E(t) entering equation (2) is an effective field that consists\nof the applied electric field, e.g. Ez(t)ez, and the internal depolarizing field EDFcreated\nby the screening charges (SC) at the interface.\nGenerally, the depolarizing field may well be sizable and affects thus th e dynamics\n[30, 31, 32]. Here we estimate the strength of EDFby introducing a one-dimensional\nSC at the interface QSC=PSa2(Table 1). As a result, the electric field induced\nby the SC is opposite to the local polarization (Fig. 1) and can be writt en as\nEDF=−QSC/(4πε0εFEa2n2\nj)ez, whereε0= 8.85·10−12A·s/(V·m) is the permittivity of\nfree space, εFE≈2000 is the dielectric constant in barium titanate and njis the index\nnumbering the polarization sites starting from the interface, e.g. nj=0= 1. Thus, the\nstrength of the depolarizing field calculated for nj=0= 1 and upon the other parameters\nisEDF≈2·106V/m, which is at least one order of magnitude smaller than the\namplitudes of the applied electric field ( ≈4·107V/m). Keeping in mind that EDF\ndecays in the FE away from the interface we can neglect the depolar izing field.\n4.2. Effect of induced electric and magnetic fields\nAccording to the Maxwell’s equations an oscillating magnetic field induce s an oscillating\nelectric field, and an alternating voltage produces an oscillating magn etic field.\nThe situation becomes especially important for the second case, sin ce even small\ninduced magnetic fields aligned perpendicularly to the inducing field and hence to the\ninitial state of the magnetization can sufficiently assist the switching at appropriate\nfrequencies[33, 34].\nFrom the Faraday’s law of the Maxwell’s equations ∇ ×E=−µFEµ0∂H\n∂tand for the\ngiven applied magnetic field Bz(t) =B0zcosωtthe induced electric field acting on the\nFE-polarization is oriented perpendicularly to the inducing field ( XY-plane, Fig. 1).\nIts amplitude for relative magnetic permittivity in BaTiO 3[35]µFE≈1 andµFM≈5000\nscales as Eind\n0=aFEµFE/(µFM)B0zω≈2V/m.\nLikewise, when an external electric field Ez(t) =E0zcosωtis applied, according to the\nAmpere’s law of the Maxwell’s equations ∇ ×B=µFMµ0εFMε0∂E\n∂t, the direction ofPolarization and magnetization dynamics of a field-driven m ultiferroic structure 6\nthe induced field is perpendicular to the inducing field. The amplitude of the induced\nmagnetic field in iron for aFM= 0.28·10−9m andεFM≈1 (since iron is assumed as an\nideal metal) is Bind\n0=aFMµFMµ0εFMε0E0zω≈2·10−3T.\nOur calculations with the estimated (and even larger) amplitudes of t he induced electric\nfield show no influence onthe Z-projection of the FE-polarization. T his is a consequence\nof the uncoupled nature for the projections of the FE-polarizatio n (cf. equation (5)).\nBoth numerical methods give a slightly enhanced (less than 1%) ME-r esponse in the\npresence of the induced magnetic field. Therefore, the effect of t he induced electric and\nmagnetic fields can be deemed irrelevant for the considered multifer roic chain and for\nthe chosen range of frequencies.\n4.3. Frequency dependence of the magnetoelectric response\nA variation of the frequency ωof the external electric and magnetic fields can also affect\nthe ME-response of the multiferroic chain.\nFig. 4 demonstrates the response of the multiferroic interface to an external magnetic\nfield. Theperiodsoftheexternalmagneticfieldshouldbecompared withacharacteristic\nfield-free precessional time Tprec\nFM≈4 ps which is valid for bulk iron. As one expects,\nmagnetic fields with longer periods favor better saturation of the m agnetization (Fig.\n4b). The response of the total electric polarization to the extern al magnetic field\nbecomesenhancedwithincreasingperiodofB-field. Thisisconfirmed bybothnumerical\nmethods (Fig. 4a, c).\nThe multiferroic response to an external electric field is shown in Fig. 5 for which\nthe situation of very short electric fields (less or around Tprec\nFM) is addressed. The\nmagnetization does not relax quick enough resulting in the form of a h ysteresis which\nis similar to the ferroelectric one. Additionally, we obtain an increase o f the net\nmagnetization response (Fig. 5b). This feature can also be observ ed using the MC-\nmethod (Fig. 5d).\n5. Summary\nThe main result obtained using two independent methods - the direct solution of\nthe LKh and the LLG equations (5, 6) and the kinetic MC method - is th at due to\nthe coupling at the interface of FE/FM the ferromagnetic subsyst em responds to an\nexternal electric field and the ferroelectric subsystem responds to an external magnetic\nfield. A use of both methods allowed a comparison of dynamical and st atistical\napproaches for studying coupling phenomena at the FE/FM interfa ce. 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Alloys Compd. 454, 340 (2008).\n... ...FE□(BaTiO )3 FM□(Fe)\np0 p1 p49 S0S1S2S49Z\nYX\nFigure 1. Alignment ofelectric dipoles pjand magnetic moments Siin the considered\none-dimensional chain. The directions of Si(t= 0)≈ {0,0,1}andpj(t= 0) ={0,0,1}\nrepresent the initial configuration. The easy axis in the FM part is alo ng the Z-\ndirection.\nTable 1. Parameters used in the numerical calculations.\nFE-material (BaTiO 3)\nNumber of sites NFE 50\nPolarization[26] PS, [C/m2] 0.265\nInitial state pj(t=0), [P S] {0.0,0.0,1.0 }\nConstant[20] γν, [Vms/C] 2.5 ·10−5\nConstant[26] αFE, [Vm/C] -2.77 ·107\nConstant[26] βFE, [Vm5/C3] 1.70 ·108\nFE-interaction κFE, [Vm/C] 1.0 ·108\nLattice constant[27] a, [m] 0.4 ·10−9\nFE/FM-coupling λ, [s/F] parameter\nFM-material (Fe)\nNumber of sites NFM 50\nGyromagn. ratio γ, [(Ts)−1] 1 .76·1011\nMoment per site[28] µS, [µB] 2.2\nInitial state Si(t=0), [µS] {0.14,0.14,0.98 }\nAnisotropy strength[29] D, [J] 1.0 ·10−22\nExchange strength[29] J, [J] 1.33 ·10−21\nDamping αFM parameterPolarization and magnetization dynamics of a field-driven m ultiferroic structure 9\n0.511.5pΣz(t)\nT0=0 K\nT1=1 K\n0.511.5\npΣz(t)T0=0 K\nT1=1 K\n-1-0.500.51\nBz(t), [J]-101SΣz(t)T0=0 K\nT1=1 K\n-1-0.500.51\nBz(t), [J]-1-0.500.51\nSΣz(t)T0=0 K\nT1=1 K(a)\n(b)(c)\n(d)\nFigure 2. The reduced total polarization/magnetization response to exter nal\nmagnetic field of the form Bz(t) =B0zcosωt. The loops a) and b) are obtained\nby using the LKh and the LLG equations; c) and d) are calculated usin g the MC\nmethod. In the both methods parameters are chosen such that Ec≈µSBz≈J, i.e.:\nλ= 240 s/F, B0z= 6.65BA,ω= 3.61·1011s−1,E0z= 0 V/m, α= 0.5. 20 first\nperiods (1 /ω) are omitted; the hysteresis curves are averaged over 100 (a, b ) and 200\n(c, d) subsequent periods.Polarization and magnetization dynamics of a field-driven m ultiferroic structure 10\n-2-10123pΣz(t)T0=0 K\nT1=1 K\n-1-0.500.51\npΣz(t) T0=0 K\nT1=1 K\n-1-0.500.51\nEz(t), [J]0.60.70.80.91SΣz(t)T0=0 K\nT1=1 K\n-1-0.500.51\nEz(t), [J]0.60.70.80.91\nSΣz(t)T0=0 K\nT1=1 K(a)\n(b)(c)\n(d)\nFigure 3. Hysteresis loops of the total reduced polarization/magnetization as a\nfunction of external electric field of the form Ez(t) =E0zcosωt. The curves a) and\nb) are obtained by using the LKh and the LLG equations; c) and d) ar e calculated\nusing the MC method. Parameters are chosen such that Ec≈a3PSEz≈J, i.e.:\nλ= 240 s/F, B0z= 0 T,E0z= 4.07·107V/m,ω= 3.61·1011s−1,α= 0.5. 20 first\nperiods (1 /ω) are omitted; the hysteresis loops are averaged over 100 (a, b) a nd 200\n(c, d) subsequent periods.Polarization and magnetization dynamics of a field-driven m ultiferroic structure 11\n0.511.5pΣz(t)\n0.70.80.91\npΣz(t)\n-1-0.500.51\nBz(t), [J]-1-0.500.51SΣz(t)2π/ω · 1/Tprec = 4\n2π/ω · 1/Tprec = 10\n2π/ω · 1/Tprec = 16\n2π/ω · 1/Tprec = 22\n-1-0.500.51\nBz(t), [J]-1-0.500.51\nSΣz(t)2π/ω = 200 MC steps\n2π/ω = 400 MC steps\n2π/ω = 1000 MC steps\n2π/ω = 2000 MC steps(a)\n(b)(c)\n(d)\nFigure 4. Response of the one-dimensional multiferroic structure to the tim e-\ndependent magnetic field Bz(t) =B0zcosωtplotted for various frequencies ω. The\nfield-free precessional time in the FM part is Tprec\nFM=Tprec≈4 ps. The curves a) and\nb)areobtainedbyusingtheLKhandtheLLGequations; c)andd)ar ecalculatedusing\nthe MC method. Parameters are T0= 0 K,α= 0.5,B0z= 6.65BAandλ= 240 s/F.\nSeveral first periods are omitted.Polarization and magnetization dynamics of a field-driven m ultiferroic structure 12\n-2-10123pΣz(t)\n-1-0.500.51\npΣz(t)\n-1-0.500.51\nEz(t), [J]0.80.91SΣz(t)\n2π/ω · 1/Tprec = 1/4\n2π/ω · 1/Tprec = 3/4\n2π/ω · 1/Tprec = 1\n2π/ω · 1/Tprec = 3/2\n-1-0.500.51\nEz(t), [J]0.80.91\nSΣz(t)2π/ω = 200 MC steps\n2π/ω = 400 MC steps\n2π/ω = 1000 MC steps\n2π/ω = 2000 MC steps(a)\n(b)(c)\n(d)\nFigure 5. Response of the one-dimensional multiferroic structure to the tim e-\ndependent electric field Ez(t) =E0zcosωtplotted for various frequencies ω. The\nfield-free precessional time in the FM part is Tprec\nFM=Tprec≈4 ps. The curves a) and\nb) are obtained by using the LKh and the LLG equations; c) and d) ar e calculated\nusing the MC method. Parameters are T0= 0 K,α= 0.5,E0z= 4.07·107V/m and\nλ= 240 s/F. Several first periods are omitted."    },    {        "title": "2003.13816v2.Magnetic_Noise_Enabled_Biocompass.pdf",        "content": "Magnetic Noise Enabled Biocompass\nDa-Wu Xiao,1Wen-Hui Hu,1Yunfeng Cai,2and Nan Zhao1,∗\n1Beijing Computational Science Research Center, Beijing 100193, China\n2Cognitive Computing Lab, Baidu Research, Beijing 100085, China\n(Dated: April 2, 2020)\nThe discovery of magnetic protein provides a new understanding of a biocompass at the molecular level.\nHowever, the mechanism by which magnetic protein enables a biocompass is still under debate, mainly because\nof the absence of permanent magnetism in the magnetic protein at room temperature. Here, based on a widely\naccepted radical pair model of a biocompass, we propose a microscopic mechanism that allows the biocom-\npass to operate without a finite magnetization of the magnetic protein in a biological environment. With the\nstructure of the magnetic protein, we show that the magnetic fluctuation, rather than the permanent magnetism,\nof the magnetic protein can enable geomagnetic field sensing. An analysis of the quantum dynamics of our\nmicroscopic model reveals the necessary conditions for optimal sensitivity. Our work clarifies the mechanism\nby which magnetic protein enables a biocompass.\nIntroduction. — Experiments have shown that migrating\nbirds employ the geomagnetic field for orientation and nav-\nigation [1, 2]. To understand the physical origin of the nav-\nigation of animals, several physical models [3, 4] have been\nproposed. A widely accepted model is the radical pair model,\nsuggested by Ritz et al. in Ref. [5]. This model assumes that\nthe navigation process is governed by radical pairs, with each\npair consisting of an unpaired electron spin [6]. The pairs are\nusually created via photon excitation and form a spin singlet\nstate [7–9]. In the geomagnetic field and the magnetic field\nprovided by the local molecular environment, the spin singlet\nstate undergoes a transition to spin triplet states [9, 10]. The\nradical pair is metastable and eventually produces di fferent\nchemical products according to the spin states of the radical\npair [11–13], and the chemical products determine the subse-\nquent navigation behavior [5, 11].\nIn the radical pair model, we focus on the singlet-triplet\ninterconversion mechanism at the molecular level. Ho-\nmogeneous geomagnetic fields cannot change the spin sin-\nglet/triplet state because of the conservation of the total spin\nangular momentum. Only inhomogeneous magnetic fields can\ncause transitions between the spin singlet and the spin triplet\nstates. Microscopically, inhomogeneous magnetic fields are\nprovided by the surrounding magnetic moments (either nu-\nclear spins or electron spins) in biological molecules. Through\nthe interaction between these spins, the radical pair can feel\nan effective magnetic field. Nevertheless, the detailed micro-\nscopic origin of the singlet-triplet interconversion process re-\nmains unclear.\nPrevious studies have focused on the nuclear spin environ-\nment around the radical pair. For example, experiments found\nthat the FADH•−O•−\n2molecule (which couples the radical\npair via the hyperfine interaction) is relevant to animal nav-\nigation [14–17]. Theoretically, studies in Refs. [13, 18–21]\nshowed that the nuclear spin environment is capable of pro-\nviding local magnetic fields and enabling a biocompass. Our\nanalysis shows that the nuclear spin concentration and the\nanisotropic dipolar coupling between the radical pair spins\nand the bath nuclear spins play important roles to enable a\nbiocompass [27]. In addition to progress on the nuclear spinbath, Ref. [22] reported a new putative magnetic receptor\n(MagR) and showed that the MagR forms a rod-like magne-\ntosensor complex with the radical pair in the photoreceptive\ncryptochromes. The MagR consists of an Fe-S cluster protein,\nwith the d electrons in the Fe atom contributing the electron\nspins [23, 24]. It is reasonable to assume that the navigation\nbehavior arises from the e ffect of an electronic spin bath.\nHowever, the microscopic role of the magnetic protein is\nunder debate. In Ref. [25], the author pointed out that elec-\ntron spins are hardly polarized at room temperature and cannot\nproduce a significant single-triplet transition process. There-\nfore, it is crucial to elucidate what makes the biocompass pos-\nsible in the absence of a finite magnetization in the MagR.\nIn this paper, we propose that the magnetic field fluctuation,\nrather than the mean magnetization, is capable of producing\nthe spin singlet and triplet transition. The electron spin bath\nof the MagR introduces a fluctuating local magnetic field to\nthe nearby radical pairs via the magnetic dipole-dipole inter-\naction, and this local magnetic field actually enables singlet-\ntriplet interconversion.\nFirst, with a semiquantitative analysis of a radical pair cou-\npling to an electron spin bath in the geomagnetic field, we find\ntwo necessary intuitive requirements for the local magnetic\nfield that need to be satisfied: i) The strength of the noise\nmagnetic field must be comparable to the geomagnetic field\n(∼10−1Gauss), and ii) The local magnetic field should have\ndirectional dependence. Then, we establish a microscopic\nmodel that describes the spin dynamics of the radical pair in\nan electron spin bath. With theoretical analysis and numeri-\ncal calculations, we find that the singlet fidelity of the radical\npair can exhibit a sensitive geomagnetic field direction depen-\ndence. Our work provides new insights into the understanding\nof the biocompass mechanism.\nTheoretical Model. — We consider a radical pair interacting\nwith a spin bath described by the following Hamiltonian:\nH=Hrp+Hbath+Hint, (1)\nwhere Hrp,HbathandHintare the Hamiltonians of the radi-\ncal pair, the bath spin and their interaction, respectively. ThearXiv:2003.13816v2  [physics.bio-ph]  1 Apr 20202\nradical pair consists of two electron spins S1andS2, form-\ning the singlet state |S/angbracketright=(|↑↓/angbracketright−|↓↑/angbracketright )/√\n2 and triplet states\n|T0/angbracketright=(|↑↓/angbracketright+|↓↑/angbracketright)/√\n2,|T+/angbracketright=|↑↑/angbracketright , and|T−/angbracketright=|↓↓/angbracketright [5].\nIn the singlet-triplet representation, the radical pair Hamilto-\nnian is diagonalized as\nHrp=/summationdisplay\nkωk|φk/angbracketright/angbracketleftφk|,for|φk/angbracketright∈{|S/angbracketright,|T0/angbracketright,|T+/angbracketright,|T−/angbracketright},(2)\nwhereωkis the energy of the singlet /triplet state|φk/angbracketright. The\nradical pair is subjected to a magnetic environment consisting\nofNinteracting spins{Ji}N\ni=1.\nHbath=N/summationdisplay\ni=1γiB·Ji+N/summationdisplay\ni>j=1Ji·Di j·Jj, (3)\nwhere Bis the geomagnetic field, γiis the gyromagnetic ratio\nof the i-th bath spin, and Di jis the coupling tensor between Ji\nandJj. The radical pair spins couple to the bath spins through\nthe interaction Hamiltonian\nHint=/summationdisplay\nk,iSk·Aki·Ji≡/summationdisplay\nk=1,2γeSk·bk, (4)\nwhereγeis the electron spin gyromagnetic ratio, Akiis the\ncoupling tensor and bk, as seen by the radical pair spin Sk, is\nthe effective magnetic field caused by the bath spins.\nIndeed, Eqs. (1)–(4) are quite general Hamiltonians de-\nscribing the interacting spins. Since the precise electronic\nstructure of the radical pair and the bath spins of the biocom-\npass system is still unclear, we did not specify the details of\nthe singlet /triplet energies ωkand the concrete forms of the\ncoupling tensors AkiandDi jin Eqs. (1)–(4). However, we\nassume that the random motion of the spins (typically with a\ntime scale>ms [26]) is not fast enough to average out the\nspin dynamics of the radical pair (typically ∼µs [11–13]).\nNevertheless, we will show that we still need some reasonable\nassumptions for ωk,AkiandDi jbased on the known structure\nof the magnetic protein to make the coupled system described\nby Eqs. (1)–(4) exhibit strong sensitivity to the geomagnetic\nfield direction.\nMagnetic Fluctuation. — We study the quantum dynamics\nof the radical pair in an unpolarized spin bath. The radical pair\nis initially prepared in a singlet state with ρrp(0)=|S/angbracketright/angbracketleftS|, and\nthe bath spins are in a high-temperature mixed state\nρbath(0)=N/circlemultiplydisplay\ni=1Ii\nTr[Ii], (5)\nwhereIiis the identity operator for the i-th spin. Starting from\nthe initial state ρ(0)=ρrp(0)⊗ρbath(0), the system evolves to\nρ(t) driven by the Hamiltonians in Eqs. (1)–(4). We focus on\nthe singlet state fidelity PS(t)=Tr/bracketleftbig|S/angbracketright/angbracketleftS|ρ(t)/bracketrightbigof the radical\npair and its dependence on the geomagnetic field direction [5].\nThe field di fferenceδb=b1−b2experienced by the two\nspins of the radical pair causes the singlet-triplet conversion.\nBefore presenting full quantum mechanical calculations of the\n%\n7KH\u0003UDGLFDO\u0003SDLU\u0003HOHFWURQ\u0003VSLQ7KH\u0003PDJ5\u0003HOHFWURQ\u0003VSLQ\u0003EDWK\u0003\u0003\u000bD\f]\\%Q\n\u0014\u0011\u0018\u0015\u0011\u0013\u0015\u0011\u0018\n\u0013 \u0017S \u0015SS\u0017\u0016 S\u000bE\fFIG. 1. (color online). (a) Illustration of the structure of the bath\nelectron spins and the radical pair. The spins are fixed in the proteins.\n(b) Magnetic field fluctuation along the geomagnetic field direction\nδbnBas a function of the geomagnetic field direction θ.\nsinglet fidelity PS(t), we present a qualitative analysis of the\neffect ofδb. One of the key concerns is that the mean value\nof the field di fferenceδbvanishes at room temperature, i.e.,\nTr[ρbath(0)δb]≡0. This condition strongly challenges the\nrole of the MagR in the biocompass mechanism. However,\nthe fluctuation of δbcan also cause the singlet-triplet conver-\nsion. Specifically, in the following, we consider the variation\nin the projection of δbalong the direction nBof the external\nmagnetic field B, i.e.,δb2\nnB=Tr[ρbath(0)(nB·δb)2]. Here, nB\nis given by the Euler angle θ,φofB. We will present require-\nments for the field di fferenceδbto play an important role in\nthe biocompass.\nFirst, the fluctuations δbnBshould have comparable\nstrengths to the geomagnetic field B. In the weak fluctuation\nlimit|δbnB| /lessmuch | B|, the system evolution will be dominated\nby the homogeneous geomagnetic field B, and singlet-triplet\nconversion can hardly occur. However, in the opposite limit\n|δbnB|/greatermuch| B|, the geomagnetic field Bwill have a negligible\ninfluence on the dynamics of PS(t). In both limiting cases,\nthe system does not exhibit a biocompass function. Using the\nstructure obtained in Ref. [22, 27] and assuming electronic\ndipolar coupling between the radical pair spins and the bath\nspins in Eq. (4), we find that the magnitude of the coupling\ntensor Aki∼101MHz, corresponding to the strength of the\nfluctuations δbnB∼10−1Gauss [see Fig. 1(b)], is on the same\norder as the geomagnetic field.\nSecond, the fluctuation of δbshould be sensitive to the\ndirection of the geomagnetic field. This condition requires\nthe coupling Akibetween the radical pair and spin bath to be\nanisotropic. Indeed, the dipolar coupling between the elec-\ntron spins satisfies this requirement. Furthermore, the rod-like\nstructure also enhances the anisotropicity of the field fluctua-\ntion, since the axial and azimuthal directions are obviously\ninequivalent. As an example, Fig. 1(b) shows that the fluctua-\ntion magnitude changes by a factor of ∼2 as the geomagnetic\nfield direction varies by π.\nWith these two intuitive requirements, we find that the dipo-\nlar coupling between the radical pair and the MagR spins is a\npromising candidate to explain the microscopic mechanism of\nthe biocompass. In the following, we discuss the optimal con-\nditions of magnetosensation through the quantum dynamics3\nof the system.\nOptimization of magnetosensation — In the system defined\nby Eqs. (1)-(4), we focus on the dynamics of the singlet fi-\ndelity PS(t)of the radical pair spins. A full analytical cal-\nculation is usually not available for a system of interacting\nelectron spins. Here, we first analyze the short-time behavior\nofPS(t). With the short-time approximation, we obtain the\nqualitative requirements for ωk,AkiandDi jto achieve opti-\nmal magnetosensation of the singlet fidelity, which are further\nconfirmed by numerical simulations.\nThe e ffective field di fferenceδbinduces transitions from\nthe singlet state|S/angbracketrightto the triplet states |T0/angbracketrightand|T±/angbracketrightand causes\na loss of the singlet fidelity. Specifically, we choose the quan-\ntization axis (the zaxis) along the direction of the geomag-\nnetic field. As shown in Fig. 2(a), the longitudinal component\nδbz=δb·ezinduces the transition |S/angbracketright→| T0/angbracketright, while the trans-\nverse components δb±=δb·n±withn±=(∓ex−iey)/√\n2\nresult in the transitions |S/angbracketright→| T±/angbracketright. The field di fferenceδb\nis nonstatic in a full quantum mechanical treatment. The dy-\nnamics of the e ffective field di fference is determined by the\ninteraction within the bath spins as\nδb(t)=eiHbathtδb(0)e−iHbatht. (6)\nThe expectation value of the field di fference/angbracketleftδb(t)/angbracketrightvan-\nishes. However, δb(t) has finite fluctuations: /angbracketleftδb(t)δb(t/prime)/angbracketright=\nTr[δb(t)δb(t/prime)ρbath(0)]/nequal0.\nTo understand the dynamic properties of the field di fference\nδb(t), it is necessary to investigate the interaction within the\nbath spins. As illustrated in Fig. 1(a), the MagR spin bath\nconsists of several ring structures. Within a ring, the distance\nbetween the spins is approximately 1-2 nm, while the inter-\nring distance is greater than 5 nm (see [27] for the coordi-\nnates of the electron spins). Suppose that the spins are all\nS0TT\u000e\nT\u00100GnbG\n\u000enb\nG\u0010nb\n/MHzm\n0200\n100\n-100(a)\n(b)(c)\n~32~63\n~94~103\n-94 -63 -32 0 32 63 940.000.050.100.150.20-5 0 50.00.10.2\n94 1030.00.1\nFIG. 2. (color online). (a) Illustration of the radical pair’s energy\nspectrum and the corresponding noise component during the transi-\ntion. (b) The energy spectrum of the two rings in the MagR (see\ntext). (c) The noise spectrum S0(ω) for the two rings in di fferent geo-\nmagnetic directions, which shows broadened peaks near 0 ,32,63,94,\nand±103 MHz. Here, for the convenience of illustration, we add\nLorentzian broadening of 0 .3 MHz/lessmuch∆ωpeak.coupled through the magnetic dipole-dipole interaction, i.e.,\nDi j=µ0γeγe/planckover2pi1\nr3\ni j/parenleftBig\n1−3ˆri jˆri j/parenrightBig\n, with ri jbeing the distance between\ntwo spins and ˆri jbeing the unit coordinate vector. Since the\ndipolar interaction strength decays as r−3\ni j, the coupling of the\nspins within a ring ( ∼101MHz) is much stronger than that\nin different rings ( <100MHz). Figure 2(b) shows the energy\nspectrum/epsilon1mofHbathobtained by diagonalizing the Schrdinger\nequation\nHbath|ψm/angbracketright=/epsilon1m|ψm/angbracketright, (7)\nwith|ψm/angbracketrightbeing the eigenstate. The energy spectrum /epsilon1m\nforms several discrete bands around 0 ,±32,±63,±94,and\n±103 MHz, resulting from the strong interaction of the elec-\ntron spins within a ring. Each band is further broadened due\nto the weak interaction of the electron spins between the rings.\nWith the transition probability PS→Tα(t) from the singlet\nstate|S/angbracketrightto the triplet states |Tα/angbracketright(forα=0 or±), the sin-\nglet fidelity is expressed as\nPS(t)=1−/summationdisplay\nαPS→Tα(t). (8)\nIn the short-time limit, the transition probability PS→Tα(t) is\napproximated as [27–29]\nPS→Tα(t)=γ2\ne/integraldisplay∞\n−∞Sα(ω)F(t,ω;ωS Tα)dω (9)\nwithωS Tα=ωS−ωTα. In Eq. (9), the function Sα(ω) is the\npower spectrum of the e ffective field di fferenceδbα\nSα(ω)=1\n2N/summationdisplay\nm,n|/angbracketleftψm|δbα|ψn/angbracketright|2δ(ω−/epsilon1mn) (10)\nwith/epsilon1mn=/epsilon1m−/epsilon1n; the function F(t,ω;ωS Tα), defined as\nF(t,ω;ωS Tα)=sin2/parenleftBigωt+ωS Tαt\n2/parenrightBig\n(ω+ωS Tα)2, (11)\nis regarded as a spectrum filter function in the frequency do-\nmain, which exhibits a peak centered at ωS Tαwith width\n∆ωfilter=1/t[28]. Moreover, to directly relate the singlet\nfidelity PS(t) to the biochemical process, we define the singlet\nproductivity on a relevant time scale\nΦS(τ;θ;φ)=1\nτ/integraldisplayτ\n0PS(t;θ;φ)dt, (12)\nwhich is a function of the geomagnetic direction. Here, τis\nthe relevant time scale in the radical pair model, chosen to be\n1µsin the subsequent discussion [11, 13].\nThe power spectrum Sα(ω) describes the dynamic prop-\nerty of the field di fferenceδbαin the frequency domain. As\nan example, Fig. 2(c) shows the power spectrum S0(ω) [see\n[27] for more results for S±(ω)]. Due to the band structure\nof the eigenenergies /epsilon1m[see Fig. 2(b)], the power spectrum\nexhibits broadened discrete peaks around specific transition4\n00.2 0.4 0.6 0.8 1.00.940.960.981.00\n0 0.2 0.4 0.6 0.8 1.00.40.60.81.00 0.2 0.4 0.6 0.8 1.00.40.60.81.0\n0.40.60.81.0(a) (c)\n(b) (d)\n04S2S SS43\nFIG. 3. (color online). The numerical result of the singlet fidelity\nas a function of time for di fferent representative geomagnetic di-\nrections: (a) when all ωSTido not overlap with the filter spectrum\n(ωST0=20 MHz,ωST+=23 MHz,ωST−=17 MHz), (b) when\nallωSTioverlap with the filter spectrum but all the triplets are de-\ngenerate (ωST0=0.1 MHz,ωST±=±3.0 MHz), and (c) when\nωST0overlaps with the noise spectrum S0(ω) while the others do\nnot(ωST0=0.01 MHz,ωST±=±20.0 MHz). (d) The singlet pro-\nductivity as a function of the geomagnetic field direction θforφ=0.\n[30]\nfrequencies [e.g., ωpeak=0,±32,±63,±94, and±103 MHz\nforS0(ω) shown in Fig. 2(c)], with a typical peak width\n∆ωbath∼101MHz. Furthermore, due to the anisotropic dipo-\nlar coupling between the spins and the rod-like geometric con-\nfiguration of the MagR, the power spectrum exhibits a depen-\ndence on the geomagnetic field direction. Figure 2(c) shows\nthat the amplitudes of the power spectrum peaks of S0(ω) are\nvery sensitive to the di fferent geomagnetic field directions.\nThe overlap between the power spectrum Sα(ω) and the fil-\nter function F(t,ω;ωS Tα) determines the loss of the singlet fi-\ndelity, as shown in Eq. (9). With this observation, we propose\nthe following necessary conditions for a robust biocompass to\nexhibit a strong dependence on the geomagnetic field direc-\ntion.\nFirst, at least one of the peaks of the power spectrum\nmust be in resonance with the singlet-triplet transition, i.e.,\n|ωS Ti−ωpeak|<∆ωbath. Essentially, the singlet fidelity loss in\nthis case can be understood by the Fermi Golden rule, where\nthe MagR spins provide resonant perturbations that cause the\nsinglet-triplet transition of the radical pair [28]. Figure 3(a)\nshows an opposite example, in which the frequencies of the\npower spectrum peaks and the singlet-triplet transition are\nmismatched. In this case, the radical pair spins can hardly\ntransition from the singlet state to the triplet states. Thus, the\nsinglet productivity is very close to unity and has a negligible\ngeomagnetic direction dependence [Fig. 3(d)].\nSecond, the energy splittings of the triplet states are crucial\nto the biocompass. Assuming that the resonance condition\nmentioned above is satisfied, and the three triplet states arenearly degenerate ( ωTα≈ωT). In this case, Eq. (8) becomes\nPS(t)≈1−γ2\ne/integraldisplay∞\n−∞/summationdisplay\nαSα(ω)F(t,ω,ω S T)dω, (13)\nwhereωS T=ωS−ωTand the total power spectrum is\n/summationdisplay\nαSα(ω)=1\n2N/summationdisplay\nm,n|/angbracketleftψm|δb|ψn/angbracketright|2δ(ω−/epsilon1mn). (14)\nNote that the total power spectrum depends on the magnitude\nof the field di fference, which is insensitive to the geomag-\nnetic field direction. Although the eigenstates |ψm/angbracketrightand|ψn/angbracketright\nin Eq. (14) depend on the geomagnetic field direction, this de-\npendence could be rather weak, particularly when averaging\nover all eigenstates. This result is verified by our numerical\ncalculations [see [27] regarding/summationtext\nαSα(ω)]. Figure 3(b) shows\nthe singlet fidelity of the radical pair when the three triplets\nare degenerate. The MagR spins cause a remarkable transi-\ntion from the singlet state to the triplet states. However, the\nsensitivity to the field direction is significantly reduced [see\nFig. 3(d)]. In sharp contrast, the nondegenerate case shows a\nstrong magnetosensation ability [see Fig. 3(c)].\nIncoherent e ffect.— Thus far, we have focused on the co-\nherent dynamics of the radical pair spins and the MagR spins.\nHowever, since the whole magnetosensation system is in-\nevitably subjected to a biological environment (mainly via\nthe electron-phonon interactions at the room temperature),\nenvironment-induced decoherence must be considered. Ac-\ncordingly, we include the relaxation and decoherence of the\nMagR bath spins, which are governed by the Lindblad equa-\ntion\n˙ρ=−i[H,ρ]+3N/summationdisplay\ni=1γi/braceleftBigg\nˆD†\niρˆDi−1\n2ˆD†\niˆDiρ−1\n2ρˆD†\niˆDi/bracerightBigg\n,(15)\nwhere ˆDi=σ(z)\niandσ(±)\ni(the Pauli matrices) for spin de-\nphasing and spin relaxation processes, respectively, and γi\nrepresents the corresponding relaxation and dephasing rates\n[31, 32]. For simplicity, we set γi≡γfor all of the MagR\nspins. Figure 4(a) compares the singlet fidelity with and with-\nout the e ffect of environmental decoherence. With the deco-\nherence process described in Eq. (15), the radical pair can still\n0 0.20.40.6 0.8 1.00.00.20.40.60.81.0\n0.50.60.70.8\n04S2S SS43(a) (b)\nFIG. 4. (color online) (a) The singlet fidelity for di fferent relaxation\nand dephasing rates γfor di fferent geomagnetic field directions. (b)\nThe singlet productivity as a function of the geomagnetic field direc-\ntionθfor di fferent relaxation and dephasing rates.5\nundergo a singlet-triplet transition. However, the geomag-\nnetic field direction sensitivity is significantly reduced in an\nenvironment with strong decoherence, as shown in Fig. 4(b).\nIn this sense, the result indicates that the quantum coherence\nwithin the MagR is crucial to the biocompass.\nConclusion. — In summary, we establish a microscopic\nmodel of the magnetic-protein-assisted biocompass and an-\nalyze the physical origin of the magnetosensation. With quan-\ntum mechanical calculations, we show that the magnetosen-\nsation of the radical pair is the consequence of the magnetic\nfluctuation of the MagR rather than the mean magnetization.\nFurthermore, we discover that microscopic spin coupling and\nthe level structure of the MagR and radical pair spins are es-\nsential to the magnetosensation. We propose two general nec-\nessary conditions, a resonance condition and a nondegeneracy\ncondition, for the biocompass. These conditions provide more\nquantitative criteria for candidate biocompass systems and can\nbe examined in future biophysical experiments at the molec-\nular level with well-developed experimental electron spin res-\nonance (ESR) and nuclear magnetic resonance (NMR) tech-\nniques. We also find that quantum coherence plays an im-\nportant role in the geomagnetic field navigation process. This\nfinding could inspire studies of various quantum e ffects in bi-\nological systems and bionic applications of artificial quantum\nsystems.\nWe thank Prof. Chang-Pu Sun, Prof. Can Xie, Prof. Ren-\nBao Liu and Dr. Yi-Nan Fang for their inspiring discussions.\nWe thank Prof. Hai-Guang Liu for sharing the knowledge of\nthe MagR structure. We also thank Prof. Hai-Guang Liu,\nProf. Jin Yu, Prof. Peng Zhang for their comments on the\nmanuscript. 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Wang, X. Rong, R.-B.\nLiu, and J. Du, Nature Commun. 2(2011).Supplemental Materials: Magnetic Noise Enabled Biocompass\nDa-Wu Xiao,1Wen-Hui Hu,1Yunfeng Cai,2and Nan Zhao1,∗\n1Beijing Computational Science Research Center, Beijing 100193, China\n2Cognitive Computing Lab, Baidu Research, Beijing 100085, China\n(Dated: April 2, 2020)\nSpinCoordinates\nSpinCoordinates\nx(Å) y(Å) z(Å) x(Å) y(Å) z(Å)\nJ111.4028 -3.63309 83.1636 J210.7877 4.58341 77.0411\nJ3-10.8003 3.79766 82.7676 J4-11.0965 -4.15934 77.1673\nJ5 -2.117 11.5884 28.8328 J6 -8.695 7.23216 24.3528\nJ7 2.187 -11.5131 29.0298 J8 8.9895 -7.46034 23.9188\nJ98.68875 7.76766 -23.8662 J10 1.9615 11.3649 -29.3439\nJ11 -8.9735 -7.76709 -23.8702 J12-2.37825 -11.4266 -29.3517\nJ13 -10.687 3.63041 -77.2724 J14 -11.286 -4.44509 -82.5264\nJ15 11.2882 -3.62059 -77.4357 J16 10.728 4.06066 -82.6077\nS1 20.8 12 0 S2 20.8 -12 0\nTABLE S1: The coordinates of the MagR bath spins [S1] and the radical pair spins. Here, Jiare the bath spins, and Siare the radical pair\nspins. In our main text, we only consider the spins J5−12for simplicity.\n(a)\n(b)\n(a): From y view-point\n(b): From z view-point\nThe magR bath spin\nThe Radical Pair spinzx\niJ\niS\nFIG. S1: (color online). Coordinate demonstration of the bath spins and the radical pair spins in Table (S1). The MagR spin bath consists of\nseveral rings. Within a ring, the distance between two spins is approximately 1 ∼2 nm, while the distance between two spins is greater than\n5 nm. The average distance between the radical pair and the MagR spin bath is approximately ∼101nm, which is neither too far nor too close.\nThe distance between two radical pair spins is approximately 2 .4 nm. We also study the e ffect of the radical pair spin positions in Fig. (S7).\n0200\n100\n-100(b)\n~32~63\n~94~103\n0200\n100\n-100(a)\n~33~99~58106~/MHzm\n/MHzm\nFIG. S2: (color online). (a) The energy spectrum of the MagR spin bath for one ring, i.e., J5−8. (b) The energy spectrum of the MagR spin\nbath for two rings, i.e., J5−12. We show that the bath spin energy spectrum forms several bands, which correspond to the strong interaction\nwithin the rings. The weak interaction between the rings splits the energy bands further. Additionally, we notice that the characteristic splitting\nwidth of each energy band is larger than >1 MHz, which sets the linewidth of the noise spectrum.arXiv:2003.13816v2  [physics.bio-ph]  1 Apr 20202\nSHORT-TIME ANALYSIS OF THE SINGLET FIDELITY DYNAMICS\nIn this section, we provide a short-time analysis of the singlet fidelity dynamics [S2–S4]. As shown in the main text, we focus\non the dynamics of the singlet fidelity (most symbols are defined in the main text)\nPS(t)=Tr/bracketleftBig\n|S/angbracketright/angbracketleftS|e−iHtρrp(0)⊗ρbath(0)eiHt/bracketrightBig\n=1\n2NTr/bracketleftBig\n|S/angbracketright/angbracketleftS|e−iHt|S/angbracketright/angbracketleftS|eiHt/bracketrightBig\n, (S1)\nwhere 1/2N=/circlemultiplytextN\ni=11/Tr[Ii] andIi=1/2 is the multiplicity of the bath spins. In the interaction picture with respect to Hrp+Hbath,\nthe singlet fidelity can be approximated as\nPS(t)≈1−1\n2N/integraldisplayt\n0/integraldisplayt1\n0Trbath[/angbracketleftS|V(t 2)V(t 1)+V(t 1)V(t 2)|S/angbracketright]dt1dt2+1\n2N/integraldisplayt\n0/integraldisplayt\n0Trbath[/angbracketleftS|V(t2)|S/angbracketright/angbracketleftS|V(t1)|S/angbracketright], (S2)\nwhere V(t)isHintin the interaction picture, defined by\nV(t)=/summationdisplay\nk=1,2γeSk(t)·bk(t)=/summationdisplay\nk=1,2γe/parenleftBig\neiHrptSke−iHrpt/parenrightBig\n·/parenleftBig\neiHbathtbke−iHbatht/parenrightBig\n. (S3)\nThe singlet fidelity can be simplified as\nPS(t)≈1−1\n2N/summationdisplay\nα=0,±/integraldisplayt\n0/integraldisplayt1\n0Trbath[(/angbracketleftS|V(t2)|Tα/angbracketright/angbracketleftTα|V(t1)|S/angbracketright+/angbracketleftS|V(t1)|Tα/angbracketright/angbracketleftTα|V(t2)|S/angbracketright)]dt1dt2\n=1−1\n41\n2Nγ2\ne/summationdisplay\nα=0,±/integraldisplayt\n0/integraldisplayt1\n0Trbath/bracketleftBig\neiωSTα(t2−t1)(nα·δb(t2)) (nα·δb(t1))†+h.c./bracketrightBig\ndt1dt2, (S4)\nwhereωSTα=ωS−ωTα,n0=nB≡ez, and n±=(∓ex−iey)/√\n2. In the eigenspace of Hbath,\nHbath|ψm/angbracketright=/epsilon1m|ψm/angbracketright, (S5)\nthe singlet fidelity can be further simplified as\nPS(t)≈1−1\n2γ2\ne/summationdisplay\nα=0,±/integraldisplayt\n0/integraldisplayt1\n0/summationdisplay\nm,ncos/parenleftbig/parenleftbig/epsilon1mn+ωSTα/parenrightbig(t2−t1)/parenrightbig|/angbracketleftψm|nα·δb(0)|ψn/angbracketright|2dt1dt2\n=1−γ2\ne/summationdisplay\nα=0,±/summationdisplay\nm,n|/angbracketleftψm|nα·δb(0)|ψn/angbracketright|2sin2/parenleftBig/epsilon1mnt+ωSTαt\n2/parenrightBig\n(/epsilon1mn+ωSTα)2, (S6)\nwhere/epsilon1mn=/epsilon1m−/epsilon1n. Define the noise spectrum as\nSα(ω)=1\n2N/summationdisplay\nm,n|/angbracketleftψm|nα·δb(0)|ψn/angbracketright|2δ(ω−/epsilon1mn), (S7)\nand we have\nPS(t)=1−/summationdisplay\nα=0,±γ2\ne/integraldisplay∞\n−∞Sα(ω)F(t,ω,ω STα)dω (S8)\nwith\nF(t,ω,ω STα)=sin2/parenleftBigωt+ωSTαt\n2/parenrightBig\n(ω+ωSTα)2. (S9)\nFig. (S3), Fig. (S4) and Fig. (2c) in our main text study the properties of the noise spectrum in detail. In Fig. (S5), we study\nthe properties of the filter spectrum F(t,ω,ω STα) in detail. We also compare the approximation result with the exact numerical\nresult of the singlet fidelity in Fig. (S6), which shows that the approximation works very well on a short time scale.3\n-94 -63 -32 0 32 63 940.00.10.20.30.4-5 0 50.00.20.4\n94 1030.00.1\n0 /4 /2 3 /40.00.10.20.30.4(a) (b)\nFIG. S3: (color online). (a) The noise spectrum component S±(ω) as a function of ω.S±(ω) has similar behavior as S0(ω) shown in the main\ntext; i.e., it has discrete peaks near 0 ,±32,±63,±64,and±103 MHz. The strength of the noise spectrum is also dependent on the geomagnetic\ndirection. (b) We show, when ω=0,94MHz, the noise spectrum S±,0(ω=0,94MHz) as a function of the geomagnetic direction θto clarify the\ngeomagnetic direction dependence of the noise spectrum. The noise spectrum strength at the specified ωis very sensitive to the geomagnetic\ndirection. This signature is very important for a biocompass, which we have pointed out in the relevant discussion of our main text. This result\nalso sheds light on the mechanism of the radical pair biocompass model. More attention should be paid to the noise spectrum for a realistic\nbiological system, rather than the mean magnetization.\n-94 -63 -32 0 32 63 940.00.20.40.60.81.0-5 0 50.00.51.0\n94 1030.00.10.2\n0 /4 /2 3 /40.10.30.50.70.9(a) (b)\nFIG. S4: (color online). In the main text, we have shown that the total noise spectrum Stot(ω)≡/summationtext\nα=0,±Sα(ω) is irrelevant with respect\nto the geomagnetic direction, which we demonstrate in Fig. (S4). (a) Stot(ω) as a function of ωfor di fferent geomagnetic directions. (b)\nStot(ω=0,94 MHz) as a function of the geomagnetic field direction θ. The geomagnetic direction dependence of Stot(ω) is canceled.\nHowever, one can still observe a very small dependence of the geomagnetic direction. The geomagnetic field changes the Zeeman energy of\nHbath, which will consequently have a small e ffect on|ψm/angbracketrightandStot(ω).\n0 5 10 15 2000.10.2filterZ' 1\nFIG. S5: (color online). The filter spectrum for ωSTα=−10 MHz and t=1µshas sharp peaks near ω=−ωSTα, and the linewidth of the noise\nspectrum is approximately 1 MHz.4\n0 0.2 0.4 0.6 0.8 10.940.960.981.00\n0 0.2 0.4 0.6 0.80.940.960.981.000 0.2 0.4 0.6 0.8 10.40.60.81.0\n0 0.2 0.4 0.6 0.8 10.40.60.81.00 0.2 0.4 0.6 0.8 10.40.60.81.0\n0 0.2 0.4 0.6 0.8 10.40.60.81.0(f)(e)\n(d)(c)\n(b)(a)\nFIG. S6: (color online). We compare the approximation result and the exact numerical calculation of the singlet fidelity for all of the cases\nin Fig. (3) of our main text. Here, (a,b), (c,d), and (e,f) correspond to the cases of Fig. (3a), Fig. (3b), and Fig. (3c) in our main text. Our\ncomparison shows that the short-time approximation fit very well with our exact numerical calculation.\n0 /4 /2 3 /40.50.60.70.80.9\n0 /4 /2 3 /40.20.40.60.8(b) (a)\nFIG. S7: (color online). In our model, the position of the radical pair electron spins is arbitrarily chosen because of the lack of in-\nformation about the radical pair spin positions. Here, we study the e ffect of the radical pair spin positions on the biocompass sen-\nsitivity (which is the singlet productivity in our model). (a) The singlet productivity behavior when the distance between two radi-\ncal pair spins changes, where RP 1,RP2,RP3represents the position of the radical pair spins rS1=(20.8,6,0),(20.8,12,0),(20.8,18,0)\nandrS2=(20.8,−6,0),(20.8,−12,0),(20.8,−18,0) in units of Å. (b) The singlet productivity behavior when the orientation (with the\nsame distance) between two radical pair spins changes, where RP 4,RP5,RP6represents the position of the radical pair spins rS1=\n(20.8,12,0),(20.8,8.48,8.48),(20.8,0,12) and rS2=(20.8,−12,0),(20.8,−8.48,−8.48),(20.8,0,−12) in units of Å. Our results show that\nthe behavior of ΦS(τ;θ;φ) is very sensitive to the orientation of the radical pair, while the distance between the two radical pair spins only\nslightly adjusts the biocompass sensitivity. Our results also suggest that detailed information about the radical pair spin positions is very\nimportant for a biocompass model.\n0 /2 3/2 20.60.70.8\n0/23/22\n0/2 3/220.600.650.700.75(b) (a) (c)\n0.0 0.2 0.4 0.6 0.8 1.00.40.60.81\nFIG. S8: (color online). In the main text, we only consider the singlet fidelity as a function of the axial angle of the geomagnetic field, θ.\nNow, we study the dynamic sensitivity and biocompass sensitivity as a function of the azimuthal angle of the geomagnetic field, φ. Here, the\nenergy structures of the radical pair spins are the same as those in Fig. (3c). (a) The singlet fidelity behavior for di fferentφ. (b) The singlet\nproductivity as a function of φfor di fferentθ. (c) The singlet productivity density plot as a function of θandφ. Our result shows that the\nbiocompass also works for sensing the azimuthal angles φ. These results are very important to the radical pair model in terms of explaining\nthe navigation mechanism of immigrating animals, because sensitivity to φmeans that animals can distinguish back and forth in immigration.5\nTHE EFFECT OF THE NUCLEAR SPIN\nGenerally speaking, the nuclear spins provide a fluctuating magnetic field via the hyperfine couplings to the radical pair. Since\nthe nuclear spins are unpolarized, the mean value of the fluctuating field is zero, but the variance of the field is finite. In this\nsense, the e ffect of the nuclei can be understood by the similar picture of MagR electron spins as we studied in the manuscript,\ni.e.,the magnetic noise enabled singlet-triplet transition . Nevertheless, the nuclear spins di ffer from the electron spins by their\n∼10−3times smaller magnetic moment, their hyperfine couplings to the radical pair and their spatial distribution. Indeed, the\nnuclear spin e ffect was studied both theoretically and experimentally [Refs. [S5–S7]]. Here, we provide an analysis on the\nnuclear spin e ffect on the biocompass, with a comparison with the electron spin bath of MagR.\nIn principle, a radical pair coupled to a nuclear spin bath can be described by the following Hamiltonians:\nH=Hrp+Hbath+Hhf. (S10)\nThe Hamiltonian of the radical pair Hrpis the same as Eq. (2) in the main text. The spin bath Hamiltonian is\nHbath=N/summationdisplay\ni=1γiB·Ii+N/summationdisplay\ni>j=1Ii·Di j·Ij, (S11)\nwhere{Ii}N\ni=1are nuclear spins, Di jis the dipolar coupling tensor. The hyperfine coupling is described by\nHhf=N/summationdisplay\nk,iSk·Aki·Ii≡/summationdisplay\nkSk·b(n)\nk, (S12)\nwhereAkiis the hyperfine tensor, b(n)\nkis the nuclei-induced magnetic field to Sk. The hyperfine interaction Hhfconsists of two\nparts: the Fermi contact interaction\nHF=N/summationdisplay\nk,iA(F)\nkiSk·Ii≡/summationdisplay\nkb(F)\nk·Sk (S13)\nand the dipole-dipole coupling\nHD=N/summationdisplay\nk,iSk·A(D)\nki·Ii≡/summationdisplay\nkb(D)\nk·Sk. (S14)\nHere, we define the Fermi contant interaction induced magnetic field b(F)\nkand the dipole-dipole interaction induced magnetic\nfieldb(D)\nk. Note that the Fermi contact interaction is isotropic, while the dipole-dipole interaction is anisotropic.\nAlthough the Hamiltonians above have similar forms to the Hamiltonians of an electron spin bath [i.e., Eqs. (1)-(4) in the\nmain text], the relative strength of Hhf[or the Hamiltoinan Hintfor the electron spin bath, see Eq.(4) in the main text] and Hbath\nis quite di fferent. To characterize this di fference, we define two ratios ηn=||Hhf||\n||Hbath||andηe=||Hint||\n||Hbath||, where||·||denotes the matrix\nnorm.\n•For nuclear spin bath, the interactions within the bath Hamiltonian Hbathis much weaker than the hyperfine coupling Hhf\nto the radical pair, i.e., ηn/greatermuch1.\nIn the bath Hamiltonian Hbath, the Zeeman energy of a nuclear spin in geomagnetic field is in the order of ∼kHz, and\nthe typical dipole-dipole interaction between two nuclear spins (e.g., two hydrogen nuclear spins separated by 2 Å) is\n∼101kHz at the most. However, the Fermi contact coupling in HFcould reach∼103kHz or even stronger (depending on\nthe electron spin density at the nuclei), and the dipole-dipole interaction in HDcould be∼102kHz for neighboring nuclei.\n•For electron spin bath, the interaction within the bath Hamiltonian Hbathis typically stronger than the coupling Hintto the\nradical pair, i.e., ηe/lessmuch1.\nNote that both interactions within HbathandHintare dipole-dipole couplings, which depends on the distance ri jbetween\ntwo electron spins JiandJjasr−3\ni j. The distance between spins within the MagR ( ∼2 nm for a Fe-S ring) is much smaller\nthan the distance between the MagR spins and the radical pair spins ( ∼10 nm in our model). Consequently, the interaction\nwithin Hbathcan reach∼103kHz, while the strength of Hintis in the order of∼102kHz.6\n0.0 0.2 0.4 0.6 0.8 1.00.40.60.81.0\n0 /4 /2 3 /40.60.650.70.0 0.2 0.4 0.6 0.8 1.00.40.60.81.0\n0 /4 /2 3 /40.50.60.7(a) (b)\n(c) (d)\nFIG. S9: (color online). (a) The dynamics of the singlet fidelity PS(t) between the nuclear spin bath with /without the dipole-dipole interaction\nwithin the spin bath. The dipole-dipole interaction within the nuclear spin bath has negelible e ffect on PS(t) in 1µs. (b) The singlet productivity\nΦ(τ,θ,φ ) as a function of the geomagnetic field direction θwithφ=0. The dipole-dipole interaction within the nuclear spin bath is unimportant\nfor the biocompass. (c) The same as (a) but for the electron spins bath. The dipole-dipole interaction is strongly a ffects the dynamics of the\nsinglet fidelity. (d) The same as (c) but for the electron spins bath. The dipole-dipole interaction is crutial for the biocompass.\nWith the analysis of the coupling strength ratios, an obvious di fference between nuclear spin bath and electron spin bath is the\nimportance of interactions within the bath. Figure (S9 (a) & (c)) shows the numerical results of the evolution of the singlet\npopulation with and without the nuclear spin dipolar interaction. In this case, the dipole-dipole coupling between the bath spins\n(<10 kHz) is too week to produce any observable e ffect in the relevant time scale ( ∼1µs), which agrees with the previous\nstudies [Refs. [S8, S13, S14]]. While, in the case of electron spin bath, the dipole-dipole coupling between bath spins strongly\naffect the dynamics of the singlet population and total singlet productivity [see Fig. S9(b) & (d)].\nBecause of the di fferent relative strength between the nuclear spin bath and the electron spin bath, we need di fferent theoretical\ntools.. The small ratio ηe/lessmuch1 for the electron spin bath allows the perturbation treatment of the interaction Hint. As we have\nshown in our manuscript, the singlet-triplet transition caused by MagR can be well understood by the picture of noise spectrum\nand the Fermi golden rule, which is essentially a perturbation treatment. However, the fact ηn/greatermuch1 for the nuclear spin bath\nindicates the dynamics of the nuclear spins are strongly a ffected by their coupling to the radical pair. A simple perturbative\npicture in this case is usually not available, and a full quantum mechanical calculation has to be performed as shown in Refs.\n[S5–S8].\nIndeed, the sharp contrast between electron spin bath and nuclear spin bath was studied in the system of solid-state spin qubit.\nExperimental [Ref. [S9]] and theoretical [Ref. [S10]] works have shown that the e ffect of electron spin bath can be well modeled\nby a stochastic process. While, in the nuclear spin bath case, the back-action of the hyperfine coupling causes counter-intuitive\nbehavior of the central spin [Refs. [S11, S12]]. Here, the biocompass system and the radical pair model provides a second\nexample to elucidate the di fferent environmental e ffect between electron and nuclear spin baths.\nBesides the di fference of Hbathbetween the nuclear spin bath and the electron spin bath, the interaction Hhfin the nuclear spin\nbath is also crucial to the magnetic biocompass. As we have shown in Eq. (S12), in additional to the dipole-dipole coupling HD,\nthe hyperfine interaction Hhfbetween the nuclear spin bath and the radical pair contains the Fermi contact coupling HF, which\nis not present in the electron spin bath case. To study the role of HDandHFof the nuclear spin bath, we first consider a toy\nmodel with a single nuclear spin with the following Hamiltonian:\nHtoy=γB·(S1+S2)+(AFS1+S1·D)·I, (S15)\nwhere AFis the Fermi contact interaction strength and D=µ0γeγn/planckover2pi1(1−3ˆrˆr)\n4πr3 is the dipole-dipole coupling tensor. We ignored\nthe finite size of the electron wavefunction, and use the point dipole approximation to model the anisotropic coupling. For\nneighboring nuclei, the Fermi contact interaction could be much stronger than the dipole-dipole coupling. However, note that7\n\u0012\u0017 \u0012\u0015 \u0016 \u0012\u0017\u0013\u0011\u0016\u0018\u0013\u0011\u0017\u0013\u0011\u0017\u0018\u0013\u0011\u0018\u0013\u0011\u0018\u0018\n\u000bD\f\n\u0014\u0013\u0010\u0014\u0014\u0013\u0013\u0014\u0013\u0014\u0014\u0013\u0015\u0014\u0013\u0010\u0015\u0014\u0013\u0013\nFIG. S10: (Color Online) (a) The singlet fidelity Φ(τ,θ,φ ) for di fferent di fferent dipolar coupling strength of HD. The Fermi contact coupling\nstrength is 2 .8 MHz. (b) The average dipole field b(D)\nkalong the geomagnetic field direction as a function of the concentration of the nuclear\nspin bath. The horizontal line indicates the typical fluctuating field strength provided by the MagR electron spins as discussed in the main text.\nthe Fermi contact coupling is of isotropic nature. With the Fermi contact coupling only, the nuclear spin bath cannot provide the\nmagnetosensation of the direction of geomagnetic field.\nFigure (S10a) shows the singlet productivity as a function of the magnetic field direction for a given Fermi contact strength\nAF=2.8 MHz and various dipole-dipole coupling strength. In the absence of dipole coupling HD=0, the singlet productivity\nisindependent of the field direction, and increasing the dipole coupling strength enhances the magnetosenstation ability. With\nthis toy model, we find that, despite of the strong Fermi contact coupling, the dipole-dipole coupling plays a key role in the\nbiocompass process. For the radical pair in a nuclear spin bath, the magnetosensation of the geomagnetic field originates from\nthe anisotropicity of the spin dipole-dipole interaciton.\nTo have a more quantitative understanding of the dipole-dipole interaction strength of the nuclear spins, we calculate the\nprojection of the dipole field b(D)\nk(with the dipole-approximation Dki=µ0γeγn/planckover2pi1(1−3ˆrkiˆrki)\n4πr3\nki) along the geomagnetic field direction,\nand average the results over di fferent random bath configurations of a given spin concentration (the number of nuclear spins per\nunit volume). Figure (S10b) shows that the averaged dipole field strength is proportional to the bath spin concentration. This is\nbecause the dipole field strength and the bath spin concentration both scale with the inter-spin distance rasr3.\nComparing the strength of the dipole field produced by the MagR electron spins and the dipole field strength of nuclear\nspins, we find that, with a concentration of the order ∼101nm3, the nuclear spin bath provides a dipole field with comparable\nstrength to the MagR electron spins ( ∼10−1−100Gauss ). The concentration of ∼101nm3is close to the possible nuclear spin\nconcentration in biology systems, which implies that both MagR electron spin bath and the nuclear spin bath around the radical\npair can have comparable contribution to the biocompass process.\nIn summary, we investigate the role of nuclear spin bath by the full quantum mechanical calculations and model analysis.\nParticularly, we clarify the key di fference between the nuclear spin bath and the electron spin in the biocompass process. We\nshow that, for nuclear spin bath, the hyperfine coupling Hhfdominates the quantum dynamics; While, for the electron spin bath,\nthe spin interaction within the bath (i.e., the bath Hamiltonian Hbath) is crucial to understand noise enabled biocompass behavior.\nFurthermore, we analyze the di fferent role of the isotropic Fermi contact coupling HFand the anisotropic dipolar coupling HDof\nthe nuclear spin bath. We show that the nuclear spin bath could provide a comparable dipole field strength to the MagR electron\nspin bath.\n∗Electronic address: nzhao@csrc.ac.cn\n[S1] Qin S, Yin H, Yang C, et al. A magnetic protein biocompass.[J]. Nature Materials, 2016, 15(2):217-226.\n[S2] Cohen-Tannoudji C, Diu B, Laloe F, et al. Quantum Mechanics (2 vol. set)[J]. 2006.\n[S3] Zhao N, Hu J L, Ho S W, et al. Atomic-scale magnetometry of distant nuclear spin clusters via nitrogen-vacancy spin in diamond[J].\nNature nanotechnology, 2011, 6(4): 242.\n[S4] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, 2003).\n[S5] Cai, J., 2011. Quantum Probe and Design for a Chemical Compass with Magnetic Nanostructures. Physical Review Letters, 106(10).\n[S6] Solov’yov, I.A., Chandler, D.E., and Schulten, K., 2007. Magnetic Field E ffects in Arabidopsis thaliana Cryptochrome-1. Biophysical\nJournal, 92(8), pp.27112726.\n[S7] Maeda, K. et al., 2012. Magnetically sensitive light-induced reactions in cryptochrome are consistent with its proposed role as a mag-\nnetoreceptor. Proceedings of the National Academy of Sciences, 109(13), pp.47744779.\n[S8] Cai, J., Guerreschi, G.G., and Briegel, H.J., 2010. Quantum Control and Entanglement in a Chemical Compass. Physical Review Letters,\n104(22).8\n[S9] De Lange, G. et al., 2010. Universal Dynamical Decoupling of a Single Solid-State Spin from a Spin Bath. Science, 330(6000), pp.6063.\n[S10] Wang, Z.-H. et al., 2012. Comparison of dynamical decoupling protocols for a nitrogen-vacancy center in diamond. Physical Review B,\n85(15).\n[S11] Zhao, N., Wang, Z.-Y . & Liu, R.-B., 2011. Anomalous Decoherence E ffect in a Quantum Bath. Physical Review Letters, 106(21).\n[S12] Huang, P. et al., 2011. Observation of an anomalous decoherence e ffect in a quantum bath at room temperature. Nature Communications,\n2(1).\n[S13] Hore, P.J. & Mouritsen, H., 2016. The Radical-Pair Mechanism of Magnetoreception. Annual Review of Biophysics, 45(1), pp.299344.\n[S14] Maeda, K. et al., 2008. Chemical compass model of avian magnetoreception. Nature, 453(7193), pp.387390."    },    {        "title": "1903.07180v2.Dynamical_symmetry_breaking__magnetization_and_induced_charge_in_graphene__Interplay_between_magnetic_and_pseudomagnetic_fields.pdf",        "content": "Noname manuscript No.\n(will be inserted by the editor)\nDynamical symmetry breaking, magnetization and induced charge in\ngraphene: Interplay between magnetic and pseudomagnetic fields\nJ. A. Sánchez-Monroy,a,1,2, C. J. Quimbayb,1\n1Departamento de Física, Universidad Nacional de Colombia, Bogotá, D. C., Colombia\n2Instituto de Física, Universidade de São Paulo, 05508-090, São Paulo, SP, Brazil\nthe date of receipt and acceptance should be inserted later\nAbstract In this paper, we investigate the two competing\neffects of strains and magnetic fields in single-layer gra-\nphene to explore its impact on various phenomena of quan-\ntum field theory, such as induced charge density, magnetic\ncatalysis, symmetry breaking, dynamical mass generation\nand magnetization. We show that the interplay between stra-\nins and magnetic fields produces not only a breaking of chi-\nral symmetry, as it happens in QED 2+1, but also parity and\ntime-reversal symmetry breaking. The last two symmetry\nbreakings are related to the dynamical generation of a Hal-\ndane mass term. We find that it is possible to modify the\nmagnetization and the dynamical mass independently for\neach valley, by strain and varying the external magnetic field.\nFurthermore, we discover that the presence of a non-zero\npseudomagnetic field, unlike the magnetic one, allows us to\nobserve an induced “vacuum” charge and a parity anomaly\nin strained graphene. Finally, because the combined effect\nof real and pseudomagnetic fields produces an induced val-\nley polarization, the results presented here may provide new\ntools to design valleytronic devices.\n1 Introduction\nIn recent times, a number of materials for which quasiparti-\ncle excitations behave like relativistic two-dimensional fermions\nhave appeared in condensed matter. One of the most fasci-\nnating examples of such materials is single-layer graphene,\nwhich is a material consisting of a single-layer of graphite.\nGraphene exhibits many interesting features, among which\nare the anomalous quantum Hall effect [1], a record high\nYoung’s modulus [2], ultrahigh electron mobility [3], as well\nas very high thermal conductivity [4]. The band structure of\nsingle-layer graphene has two inequivalent and degenerate\nae-mail: jasanchezm@unal.edu.co\nbe-mail: cjquimbayh@unal.edu.covalleys, /uni20D7Kand/uni20D7K′, at opposite corners of the Brillouin zone.\nThe possibility to manipulate the valley to store and carry in-\nformation defines the field of “valleytronics”, in a analogous\nway as the role played by spin in spintronics.\nOne of the most exciting aspects of the physics of single-\nlayer graphene is that several unobservable phenomena in\nexperiments of high energy physics may be observed, such\nas Klein tunneling [5] and “Zitterbewegung” [6]. From the\npoint of view of quantum field theory, graphene exhibits\nsimilar features with quantum electrodynamics in three di-\nmensions (QED 2+1)1in such a way that it is possible to\nexplore sophisticated aspects of three-dimensional quantum\nfield theory; for instance, magnetic catalysis, symmetry break-\ning, dynamical mass generation, and anomalies, among oth-\ners. It is possible, in the first place, because the two val-\nleys can be associated with two irreducible representations\nof the Clifford algebra in three dimensions. In the second\nplace, because the low-energy regime quasiparticles behave\nlike massless relativistic fermions, where the speed of light\nis replaced by the Fermi velocity vF, which is about 300\ntimes smaller than the speed of light.\nThe presence of a mass gap may turn single-layer gra-\nphene from a semimetal into a semiconductor. This can be\naccomplished, for example, when a single-layer graphene\nsheet is placed on a hexagonal boron nitride substrate [10,\n11], or deposited on a SiO 2surface [12]. Additionally, it has\nbeen proposed that a bandgap can be induced by vacuum\nfluctuations [13]. Significantly, a mass gap suppresses the\n1Since the electrostatic potential between two electrons on a plane is\nthe usual 1 /slash.leftrCoulomb potential instead of a logarithmic potential,\nwhich is distinctive of quantum electrodynamics in 2 +1 dimensions\n(QED 2+1), the theory that describes graphene at low energies is known\nasreduced quantum electrodynamics (RQED 4;3) [7–9]. In RQED 4;3,\nthe fermions are confined to a plane; nevertheless, the electromagnetic\ninteraction between them is three-dimensional.arXiv:1903.07180v2  [hep-th]  3 Jan 20212\nKlein tunneling so that this fact could be useful in the de-\nsign of devices based on single-layer graphene [14].\nThe phenomenon known as magnetic catalysis appears\nwhen a dynamical symmetry breaking occurs in the pres-\nence of an external magnetic field, independent of its inten-\nsity [15, 16]. Since in QED 2+1the mass term breaks the chi-\nral symmetry in a reducible representation2, magnetic catal-\nysis rise as m→0. Dynamical symmetry breaking is a conse-\nquence of the appearance of a nonvanishing chiral conden-\nsate/uni27E80/divides.alt0T[Y;¯Y]/divides.alt00/uni27E9, which leads to the generation of a fer-\nmion dynamical mass [15]. In particular, when single-layer\ngraphene is subjected to an external magnetic field, a nonva-\nnishing chiral condensate ensures that there will be a dynam-\nical chiral symmetry breaking [15], as well as a dynamical\nmass equal for each valley [17, 18].\nWhen a sample of single-layer graphene presents stra-\nins, ripples or curvature, the dispersion relation is modi-\nfied in such a way that an effective gauge vector field cou-\npling is induced in the low energy Dirac spectrum (the so-\ncalled pseudomagnetic field [19]). The mechanical control\nover the electronic structure of graphene has been explored\nas a potential approach to “strain engineering” [20, 21].\nOriginally, it was observed that strain produces a strong gauge\nfield that effectively acts as a uniform pseudomagnetic field\nwhose intensity is greater than 10 T [22], a pseudomagnetic\nfield greater than 300 T was experimentally reported later\n[20]. This pseudomagnetic field opens the door to previously\ninaccessible high magnetic field regimes.\nIn contrast to the case of a real external magnetic field,\nthe pseudomagnetic field experienced by the particles in the\nvalleys /uni20D7Kand/uni20D7K′have opposite signs. Hence, when a sample\nof strain single-layer graphene is placed in a perpendicular\nmagnetic field, the energy levels suffer a different separation\nfor each valley, which results in an induced valley polariza-\ntion [23]. The previous one is precisely the key requirement\nfor valleytronic devices. Beyond theoretical calculations, the\npresence of Landau Levels in graphene has been experi-\nmentally observed in external magnetic fields [24], strain-\ninduced pseudomagnetic fields [20, 25] and in the coexis-\ntence of pseudomagnetic fields and external magnetic fields\n[26]. Moreover, the effects of the combination of an external\nmagnetic field and a strain-induced pseudomagnetic field in\ndifferent configurations were studied in order to construct a\nvalley filter [27–30].\nIn the first part of this paper, we study how an interplay\nbetween real and pseudomagnetic fields affects the symme-\ntry breaking and the dynamical mass generation. As we will\nsee, the presence of these two fields produces not only a\nbreaking of chiral symmetry but also parity and time-reversal\nsymmetry breaking. Furthermore, we will show that there\n2Because the chiral symmetry cannot be defined for irreducible repre-\nsentations, it does not make sense to talk about chiral symmetry break-\ning.will be a dynamical mass generation of two types, the usual\nmass ( m¯yy) and another known as Haldane mass [31], un-\nlike QED 2+1where only the usual mass term is dynamically\ngenerated. As a result of this, the dynamical fermion masses\nwill be different for each valley. In this paper, we will use\na non-perturbative method based on the quantized solutions\nof the Dirac equation, the so-called Furry picture . The rea-\nson for using this method is to obtain nonperturbative results\nsince the effective coupling constant in graphene is of order\nunity, a≈2:5, raising serious questions about the validity of\nthe perturbation expansion in graphene [32]. However, the\nlatter has been a matter of controversy, since at low energy it\nwas experimentally observed that the effective fine-structure\nconstant approaches 1 /slash.left7 [33].\nIn the second part, we investigate how the presence of\nreal and pseudomagnetic fields affects the magnetization. In\nconventional metals, the magnetism receives contributions\nfrom spin (Pauli paramagnetism) and orbitals (Landau dia-\nmagnetism). In particular, the orbital magnetization of gra-\nphene in a magnetic field has shown a non-linear behavior\nas a function of the applied field [34]. In order to exam-\nine how the magnetization and the susceptibility behave for\neach valley in the presence of constant magnetic and pseu-\ndomagnetic fields, we will first obtain the one-loop effec-\ntive action. The one-loop effective action without strains in\n2+1 dimensions had been previously calculated within the\nSchwinger’s proper time formalism [35–37] and using the\nfermion propagator expanded over the Landau levels [38].\nTaking into account that in static background fields the one-\nloop effective action is proportional to the vacuum energy\n[39], which can be calculated in a direct way employing the\nfurry picture, we use this method to study the most general\ncase. As we will show, the presence of magnetic and pseu-\ndomagnetic fields allows us to manipulate the magnetization\nand the susceptibility of each valley independently.\nFinally, we study the parity anomaly and the induced\nvacuum charge in strained single-layer graphene. In quan-\ntum field theory, if a classical symmetry is not conserved at\na quantum level, it is then said that the theory suffers from an\nanomaly. For instance, in QED 2+1if one maintains an invari-\nant gauge regularization in all the calculations, with an odd\nnumber of fermion species, the parity symmetry is not pre-\nserved by quantum corrections, i.e.it has a parity anomaly\n[35, 40, 41]. In this way, the quantum correction to the vac-\nuum expectation value of the current can be computed to\ncharacterize the parity anomaly. As it was pointed out by Se-\nmenoff [41], external magnetic fields induce a current ( jm)\nof abnormal parity in the vacuum for each fermion species.\nUnfortunately, for a even number of fermion species, the to-\ntal current is canceled: Jm\n+=jm\n1+jm\n2=0, and therefore the in-\nduced vacuum current is not directly observable. Therefore,\nit is possible to maintain the gauge and parity symmetries\neven at the quantum level [35]. In the literature, a number of3\nscenarios were proposed to realize parity anomaly in 2 +1\ndimensions. Haldane introduces a condensed matter lattice\nmodel in which parity anomaly takes place when the pa-\nrameters reach critical values [31]. Obispo and Hott [42, 43]\nshow that graphene coupling to an axial-vector gauge poten-\ntially exhibits parity anomaly and fermion charge fraction-\nalization. Zhang and Qiu [44] show that in a graphene-like\nsystem, with a finite bare mass, a parity anomaly related r-\nexciton can be generated by absorbing a specific photon. Al-\nternatively, as Semenoff remarked [41], one could consider\nan “unphysical” field with abnormal parity coupled to the\nfermions, since for such field the total induced vacuum cur-\nrent should be different from zero and hence observable. As\nwe will show, this field is physical, and it is just a pseudo-\nmagnetic field with a simple uniform field profile.\nThis paper is organized as follows: In section 2, we intro-\nduce the Dirac Hamiltonian and the symmetries for single-\nlayer graphene in a finite mass gap. In section 3, we present\nthe Furry picture for fermions in the presence of real and\npseudomagnetic fields. In section 4, we compute the mag-\nnetic condensate and discuss how this characterizes the sym-\nmetry breaking and its connection with the dynamical mass\nof each valley. In section 5, the one-loop effective action\nand the magnetization are calculate for each valley. In sec-\ntion 6, we calculate the total induced vacuum charge density\nto show that graphene in the presence of a pseudomagnetic\nfield exhibits a parity anomaly. In appendix Appendix A, we\nobtain the exact solution of the Dirac equation for uniform\nreal and pseudomagnetic fields. In appendix Appendix B,\nwe compare the calculation of fermionic condensate in the\nFurry picture with the method via fermion propagator and\nprove that the trace of the fermion propagator evaluated at\nequal space-time points must be understood as the expecta-\ntion value of the commutator of two field operators. Finally,\nsection 7 contains our conclusions.\n2 Dirac Hamiltonian for graphene\nIn a vicinity of the Fermi points, the Dirac Hamiltonian in\nthe presence of real ( A) and pseudo ( a) magnetic potentials\nreads (/uni0335h=vF=1) [45, 46]3\nHD[A;a]=G0Gi(pi+eAi+ieai)+G0m; (1)\nwhere mis a mass gap, ai=a35\niG35,G35=iG3G54. This 4×\n4 Hamiltonian acts on the four-component “spinor”, yT=\n3The electric charge eis multiplying the term aionly for dimensional\nreasons; strictly speaking, one could write eAi=˜Aito emphasize that it\nis independent of e.\n4It turns out that one can identify a35\nias one component of a non-\nAbelian SU(2)gauge field within the low-energy theory of graphene\n[47, 48]. The other two components of this non-Abelian SU(2)gauge\nfield are proportional to G3andG5, since they are off-diagonal in val-\nley index mixing the two inequivalent valleys [47, 48]. In this case,\nthe pseudo-gauge potential is ai=a3\niG3+a5\niG5+a35\niG35. Assuming a(yK\nA;yK\nB;yK′\nA;yK′\nB), where the components take into account\nboth two valleys (/uni20D7Kand/uni20D7K′) and the two sublattices (A and\nB) [49], the quantum number associated with the two sub-\nlattices is usually referred to as pseudospin . If one wants to\ninclude the real spin, the spinor will have eight-components,\nand the Dirac Hamiltonian will be HD(8×8)=I2⊗HD[A;a]\n[47]. For subsequent calculations, it is sufficient to consider\nHD[A;a], given that including the real spin only increases\nthe degeneration of the Landau levels by two ( gs=2). Since\nthe difference between QED 2+1and RQED 4;3lies in the ki-\nnetic term of the gauge fields, and the magnetic field here\nis considered as an external field and the pseudomagnetic\nis a non-dynamical field. Then Eq. (1) is an appropriate de-\nscription for strained graphene in the presence of an external\nmagnetic field.\nAs a matter of convenience, we choose here the G−ma-\ntrices as [50]\nG0=s3⊗s3=/parenleft.alt4s30\n0−s3/parenright.alt4;\nG1=s3⊗is1=/parenleft.alt4is10\n0−is1/parenright.alt4;\nG2=s3⊗is2=/parenleft.alt4is20\n0−is2/parenright.alt4;\nG3=is1⊗I2×2=/parenleft.alt40iI\niI0/parenright.alt4;\nG5=−s2⊗I2×2=/parenleft.alt40iI\n−iI0/parenright.alt4;\nG35=is3⊗I2×2=/parenleft.alt4iI0\n0−iI/parenright.alt4; (2)\nso that(G3)2=−1,(G5)2=1,G3andG5anticommute with\nGm, while G35commutes with Gmand anticommutes with\nG3andG5. Note that Gm(m=0;1;2) are block-diagonal,\nwhere each block is one of two inequivalent irreducible rep-\nresentations of the Clifford algebra in 2 +1 dimensions. For\nodd dimensions, there are two inequivalent irreducible rep-\nresentations of the Dirac matrices that we denote as R1and\nR2. The two inequivalent representations were chosen as gm\nand−gmforR1andR2, respectively, where\ng0=s3;g1=is1;g2=is2: (3)\nGiven that there is no intervalley coupling, we can rewrite\nthe Dirac Hamiltonian as HD=H+[A;a]⊕H−[A;a], thus\nH±=is3si(pi+eAi∓ea35\ni)±s3m; (4)\nwhere H+andH−represent the Hamiltonian near of valley\n/uni20D7K(representation R1) and/uni20D7K′(representation R2), respec-\ntively. H+acts in a two-component spinor that describes a\nsmooth enough deformation in the graphene sheet, one can keep only\nthe component a35\ni, which does not mix the two inequivalent valleys\n[47]. Hence, Eq. (1) captured the physics of low-energy strained gra-\nphene.4\nfermion with pseudospin up and an antifermion with pseu-\ndospin down, while H−acts in a two-component spinor that\ndescribes a fermion with pseudospin down and an antifermion\nwith pseudospin up. Thus, we obtain two decoupled Dirac\nequations in (2+1)-dimensions\ni¶y(x;y;t)\n¶t=H±y(x;y;t): (5)\nFinally, we can write the Lagrangian density for this system\nas the sum of two Lagrangian densities for each valley\nL=L+/ucurlyleft/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/ucurlymid/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/ucurlyright\ni¯yKgmD+\nmyK−m¯yKyK+L−/ucurlyleft/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/ucurlymid/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/ucurlyright\ni¯yK′gmD−\nmyK′+m¯yK′yK′;\n(6)\nwhere D±\ni=¶i+ieAi∓iea35\ni, which can be interpreted as a\nsystem describing two species of two-component spinors,\none with mass +mand coupled to eAi−ea35\niand the other\nwith mass−mand coupled to eAi+ea35\ni. Hence, in the vicin-\nity of the Fermi points, graphene monolayers constitute an\nideal scenario to simulate the matter sector of QED 2+1. In\nthe following, we will neglect the corrections due to the ef-\nfects of Coulomb interactions between the charge carriers.\nHowever, we point out that the model given by Eq. (1) is\nin good agreement with the experiments carried out in gra-\nphene in the presence of external magnetic fields [24, 51],\npseudomagnetic fields [20, 25], and in the combination of\nmagnetic and pseudomagnetic fields [26].\n2.1 Symmetries in the irreducible and reducible\nrepresentations\nIrreducible representations: For irreducible representations,\nit is possible to define the parity ( P), charge conjugation ( C),\nand time-reversal ( T) transformations as follows:\nPy(t;x;y)P−1=−ig1y(t;−x;y); (7)\nTy(t;x;y)T−1=−ig2y(−t;x;y); (8)\nCy(t;x;y)C−1=−g2(¯y(t;x;y))T: (9)\nHerePandCare unitary operators and Tis an anti-linear\noperator [52], i.e.T(c−number)T−1=(c−number)∗. One\ncan check that the mass terms in the Dirac Lagrangian is\nnot invariant under PorT. However, the combined trans-\nformation PTleaves the mass terms invariant, so CPT is a\nsymmetry of the Dirac Lagrangian [53]. Since the gmform\nthree 2×2 matrices and no other matrix anticommutes with\nthem, the chiral symmetry cannot be defined for irreducible\nrepresentations.\nReducible representation: For a reducible representation,\nlet us take the four-component spinor yT=(yK;yK′)T. As\nit has been pointed out in Refs. [50, 54], because the free\nLagrangian uses only three Dirac matrices, parity, chargeconjugation and time-reversal transformations can be imple-\nmented by more than one operator\nPy(t;r)P−1=Pjy(t;r′); (10)\nTy(t;r)T−1=Tjy(−t;r); (11)\nCy(t;r)C−1=Cj(¯y(t;r))T; (12)\nwhere\nP1=−iG1G3; P2=−G1G5; (13)\nT1=−G2G3; T2=−iG2G5; (14)\nC1=−iG0G1; C2=−G2; (15)\nwith r=(x;y)andr′=(−x;y).\nWe present the transformation properties of some bilin-\nears under P,CandTtransformations in Tab. 1 . Using\nthese properties, one can easily prove that the massive (or\nmassless) Dirac Lagrangian in 2 +1 dimensions for the re-\nducible representation is invariant under P,CandT, re-\ngardless of which transformation is used, i.e.the Dirac La-\ngrangian is invariant under P1andP2,C1andC2,T1andT2,\nor even a linear combination of this operator could be used,\nwith some restrictions [54]. In our Lagrangian, , there are\nbilinears given by ¯yy,¯yGmyand ¯yGmG35y, which are\nindependent of j. Therefore, any of the operators Pj,Cjand\nTjcan be used to implement P,CandT, respectively. Sur-\nprisingly, the transformation properties of ¯yG3y,¯yG5y,\n¯yGmG3yand ¯yGmG5ydepends on j. As a result, we find\ntwo non-equivalent realizations of parity, charge conjuga-\ntion and time-reversal, which is unusual5. For example, the\nterms ¯yGmG3yand ¯yGmG5yappear when a non-Abelian\nSU(2)gauge field is introduced in graphene [47, 48]. In\nwhat follows we will be interested only in the Lagrangian\ndensity (6).\nIt should be noted that in the literature one can find two\ndifferent transformations which are defined as time-reversal.\nOne, ˆT, which acts as ˆTy(t;x;y)ˆT−1=ˆTy∗(−t;r). The\nother, T, which is the one considered here, Eq. (11), and is\nreferred as Wigner time-reversal. The latter transformation\nwas defined consistently with what has been done in four\ndimensions by Weinberg (Ch. 5. in [39]), and Peskin and\nSchroeder (Ch. 3. in [52]). Moreover, the one that concerns\ntheCPT theorem is T(for a detailed discussion, see sec.\n11.6. in [55]).\n5In fact, there is an infinite number of non-equivalent realizations of\nP,CandTsince any linear combination of the two realizations found\nis an inequivalent realization.5\nPj Tj Cj\n¯yy(t;r) ¯yy(t;r′) ¯yy(−t;r) ¯yy(t;r)\n¯yGmy(t;r) ¯y˜Gmy(t;r′) ¯y¯Gmy(−t;r) −¯yGmy(t;r)\n¯yGmG35y(t;r) −¯y˜GmG35y(t;r′) −¯y¯GmG35y(−t;r) ¯yGmG35y(t;r)\n¯yiG35y(t;r) −¯yiG35y(t;r′) −¯yiG35y(−t;r) ¯yiG35y(t;r)\n¯yG3y(t;r) (−1)j¯yG3y(t;r′) (−1)j+1¯yG3y(−t;r) (−1)j+1¯yG3y(t;r)\n¯yG5y(t;r) (−1)j+1¯yG5y(t;r′) (−1)j¯yG5y(−t;r) (−1)j¯yG5y(t;r)\n¯yGmG3y(t;r)(−1)j¯y˜GmG3y(t;r′)(−1)j+1¯y¯GmG3y(−t;r)(−1)j+1¯yGmG3y(t;r)\n¯yGmG5y(t;r)(−1)j+1¯y˜GmG3y(t;r′)(−1)j¯y¯GmG3y(−t;r)(−1)j¯yGmG3y(t;r)\nTable 1 P,CandTtransformation properties of some bilinears, here ˜Gm={G0;−G1;G2}and ¯Gm={G0;−Gi}.\nThe transformation properties of the electromagnetic po-\ntential are [53]\nPA0(t;x;y)P−1=A0(t;−x;y);\nPA1(t;x;y)P−1=−A1(t;−x;y);\nPA2(t;x;y)P−1=A2(t;−x;y);\nTA0(t;x;y)T−1=A0(−t;x;y);\nT/uni20D7A(t;x;y)T−1=−/uni20D7A(−t;x;y);\nCAm(t;x;y)C−1=−Am(t;x;y); (16)\nwhich leave the Lagrangian invariant. For pseudo-magnetic\npotential we should have\nPa35\n1(t;x;y)P−1=a35\n1(t;−x;y);\nPa35\n2(t;x;y)P−1=−a35\n2(t;−x;y);\nT/uni20D7a35(t;x;y)T−1=/uni20D7a35(−t;x;y);\nC/uni20D7a35(t;x;y)C−1=/uni20D7a35(t;x;y): (17)\nso that the interaction ¯yGiG35a35\niyin the Lagrangian is in-\nvariant.\nFor reducible representations, the transformation y→\neiam˜smyleaves invariant the kinetic term, where the gener-\nators ˜sm=sm⊗I2×2={I4×4;−iG3;−G5;−iG35}are the gen-\nerators of a global U(2)symmetry, with s0≡I2×2and the\namare taken as constants. The mass term m¯yybreaks this\nglobal symmetry down to U(1)×U(1)symmetry, whose\ngenerator are I4×4and−iG35[56]. However, when the mass\nvanishes, the quantum corrections generate a vacuum expec-\ntation value of ¯yy(to be precise [¯y;y]/slash.left2, see below), then,\nthe symmetry would have broken down to U(1)×U(1).\nIt should be noted that besides the usual mass term m¯yy,\nthere is a mass term mt¯y[G3;G5]\n2y=mt¯yiG35yknown as\nHaldane mass term [31], which is invariant under the U(2)\nsymmetry. However, this term breaks parity and time-reversal\nsymmetries (see Tab. 1).3 Furry picture\nIn this section we present the Furry picture based on the\nquantized solutions of the Dirac equation, and we generalize\nwhat has been done in Refs. [57, 58] for a Dirac equation in\nthe presence of a real magnetic field to the case in which the\nDirac equation is in the presence of real and pseudomagnetic\nfields. In static background gauge fields, the Dirac equation\n(5) can be rewritten as\n/uni239B\n/uni239DE∓m−(D±\n1−iD±\n2)\n(D±\n1+iD±\n2) E±m/uni239E\n/uni23A0y=0: (18)\nThere are two possible solutions depending on the threshold\nstates (/divides.alt0E/divides.alt0=±m). The positive-energy solutions ( y(+)) are\ny(+)\n±;1=e−i/divides.alt0E/divides.alt0t/roottop\n/rootmod/rootmod/rootbot/divides.alt0E/divides.alt0±m\n2/divides.alt0E/divides.alt0/uni239B\n/uni239Df\n−D±\n1+iD±\n2\n/divides.alt0E/divides.alt0±mf/uni239E\n/uni23A0;or\ny(+)\n±;2=e−i/divides.alt0E/divides.alt0t/roottop\n/rootmod/rootmod/rootbot/divides.alt0E/divides.alt0∓m\n2/divides.alt0E/divides.alt0/uni239B\n/uni239DD±\n1−iD±\n2\n/divides.alt0E/divides.alt0∓mg\ng/uni239E\n/uni23A0; (19)\nwhere y(+)\n+;irefers to the positive-energy solution in the rep-\nresentation R1andy(+)\n−;irefers to the positive-energy solu-\ntion in the representation R2. The negative-energy solutions\n(y(−)) are\ny(−)\n±;1=e+i/divides.alt0E/divides.alt0t/roottop\n/rootmod/rootmod/rootbot/divides.alt0E/divides.alt0∓m\n2/divides.alt0E/divides.alt0/uni239B\n/uni239Df\n−D±\n1+iD±\n2\n/divides.alt0E/divides.alt0∓mf/uni239E\n/uni23A0;or\ny(−)\n±;2=e+i/divides.alt0E/divides.alt0t/roottop\n/rootmod/rootmod/rootbot/divides.alt0E/divides.alt0±m\n2/divides.alt0E/divides.alt0/uni239B\n/uni239DD±\n1−iD±\n2\n/divides.alt0E/divides.alt0±mg\ng/uni239E\n/uni23A0; (20)\nwhere fandgare two functions such that\n−(D±\n1−iD±\n2)(D±\n1+iD±\n2)f=(E2−m2)f; (21)\n−(D±\n1+iD±\n2)(D±\n1−iD±\n2)g=(E2−m2)g: (22)\nNote that the threshold states /divides.alt0E/divides.alt0=mand/divides.alt0E/divides.alt0=−mmust be\nspecified separately. When /divides.alt0E/divides.alt0=mis a positive (negative)\nenergy solution, the negative (positive) energy threshold is\nexcluded, because of the factor 1 /slash.left/radical.alt1\n/divides.alt0E/divides.alt0−m[57]. For exam-\nple, for the valley /uni20D7K, or equivalently the representation R1,6\nthe positive-energy solutions for /divides.alt0E/divides.alt0=m>0 and/divides.alt0E/divides.alt0=m<0\nare respectively\ny(+0)\n+;1=e−i/divides.alt0m/divides.alt0t/uni239B\n/uni239Df(0)\n0/uni239E\n/uni23A0;y(+0)\n+;2=e−i/divides.alt0m/divides.alt0t/uni239B\n/uni239D0\ng(0)/uni239E\n/uni23A0; (23)\nwhere f(0)(x;y)satisfies the first-order threshold equation\n(D+\n1+iD+\n2)f(0)=0; (24)\nandg(0)(x;y)satisfies\n(D+\n1−iD+\n2)g(0)=0: (25)\nIt turns out that if the solutions of (24) are normalizable, then\nthe solutions of (25) are not, and vice versa [57, 59]. Now, in\nthe absence of pseudo-magnetic field ( a35\ni=0), one has that\nD=D+=D−. Thus, if y(+0)\n+;1/parenleft.alt2y(+0)\n+;2/parenright.alt2is a positive-energy\nsolution for the valley /uni20D7K, then the valley /uni20D7K′only has the\nnegative-energy solution y(−0)\n−;2/parenleft.alt2y(−0)\n−;1/parenright.alt2. This leads to the\nwell-known asymmetry in the spectrum of the states. Re-\nmarkably, this does not necessarily happen when there is a\npseudo-magnetic field, since D+≠D−. Additionally, if the\nsolutions of (24) are normalizable, this does not imply that\nthe solutions of (D+\n1−iD+\n2)g(0)=0 are not. Therefore, both\nvalleys may have positive (or negative) energy states simul-\ntaneously. The Appendix (Appendix A) illustrates this point\nin the case of constant real and pseudomagnetic fields.\nOne can calculate the vacuum condensate /uni27E8¯yy/uni27E9(pairing\nbetween fermions and antifermions in the vacuum) in 2 +1\ndimensions by expanding out the fermion field in a com-\nplete orthonormal set of the positive- and negative-energy\nsolutions ( i=R1;R2)\nYi(/uni20D7x;t)=/summation.disp/integral.disp\nn/summation.disp/integral.disp\np[ai;n;py(+)\ni;n;p+b†\ni;n;py(−)\ni;n;p]: (26)\nThe solutions are labeled by two quantum numbers (n;p), in\nwhich the label nrefers to the eigenvalue En, whilst the label\npdistinguishes between degenerate states. In general, both\nnand pmay take discrete and/or continuous values [57].\nTheai;n;pandb†\ni;n;pare the fermion annihilation operator and\nantifermion creation operator, respectively, which obey the\nanticommutation relations\n{ai;n;p;a†\nj;n′;p′}={bi;n;p;b†\nj;n′;p′}=di j¯dnn′¯dpp′; : (27)\nwhere ¯da;a′, is the Kronecker delta if atakes discrete val-\nues, or is the Dirac delta if it takes continuous values. Us-\ning the commutation relations (27), the vacuum expectation\nvalue/uni27E8¯YiYi/uni27E9can be written as\n/uni27E80/divides.alt0¯Yi(x)Yi(x)/divides.alt00/uni27E9≡tr/uni27E8¯Yi;aYi;b/uni27E9=/summation.disp/integral.disp\nn/summation.disp/integral.disp\np¯y(−)\ni;n;p(x)y(−)\ni;n;p(x);\n(28)\ni.ethe fermion condensate is a sum over occupied negative-\nenergy states. The tr is over the spinorial indices {a;b}. Letus also write the condensate /uni27E8Yi¯Yi/uni27E9, which will be relevant\nto what follows, as\n/uni27E80/divides.alt0Yi(x)¯Yi(x)/divides.alt00/uni27E9≡tr/uni27E8Yi;b¯Yi;a/uni27E9=/summation.disp/integral.disp\nn/summation.disp/integral.disp\np¯y(+)\ni;n;p(x)y(+)\ni;n;p(x);\n(29)\ni.ethis condensate is a sum over occupied positive-energy\nstates.\nIn the Appendix B, we compare the calculation of fermi-\nonic condensate in the Furry picture via fermion propagator.\nWe prove that the trace of the fermion propagator evaluated\nat equal space-time points must be understood as the expec-\ntation value of the commutator of two field operators. Thus,\nthe Schwinger’s choice [60], which is equivalent to perform\nthe coincidence limit symmetrically on the time coordinate\n[61], i.e.\n/uni27E80/divides.alt0[¯Yi;Yi]/divides.alt00/uni27E9\n2=−trSF(x;x)\n≡−/uni239B\n/uni239C/uni239C\n/uni239Dtr lim\nx0→y+\n0x→ySF(x;y)−tr lim\nx0→y−\n0x→ySF(x;y)/uni239E\n/uni239F/uni239F\n/uni23A0:(30)\nBesides, we argue that this must be the order parameter for\nchiral (or parity) symmetry breaking and not the fermionic\ncondensate, as is commonly assumed in the literature.\n4 Fermion condensate\nChiral symmetry breaking in (2 +1)- and (3+1)-dimensional\ntheories has been a subject of intense scrutiny over the past\ntwo decades [15, 17, 18, 37, 57, 62–78]. In the presence of\na uniform magnetic field, the appearance of a nonvanish-\ning chiral condensate /uni27E8¯YY/uni27E9≠0 in the limit m→0, produce\nspontaneous chiral symmetry breaking [15, 62, 63]. For ex-\nample [15, 64], in the Nambu-Jona-Lasinio (NJL) model,\nthe spontaneous symmetry breaking occurs when the cou-\npling constant exceeds some critical value, i.e.when l>lc.\nWith an external uniform magnetic field lc→0, independent\nof the intensity of the magnetic field B, the magnetic field is\na strong catalyst of chiral symmetry breaking (see Ref. [16]\nfor review).\nThe exact expression for a fermion propagator in an ex-\nternal magnetic field in 3 +1 dimensions was found for the\nfirst time by Schwinger using the proper-time formalism [60].\nFor 2+1 dimensions, the fermion propagator was presented\nin the momentum representation by Gusynin et al. in [15].\nIn [15, 62, 65], the vacuum condensate was computed in the\nreducible representation (fourcomponent spinor) using the\nexpression of the fermion propagator in the presence of an\nuniform magnetic field. On the other hand, Das and Hott\nintroduced an alternative derivation of the magnetic conden-\nsate using the Furry Picture. This method has been used to7\ncalculate the magnetic vacuum condensate in 3 +1 dimen-\nsions [79], at finite temperature [56, 58, 68, 71], and as well\nas in the presence of parity-violating mass terms [79]. In ap-\npendix Appendix B.1, we compute the vacuum condensate\nin the presence of an external magnetic field for the two irre-\nducible representations and show that these two methods are\nconsistent if we take the definition of the propagator in equal\ntimes, as the one introduced by Schwinger in [60]. Further-\nmore, we discuss why the vacuum expectation value of the\ncommutator of two field operators is the appropriate order\nparameter to describe chiral (or parity) symmetry breaking.\nIn order to study the effect of strains, in the following we\nconsider a sample of graphene in the presence of constant\nreal ( B) and pseudo ( b) magnetic fields6. In this case, we\nchoose /uni20D7A=(0;Bx)and/uni20D7a35=(0;bx). The explicit solution\ncan be found in the (Appendix A). In a similar way shown\nin Appendix B.1, we compute the vacuum expectation value\nof the commutator in the two irreducible representations for\narbitrary values of m,eBandeb, i.e.,\n1\n2/uni27E8[¯Y±;Y±]/uni27E9B=−sgn(m)/divides.alt0eB±eb/divides.alt0\n4p−/divides.alt0eB±eb/divides.alt0\n2p∞\n/summation.disp\nn=1m\n/divides.alt0E±n/divides.alt0\n=sgn(m)/divides.alt0eB±eb/divides.alt0\n4p−m/radical.alt1\n2/divides.alt0eB±eb/divides.alt0\n4pz/parenleft.alt41\n2;m2\n2/divides.alt0eB±eb/divides.alt0/parenright.alt4;(31)\nwith/divides.alt0E±\nn/divides.alt0=/radical.alt1\nm2+2/divides.alt0eB±eb/divides.alt0n, here the (−)sign refers to\nthe representation R1(valley /uni20D7K), whereas (+)refers to the\nrepresentation R2(valley /uni20D7K′).z(s;q)is the Hurwitz zeta\nfunction defined by\nz(s;q)=∞\n/summation.disp\nn=01\n(n+q)s: (32)\nThe commutator can be rewritten in an integral representa-\ntion as\n1\n2/uni27E8[¯Y±;Y±]/uni27E9B;b\n=−m\n4p3\n2/integral.disp∞\n0dte−m2tt−1\n2/divides.alt0eB±eb/divides.alt0coth(/divides.alt0eB±eb/divides.alt0t); (33)\nwhere we have used that\n/integral.disp∞\n0dte−m2tt−1\n2/divides.alt0w/divides.alt0coth(/divides.alt0w/divides.alt0t)\n=(2/divides.alt0w/divides.alt0p)1\n2/uni239B\n/uni239Dz/parenleft.alt41\n2;m2\n2/divides.alt0w/divides.alt0/parenright.alt4−/divides.alt0w/divides.alt01\n2\n21\n2/divides.alt0m/divides.alt0/uni239E\n/uni23A0; (34)\nwhich can be obtained after regularization with the e−inte-\ngration technique [81]. Although Eq. (33) is divergent, the\ndivergences are already present for zero external field\n1\n2/uni27E8[¯Y±;Y±]/uni27E90;0=−m\n4p3\n2/integral.disp∞\n0dte−m2tt−3\n2: (35)\n6We noted that the pseudomagnetic field study here is mathemati-\ncally equivalent to considering a Dirac oscillator potential in (2+1)-\ndimensions [80]. Consequently, a constant pseudomagnetic field can be\nseen as a physical realization of the two-dimensional Dirac oscillator.\n-10 -5 0 5 10-0.8-0.6-0.4-0.20.0Fig. 1 (color online). The c−condensates as a function of a external\nmagnetic field, m−/slash.left(m/divides.alt0m/divides.alt0)(dashed line) and m+/slash.left(m/divides.alt0m/divides.alt0)(continuous\nline), for eb/slash.leftm2=5. Since m±(eB±eb;m)/slash.left(m/divides.alt0m/divides.alt0)=m±/parenleft.alt1eB±eb\nm2 ;1/parenright.alt1it is\nsatisfied, when the mass is changed only a widening of the “parabolic”\nform of the function occurs.\nTherefore, by subtracting out the vacuum part, a finite result\nis obtained [17, 77, 81]\nm±=1\n2/uni27E8[¯Y±;Y±]/uni27E9B;b−1\n2/uni27E8[¯Y±;Y±]/uni27E90;0\n=−m\n4p3\n2/integral.disp∞\n0dte−m2tt−1\n2/parenleft.alt3/divides.alt0eB±eb/divides.alt0coth(/divides.alt0eB±eb/divides.alt0t)−1\nt/parenright.alt3:\n(36)\nTo evaluate in a simple form Eq. (35), first note that al-\nthough this integral is divergent, Eq. (31) has a finite limit as\n/divides.alt0eB±eb/divides.alt0→0 if we used the analytic continuation of the Hur-\nwitz zeta function so1\n2/uni27E8[¯Y±;Y±]/uni27E90;0=sgn(m)m\n2p, which co-\nincides with the regularization using the e-integration tech-\nnique [81]. Henceforth, we will refer to m±as the c−con-\ndensates, which in terms of the analytically continue Hur-\nwitz zeta function are given by7\nm±=−sgn(m)/divides.alt0eB±eb/divides.alt0\n4p\n−m/radical.alt1\n2/divides.alt0eB±eb/divides.alt0\n4pz/parenleft.alt41\n2;1+m2\n2/divides.alt0eB±eb/divides.alt0/parenright.alt4−/divides.alt0m/divides.alt0m\n2p: (37)\nOne can show that there is no critical value of the fields in\nwhich the m±/slash.leftmchanges sign. Notably, if B≠0 or b≠0,\nthe values of the m±for each valley are different and when\n/divides.alt0B/divides.alt0=/divides.alt0b/divides.alt0one of the two is zero, as showed in Fig. 1. Finally,\nlet us take the limit m→0\nm±=−sgn(m)/divides.alt0eB±eb/divides.alt0\n4p: (38)\nAs we will see below, this is related to a breaking of chiral,\nparity and time-reversal symmetries.8\n-40 -20 0 20 4005101520253035\nFig. 2 (color online) Dynamical masses in the constant-mass approx-\nimation versus external magnetic field, m+\ndyn/slash.left(ma)(continuous line)\nandm−\ndyn/slash.left(ma)(dashed line), for eb/slash.leftm2=10.\n4.1 Dynamical mass\nDynamical mass generation in QED 2+1has been a subject\nof study in the past three decades [15, 17, 18, 69, 82–84].\nAs shown in Ref. [18], the dynamical mass with a two-\ncomponent fermion in a uniform magnetic field, in the so-\ncalled constant-mass approximation, is\nm2+1\ndyn=2aW/uni239B\n/uni239De−gE/slash.left2/radical.alt1\n2/divides.alt0eB/divides.alt0\n2a/uni239E\n/uni23A0; (39)\nwithgEthe Euler constant, a=e2/slash.left(4p)andW(x)the Lam-\nbert Wfunction. In the latter formula, it is necessary that\n/divides.alt0eB/divides.alt0/uni226Bm2\ndynfor consistency. For weak magnetic fields, the\ndynamical mass has a quadratic behavior in the magnetic\nfield, m2+1\ndyn=m0+m2(/divides.alt0eB/divides.alt0)2+: : :[85]. Futhermore, the ra-\ndiative corrections to the mass of a charged fermion when it\noccupies the lowest Landau level in RQED 4;3was recently\ncomputed in the one-loop approximation in Refs. [84, 86].\nIts associated equation reads\nmRQED\ndyn=e2\n4p/radical.alt1\n/divides.alt0eB/divides.alt0/bracketleft.alt4√\n2p3/slash.left2erfc/parenleft.alt41√\nl/parenright.alt4−10\n3√\nlG/parenleft.alt30;1\nl/parenright.alt3/bracketright.alt4 ;\n(40)\nwith l=/divides.alt0eB/divides.alt0\nm2and where erfc (z)andG(z)are the complemen-\ntary error function and upper incomplete gamma function,\nrespectively. Significantly, the dynamical mass does not van-\nish even at the limit of zero bare mass ( m→0) [84, 86]\nmRQED\ndyn=e2\n2√\n2/radical.alt1\np/divides.alt0eB/divides.alt0: (41)\nIn graphene, while the photon propagates in 3 spatial dimen-\nsions, the fermions are localized on 2 spatial dimensions,\n7For numerical calculations, instead of utilizing the integral (36), the\nc−condensates can be evaluated in a much more efficient way using\nthe Hurwitz zeta functions.because of this, RQED 4;3is an appropriate model to de-\nscribe the low-energy physics for this system. Nevertheless,\nas mentioned above, the coupling constant is large, hence,\nthis perturbative result is not necessarily accurate [32, 84].\nIn the following, because of the lack of a better estimate, we\nuse this result to determine the dynamical mass in graphene.\nIn general, the dynamical mass should be given by mdyn=\ng(/divides.alt0eB/divides.alt0), where gis a general function. It is straightforward\nto extend this result to include the pseudomagnetic field.\nHence, the dynamical mass would read as mdyn=g(/divides.alt0eB±eb/divides.alt0)\nand we thus obtain\nm±\ndyn=a/radical.alt1\n/divides.alt0eB±eb/divides.alt0/uni23A1/uni23A2/uni23A2/uni23A2/uni23A2/uni23A3√\n2p3/slash.left2erfc/uni239B\n/uni239Dm/radical.alt1\n/divides.alt0eB±eb/divides.alt0/uni239E\n/uni23A0\n−10m\n3/radical.alt1\n/divides.alt0eB±eb/divides.alt0G/parenleft.alt40;m2\n/divides.alt0eB±eb/divides.alt0/parenright.alt4/uni23A4/uni23A5/uni23A5/uni23A5/uni23A6; (42)\nwitha=e2/slash.left(4p). According to these findings, the dynam-\nical fermion mass is different for each valley, m+\ndynfor/uni20D7K′\nandm−\ndynfor/uni20D7K(see Fig. 2). This is not surprising since the\nc−condensates, m+andm−, are different if a pseudomag-\nnetic field is included. Furthermore, it is possible to con-\nstruct a Lagrangian that describes two species of fermions,\neach with different mass, introducing the usual mass term\n(m) and a Haldane mass term ( mt)\nLm=m¯yy+mt¯yiG35y: (43)\nIn this case, the two masses will be m±mtandyis taken as\na four-component spinor. This result implies that the inter-\nplay between the real and pseudomagnetic fields allows us\nto dynamically generate these two terms. With the help of\nEq. (38), one can realize that\n1\n2/uni27E8[¯y;y]/uni27E9B;b−1\n2/uni27E8[¯y;y]/uni27E90;0\n=−sgn(m)\n4p(/divides.alt0eB+eb/divides.alt0+/divides.alt0eB−eb/divides.alt0); (44)\nwhile in the limit m→0, we have\n1\n2/uni27E8[¯y;iG35y]/uni27E9B;b−1\n2/uni27E8[¯y;iG35y]/uni27E90;0\n=−sgn(m)\n4p(/divides.alt0eB+eb/divides.alt0−/divides.alt0eB−eb/divides.alt0): (45)\nTherefore, the usual mass term is always generated indepen-\ndently of Bandb, whereas the Haldane mass term is only\ngenerated if B≠0 and b≠0 simultaneously. Note that when\nB=b(orB=−b), one of the c−condensates is zero and the\ndynamical mass of this valley will be independent of Band\nb.\nFinally, it is important to realize that for zero pseudo-\nmagnetic field, the mass term in irreducible representations\nbreaks parity and time-reversal symmetries, while in a re-\nducible representation chirality is broken. Thus, for irredu-\ncible representations, the c−condensate ( m=m+=m−) is9\nthe order parameter of the dynamical parity and there is\na time-reversal symmetry breaking. In contrast, dynamical\nsymmetry breaking occurs in reducible representations. In\nreducible representations, however, non-zero magnetic and\npseudo magnetic fields produce a dynamical symmetry break-\ning, not only of the chiral symmetry but also of the parity\nand time-reversal symmetries8. The reason for this is that\nby including pseudomagnetic field, the dynamical mass is\ndifferent for each valley since a Haldane mass term (which\nbreaks parity and time reversal) is generated9.\n5 One-loop effective action and magnetization\nIn this section we compute the effective action and the mag-\nnetization in the presence of uniform real and pseudomag-\nnetic fields. We consider the fermionic part of the generating\nfunctional for each valley\nZ±=eiW±(46)\n=/integral.dispD¯Y±DY±exp/parenleft.alt3i/integral.dispd3x[¯Y±(i/slash.left¶+e/slash.leftA∓e/slash.lefta35−m)Y±]/parenright.alt3:\nThen, we introduce the one-loop effective Lagrangian L(1)\n±\nvia ln Z±=i∫d3xL(1)\n±(x). In the presence of a static back-\nground field, the one-loop effective action is proportional to\nthe vacuum energy (Ch. 16. in [39]). The vacuum energy of\nthe Dirac energy field can be computed using the formula\n[88]\nEvac=1\n2/uni239B\n/uni239D−/summation.disp\nEn>0En+/summation.disp\nEn<0En/uni239E\n/uni23A0; (47)\nwhich depends upon the zero-point energies of both positive-\nand negative-energy states10. In our case, it is straightfor-\nward to obtain the density vacuum energy for each valley\nE±\nvac(B;b)=E±\nvac\nA\n=−/divides.alt0eB±eb/divides.alt0\n4p/divides.alt0m/divides.alt0−/divides.alt0eB±eb/divides.alt0\n2p∞\n/summation.disp\nn=1/radical.alt1\nm2+2/divides.alt0eB±eb/divides.alt0n (48)\n=/divides.alt0eB±eb/divides.alt0\n4p/divides.alt0m/divides.alt0−/divides.alt0eB±eb/divides.alt03\n2\n21\n2pz/parenleft.alt4−1\n2;m2\n2/divides.alt0eB±eb/divides.alt0/parenright.alt4;\n8In Ref. [45], it had been suggested that the flux of the non-Abelian\npseudomagnetic field catalyzes the time-reversal symmetry breaking.\n9The Haldane mass term, for example, can also be dynamically gen-\nerated in graphene at sufficiently large strength of the long-range\nCoulomb interaction [87].\n10Provided that for each eigenvalue En, there is an eigenvalue −En,\nthen the two sums in Eq. (47) reduces to the sum over the Dirac sea\nEvac=/summation.disp\nEn<0En:\nThis equation is always satisfied by a charge conjugation invariant\nbackground; however, this is not our case. The use of this equation in\na magnetic field background has led to erroneous conclusions in Ref.\n[71, 89].where we have used that the Landau degeneracy per unit\narea is/divides.alt0eB±eb/divides.alt0/slash.left(2p). In order to calculate the purely mag-\nnetic field effect, we need to subtract the zero-field part.\nThus, the one-loop effective Lagrangian density is11\nL(1)\n±=−(E±\nvac(B;b)−E±\nvac(0;0))\n=−/divides.alt0eB±eb/divides.alt0\n4p/divides.alt0m/divides.alt0+/divides.alt0eB±eb/divides.alt03\n2\n21\n2pz/parenleft.alt4−1\n2;m2\n2/divides.alt0eB±eb/divides.alt0/parenright.alt4+/divides.alt0m/divides.alt03\n6p:\n(49)\nUsing Eq. (34), we can rewritten the one-loop effective La-\ngrangian in an integral representation such as\nL(1)\n±=−/integral.disp∞\n0dte−m2tt−5\n2\n8p3/slash.left2(/divides.alt0eB±eb/divides.alt0tcoth(/divides.alt0eB±eb/divides.alt0t)−1):\n(50)\nForb=0, the result is in agreement with what was found in\nRef. [35–38]. In particular, when m→0, we arrive to\nL(1)\n±;m=0=/divides.alt0eB±eb/divides.alt03\n2\n21\n2pz/parenleft.alt3−1\n2/parenright.alt3: (51)\nHere z(x)is the Riemann-zeta function. One can compute\nthe orbital magnetization for each valley ( M±) employing\nthe one-loop effective Lagrangian, namely M±=¶L(1)\n±\n¶B. A\nstraightforward calculation gives\nM±=−e\n8p/integral.disp∞\n0dt\np1/slash.left2e−m2tt−3\n2(coth(/divides.alt0eB±eb/divides.alt0t)\n−/divides.alt0eB±eb/divides.alt0t\nsinh2(/divides.alt0eB±eb/divides.alt0t)/parenright.alt4sgn(eB±eb); (52)\nwhich can also be written as\nM±=/bracketleft.alt4−e/divides.alt0m/divides.alt0\n4p−em2\n√\n32p/divides.alt0eB±eb/divides.alt01/slash.left2z/parenleft.alt41\n2;m2\n2/divides.alt0eB±eb/divides.alt0/parenright.alt4\n+3e/divides.alt0eB±eb/divides.alt01\n2√\n8pz/parenleft.alt4−1\n2;m2\n2/divides.alt0eB±eb/divides.alt0/parenright.alt4/uni23A4/uni23A5/uni23A5/uni23A5/uni23A5/uni23A6sgn(eB±eb):(53)\nTherefore, the orbital magnetization displays nonlinear be-\nhavior in the magnetic and pseudomagnetic fields (see Fig.\n3a). For b=0, the result is in agreement with Refs. [34, 36,\n38] and in the limit m→0 the magnetization is\nM±=3e/divides.alt0eB±eb/divides.alt01\n2√\n8pz/parenleft.alt3−1\n2/parenright.alt3sgn(eB±eb); (54)\nfor each valley12. Notably, the value and sign of magnetiza-\ntion are different for each valley. Thus, they can be modified\n11This approach is equivalent to compute an infinite series of one–loop\ndiagrams with the insertion of one, two, . . . external lines (see for in-\nstance Ref. [61]).\n12It should be noted that Eq. (54) also corresponds to the dominant\nterm in the strong field expansion.10\n-30 -20 -10 0 10 20 30-0.6-0.4-0.20.00.20.40.6\n-30 -20 -10 0 10 20 30-0.005\n-0.015\n-0.025\n-0.035\nFig. 3 (color online) (a) Magnetization vs the external magnetic field, M+(continuous line), M−(dashed line) and the total magnetization M=\nM++M−(dotted line). The magnetization ( M+) in the valley /uni20D7Kis zero in eB/slash.leftm2=−10 while the magnetization ( M−) in the valley /uni20D7K′is zero\nineB/slash.leftm2=10. (b) Magnetic susceptibility vs the external magnetic field, c+(continuous line), c−(dashed line) and the total magnetization\nc=c++c−(dotted line). Here we taken eb/slash.leftm2=10.\nby the strains or varying the applied magnetic field. In par-\nticular, the magnetization of one valley could be zero while\nthe other does not, as it can be seen in Fig. 3a.\nHaving calculated M±, one can compute the magnetic\nsusceptibility for each valley in the presence of magnetic and\npseudomagnetic fields, which is simply given by c±=¶M±\n¶B,\nthus\nc±=e\n16√\n2p/divides.alt0eB±eb/divides.alt05\n2/bracketleft.alt412/divides.alt0eB±eb/divides.alt02z/parenleft.alt4−1\n2;m2\n2/divides.alt0eB±eb/divides.alt0/parenright.alt4\n−4m2/divides.alt0eB±eb/divides.alt0z/parenleft.alt41\n2;m2\n2/divides.alt0eB±eb/divides.alt0/parenright.alt4−m4z/parenleft.alt43\n2;m2\n2/divides.alt0eB±eb/divides.alt0/parenright.alt4/bracketright.alt4\n(55)\nIn the presence of magnetic and pseudomagnetic fields, the\ntotal susceptibility has two minimums, see Fig. 3b, which\nis a distinctive feature compared with the case eb=0 [34].\nFinally, in the limit m→0 the susceptibility is\nc±=3ez(−1/slash.left2)√\n32p/divides.alt0eB±eb/divides.alt01/slash.left2; (56)\nwhich is divergent when B=±b. Since c±(B±b=0;m≠0)\nis finite, it can be concluded that the mass acts as a regulator.\n6 Induced charge density\nIn this section, we derive an expression for the induced charge\ndensity in the presence of uniform real and pseudomagnetic\nfields. We first noticed, that as in the case of the fermi-\nonic condensate, the formula −trgmS(x;x)=/uni27E80/divides.alt0jm/divides.alt00/uni27E9also de-\nserves to be revised. The current must be understood as\njm→∶jm(x)∶=∶e¯Y(x)gmY(x)∶\n=e\n2[¯Y(x);gmY(x)]\n=1\n2gm\n˙aa[¯Y˙a(x);Ya(x)]; (57)which is the correct definition of the current operator, where\nit has been subtracted an infinite charge of the vacuum state\n[61]. In an analogous way to what was done before, we can\nfind that in the Furry picture the vacuum expectation value\nof the current operator is\n/uni27E80/divides.alt0∶jm\ni(x)∶/divides.alt00/uni27E9=e\n2/uni27E80/divides.alt0[¯Yi(x);gmYi(x)]/divides.alt00/uni27E9\n=e\n2/summation.disp/integral.disp\nn/summation.disp/integral.disp\np/parenleft.alt2¯y(−)\ni;n;p(x)gmy(−)\ni;n;p(x)−¯y(+)\ni;n;p(x)gmy(+)\ni;n;p(x)/parenright.alt2:\n(58)\nFor constant real and pseudomagnetic fields, using the or-\nthogonality of Hermite polynomials, one can show that\n/uni27E80/divides.alt0∶ji\n±(x)∶/divides.alt00/uni27E9=0;(i=1;2); (59)\ni.ethe induced current vanishes. A nonvanishing vacuum\ncurrent would arise in the presence of an external electric\nfield [36]. On the other hand, the induced charge density is\ngiven by\n/uni27E80/divides.alt0r(x)±/divides.alt00/uni27E9=/uni27E80/divides.alt0∶j0\n±(x)∶/divides.alt00/uni27E9=±sgn(m)\n4pe2(B±b): (60)\nCuriously, in contrast to m±, the induced charge density only\nreceives contributions from the lowest Landau level (LLL),\neven if m≠0. An alternative technique to compute the charge\ndensity is through spectral function. It can be shown that the\ninduced charge is [90, 91]\nQ±=/integral.dispd2x/uni27E80/divides.alt0r(x)±/divides.alt00/uni27E9=−e\n2lim\ns→0+h(H±;s); (61)\nwhere\nh(H±;s)=/summation.disp\nn/divides.alt0E±\nn/divides.alt0−ssgn(E±\nn): (62)\nis the hinvariant of Atiyah, Patodi, and Singer. Using the\nLandau degeneracy per unit area and noting that except for11\nthe LLL, for each eigenvalue E±\nnthere is an eigenvalue −E±\nn,\nthen we obtain Eq. (60) as we should. Therefore, the total\ninduced charge density is\n/uni27E80/divides.alt0r(x)/divides.alt00/uni27E9=/uni27E80/divides.alt0r(x)+/divides.alt00/uni27E9+/uni27E80/divides.alt0r(x)−/divides.alt00/uni27E9\n=sgn(m)\n4pe2(B+b)−sgn(m)\n4pe2(B−b)\n=sgn(m)\n2pe2b; (63)\nwhich is observable (and measurable) when taking a nonzero\npseudomagnetic field. This is remarkable since this is not\npossible in the case of a pure magnetic field [41]. Eq. (63)\nshows that in the presence of pseudomagnetic fields, the sys-\ntem has a parity anomaly, even with two fermionic species.\nEqs. (59) and (60) are related with the Chern-Simons rela-\ntion, which in the case of zero pseudomagnetic field reads\n[41]\n/uni27E80/divides.alt0∶jm\n±(x)∶/divides.alt00/uni27E9=±e2\n8psgn(m)emnlFnl: (64)\nIt is clear from this equation equation that when the two rep-\nresentations are present, the vacuum expectation value of the\ntotal current is always zero.\n7 Conclusions\nIn summary, we have examined the Dirac Hamiltonian in 2 +\n1 dimensions and found an infinite number of non-equivalent\nrealizations of parity, charge conjugation, and time-reversal\ntransformations for the reducible representation. We have\nthen explored how the interplay between real and pseudo-\nmagnetic fields affects some aspects of the three-dimensional\nquantum field theory. For the case of uniform magnetic and\npseudomagnetic fields, by employing a non-perturbative ap-\nproach we have found that: (i) The c−condensate is the ap-\npropriate order parameter for studying the breaking of chi-\nral, parity and time-reversal symmetries. (ii) One can control\nthe magnetization, susceptibility, and the dynamical mass\nindependently for each valley by straining and varying the\napplied magnetic field. (iii) The dynamical mass generated\nis due to two terms, the usual mass term ( m¯yy) and a Hal-\ndane mass term ( mt¯yiG35y), being the latter, for the case\nin which the two fields are simultaneously different from\nzero, the one that breaks parity and time-reversal symme-\ntries. (iv) For non-zero pseudomagnetic field, the total in-\nduced “vacuum” charge density is not null. This last result\nimplies that strained single-layer graphene exhibits a par-\nity anomaly. Therefore, strained graphene in the presence\nof an external magnetic field has distinctive features com-\npared with QED 2+1, which lacks the aforementioned conse-\nquences (i)-(iv). Finally, it would be interesting to extend our\ncalculations to include the effect of Coulomb interactions on\nthe magnetic catalysis, symmetry breaking, dynamical massgeneration, etc. See, for instance, the study of the magnetic\ncatalysis in unstrained graphene in the weak-coupling limit\n[92].\nAcknowledgements J. A. Sánchez is grateful to F. T. Brandt, C. M.\nAcosta and J. S. Cortés for useful comments.\nAppendix A: Exact solutions of the Dirac equation in\nthe presence of uniform real and pseudomagnetic fields\nIn the presence of a constant real ( B) and pseudo ( b) mag-\nnetic fields, one can choose the real potential and the pseudo-\npotential as /uni20D7A=(0;Bx)and/uni20D7a35=(0;bx), respectively. For\nthe valley /uni20D7Kand with m≥0, it is straightforward to find the\nspectrum and the solutions of Eq. (5). However, we need\nto find them independently for the cases e(B−b)>0 and\ne(B−b)<0. For the first case e(B−b)>0, the spectrum and\nsolutions are given by\nE−\nn=±/radical.alt1\nm2+2/divides.alt0eB−eb/divides.alt0n; (A.1)\ny(+)\nK=N−\nne−i/divides.alt0E−\nn/divides.alt0t+ipy/uni239B\n/uni239D(/divides.alt0E−\nn/divides.alt0+m)I−(n;p;x)\n−/radical.alt1\n2/divides.alt0eB−eb/divides.alt0nI−(n−1;p;x)/uni239E\n/uni23A0;\n(A.2)\ny(−)\nK=N−\nnei/divides.alt0E−\nn/divides.alt0t−ipy/uni239B\n/uni239D/radical.alt1\n2/divides.alt0eB−eb/divides.alt0nI−(n;−p;x)\n(/divides.alt0E−\nn/divides.alt0+m)I−(n−1;−p;x)/uni239E\n/uni23A0; (A.3)\nwhere\nN±\nn=1/radical.alt1\n2/divides.alt0E±n/divides.alt0(/divides.alt0E±n/divides.alt0+m); (A.4)\nI±(n;x;p)=/divides.alt0eB±eb/divides.alt01\n4\n√\n2nn!p1\n4Hn/bracketleft.alt4/radical.alt1\n/divides.alt0eB±eb/divides.alt0/parenleft.alt4x−p\n/divides.alt0eB±eb/divides.alt0/parenright.alt4/bracketright.alt4\n×exp/uni239B\n/uni239D−/divides.alt0eB±eb/divides.alt0\n2/parenleft.alt4x−p\n/divides.alt0eB±eb/divides.alt0/parenright.alt42/uni239E\n/uni23A0(A.5)\nwith I(n=−1;p;x)=0 and E+\nn=±/radical.alt1\nm2+2/divides.alt0eB+eb/divides.alt0n. Note\nthat the lowest Landau level (LLL) describes particle (fer-\nmion) states with energy E0=m≥0. In order to find the\nsolutions for the case e(B−b)<0, one can use the charge-\nconjugate operator, Eq. (9), Cy=−g2(¯y)T=s1y∗[93], so,\nthe solutions are given by\ny(+)\nK=−N−\nne−i/divides.alt0E−\nn/divides.alt0t+ipy/uni239B\n/uni239D(/divides.alt0E−\nn/divides.alt0+m)I−(n−1;p;x)/radical.alt1\n2/divides.alt0eB−eb/divides.alt0nI−(n;p;x)/uni239E\n/uni23A0;(A.6)\ny(−)\nK=−N−\nnei/divides.alt0E−\nn/divides.alt0t−ipy/uni239B\n/uni239D−/radical.alt1\n2/divides.alt0eB−eb/divides.alt0nI−(n−1;−p;x)\n(/divides.alt0E−\nn/divides.alt0+m)I−(n;−p;x)/uni239E\n/uni23A0;12\n(A.7)\nNow the LLL describes hole (antifermion) states.\nOne can find the solution for the valley /uni20D7K′noting that\nthe change of representation is equivalent to change m→\n−mandB−b→B+bin the solutions (A.2), (A.3), (A.6)\nand (A.7). For the point /uni20D7K′, the LLL describes a hole (an-\ntifermion) state for e(B+b)>0 and a particle (fermion) state\nfore(B+b)<0. Hence, we can have a fermion (antifermion)\nstate in the LLL simultaneously to /uni20D7Kand/uni20D7K′if the condition\neb<eB<−eb(eb>eB>−eb) is satisfied.\nAppendix B: Magnetic condensate\nIn this Appendix, we will compare the calculation of fermi-\nonic condensate in the Furry picture with the method via\nfermion propagator. We then demonstrate that the method\nvia the fermion propagator does not calculate the vacuum\nexpectation value of the product of two field operators (mag-\nnetic condensate), but rather a vacuum expectation value of\nthe commutator of two field operators. Finally, we show that\nthe commutator is the order parameter is for chiral (or par-\nity) symmetry breaking, instead of the magnetic condensate,\nas it is often asserted in the literature.\nAppendix B.1: Magnetic condensate via furry picture\nLet us consider the fermionic condensate via the Furry pic-\nture in a constant background magnetic field. The explicit\nsolution can be found in the Appendix (Appendix A), taking\nb=0. Inserting Eq. (A.3) into Eq. (28), we obtain that for\nthe irreducible representation R1(valley /uni20D7K), for eB>0 and\nm>0, the fermion condensate is\n/uni27E80/divides.alt0¯YKYK/divides.alt00/uni27E9=∞\n/summation.disp\nn=0/integral.dispd pN2\nn\n2p/bracketleft.alt12/divides.alt0eB/divides.alt0nI(n;−p;x)2:\n−(/divides.alt0En/divides.alt0+m)2I(n−1;−p;x)2/bracketright.alt\n=−/divides.alt0eB/divides.alt0\n2p∞\n/summation.disp\nn=1m\n/divides.alt0En/divides.alt0; (B.8)\nwith/divides.alt0En/divides.alt0=/radical.alt1\nm2+2e/divides.alt0B/divides.alt0nand where we used that\n/integral.dispd pI(n−1;p;x)2=/uni23A7/uni23AA/uni23AA/uni23A8/uni23AA/uni23AA/uni23A90 if n=0;\n/divides.alt0eB/divides.alt0ifn>0;(B.9)\nand that for n=0 all the terms are zero. Now, for eB<0,\ninserting Eq. (A.7) into Eq. (28) yields\n/uni27E80/divides.alt0¯YKYK/divides.alt00/uni27E9= =∞\n/summation.disp\nn=0/integral.dispd pN2\nn\n2p/bracketleft.alt12/divides.alt0eB/divides.alt0nI(n−1;−p;x)2\n−(/divides.alt0En/divides.alt0+m)2I(n;−p;x)2/bracketright.alt\n=−/divides.alt0eB/divides.alt0\n2p−/divides.alt0eB/divides.alt0\n2p∞\n/summation.disp\nn=1m\n/divides.alt0En/divides.alt0: (B.10)It is clear now that n=0 contribute to the condensate. In a\nsimilar way, the condensates in the irreducible representa-\ntionR2(valley/uni20D7K′) are\n/uni27E80/divides.alt0¯YK′YK′/divides.alt00/uni27E9=−/divides.alt0eB/divides.alt0\n2p−/divides.alt0eB/divides.alt0\n2p∞\n/summation.disp\nn=1m\n/divides.alt0En/divides.alt0; (B.11)\n/uni27E80/divides.alt0¯YK′YK′/divides.alt00/uni27E9=−/divides.alt0eB/divides.alt0\n2p∞\n/summation.disp\nn=1m\n/divides.alt0En/divides.alt0; (B.12)\nforeB>0 and eB<0, respectively. In the fermion condensa-\ntes, the term ∑∞\nn=11\n/divides.alt0En/divides.alt0is in general divergent. However, the\nresults are understood by means of an appropriate analytic\ncontinuation.\nEqs. (B.8) and (B.11) are in agreement with what was\nfound in Ref. [71]. When m<0, one just need to exchange\nresults in Eq. (B.8) with the one of Eq. (B.10), and Eq.\n(B.11) with Eq. (B.12). In the limit m→0+,i:e:for mass-\nless fermions, the fermionic condensates are\n/uni27E80/divides.alt0¯YKYK/divides.alt00/uni27E9=/uni23A7/uni23AA/uni23AA/uni23A8/uni23AA/uni23AA/uni23A9−/divides.alt0eB/divides.alt0/slash.left(2p)ifeB<0;\n0 if eB>0;(B.13)\nand\n/uni27E80/divides.alt0¯YK′YK′/divides.alt00/uni27E9=/uni23A7/uni23AA/uni23AA/uni23A8/uni23AA/uni23AA/uni23A90 if eB<0;\n−/divides.alt0eB/divides.alt0/slash.left(2p)ifeB>0:(B.14)\nNotably, only if the LLL has negative energy states there\nis a non-vanished magnetic condensate. The condensate in\nthe 4×4 reducible representation is simply the sum of the\nirreducible representations\n/uni27E80/divides.alt0¯YY/divides.alt00/uni27E9=/uni27E80/divides.alt0¯YKYK/divides.alt00/uni27E9+/uni27E80/divides.alt0¯YK′YK′/divides.alt00/uni27E9=−/divides.alt0eB/divides.alt0\n2p; (B.15)\nwhich is in agreement with the results of Refs. [17, 56–58,\n71, 79].\nAppendix B.2: Magnetic condensate via the fermion\npropagator\nNow we consider the usual way to calculate the vacuum con-\ndensate using the fermion propagator [15]\nSF(x;y)=/uni27E80/divides.alt0T[Y(x)¯Y(y)]/divides.alt00/uni27E9; (B.16)\nwhere Tis the time-ordering operator\n/uni27E80/divides.alt0T[Y(x)¯Y(y)]/divides.alt00/uni27E9=q(x0−y0)/uni27E80/divides.alt0Y(x)¯Y(y)/divides.alt00/uni27E9\n−q(y0−x0)/uni27E80/divides.alt0¯Y(y)Y(x)/divides.alt00/uni27E9;(B.17)\nwithq(x)the Heaviside step function. In the presence of\na magnetic field, SF(x;y)can be calculated by using the\nSchwinger (proper time) approach [60]\nSF(x;y)=exp/bracketleft.alt3/integral.dispy\nxdxmAm\next/bracketright.alt3˜S(x−y); (B.18)13\nwhere the integral is calculated along the straight line, and\nthe Fourier transform of SF(x)(in Euclidean space) is ( k3=\n−ik0)\n˜S(k)=−i/integral.disp∞\n0dsexp/bracketleft.alt4−s/parenleft.alt4m2+k2\n3+k2tanh(eBs)\neBs/parenright.alt4/bracketright.alt4\n×/parenleft.alt1−kmgm+m1−i(k2g2−k1g2)tanh(eBs)/parenright.alt1\n×(1−ig1g2tanh(eBs)); (B.19)\nwith 1andgmthe 4×4 (2×2) identity matrix and the 4 ×4\n(2×2)g-matrices in the reducible (irreducible) representa-\ntions. The magnetic condensate has been calculated as [15,\n62, 65]\n/uni27E80/divides.alt0¯Y(x)Y(x)/divides.alt00/uni27E9S=−lim\nx→ytrSF(x;y); (B.20)\nwhere the superscript Semphasizes that the condensate is\ncomputed via the propagator. For the reducible representa-\ntions, in the limit m→0, the condensate reads [15]\n/uni27E80/divides.alt0¯Y(x)Y(x)/divides.alt00/uni27E9S=−/divides.alt0eB/divides.alt0\n2p; (B.21)\nwhich is in agreement with Eq. (B.15). However, for the ir-\nreducible representations the magnetic condensate is\n/uni27E80/divides.alt0¯YK(K′)(x)YK(K′)(x)/divides.alt00/uni27E9S=−/divides.alt0eB/divides.alt0\n4p; (B.22)\nwhich is the same for both representations, but it differs from\nwhat was obtained previously in Eqs. (B.13) and (B.14). To\nexplain why the condensates are different, let us critically\nreview the calculation through propagator13. If one part of\nthe definition of the propagator Eqs. (B.16) and (B.17), it\nis clear that lim x→ytrS(x;y)depends on how one takes the\nlimit, namely\ntr lim\nx0→y+\n0x→ySF(x;y)=/uni27E80/divides.alt0Yi(x)¯Yi(x)/divides.alt00/uni27E9S; (B.23)\ntr lim\nx0→y−\n0x→ySF(x;y)=−/uni27E80/divides.alt0¯Yi(x)Yi(x)/divides.alt00/uni27E9S: (B.24)\nUsing Eq. (29) one can show that, in the limit m→0+, the\nfermion condensates /uni27E8Yi¯Yi/uni27E9for the irreducible representa-\ntions are\n/uni27E80/divides.alt0YK(x)¯YK(x)/divides.alt00/uni27E9=/uni23A7/uni23AA/uni23AA/uni23A8/uni23AA/uni23AA/uni23A90 if eB<0;\n/divides.alt0eB/divides.alt0/slash.left(2p)ifeB>0;(B.25)\n/uni27E80/divides.alt0YK′(x)¯YK′(x)/divides.alt00/uni27E9=/uni23A7/uni23AA/uni23AA/uni23A8/uni23AA/uni23AA/uni23A9/divides.alt0eB/divides.alt0/slash.left(2p)ifeB<0;\n0 if eB>0;(B.26)\nand\n/uni27E80/divides.alt0Y(x)¯Y(x)/divides.alt00/uni27E9=/uni27E80/divides.alt0YK(x)¯YK(x)/divides.alt00/uni27E9+/uni27E80/divides.alt0YK′(x)¯YK′(x)/divides.alt00/uni27E9\n=/divides.alt0eB/divides.alt0\n2p; (B.27)\n13Using the so-called Ritus eigenfunctions method, it has recently been\nfound Eq. (B.22) [18], however, this method also uses the propagator\nto calculate the condensate.for the reducible representation. It is clear that for irredu-\ncible representations neither /uni27E8¯YK(K′)YK(K′)/uni27E9nor/uni27E8YK(K′)¯YK(K′)/uni27E9\ncoincide with /uni27E8¯YK(K′)¯YK(K′)/uni27E9S. The calculation of the con-\ndensate in Refs. [15, 62] was actually calculated by taking\nx=y. This will formally lead to calculate the propagator in\nx0=y0, which depends on q(0). In the literature there is not\na consensus of what should be the value of q(0). For in-\nstance, q(0)can be 1, 0 or 1 /slash.left2 (page 24 in [94]). However,\nthe only consistent value with Eqs. (B.13), (B.14), (B.15),\n(B.25), (B.26) and (B.27) is the value of 1 /slash.left2, since\ntrSF(x;x)=/uni27E80/divides.alt0Y(x)¯Y(x)/divides.alt00/uni27E9\n2−/uni27E80/divides.alt0¯Y(x)Y(x)/divides.alt00/uni27E9\n2\n=1\n2/uni27E80/divides.alt0[Y(x);¯Y(x)]/divides.alt00/uni27E9 (B.28)\nso,\n/uni27E80/divides.alt0¯Y(x)Y(x)/divides.alt00/uni27E9=/uni27E80/divides.alt0Y(x)¯Y(x)/divides.alt00/uni27E9−2trSF(x;x); (B.29)\nwhich gives us the correct value for all condensates. In fact\nthis last choice was the Schwinger’s choice: “the average\nof the forms obtained by letting y approach x from the fu-\nture, and from the past” [60]. Therefore, the Heaviside step\nfunction in the fermion propagator should be understood as\nq(t)=/uni23A7/uni23AA/uni23AA/uni23AA/uni23AA/uni23AA/uni23A8/uni23AA/uni23AA/uni23AA/uni23AA/uni23AA/uni23A90 if t<0\n1/slash.left2 ift=0\n1 if t>0:(B.30)\nIn summary, −trSF(x;x)≠/uni27E80/divides.alt0¯Y(x)Y(x)/divides.alt00/uni27E9. We believe that\nthis has not been noticed before since, in particular, when the\ncondensate is computed in 2 +1 dimensions in a reducible\nrepresentation, the two values match, which as we showed\nis just a coincidence14.\nAppendix B.3: Order parameter\nIt is widely claimed in the literature that in the reducible\nrepresentations the fermion condensate /uni27E8¯YY/uni27E9is the order\nparameter of dynamical chiral symmetry breaking [15, 57,\n58, 62, 63]. However, as we have just showed, a part of\nthe literature calculates /uni27E8¯YY/uni27E9and another part calculates\n/uni27E8[¯Y;Y]/uni27E9/slash.left2 as the order parameter, always assuming that\n/uni27E8¯YY/uni27E9is being calculated. Thus, we may ask what is the\norder parameter for chiral (or parity) symmetry breaking.\nSince in reducible representations the two values match, we\nwill use the irreducible representations to solve this puzzle.\nAlthough in irreducible representations we have not a\nchiral symmetry, as we mentioned above, the mass term bre-\naks parity and time-reversal symmetry. If the fermion con-\ndensate is the order parameter of dynamical parity and time-\nreversal symmetry breaking, Eqs. (B.13) and (B.14) would\n14One can show that in 3 +1 dimensions the two values also match, see\nRef. [64] for the calculation via fermion propagator and reference [79]\nfor the calculation in the Furry picture.14\nlead us to conclude that these broken symmetries depend on\nthe magnetic field orientation. Moreover, since the fermi-\nonic condensate is zero for eB>0 (eB<0) in the represen-\ntationR1(R2) there would not be a dynamical symmetry\nbreaking in this case and therefore would not have a mag-\nnetically induced mass. However, if the order parameter is\nthe commutator, which is independent of the orientation of\nthe magnetic field and the representation, we would obtain a\nmagnetically induced mass for eB≠0.\nIn QED 2+1and RQED 4;3the dynamical mass genera-\ntion has already been investigated [15, 17, 18, 69, 82–84].\nIn particular, the dynamical mass generation with a two-\ncomponent fermion was examined in [18, 82, 84]. From the\nresults of these works, it can be concluded that there is no ev-\nidence that the dynamical mass depends on the orientation of\nthe magnetic field. Moreover, under some assumptions, an\nexplicit formula was found for the dynamical mass when it\nis much smaller than the magnetic field [18]. 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A general mathematical\nformulation of magnetic stochasticity in turbulence has been developed in previous work in terms\nof theLp-normSp(t) =1\n2jj1\u0000^Bl:^BLjjp,pth order magnetic stochasticity of the stochastic \feld\nB(x;t), based on the coarse-grained \felds, BlandBL, at di\u000berent scales, l6=L. For laminar \rows,\nstochasticity level becomes the level of \feld self-entanglement or spatial complexity. In this paper,\nwe establish a connection between magnetic stochasticity Sp(t) and magnetic di\u000busion in magneto-\nhydrodynamic (MHD) turbulence and use a homogeneous, incompressible MHD simulation to test\nthis prediction. Our results agree with the well-known fact that magnetic di\u000busion in turbulent\nmedia follows the super-linear Richardson dispersion scheme. This is intimately related to stochas-\ntic magnetic reconnection in which super-linear Richardson di\u000busion broadens the matter out\row\nwidth and accelerates the reconnection process.\nI. INTRODUCTION\nIn the early 1940s, Onsager pointed out, but never\npublished, the remarkable fact that the velocity \feld in a\nturbulent \ruid becomes H older singular1in the limit of\nvanishing viscosity; \u0017!0 ([1]; [2]; [3]). This approach\nwas based on an exact mathematical analysis of the\nhigh Reynolds-number regime of incompressible hydro-\ndynamic turbulence. Such an analysis can be called, us-\ning a slightly more modern language, a non-perturbative\nrenormalization group analysis [2]. Both laboratory ex-\nperiments and numerical simulations (see e.g., [4]; [5];\nand also [1]; [2] and references therein) have con\frmed\nthat the kinetic energy dissipation rate in a \ruid does\nnot vanish in the limit of vanishing viscosity \u0017. Such\na non-vanishing limit of energy dissipation requires that\nspace-gradients of velocity diverge in the limit \u0017!0,\ni.e.,ru!1 ( [2]; [3]). This blow-up of velocity gra-\ndients resembles ultra-violet singularities encountered in\nquantum \feld theory. Therefore, hydrodynamic equa-\ntions will become ill-de\fned in this limit as they contain\nvelocity gradients. Turbulent magnetic \felds as well face\nthe same singularity problem in the limit when viscos-\nity and resistivity tend to zero \u0017;\u0011!0 simultaneously.\nConsequently, MHD equations become ill-de\fned in this\nlimit when the \row is turbulent. All in all, this sug-\ngests that the conventional ideal hydrodynamics (HD)\nand ideal MHD may be applied only if the \rows remain\nlaminar and all quantities Lipschitz-continuous.\n\u0003ajafari4@alumni.jh.edu\nyevishni1@jhu.edu\nzvvaikun1@jhu.edu\n1The complex (or real) valued function ginRnis H older con-\ntinuous if two non-negative and real constants Candhexist\nsuch thatjg(x)\u0000g(y)j\u0014Ckx\u0000ykhfor allx;y2Domain (g).\nIf the H older exponent his equal to unity, then gis Lipschitz\ncontinuous. Also gis called H older singular if h<1.In a magnetized \ruid, the magnetic di\u000busivity (resis-\ntivity) and viscosity may be small but \fnite. In the\nlimit of vanishing magnetic di\u000busivity, the magnetic \feld\nseems to be frozen into the \ruid. This magnetic \rux-\nfreezing principle is widely applied as an estimate to\nMHD equations in the laboratory and astrophysical sys-\ntems with the presumption that ideal MHD holds to a\ngood accuracy. With turbulence, ubiquitous in astro-\nphysical and laboratory systems (see e.g., [6]; [7]; [8]; [9]\nand references therein), the velocity and magnetic \felds\nbecome singular in the limit \u0017;\u0011!0 and ideal MHD\ncannot be applied. For instance, magnetic (and velocity)\n\feld lines are usually assumed to be well-de\fned in such\napproaches, however, mathematically, the existence and\nuniqueness of integral curves (\feld lines) is guaranteed\nonly for Lipschitz continuous vector \felds. What does a\nmagnetic \feld line mean if the \feld is (H older) singular\nrather than Lipschitz-continuous?\nIt has been shown that in the limit when viscosity\nof a turbulent \ruid tends to zero, its Lagrangian par-\nticle trajectories become stochastic (see e.g., [10]; [11];\n[12]). Also, it turns out that magnetic \feld lines be-\ncome stochastic in turbulent, magnetized \ruids in the\nlimit when resistivity and viscosity tend to zero simulta-\nneously ([13]; [6]; [14]; [15]). Under such circumstances,\ninstead of the conventional magnetic \rux freezing [16] a\nstochastic version is introduced [13]. Although the con-\ncept of a stochastic magnetic \feld is used in such con-\ntexts, but it is not mathematically obvious at all what a\nstochastic vector \feld really means. In other words, the\nnotion of a stochastic variable is well-known for scalar\nquantities such as a \ructuating temperature or the price\nof certain goods in the market. However, a vector \feld\nassigns a vector, with magnitude and direction, to ev-\nery point in space and time and we need a more general\nstatistical formulation to study the randomness of a vec-\ntor \feld and its relationship with the topology and other\nfeatures of the \feld.\nJafari and Vishniac [3] presented a mathematical for-arXiv:1908.06474v2  [astro-ph.HE]  20 Aug 20192\nmulation for the stochasticity level of magnetic \felds in\nterms of the unit vectors tangent to the renormalized\n\felds at di\u000berent coarse-graining scales. The time depen-\ndent angle between such two unit vectors at a space-time\npoint ( x;t) provides a means to de\fne a local stochas-\nticity level; seexII. The average stochasticity level in an\narbitrary volume Vcan then be de\fned using Lpnorms.\nThe time evolution of the stochasticity level, de\fned in\nthis way, would then be associated with the topological\ndeformations of the magnetic \feld.\nIn the present paper, \frst we brie\ry review the con-\ncept of vector \feld stochasticity developed by [3] in xII. In\nxIII, we relate magnetic di\u000busion to magnetic stochastic-\nity and test the theoretically predicted relationship using\nthe data extracted from an incompressible, homogenous\nMHD simulation, archived in an online, web-accessible\ndatabase ([17];[18];[19]). In xIV, we summarize and dis-\ncuss our results. In order to present a more complete\ndiscussion on magnetic di\u000busion, we have also added an\nappendix to discuss the 2-particle Richardson di\u000busion\nand the related scaling laws in MHD turbulence.\nII. VECTOR FIELD STOCHASTICITY\nIn order to remove the singularities of the velocity \feld\nu(x;t) or magnetic \feld B(x;t) in a turbulent \row, we\ncan renormalize (coarse-grain) it at a length scale lby\nmultiplying it by a rapidly decaying function and inte-\ngrating over a volume V. For example, for magnetic \feld\nB, we have\nBl(x;t) =Z\nVGl(r)B(x+r;t)d3r; (1)\nwhereGl(r) =l\u00003G(r=l) withG(r) being a smooth,\nrapidly decaying kernel, e.g., the Gaussian kernel scales\nase\u0000r2=l2. Without loss of generality, we may assume\nG(r)\u00150; (2)\nLimjrj!1G(r)!0; (3)\nZ\nVd3rG(r) = 1; (4)\nZ\nVd3rrG(r) = 0; (5)\nand\nZ\nVd3rjrj2G(r) = 1: (6)The renormalized \feld Blrepresents the average \feld in\na parcel of \ruid of length scale l. It is non-singular and\nits spatial gradients are well-de\fned [3].\nThe scale-split magnetic energy density,  (x;t) is de-\n\fned [3] as\n (x;t) =1\n2Bl(x;t):BL(x;t): (7)\nwhich is divided into two scalar \felds as  (x;t) =\n\u001e(x;t)\u001f(x;t) such that\n\u001el;L(x;t) =(^Bl(x;t):^BL(x;t)BL6= 0 &Bl6= 0;\n0 otherwise;\n(8)\nwhich is called magnetic topology \feld and\n\u001f(x;t) =1\n2Bl(x;t)BL(x;t); (9)\nwhich is called magnetic energy \feld. The quantity\n^Bl(x;t):^BL(x;t) is in fact the cosine of the angle be-\ntween two coarse-grained components BlandBL, hence\nit is a local measure of the \feld's stochasticity level. In\norder to develop a statistical measure, we can take the\nvolume average of this quantity in a volume Vat timet\nwhich de\fnes magnetic stochasticity level Sp(t) given by\nSp(t) =1\n2jj\u001e(x;t)\u00001jjp; (10)\nwhere we have used the Lpnorms for averaging2. The\ncross energy is de\fned using the energy \feld \u001f(x;t) as\nEp(t) =jj\u001f(x;t)jjp; (11)\nWithp= 2, the second order magnetic stochasticity\nlevelS2, magnetic topological deformation T2=@tS2(t),\nmagnetic cross energy density E2(t), and magnetic \feld\ndissipation D2=@tE2(t) are given by\nS2(t) =1\n2(\u001e\u00001)rms; (12)\nT2(t) =1\n4S2(t)Z\nV(\u001e\u00001)@\u001e\n@td3x\nV; (13)\nE2(t) =\u001frms; (14)\nand\nD2(t) =1\nE2(t)Z\nV\u001f@t\u001fd3x\nV: (15)\n2TheLpnorm of f:Rm!Rmis the mapping f!jjfjjp=\n[R\nVjf(x)jp(dmx=V)]1=p. In this paper, we will take p= 2 for\nsimplicity,jjfjj2=frms, which is the root-mean-square (rms)\nvalue of f.3\nIII. DIFFUSION IN TURBULENCE\nIn a resistive \ruid, magnetic \feld lines will di\u000buse away\nas a result of a non-zero magnetic di\u000busivity \u0011(which is\nproportional to electrical resistivity). This phenomenon\nis similar to the di\u000busion of a passive scalar such as dye in\na \ruid like water. In Taylor di\u000busion (also called normal\ndi\u000busion; the di\u000busion scheme present also in Brownian\nmotion), the average (rms) distance of a particle from a\n\fxed point, \u000e(t), increases with time tas\n\u000e2(t) =DTt; (16)\nwhereDTis the (constant) di\u000busion coe\u000ecient. Note\nthat no matter the medium is turbulent or not, this dif-\nfusion scheme will apply but with di\u000berent di\u000busion coef-\n\fcients. Turbulence will in general increase the di\u000busion\ncoe\u000ecientDTmaking the di\u000busion process more e\u000ecient\nbut the nature of this normal di\u000busion will remain lin-\near in time (see below) at scales much larger than the\nturbulent inertial range. Because by de\fnition \u000e2\u0018t\r\nis sub-linear if \r <1, linear if \r= 1 and super linear if\n\r >1, hence normal di\u000busion is a linear di\u000busion.\nIn the presence of turbulence, although the normal dif-\nfusion scheme is still valid at scales much larger than the\nlarge eddies in the inertial range but it cannot be ap-\nplied in the inertial range of turbulence. In the inertial\nrange, the average (rms) separation of two di\u000busing par-\nticles grows super-linearly with time. This corresponds\nto 2-particle Richardson di\u000busion;\n\u00012(t) =DRt3; (17)\nwith di\u000busion coe\u000ecient DR. This result can be obtained\nin several ways discussed in the Appendix (see also [6];\n[7]). The power of 3 indicates, of course, a super-linear\ndi\u000busion. It is important to emphasize that the Richard-\nson di\u000busion is a 2-particle di\u000busion (i.e., it is concerned\nwith the average separation of two particles undergoing\ndi\u000busion in turbulence) while the normal (Taylor) di\u000bu-\nsion is a one-particle di\u000busion scheme (i.e., it is concerned\nwith the average distance of a di\u000busing particle from a\n\fxed point). It turns out, as it might be expected, that\nmagnetic \feld lines undergo Richardson di\u000busion in the\nturbulence inertial range [6]; see Fig.(1).\nSpectral analysis (Fourier decomposition) is often used\nto study turbulent magnetic \felds in which one speaks\nof parallel\u0015k=k\u00001\nkand perpendicular \u0015?=k\u00001\n?wave-\nlengths and wave-numbers ( kkandk?) with respect to\nthe local magnetic \feld. In such an approach, general\nequations (16) and (17) are translated into the following\nrelationship:\n\u00152\n?\u0019\u000b\u0015\f\nk; (18)\nwith \fxed \u000b(for a given turbulence inertial range) as\nthe di\u000busion coe\u000ecient; see Appendix. Note that \f= 3\ncorresponds to the super-linear Richardson di\u000busion and\n\f= 1 to normal, sub-linear dissipative di\u000busion. How\nFIG. 1. Evidence of super-linear magnetic dispersion in turbu-\nlence. Top: Mean squared dispersion of \feld lines backwards in\ntime with the variable t\u0003=tf\u0000t, in directions both parallel (red)\nand perpendicular (blue) to the local magnetic \feld. Times are\nnormalized by the inverse r.m.s. current, 1 =jrms, and distances\nare normalized by the resistive length, ( \u0015=jrms)1=2. Error bars,\ns.e.m. The dashed line shows the conventional di\u000busive estimate,\nhr2(t\u0003)i= 4\u0015t\u0003, and the solid line is /t8=3\n\u0003. For details see [6].\nFIG. 2. Parallel,\u0015k, and perpendicular, \u0015?, wavelengths with\nrespect to the large scale magnetic \feld BL(coarse-grained Bon\nscaleL), in a region of scale l<L with local \feld Bl(coarse-grained\nBon scalel<L ). The angle \u0012is a stochastic variable because of\nthe randomness inherent in B, hence it can be used to quantify\nmagnetic stochasticity level. Its local relationship with \u0015kand\u0015?,\non the other hand, provides a means to relate the stochasticity\nlevelS2(t), de\fned by eq.(12), to magnetic di\u000busion, eq.(18). This\nrelationship is quanti\fed by eq.(21).\ncan we relate magnetic di\u000busion to the level of random-\nness in magnetic \feld?\nConsider the coarse-grained magnetic \feld Blin a re-\ngion of length scale l. Denote by \u0015?and\u0015k, respectively,\nthe perpendicular and parallel components of Blwith re-\nspect to BL; see Fig.(2). The assumption is that we are\nin the inertial range of turbulence and Lis few times\nlarger than l. From eq.(18) one can write\n\u0015k=l\u001e;&\u0015?=l(1\u0000\u001e2)1=2;\nwhere\u001e(x;t) = cos\u0012=^Bl:^BL. We have\nl2(1\u0000\u001e2) =\u000bl\f\u001e\f: (19)4\nOne can relate the stochasticity level, given by eq.(12),\nto the magnetic di\u000busion, which is related to eq.(19), by\nwriting the latter expression as\n1\n2(1\u0000\u001e) =\u000b\n2l\f\u00002\u001e\f\n1 +\u001e;\nwhich upon taking Lp-norm and using eq.(12) gives us\nSp(t) =\u000b\n2l\f\u00002k\u001e\f(x;t)\n1 +\u001e(x;t)kp: (20)\nTakingp= 2, for simplicity, we \fnd\nS2(t) =\u000b\n2l\f\u00002\u0010\u001e\f(x;t)\n1 +\u001e(x;t)\u0011\nrms: (21)\nTABLE I. Numerical values of hf(t)iTfor di\u000berent scales\nlandL, which is assumed to be few times larger than l,\nfor\f= 1;3;5;7 in a randomly selected sub-volume of size\n194\u000242\u000233 in grid units; see eq.(23). The mean, standard\ndeviation (STD) and relative standard deviation of hf(t)iT\nare calculated for di\u000berent scales landL. These data, and\nsimilar ones for other randomly selected sub-volumes, indicate\nthat\f= 3 gives the smallest relative standard deviation for\nhf(t)iT. Physically, this means that only for super-di\u000busion\nwith\f\u00183 an almost constant di\u000busion coe\u000ecient can be\nobtained as \u000b= 2hf(t)iT.\nl; L \f = 1\f= 3\f= 5\f= 7\n3;7 0.2774 0.0317 0.0036 0.0004\n3;9 0.2128 0.0244 0.0028 0.0003\n5;9 0.5834 0.0241 0.0009 0.0000\n3;11 0.1590 0.0180 0.0020 0.0002\n5;11 0.5087 0.0212 0.0008 0.0000\n7;11 0.8985 0.0189 0.0004 0.0000\nMean 0.4400 0.0230 0.0018 0.0002\nSTD 0.2802 0.0050 0.0012 0.0002\nRelative STD 0.6368 0.2174 0.6667 1.000\nFirst, we note that the scale dependence decreases with\nincreasing the scale since we are coarse-graining using a\nrapidly decaying kernel G. Hence with L>l , we expect\na weaker dependence on Lin eq.(21), which is based on\nthe assumption that Lis few time larger than l. The\nrelationship given by eq.(21) should hold for the right\nchoice of\ffor magnetic di\u000busion (i.e., \f= 1 for resistive\ndi\u000busion and \f= 3 for Richardson di\u000busion). With such\na choice, the di\u000busion coe\u000ecient \u000bcan be obtained from\nthis expression by time-averaging. If the di\u000busion scheme\nis super-linear Richardson di\u000busion in the inertial range,\nas discussed above, the evaluation of the above expression\nshould lead to \f= 3 with a \fxed di\u000busion coe\u000ecient \u000b.\nIn order to examine eq.(21) numerically, we use\na homogeneous, incompressible MHD numerical simu-\nlation archived in an online, web-accessible database\n([17];[18];[19]). This is a direct numerical simulation\n(DNS), using 10243nodes, which solves incompressible\nFIG. 3. Plots off(t) (curves) de\fned by eq.(22) with respect to\ntime and its time-average hf(t)iT(horizontal lines) for l= 3;L=\n11 (dotted, cyan), l= 5;L= 11 (dashed, red), l= 7;L= 11 (solid,\nyellow). For the correct value of \f, the time average of this function,\nhf(t)iT, should be almost independent of scale and approximately\nequal to the half of the di\u000busion coe\u000ecient \u000b, de\fned by eq.(18).\nFor di\u000berent scales, the standard deviation and relative standard\ndeviation corresponding to \f= 1 (A) are much larger that their\ncounterparts for \f= 3 (B). The numerical values of \u000b, in the same\nsub-volume, are shown in Table.(I). This result holds in di\u000berent\nsub-volumes of the simulation box.\nMHD equations using pseudo-spectral method. The sim-\nulation time is 2 :56sand 1024 time-steps are avail-\nable (the frames are stored at every 10 time-steps of the\nDNS). Energy is injected using a Taylor-Green \row stir-\nring force. Let us de\fne\nf(t) :=1\nl\f\u00002S2(t)\u0010\n\u001e\f(x;t)\n1+\u001e(x;t)\u0011\nrms; (22)\nand evaluate the (rms) time average of this function,\nhf(t)iTover the time interval T(simulation time). For\nthe right choice of \ffor magnetic di\u000busion in turbulence,\nif eq.(22) holds,hf(t)iTshould be a constant and inde-\npendent of l:5\nFIG. 4. Same as Fig.(3) but for \f= 5 (A) and \f= 7 (B); see\nalso Table.(I). The value of f(t) becomes smaller by increasing \f,\nhowever, its relative standard deviation ramps up indicating that\nf(t) is not a constant for \f > 3 and thus no constant di\u000busion\ncoe\u000ecient can be de\fned. For \f6= 3, in general, a slight change in\nscale leads to a large relative change in di\u000busion coe\u000ecient.\nhf(t)iT=\u000b\n2=const: (23)\nIn Fig.(3), we have plotted f(t) as a function of time,\nfor di\u000berent values of landLfor both\f= 1 and\n\f= 3 in a randomly selected sub-volume of the simu-\nlation box with size 194 \u000242\u000233 in grid units equivalent\nto 1:2\u00020:26\u00020:20 in physical units with coordinates\n[400;733;300]\u0000[596;775;333] in the simulation box. Re-\npeating this computation in several other randomly se-\nlected sub-volumes of the simulation box leads to similar\nresults. We have also evaluated other values of \fin the\nsame sub-volume; e.g., see Fig.(4) for \f= 5;7. The\nnumerical values of hf(t)iTfor di\u000berent scales landL,\nalong with the mean value, standard deviation and rela-\ntive standard deviation, are presented in Table.(I). These\nresults indicate that \f= 3 (Richardson di\u000busion) leadsto an almost constant (with respect to scales landL>l )\ndi\u000busion coe\u000ecient DR:=\u000bwhile the relative standard\ndeviation of the time-averaged quantity hf(t)iTbecomes\nincreasingly larger as \fincreases and no constant di\u000bu-\nsion coe\u000ecient, with respect to scale, can be de\fned in\nthese cases.\nIV. SUMMARY AND CONCLUSIONS\nIn this paper, we have advanced physical arguments to\nrelate magnetic stochasticity level Sp(t) to magnetic dis-\npersion in MHD turbulence. We have also tested this the-\noretical prediction using an incompressible, homogeneous\nMHD numerical simulation stored online ([17];[18];[19]).\nOur results agree with the super-linear, Richardson dif-\nfusion scheme for turbulent magnetic \felds.\nStochasticity level of turbulent magnetic \felds\nis quanti\fed by volume-averaging the scalar \feld\n\u001e(x;t) = ^Bl:^BLwhere ^Bl=Bl=jBljandBlis\nthe coarse-grained magnetic \feld at scale l;Bl=\nl\u00003R\nVG(r=l)B(x+r;t)d3rwith a rapidly decaying ker-\nnelG(r) =G(r). Likewise ^BLcan be computed for\na larger scale L > l . More speci\fcally, p-order mag-\nnetic stochasticity is de\fned as the Lp-norm (volume-\naverage)jj\u001e\u00001jjp=2. Hence for p= 2, the second order\nmagnetic stochasticity level is given by the rms-average\nS2(t) =1\n2(\u001e\u00001)rms[3].\nWe have shown, using simple scaling laws of MHD tur-\nbulence, that stochasticity level Sp(t) is related to the dif-\nfusion power \fand di\u000busion coe\u000ecient \u000bin\u00152\n?\u0019\u000b\u0015\f\nk\nby eq.(20);\nSp(t) =\u000b\n2l\f\u00002k\u001e\f(x;t)\n1 +\u001e(x;t)kp:\nWe have used the second order stochasticity level S2(t)\nto numerically check the Richardson value \f= 3 against\nnormal di\u000busion associated with \f= 1. In our statis-\ntical analyses of several sub-volumes of the simulation\nbox, super-linear di\u000busion with \f= 3 leads to the small-\nest standard deviation, which is at least an order of\nmagnitude smaller than that corresponding to \f= 1.\nIt should be emphasized that we have related magnetic\nstochasticity to its di\u000busion scheme in turbulence, whose\nsuper-linear nature can be inferred using any MHD tur-\nbulence model such as Goldreich-Sridhar model [20] or\nKolmogorov scaling laws [21], discussed in the Appendix.\nOur arguments in this paper are thus quite generally ap-\nplicable to magnetic \felds in turbulence inertial range\nand are independent of any MHD turbulence model.\nAppendix A: Richardson Di\u000busion and MHD\nTurbulence\nIn this Appendix, we present well-known theoretical\nevidence in support of the super-linear nature of mag-6\nnetic di\u000busion in the presence of turbulence invoking dif-\nferent methodologies. The super-linear nature of mag-\nnetic di\u000busion does not depend on any MHD turbulence\nmodel, rather the implication is that any successful MHD\nturbulence model would agree with Richardson 2-particle\ndi\u000busion scheme. Super-linear magnetic di\u000busion in tur-\nbulence inertial range is a model-independent, universal\nfeature of turbulent magnetic \feld. Our arguments in\nthis paper relate this phenomenon to the stochasticity\nlevel of magnetic \felds. Analytical and numerical stud-\nies of magnetic di\u000busion in turbulence inertial range can\nbe found in e.g., [13]; [6]; [15]; [7] and [22] and references\ntherein.\nTo estimate the 2-particle separation in a turbulent\n\row (for a detailed discussion see [22]), we write the dis-\ntance between two arbitrary particles, initially separated\nby \u0001(t= 0)\u0011\u00010= \u0001\u000b0, at timetas\n\u0001(t) =jX(\u000b0;t)\u0000X(\u000b;t)j; (A1)\nwhere\u000b0=\u000b+\u0001\u000b0. Assuming a H older singular veloc-\nity \feld u, we have\nju(x0;t)\u0000u(x;t)j\u0014Ajx0\u0000xjh; (A2)\nwhereAis a constant and h < 1 is the H older expo-\nnent. Taking the time derivative of eq.(A1), using the\ntriangular inequality and eq.(A2), we \fnd\nd\u0001(t)\ndt\u0014AjX(\u000b0;t)\u0000X(\u000b;t)jh=A[\u0001(t)]h;(A3)\nwith the solution\n\u0001(t)\u0014h\n\u00011\u0000h\n0+A(1\u0000h)(t\u0000t0)i1\n1\u0000h: (A4)\nThere is a remarkable di\u000berence in the above expres-\nsion for two di\u000berent choices 0 <h< 1 andh!1. It is\nsimple calculus to show that the latter case leads to\n\u0001(t)\u0014\u00010exp [A(t\u0000t0)]; h!1: (A5)\nThis result implies that for t! 1 , we have \u0001( t)'\n\u00010eA(t\u0000t0). Therefore we arrive at the important result\nthat in this case the initial conditions are never forgotten!\nHowever, the choice 0 <h< 1 leads to\n\u0001(t)'h\nA(1\u0000h)(t\u0000t0)i1\n1\u0000h; t!1: (A6)\nThis implies that the information about initial condi-\ntions, i.e., the initial separation of particles, is lost for\nlong times. Using the Kolmogorov's ([21]) theory of tur-\nbulence in the inertial range, we know that h= 1=3.\nTherefore, the above expression yields\n\u00012(t)/(t\u0000t0)3: (A7)The power of 3 indicates, of course, a super-linear dif-\nfusion. At su\u000eciently large times t\u001dt0, we recover\neq.(16): \u00012(t)\u0018t3.\nHistorically, however, Richardson took a di\u000berent ap-\nproach to get this result, an understanding of which is\nboth instructive and also important for many other prob-\nlems (see also [14]). Richardson's probability density for\nparticle separation vector l=x1\u0000x2, with a scale-\ndependent di\u000busion coe\u000ecient K(l)\u0018K0l4=3, satis\fes\n@tP(l;t) =rli\u0010\nK(l)rliP(l;t)\u0011\nwith a similarity solu-\ntion [13],\nP(l;t) =A\n(K0t)9=2exp\u0010\n\u00009l2=3\n4K0t\u0011\n: (A8)\nUsing this probability density to average l2, one \fnds\nhl2(t)i= (1144=81)K3\n0t3. This is intimately related to\nKolmogorov's relation\nl2(t)\u0018(g0\u000f)t3; (A9)\nwhich is a solution to the initial value problem dl(t)=dt=\n\u000eu(l) = (3=2)(g0\u000fl)1=3,l(0) =l0for su\u000eciently long times\nt\u001dt0. Hereg0is Richardson-Obukhov constant and \u000f\nthe mean energy dissipation rate.\nIn the following we also present a brief discussion of\nMHD turbulence scaling laws related to magnetic di\u000bu-\nsion in the turbulence inertial range. We shall see that\nsuch considerations generally agree with the reulst given\nby eq.(A9).\nIn the Kolmogorov's hydrodynamic turbulence [21],\nthe statistics of turbulent motions is determined by the\nenergy transfer rate \u000f=v3\ny=yand length scale yat the\ninertial range, where the energy is supposedly transferred\nwith no dissipation, and the energy transfer rate and vis-\ncosity\u0017at the dissipative range, where the energy is dis-\nsipated by viscosity. Dimensional analysis leads to the\nturbulent velocity vy\u0018\u000f1=3y1=3and the turbulent eddy\ntime scale\u001cy\u0018\u000f\u00001=3y2=3in the inertial range. This scal-\ning leads to the famous velocity spectrum vk\u0018k\u00001=3and\nenergy power spectrum\nEKol(k)\u0018k\u00005=3: (A10)\nAt the dissipative range, a similar dimensional anal-\nysis shows that vd\u0018\u000f1=4\u00171=4,\u001cd\u0018\u000f\u00001=2\u00171=2and\nyd\u0018\u00173=4\u000f\u00001=4\u0011y(\u0017=yvy)3=4.\nKolmogorov scaling, for hydrodynamic turbulence,\ncannot be applied directly to MHD turbulence because\nof the complications that magnetic \feld introduces.\nIroshinkov [23], and Kraichnan [24] independently devel-\noped a model for incompressible MHD turbulence. Since\ntwo-wave interactions behave as elastic collisions, this\npicture relies on the interactions of triads of waves [25].\nThe Iroshinkov-Kraichnan (IK) model assumes that (a)\nonly oppositely directed waves interact, (b) turbulence\nis isotropic and (c) energy cascades from long to short7\nwavelengths, and \fnally (d) dominant interactions in-\nvolve three wave-couplings (triads). The assumption that\ninteractions are local, i.e. only \ructuations of compara-\nble sizes interact, then propagating \ructuations behave\nas Alfv\u0013 en wave packets of parallel extent lkand perpen-\ndicular extent l?. Suppose \u000eul\u0018\u000eBl, where\u000euland\n\u000eBlare, respectively, the \ructuations in the velocity and\nmagnetic \felds. Two counter-propagating wave pack-\nets would require an Alfv\u0013 en time, \u001cA\u0018lk=VA, to pass\nthrough each other. During this time, the amplitude of\neach wave packet will su\u000ber a change \u0001 \u000eul,\n\u0001\u000eul\n\u001cA\u0018\u000eul\n\u001cl; (A11)\nwhere\u001cl\u0018l=\u000eulis the eddy turnover time. The as-\nsumption of the local, weak interactions means\n\u0001\u000eul\u001c\u000eul,\u001cA\u001c\u001cl: (A12)\nThe cascade time, \u001cnlis de\fned as the time that it takes\nto change\u000eulby an amount comparable to itself assum-\ning that the changes accumulate in a random walk man-\nner3;\n\u001cnlX\nt=0\u0001\u000eul\u0018\u000eul\u001cA\n\u001cl\u0012\u001cnl\n\u001cA\u00131=2\n\u0018\u000eul: (A13)\nThus, we obtain the energy transfer rate\n\u001cnl\u0018l2VA\nlk\u000eu2\nl: (A14)\nThe isotropy assumption means that the power is dis-\ntributed isotropically in wave space. Assuming lk\u0018l, we\nobtain\n\u001cIK'lVA\nv2\nk; (A15)\nwhere we have simply used vk, to refer to the rms velocity\non the eddy scale k\u00001, instead of \u000eul. Whenvk<VA, we\nhave\u001cnl>kvk, which implies the reduction of the energy\ncascade to higher wavenumbers by the magnetic \feld.\nConstancy of the energy transfer rate v2\nk=\u001cIK=const:\nleads tovk\u0018k\u00001=4. Therefore the Kraichnan-Iroshinkov\nenergy power spectrum is given by\n3The average total length in a random walk with average step size\nlvanishesh\u0006ilii= 0, buth(\u0006ili)2i=h\u0006il2\nii=Nl2. The total\nnumber of steps Nis the total time ttotdivided by the aver-\nage time of one step t, that isN=ttot=t. Thush(\u0006ili)2i1=2=\n(ttot=t)1=2l. For the velocity \ructuations, eq.(A11) gives \u0001 \u000eul\u0018\n\u000eul(\u001cA=\u001cl). To add up \u0001 \u000eul's during total time \u001cnl, that is\n\u001cnlP\nt=0\u0001\u000eul= (\u001cA=\u001cl)\u001cnlP\nt=0\u000eul, one replaces the step size with \u000eul,\naverage step time with \u001cA. This leads to the \frst part of\neq.(A13).EIK(k)\u0018k\u00003=2: (A16)\nIK theory was the most popular model accepted as\nMHD generalization of Kolmogorov's ideas for about\n30 years. In 1970's, measurements showed strong\nanisotropies in the solar wind with lk> l?. Goldreich\nand Sridhar ([20] henceforth GS95; [26]) suggested that\nthe e\u000bect of residual three wave couplings is consistent\nwith eq.(A14), for the basic nonlinear timescale, but an\nanisotropic spectrum should be considered in which the\ntransfer of power between modes moves energy toward\nlargerk?with no e\u000bect on kk. Here,k?is the wavevec-\ntor component perpendicular, and kkis the wavevector\ncomponent parallel, to the direction of magnetic \feld.\nTherefore, using eq.(A14), the basic nonlinear timescale\ncan be written as\n\u001cnl'kkVA\nk2\n?v2\nk; (A17)\nwhere!A=kkVAis Alfv\u0013 en wave frequency.\nThe critical balance requires that kkandk?are related\nas\nkkVA\u0019k?vk; (A18)\nwhereVAis the Alfv\u0013 en speed and vk. This is translated\ninto the requirement that the \feld couples to a typical\neddy at a rate approximately equal to the eddy turnover\nrate. The second assumption in the GS95 model is that\nthe nonlinear energy transfer rate is \u0018k?vk, similar to\nthat of hydrodynamic turbulence [21]. These assump-\ntions together lead to a power spectrum which behaves\nlike hydrodynamic turbulence, i.e. vk/k\u00001=3\n?. Con-\nsequently, the energy power spectrum of GS95 is given\nby\nEGS(k?)\u0018k\u00005=3\n?: (A19)\nSuppose that we inject energy into the medium with a\nparallel length scale l, with the corresponding perpendic-\nular scalel?=lkMA(whereMA=VT=VAis the Alfv\u0013 en\nMach number), that creates an rms velocity VT. The re-\nsulting inertial turbulent cascade satis\fes critical balance\nat all smaller scales, therefore\n\u001c\u00001\nnl'kkVA'k?vk; (A20)\nand the constant \row of energy through the sub-Alfv\u0013 enic\ncascade is given by\n\u000f'v4\nl\nlkVA'V2\nT\nlk=VA'v2\nk\n\u001cnl'k?v3\nk; (A21)\nwherevl=pVTVAis sometimes de\fned as the velocity\nfor isotropic injection of energy which undergoes a weakly\nturbulent cascade and ends up with a strongly turbulent8\ncascade (see e.g., [27]; [28]). Putting all this together, we\n\fnd the following relationship:\nkk'l\u00001\nk\u0012k?lVT\nVA\u00132=3\n; (A22)\nbetween parallel and perpendicular wave-numbers. Notethat from here we also get \u001c\u00001\nnl'kkVA'VA\nlk\u0010\nk?lVT\nVA\u00112=3\nand the rms velocity in the large scale eddies, vk'\nVT\u0010\nk?lVT\nVA\u0011\u00001=3\n.\nThus the perpendicular wave-number k?scales as\nk?/k3=2\nkin terms of the parallel wave-number kk. This\nscaling is exactly similar to the Richardson scaling given\nby eq.(A9). This result can be presented in terms of\nthe wave-lengths parallel and perpendicular to the local\nmagnetic \feld as \u00152\n?\u0018\u00153\nk.\n[1] G. L. Eyink and K. R. Sreenivasan, Reviews of Modern\nPhysics 78, 87 (2006).\n[2] G. L. Eyink, arXiv e-prints (2018), arXiv:1803.02223\n[physics.\ru-dyn].\n[3] A. Jafari and E. Vishniac, Phys. Rev. E 100, 013201\n(2019).\n[4] K. R. Sreenivasan, Physics of Fluids 27, 1048 (1984).\n[5] K. R. Sreenivasan, Physics of Fluids 10, 528 (1998).\n[6] G. Eyink, E. Vishniac, C. Lalescu, H. Aluie, K. Kanov,\nK. B urger, R. Burns, C. Meneveau, and A. Szalay, Na-\nture (London) 497, 466 (2013).\n[7] A. Jafari and E. Vishniac, arXiv e-prints (2018),\narXiv:1805.01347 [astro-ph.HE].\n[8] A. Jafari and E. T. Vishniac, Astrophys. J. 854, 2 (2018).\n[9] A. Jafari, arXiv e-prints (2019), arXiv:1904.09677 [astro-\nph.HE].\n[10] D. Bernard, K. Gawedzki, and A. Kupiainen, Journal of\nStatistical Physics 90, 519 (1998), cond-mat/9706035.\n[11] K. Gaw\u0018 edzki and M. Vergassola, Physica D Nonlinear\nPhenomena 138, 63 (2000).\n[12] W. E. Vanden Eijnden and V. Eijnden, Proc. Natl. Acad.\nSci.97, 8200 (2000).\n[13] G. L. Eyink, Phys. Rev. E 83, 056405 (2011),\narXiv:1008.4959 [physics.plasm-ph].\n[14] G. L. Eyink, Astrophys. J. 807, 137 (2015),\narXiv:1412.2254 [astro-ph.SR].\n[15] C. C. Lalescu, Y.-K. Shi, G. L. Eyink, T. D. Drivas, E. T.\nVishniac, and A. Lazarian, Physical Review Letters 115,\n025001 (2015), arXiv:1503.00509 [physics.plasm-ph].[16] H. Alfv\u0013 en, Ark. Mat., Astron. Fys. 29B, 1 (1942).\n[17] Forced MHD Turbulence Dataset, Johns Hopkins Turbu-\nlence Databases, https://doi.org/10.7281/T1930RBS\n(2008(Accessed April, 2019)).\n[18] Y. Li, E. Perlman, M. Wan, Y. Yang, R. Burns, C. Mene-\nveau, R. Burns, S. Chen, A. Szalay, and G. Eyink., Jour-\nnal of Turbulence 31(2008).\n[19] E. Perlman, R. Burns, Y. Li, and C. Meneveau, in Pro-\nceedings of the 2007 ACM/IEEE Conference on Super-\ncomputing , SC '07 (ACM, 2007) pp. 23:1{23:11.\n[20] P. Goldreich and S. Sridhar, Astrophys. J. 438, 763\n(1995).\n[21] A. Kolmogorov, Akademiia Nauk SSSR Doklady 30, 301\n(1941).\n[22] G. L. Eyink, Turbulence theory, course notes, http:\n//www.ams.jhu.edu/ ~eyink/Turbulence/notes.html\n(2019).\n[23] P. S. Iroshnikov, Astronomicheskii Zhurnal 40, 742\n(1963).\n[24] R. H. Kraichnan, Physics of Fluids 8, 1385 (1965).\n[25] P. H. Diamond and G. Craddock, Comments on Plasma\nPhysics and Controlled Fusion 22, 287 (1990).\n[26] P. Goldreich and S. Sridhar, Astrophys. J. 485, 680\n(1997), astro-ph/9612243.\n[27] A. Lazarian and E. T. Vishniac, Astrophys. J. 517, 700\n(1999), astro-ph/9811037.\n[28] A. Jafari, E. T. Vishniac, G. Kowal, and A. Lazarian,\nAstrophys. J. 860, 52 (2018)."    },    {        "title": "2202.05622v1.Mutual_conversion_between_a_magnetic_Neel_hopfion_and_a_Neel_toron.pdf",        "content": "Mutual conversion between a magnetic Néel hopfion and a Néel toron\nShuang Li,1Jing Xia,2Laichuan Shen,3Xichao Zhang,4Motohiko Ezawa,5,\u0003and Yan Zhou1,y\n1School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China\n2College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China\n3The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, Beijing 100875, China\n4Department of Electrical and Computer Engineering,\nShinshu University, Wakasato 4-17-1, Nagano 380-8553, Japan\n5Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-8656, Japan\n(Dated: February 14, 2022)\nThree-dimensional (3D) magnetic textures attract great attention from researchers due to their fascinating\nstructures and dynamic behaviors. Magnetic hopfion is a prominent example of 3D magnetic textures. Here, we\nnumerically study the mutual conversion between a Néel-type hopfion and a Néel-type toron under an external\nmagnetic field. We also investigate the excitation modes of hopfions and torons in a film with strong perpen-\ndicular magnetic anisotropy. It is found that the Néel-type hopfion could be a stable state in the absence of the\nexternal magnetic field, and its diameter varies with the out-of-plane magnetic field. The Néel-type hopfion\nmay transform to a Néel-type toron at an out-of-plane magnetic field of about 20 mT, where the cross section\nstructure is a Néel-type skyrmion. The hopfion and toron show different excitation modes in the presence of an\nin-plane microwave magnetic field. Our results provide a method to realize the conversion between a Néel-type\nhopfion and a Néel-type toron, which also gives a way to distinguish Bloch-type and Néel-type hopfions.\nI. INTRODUCTION\nTopological solitons are stable particle-like configurations\noriginally proposed in continuous-field theory [1]. The past\nfew decades witnessed a continuous growth about topolog-\nical spin textures in magnetic systems. A nanoscale exam-\nple in one space dimension is the chiral domain wall [2–4].\nMagnetic skyrmions [5–13] and magnetic vortices [14, 15]\nare examples of two-dimensional (2D) topological solitons.\nThree-dimensional (3D) topological spin textures in magnetic\nmaterials have been observed recently including skyrmion\nstrings [16], target skyrmions [17], chiral bobbers [18], and\nhopfions [19]. Among these topological solitons, the 2D mag-\nnetic skyrmions in magnetic materials with Dzyaloshinskii-\nMoriya interactions (DMIs) have aroused much interest [20,\n21] due to their prominent properties, such as their nanoscale\nsize and low driving current density [22]. Besides, magnetic\nskyrmions can be used as information carrier in future spin-\ntronic applications [23–28].\nThe magnetic skyrmion has a natural generalization to a\ntopological soliton with a 3D topology, known as a magnetic\nhopfion. Similar with the clarifications of skyrmions, hopfions\nare identified with two types, that is, the Bloch-type and Néel-\ntype hopfions. The cross section structure of a Bloch-type\nhopfions is a Bloch-type skyrmionium cite2015skyrmionic,\nwhile the cross section structure of a Néel-type hopfion is\na Néel-type skyrmionium [30]. The hopfion was originally\nproposed in the Skyrme-Faddeev model [31], where the topo-\nlogical number is not a generalized winding number but in-\nstead is a linking number of field lines, given by the integer-\nvalued Hopf invariant. Several works have numerically pre-\ndicted stable hopfions in the chiral magnetic system with the\n\u0003Email: ezawa@ap.t.u-tokyo.ac.jp\nyEmail: zhouyan@cuhk.edu.cnDzyaloshinskii-Moriya interaction and interfacial perpendicu-\nlar magnetic anisotropy (PMA) [30, 33, 34] and reported that\na Bloch hopfion can transform to a monopole-antimonopole\npair (MAP) [34] or a toron [35]. Most recently, magnetic\nhopfions were experimentally created by adapting the PMA\nin Ir/Co/Pt multilayered systems [32].\nThe basic element of hopfions and torons is a double twist\ntorus [37, 38]. Torons were initially observed in liquid crys-\ntals and also found in chiral magnets later [39–42]. The tran-\nsition between skyrmions and torons has been recently ob-\nserved [41, 43]. A magnetic toron is a spatially localized\n3D spin texture composed of Bloch-type skyrmion layers with\ntwo Bloch points at its two ends [37, 44]. Monopoles or Bloch\npoints are topologically nontrivial point singularities, where\nmagnetization suffers a discontinuity [45].\nThe research of various dynamics of topological solitons is\nvital part in real applications and the studies of dynamic exci-\ntations could guide further design of microwave devices. The\nhigh-frequency gyrotropic modes of vortices have been stud-\nied during the past decades and the excitations of magnetic\nvortex can be applied in spin-torque oscillators [46–48] and\nspin-wave emitters [26]. Spin excitation of skyrmions may\noffer more promising prospects toward the design of novel de-\nvices. Three characteristic resonant modes, which are breath-\ning, clockwise (CW) and counterclockwise (CCW) modes,\nhave been predicted theoretically in the year 2012 [49], and\nlater on, these three modes were observed experimentally in\nthe helimagnetic insulator using microwave absorption spec-\ntroscopy [50]. Apart from the three aforementioned resonance\nmodes, theoretical studies predict the existence of more col-\nlective excitations of skyrmions which have not been observed\nin experiments up until now [51–53].\nAlthough numerous studies have been made on the dy-\nnamic excitation of low-dimensional topological solitons in-\ncluding vortices and magnetic skyrmions, 3D topological soli-\ntons have still not been well explored in this yield. The collec-\ntive spin wave modes of the target skyrmion with the externalarXiv:2202.05622v1  [cond-mat.mes-hall]  11 Feb 20222\n𝑥𝑥\n𝑦𝑦𝑧𝑧(a)\n(b) (c)\n𝑥𝑥𝑦𝑦\n𝑧𝑧𝑦𝑦\n𝑧𝑧\n𝑥𝑥(d)\n𝑚𝑚𝑥𝑥𝑚𝑚𝑦𝑦\nFIG. 1. Schematic of the proposed structure. (a) The thin cuboids\nat the top and bottom represent the magnetic films with strong PMA.\nThe transparent region in the middle is the chiral magnet cuboid. The\nring at the center represents the Néel-type hopfion. (b), (c), (d) Mid-\nplane cross-sectional spin configurations in the xyplaneyzplane,\nandxzplane. The color sphere and the coordinate system are shown\nin the insets.\nfield have been investigated [53, 54]. Recently, the resonant\nspin wave modes of the Bloch-type hopfion were studied by\nmicromagnetic simulations [35], while the excitation modes\nof the Néel-type hopfion have not been reported.\nIn this work, we study the Néel-type hopfion and show that\nit is stable in a magnetic film in the absence of an external\nmagnetic field. The film is sandwiched by two layers with\nhigh PMA. We also study another nontrivial state, the Néel-\ntype toron consisting of Néel-type skyrmion layers with two\npoint singularities at two ends, is also stabilized in this struc-\nture. We show that the Néel-type hopfion could transform to\na Néel-type toron by an applied magnetic field. Furthermore,\nwe computationally study the resonant dynamics of the Néel-\ntype hopfion and the Néel-type toron, which are excited by\nmicrowave magnetic field. The excitation modes are distinct\nbetween these topological spin textures. Especially, the res-\nonant modes of Néel-type hopfions are different from that of\nBloch-type hopfions [35]. We find a new type of 3D topolog-\nical solitons called the Néel-type toron and our results offer a\nnew observable way to clarify Bloch-type hopfions and Néel-\ntype hopfions based on their excitation mode dynamics.\nII. MODEL AND METHODS\nTo stabilize a Néel-type hopfion in ferromagnetic material,\nwe consider a magnetic film sandwiched by two PMA mag-\nnetic thin layers, as shown in Fig. 1(a). In the continuumapproximation, the free energy is given by\nE=Z\n[Aex(rm)2+wD+Kb(1\u0000m2\nz) +\u00160\nHzMs(1\u0000mz)]dV+Z\nKs(1\u0000m2\nz)dS;(1)\nwhereAexis the exchange constant. wD=Di[mz(r\u0001m)\u0000\n(m\u0001r)mz]is the interface-induced DMI energy density,\nwhich is favorable to hosting a stable Néel-type hopfion. Kb\nandKsare the bulk and the interfacial PMA constant, respec-\ntively.Hzis external magnetic field and Msis the saturation\nmagnetization. The film has a length of l= 128 nm and a\nheight ofh= 15 nm, with each PMA layer on the top and bot-\ntom measuring 0:5nm. The meshed cell size is 0:5nm\u00020.5\nnm\u00020.5 nm for total 2097152 cells per simulation. The non-\nlocal demagnetization energy is excluded for computational\nefficiency.\nWe minimize the free energy (1) with an initial state com-\nputing from an ansatz [30]. A QH= 1 Néel-type hopfion is\nenabled in the structure in the absence of an external magnetic\nfield and the midplane cross-sectional spin configurations are\nshown in Figs. 1(b)-(c) in three directions where the cross sec-\ntion structure in the xyplane is a typical Néel-type skyrmio-\nnium and in the yzorxzplane is a skyrmion-antiskyrmion\npair. The Hopf invariant is defined as\nQH=\u0000Z\nB\u0001Ad3r; (2)\nwhereBi=1\n8\u0019\u000fijkm\u0001(@jm\u0002@km)is the emergent magnetic\nfield calculating from spin texture, where i;j;k =x;y;z and\n\u000fis the Levi-Civita tensor. And Ais a magnetic vector po-\ntential satisfyingr\u0002A=B[55]. Using this model, the\nrelaxed states of the Néel-type hopfion under various external\nmagnetic fields and the corresponding dynamic response to\na microwave magnetic field are simulated by using the GPU\naccelerated micromagnetic simulation software package Mu-\nmax3 [56]. Mumax3 performs a numerical time integration of\nthe Landau-Lifshitz-Gilbert equation at zero temperature\ndm\ndt=\u0000\rm\u0002He\u000b+\u000bm\u0002dm\ndt; (3)\nwhere\ris the gyromagnetic constant, \u000bis the phenomenolog-\nical Gilbert damping constant and He\u000bis the effective field.\nOther parameters are based on previous numerical current-\ndriven dynamics of hopfions in confined thin film [30]. Ex-\nplicitly,Aex= 0:16pJm\u00001, DMI strength of Di= 0:115\nmJm\u00002,Ms= 151 kAm\u00001,Ks= 1 mJm\u00002andKb= 20\nkJm\u00003.\nTo study the dynamic response to magnetic field system-\natically, calculations in our simulations consist of two steps.\nFirstly, the relaxed states in the model are determined by static\nexternal magnetic fields along the zaxis with interval of 1 mT\nfrom\u000020mT to 20 mT. The energy is minimized to get an\nequilibrium state with a large damping constant of \u000b= 0:05\nfor each value of the applied static field Hz. Then the magne-\ntization dynamics is computed about this equilibrium state by\nadding an additional microwave magnetic field in the plane.3\n−10 −5 0𝐵𝐵(mT)(a)𝑙𝑙𝑥𝑥/𝑙𝑙𝑦𝑦=1.44\n𝑙𝑙𝑥𝑥𝑙𝑙𝑦𝑦\n(b)\n (c) (d)\n𝑚𝑚𝑧𝑧\n𝑥𝑥𝑧𝑧𝑦𝑦\n𝑥𝑥𝑙𝑙𝑥𝑥/𝑙𝑙𝑦𝑦=1.08𝑙𝑙𝑥𝑥/𝑙𝑙𝑦𝑦=1\n𝑚𝑚𝑥𝑥𝑚𝑚𝑦𝑦\nFIG. 2. Spin configurations of enlarged Néel-type hopfions and a\nNéel-type toron. (a) The set of preimages for the spin configurations\nof enlarged Néel-type hopfions with different aspect ratio with sev-\neral external magnetic fields. lxandlyare the lengths of outer torus\nof Néel-type hopfion in the xandydirections. (b) The preimage of a\nNéel-type toron in the presence of out-of-plane magnetic field of 20\nmT; (c) and (d) Midplane cross-sectional structures of a Néel-type\ntoron in thexyplane andxzplane.\nA sinc function, hxsin(2\u0019fmaxt)\n2\u0019fmaxt, with a cutoff frequency of\nfmax= 15 GHz and amplitude of hx= 0:5mT, is used to\nexcite the equilibrium states as its Fourier transform is a rect-\nangular function. The simulation runs for 20ns with data sam-\npled every 5ps, using a smaller value of damping constant,\n\u000b= 0:002. The pulse is arbitrarily offset in time to peak at 1\nps.\nIII. RESULTS AND DISCUSSIONS\nThe spin configurations of the relaxed states are found to\nvary with the static external magnetic fields. When a static\nmagnetic field in the \u0000zdirection is applied, the diameter\nof the Néel-type hopfion expands as the magnetic field in-\ncreases. As a result, enlarged hopfions with different aspect\nratioslx=lyare obtained as shown in Fig. 2(a). lxandlyare\nthe lengths of outer torus of the enlarged hopfion in the xandy\ndirections, respectively. Obviously, the enlarged hopfion will\ndeform when the magnetic field is large, resulting from the\ninteraction with boundaries. The preimages of spin configu-\nrations are plotted using Spirit [57] for further visualization.\nWith applied field increasing in the +zdirection, the core\nspins of the Néel-type hopfion gradually shrink, and the cross\nsection structure of the Néel-type hopfion transforms from the\nNéel-type skyrmionium [29, 53] to the Néel-type skyrmion\nuntil 20 mT [Fig. 2(c)]. The preimage of a Néel-type toron is\nshown in Fig. 2(b). The Néel-type toron is similar with toron\nthat previous works studied, while its cross section structure\nis not vortex-like but hedgehog-like. Thus, we name it Néel-\ntype toron, and it is stable with wide range of external fields.\nThe cross section structure of a Bloch-type toron is a Bloch-\ntype skyrmion, while the cross section structure of a Néel-type\nEnergy (J)1e−182.2\n1.01.21.41.61.82.0\n0.80.00000    0.00002    0.00004      0.00006     0.00008      0.00010  \n Position in path (a.u.)ab\ndc\nFIG. 3. Minimal energy path between the Néel-type hopfion and\nthe Néel-type toron. The reaction coordinate is an order parameter\nthat represents the relative distance between two neighboring states.\nPoints a and d represent the Néel-type hopfion and the Néel-type\ntoron, respectively. Point b is a saddle point, and the Néel-type toron\nis formed at point c. The inset is the Hopf invariant QHfrom equa-\ntion (2) of each state along the minimal energy path.\ntoron is a Néel-type skyrmion. They are both ends with two\npoint singularities [Fig. 2(d)]. The topological charge for sin-\ngularities is defined as [58]\nq=1\n8\u0019Z\ndSi\u000fijkm\u0001@jm\u0002@km; (4)\nEquation (4) is also rewritten as q=1\n4\u0019R\nG\u0001dS, where Gis\ngyrovector. The topological charge density is defined accord-\ning to the divergence theorem as\n\u001a=1\n4\u0019r\u0001G=1\n4\u0019[@xGx+@yGy+@zGz]; (5)\nwhereGi=1\n2\u000fijkm\u0001@jm\u0002@km. For a toron, only theR\nV@zGzdVis nonzero, and thus, the topological charge of\na toron is\u00061.\nTo further understand the topological conversion between\nthe Néel-type hopfion and the Néel-type toron, a minimal\nenergy path (MEP) calculation is performed between two\nstates [59, 60]. The MEP calculation is carried out using the\ngeodesic nudged elastic band (GNEB) method. Results from\nthe MEP calculation are shown in Fig. 3. It exists an en-\nergy barrier between the Néel-type hopfion and the Néel-type\ntoron, and an active energy is required to trigger the conver-\nsion between the Néel-type hopfion and the Néel-type toron.\nThe initial state, the barrier peak, the intermediate state and\nthe final state are plotted in Figs. 3(a)-3(d) for visualizing the\nconversion process. Transformation from the Néel-type hop-\nfion state (point a) to the intermediate state (point c) is caused\nby the slip of spins and reconnection of the preimages. And\nthen, the intermediate state (point c) is relaxed to the Néel-\ntype toron state (point d). Reverse process occurs with the4\n(b)\n𝐵𝐵=−20mT𝐵𝐵=0mT𝐵𝐵=20mT\n𝑡𝑡=1\n8𝑇𝑇 𝑡𝑡=14𝑇𝑇\n𝑡𝑡=58𝑇𝑇\n𝑡𝑡=34𝑇𝑇−𝛿𝛿𝑚𝑚𝑥𝑥𝛿𝛿𝑚𝑚𝑥𝑥\nHopfionToronEnlarged \nhopfion(a)\nFIG. 4. (a) Power spectral density (PSD) response of the spin texture as a function of the external static magnetic field. Two vertical dashed\nlines divide the spectrum area into three regions, each of which represents the enlarged hopfion, the Néel-type hopfion, and the Néel-type\ntoron. (b) Midplane snapshots of the real-time dynamics of \u000emxresolved at \u000020mT (enlarged hopfion), 0 mT (Néel-type hopfion), and 20\nmT (Néel-type toron). Each profile shows the results of averaging over the same instant of last ten periods.\nmagnetic field removed. In detail, the spins of central part\nrotate reversely, and also the point singularities move toward\neach other until they annihilate.\nIn the following, we discuss the difference of collective\nmodes of the enlarged hopfion, the Néel-type hopfion and the\nNéel-type toron. The dynamic resonances are calculated by\nexciting the equilibrium states with a time-varying field de-\nscribed by the sinc function in the presence of static perpen-\ndicular magnetic field. It is difficult to excite the Néel-type\nhopfion in this system by applying out-of-plane microwave\nmagnetic field because of the strong boundary condition on\nthe top and bottom layers. Furthermore, the finite thickness of\nthis system matters, which is similar to the spin wave modes of\n3D target skyrmions [54]. When the thickness is comparable\nwith or even larger than the helical period, it is much easier\nto develop these collective modes along the disk normal di-\nrection due to the nature of spin waves and helimagnets. The\nthickness in this model is not comparable to the helical period\nof the material to get visible effect of the collective modes\nalong the normal direction. Collective modes along the nor-\nmal direction of the Bloch-type hopfion in a disk have been in-\nvestigated [35, 36]. Therefore, only the resonances excited by\nin-plane microwave magnetic field are discussed here. Simu-\nlations are performed in steps of 1 mT from \u000020mT to 20 mT\nand the spatially average fluctuation in xcomponents of the\nmagnetization are considered, \u000emx(t) =hmx;0i\u0000hmx(t)i,\nwheremx;0represents the xcomponents of the magnetization\nof the equilibrium state. The power spectrum density is com-\nputed from the Fourier transformation of this fluctuation. In\nFig. 4(a), we show calculated spectrum as a function of ex-\nternal perpendicular field Hz. Three spin textures are all reso-\nnant at around 5.4 GHz. For enlarged hopfion, a new peak oc-\ncurs when the external magnetic field is larger than \u000010mT,\nbecause the spin texture has slightly deformation. Over the\ncritical transition field, that is 20 mT, the resonant frequencyfor toron increases slightly as Hzincreases.\nTo further clarify the collective modes of each spin texture\nin details, the dynamics of the relaxed states is resolved with\napplying a stationary oscillating magnetic field, which is de-\nscribed byh(t) = (hxsin(!Rt);0;0)withhx= 0:5mT. The\nresonance frequency !Ris fixed at 5:4GHz for the common\nresonance frequency of three spin textures, and the period T\nis2\u0019=! R. Figure 4(b) shows the resulting profiles of fluctua-\ntions in thexcomponents of the magnetization. Each profile\nis obtained as follows. Running with the uniform sinusoidal\nmagnetic field applied along the xaxis over 20 periods Tof\nthe excitation, the state is saved at time intervals of T=8over\nthe last ten periods of the simulations. Thus eight snapshots\nof the state for each instant, ti= 0, 1/8T, 1/4T,\u0001\u0001\u0001, 7/8T,\nrelative to the phase of the field excitation are obtained. The\nfinal profile at each tiis then obtained by averaging over the\nten snapshots, which allows artifacts to be averaged out over\na single period of excitation [61]. This method is called stro-\nboscopic method.\nFor enlarged hopfions as shown in Fig. 4(b), the relaxed\nstate is obtained when a magnetic field of \u000020mT is applied.\nThe fluctuation of the xcomponents of magnetization shows\nexpanding trend, which is similar with breathing mode of\nskyrmions. The more snapshots for detailed modes are shown\nin Ref. [62]. The strong PMA layers confine the motion of\nthe magnetization of the enlarged hopfion, and therefore there\nis no vertical mode. A Néel-type hopfion exists stably when\nmagnetic field equals zero, and the resonant modes are col-\nlective modes. It shows the rotation of the outer torus and\nthe expansion of the core of the Néel-type hopfion. At the\nsame time, the spins of the torus of the hopfion flip reversely\nwith the in-plane microwave magnetic field [62]. With 20 mT\nmagnetic field applied, the situation is different for the Néel-\ntype toron. It shows clearly the rotation mode with excita-\ntion, because the cross section structure is a typical Néel-type5\nskyrmion [62]. Relevant studies reported that the in-plane mi-\ncrowave magnetic field excites the rotation mode and the out-\nof-plane microwave magnetic field excites the breathing mode\nof skyrmions [63]. Unlike the resonant modes of the Néel-\ntype hopfion, the Bloch-type hopfion show multiple resonant\npeaks [35, 36]. And the modes of Bloch-type hopfion show\nhybridized modes with breathing and rotating characters.\nIV . CONCLUSION\nIn conclusion, we have studied numerically the Néel-type\nhopfion and the Néel-type toron in a film, and investigated\ntheir dynamic response to static magnetic field and microwave\nmagnetic field. This work shows the topological conversion\nbetween a Néel-type hopfion and a Néel-type toron with an\nactive energy, which results from the reversal of spins and the\nreconnection of a torus. Furthermore, dynamic simulations\nare employed to resolve individual resonant modes of the en-\nlarged hopfion, the Néel-type hopfion and the Néel-type toron,\nand the power spectrum is calculated. The collective modes\nin 3D spin textures are hybridized modes due to 3D struc-\ntures including breathing mode and rotation mode. Compared\nto the resonant modes in 2D skyrmions, breathing and rota-\ntion modes are not sufficient to characterize the modes in 3D\ntopological solitons. Collective modes of enlarged hopfions,\nNéel-type hopfion and Néel-type toron show distinct behav-iors. We find a new type of 3D topological soliton called\nNéel-type toron. In addition, our results provide a method to\nrealize the conversion between Néel-type hopfion and Néel-\ntype toron, and it also gives evidence of clarifying the Néel-\ntype hopfion and Bloch-type hopfion based on their different\nexcitation modes.\nACKNOWLEDGMENTS\nThis study is supported by Guangdong Spe-\ncial Support Project (Grant No. 2019BT02X030),\nShenzhen Fundamental Research Fund (Grant No.\nJCYJ20210324120213037), Shenzhen Peacock Group\nPlan (Grant No. KQTD20180413181702403), Pearl River\nRecruitment Program of Talents (Grant No. 2017GC010293)\nand National Natural Science Foundation of China (Grant\nNos. 11974298, 61961136006). J.X. acknowledges the\nsupport by the National Natural Science Foundation of China\n(Grant No. 12104327). X.Z. was an International Research\nFellow of the Japan Society for the Promotion of Science\n(JSPS). X.Z. was supported by JSPS KAKENHI (Grant No.\nJP20F20363). 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B 90, 064410 (2014).\n[62] See Supplementary Material for the animations of the details of\nexcitation modes of the enlarged hopfion, the Néel hopfion, and\nthe Néel toron, respectively.\n[63] M. Mochizuki, Phys. Rev. Lett. 108, 017601 (2012)."    },    {        "title": "2001.09774v1.Non_exponential_magnetic_relaxation_in_magnetic_nanoparticles_for_hyperthermia.pdf",        "content": "arXiv:2001.09774v1  [physics.app-ph]  27 Jan 2020Non-exponential magnetic relaxation in magnetic nanopart icles for hyperthermia\nI. Gresits,1, 2Gy. Thur´ oczy,3O. S´ agi,2S. Kollarics,2G. Cs˝ osz,2B. G.\nM´ arkus,2N. M. Nemes,4, 5M. Garc´ ıa Hern´ andez,4,5and F. Simon2, 6\n1Department of Non-Ionizing Radiation, National Public Hea lth Center, Budapest, Hungary\n2Department of Physics, Budapest University of Technology a nd Economics and MTA-BME\nLend¨ ulet Spintronics Research Group (PROSPIN), Po. Box 91 , H-1521 Budapest, Hungary\n3Department of Non-Ionizing Radiation, National Public Hea lth Institute, Budapest, Hungary\n4GFMC, Unidad Asociada ICMM-CSIC ”Laboratorio de Heteroest ructuras con Aplicaci´ on en Espintronica”,\nDepartamento de Fisica de Materiales Universidad Complute nse de Madrid, 28040\n5Instituto de Ciencia de Materiales de Madrid, 28049 Madrid, Spain\n6Laboratory of Physics of Complex Matter, ´Ecole Polytechnique F´ ed´ erale de Lausanne, Lausanne CH-1 015, Switzerland\nMagnetic nanoparticle based hyperthermia emerged as a pote ntial tool for treating malignant tumours. The\nefficiency of the method relies on the knowledge of magnetic p roperties of the samples; in particular, knowledge\nof the frequency dependent complex magnetic susceptibilit y is vital to optimize the irradiation conditions and to\nprovide feedback for material science developments. We stu dy the frequency-dependent magnetic susceptibility\nof an aqueous ferrite suspension for the first time using non- resonant and resonant radiofrequency reflectometry.\nWe identify the optimal measurement conditions using a stan dard solenoid coil, which is capable of providing\nthe complex magnetic susceptibility up to 150 MHz. The resul t matches those obtained from a radiofrequency\nresonator for a few discrete frequencies. The agreement bet ween the two different methods validates our ap-\nproach. Surprisingly, the dynamic magnetic susceptibilit y cannot be explained by an exponential magnetic\nrelaxation behavior even when we consider a particle size-d ependent distribution of the relaxation parameter.\nPACS numbers:\nIntroduction\nNanomagnetic hyperthermia, NMH,1–13is intensively stud-\nied due to its potential in tumor treatment. The prospective\nmethod involves the delivery of ferrite nanoparticles to th e ma-\nlignant tissue and a localized heating by an external radiof re-\nquency (RF) magnetic field affects the surrounding tissue on ly.\nThe key medical factors in the success of NMH4–6,14include\nthe affinity of tumour tissue to heating and the specificity of\nthe targeted delivery.\nConcerning the physics and material science challenges, i)\nthe efficiency of the heat delivery, ii) its accurate control and\niii) its precise characterization are the most important on es.\nConcerning the latter, various solutions exists which incl udes\nmodeling the exciting RF magnetic field with some knowledge\nabout the magnetic properties of the ferrite15–19, measure-\nment of the delivered heat from calorimetry15,17,20–22, or deter-\nmining the dissipated power by monitoring the quality facto r\nchange of a resonator in which the tissue is embedded23,24.\nAll three challenges are related to the accurate knowledge\nof the frequency-dependent complex magnetic susceptibili ty,\n/tildewideχ=χ′−iχ′′, of the nanomagnetic ferrite material. The dis-\nsipated power per unit volume, Pis proportional to the value\nofχ′′at the working frequency, ω, as:P= 0.5µ0ωχ′′H2\nAC,\nwhereµ0is the vacuum permeability, HACis the AC mag-\nnetic field strength. Although measurement of /tildewideχ(ω)is a well\nadvanced field due to e.g. the extensive filter or transformer\napplications10,25–34, we are not aware of any such attempts for\nnanomagnetic particles which are candidates for hyperther -\nmia.\nKnowledge of /tildewideχ(ω)would allow to determine the optimal\nworking frequency, which is crucial to avoid interference d ue\nto undesired heating of nearby tissue e.g. by eddy currents1,3,9.\nIn addition, an accurate characterization of /tildewideχ(ω)can provide\nan important feedback to material science to improve the fer -\nrite properties. Last but not least, measurement of /tildewideχ(ω)wouldallow for a better theoretical description of the high frequ ency\nmagnetic behavior of ferrites. Most reports suggest1,3,9,35that\na single relaxation time, τ, governs the frequency dependence\nof/tildewideχ(ω). The magnetic relaxation time, τ, is given by to\nthe Brown and N´ eel processes; these two processes describe\nthe magnetic relaxation due to the motion of the nanomag-\nnetic particle and the magnetization of the nanoparticle it -\nself (while the particle is stationary). When the two pro-\ncesses are uncorrelated, the magnetic relaxation time is gi ven\nas1/τ= 1/τB+1/τN, whereτBandτNare the respective re-\nlaxation times. These two relaxation types have very differ ent\nparticle size and temperature dependence, which would allo w\nfor a control of the dissipation. Nevertheless, the major op en\nquestions remain, i) whether the single exponential descri p-\ntion is valid, and ii) what the accurate frequency dependenc e\nof the magnetic susceptibility is.\nMotivated by these open questions, we study the frequency\ndependence of /tildewideχon a commercial ferrite suspension up to\n150 MHz. We used two types of methods: a broadband non-\nresonant one with a single solenoid combined with a network\nanalyzer and a radiofrequency resonator based approach. Th e\nlatter method yields the ratio of χ′′andχ′for a few discrete\nfrequencies. The two methods give a good agreement for\nthe frequency-dependent ratio of χ′′/χ′which validates both\nmeasurement techniques. We find that the data cannot be ex-\nplained by assuming that each magnetic nanoparticle follow s\na magnetic relaxation with a single exponent even when the\nparticle size distribution is taken into account. Our work n ot\nonly presents a viable set of methods for the characterizati on\nof/tildewideχbut it provides input to the theories aimed at describing\nthe magnetic relaxation in nanomagnetic particles and also a\nfeedback for future material science developments.2\nI. THEORETICAL BACKGROUND AND METHODS\nThe physically relevant quantity in hyperthermia is the\nimaginary part of the complex magnetic susceptibility, /tildewideχ, i.e.\nχ′′as the absorbed power is proportional to it. Although,\nwe recently developed a method to directly determine the ab-\nsorbed power during hyperthermia23, a method is desired to\ndetermine the full frequency dependence of /tildewideχ. This would\nnot only lead to finding the optimal irradiation frequency du r-\ning hyperthermia but it could also provide an important feed -\nback to materials development and for the understanding of\nthe physical phenomena behind the complex susceptibility i n\nferrite suspensions.\nThe generic form of the complex magnetic susceptibility of\na material reads:\n/tildewideχ(ω) =χ′(ω)−iχ′′(ω). (1)\nLinear response theory dictates that these can be transform ed\nto one another by a Hilbert transform36,37as:\nχ′(ω) =1\nπP/integraldisplay∞\n−∞χ′′(ω′)\nω′−ωdω′, (2)\nχ′′(ω) =−1\nπP/integraldisplay∞\n−∞χ′(ω′)\nω′−ωdω′, (3)\nwherePdenotes the principal value integral.\nWe note that we use a dimensionless volume susceptibility\n(invoking SI units) throughout. If a single relaxation proc ess is\npresent (similar to dielectric relaxation or to the Drude mo del\nof conduction, which yield /tildewideǫ(ω)and/tildewideσ(ω), respectively), the\ncomplex magnetic susceptibility takes the form:\nχ′(ω) =χ01\n1+ω2τ2, (4)\nχ′′(ω) =χ0ωτ\n1+ω2τ2, (5)\nwhereχ0is the static susceptibility.\nThe corresponding χ′andχ′′pairs can be constructed\nwhen multiple relaxation times are present in the descrip-\ntion of their frequency dependence. There is a general\nconsensus1,4,8,14,17,19,38–42although experiments are yet lack-\ning, that the single relaxation time description approxima tes\nwell the frequency dependence of the magnetic nanoparticle s.\nThe frequency dependence of χ′′is though to be described by\nthe relaxation time of the nanoparticles: 1/τ= 1/τN+1/τB,\nwhere the N´ eel and Brown relaxation times are related to the\nmotion of the magnetization with respect to the particles an d\nthe motion of the particle itself, respectively.\nWe used a commercial sample (Ferrotec EMG 705, nominal\ndiameter 10 nm) which contains aqueous suspensions of sin-\ngle domain magnetite (Fe 3O4) nanoparticles. We verified the\nmagnetic properties of the sample using static SQUID mag-\nnetometry; it showed the absence of a sizable magnetic hys-\nteresis (data shown in the Supplementary Information), whi ch\nproves that the material indeed contains magnetic mono-\ndomains.A. Measurements with non-resonant circuit\nAt frequencies below ∼5−10MHz the conventional\nmethods of measuring the current-voltage characteristics can\nbe used for which several commercial solutions exist. This\nmethod could e.g. yield the inductivity change for an induct or\nin which a ferrite sample is placed. However, above these fre -\nquencies the typical circuit size starts to become comparab le\nto the electromagnetic radiation wavelength thus wave effe cts\ncannot be neglected. The arising complications can be con-\nveniently handled with measurement of the Sparameters, i.e.\nthe reflection or transmission for the device under test.\nObtaining /tildewideχ(ω)is possible by perturbing the circuit prop-\nerties of some broadband antennas or waveguides while mon-\nitoring the corresponding Sparameters43(the reflected am-\nplitude,S11, and the transmitted one, S21) with a vector net-\nwork analyzer (VNA). We used two approaches: i) a droplet\nof the ferrite suspension on a coplanar waveguide (CPW) was\nmeasured and ii) about a 100 µl suspension was placed in a\nsolenoid. It is crucial in both cases to properly obtain the n ull\nmeasurement, i.e. to obtain the perturbation of the circuit due\nto the ferrite only. For the solenoid, we found that a sam-\nple holder filled with water gives no perturbation to the cir-\ncuit parameters as expected. In contrast, the CPW parameter s\nare strongly influenced by a droplet of distilled water whose\nquantity can be hardly controlled therefore performing the null\nmeasurement was impossible and as a result, the use of the\nCPW turned out to be impractical. Additional details about\nthe VNA measurements, including details of the failure with\nthe CPW based approach, are provided in the Supplementary\nInformation.\nIn the second approach, we used a conventional solenoid\n(shown in Fig. 2) made from 1 mm thick enameled copper\nwire, its inner diameter is 6 mm and it has a length of 23 mm\nwith 23 turns. The coil is soldered onto a semi-rigid copper R F\ncable that has a male SMA connector. Fig. 2. shows the equiv-\nalent circuit which was found to well explain the reflection c o-\nefficient in the DC-150 MHz frequency range (more precisely\nfrom 100 kHz which is the lowest limit of our VNA model\nRohde & Schwarz ZNB-20). The frequency dependence of\nthe wire re ce due to the skin-effect was also taken into ac-\ncount in the analysis. The parallel capacitor arises from th e\nparasitic self capacitance of the inductor and from the smal l\ncoaxial cable section. Further details about the validatio n of\nthe equivalent circuit (i.e. our fitting procedure) are prov ided\nin the Supplementary Information.\nThe frequency dependent complex reflection coefficient, Γ\n(same asS11this case), and Zof the studied circuit are related\nby44:\nΓ =Z−Z0\nZ+Z0, (6)\nwhereZ0is the 50 Ωwave impedance of the cables and Z\nis the complex, frequency dependent impedance of the non-\nresonant circuit. It can be inverted to yield Zas:Z=Z01+Γ\n1−Γ.\nThe admittance for the empty solenoid reads:\n1\nZempty=1\nR(ω)+iωL+iωC, (7)\nThe analysis yields fixed parameters for R(ω)andC, whereas\nthe effect of the sample is a perturbation of the inductivity :3\nVNA \nC\nCR\nL\nFIG. 1: Upper panel: photograph of the solenoid used in the no n-\nresonant susceptibility measurements. Lower panel: the eq uivalent\ncircuit model including a parasitic capacitor, Cdue to the small coax-\nial cable section and the self capacitance of the inductor. Rhas a\nfrequency dependence due to the skin-effect.\nFIG. 2: Upper panel: photograph of the solenoid used in the no n-\nresonant susceptibility measurements. Lower panel: the eq uivalent\ncircuit model including a parasitic capacitor, Cdue to the small coax-\nial cable section and the self capacitance of the inductor. Rhas a\nfrequency dependence due to the skin-effect.\nL→L(1+η/tildewideχ(ω)). We introduced the dimensionless filling\nfactor parameter, η, which is proportional to the volume of the\nsample per the volume of the solenoid, albeit does not equal t o\nthis exactly due to the presence of stray magnetic fields near\nthe ends of the solenoid. This parameter, η, also describes\nthat the susceptibility can only be determined up to a linear\nscaling constant with this type of measurement. In principl e,\nthe absolute value of /tildewideχ(ω)could be determined by calibrating\nthe result by a static susceptibility measurement (e.g. wit h a\nSQUID magnetometer) and by extrapolating the dynamic sus-\nceptibility to DC. It is however not possible with our presen t\nsetup asχshows a strong frequency dependence down to our\nlowest measurement frequency of 100 kHz.\nA straightforward calculation using Eq. (7) yields that\nη/tildewideχ(ω)can be obtained from the measurement of the admit-\ntance in the presence of the sample, 1/Zsample as:\nη/tildewideχ=/parenleftBig\n1\nZsample−iωC/parenrightBig−1\n−/parenleftBig\n1\nZempty−iωC/parenrightBig−1\niωL(8)\nB. Measurements with resonant circuit\nFig. 3 shows the block diagram of the resonant circuit mea-\nsurements which is the same as in the previous studies23,24.\nThis type of measurement is based on detecting the changesSource\nLR\nCT50 ΩHybrid\njunctionDetectorCM\nResonator\nFIG. 3: Left: Block diagram of the resonant measurement meth od.\nRight: The schematics of the resonator circuit. It has 2 vari able ca-\npacitors, the tuning ( CT) is used for setting the resonant frequency\nand the matching ( CM) is for setting the impedance of the resonator\nto 50Ωat resonant frequency. Detailed description is in Ref. 23\nin the resonator parameters, resonance frequency ω0and qual-\nity factor, Q. The presence of a magnetic material induces a\nchange in these parameters43–45as:\n∆ω0\nω0+i∆/parenleftbigg1\n2Q/parenrightbigg\n=−η/tildewideχ (9)\nHerein,∆ω0and∆/parenleftBig\n1\n2Q/parenrightBig\nare changes in resonator eigenfre-\nquency and the quality factor. The signs in Eq. (9) express\nthat in the presence of a paramagnetic material, the resonan ce\nfrequency downshifts (i.e. ∆ω0<0) and that it broadens (i.e.\n∆/parenleftBig\n1\n2Q/parenrightBig\n>0) when both χ′andχ′′are positive.\nThe resonator measurement has a high sensitivity to minute\namounts of samples43however its disadvantage is that its re-\nsult is limited to the resonance frequency only. Eq. (9) is\nremarkable, as it shows that the ratio ofχ′′andχ′can be\ndirectly determined at a given ω0(we use throughout the ap-\nproximation that Qis larger than 10, thus any change to ω0\ncan be considered to the first order only). Namely:\nχ′′\nχ′=−ω0∆/parenleftBig\n1\n2Q/parenrightBig\n∆ω0=−∆HWHM\n∆ω0(10)\nwhere we used that the half width at half maximum, HWHM\nis: HWHM =ω0/2Q. The broadening of the resonator profile\nmeans that ∆HWHM is positive.\nThis expression provides additional microscopic informa-\ntion when the magnetic susceptibility can be described by a\nsingle relaxation time:\n−∆HWHM\n∆ω0=χ′′\nχ′=ω0τ (11)\nE.g. when a measurement at 50 MHz returns −∆HWHM\n∆ω0= 1,\nwe then obtain directly a relaxation time of τ= 3ns. The right\nhand side of Eq. (11) can be also rewritten as ω0τ=f0/fc,\nwhere we introduced a characteristic frequency of the parti cle\nabsorption process, i.e. where χ′′has its maximum.\nThis description has an interesting consequence: it makes\nlittle sense to use tiny nanoparticles, i.e. to push τto an exces-\nsively short value (or fcto a too high value). The net absorbed4\npower reads: P∝fχ′′and when the full expression is sub-\nstituted into it, we obtain P∝f2\nfc/parenleftbigg\n1+f2\nf2c/parenrightbigg. This function is\nroughly linear with fbelowfcand saturates above it to a con-\nstant value. This means that an optimal irradiation frequen cy\nshould be at least as large as fc.\nII. RESULTS AND DISCUSSION\n/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54\n/s32/s82/s101/s97/s108\n/s32/s73/s109 /s104/s99 /s39/s40 /s119 /s41/s32/s44/s32 /s104/s99 /s39/s39/s40 /s119 /s41 /s71\n/s115/s97/s109/s112/s108/s101/s45/s71\n/s101/s109/s112/s116/s121\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s32/s104/s99 /s39/s40 /s119 /s41\n/s32/s104/s99 /s39/s39/s40 /s119 /s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s77/s72/s122/s41\nFIG. 4: The reflection coefficient for the solenoid with the sa mple\ninside, relative to the empty solenoid (upper panel). The re al and\nimaginary parts of the dynamic susceptibility as obtained f rom the\nreflection coefficients according to Eqs. (7) and (8). Note th at neither\ncomponent of /tildewideχfollows the expected Lorentzian forms.\nWe measured the reflection coefficient for the non-resonant\ncircuit,Γempty, i.e. for an empty solenoid in the 100 kHz-\n150 MHz range. The lower frequency limit value is set by\nvector network analyzer and values higher than 150 MHz are\nthought to be impractical due to water dielectric losses and\neddy current related losses in a physiological environment1–3.\nWe also measured the corresponding reflection coefficients\nwhen the ferrite suspension sample, Γsample and only distilled\nwater was inserted into it. The presence of the water refer-\nence does not give an appreciable change to Γ(data shown\nin the Supplementary Information) as expected. The differ-\nenceΓsample−Γempty is already sizeable and is shown in Fig.\n4. The dynamic susceptibility is obtained by first determini ng\nthe empty circuit parameters (details are given in the SM) an d\nthese fixed R,LandCare used together with Eqs. (7) and (8)\nto calculate /tildewideχ(ω). The result is shown in the lower panel of\nFig. 4.\nWe note that the use of Eq. (8) eliminates Rand we have\nalso checked that the result is little sensitive to about 10 %change in the value of LandC, therefore the result is robust\nand it does not depend much on the details of the measurement\ncircuit parameters. The ratio of the two components is parti c-\nularly insensitive to the parameters: Lcancels out formally\naccording to Eq. (8) but we also verified that a 20 % change\ninCleaves the ratio unaffected.\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s77/s72/s122/s41/s99 /s39/s39/s47 /s99 /s39/s32/s110/s111/s110/s45/s114/s101/s115/s111/s110/s97/s110/s116\n/s32/s114/s101/s115/s111/s110/s97/s110/s116\nFIG. 5: The comparison of the ratios of χusing the non-resonant\nbroadband and the resonator based (at some discrete frequen cies) ap-\nproaches.\nFig. 5 shows the ratio of the two terms of the dynamics\nmagnetic susceptibility, χ′′(ω)/χ′(ω), as determined by the\nbroadband method and using the resonator based approach.\nThe latter data is presented for a few discrete frequencies.\nThe two kinds of data are surprisingly close to each other,\ngiven the quite different methods as these were obtained. Th is\nin fact validates both approaches and is a strong proof that\nwe are indeed capable of determining the complex magnetic\nsusceptibility of the ferrofluid sample up to a high frequenc y.\nOne expects that the signal to noise performance of the res-\nonator based approach is superior to that obtained with the\nnon-resonant method by the quality factor of the resonator46,\nwhich is about 100. In fact, the data shows just the opposite o f\nthat and the resonator based data point show a larger scatter ing\nthan the broadband approach. This indicates that the accura cy\nof the resonator method is limited by a systematic error, whi ch\nis most probably related to the inevitable retuning of the re s-\nonator and the reproducibility of the sample placement into\nthe resonator.\nThe experimentally observed dynamic susceptibility has\nimportant consequences for the practical application of hy per-\nthermia. Given that the net absorbed power: P∝fχ′′, it\nsuggests that a reasonably high frequency, f, should be used\nfor the irradiation, until other types of absorption, e.g. d ue to\neddy currents1–3, limit the operation.\nWe finally argue that the experimental observation cannot\nbe explained by an exponential magnetic relaxation either d ue\nto the rotation of magnetization (the Ne´ el relaxation) or d ue5\nto the rotation of the particle itself (the Brown relaxation ). In\nprinciple, both relaxation processes are particle size dep en-\ndent; in the nanometer particle size domain the Brown proces s\nprevails and it was calculated in Ref. 3 that for a particle di -\nameter of d= 10 nm we get τ= 300 ns (fc= 21 MHz) for\nd= 11 nm,τ= 2µs (fc= 3 MHz), and for d= 9 nm,\nτ= 50 ns (fc= 125 MHz). These frequencies would in prin-\nciple explain a significant χ′′(ω)in the1−100MHz range,\nsuch as we observe.\nHowever, a simple consideration reveals from Eqs. (4) and\n(5) that for a single exponential magnetic relaxation for ea ch\nmagnetic nanoparticle, the ratio of χ′′/χ′is astraight line as\na function of the frequency, which starts from the origin wit h\na slope depending on the distribution of the different τparam-\neters and particle sizes. Similarly, a single exponential r elax-\nation would always give a monotonously decreasing χ′(ω),\nirrespective of the particle size and τdistribution. Clearly,\nour experimental result contradicts both expectations: χ′′/χ′\nis not a straight line intersecting the origin and χ′(ω)signif-\nicantly increases rather than decreases above 20 MHz. We\ndo not have a consistent explanation for this unexpected, no n-\nexponential magnetic relaxation, which should motivate fu r-\nther experimental and theoretical efforts on ferrofluids. W e\ncan only speculate that a subtle interplay between the Ne´ el\nand Brown processes could cause this effect, whose explana-\ntion would eventually require the full solution of the equat ion\nof motion of the magnetic moment and the nanoparticles, such\nas it was attempted in Ref. 35.\nSummary\nIn summary, we studied the frequency-dependent dynamic\nmagnetic susceptibility of a commercially available ferro fluid.\nKnowledge of this quantity is important for i) determining\nthe optimal irradiation frequency in hyperthermia, ii) pro vid-\ning feedback for the material synthesis. We compare the re-\nsult of two fully independent approaches, one which is based\non measuring the broadband radiofrequency reflection from\na solenoid and the other, which is based on using radiofre-\nquency resonators. The two approaches give remarkably sim-\nilar results for the ratio of the imaginary and real parts of t he\nsusceptibility, which validates the approach. We observe a sur-\nprisingly non-exponential magnetic relaxation for the ens em-\nble of nanoparticles, which cannot be explained by the distr i-\nbution of the magnetic relaxation time in the nanoparticles .\nAcknowledgements\nThe authors are grateful to G. F¨ ul¨ op, P. Makk, and Sz.\nCsonka for the possibility of the VNA measurements and for\nthe technical assistance. Jose L. Martinez is gratefully ac -\nknowledged for the contribution to the SQUID measurements.\nSupport by the National Research, Development and Innova-\ntion Office of Hungary (NKFIH) Grant Nrs. K119442, 2017-\n1.2.1-NKP-2017-00001, and VKSZ-14-1-2015-0151 and by\nthe BME Nanonotechnology FIKP grant of EMMI (BME\nFIKP-NAT) are acknowledged. The authors also acknowledge\nthe COST CA 17115 MyWA VE action.6\n1Q. Pankhurst, J. Connolly, S. Jones, and J. Dobson, Journal o f\nPhysics D-Applied Physics 36, R167 (2003).\n2M. Kallumadil, M. Tada, T. Nakagawa, M. Abe, P. Southern, and\nQ. A. 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Kostylev, Physica E Low-Dimensional\nSystems and Nanostructures 69, 253 (2015).7\nAppendix A: Magnetic properties of the sample\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s109/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32/s40/s101/s109/s117/s41\n/s45/s48/s46/s48/s53/s48 /s45/s48/s46/s48/s50/s53 /s48/s46/s48/s48/s48 /s48/s46/s48/s50/s53 /s48/s46/s48/s53/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s109/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32/s40/s101/s109/s117/s41\n/s109\n/s48/s72 /s32/s40/s84/s41/s72/s121 /s115/s116/s101/s114/s101/s115/s105/s115/s61/s32/s50/s32/s109/s84\nFIG. 6: The magnetic moment of the sample, m, versus the mag-\nnetic field strength, µ0H, curve for the ferrite particle suspension.\nThe absence of a sizeable magnetic hysteresis indicates tha t this is a\nmonodomain sample. The estimate for the maximum hysteresis value\nis about 2 mT.\nThe magnetic moment versus the magnetic field strength,\nµ0H, is shown in Fig. 6 as measured with a SQUID magne-\ntometer. Notably, the sample magnetism shows a saturation\nabove 0.1 T, however it has a very small hysteresis of about\n2 mT. Common hard, multidomain ferromagnetic materials,\nwhich saturate is small magnetic fields, usually display a si g-\nnificant hysteresis. Our observation agrees with the expect ed\nbehavior of the sample, i.e. that it consists of mono-domain\nnanoparticle, which can easily align with the external mag-\nnetic field.\nAppendix B: Details of the non-resonant susceptibility\nmeasurement\nWe discuss herein how the solenoid based broadband sus-\nceptibility measurement can be performed. We first prove tha t\nthe equivalent circuit, presented in the main text, provide s an\naccurate description. The reflectivity data is shown in Fig. 7.\nWe obtain a perfect fit (i.e. the measured and fitted curves\noverlap) when we consider the equivalent circuit in the main/s45/s49/s48/s49/s82/s101/s40 /s71 /s41\n/s32/s77/s101/s97/s115/s117/s114/s101/s100\n/s32/s77/s111/s100/s101/s108/s105/s110/s103/s44/s32 /s82 /s40/s119 /s41/s44/s32 /s76 /s44/s32 /s67\n/s32/s77/s111/s100/s101/s108/s105/s110/s103/s44/s32 /s82 /s40/s119 /s41/s44/s32 /s76\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s45/s49/s48/s49\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s77/s72/s122/s41/s73/s109/s32/s40 /s71 /s41\nFIG. 7: The reflection coefficient, Γ, and its modelling with vari-\nous equivalent circuit assumptions. The best fit is obtained when the\nequivalent circuit containing a frequency dependent wire r esistance\n(with a small DC value) was considered in addition to an induc tor\nand a capacitor. The absence of the capacitor does not give an appro-\npriate fit (dotted red curve). We note that a constant wire res istance,\nhowever with an unphysically large value, gives also an appr opriate\nfit.\ntext with parameters RDC= 15.5(2) mΩ,L= 0.62(1)µH,\nandC= 4.65(1) pF. This fit also considered the frequency\ndependency of the coil resistance due to the skin effect, who se\nDC value is RDC= 13 mΩ. We also performed the fit without\nconsidering the skin effect, which gave an unrealistically large\nRDC= 150 mΩwhile the fit being seemingly proper. A fit\nwithout considering a capacitor does not give a proper fit (do t-\nted curve in the figure): its major limitation is that it canno t\nreproduce the zero crossing of Γ, i.e. a resonant behavior in\nthe impedance of the circuit. As a result, we conclude that th e\nequivalent circuit in the main text provides a proper descri p-\ntion of the measurement circuit and that the fitted parameter s\ncan be used to obtain the complex susceptibility of the sampl e,\nas we described in the main text.\nFig. 8. demonstrates that the presence of the sample gives\nrise to a significant change in the reflection coefficient, Γ,\nwhereas the reflection is only slightly affected by the prese nce\nof the water (maximum Γchange is about 0.2 % below 150\nMHz) and its effect is limited to frequencies above 150 MHz.\nProbably, the inevitably present stray electric fields (due to the\nparasitic capacitance of the solenoid) interact with the wa ter\ndielectric, which results in this effect. The stray electri c fields\nand the parasitic capacitance become significant at higher f re-\nquencies: then there is a significant voltage drop across the\nsolenoid inductor coil, thus its windings are no longer equi -\npotential and an electric field emerges.\nAppendix C: Details of the susceptibility detection using a CPW\nThe coplanar waveguide or CPW is a planar RF and mi-\ncrowave transmission line whose impedance is 50 Ωat a wide\nfrequency range43,44. The CPW can be thought of as a halved\ncoaxial cable which makes the otherwise buried electric and8\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s71\n/s119/s97/s116/s101/s114/s47/s115/s97/s109/s112/s108/s101/s45/s71\n/s101/s109/s112/s116/s121\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s77/s72/s122/s41/s71\n/s115/s97/s109/s112/s108/s101/s45/s71\n/s101/s109/s112/s116/s121 \n/s32/s82/s101/s97/s108\n/s32/s73/s109\n/s71\n/s119/s97/s116/s101/s114/s45/s71\n/s101/s109/s112/s116/s121 \n/s32/s82/s101/s97/s108\n/s32/s73/s109\nFIG. 8: Reflection coefficients, Γ, with respect to the empty circuit.\nΓwater denotes the reflection coefficient when the solenoid is filled\nwith water in a quartz tube. Note that the sample gives rise to a signif-\nicant change in the reflection below 100 MHz, whereas the pres ence\nof water (dashed lines) only slightly changes it above this f requency.\nmagnetic fields available to study material parameters, ess en-\ntially as a small piece of an irradiating antenna43,44. However\nas we show below, the inevitable simultaneous presence of th e\nelectric and magnetic field hinders a meaningful analysis.\nS11 S21R* RS R* L* LS L* \nC* CS C* G* GS G* CPW section CPW section Sample drop\nPort 2 Port 1 \nFIG. 9: Upper panel: The equivalent circuit of a CPW section w ith\na sample on the top. LS,RS,CS, andGSare effective inductance,\nseries resistance, capacitance, and shunt conductance of t he small\nCPW section which contains the sample, respectively. L∗,R∗,C∗,\nandG∗are corresponding distributed circuit parameters which ar e\nnormalized to unit length. In an ideal waveguide R∗= 0 andG∗=\n0, and also Z0=/radicalbig\nL∗/C∗. Lower panel: Photo of CPW with a\ndroplet of the sample. Port 1 is labeled with green and Port 2 i s with\nred tape.\nWe show the U shaped CPW section use in our experiments\nalong with the equivalent circuit of the CPW in Fig. 9. As\nthis device has two ports, one can measure the frequency de-\npendent complex reflection ( S11) and transmission ( S21) co-\nefficients simultaneously with a VNA. We placed the sample\ndroplet on the top of the gap section of the CPW between the\ncentral conductor and the grounding side plate, where the RF\nmagnetic field component is the strongest. The presence ofthe sample influences all parameters for the waveguide sec-\ntion where it is placed: the inductance Ls, capacitance Cs, the\nseries resistance Rs, and the shunt inductance Gs. All 4 pa-\nrameters are extensive, i.e. these depend on the quantity of the\nsample and one can express the inductivity as Ls=L0(1+η/tildewideχ)\nwhereL0is the inductivity of the CPW section which is af-\nfected by the sample, ηis the relevant filing factor that is di-\nmensionless and /tildewideχis the complex magnetic susceptibility.\nTheSparameters for such a device read47:\nS11,sample=Rs+iωL s+Z0\n1+Z0(Gs+iωC s)−Z0\nRs+iωL s+Z0\n1+Z0(Gs+iωC s)+Z0(C1)\nand\nS21,sample=2Z0\n1+Z0(Gs+iωC)\nRs+iωL s+Z0\n1+Z0(Gs+iωC s)+Z0(C2)\nWe calibrated the system that without sample (empty case)\nso that the VNA shows 0 for S11and real 1 to S21on the\nentire frequency range. During the calibration, port 1 of th e\nVNA was connected to the CPW and we assembled and disas-\nsembled the necessary calibrating elements (OPEN, SHORT,\nMATCH) onto port 2 and the second end of the CPW. There-\nfore the VNA reference plane was this end of the CPW. The\ncalibration could be achieved down to Γ<5·10−4(not\nshown).\nMagnetic field \nElectric fieldε0εr ceramic\nFIG. 10: Cross section of a coplanar waveguide showing the el ectric\nand magnetic field. Note that magnetic field is present around the\ncentral conductor and that there is a significant electric fie ld in the\ntwo gaps between the central conductor and the neighboring g round\nplates.\nWe first measured the reflection transmission coefficient\nchange under the influence of a small distilled water droplet\nwith approximately the same size as that of the sample. We\nobserve a Γchange up to about 5 % (maximum value at 150\nMHz, data not shown) for both coefficients, which is a size-\nable value. We note that the solenoid investigation, which w e\ndiscussed in the main paper, gave a change in Γfor the influ-\nence of water of about 0.2 %. Clearly, this larger sensitivit y of\nthe measurement for water is due to the electric field which is\nsignificant for the CPW and is much smaller for the solenoid.\nIt is even more intriguing that the effect of the sample is\nprimarily to shift the real parts of both S11andS21by the same\namount even at DC, while leaving the imaginary components\nunchanged (data not shown). For our typical droplet size, su ch9\nas this shown in Fig. 9, this amount is ∆Γ≈0.04. Rewriting\nEqs. (C1) and (C2) in the zero frequency limit, yields:\nS11,sample, DC=Rs+Z0\n1+Z0Gs−Z0\nRs+Z0\n1+Z0Gs+Z0, (C3)\nS21,sample, DC=2Z0\n1+Z0Gs\nR+Z0\n1+Z0Gs+Z0. (C4)\nWe find that in the reasonable limit of Rsample/lessorsimilarZ0, the\ninfluence of Gsdominates and that the experimental finding\nimplies the presence of a significant shunt conductance due t o\nthe sample. We speculate that this may be due to the presence\nof excess OH−ions in the ferrofluid (the Ferrotec EMG 705\nhas a pH of 8-9), which conduct the electric current. Again,\nthis effect is the result of the finite electric field across th e gap\nof the CPW, where we place the sample.\nThe two effects, the presence of a significant capacitance\ndue to water and a shunt inductance due to the conductivity\nof the ferrofluid, occur simultaneously when using a CPW for\nthe measurement. In fact, the effect of these factors domina te\nthe reflection/transmission. This means that determining t he\nmagnetic susceptibility for a case when a finite electric fiel d is\npresent, proves to be impractical.\nAppendix D: Additional details on the theory of resonators\nCoil+Sample Trimmer \ncapacitors \nFIG. 11: Photograph of the radiofrequency resonator. The sa mple,\nmeasurement coil and the two trimmer capacitors are indicat ed.\nFig. 11. shows a photograph of the radiofrequency res-\nonator circuit which was used in the studies. Note the presen ce\nof the two trimmer capacitors, which act as frequency tuning\nand impedance matching elements.\nThe following equation was used in the main text to deter-\nmine the relation between resonator parameters and the mate -\nrial properties:\n∆ω0\nω0+i∆/parenleftbigg1\n2Q/parenrightbigg\n=−η/tildewideχ (D1)We note that the - sign before the imaginary term on the left\nhand side varies depending on the definition of the sign in the\ncomplex response function /tildewideχ. We use the convention of Ref.\n43 where /tildewideχ=χ′−iχ′′which results in the + sign in Eq. (D1).\nThe factor 2 in Eq. (9) may seem disturbing but it is the di-\nrect consequence of the Qfactor definition: Q=FWHM/ω0,\nwhere FWHM is the full width at half maximum of the res-\nonance curve (in angular frequency units). Thus 1/2Q=\nHWHM/ω0, where HWHM is the half width at half maximum\nof the resonance curve. We also recognize that a Lorentzian\nshaped resonator profile can be expressed as1\n(ω−ω0)2+1/τ2,\nwhereτis the time constant of the resonator and τ= 2Q/ω0.\nThis also means that HWHM = 1/τ.\nThis allows to express the above equation in a more com-\npact way by introducing the complex angular frequency of the\nresonator:\n/tildewideω=ω0+iω0\n2Q=ω0+i1\nτ. (D2)\nIt is interesting to note that the complex Lorentzian linesh ape\nprofile is proportional to 1/i/tildewideω. It then follows from Eq. (D2)\nthat Eq. (D1) can be expressed as:\n∆/tildewideω\nω0=−η/tildewideχ (D3)\nwhere∆/tildewideωis the shift (or change) of (the complex) /tildewideω.\nFig. 12 shows the changes in the reflection curves at the\nresonant method.\nFrequency (MHz)60 60.5 61 61.5 62|Γ|2\n00.20.40.60.81\nEmpty\nSample\nFIG. 12: The reflection curves at the resonant method. The shi ft of\nthe resonance frequency due to the sample (red) is clearly vi sible."    },    {        "title": "1412.5396v1.Revealing_the_role_of_orbital_magnetism_in_ultrafast_laser_induced_demagnetization_in_multisublattice_metallic_magnets_by_terahertz_spectroscopy.pdf",        "content": "Revealing the role of orbital magnetism in ultrafast laser-ind uced demagnetization in \nmultisublattice metallic magnets by terahertz spectroscopy \n \nT. J. Huisman1, R. V. Mikhaylovskiy1, A. Tsukamoto2, Th. Rasing1 and A. V. Kimel1 \n1Radboud University Nijmegen, Institute for Molecules and Materials, 6525 AJ Nijmegen, The \nNetherlands \n2College of Science and Technology, Nihon University, 7- 24-1 Funabashi, Chiba, Japan \nSimultaneous detection of THz emission and transient magneto-optica l response is employed to \nstudy ultrafast laser-induced magnetization dynamics in multisub lattice magnets NdFeCo and \nGdFeCo amorphous alloys with in-plane magnetic anisotropy. A satisfactory quantitative agreement \nbetween the dynamics revealed with the help of these two tec hniques is obtained for GdFeCo. For \nNdFeCo the THz emission reveals faster dynamics than the mag neto-optical response. This indicates \nthat in addition to spin dynamics of Fe ultrafast laser excitati on triggers faster magnetization \ndynamics of Nd. \nSpin -orbit coupling is a key mechanism exploited by the fast expanding fi eld of spintronics, \nwhich ties charge and spin degrees of freedom together [1,2,3]. Besides spin or charg e manipulation, \none could try to manipulate the orbital degree of freedom. This leads to the intrigu ing question about \nthe role of orbital magnetization in magnetization dynamics. The seminal o bservation of \nsubpicosecond demagnetization in ferromagnetic nickel by a laser pulse pu blished almost two decades \nago [4] triggered the field of ultrafast magnetization dynamics - a topic that has been continuously \nfueled by intriguing observations as well as controversies in the scientific co mmunity [5- 11]. One of \nthe main reasons for the controversies is the lack of an artifact-free technique capable of observing \nthe magnetization dynamics at the subpicosecond timescale. \nMost experimental studies of ultrafast magnetism performed so far employ an all-optical \npump-probe technique in which the magnetization is probed indirectly via the magneto-optical \nFaraday or Kerr effect. However it was noticed that at the subpicosecond time- scale the magneto-\noptical probes are subjective to artifacts [ 12]. In particular, it was argued that if the temporal behavior \nof the Kerr ellipticity is different from the one of the Kerr rotation, the dynamics of the magneto-optical \nsignal cannot be directly associated with the true magnetization dynamics [ 13]. However, the opposite \nstatement is not obviously true and similar behavior of the ellipticity and the r otation cannot be used \nas a proof that the dynamics of the magneto-optical Kerr effect (MOKE) adequately reflects the  \nmagnetization dynamics. Additional complications in the interpretation of time-resolved magneto-\noptical experiments arise in multi-sublattice magnets, especially when magnetic sublattice s possess \nboth spin and orbital magnetization [ 14-19]. Alternatively, in order to deduce information about \nsubpicosecond magnetization dynamics one can employ the fact that such dynami cs is accompanied \nby the emission of terahertz (THz) electro-magnetic radiation [ 20-23].  \nTo reveal how orbital magnetization affects ultrafast laser induced demagnetizatio n \nmeasurements, we used simultaneous measurements of the MOKE and the electric field of the emitted \nTHz radiation and applied both probes of demagnetization to NdFeCo and GdFeCo alloys. The orbital \nmomenta of Fe and Gd in the GdFeCo alloys are expected to be quenched, while in Nd the \nmagnetization is dominantly determined by its orbital momentum. This makes these sampl es suitable \nfor studying the role of orbital momentum in ultrafast magnetization dynamics [ 15,19].  \nWe show that nonlinear optical effects of non-magnetic origin or the inverse spin -Hall effect \ncannot be responsible for the observed emission of the THz radiation, and thus th e emission originates \nfrom the ultrafast demagnetization. Assuming that the MOKE represents the dynamics of the net \nmagnetization, we deduce the spectrum of the THz emission which is generated by such magnetizatio n \ndynamics by solving the Maxwell equations for a thin magnetic film, takin g into account the \npropagation of the THz radiation from the emitter to the de tector.  Comparing these calculated spectra with the actually measured ones we reveal that there is a very good match between the spectra fo r a \npure Co film, which was used as a reference. While qualitative similarities between the spectra are \nobtained for GdFeCo alloys, in NdFeCo alloys there are clear discrepancies between the calculated and \nthe actually measured THz spectra. This finding shows that the actual laser-induced magnetization \ndynamics in NdFeCo alloys is faster and more complicated than time-resolved magneto-optical \nmeasurements might reveal, providing indication of the Nd orbital magnetiza tion role in ultrafast \ndemagnetization .  \nThe multisublattice magnetic materials studied in this work are rare-earth transition-metal  \namorphous alloys: NdFeCo and GdFeCo. The results presented here are for Nd 0.2(Fe 0.87Co0.13)0.8 and    \nGd 0.3(Fe 0.87Co0.13)0.7, measured at room temperature. We note that temperature dependent \nmeasurements in the range 150K to 300K as well as similar measurements on Nd 0.5(Fe 0.87Co0.13)0.5 and \nGd 0.18(Fe 0.87Co0.13)0.82 show similar results. The GdFeCo alloys are ferrimagnetic and the properties of \nthese alloys are essentially determined by the fact that the Gd spin moments (4 f and 5 d) are aligned \noppositely to the spin moments of Fe (3 d) and Co . Importantly, in Gd 0.18(Fe 0.87Co0.13)0.82 the FeCo \nmagnetization is larger than the one of Gd at all temperatures, while in Gd 0.3(Fe 0.87Co0.13)0.7 the net \nmagnetization is dominated by the Gd sublattice. In the NdFeCo alloy, the orbital momentum of N d is \nlarger than the spin of Nd and aligned antiparallel with respect to it. Hence, despite the \nantiferromagnetic coupling of the Fe and Nd spins, the alloy is effectively ferromagnetic. The NdFeCo \nand GdFeCo alloys were incorporated into a layered structure with different layers to prevent oxidation \n(capping layer SiN), reduce laser-induced heating (heat sink AlTi) and facilitate the growth (buffer layer \nSiN). The layered structure is shown in Fig. 1 (a). As a reference we used a 12 nm thick pure Co film \ndeposited on a 0.5 mm thick glass substrate. All the magnetic films have in-plane mag netic anisotropy \nwith coercive fields below 150 Gauss at room temperature .  \nOur experimental approach uniquely combines THz time-domain spectroscopy (THz -TDS) with \nan optical pump-probe sche me, as sketched in Fig. 1 (a). An amplified titanium-sapphire lase r system \nis used, producing light pulses with a duration of 50 fs at a repetitio n rate of 1 kHz and with a center \nwavelength of 800 nm. The laser beam is divided in three parts: pump, probe and  gate . The pump \nfluence was approximately 1 mJ/cm2. The pump was focused onto an area on the sample with a \ndiameter of approximately 1 mm . THz radiation emitted from the sample is collected and focused onto \na ZnTe crystal using two parabolic mirrors. The THz radiation induces birefringen ce inside the ZnTe \ncrystal due to the electro-optic Pockels effect. By probing this birefringence using the  gate pulse and a \nbalanced bridge detection scheme, we are able to reconstruct the electric field of the emitted THz \nradiation as a function of time . Simultaneously, the probe pulse with a fluence of approximately 10 \ntimes less than the pump is focused on the sample to a spot which is approximately twice smaller in \ndiameter than the pump. The angle of incidence of the probe beam is 25 degrees. By measuring the \npolarization rotation of the reflected probe pulse , information about the ultrafast laser-induced \nmagnetization dynamics is obtained by means of the MOKE .  The measurements were performed in \nmagnetic fields up to 1 kG applied in-plane of the samples. Performing the me asurements for tw o \npolarities of the magnetic field and taking the difference between the measurements, one can deduce \nsignals odd with respect to the field and thus minimize any influence of artifacts of non-magnetic \norigin. Simultaneous MOKE and THz detection of the laser-induced dynamic s minimize potential \nambiguities, which may arise when comparing two different experiments.    \n  0,60,70,80,91,0\n-1 0 1 2 3 4024\nNd0.2(Fe0,9Co0,1)0,8(b)M/M\nCoGd0.3(Fe0.9Co0.1)0.7\n(c)\nGd0.3(Fe0.9Co0.1)0.7Nd0.2(Fe0,9Co0,1)0,8E-field (V/cm)\nTime (ps)Co\n \nFIG. 1. (color online). (a) Layered structure under study and the scheme of the experi ment. Re is rare-\nearth being either Nd or Gd. The reflected probe pulse is used to measure the MOKE signal. The electric \nfield of the THz emission is detected with the help of electro-optic (EO) detection. A field of +/- 1 kG is \napplied in-plane. (b) Ultrafast magnetization dynamics as deduced from time-resolved MOK E \nmeasurements. The solid lines are fitting functions as described in the supplementary info rmation. (c)  \nElectric field of the emitted THz radiation  as a function of time. The position of zero-time is chosen \narbitrary so that the position of the peak of the pump pulse is close to zero. The traces are vertically \nshifted with respect to each other. \nFigure 1 (b ) show s the results of all-optical pump-probe experiments in which the laser-\ninduced dynamics is probed with the help of the MOKE. Using static MOKE measurements the transient \nsignals were calibrated. The figure shows the data in units of the relative change of the ma gnetization. \nNdFeCo shows clear ly distinct dynamics from that of GdFeCo and the reference Co-sample.   \nFigure 1 (c ) shows the temporal evolution of the electric field of the emitted radiation for th e \nthree samples . With the help of wire-grid polarizers, we found that the electric field of the THz emi ssion \nfrom all the samples is linearly polarized, perpendicular to the magnetization . Figure 2 shows the THz \nwaveforms measured for the Nd 0.2(Fe 0.87Co0.13)0.8 alloy , which is representative for all the studied \nsamples. The traces clearly change sign when the applied field is reversed . Moreover, measuring the \npeak amplitude as a function of the applied magnetic field reveals a hysteresis behavior as shown on \nthe inset of Fig. 2. This observation is a clear demonstration of the fact that the electric fiel d of the THz \nemission is proportional to the magnetization of the magnetic layer. \nOne can argue that the THz emission is not due to ultrafast laser-induced demagnetization, but \nhas an electric-dipole nature [ 21] and occurs because of effective symmetry breaking in the studied \nheterostructure. Such a symmetry breaking is especially efficient at the interfaces of the magn etic film . \nOne of the microscopic realizations of this mechanism is suggested in Ref. [23] showing that the THz \nemission is generated due to the inverse spin Hall-effect experienced by a spin-polarized current from \na magnetic to a non-magnetic layer. To check for emission generated by such a spin current from the \nmagnetic layer, we inverted the sample by turning it around while keeping th e applied magnetic field \nfixed. In this way the direction of a potential spin current between adjacent layers has to reverse sign . \nTherefore the sign of the THz radiation which would originate from this current, should be reversed as \nwell. Figure 2 shows the THz waveforms after turning the sample around. Besides different pulse \nwidths and amplitude changes, which are mostly due to absorption in the glass substrate, no sign \nchange is observed. The measurements on all other samples showed similar results. Hence, the \nexperiments have revealed no indication that the emitted THz radiation is due to a spin-current o r \nsimilar symmetry breaking effects . From all these features we conclude that ultrafast laser-induced \ndemagnetization can be reliably assigned to be the main source of the observed THz emission from \nthe studied samples.    \n -2 0 2 4 6 8-3036912\n-B -B+BE-field (V/cm)\nTime (ps)+B\n-0,3 0,0 0,3-404E-field (V/cm)\nField (kG)\n \nFIG. 2. (color online). Electric field of the emitted THz radiation from the Nd 0.2(Fe 0.87Co0.13)0.8 sample as \na function of time. The measurements have been performed in an external magnetic field of 1 kG and \n-1 kG (+ B and – B). The waveforms are measured for two orientations of the sample as indicated next \nto curves . The inset shows the hysteresis of the peak amplitude when the pump pulse excites the \nsample from the side of the substrate. The position of zero time for the upper curves is arbitrary \nchosen. The additional delay of the data shown in the lower curves is due to d ifferent propagation \nspeeds of the THz and visible radiation inside the substrate. \nTo compare the THz emission and the MOKE measurements, we assumed that the dynamics \nof the MOKE is fully defined by the dynamics of the net magnetization . In the simplest approximation, \nthe ultrafast demagnetization emission and magneto-optical probe data are co mpared with each other \nassuming coherent excitation of magnetic dipoles and observing them in far-field [ 20]. This gives \n22\ntME , where E the electric field of the emitted THz radiation in the time-domain, M is the \nnet magnetization in time-domain and t is time. This expression, however, disregards any effect of \noptical components on the propagation of the radiation and significant si ze of the source, which \ndeform the observed terahertz radiation significant ly both in time and space. To derive a more precise \nsolution, we solve the Maxwell equations with corresponding boundary condition s for an infinite film \nwith homogenous magnetization dynamics. We use Faraday’s aŶd Aŵpère’s law s to come to an \nexpression for the spectrum of the electric field (in Gaussian units) generated at th e surfaces of the \nferromagnet:  \nnzci\nx yndm icE \n e11~4~,  (1) \nwhere yE~is the y-component of the electric field of the THz emission in the frequency domain ,  is \nthe angular frequency, c the speed of light, xm~ the x-component of the magnetization in the \nfrequency domain, d the thickness of the magnetic layer and n the refractive index of the substrate. \nFor these calculations we used information about the net magnetization of the al loys in \nthermodynamic equilibrium . The derivation of Eq. 1 as well as details of the calculations of the \nanticipated spectra of the THz emission from the time-dependencies of the MOKE are presented in the \nsupplementary information.  \nIn Fig. 3 we compare our measured spectra (solid lines) with spectra calculated from the  time-\nresolved MOKE data (dashed lines) for different samples. For the calculated spectra of Co we assumed \nthat the net magnetization is equal to 1400 emu/cm3 [24]. For Gd 0.3(Fe 0.87Co0.13)0.7 we took the \nmagnetization equal to 150 emu/cm3 [25]. For Nd 0.2(Fe 0.87Co0.13)0.8 we used 462 emu/cm3 as measured \nwith the help of a vibrating magnetometer and consistent with literature [ 26-25]. The spectra obtained \nfor the Co sample show excellent mutual agreement. These observations show that THz emission and  \nMOKE for a pure ferromagnetic material provide identical information of the ul trafast magnetization \ndynamics. Hence, regarding the totally different nature of the probes, such an agreement is a good \nindication of the viability of the two techniques for studying ultrafast mag netization dynamics.  \n0,00,51,0\n0,00,5\n0,0 0,5 1,0 1,5 2,00,00,5x5.52 = MOKE 1 = THz emission\n21Nd0.2(Fe0.9Co0.1)0.8\nGd0.3(Fe0.9Co0.1)0.7 \nCo1\n12E-field (Vs/cm)\nx2\n2 \nFrequency (THz)\n \nFIG. 3. (color online). Spectra of the THz emission.  The solid lines (1) are spectra obtained with the help \nof the Fourier transform of the measured THz waveforms. The dashed lines (2) indicate the calculat ed \nspectra inferred from the time-resolved MOKE data. The calculated spectra of GdFeCo and NdFeCo are  \nshown multiplied by 2 and 5.5 respectively. The shaded areas are meant to highligh t the difference \nbetween the calculated and measured spectra.  \nFor GdFeCo the spectrum calculated from the MOKE and the actually measured spectrum of \nthe THz emissi on show a mismatch in amplitudes by a factor of 2 . Simultaneous ly, we observe that the \nfull-width at half maximum (FWHM) of the spectrum obtained using the MOKE agrees well with the \nFWHM of the actually measured THz spectrum (see Table 1). The expectation values of the spectra are \nalso similar . Hence there is a satisfactory quantitative and a good qualitative agreement between the \nspectrum calculated from the MOKE and the actually measured spectrum of the THz emission during \nultrafast laser induced demagnetization of GdFeCo .   \nNeither quantitative nor qualitative agreement between the spectrum calculated from the  \nMOKE and the actually measured spectrum of the THz emission was obtained in the case of ul trafast \nlaser-induced demagnetization of NdFeCo. In particular, Fig. 3 clearly indicates  the lack of higher \nfrequencies in the spectrum calculated from the MOKE response of NdFeCo, see also table 1. This implies that the dynamics revealed by the MOKE is slower than the magnetization dyn amics revealed \nwith the help of THz emission spectroscopy.  \n Co GdFeCo NdFeCo \nFWHM MOKE 0.62 THz 0.76 THz 0.70 THz \nFWHM THz 0.58 THz 0.73 THz 0.81 THz \n|FWHM THz –FWHM MOKE| / FWHM MOKE 6.1 % 4.4 % 16 % \nmean MOKE 0.65 THz 0.76 THz 0.67 THz \nmean THz 0.64 THz 0.79 THz 0.77 THz \n|mean THz –mean MOKE| / mean MOKE 1.1 % 4.2 % 14 % \nTable 1.  Full-width at half maximum (FWHM) and expectation value (i.e. center of mass) of th e spectra \npresented in F ig. 3.   \nTo determine the origin of the observed disagreements in the spectra, we note that the \nGdFeCo and NdFeCo samples have similar structures. Hence the layered structure of the studied \nsamples cannot explain the qualitatively different results obtained for these two types of alloys. \n One may argue that the observed discrepancies between the calculated and measured spectra \nfor Nd FeCo can be due to elemental specificity of the MOKE, which at the wavelength of 800 nm is \nmainly sensitive to the magnetization of the Fe-sublattice. Note that the THz emissi on is characterized \nby a different sensitivity, which follows the net magnetization dynamics. As t he MOKE is mainly \nsensitive to the magnetization of the Fe-sublattice, the MOKE signals measured for \nGd 0.18(Fe 0.87Co0.13)0.82  and Gd 0.3(Fe 0.87Co0.13)0.7 in an external magnetic field should have different signs. \nThis sign change was indeed observed in the MOKE measurements, but not in the THz e mission \nmeasurements (see supplementary information ) revealing that at least the sign of the THz emission is \ndefined by the net magnetization in these alloys.   \nHowever, if in NdFeCo the dynamics measured with the MOKE is associated with the d ynamics \nof the FeCo sublattice, the mismatch in spectral distribution shows that the Nd-sublattice exhibits \nfaster magnetization dynamics compared to the FeCo sublattice.  This is in strong contrast to GdFeCo \nin which X-ray measurements revealed that the dynamics of the Gd sublattice in the GdFeCo alloy \ndemagnetizes slower than the FeCo sublattice [ 16]. Analyzing the similarities and differences of \nGdFeCo and NdFeCo alloys, indicates that the orbital magnetization of Nd play s a significant role in the \nultrafast demagnetization process. However, our experiments cannot resolve the exact roles o f spin \nmomentum, orbital momentum and spin-orbit interaction in the process of u ltrafast laser induced \ndemagnetization. Resolving this issue is an intriguing subject for future  studies with the help of X-ray \ntechniques. \nTo summarize we have combined pump-probe MOKE and THz spectroscopy techni ques for \nsimultaneous observation of the ultrafast magnetization dynamics and direc tly compare them with \neach other. For NdFeCo we observed clear differences in the MOKE and THz responses , showing that \nthe MOKE measures slow er dynamics than the magnetization dynamics that actually takes place .  \n \nWe would like to thank T. Toonen and A. van Etteger for technical support. We would  like to \nthank J. D. Costa for providing the Co sample, M. Huijben for the vibrating m agnetometer \nmeasurements and J. Becker for fruitful discussions. This work was supported by the Fou ndation for \nFundamental Research on Matter (FOM), the European Unions Seventh Framework Program \n(FP7/2007-2013) grant No. 280555 (Go-Fast), and European Research Council g rant No. 257280 \n(Femtomagnetism ).  \n1. I. Žutić , J. Fabian, S. Das Sarma , Spintronics: Fundamentals and applications,  Reviews of Modern Physics, 76, \n323 (2004)  \n2. A. Fert, Nobel Lecture: Origin, development and future of spintronics , Reviews of Modern Physics, 80, 1517 \n(2008)  \n3. T. Kuschel, and Günter Reiss, Spin Orbitronics: Charges ride the spin wave , Nature Nanotechnology, \ndoi:10.1038/nnano.2014.279 (2014)  \n4. E. Beaurepaire, J. -C. Merle, A. Daunois, and J. –Y. Bigot, Ultrafast Spin Dynamics in Ferromagnetic Nickel , \nPhysical Review Letters, 76, 4250 (1996) \n5. L. Guidoni, E. Beaurepaire, and J. –Y. 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Oh, Comparisons of Magnetic and Magneto-Optic Properties between Fe-ri ch \nand Nd-rich Amorphous Nd xFe1-x Alloys , Journal of Magnetics 3, 49 (1998) SUPPLEMENTARY INFORMATION: \nRevealing the role of orbital magnetism in ultrafast laser-induced d emagnetization in \nmultisublattice metallic magnets by terahertz spectroscopy \n \nT. J. Huisman1, R. V. Mikhaylovskiy1, A. Tsukamoto2, Th. Rasing1 and A. V. Kimel1 \n1Radboud University Nijmegen, Institute for Molecules and Materials, 6525 AJ Nijmegen, The \nNetherlands \n2College of Science and Technology, Nihon University, 7- 24-1 Funabashi, Chiba, Japan \n \nI. THz emission derived from the Maxwell equations \n \nHere we derive mathematical expressions describing how a fast laser induced demagneti zation gives \nrise to the emitted THz radiation.  \nWe assume a thin metal film is in the xy plane on top of a substrate as indicated in F ig. S1. The \nfilm is assumed to be magnetized along the x-axis and exhibits homogeneous magnetization dy namics. \nFroŵ Aŵpere’s aŶd Faraday’s law s we can relate the magnetic field of the electromagnetic radiation \nto the magnetization of the medium as: \n x xxMcHczH\nzz~4~~1\n2 22 \n\n\n\n\n\n,  (S 1) \nwher e xH~ is the x-component of the magnetic field in the frequency domain ,  z  is the dielectric \npermittivity,  is the angular frequency, c is the speed of light and  zm dMx x~~  is the x-\ncomponent of the magnetization in the frequency domain, d is the thickness of the magnetic film and \n z  is the Dirac function. We are looking for the plane wave solutions obeying to  \n\n\n0,e0,e~\nzDzCH\nnz\nciz\nci\nx\n.    (S 2) \nHere n is the refractive index of the substrate. Continuity of xH~ implies DC , while integration of \nEq. S1 over the thickness of the film implies  dmczH\nzxx ~4~1\n2\n \n\n\n\n, which results in\n1~4\nnndm icDx. Froŵ Faraday’s law it follows (in Gaussian units): \nnzci\nx yndm icE \n e11~4~.    (S 3)  \nFIG. S1. (color online). Geometry of the modelled experiment. A laser pump pu lse initiates \nmagnetization dynamics in the sample which is accompanied by the emission of electromagnetic \nradiation. \nTo determine the generated electromagnetic radiation, the spectrum xm~ is required. The \nmagnetization dynamics is inferred from time-resolved magneto-optical Kerr effect  (MOKE) probe \ndata. To prevent numerical errors the observed MOKE rotation is fitted with an empiri cal function, \nwhich is either: \n  Ft DtEeCeBt AMM \n\n\n\n\n \n21erf21, (S 4) \nor \n  EteDtCBt AMM\n\n\n\n\n )ln(21erf21,  (S5) \nwith A, B, C, D, E and F as fitting parameters. The choice of the fitting function depends on \nthe shape of MOKE rotation in time. Figure 1 (b ) in the main text of the paper shows the magneto-\noptical probe data fitted using Eqs . S4 and S5 . Figure S2 shows the electromagnetic radiation emitted \nas a result of the magnetization dynamics inferred from the time-resolved MOKE data. \n012\n01\n0,0 0,5 1,0 1,5 2,001CoGd0.3(Fe0.9Co0.1)0.7 Nd0.2(Fe0.9Co0.1)0.8\n E-field (Vs/cm)\nFrequency (THz)\n \nFIG. S2. (color online). The calculated spectra of the emitted electromagnetic radiation inferred from \nthe time-resolved magneto-optical data shown in Fig. 1 (b ) of the main text of the paper.  \nII. Influence of the spectrometer on the generated THz emission \nThe observed spectrum of the emitted electromagnetic radiation can be related to the \nspectrum of the generated radiation by a linear relation: \n source observed EK E  ,    (S 6) \nwhere  K  is the transfer function which accounts for the substrate transmission, propagation \neffects and response of the ZnTe crystal. The transmission through a substrate much thicker than the \nwavelength of the electromagnetic wave is given by the Fresnell transmission equation: \ntii\nnnnt2      (S 7) \nwhere in is the refractive index of the medium from which the radiation originates and  tn is the \nrefractive index of the medium to which the radiation is transmitted . The refractive index of the glass \nsubstrate is obtained experimentally with the help of THz transmission spectroscopy as described in \n[1]. Our results are comparable with the observations in [2]; the absorption of glass can be \napproximated by an increasing quadratic function with increasing THz frequencies, which eff ectively \nsuppresses high frequencies of the emitted radiation. \nTo come from a near-field solution to the THz radiation at the detecting crystal, we apply \nGaussian propagation similar to the one described in [3] . In order to apply Gaussian propagation, one \nneeds to provide an initial diameter for the Gaussian beam in the model. Our initial diameter used in \nthe calculations is taken equal to the diameter of the pump beam at the sample. In our setup, t he \neffects of propagation remove low frequencies in the spectrum. Frequencies above 2 THz are  all \nenhanced by a factor equal to the ratio of the foc al lengths of the two parabolic mirrors used to collect \nand refocus the THz emission.  \nFor the ZnTe response we applied the methods mentioned in [4], which shows that hig her \nfrequencies are suppressed. To compare quantitatively the calculated spectra with the experimentally \nobtained ones, we used E q. 9 from [5] which shows how the observed ellipticity in the ZnTe crystal is  \nrelated to the electric field amplitude of the THz radiation. \nTaking into account both the propagation effects and the ZnTe response defines the \nspectrometer response, which can be visualized as a bandpass filter centered around 1.7 THz (see Fig.  \nS3). The ZnTe crystal attenuates frequencies above 1.7 THz, while effects of propagation attenuate \nfrequencies below 1.7 THz.  \n0 1 2 3 40100200Spectrometer response\nFrequency (THz) \nFIG. S3. The spectrometer response as function of frequency. \n III. Additional data for GdFeCo  \n \nThe two used GdFeCo samples, Gd 0.18(Fe 0.87Co0.13)0.82 and  Gd 0.3(Fe 0.87Co0.13)0.7, have different ratios \nbetween the magnetizations of the sublattices. For Gd 0.18(Fe 0.87Co0.13)0.82 the magnetization of the FeCo \nsublattice dominates the net magnetization, while for Gd 0.3(Fe 0.87Co0.13)0.7 the magnetization of the Gd \nsublattice dominates the net magnetization. Figure S4 (a) shows the dynamics of the MO KE measured \nfor these two samples. It is seen that the dynamics have opposite signs, which  is the result of different \nsublattices dominating the magnetization. Figure S4 (b) shows that the signs of the THz emission are \nthe same for the two GdFeCo samples. The difference in amplitudes can be related to the difference \nin the net magnetizations. \n-0.10.00.1\n-1 0 1 2 3-0.50.00.51.01.5\nGd0.2(Fe0.9Co0.1)0.8Gd0.2(Fe0.9Co0.1)0.8Gd0.3(Fe0.9Co0.1)0.7M/M(a)\nGd0.3(Fe0.9Co0.1)0.7E-field (V/cm)\nTime (ps)(b)\n \nFIG. S4. (color online). (a) MOKE measured demagnetization for Gd 0.18(Fe 0.87Co0.13)0.82 and \nGd 0.3(Fe 0.87Co0.13)0.7. The solid lines are fitting functions as expressed in Eq. S4. ( b) Measured \ndemagnetization emission of Gd 0.18(Fe 0.87Co0.13)0.82 and Gd 0.3(Fe 0.87Co0.13)0.7. The traces are vertically \nshifted with respect to each other. The position of zero-time is chosen arbitrary so tha t the position of \nthe peak of the pump pulse is close to zero.  \n \n \n \n \n1. L. Duvillaret, F. Garet, and J. -L. Coutaz, A Reliable Method for Extraction of Material Parameters in Terah ertz \nTime-Domain Spectroscopy , Slected Topics in Quantum Electroncs, IEEE Journal of, 2, 739 (1996) \n2. M. Naftaly, and R. E. Miles, Terahertz time-domain spectroscopy of silicate glasses and the relationship to \nmaterial properties , Journal of Applied Physics, 102, 043517 (2007) \n3. P. Kužel, M. A. KhazaŶ, aŶd J. Kroupa, Spatiotemporal transformations of ultrashort terahertz puls es, Journal \nof Optical Society of America B, 16, 1795 (1999) \n4. G. Gallot, J. Zhang, R. W. McGowan, T. –I. Jeon, and D. Grischkowsky, Measurements of the THz absorption and \ndispersion of ZnTe and their relevance to the electro-optic detection of THz radiation , Applied Physics Letters, 74, \n3450 (1999) \n5. P. C. M. Planken, H. -K. Nienhuys and H. J. Bakker, T. Wenckebach, Measurement and calculation of the \norientation dependence of terahertz pulse detection in ZnTe , Journal of the Optical Society of America B, 18, 313 \n(2001)                                                             "    },    {        "title": "1505.07035v1.Vacancy_defects_and_monopole_dynamics_in_oxygen_deficient_pyrochlores.pdf",        "content": "Vacancy defects and monopole dynamics in oxygen -deficient pyrochlores  \nG. Sala1, M. J. Gutmann2, D. Prabhakaran3, D. Poma ranski4,5, C. Mitchelitis4,5, J. B. Kycia4,5, D. \nG. Porter1, C. Castelnovo6 and J. P. Goff1* \n1Department of Physics, Royal Holloway, University of London, Egham, TW20 0EX, UK  \n2ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UK  \n3Department of Physics, University of Oxford, Oxford OX1 3PU, UK  \n4Department of Physics and Astronomy and Guelph -Waterloo Physics Ins titute, University of \nWaterloo, Waterloo, Ontario N2L3G1, Canada  \n5Institute for Quantum Computing, University of Waterloo, Waterloo , Ontario N2L 3G1, Canada  \n6Theory of Condensed Matter group, Cavendish Laboratory, University of Cambridge, \nCambridge CB3 0HE, UK  \nThe idea of magnetic monopoles in spin ice has enjoyed much success at intermediate \ntemperatures, but at low temperatures a description in terms of monopole dynamics  alone \nis insufficient. Recently, numerical simulations were used to arg ue that magnetic impurities \naccount for this discrepancy by introducing a magnetic equivalent of residual resistance in \nthe system . Here we propose that oxygen deficiency is the leading cause of magnetic \nimpurities  in as -grown samples, and we  determine the  defect structure  and magnetism in  \nY2Ti2O7-δ using diffuse  neutron scattering and magnetiz ation measurements. These defects  \nare eliminated by oxygen annealing. The introduction of oxygen vacanci es causes Ti4+ to \ntransform  to magnetic Ti3+ with quenched orbital magnetism , but the concentration is \nanomalously low . In the spin -ice material  Dy2Ti2O7 we find that the same oxygen -vacancy \ndefects suppress  moments on neighbouring rare -earth sites, and that these magnetic  \ndistortions  dramatically slow down  the long -time  monopole  dynamics  at sub -Kelvin \ntemperatures.  \nThe frustrated geometry of the pyrochlore lattice of corner -sharing tetrahedra , and the \ncrystal electric field effects that constrain the rare -earth magnetic moments to point along the \ntetrahedron axes,  are responsible in great part for the behaviour of the spin -ice materials1. The \nleading effect is the appearance of “2in-2out” ice rules2-5, analogous to the proton arrangement in \nwater ice6. These local constraints give rise to an intrig uing display of topological properties7. \nThe frustrated geometry  in these systems  provides the key to a longstanding problem in \ntheoretical physics: how to stabilize fractional excitations in three dimensions. The proposal that \nexcitations in spin ice fractionalize into de -confined  magnetic monopol es interacting  via the \nmagnetic Coulom b inter action8 has led to intense experimental activity in recent years. In the \ncase of diffuse neutron scattering it was po ssible to observe the “ Dirac strings ” that trace the \nrandom walk of monopoles9, and to infer the presence of mon opoles from the broa dening of  \n“pinch -point” features10-11. Magnetic currents have been observed in a magnetic field12, and \nmagnetic relaxation measurements have been interpreted in terms of the diffusive dy namics of \nfree monopole s13-15.  However, r ecent experiments conducted at sub -Kelvin  temperatu res show that \nmagnetization dynamics in spin ice samples occurs on far longer time scales than one could \nexplain using straightforward monopole hydrodynamics, even accounting for Coulomb \ninteractions16-18. In an  attempt to explain th is discrepancy, magnetic impurities were shown to be \ncapable of dramatically reducing the flow of magnetic monopoles,  similarly to electrical \nconductors in which local impuritie s can decre ase the electrical conductivity19. To date, magnetic \nimpurit ies in s pin ice have been model led based on the assumption  that they resemble “stuffed \nspin ice”20. The determination of the nature of the defects  in as -grown samples , whether they \ncompri se substitutions or vacancies, th e extent of the distortion of the surrounding lattice , and the \neffects  on the magnetic properties, has become a pressing issue . Understanding thes e defects is \ncrucial for experiments directed at single monopole detection, the observation of monopole \ncurrents, and the design of potential spin-ice devices.  \nWe answer this  question  by first investigating defect structures in oxygen -depleted \npyrochlores and then studying how the oxygen vacancies affect the magnetism of spin -ice \nmaterials.  The absence of a large -moment rare -earth ion in Y2Ti2O7-δ allows us to focus on the \nstructure and on the magnetism of  the Ti sites. The structure of the defect clusters in  oxygen -\ndeficient Y 2Ti2O7-δ was determined using  diffuse neutron scattering , which is particularly \nsensitive to vacancies and displacements  of oxygen ion s. We also find that oxygen vacancies are  \nthe dominant defects in as -grown “stoichiometric” samples , rather than the s tuffing of Ti sites \nwith Y . The removal of O2- ions results in the r eplacement of Ti4+ ions by Ti3+, and this leads to \nthe introduction of magnetic moments on the Ti sites. Our results imply that Ti3+ ions are far \nfrom being the leading magnetic perturbation in spin -ice materials. We relate our results to \nDy2Ti2O7-δ and we find that oxygen vacancies cha nge the nature of the magnetism on  the \nsurrounding four Dy3+ ions. We find  that these magnetic impurities dramatically change the low-\ntemperature monopole dynamics.  \nAll of our  x-ray diffraction data for the black oxygen -depleted Y2Ti2O7-δ , the yellow as-\ngrown and transparent annealed Y 2Ti2O7 refine in the cubic pyrochlore structure, space group Fd\n3\nm. Y and Ti ions are located on pyrochlore lattices, and there are two inequivalent O sites: O(1) \nlocated a t the centre of the Y  tetrahedra , and O(2) filling interstitial regions. Refinement of x-ray \ndiffraction data  reveal s equal concentrations o n the Y a nd Ti sites to within 2%, see Table 1 , so \nthat stuffing of Ti sites by Y is minimal . The stoichio metry  of the oxygen -depleted sample \ndetermined by thermo -gravimetric analysis  (TGA)  was Y2Ti2O6.79, and the diffraction data are \nconsistent with these values . The vacancies were found to be  mainly on the O(1) s ites, in  \nagreement with the literature that  reports  that the O(1) are not as strongly bound to the latti ce as \nthe O(2) ions21,22. The length  of the unit cell in creases monotonically with  decreasing oxygen \nconcentration , as expected .  \nDiffuse neutron scattering is very sensitive to  departures from ideal stoichiometry. \nY2Ti2O6.79 exhibits strong, highly anisotropic diffuse scattering throughout reciprocal space. Figure 1 shows the diffuse scattering from Y2Ti2O6.79 in the ( hk7) plane , which is particularly \ninformative. The diffuse scattering in this plane comprises a distinctive pattern of four rods that \ncreate a cross at the centre of the plane and four arcs that link the rods. Figure 2 shows that the \nas-grown sample has qualitatively very similar,  but lower intensity scattering. T hus we conclude \nthat the oxygen -depleted and as -grown samples have very similar defect structures. The fact that \nno diffuse scattering was detected for the anneal ed sample shows that it has very few  defects, \nand that it is possible to obtain a sample much closer to  ideal stoichiometry by annealing in \noxygen.  It also suggests that oxygen vacancies are the main defects in the as -grown sample, and \nseems to rule out the possibility of stuffing of the Ti lattice by Y ions on the same scale as \noxygen vacancies.  On the basis of the diffuse scattering intensities, we estimate compositions \nY2Ti2O6.97 in Y2Ti2O7.00 for the as -grown and annealed samples.  \nWe developed a Monte Carlo code23,24 that is able to reproduce qualitatively the main \nfeatures of the diffuse scattering. We are only able to  reproduce the observed crosses when there \nare relatively large relaxations of Y ions away from isolated O(1) vacancies along <111>  \ndirections. The phy sical origin of this is the Coulomb repulsion between the O(1) vacancies and \nthe Y ions that leads to the expansion of the Y cage. We replace two Ti4+ in neighbouring \ntetrahedra  by Ti3+ for charge compensation, and move neighbouring O(2) towards Ti4+ sites so \nthat Ti3+-O and Ti4+-O bond lengths agree with those in the literature. The smaller displacements \nof the surrounding ions were simulated using the “balls and springs” model25, and this \nsuccessfully reproduces the four arcs that link the branches of the  cross, see Fig. 1(b ). \nNeighbouring O(1) are pushed away along <111> directions by Y ions . We found that t here is no \nevidence for correlations between the O(1) vacancies. The defect structure  around an O(1)  \nvacancy is  shown in Fig. 1(a ). Replacement of the  Ti3+ ions by Y3+ ions would give the stuffed \nspin ice Y 2(Ti2-xYx)O7-x/2. Although the diffuse neutron scattering is consistent with a low level \nof stuffing, the x -ray diffraction effectively rules it out. This is in contrast to Yb 2Ti2O7, where the \nsmaller  Yb ions are found to substitute for Ti at the percent level26. Even where stuffing does \noccur, it is highly likely that charge compensating O(1) vacancies are important defects.  \nFor Y2Ti2O7-δ, the removal of O2- ions cha nges the oxidation state of Ti ions in order to \npreserve charge neutrality. Ti3+ ions are magnetic and for modest concentrations of vacancies \nY2Ti2O7-δ becomes paramagnetic, as the leading dipolar interaction between them is negligible at \nthe temperatures  of interest . Consistently, w e did not detect any magnetic diffuse scattering from \nY2Ti2O6.79 using unpolarised  neutrons on SXD  down to 0.3K . The SQUID data presented in Fig. \n3 are fitted with a Brillouin function with S = ½, and this suggests that t he orbital moment is \nquenched. However, t he concentration of defects δ = 0.023 is much lower than the values \nobtained using structural refinement or TGA . One possible explanation is that neighbouring Ti3+ \nform strong antiferromagnetic bonds and the observed signal is from isolated Ti3+ ions. Another \npossibility is that a sizable proportion of the Ti ions form Ti2+, which is expected to have a \nsinglet ground state. It is even possible that partial charge compensation may be achieved \nthrough trapped electrons on vacancies, via the so -calle d F-centres often responsible for colour in this class of material27. Incidentally, t he presence of almost -free Ti3+ ions may be the rapidly \nfluctuating magnetic impurities required to understand the NMR relaxation at low temperature28. \nAnnealing as -grown Dy2Ti2O7 in oxygen leads to defect -free transparent yellow crystals \nand, therefore, we conclude that the dominant defects in this case are also oxygen vacancies. Our \ndiffuse neutron scattering studies show that oxygen vacancies are located on the O(1) site s for \nDy2Ti2O7, see Fig. 4(a) . The oxygen concentration measured using TGA is δ = 0.02 for the as -\ngrown sample. In Fig. 4(b) we compare the static magnetic susceptibility of an as -grown sample \nbefore and after annealing in oxygen. We are able to clearly re solve a reduction in saturation \nmagnetization that implies a reduced moment on the defective Dy sites. Our crystal electric field \ncalculations show that Dy3+ ions have a reduced moment in the presence of an O(1) vacancy, and \nthe anisotropy changes from easy axis along <111> for the stoichiometric compound to easy \nplane perpendi cular to the local <111>, see section C of the Supplementary Information .  \nAC susceptibility measurements were conducted on an as -grown and annealed Dy 2Ti2O7 \ncrystal at the Univers ity of Waterloo using a SQUID Susceptometer on a dilution refrigerator16.  \nThese samples are both needle -shaped to reduce the demagnetization correction, they have the \nsame crystallographic orientation, and the results were obtained at the same temperature , T = \n800mK. The measured imaginary portion of the AC susceptibility, χ’’(ω), was transformed to the \ndynamic correlation function C(t) = < M(0)M(t)>, where M(t) is the time -dependent \nmagnetization of the sample29. The dynamic correlation function results  are presented in Fig. \n4(c), where they are compared with previous results  on a different, non -annealed sample of \nDy2Ti2O7, at T = 800 mK19. The very slow long -time tail in C(t) associated with magnetic \ndefects is observed for the as -grown sample, but it is entirely suppressed for the annealed \nsample. The as -grown sample characteristics match cl osely with the previous results .  Revell et \nal. attributed the long -time tail as being a result of interactions of monopoles with magnetic \nimpurities from a slight level of stuffed sites (substitution of Dy for Ti)19.  The fact tha t \nannealing in oxygen eliminates the long -time tale suggests that the magnetic impurity sites in th e \nas-grown sample results from o xygen vacancies. The stretched exponentials seen in the \ncorrelation function at early times for all samples were very similar.  \nWe have  conducted preliminary investigations of the effects of the magnetic defects \nintroduced by isolated oxygen vacancies on the monopole dynamics. Figure 5(a) shows a \ntetrahedron with 4 easy -plane spins surrounding an O(1) vacancy, as suggested by our crystal \nelectric field calculations. The 4 neighbouring tetrahedra  have, therefore, 3 easy -axis and 1 easy -\nplane spin each . The concept of ice rules is no longer valid for any of these defective tetrahedra. \nOn the other hand, farther tetrahedra  are minimally pertu rbed and are expected to behave as in \nstoichiometric spin ice. Let us consider what happens to the system when a monopole hops from \none of the farther stoichiometric tetrahedra onto one of the tetrahedra directly affected by the \nvacancy, as illustrated in the figure , where an adjacent tetrahedron is shown initially in an \nexcited 3out -1in monopole state. Flipping the spin that joins the stoichiometric tetrahedron to a \ndefective tetrahedron results in the monopole hopping onto one of the 3 -easy-axis, 1 -easy-plane tetrahedra, see Fig. 5(b). The direction of the 4 easy -plane spins relaxes  to minimise the energy \nof the system given the orientation of the surrounding easy -axis spins, and this is responsible for \na reduction in the energy of the system.  Pictorially,  the monopole cha rge has “delocalised”  over \nthe defective tetrahedra and one can no longer identify  a specific tetrahedron where the \nmonopole  resides.  \nIn section D of the Supplementary Information we have calculated the energy changes for \nall configuratio ns of this cluster of spins , for exchange and dipolar interactions truncated at \nnearest -neighbour distance, with various working assumptions on t he exchange interaction \ninvolving  easy-plane spins. In all cases, we obtained a broad distribution of energies down to \nlarge negative values, large enough to be comparable to the bare energy cost of an isolated \nmonopole. This means that a monopole coming into contact with a vacancy cluste r can become \nstrongly pinned to it. It is remarkable to notice the similarity in the way that mo nopoles interact \nwith these o xygen vacancies compared to the static stuffed moments discussed by Revell et al19. \nDepending on the direction of approach, the mag netic defects can either attract or repel a \nmonopole. Since the static magnetic moments introduced in the simul ations were able to explain \nthe long -time tail in the magnetic relaxation of the system19, one can reasonably expect that local \nmagnetic dis tortions introduced by o xygen vacancies can lead to similar long -time tails. The \nexperimental results in Fig. 4(c) demonstrate that this is the case.  \nOne must further notice that, from the very same energetic arguments presented above, \nthese magnetic defects ca n also act as nucleation centres for monopoles. Hence they may be \nresponsible for an increase in monopole density in thermodynamic equilibrium. In turn, a \nheightened monopole density means faster (linear -response) dynamics14. This is precisely the \nopposite  effect of a pinning centre. Understanding the interplay between these two effects \nrequires careful studies that are beyond the scope of the present article. One might nonetheless \nspeculate that the speedup due to an increase in monopole density is more li kely to dominate at \nshort time scales (close to equilibrium), whereas pinning effects become important when the \nsystem is driven far from equilibrium at long times. It is intriguing to note  that this expectation \nappears to match the qualitative differences  between as -grown and annealed behaviour of C(t) in \nFig.4(c) , whereby the removal of o xygen vacancies is seen to simultaneously suppress the long -\ntime tail and slow down the short -time magnetic relaxation at the same time.  \nWe have clearly demonstrated  the important role of o xygen vacancies in the monopole \nphysics at low temperature. Further studies are required to understand why the magnetic \nresponse of the Ti sites is suppressed . The magnetism on the rare -earth sites makes a substantial \ncontribution t o the monopole dynamics, but it would be of great interest to explore further the \neffects on the system . Our results make i t apparent that the density of o xygen vacancies in spin \nice samples is of key importance  in the interpretation and understanding of relaxation and \nresponse properties, and far from equilibrium behaviour in general. Many experimental results in \nthis direction are already available13,17 -19,30, and in some cases it may be that further work is \nneeded  to distinguish the effects of o xygen vacancies from the stoichiometric behaviour. Moreover, theoretical work to date has largely focused on stoichiometric models14-15,28,31 -34, and \nit will be interesting to see which new phenomena may emerge when a tu neable density o f \nimpurities is introduced via o xygen depletion.  \nMethods  \nSingle crystals of Y 2Ti2O7-δ were grown at the Clarendon Laboratory by the floating zone \ntechnique . A few pieces of single crystal from the same batch as the as -grown sample were \nannealed in O 2 at a flow rate of 50 mil min-1 and a temperature of 1200 ºC for 2 days . The as -\ngrown, nominally stoi chiometric samples are yellow, and a nnealing in oxyg en produc es \ntransparent samples. The oxygen -depleted samples were grown and annealed in flowing mixed \ngas of hydrogen  and argon , and this  leads to crystals that a re black. The D y2Ti2O7-δ crysta l \ngrowth is described in Ref [35 ]. The average structure was determined using x -ray diffraction at \nRoyal Holloway, and the defe ct structures were studied by  diffuse neutron scattering using the \nsingle -crystal diffractometer (SXD) at the ISIS pulsed neutron source at the Rutherford Appleton \nLaboratory. SXD combines the white -beam Laue technique with area detectors covering a solid \nangle of 2π steradians, allowing comprehensive diffraction and diffuse scattering data sets to be \ncollected. Samples were mounted on aluminium pins and  cooled to 5K using a closed -cycle \nhelium refrigerator in order to suppress the phonon contribution to the diffuse scattering. A \ntypical data set required four orientations to be collected for 4 hours per orientation. Data were \ncorrected for incident flux using a null scattering V/Nb sphere. These  data were then combined \nto a volume of reciprocal space and sliced to obtain single planar and linear cuts. The DC \nmagnetization was measured using a Quantum Design 7T SQUID magnetometer  at the \nClarendon Laboratory , and the AC magnetization was measured u sing a SQUID in a dilution \nrefrigerator at the University of Waterloo . The as -grown Dy 2Ti2O7 crystal had dimensions \n1.0×1.0× 4.0mm3, and the annealed Dy 2Ti2O7 crystal had dimensions 1.0× 0.32×4.0mm3, where \nthe long axis was directed along the magnetic field.  The data from Revell et al. was obtained on \na different, non -annealed sample of Dy 2Ti2O7 with dimensions 1 .0×1.0×3.9mm3. \nFor the Monte Carlo code , a crystal comprising 64 × 64 × 64 unit cells was generated \nfrom the average structure obtained fro m the refinement of the diffraction data. From a statistical \nperspective, the use of a large supercell helps us to average over the disorder in the system (self \naverage), and suppress es the background noise. O(1) ions are removed at random until we obtain \nthe depletion concentration. Large  displacements of surrounding Y ions and smaller \ndisplacements of O(2) ions next to Ti3+ ions are introduc ed by hand. The distortion of the \nsurrounding lattice  is simulated using the “balls and spring” model in which hard spheres are \nconnected to neighbouring ions by springs, and the simulation randomly displaces ions in order \nto minimize the elastic energy. Further details of the  model are presented in sections A and B of \nthe S upplementary Information . The crystal electric field was calculated using a point -charge \nmodel36, and the monopole dynamics was investigated using cluster calculation s, see sections C \nand D of the Supplementary Information  for further details. T he magnetization data we re \nmodelled using a Brill ouin function .  1. Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske, T., Godfrey, K. W. Geometrical \nfrustration in the ferromagnetic pyrochlore Ho 2Ti2O7. Phys. Rev. Lett.  79, 2554 -2557 \n(1997).  \n2. Bramwell, S. T., Gingras. 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Seitz, F., Thurnbull, \nD. (Academic Press, New York and London, 1964).  \n \n Acknowledgements  \nWe thank M. Ju ra and T. Lehner  for their help, and B. Gaulin, M. T. Hutchings, R. Moessner, S. \nL. Sondi, S. Dutton and U.Karahasanovic  for helpful discussions. We acknowledge support from \nthe South East Physics Network and the Hubbar d Theory Consortium, and we are  grateful for the \nfinancial support and hospitality of ISIS.  This work was supported in part by EPSRC grants \nEP/G049394/1 and EP/K02896 0/1, NSERC,  the Helmholtz Virtual Institute \"New States of \nMatter  and Their Excitations\", an d the EPSRC NetworkPlus on \"Emergence and Physics far \nfrom Equilibrium.  \nAuthor contributions  \nJ.P.G., C.C. and D.P r. designed the research . The neutron measurements were performed by \nG.S., M.J.G. and J.P.G. , the x -ray diffraction was performed by D.G.P. The  crystals were grown \nby D.P r. The DC magnetisation was measured by D.Pr. and G.S. and the magnetic relaxation \nmeasurements were performed by D.Po. , C.M.  and J.B.K. Theoretical modelling and simulation \nwas by G.S. , M.J.G.  and C.C. The manuscript was drafted  by J.P.G., C.C. and G.S. and all \nauthors participated in the writing and review of the final draft.  \nAdditional information  \nSupplementary Information accompanies this paper at http://www.nature.com/nmat/index.html . \n \n \n \n \n \n \n \n \n \n \n \n \n  Depleted  As-grown  Annealed  \nColour  Black  Yellow  Transparent  \nSpace group  Fd\n3m Fd\n3m Fd\n3m \nLattice parameter  10.123(3 )Å 10.111(3 )Å 10.102(2 )Å \nY 0.992(9)  1.016(5)  1.01(1)  \nTi 1 1 1 \nO(1)  0.88(2)  1.01(2)  0.97(3)  \nO(2)  0.96(2)  1.06(1)  1.10(2)  \nRw 9.38 6.58 11.71  \n \nTable 1. Refinement of the average structure of Y2Ti2O7-δ from the x -ray structure factors \nmeasured at T = 300K. The Y and Ti occupancies are equal within experimental uncertainty, and \nthe oxygen vacancies in the oxygen -depleted sample are primarily located on O(1) sites.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure captions  \nFig. 1.  Oxygen -vacancy defect structure in Y 2Ti2O7-δ. (a) Schematic diagram of O(1)  \nvacancies and the  associate d distortion of the surrounding lattice,  with displacements indicated \nby green arrows. The four neighbouring Y ions relax away from the vacancy along <111> due to \nCoulomb repulsion, and they push ne ighbouring O(1) ions in the same direction. Two Ti4+ ions \ntransform to Ti3+ to preserve charge neutrality , and nearest neighbour O(2) ions move toward \nTi4+ ions to give the correct bond lengths . (b) The diffuse neutron scattering in the ( hk7) plane \nfrom Y 2Ti2O6.79 measured at T = 5 K (upper half) compared to the Monte Carlo simulation \n(lower half).  The relatively large displacement of the Y ions is needed to obtain the  cross, and \nthe O(2) movement  is required to reproduce the arcs.   \nFig. 2. Composition dep endence of the diffuse scattering.  (a) Cuts through the diffuse \nscattering from oxygen -depleted (red), as -grown (green) and annealed (blue ) Y2Ti2O7-δ along the \nline highlighted in white in (b). The diffuse neutron scattering in the ( hk7) plane from as-grown \n(b) and annealed (c) Y2Ti2O7 measured at T = 5 K. The diffuse scattering from the as -grown \nsample closely resembles that from oxygen -depleted Y2Ti2O6.79 in Fig. 1(b). There is no diffuse \nscattering from the sample annealed in oxygen.  \nFig. 3. M agnetism on the Ti ions in Y2Ti2O7-δ. (a) The DC magnetization as a function of field \nin the [111] direction at T = 2K  (a) for oxygen depleted Y2Ti2O6.79 and (b) for as -grown and \nannealed samples . The experimental data are sho wn as red circles, and the lines show  the fit s of \nBrillouin function s for the Hund’s -rule value J = 3/2 (blue) and the orbitally -quenched  value J = \n½ (green) . The saturation levels for the as -grown and annealed Y 2Ti2O7 samples are a few \nthousandths of a percent of the value for Dy 2Ti2O7, suggesting that the induced Ti magnetism \ndoes not significantly affect the monopole dynamics in spin ice.  \nFig. 4. Defect structure and m agnetism on the Dy ions in  Dy2Ti2O7-δ. (a) Diffuse neutron \nscattering from  Dy2Ti2O7-δ. reveals the same defect structure as Fig. 1(a) with Y ions replaced by \nDy ions.  (b ) DC magnetization as  a function of field in the [100 ] direction at  T = 2K for an as -\ngrown sample of Dy2Ti2O7 before and after annealing in oxygen.  The presen ce of oxygen \nvacancies reduce s the saturation magnetization. (c) The dynamic c orrelation functions from the \nAC susceptibility. The as -grown sample exhibits the long -time tail seen  previously and attributed \nto magnetic defects19, but this tail is entirely suppressed by anne aling in oxy gen.  The inset \nshows Log[ C(t)] versus time on  a  log -log scale, where a line ar regime identifies stretched \nexponential behaviour .  \nFig. 5. Monopole trapping by oxygen vacancies in Dy 2Ti2O7. An O(1) vacancy is surrounded \nby four  easy-plane spins (green -tip arrows) that are free to rotate in the plane perpendicular to the \nlocal [111] directions (green semi -transparent discs). E asy-axis s pins are shown with red -tip \narrows. The nearest -neighbour tetrahedr a are in 2in –1out and 2out –1in easy-axis spin \nconfigurations.  (a) A 3out -1in monopole is located in the next -nearest -neighbour tetrahedron. (b) When the yellow  arrow is flipped, the monopole hops to the nearest -neighbour tetrahedron , \nwhich  changes from 2in–1out to 2out–1in. The easy -plane spins a re able  to relax in response to \nthe change in orientation of the neighbouring easy -axis spin s. The energy of (b) is substantially \nlower than (a) , strongly pinning the monopole to the vacancy .   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig. 1   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \nFig. 2   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Fig. 3   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4   \n \n \n \n \nFig. 5   \n"    },    {        "title": "0806.3756v3.A_simple_mechanism_for_the_reversals_of_Earth_s_magnetic_field.pdf",        "content": "arXiv:0806.3756v3  [physics.geo-ph]  9 Feb 2009A simple mechanism for the reversals of Earth’s magnetic fiel d\nFran¸ cois P´ etr´ elis1, St´ ephan Fauve1, Emmanuel Dormy2,3, Jean-Pierre Valet3\n(1) Ecole Normale Sup´ erieure, LPS, UMR CNRS 8550, 24 Rue Lho mond, 75005 Paris, France;\n(2) Ecole Normale Sup´ erieure, LRA, 24 rue Lhomond, 75005, P aris, France.\n(3) Institut de Physique du Globe de Paris, UMR CNRS 7154,\n4, Place Jussieu, F-75252 Paris Cedex 05, France.\n(Dated: July 12, 2018)\nWe show that a model, recently used to describe all the dynami cal regimes of the magnetic field\ngenerated by the dynamo effect in the VKS experiment[1], also provides a simple explanation of the\nreversals of Earth’s magnetic field, despite strong differen ces between both systems.\nPACS numbers: 47.65.+a, 52.65.Kj, 91.25.Cw\nThe Earth’smagnetic field can be roughly described as\na strong axial dipole when averaged on a few thousands\nyears. As shown by paleomagnetic records, it has fre-\nquently reversed its polarity on geological time scales.\nField reversals have also been reported in several nu-\nmerical simulations of the geodynamo [2] and more re-\ncently, in a laboratory experiment involving a von Kar-\nman swirling flow of liquid sodium (VKS [3]). It is worth\npointing out that numerical simulations are performed in\na parameter range orders of magnitude away from real-\nistic values, and that both the parameter range and the\nsymmetries of the flow in the VKS experiment strongly\ndiffer from the ones of the Earth’s core. We thus ex-\npect that if a general mechanism for field reversals ex-\nists, it should not depend on details of the velocity field.\nThis is expected in the vicinity of the dynamo threshold\nwhere nonlinear equations govern the amplitudes of the\nunstable magnetic modes. We assume that two modes\nhave comparable thresholds. This has been observed for\ndipolar and quadrupolar dynamo modes [4] and has been\nused to model the dynamics of the magnetic fields of the\nEarth or the Sun [5]. However, in contrast to these pre-\nvious models, we consider two axisymmetric stationary\nmodes and expand the magnetic field B(r,t) as\nB(r,t) =a(t)B1(r)+b(t)B2(r)+... . (1)\nWe define A(t) =a+iband write the evolution equa-\ntion for Ausing the symmetry constraint provided by\nthe invariance B→ −Bof the equations of magnetohy-\ndrodynamics. Thisimposes A→ −A, thustheamplitude\nequation for Ais to leading nonlinear order\n˙A=µA+ν¯A+γ1A3+γ2A2¯A+γ3A¯A2+γ4¯A3(2)\nwhereµ,νandγiare complex coefficients. Equations\nof the form (2) arise in different contexts, for instance\nfor strong resonances, and their bifurcation diagrams are\nwell documented [6]. Defining A=Rexpiθ, a further\nsimplification can be made when the amplitude Rhas a\nshorter time scale than the phase θand can be adiabati-\ncally eliminated. In that case, θobeys an equation of the\nform\n˙θ=α0+/summationdisplay\nn≥1(αncos2nθ+βnsin2nθ).(3)a.BsBu\n−Bu−Bsab\nb.ab\nFIG. 1: Phase space of a system invariant under B→ −B\nand displaying a saddle-node bifurcation: (a) below the on-\nset of the saddle-node bifurcation, (square blue): stable fi xed\npoints; (red circle): unstable fixed points. (b) Above the\nthreshold of the bifurcation, the fixed points (empty symbol s)\nhave collided and disappeared, the solution describes a lim it\ncycle. Note that in (a), below the onset of the saddle-node\nbifurcation, fluctuations can drive the system from BstoBu\n(phaseBs→Bu) and initiate a reversal (phase Bu→ −Bs)\nor an excursion (phase Bu→Bs).\nThe absence of odd Fourier terms results from the invari-\nanceB→ −Bthat implies θ→θ+π. Stationary solu-\ntions of (3) disappear by saddle-node bifurcations when\nparameters are varied. When no stationary solution ex-\nist any more, a limit cycle is generated which connects\nthe former stable point θstoθs+π, i.e.,Bsto−Bs(see\nFig. 1). This elementary mechanism for reversals is not\nrestricted to the validity of (3) but results from the two\ndimensional phase space of (2) [1]. Thus, the qualitative\nfeatures of the dynamics can be captured using the sim-\nplest possible model keeping the leading order Fourier\ncoefficients α0andβ1(α1can be eliminated by changing\nthe origin θ→θ+θ0).\nSo far, we did not consider possible effects of fluctua-\ntions. The flow in the Earth’s core, as well as in the VKS\nexperiment, is far from being laminar. We can therefore\nassume that turbulent fluctuations act as noisy terms in\nthe low dimensional system that describes the coupling\nbetween the two magnetic modes. In Fig. 1a, the system\nis below the threshold of the saddle-node bifurcation and\nin the absence of fluctuation exhibits two stable (mixed)\nsolutions. If the solution is initially located close to one\nof the stable fixed points, say Bs, fluctuations can push2\nthe system away from Bs. If it goes beyond the unsta-\nble fixed point Bu, it is attracted by the opposite fixed\npoint−Bs, and thus achieves a polarity reversal. A re-\nversal is made of two successive phases. The first phase\nBs→Buin Fig. 1a is the approach toward an unsta-\nble fixed point. The deterministic dynamics acts against\nthe evolution and this phase is slow. The second phase\nBu→ −Bs, is fast since the deterministic dynamics fa-\nvors the motion.\nAt the end of the first phase, the system may return\ntoward the initial stable fixed point (phase Bu→Bs),\nwhich corresponds to an excursion. We emphasize that,\nclose enough to the saddle-node bifurcation, reversals re-\nquire vanishingly small fluctuations. To take them into\naccount, we modify the equation for θinto\n˙θ=α0+α1sin(2θ)+∆ζ(t), (4)\nfrom which we derive the evolution of the dipole by\nd=Rcos(θ+θ0).ζis a Gaussian white noise and ∆\nis its amplitude. We have computed a time series of the\ndipole amplitude for a system below the threshold of the\nbifurcation ( α1=−185Myr−1,α0/α1=−0.9, θ0= 0.3)\nand with noise amplitude ∆ //radicalbig\n|α1|= 0.2.Note that α1\nis arbitrary at this stage. Its value results from a fit of\npaleomagnetic data (see below). The dipole amplitude\nis displayed in Fig. 2 together with a time series of the\nmagnetic field measured in the VKS experiment and the\ncomposite record of the geomagnetic dipole for the past\n2 Myr. The three curves display very similar behaviors\nwith abrupt reversals and large fluctuations. We have\nchecked that similar dynamics are obtained when equa-\ntion (2) with noisy coefficients is numerically integrated.\nOne of the most noticeable features common to these\nthree curves is the existence of a significant overshoot\nthat immediately follows the reversals. In Fig. 3, the\nenlarged views of the period surrounding reversals and\nexcursions also show that this is not the case for the\nexcursions. In fact, the relativeposition of the stable and\nunstable fixed points (Fig. 1) controls the evolution of\nthe field. During the first phase, reversalsand excursions\nare similar, but they differ during the second phase. The\nsynopsisshows that the reversalsreachthe opposite fixed\npoint from a larger value and thus display an overshoot\nwhile excursions do not.\nBelow the onset of bifurcation, reversals occur very\nseldomly, which indicates that their occurence requires\nrarely cooperative fluctuations. The evolution from the\nstable to the unstable fixed point (phase Bs→Buin\nFig. 1) can be described as the noise driven escape of the\nsystem from a metastable potential well. The durations\nof polarity intervals are equivalent to the exit time and\nare exponentially distributed [7] according to\nP[T]∝exp(−T/∝angbracketleftT∝angbracketright). (5)\nThe averaged duration ∝angbracketleftT∝angbracketrightdepends on the intensity of\nthe fluctuations and on the distance to the saddle-node0  0.5 1 1.5 2−1−0.500.51\nt (Myr)D\n0 100 200 300 400−300−200−1000100200300\nt (s)B (G)\n−2 −1.5 −1 −0.5 0−2−1012\nt (Myr)Relative Dipole Intensity\nFIG.2: (Top)Time series ofthedipole amplitudefor asystem\nbelow the threshold of the bifurcation (see text for the valu es\nof the parameters) (Middle) Time series of the magnetic field\nmeasured in the VKS experiment for the two impellers ro-\ntating with different frequencies F1=22 Hz, F2=16 Hz (data\nfrom [3]). (Bottom) Composite paleointensity curve for the\npast 2 millions years, present corresponds to t=0 (data from\n[8]).\nbifurcation. For the model studied, we obtain\n∝angbracketleftT∝angbracketright=π/radicalbig\nα2\n1−α2\n0exp/bracketleftbigg\n2|α1|(2(α1+α0)/α1)3/2\n3∆2/bracketrightbigg\n,(6)\nwhich corresponds to ∝angbracketleftT∝angbracketright ≃170 kyr for the parame-\nters used in Fig. 2. We observe that deterministic pa-\nrameters are of the order of the Ohmic dissipation time\n(π/|α1| ≃17000 years) whereas much larger time scales\nare measured for ∝angbracketleftT∝angbracketrightbecause of the low noise inten-\nsity. This explainsthat the mean duration ofphases with\ngiven polarity is much larger than the one of a reversal.\nThe above predictions assume that the noise intensity\nand the deterministic dynamics do not vary in time. It is\nlikely that the Rayleigh number in the core and the effi-\nciency of coupling processesbetween the magnetic modes\nhave evolved throughout the Earth’s history. The expo-\nnential dependence of the mean polarity duration ∝angbracketleftT∝angbracketrighton\nnoise intensity implies that a moderate change in convec-\ntion can result in a very large change of ∝angbracketleftT∝angbracketright. This might\naccount for changes in the rate of geomagnetic reversals3\n−50−25 0  2500.20.40.60.81\nt (kyr)|D|\n−5 0 550100150200250300\nt (s)|B| (G)\n−80−60−40−20 0204000.511.5\nt (kyr)Relative Intensity (a.u.)\n  \nBrunnes−Matuyama (0.78 Ma)\nUpper Jaramillo (0.99 Ma)\nLower Jaramillo (1.07 Ma)\nUpper Olduvai (1.77 Ma)\nLower Olduvai (1.94 Ma)\n−50−25 0  255000.20.40.60.81\nt (kyr)|D|\n−5 0 5050100150200\nt (s)|B| (G)\n−40 −20 0 20 4000.511.5\nt (kyr)Relative Intensity (a.u.)\n  \nLachamp (40 ka)\nBlake (115 ka)\nPringle Falls (191 ka)\nLa Palma (590 ka)\nFIG. 3: Comparison of reversals and excursions in the soluti ons of the equation for θ(left), the VKS experiment (middle) (data\nfrom [3]), and in paleomagnetic data (right) (data from [8]) . Black curves represent the averaged curve, each realizati on being\nrepresented in grey.\nand very long periods without reversals (so called super-\nchrons). The reversal rate reported in [9] is displayed in\nFig. 4 together with a fit using Eq. 6 assuming a linear\nvariationintimeofthecoefficientsgoverningthedistance\nbetween BsandBu. A simple variation of one parame-\nter captures the temporal evolution of the reversal rate.\nThus, although it can be claimed that there are several\nfitting parameters, this quantitative agreement strength-\nens the validity of our model of reversals.\n0 50 100 1500123456\nCretaceous\nSuperchron\nTime τ (Myr)Estimate of reversal rate <T>−1\nFIG. 4: Variation of the reversal rate /angbracketleftT/angbracketright−1close to a super-\nchron together with the fit with equation (6). Data ( •) have\nbeen extracted from [9]. |α1|= 185 Myr−1, ∆/p\n|α1|= 0.2,\nα0/α1=−0.9(1−τ/650) for τ <80,α0/α1=−0.9(0.55 +\nτ/360) for τ >120, where τis time in millions of years.\nWenowconsiderthedifferentpossibilitiesforthemode\nB2(r) coupled to the Earth’s dipolar field B1(r). As-\nsuming that the equator is a plane of mirror symmetry,the different modes can be classified as follows: dipo-\nlar modes are the ones unchanged by mirror symme-\ntry,D→D, whereas quadrupolar modes change sign,\nQ→ −Q. From an analysis of paleomagnetic data, Mc-\nFadden et al. have proposed that reversals involve an\ninteraction between dipolar and quadrupolar modes [10].\nIn that case, B1(r) andB2(r) change differently by mir-\nror symmetry. If the flow is mirror symmetric, this im-\nplies that Eq. (2) should be invariant under A→¯A\nwhich amounts to θ→ −θ. Consequently, αn= 0 in (3)\nand no limit cycle can be generated. We thus obtain an\ninteresting prediction in that case: if reversals involve a\ncoupling of the Earth’s dipole with a quadrupolar mode,\nthen this requires that the flow in the core has broken\nmirror symmetry. This mechanism explains several ob-\nservations made in numerical simulations: reversals of\ntheaxialdipole, simultaneouswith theincreaseoftheax-\nial quadrupole, have been found when the North-South\nsymmetry of the convective flow is broken [12]. It has\nbeen shown that if the flow or the magnetic field is forced\nto remain equatorially symmetric, then reversals do not\noccur[13]. Thepossibleeffectsofheterogeneousheatflux\nat the core-mantle boundary (CMB) on the dynamics of\nthe Earth’s magnetic field have also been investigated\nnumerically [14]. Compared to the homogeneous heat\nflux, patterns of antisymmetric heterogeneous heat flux\nwere shown to yield more frequent reversals. Within our\ndescription, this appears as a direct consequence of the\nbreaking of hydrodynamic equatorial symmetry driven\nby the thermal boundary conditions. From the point\nof view of the observations, little is known on the ac-\ntual flow inside the Earth’s core. It has recently been4\nnoted that the ends of superchrons are followed by ma-\njor flood basalt eruptions and massive faunal depletions\n[15]. The authors suggested that large thermal plumes\nascending through the mantle favor reversals and sub-\nsequently produce large eruptions. In the light of our\nwork, it is tempting to associate to the thermal plumes\n(which provide a localized thermal forcing at the sur-\nface of the CMB) with an enhanced deviation from the\nflow equatorial symmetry, which results according to the\nabovedescription in anincreaseofreversalfrequency and\ntherefore ends superchrons.\nIn contrast, another scenario has been proposed in\nwhich the Earth’s dipole is coupled to an octupole, i.e.,\nanother mode with a dipolar symmetry [11]. This does\nnot require additional constraint on the flow in the core\nin the frameworkof ourmodel. In any case, the existence\nof two coupled modes allows the system to evolve along\na path that avoids B=0. In physical space, this means\nthat the total magnetic field does not vanish during a\nreversal but that its spatial structure changes.\nNumerical simulations of MHD equations [16] or of\nmean field models have displayed reversals that seem to\ninvolve “transitions between the steady and the oscilla-\ntory branch of the same eigenmode” [17]. That situa-\ntion can be obtained in the vicinity of a codimension-two\nbifurcation with a double zero eigenvalue and only one\neigenmode. This type of bifurcation also exists in our\nmodel (2) but requires tuning of two parameters µand\nν. This is not necessary for the scenario of reversals we\nhave described. Other features of reversals observed in\nnumerical simulations at magnetic Prandtl number of or-\nder one, such as mechanisms of advection/amplification\nof the field due to localized flow processes [18], are not\ndescribed by our model which requires the limit of small\nmagnetic Prandtl number (relevant to the Earth’s core).\nEquations similar to (3) have been studied in a varietyof problems, for instance for the orientation of a rigid\nrotator subject to a torque [19], used as a toy model for\nthe toroidal and the poloidal field of a single dynamo\nmode. Indeed, symmetries constrain the form of the\nequation for θeven though the modes and the physics\ninvolved are different. We emphasize that the above\nscenario is generic and not restricted to the equation\nconsidered here. Limit cycles generated by saddle-node\nbifurcations that result from the coupling between two\nmodes occur in Rayleigh-B´ enard convection [20]. A\nsimilar mechanism can explain reversalsof the largescale\nflow generated over a turbulent background in thermal\nconvection or in periodically driven flows [21]. We\nhave proposed a scenario for reversals of the magnetic\nfield generated by dynamo action that is based on the\nsame type of bifurcation structure in the presence of\nnoise. It offers a simple and unified explanation for\nmany intriguing features of the Earth’s magnetic field.\nThe most significant output is that the mechanism\npredicts specific characteristics of the field obtained\nfrom paleomagnetic records of reversals and from recent\nexperimental results. Other characteristic features such\nas excursions as well as the existence of superchrons\nare understood in the same framework. Below the\nthreshold of the saddle-node bifurcation, fluctuations\ndrive random reversals by excitability. We also point out\nthat above its threshold, the solution is roughly periodic.\nIt is tempting to link this regime to the evolution of\nthe large scale dipolar field of the Sun (which reverses\npolarity roughly every 11 years). Recent measurements\nof the Sun surface magnetic field have shown that two\ncomponents oscillate in phase-quadrature [22]. This\nwould be coherent with the oscillatory regime above the\nonset of the saddle-node bifurcation if these components\ncorrespond to two different modes.\nWe thank our colleagues from the VKS team with\nwhom the data published in [3] have been obtained.\n[1] F. P´ etr´ elis and S. Fauve, J. Phys. Condens. Matter 20,\n494203 (2008).\n[2] See for instance, P. H. Roberts and G. A. Glatzmaier,\nRev. Mod. Phys. 72, 1081 (2000).\n[3] M. Berhanu et al., Europhys. Lett. 77, 59001 (2007).\n[4] P. H. Roberts, Phil. Trans. Roy. Soc. A 272, 663 (1972).\n[5] E. Knobloch and A. S. Landsberg, Mon. Not. R. Astron.\nSoc.278, 294 (1996); I. Melbourne, M. R.E. Proctor and\nA. M. Rucklidge, Dynamo and dynamics, a mathematical\nchallenge , Eds. P. Chossat et al., pp. 363-370, Kluwer\nAcademic Publishers (2001).\n[6] V. Arnold, Geometrical Methods in the Theory of Ordi-\nnary Differential Equations (Springer-Verlag, 1982).\n[7] N. G. Van Kampen, Stochastic Processes in Physics and\nChemistry (North Holland, Amsterdam, 1990).\n[8] J.-P. Valet, L. Meynadier and Y. Guyodo, Nature 435,\n802 (2005).\n[9] P. L. McFadden et al., J. Geophys. Research 105, 28455\n(2000).\n[10] P. L. McFadden et al., J. Geophys. Research 96, 3923(1991).\n[11] B. M. Clement, Nature 428, 637 (2004).\n[12] J. Li, T. Sato and A. Kageyama, Science 295, 1887\n(2002).\n[13] N. Nishikawa and K. Kusano, Physics of Plasma 15,\n082903 (2008).\n[14] G. Glatzmaier et al., Nature 401885 (1999).\n[15] V. Courtillot and P. Olson, Earth and Planetary Science\nLetters260, 495 (2007).\n[16] G. R. Sarson and C. A. Jones, Phys. Earth Planet. Int.\n111, 3 (1991).\n[17] F. Stefani and G. Gerbeth, Phys. Rev. Lett. 94, 184506\n(2004); F. Stefani and al., G. A. F. D. 101, 227 (2007).\n[18] J. Aubert, J. Aurnou and J. Wicht, Geophys. J. Int. 172,\n945 (2008).\n[19] P. Hoyng, Astronomy & Astrophysics 171348, (1986).\n[20] L. S.TuckermanandD.Barkley, Phys.Rev.Lett. 61, 408\n(1988); J. H. Siggers, J. Fluid Mech. 475, 357 (2003).\n[21] R. Krishnamurti and L. N. Howard, Proc. Natl. Acad.\nSci.78, 1981 (1981); J. Sommeria, J. Fluid Mech. 170,5\n139(1986); B. Liu andJ. Zhang, Phys.Rev.Letters, 100,\n244501 (2008).\n[22] R. Knaack and J.O. Stenflo, Astronomy & Astrophysics438, 349 (2005)."    },    {        "title": "2303.15336v1.Magnetic_manipulation_of_superparamagnetic_colloids_in_droplet_based_optical_devices.pdf",        "content": " 1 Magnetic manipulation of superparamagnetic colloids in droplet -based optical devices  \nI. Mattich¹, J. Sendra2, H. Galinski2, G. Isapour3, A.F. Demirörs¹, M. Lattuada4, S. Schuerle5, A. R. \nStudart¹  \n1 Complex Materials, Department of Materials, ETH Zürich, 8093 Zürich, Switzerland  \n2 Laboratory for Nanometallurgy, Department of Materials, ETH Zürich, 8093 Zürich, Switzerland  \n3 Department of Mechanical Engineering, MIT, USA  \n4 Department of Chemistry, Uni versity of Fribourg, Switzerland  \n5 Department of Health Sciences and Technology, Institute for Translational Medicine, ETH Zürich, \n8093 Zürich, Switzerland  \n \nAbstract  \nMagnetically assembled superparamagnetic colloids have been exploited as fluid mixers, swimmers \nand delivery systems  in several  microscale  applications . The encapsulation of such colloids in droplets \nmay open new opportunities to build magnetically control led displays and optical components. Here, \nwe study the assembly of superparamagnetic colloids inside droplets under rotating magnetic fields  and \nexploit this phenomenon to create functional optical devices . Colloids are encapsulated in \nmonodisperse drople ts produced by microfluidics  and magnetically assembled into dynamic two-\ndimensional clusters. Using a n optical microscope  equipped with a magnetic control setup , we \ninvestigate the effect of the magnetic field strength and rotational frequency on the size , stability  and \ndynamics  of 2D colloidal clusters  inside droplets . Our results show that cluster size and stability depend \non the magnetic forces acting on the s tructure under the externally imposed field. By rotating the cluster \nin specific orientations, we illustrate how magnetic fields can be used to control the effective refractive \nindex  and the transmission of light through the colloid -laden droplets, thus demonstrating the potential \nof the encapsulated colloids  in optical applications . \n \nIntroduction  \nMagnetic  fields offer a powerful means to assembl e or manipulat e colloids for a broad range of fields , \nfrom microrobotics to medicine , biotechnology  and manufacturing . 1-4 In microrobotics, magnetic stimuli \nhave been used to control the locomotion of miniaturized robots 1, 5 and to design substrates for the \nremote manipulation of small -scale objects. 6-8 Locally induced  hyperthermia  and directed transport \nusing iron oxide particles are well-known example s of potential  application s in medicine, in which  a \nmagnetic stimulus is employed as a non-invasive tool for drug delivery and cancer therapy . 2, 9 In \nmanufacturing, magnetic fields have been exploited for the fabrication of composites with controlled \norientation of reinforcing particles  3, 10 or the formulation of reversible adhesi ves. 11 Moreover, analytical \nbiotechnological assays  often rely on magnetic fields to recover surface -functionalized colloid s in  2 biological and chemical separation processes . 4, 12 In many of these applications, superparamagnetic \niron oxide nano particles  (SPIONs)  are used as the magnetically respons ive colloids. 9 \nSuspensions of superparamagnetic particles exhibit very rich phase behaviour and dynamics when \nsubjected to time -varying magnetic fields. 13, 14 Their main advantage lies on the fact that the se particles \nact as magnetic dipoles exhibiting a single m agnetic domain  only in the presence of an external \nmagnetic field. This allows for switching the colloidal state of the  suspension from a fluid dispersion to \nhierarchical assemblies of particles interacting through  attractive  and repulsive dipol ar forces . \nDepending on the concentration of particles and the type of magnetic stimulus applied, \nsuperparamagnetic colloids can assemble into chains, two -dimensional clusters or three -dimensional \nhierarchical networks. 13 Such colloidal structures have been considered for several prospective \napplications as microfluidic  mixers  15, microswimmers  and micropumps  16, cargo transporters  17, and \nartificial ciliated surfaces . 13, 18 Despite these potential  applications and our advanced understanding of \ntheir magnetic response, the directed assembly of superparamagnetic particles compartmentalized \ninside droplets remains to be investigated and technologically exploited.  \nThe compartmentalization of colloidal particles inside droplets and capsules is a n effective  approach to \ncontrol the release  of molecules in delivery systems  19 and to program the  brightness  of pixels in \ncommercially available electronic books . 20, 21 In electronic paper, the encapsulated particles are \nmanipulated using an external electrical field to change optical properties locally across large areas. \nThe applied electric al field drives the motion of black and white encapsulated particles of opposite \ncharges to different regions of the capsule , thus  allowing for electrical control of the local brightness . 20 \nIn nature , pigment  granule s compartmentalized in chromatophore cells are also used by cephalopods \nto change color on demand. 22, 23 In this case, color is generated through the displa cement of granules \nvia controlled contraction or expansion of encapsu lating sacks . 22 Such a strategy has inspired the \ndevelopment of magnetically controlled smart windows.  24 These biological and technolog ical examples \ndemonstrate the potential of controlled colloidal manipulation in compartments as an enticing approach \nto create novel functionalities. Given the increasing availability of multiferroic materials that can \ngenerate magnetic fields using low-power electrical  input,  25, 26 the use of magnetic torques and forces  \nto drive and control the assembly of colloids in droplets is a n interesting and technologically relevant \napproach that calls for furthe r scientific research . \nHere, we study the assembly and manipulation of superparamagnetic colloids inside droplets driven by \na time -varying external magnetic field. The magnetically responsive colloids are encapsulated in water -\nin-oil droplets through a microfluidic emulsification  approac h. After encapsulation in monodisperse \ndroplets, the particles are assembl ed and manipulated using  a tunable rotating magnetic field. Next, \nthe response of the colloids to the external field is studied by optical microscopy imaging of multiple \ndroplet arrays. Finally, we demonstrate how superparamagnetic particles in droplets can be potentially \nexploited as ma gnetically controlled optica l shutters  and microlens arrays with tunable focal length . \n \n  3 Results and Discussion  \nThe assembly of colloids under rotating magnetic fields is experimentally studied by encapsulating \nmonodisperse colloidal particles inside monodisperse water droplets suspended in a continuous oil \nphase  (Figure 1) . The colloids consist of polystyrene particles with an average size of  480 nm loaded  \nwith superparamagnetic iron oxide nanoparticles (SPIONs , 10-20 nm ) to render them magnetic ally \nrespons ive. Due to the high density of carboxylic acid groups (COOH) on the surface (> 50 μmol/g), the \nparticles become negatively charged in water at neutral and alkaline pHs.  1-decanol is used as t he \ncontinuous phase , whereas the aqueous droplet is prepared from do uble deionized water . The \nconcentration of colloidal particles inside the droplet is varied between 0.01 and 0.78 vol% to generate \nclusters that are  large enough for visualization by optical microscopy while keeping the system \nsufficiently dilute to facili tate assembly . \nTo study the assembly of the particles into two -dimensional clusters , we apply a time -varying  external \nmagnetic field to a monolayer of colloid -laden droplets using two types of magnetic stimuli (Figure  1a-\nc). In a first approach, an external magnetic field is used to drive the assembly of 2D colloidal clusters. \nTo this end, the applied field is rotated within the plane orthogonal to the viewing direction  (xy-plane) , \nthus facilitat ing visualization of the cluster assembly  process . In a second step, the same  rotating  \nmagnetic field is first used  to form the 2D clusters  and is  then employed to manipulate the colloidal \nassembly within the droplet. This  manipulation  is accomplished by applying the rotating  field in a plane \nparallel to the viewing angle and revolving it around the viewing axis to r otate the 2D colloidal cluster.  \nTo study the dynamics of the colloidal  assembly process  and the magnetic manipulation of the resulting \nclusters , the droplets need to be suff iciently stable against coalescence and coarsening  events . \n  4  \nFigure  1. Microfluidic e ncapsulation  and magnetically driven assembly of colloidal particles inside \ndroplets. (a) Illustration of colloids loaded with superparamagnetic nanoparticles con fined  in water -in-\noil droplet s. When no magnetic field is applied, the colloidal particles undergo random Brownian motion. \n(b) Schematic of the assembly of colloidal particles into  large two-dimensional clusters  under a rotating \nmagnetic field with a frequency of 125 Hz applied within the x y-plane. (c) After cluster formation, the \naxis of rotation  of the rotati ng field can be  slowly adjusted  until it lies parallel to the y-axis, thus shifting \nthe color of the droplet from dark to translucent . At this point, the rotating field plane is  revolved around \nthe z -axis at a certain pre-defined speed  to study the mechanical stability of the cluster . (d) Overview \nof the flow -focusing microfluidic device, highlighting the junctions and chamber where droplets are \nformed and collected.  (e) Cartoon illustrating the stabilization of the water -oil interface by surface -\nmodified silica nanoparticles and surfa ctant molecules.  (f) Magnification of the first junction, indicating \nthe laminar co -flow of the aqueous phases containing the superparamagnetic colloids (center) and the \nsilica nanoparticles (close to walls). ( g) Magnification of the second junction, where  water -in-oil droplets \nare formed by flow -induced dripping. ( h) Magnification of the observation chamber at the end of the \nserpentine channel. The droplets are stable and do not coalesce upon physical contact. The particles, \nfluids and surfactants were fal se-colored in the images to facilitate visualization.  \n \n 5 Monodisperse , stable droplets were generated in a flow -focusing microfluidic device  using Pickering \nemulsions as templates  (Figure 1d -h). The Pickering emulsions were formed by adsorbing silica \nnanoparticles at the water -oil interface. In this microfluidic emulsification approach, water droplets are \nformed in a continuous oil phase through a flow -induced dripping mechanism. 27 Emulsification occurs \nby injecting  the aqueous inner phase and the oil outer phase through separate input channels  of the \nmicrofluidic device  (Figure 1d ). The channels are designed to form a n aperture , at which the dripping \nphenomenon takes place  (Figure 1 g). The aqueous inner phase consists  of a suspension of \nsuperparamagnetic  colloidal particles in water , whereas 1-decanol  is used as the outer oil phase.  \nTo facilitate the Pickering stabilization of  the oil -water interface, a second coaxial aqueous phase  is \nadded in the device during emulsification. Such liquid phase  comprises  the silica nanoparticles \nsuspended in water  and is injected along side the innermost aqueous phase  in a co -flow configuration. \nSuch a n approach enables the delivery of the silica nanoparticles very close to the oil -water interface \nformed upon dripping.  The silica nanoparticles were partially hydrophobized with hexyl amine to favor \ntheir adsorption at the oil -water interface . Interfacial adsorption of the silica particles is aided by a \nsilicone -based surfactant  (ABIL 90 EM)  added to the oil phase.  \nDroplet microfluidics enable d precise control over the emulsion droplet size, monodispersity,  droplet \nsurface coverage , and concentration of encapsulated superpara magnetic particles . By tuning the \nconcentration of surfactant and modified silica nanoparticles, s table Pickering emulsions were obtained \nat the end of the serpentine channel  and outside  of the microfluidic device . This high stability was crucial  \nto study the assembly and manipulation of the encapsulated superparamagnetic nanoparticles using \nan external magnetic field.  \nMagnetic fields with a rotational frequenc y of 125 Hz were applied to a monolayer of droplets to study \nthe assembly and manipulation of colloidal clusters  (Figure 2 ). Because of their negative surface \ncharges, the colloids are initially electrostatically stabilized and uniformly dispersed within the droplets \n(Figure 2a ). The use of a magnetic setup with eight electromagnetic coils and five degrees of freedom \n28, 29 allowed us to gain full control over the magnetic field  applied to the colloidal particles . Comparative \nexperiments with particles suspended in water showed that their encapsulation inside droplets is crucial \nto prevent cluster -cluster interactions and thus allow  for the systematic  investigation of the assembly \nand dynamics of individual 2D clu sters. The high monodispersity of the droplets led to crystallization in \nordered domains, th ereby  facilitating the imaging of the colloidal assembly process in multiple droplets \nsimultaneously.  \nOur experiments revealed that the initially dispersed colloida l particles readily assembl e into 2D clusters \nwhen subjected to a rotating magnetic flux density  (𝐵) with amplitude  in the range of  6-11 mT and a \nfrequency  (𝑓) of 125 Hz (Figure 2 ). The magnitude of the  field applied corresponds to a magnetic field \nstrength between  1132 and 2076 A/m.  The assembly process is governed by magnetic interactions \nbetween the field -induced magnetic dipoles within individual particles. The superparamagnetic nature \nof the colloids  leads to induced dipoles within the plane of the rotating field , which attract each other to \ninitially form small clusters of colloidal particles. Over time,  the small clusters merge into a large two-\ndimensional assembly  with one or two layers of particles, depending on the initial colloid concentration.  6 While the  dipole -dipole interaction s in the bulk of the cluster  cancel each other,  the particles positioned \nat the edge of the cluster experience a net magnetic moment du e to the absence of neighboring  colloids  \noutside the cluster . 14 The dipolar -dipolar attractive interactions between such particles lead to a line \ntension  (𝜆), which is  a 2D analogue of the surface tension of three -dimensional matter . \nThe emergence of a line tension allows us to magnetically control the size and mechanical properties \nof the assembled cluster. To demonstrate this, we measured the average diameter of the clusters as a \nfunction of the applied magnetic field for a constant frequency  of 125 Hz  (Figure 2c ). The experiments \nreveal a linear decrease in the average cluster size with the applied field. This result can be interpreted \nby analyzing  the effect of the magnetically induced line tension on the structure of the colloidal cluster. \nThe line tension  arising from the rotating magnetic field pulls the colloidal particles together  into a more \ncompact closely packed structure. This compressive force is counteracted by electrostatic and/or steric \nrepulsive forces between adjacent particles , which eventually establishes a new equilibrium at smaller \ninterparticle  distances . \nThe smaller distance between colloidal particles affects the mechanical properties of the cluster. At \ncloser distances, stronger interactions are expected between the colloidal particles, enhancing the \nstiffness of the cluster. To gain an ins ight into the effect of the magnetic field on the cluster stiffness, \nwe call upon simple scaling relations previously proposed. The shear modulus of the cluster ( 𝐺′), taken \nhere as a measure of stiffness,  can be approximated by 𝜆𝑅⁄. 14 Earlier work has shown that the tension \nline 𝜆 is expected to scale with the square of the fie ld amplitude: 𝜆~𝐻2. 14 Combining these  relation s, \nwe conclude that the stiffness of the cluster should scale with  the ratio 𝐻2𝑅⁄, a quantity that can be \ndirectly calculated from our experimental data. By plotting  experimental values of  𝐻2𝑅⁄ as a function of \n𝑅, one finds that the stiffness of the cluster should increase with the reduction in cluster size , as \nillustrated in  Figure 2d . Indeed, decreasing the cluster size by a few percent results in a 5 -fold increase \nin the expected stiffness . This indicates the possibility of using magnetic fields to control the stiffness \nof the 2D colloidal cluster s.   7  \n \nFigure 2. Magnetic assembly of superparamagnetic particles into 2D colloidal clusters  inside droplets . \n(a) Optical microscopy image of an a rray of monodisperse water -in-oil droplets with superparamagnetic \nparticles homogeneously suspended in the aqueous phase. (b) Particles assemble into 2D clusters \nwhen a magnetic field  rotating at 125 Hz is applied within the xy -plane. (c) Effect of the magnetic field \nstrength  (𝐻) on the diameter ( 2𝑅) of the assembled colloidal clusters . (d) Impact of the clus ter size on \nthe ratio 𝐻2𝑅⁄, which is taken as an indicative  value for the stiffness of the colloidal cluster.  \n \nIn addition to the formation of 2D assemblies with controlled size and stiffness, t he rotating field can \nalso be used to manipulate the as -formed colloidal clusters encapsulated within the droplets. \nManipulation of the dynamically assembled 2D clusters is possible by moving the plane of rotation  while \nkeeping the in-plane frequency active (Figure 3 ). To demonstrate such magnetic control capabilities, \nwe performed experiments in which 2D clusters are first assembled with a magnetic field rotating in the \nxy-plane . Afterwards, t he axis of rotation of the clusters  is turned  to become parallel to  the y -axis. \nFinally, their stability is studied by revolving the clusters around the z -axis at increasing angular speeds, \nstarting from 0.1 rad/s (Figure 3a,b).  Optical microscopy imaging reveals that the encapsulated clusters \ncan be effectively rotated inside the drop lets without losing their two-dimensional morphology.  The \nmotion of the cluster into a new orientation changes dramatically the transmittance of light across the \ndroplet  and thereby the optical properties of the entire droplet array . \n 8 The ability of the cluster to remain stable during magnetic manipulation de pends on the applied \nrevolving speed  (𝜔) and the external magnetic field strength  (𝐻). We experimentally observed that \nrevolving speeds above a critical value (𝜔𝑐) lead to extensive fragmentation of the 2D clusters into \nsmaller colloidal assemblies ( Figure 3c  and Movie S1 ). To better understand this fragmentation \nphenomenon, we measured the critical revolving speeds for clusters of different sizes subjected to \ndistinct external rotating magnetic field strengths (Figure 3d ). Our results sho w that the critical revolving \nspeed increases non-linearly  with the strength of the rotating field applied  (𝐻). For a cluster intermediate \nsize of 70  m, we find that the 𝜔𝑐 value doubles when the magnetic field increases from 1321  to 2265  \nA/m. Moreover, smaller clusters can be revolved without fragmentation  at a higher velocity compared \nto their larger counterparts.  Under a magnetic field  strength  of 673 A/m, the threshold revolving speed \ndrops from 3.37 to 1.42 rad/s if the cluster size is enlarged from 50 to 10 0 m. \nThe effect of the magnetic field strength and cluster size on the critical revolving speed (𝜔𝑐) can be \ninterpreted by considering  the forces acting on the cluster during magnetic manipulation. Earlier studies \nhave shown that  a balance between the magnetic torque  𝑇𝑚 arising from the applied  field and the \nreactive viscous torque  𝑇𝜂 exerted by the liquid governs the dynamics of anisotropic assemblies  and \nparticles  under rotating magnetic fields .13, 30 , 31 For steady -state rotation  of our system , a net balance of \nthese forces is achieved such that 𝑇𝑚+𝑇𝜂=0, leading to the following expression : \n𝜔−𝜔𝑐𝑠𝑖𝑛2𝜃=𝑑𝜃\n𝑑𝑡     (1) \nwith \n𝜔𝑐=𝜇0𝜒𝑐𝑙𝑢𝑠𝑡𝑒𝑟2\n12𝜂0(𝑓\n𝑓0⁄)(1+𝜒𝑐𝑙𝑢𝑠𝑡𝑒𝑟 )∙𝐻02     (2) \nHere,  𝜃 represents the phase lag  between the plane of the rotating magnetic field and the cluster’s long \naxis, while 𝜔 is the fixed magnetic field angular speed  applied . 𝜇0 indicates  the magnetic permeability \nof free space, 𝜒𝑐𝑙𝑢𝑠𝑡𝑒𝑟  is the effective magnetic susceptibility of the cluster  and 𝐻0 is the applied magnetic \nfield. 𝜂0 is the viscosity of the  fluid surrounding the cluster  and (𝑓\n𝑓0⁄) represents the Perrin friction \nfactor.  The frequency of rotation of the cluster is coupled with the magnetic field rotational frequency \nwhen the phase lag is constant  over time  (𝑑𝜃\n𝑑𝑡=0). Assuming that inertial and gravitational effects can \nbe neglected, the anisotropic object will follow the rotating magnetic field if its frequency 𝜔 lies below \nthe critical value, 𝜔𝑐. When this critical condition is surpassed, the rotating object cannot  keep up with \nthe speed of the rotating field, resulting in structural instability and loss of synchronous motion.  \nThe theoretical analysis above predicts that the critical angular speed  (𝜔𝑐) measured in our experiments \nshould scale with the square of the applied magnetic field strength  (𝐻0): 𝜔𝑐=𝑘𝐻02, with 𝑘 depending on  \nthe particle magnetization, cluster diameter and fluid viscosity . We test this prediction by comparing  the \nanalytical model derived in Eq. 2 to our experimental data  (Figure 3d ). In this comparison, we estimate \nthe parameter 𝑘 assuming values for 𝜇0, 𝜒𝑐𝑙𝑢𝑠𝑡𝑒𝑟 , 𝜂0, and 𝑓\n𝑓0⁄ that are known for our system or available \nin the literature (supporting information).  The good agreement observed between experiments and the  9 analytical model suggest that the dynamics of the rotating clusters is captured well by the torque  \nbalance propo sed in the literature  (Figure 3d) . The decrease in the 𝑘 value with increasing cluster size  \nreflect s the stronger viscous forces exerted on larger clusters compared to smaller counterparts. This \nexplains the lower critical rotational speeds  experimentally needed for the fragmentation of larger \nclusters.  Fragmentation lowers the viscous forces acting on the newly formed smaller clusters, which \nare stable enough to synchronously rotate with the applied magnetic field ( Movie S 1). Catastrophic \nfragmentation events were more difficult to detect for smaller cluster sizes . This may explain the \ndiscrepancy for smaller cluster diameters with our analytical model.  \n \n \nFigure 3. Manipulation and fragmentation dynamics of magnetically assembled 2D colloidal  clusters . \n(a) Optical microscopy image depicting an a rray of encapsulated clusters formed in the presence of a \nmagnetic field rotating within  the xy -plane. (b) Reorientation of the clusters shown in (a) achieved by \nturning  the axis of rotation  around the x-axis. (c) Fragmentation of the clusters observed when the \nrevolving speed ( 𝜔) of the rotating field is increased above the critical value, 𝜔𝑐. In this experiment, the \nplane of the rotating magnetic field was revolved around the z -axis. (d) Effect of the magnetic field \nstrength on the critical revolving speed ( 𝜔𝑐) for clusters of differ ent diameters  d. The continuous lines \nrepresent the analytical solution  𝜔𝑐=𝑘𝐻02, with  𝑘 values calculated from parameters that are known for \nour system ( Table S1, supporting information) . \n \n 10 The formation and manipulation of 2D colloidal clusters under a rotating field opens new opportunities \nto magnetically control the optical properties of individual droplets. The use of magnetic fields to control \nthe refractive index and light transmittance of the droplets can potentially be exploited f or the \ndevelopment of active optical components, such as shutters and microlens arrays. To illustrate this \npotential, we first study the transmittance of light across monodisperse droplets under magnetic fields \nand later assemble similar droplets into a po lymer matrix to create magnetically controlled soft optical \nshutters.  \nIn the first demonstrator, a colloid -laden water -in-oil droplet is used as a microlens with magnetically \ncontrolled effective refractive index. The effective refractive index is affected  by the spatial distribution \nand assembly of the superparamagnetic colloids inside the droplet. By changing the effective refractive \nindex of the droplet, it is possible to actively control the focal length of the microlens.  In contrast to \ninspiring previo us research on droplet -based lens es, 32 the focal length in our demonstrator is actively \ncontrolled by the external magnetic field.  To quantify this effect, we measured the change in the focal \nlength of a single microlens  upon exposure to a magnetic field parallel or perpendicular to the incoming \nlight (Figure 4 ). For these measurements, a table -top optical setup was built around the electro -\nmagnetic coils that apply the exte rnal magnetic field  (Figure 4d,e). A sample holder was 3D printed to \nhost the monolayer of droplets loaded with superparamagnetic colloids (Figure S1).  \nFor the experiment, a monochromatic collimated laser beam with a wavelength of 532 nm illuminates \nthe sample and the refracted beam intens ity profile is mapped around the resulting focal point . To \nenable the assembly and manipulation of 2D colloidal clusters inside the individual droplets, a magnetic \nfield rotating at 125 Hz was applied following the protocol stated above. Before the magneti c field is \napplied , the droplet contains a homogeneous dispersion of particles (state 1). The change in focal length \nof the droplet -based microlens was measured for two magnetically induced conditions: 2D cluster ing \nparallel  to the input beam  (state 2) and 2D clustering perpendicular to the input beam (state 3) . \nThe experimental results show that we can discretely change the focal length depending on the state \nof the superparamagnetic colloids ( Figure 4 a-c). Because of the lower refractive index of water \ncompared to the oil, the focal point for all the investigated states is positioned between the light source \nand the droplet -based microlens. The magnetically  induced assembly of dispersed particles  into \ncluste rs directly affects the experimentally measured focal length . To evaluate this effect, we set as \nreference the focal length created by droplets with parallel -aligned clusters (state 2, z=0 in Figure 4b) \nand report the shift in focal length observed when th e magnetic field is switched off (state 1, Figure 4a) \nand when clusters are assembled perpendicular to the incoming light (state 3, Figure 4c).  In the \nreference state (2) , the cluster is oriented parallel to the beam and interacts minimally with the incoming \nlight, leading to droplets with optical properties dominated by the aqueous phas e. Switching off the field \n(state 1) leads to thermal randomization of the particles and a shift in focal length of 92 µm. The focal \nlength change reduces to 84 µm, if  the cluster is magnetically oriented perpendicular to the input beam \n(Figure 4 b,c). \nTo establish a quantitative correlation between the change of the microlens focal length and the \nassembly of superparamagnetic colloids in the droplet , we applied  a ray tracing analytical model and  11 finite element simulations to our optical system. Using the transfer matrix method, the analytical model \npredicts the inverse of the focal length (1𝑓⁄) to depend on the effective refractive index of the droplet, \nthe refractive index of the oil and the radius of the droplet (see SI  and Figure S2 ). For a given oil and \ndroplet size, the analytical model indicates that the focal length of the microlens should de crease \ncontinuously with the effective refractive of the droplet , 𝑛𝑒𝑓𝑓 (Figure 4f). To complement this theoretical \nmodel, we performed finite element simulations on a representative droplet illuminated with \nmonochromatic collimated light (Figure 4g  and Figure S 3). The simulation s confirm  that the droplet \ndiverges the incoming light , leading to  a focal point positioned between the droplet and light source, as \nobserved in the experiment s (Figure 4h and Figure S3). Changes in the refractive index of the droplet \nresults in a shift of the simulated focal length , in close agreement to the analytical model (Figure 4f). By \nexploring other materials and droplet types, the simulations provide useful guidelines for the design of \nthe droplet -based microlens (Figure S 4).  \nThe simulations and the analytical model suggest that the focal length change observed when the \nparallel -aligned clusters (state 2) disperse into homogeneous ly distributed colloids (state 1) can be \nexplai ned by a change in the effective refractive index of the droplet. Assuming that the refractive index \nof the droplet with the parallel -oriented cluster is dominated by water ( 𝑛𝑒𝑓𝑓=1.330, red circle in Figure \n4f), our simulation predicts that the effective refractive index of the droplet should  increase to 1.345 for \nthe homogeneously suspended  colloids (dashed line , Figure 4f). While the magnetic nature of the \nsuspended colloids  prevents us from estimat ing the effective refractive index of the homogeneous \ndroplet , the higher refractive index of polystyrene ( 𝑛=1.58) compared to water  makes our physical \ninterpretation  a reasonable qualitative explanation for the experimental observations . Because of the \nheterogeneous  nature of droplets with p erpendicularly aligned clusters, we expect the focal length \nchange induced  by this configuration to arise from  the complex interaction s of the beam with the \nparticles within the oriented cluster.  \n \n  12  \n \nFigure 4. Active microlenses comprising water -in-oil droplets loaded with magnetically responsive \nsuperparamagnetic colloids. (a-c) Maps of the light intensity spatial distribution resulting from the \nrefraction by droplets containing colloids in different configurations: (a) homogeneously dispersed \nsuperparamagnetic nanoparticles  (no magnetic field) , (b) colloidal cluster oriented with its plane parallel \nto the laser beam, (c) colloidal cluster oriented with its plane perpendicular to the laser beam. (d) \nCartoon showing a Pickering stabilized droplet containing a monolayer  cluster  of magnetic ally \nresponsive nanoparticles. Only half of the surface of the droplet is covered with interfa cially adsorbed \ncolloids to better visualize the encapsulated colloidal cluster . (e) Rendering of the optical setup used to \nmeasure the change in focal length of individual droplet -based lenses  under the action of a magnetic \nfield. The individual components of the setup a re: (1) magnetic coils, (2) laser beam, (3) sample, (4) UV \nlens mounted on a z -axis translational lens mount and (5) detector. (f) Effect of the effective refractive \nindex of the droplet on the change in focal length using a colloid -free water droplet as r eference . \nExperimental data is shown in red, green triangles indicate results from ray tracing simulations and the \nblack line is the analytical solution.  (g) Illustration of the three -dimensional ray optics model used for \nthe finite element simulations, including boundary and ray release conditions.  (h) Simulated ray \ntrajectories ( 𝜆=532 nm) for a homogeneously dispersed suspension of particles with effective  refractive  \nindex of 𝑛𝑒𝑓𝑓 =1.34 50, corresponding to a change in focal length of -92 µm as measured in the \nexperiment . \n 13 In addition to microlens arrays, the colloid -laden droplets can also be exploited as magnetically \ncontrolled optical shutters. We provide an example of such an application by preparing a demonstrator \nthat consists of a polymer monolith containing a monolayer of water droplets  (Figure 5 a,b). The monolith \nis made by using a n oil continuous phase that can be polymerized after the assembly of the microfluidic \ndroplets  (Figure S5) . In this case, a high concentration of superparamagnetic particles of 0.78 vol%  is \nused in order to amplify the difference in light transmission through the monolith when the sh utter is in \nthe ON  and OFF  state s. A rotating magnetic field  is applied to assemble the particles into anisotropic \nstructures  aligned either perpendicular or parallel to the incoming light , thus switching the shutter ON \nor OFF, respectively . By using a low  frequency up to  5 Hz, we expect the particles to assemble into \noriented clusters under the applied field.  \nThe performance of the optical shutter was assessed by measuring the evolution of the transmitted light \nwhile the plane of the rotating magnetic field was periodically changed between the parallel and \nperpendicular orientations . Experiments were carried out under an optical microscope using magnetic \nflux density  magnitudes in the range 1 – 20 mT . The transmitted light intensity was obtained directly \nfrom optical microscopy images using image analysis software (Figure 5 c,d). \nOur results reveal that the transmitted light intensity can change up to two -fold when the plane of the \nrotating magnetic field is switched between the perpendicular and parallel orientations ( Figure 5 c). \nImages of the droplet when exposed to the parallel field orientation confirm the assembly of the particles \ninto multiple anisotropic structures, the al ignment of which favors light transmission. When the field is \nchanged to the perpendicular orientation, the anisotropic colloidal structures are no longer visible due \nto complete blockage of the incoming light.  The oscillations in transmitted light observe d in the OFF \nstate match the frequency of the applied magnetic field  (Figure 5 d and Figure S6 ) and probably result \nfrom the oblate geometry of the colloidal clusters . This suggests that the magnetically assembled \nstructures oscillate around the plane of th e magnetic field despite being locked in the imposed  \norientation.  \nThe timescale of the light  intensity changes depends on the switching direction  and on the magnitude \nof the magnetic field . Switching the shutter from OFF to ON (darkening effect) happens at a speed that  \ndepends on the magnetic field applied.  The time it takes to switch from the maximum to the minimum \nintensity value for field flux densities of 5, 10, 15 and 20 mT are 0.07, 0.15, 0.25 and 0.25 seconds, \nrespectively . Larger applied magnetic fields lead to higher maximum intensity values, resulti ng in longer \ntime intervals  for them to return to their baseline transmittance levels.  The effect of the magnetic field \nstrength on the switching time is reversed when we consider the switching from the ON to the OFF \nstates (whitening effect). In this case , the duration necessary to increase the intensity to a certain fixed \nlevel decreases with the magnitude of the  magnetic field. When a magnetic flux density of 10 mT is \napplied, it takes 2 seconds to increase the light transmittance above  20%. However, whe n a fie ld \namplitude of 20 mT is used, the same intensity value can be attained in only 0.45 seconds. The \nstrongest whitening effect  leads to a 30% rise in light transmittance  from the baseline . This is  achieved \nthrough the application of a magnetic flux density of 20 mT and a rotational frequency of 1 Hz (s ee SI \nand Figure S6). The long  timescales needed to transition from the ON to the OFF state might result  14 from the relatively slow process of merger of small initial particle chains into larger anisotropic \nstructures . These experimental results provide insights into the several parameters governing the \ndynamics of the magnetic assembly process. Future research should be dedicated to further explore \nthis parameter space and thus tune the switching timescales to meet technological demands.  \n \n \nFigure 5. Droplet -based optical shutter controlled by magnetic fields. (a) Photograph of a polymer \nmonolith containing colloid -laden water droplets . (b) Rendered sketch showing the manufacturing steps \nof the optical shutter. (c) Optical microscopy images displaying  the color of the droplets when the shutter \nis ON (left) and O FF (right). A flux field of 20  mT is applied for the OFF and ON state s. Scale bar is 500 \nµm. (d) Intensity of light transmitted across the droplet array when the shutter is subjected to \nconsecutive ON/OFF cycles. The ON state is reached when the magnetic field is rotated within the xy -\nplane , whereas the OFF state is achieved w hen the applied field rotates within the yz -plane.  The \nexperiment al data depict the effect of the magnetic f lux density  at a fixed frequency of 2 Hz. \n \nConclusions  \nSuperparamagnetic colloids in side droplets can be harnessed to create droplet -based microlenses and \noptical shutters driven by low -magnitude  magnetic fields  (6-12 mT) . In the presence of a rotating \nmagnetic field, the colloids assem ble into tunable two -dimensional clusters  driven  by the line tension \narising from the alignmen t of magnetic dipoles at the edge of the cluster . The cluster comprises one \nlayer of superparamagnetic colloids and can easily vary between 50 and 250  𝜇m in diameter, depending \non the droplet size  and the initial particle concentration in the droplet. Increasing the applied magnetic \nfield strength enhances the line tension, resulting in smaller and stiffer clusters. Clusters can be \nmanipulated by revolving the applied rotating field below a critical fr equency. Such a critical frequency \nscales with the square of the applied field, as predicted by a balance of viscous and magnetic torques \nacting on the cluster. The ability to assemble and manipulate the 2D colloidal clusters allows one to \n 15 change the effec tive refractive index of the droplet and thereby shift the focal length of magnetically \ncontrolled droplet -based microlenses. Alternatively, the assembly of colloids into clusters can be used \nto change the transmittance of light through the droplet, thus leading to magnetically driven optical \nshutters. In addition to these proof -of-concept demonstrators, the proposed encapsulated colloids may \nfind potential applications in bioinspired colo r-changing displays,  optical devices for \ntelecommunications, camouflage skins, and magnetically writable boards.  \n \nAcknowledgements  \nThe financial support from ETH Zürich and from the Swiss National Science Foundation through the \nNational Center of Competenc e in Research (NCCR) for Bio -Inspired Materials  is gratefully \nacknowledged . The authors a lso thank  Dr. Tom Valentin and Dr.  Nima Mirkhani for the introduction to \nthe magnetic system MFG -100 from MagnetobotiX®. The glass devices were fabricated at the ETH \ncenter for micro -and nanoscience, FIRST.  \nConflict of interest: S. Schuerle is co -founder and Member of the Board of MagnebotiX AG.  \nMaterials and Methods  \nManufacturing of microfluidic devices  \nGlass microfluidic devices were fabricated using wet etching techniques previously reported in the \nliterature. 33 Briefly, two 1.0 mm thick borosilicate glass wafers (Boro float 33 Schott) were annealed for \n4 h at 580 °C to remove any internal stresses caused by the polishing during production. 50 nm Cr and \n50 nm Au thick layers were deposited on the glass substrates using electron beam physical vapor  \ndeposition. The wafers were spin -coated with a negative photoresist (AZ10XT 520 cp, thickness of 8 \nµm) and templated with a sequence of photolithography steps. Both wafers were chemically etched in \n40% HF at an etching rate of 3.2 µm/min . The flow -focusing device was etched  to a depth of 120 μm , \nwhile for the open step emulsification device channels 100 µm deep were produced . Following the \netching step, the mask was removed in succession using acetone, isopropanol, gold etchant, and \nchrome etchant. The wafers were diced into 15 ×  30 mm2 (flow -focusing) or 15 × 50 mm2 (step \nemulsification) chips and 0.7 mm inlets were drilled using a diamond -coated drill bit. Next, the wafers \nwere meticulously cleaned with acetone and isopropanol and subjected to two surface activation steps, \nfirst in piranha (1:1, sulfuric acid:hydrogen peroxide) and later in RCA1 (1:1 :5, ammonium \nhydroxide:hydrogen peroxide:water) solutions. After such treatments, the wafers were flushed with \nwater and manually aligned while still wet. Weakly bonded chips were obtained after drying for several \nhours. A stronger connection was achieved by thermal bonding of two symmetric glass wafers for 4 h \nin a furnace at a temperature of 620 °C, which is slightly above the glass transition temperature (T g) of \nborosilicate glas s. To ensure good contact between the two wafers, weight s were  placed on top of each \nchip, resulting in an applied pressure of 22 kPa  (15 × 30 mm2) and 26 kPa (15  × 50 mm2). The success \nof the bonding step was controlled by visual inspection under an optical microscope.  \n  16 Surface treatment of microfluidic channels  \nThe microfluidic device was connected to plastic syringes (BD Luer -Lok™ Syringe), mounted on \ndisplacement -controll ed pumps (Pump 33 DDS, Harvard Apparatus), using PTFE tubing (outer \ndiameter of 1.6 mm, inner diameter of 0.8 mm, Bohlender) and a 4 -way linear connector (Dolomite \nMicrofluidics).  \nTo form water -in-water -in-oil or water -in-oil emulsion droplets, the channel s of the microfluidic device s \nneed to be surface treated to become hydrophobic. To prepare for such surface treatment, the device s \nwere cleaned in a furnace (model LT 5/11/B410, Nabertherm) at 550 °C for 4 h, then flushed with 1 M \nNaOH and rinsed with wate r. The functionalization was performed by flowing a toluene solution \ncontaining 5  vol% octadecyltrimethoxysilane (ODTMS, Acros) and 0.5  vol% butyl amine (Acros) for 2 h \nat a flow rate of 100 µL/h.  \n \nMicrofluidic emulsification  \nWater droplets loaded with ma gnetic nanoparticles were prepared by microfluidic emulsification. \nMonodisperse magnetic responsive polystyrene nanoparticles were purchased from MicroParticles \nGmbH (PS -MAG -COOH) . The as -received particles show a mean diameter of 536 nm and contain \nCOOH f unctional groups on the surface. The polystyrene is loaded with an iron oxide content > 30 wt% \nand the surface  functionalized with carboxyl groups . The particles are supplied as a suspension with \n1.96 wt% solid content. Such dispersion was diluted with Mil li-Q water at different concentrations \nranging from 0.01 wt% up to 0.78 wt% and used as dispersed phase of the emulsions.  \nUnless mentioned otherwise, the water droplets were covered with silica nanoparticles to form highly \nstable Pickering emulsions. The silica nanoparticles were suspended in water and delivered as a co -\nflowing aqueous middle phase during the emulsification p rocess (Figure 2, main text). The suspensions \ncontained 10 vol% silica nanoparticles with a mean diameter 120 nm (SNOWTEX ZL, Nissan \nChemicals). The particles were partially hydrophobized with hexyl amine (Acros) to a calculated SiOR \nsurface density of 3.2 2 nm-1 before dilution to a concentration of 1.5 vol% to be used as the middle \nphase. For all the experiments, the flow rates of the dispersed and middle phases were set to 600 µl/h \nand 300 µl/h, respectively. This leads to a theoretical surface coverage o f around three monolayers. \nThe long serpentine channel in the chip was designed to give the silica nanoparticles enough time to \ndiffuse and adsorb at the water/oil interface.  \nThe continuous phase of the emulsions was formulated depending on the targeted ex periments or \ndemonstrations. For the experiments on the assembly and fragmentation dynamics of 2D colloidal \nclusters, the non -ionic surfactant cetyl PEG/PPG -10/1 dimethicone (ABIL 90 EM, kindly provided by \nEvonik) was dissolved in 1 -decanol at a concentrat ion of 2 wt% and used as continuous phase. This \nnon-aqueous surfactant solution was also employed for the demonstration of magnetically controlled \nactive lenses. For the magnetic shutter experiments, 2 wt% polyglycerol polyricinoleate (PGPR 4150, \nkindly pr ovided by Palsgaard) was used as surfactant in the oil phase, which consisted of isodecyl \nmethacrylate (IDMA, Sigma -Aldrich) and 1,6 -hexanediol dimethacrylate (HDDMA, TCI) with a weight  17 ratio of 7:3 and 1 wt% 2,4,6 -trimethylbenzoyldiphenyl phosphine oxide (TPO, Sigma -Aldrich) as \nphotoinitiator. Because the continuous phase was polymerized right after emulsification, no Pickering \nstabilization with silica nanoparticles was employed for the fabrication of the magnetic shutters.  For this \nset of experiments, a step emulsification glass microfluidic device with 100 µm step height was \nemployed.  \n \nMagnetic alignment setup  \nThe assembly and manipulation of the magnetic responsive nanoparticles was performed using the \nmagnetic system MFG -100 from MagnetobotiX®, which generates arbitrary magnetic flux densities  up \nto 20 mT at frequencies up to 2 kHz. As described in previous wo rk, 34 the arrangement of eight \nelectromagnets into two se ts allows for unrestrained rotational freedom of the magnetic field. This is \naccomplished by the superposition of multiple magnetic fields, which creates a homogeneous magnetic \nfield within a spherical workspace with a diameter of approximately 10 mm. Imag ing of the colloidal \nmanipulation was performed using an inverted optical microscope (DM IL LED, Leica) equipped with a \nLeica DFC 295 camera.  \n \nMagnetic assembly and manipulation  \nMonodisperse droplets containing defined concentrations of magnetic nanoparticles were prepared to \nstudy the assembly and fragmentation dynamics of two -dimensional superparamagnetic colloidal \nclusters. To investigate the role of the magnetic field density and rotational frequency over the size of \nthe assembled colloidal clusters, a monolayer of monodispersed droplets containing magnetic \nresponsive particles was deposited on a microscope well slide. The glass surface of the slide was \nmodified to be hydrophob ic using a solution containing 5  vol% octadecyltrimethoxysilane and 0.5  vol% \nbutyl amine in toluene. A cover slip was placed on top of the slide to seal the well and reduce the \nevaporation of 1 -decanol. The colloids were assembled into a single cluster usi ng the manual controls \nof the magnetic setup (pitch, yaw, roll) at a fixed magnetic flux density of 10 mT and a rotational \nfrequency of 30 Hz. Once shaped, a rotational magnetic field was applied within the xy -plane to \nmanipulate the colloidal clusters (Fi gure 1, main text). The magnetic flux density ranged from 6 to 12 \nmT with a rotational frequency of 125 Hz and a time step between pulses of 2 ms. The concentration \nof particles in the droplets was varied from 0.02 to 0. 078 wt%. For the fragmentation analy sis, clusters \nwith three different approximate diameters were studied: 50, 70 and 100 µm. To facilitate visualization, \nthe clusters were rotated around the x -axis, so that the edge of the clusters pointed in the direction of \nthe observation field. To study  the stability of the 2D colloidal assembly, the rotational plane of the \nclusters was revolved around the z -axis at angular velocities increasing from 0.314 to 2.93 rad/s. For \nangular velocities between 0.3 rad/s and 1.15 rad/s the increment was set to 0.1  rad/s, while from 1.15 \nrad/s to 2.93 rad/s the velocity was increased by 0.05 rad/s every revolution. The critical angular velocity \nat which fragmentation took place was recorder for each cluster inside the droplets.  \n  18 Active optical elements  \nThe active mi crolenses array was manufactured by collecting a monolayer of monodisperse droplets \non a microscope cover slip  (#1.5, 𝑛=1.523, 35 VWR ), which was previously glued on a 3D printed \nholder (Fig ure S1). A 1 mm spacer was placed around the emulsion and covered with a second cover \nslip (#1.5,  𝑛=1.523 (Hibbs A.R., 2006) , VWR ) to remove the liquid meniscus and to reduce \nevaporation. The droplets were 271 µm in diameter (coefficient of variation, CV = 1.5%) and contained \n0.1061 wt% of magnetic particles. The effective refractive index of the magnetic particles and of the \nhomogeneous particle suspension was calculated using a simple mixing rule. The profile of the light \ntransmitted through the droplets was analyzed  using an optical setup comprising a collimated laser \nbeam with a wavelength of 532 nm, a fused silica lens (focal length 50 mm, Thorlabs) mounted on a z -\naxis translational lens mount (SM1ZA, Thorlabs), a 20x magnification objective (Olympus) and a CMOS  \ncamera ( DCC3260M , Thorlabs ). The setup was aligned using 30 mm cage components (Thorlabs) and \na 3D printed connector attached to the magnetic setup.  \nThe magnetic shutter system was fabricated by depositing a monolayer of droplets on a layer of \npolymerized continuous phase ins ide a 3D printed substrate ( Figure S5). The sample was \nphotopolymerized using a UV light source (Omnicure Series 1000, wavelength range 320 -500 nm) with \nan irradiance of 93 mW/cm2 in a N 2 atmosphere for 10 minutes. The droplets were 401.7  µm in diameter \n(CV 8.3%) and contained 1.96 wt% of magnetic particles. 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Garsha, in Handbook Of Biological Confocal Microscopy , \ned. J. B. Pawley, Springer US, Boston, MA, 2006, pp. 650 -671. \n \n   21 Supporting Information  \nMagnetic  manipulation of superparamagnetic colloids in droplet -based optical devices  \nI. Mattich¹, J. Sendra2, H. Galinski2, G. Isapour3, A.F. Demirörs¹,  M. Lattuada4, S. Schuerle5, A. R. \nStudart¹  \n1 Complex Materials, Department of Materials, ETH Zürich, 8093 Zürich, Switzerland  \n2 Laboratory for Nanometallurgy, Department of Materials, ETH Zürich, 8093 Zürich, Switzerland  \n3 Department of Mechanical Engineering, MIT, USA  \n4 Department of Chemistry, Un iversity of Fribourg, Switzerland  \n5 Department of Health Sciences and Technology, Institute for Translational Medicine, ETH Zürich, \n8093 Zürich, Switzerland  \n \nContent:  \nSupporting Text  \nSupporting Movies  \nSupporting Figures  \n \n   22 Supporting Text  \nCritical frequency of the rotating magnetic field  \nThe revolving speed of the magnetic field above which the clusters undergo fragmentation can be \nestimated by balancing the magnetic and viscous torques exerted on the assembled cluster. On the \nbasis of previous work , 30, 31 we expect this critical revolving speed  (𝜔𝑐) to be given by the following \nequation  (see also main text) : \n \n𝜔𝑐=𝜇0𝜒𝑐𝑙𝑢𝑠𝑡𝑒𝑟2\n12𝜂0(𝑓\n𝑓0⁄)(1+𝜒𝑐𝑙𝑢𝑠𝑡𝑒𝑟 )∙𝐻02 (S1) \n \nwhere 𝜇0 is the magnetic permeability of free space (1.257 10-6 𝑁𝐴2⁄), 𝜒𝑐𝑙𝑢𝑠𝑡𝑒𝑟  is the effective magnetic \nsusceptibility of the cluster, 𝐻0 is the magnitude of the applied magnetic field  strength , 𝜂0 is the viscosity \nof the fluid surrounding the cluster , and  𝑓𝑓0⁄ is the Perrin friction factor.  The parameters governing the \neffect of the magnetic field on the critical revolving speed can be s ummarized by the term 𝑘: \n \n𝑘=𝜇0𝜒𝑐𝑙𝑢𝑠𝑡𝑒𝑟2\n12𝜂0(𝑓\n𝑓0⁄)(1+𝜒𝑐𝑙𝑢𝑠𝑡𝑒𝑟 )  (S2) \n \nTo theoretically predict the dependence of 𝜔𝑐 on the applied magnetic field strength (𝐻0), we calculate \nthe pre -factor, 𝑘, for clusters of different sizes and compare the predictions with the experimental results \n(Figure 3d). The parameters used to calculate 𝑘 were obtained directly from the literature or were \nestimated based on other variables of our system (Table S1).  \nThe magnetic susceptibility of the cluster depends on the susceptibility of the single magnetite \nnanoparticles, 𝜒𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑡𝑒 , and their volume fraction in the polystyrene -magnetite nanocomposite  colloid  \nΦ𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑡𝑒 : 𝜒𝑐𝑙𝑢𝑠𝑡𝑒𝑟 =𝜒𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑡𝑒 ∙Φ𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑡𝑒 . The Perrin friction factor for equatorial rotation of an oblate \nspheroid along one of the long axes  is given by : \n \n 𝑓\n𝑓0⁄=4\n3(1\n𝑝2−𝑝2)\n2−𝑆\n𝑝2,   (S3) \n \nwith the Perrin S factor for oblate spheroids 𝑆≝2tan−1𝜉\nξ, 𝜉≝√|𝑝2−1|\n𝑝 and axial ratio 𝑝=ℎ\n𝑅𝑐, where  𝑅𝑐 and \nℎ are the radius and the thickness of the cluster, respectively. Our experiments show that the clusters \ncomprise a single layer of superparamagnetic nano particles. Therefore, we assume the thickness of  23 the cluster to be equal to two times the radius of the magnetite nanoparticles ( 2𝑎). The radius of the \nclusters depends on the concentration of superpara magnetic colloids  in the droplet  (Table S1) . \nExcept for the smallest cluster size, t he predictions from the above theoretical model are in reasonable \nagreement with the experimental data obtained for the critical revolving speed ( 𝜔𝑐) as a function of the \napplied magnetic field for diff erent cluster sizes (Figure 3d) . The theoretical underestimation of the 𝜔𝑐 \nvalue s for the smallest cluster size suggests that our assumptions are not valid for this experimental \ncondition.  \n \nTable S1 : Parameters used in the calculation of the critical frequency needed for cluster fragmentation \nunder a revolving magnetic field (Figure 3d).  \nSample  Colloidal  \nconcentration  \n \nwt% Cluster  \nradius  \n \n𝑅𝑐 [𝜇𝑚] Perrin \nfactor  \n \n𝑓\n𝑓0⁄ \n Nanoparticle \nmagnetic \nsusceptibility  \n𝜒𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑡𝑒  Magnetite  \ncontent  \n \n𝑣𝑜𝑙% Cluster \nmagnetic \nsusceptibility  \n𝜒𝑐𝑙𝑢𝑠𝑡𝑒𝑟  Fluid  \nviscosity  \n \n𝜂0 [𝑃𝑎∙𝑠] \n        \n1 0.020 25.03  44.26  20 15.32  3.22 0.01 \n2 0.039 38.45  67.99  20 15.32  3.22 0.01 \n3 0.078 48.69  86.11  20 15.32  3.22 0.01 \n \n \nCalculation of focal length using analytical model  \nThe optical response of the droplet -based microlens was evaluated using an analytical ray tracing \nmodel. In this model, the incoming collimated light interacts with different optical elements  of the active \nlens system, namely two glass slides, the oil medi um and the water droplet with the magnetic \nnanoparticle s (Figure S2 ). The parallel ray originated from the collimated beam (shown in red  in Figure \nS2) goes through the first glass slide without any deviation before encounter ing the microlens. From \nthere, i t goes through four interfaces 𝐶1, 𝐶2, 𝐼𝑜𝑔 and 𝐼𝑔𝑎 and three different media 𝑆𝑤, 𝑆𝑜 and 𝑆𝑔. As such, \nthe resulting ray tracing transfer matrix is described as follows:  \n𝑀=𝐼𝑔𝑎𝑆𝑔𝐼𝑜𝑔𝑆𝑜𝐶2𝑆𝑤𝐶1=[𝐴 𝐵\n𝐶𝐷]  (S4) \nwhere the propagators through water (inside droplet), oil and glass are defined as:  \n𝑆𝑤=[1𝑡\n01]  (S5) \n𝑆𝑜=[1𝑑𝑜\n0 1]  (S6) \n𝑆𝑔=[1𝑑𝑔\n0 1]  (S7)  24 Here, 𝑡=2𝑅 is the thickness of the microlens , 𝑅 is the radius of the droplet, 𝑑0 is the distance between \nthe droplet surface and the glass slide and 𝑑𝑔 is the thickness of the glass slide (Figure S 2). \nThe refractions at the oil -glass and glass -air interface are defined as:  \n𝐼𝑜𝑔=[1 0\n0𝑛𝑜\n𝑛𝑔]  (S8) \n𝐼𝑔𝑎=[1 0\n0𝑛𝑔]  (S9) \nThe refraction at the microlens curved interfaces (here the sign convention has already been applied \nsuch that R>0) is described as:  \n𝐶1=[1 0\n−𝑛𝑤−𝑛𝑜\n𝑅𝑛𝑤𝑛𝑜\n𝑛𝑤]  (S10)  \n𝐶2=[1 0\n−𝑛𝑤−𝑛𝑜\n𝑅𝑛𝑜𝑛𝑤\n𝑛𝑜]  (S11)  \nFrom the ray transfer matrix, the effective focal l ength from the principal plane to the focal plane will be \n−1/𝐶. To better compare with the ray tracing simulations, the focal length from the focal plane to the \ncenter of the microlens is calculated assuming a small angle approximation such that  \n𝑓𝑒𝑓𝑓=𝐴\n𝐶+𝑑𝑔+𝑑𝑜+𝑅  (S12)  \nCalculating the coefficients of the transfer matrix 𝑀, one reaches the following expressions:  \n𝐴=1−𝛼𝑡\n𝑛𝑤−1\n𝑓𝑙(𝑑𝑜+𝑑𝑔𝑛𝑜\n𝑛𝑔)  (S13)  \n𝐶=−𝑛𝑜\n𝑓𝑙  (S14)  \nwhere 𝛼=𝑛𝑤−𝑛𝑜\n𝑅 and 𝑓𝑙 is the  focal length of the microlens, therefore  \n−1\n𝑓𝑙=𝛼\n𝑛𝑜(−2+𝛼𝑡\n𝑛𝑤)  (S15)  \nSubstituting 𝑛𝑤 for an effective refractive index 𝑛𝑒𝑓𝑓 yields the effective focal length in the case for a \nwater droplet loaded with magnetic nanoparticles  (𝑓𝑒𝑓𝑓). \n \n  25 Ray optics simulations  \nThe propagation of light through the microlens was also modelled using the Ray Optics Module of \nCOMSOL Multiphysics. These  numerical simulations were not intended to replicate the experimental \nresults one -to-one, but to derive an intuitive physical picture of the light -matter interaction s and test the \nvalidity of the analytical model introduced above.  \nIn the simulations, a set of rays is propagated through a three -dimensional microlens  to mimic the \nexperimental setup introduced in the manuscript ( Figure  and Figure S3 ). To add the conformal SiO 2 \nshell ( 𝑛𝑆𝑖𝑂2−𝑁𝑃=1.47) to the water droplet, a material discontinuity condition on the surface of the water \ndroplet is imposed.  The focal length of the system is determined by calculating the intercept of the rays \nwith the x -axis. Here, we calculate the change in focal length as a function of orientation and refractive \nindex for  a 4 monolaye r (2 µm) thick colloidal cluster. The cluster is assumed to have an ellipsoidal \nshape with two long semi -axes being 123 µm long and the short semi -axis being 1 µm thick.  \nWhile the focal length of the concave microlens is unchanged when the colloidal cluster is oriented \nparallel to the light rays, the focal length decreases when the cluster is oriented perpendicular. This \nleads to a defocus. By switching between these two states, the focal length can be altered by 25%. In \nthe absence of an ex ternal magnetic field, the colloidal particles are homogeneously distributed within \nthe water droplet. As such, the refractive index of the droplet loaded with colloidal particles can be \ntreated with the effective medium approximation.  \nIn agreement with ou r analytical ray tracing model, the focal length obtained from the simulation \ndecreases with increasing effective refractive index  (Figure 4f) . This validation is an important result, \nas we can then predict the focal length and focal power of the microlens  with different droplet \nconfigurations  (Figure S 4). \n \nSupporting Movies  \nMovie S1: Two-dimensional colloidal clusters undergoing f ragmentation  upon increase of the revolving \nfrequency of the applied rotating magnetic field . Experimental parameters: cluster diameter , 70 µm;  \nparticle concentration , 0.039 wt%; magnetic flux density , 12 mT; rotating frequency , 125 Hz; revolving \nfrequency range , 12 deg/s – 180 deg/s with a  frequency increase each turn of 6 deg/s . \nMovie S2: Magnetic ally driven optical  shutter  made from arrays of droplets containing \nsuperparamagnetic colloids . Experimental parameters: m agnetic flux density , 10 mT; rotating \nfrequency , 2 Hz; particle concentration , 1.96 wt% . \n   26 Supporting Figures  \n \nFigure S 1. 3D printed slide used to contain the active microlens system. (a-b) Renderings of the sample \nholder in the (a) assembled and (b) exploded views.  The rendering shown in (b) of the exploded view \nindicates (1) the 3 D printed sample holder, (2) the first glass cover slip, (3)  the layer of oil, (4) the layer \nof monodisperse droplets, and (5) the second glass cover slip . (c) Photograph of the 3D printed \nsubstrate.  Scale bar is 2 cm.  \n \n \n 27  \nFigure S 2. Schematic illustrating the different optical elements used for the analytical ray tracing  model , \nincluding two glass slides ( 𝑆𝑔) of refractive index 𝑛𝑔, the oil domain  (𝑆𝑜) with refractive index  𝑛𝑜 and the \nwater droplet  (𝑆𝑤) with refractive index  𝑛𝑤. The interfaces between the optical elements are indicated \nby 𝐶1, 𝐶2, 𝐼𝑜𝑔 and 𝐼𝑔𝑎. 𝑅, 𝑑𝑜 and 𝑑𝑔 represent the lengths and distances from the experimental system.  \n \n \nFigure S 3. (a) Illustration of the three -dimensional ray optics model including boundary and ray release \nconditions. (b) Change in focal leng th compared to a water droplet (𝑛𝐻2𝑂=1.333) as function  of platelet \n(cluster) refractive index ( 𝑛𝑝) for two different orientations of a pa rticle -made platelet (thickness of 4 \nmonolayers ). If the platelet is oriented parallel to the propagating rays, the focal length is independent \nof the platelet’s refractive index. If the platelet is oriented perpendicular to t he light rays, the focal length \n 28 decreases with increasing refractive index. Panels (c) -(d) display  the ray trajectories  (𝜆=532  𝑛𝑚) for \nplatelet index of 𝑛𝑝=2.37 for parallel or perpendicular alignment.  \n \n \nFigure S 4. (a-b) Ray trajectories ( 𝜆=532  𝑛𝑚) for a water droplet with index of 𝑛𝐻2𝑂=1.333 in oil \n(𝑛𝑜𝑖𝑙=1.420). The microlens acts as a concave lens with negative focal length and negative focal \npower. (c -d) Ray trajectories ( 𝜆=532  𝑛𝑚) for an oil droplet with index of 𝑛𝑜𝑖𝑙=1.42 in water 𝑛𝐻2𝑂=\n1.333. The microlens acts as a convex lens with positive focal length and positive focal power.  \n \n 29  \nFigure S5 . 3D printed emulsion container used to manufacture the active shutter elements. (a) \nRendering of the (1) 3D printed container showing (2) two PTFE tubes partially filled with \npoly(dimethylsiloxane) (PDMS, Sylgard 184, Dow Corning). (b) Rendering of the printed holder with (1) \nthe microfluidic device  attached to (2) the Dolomite connec tor immersed in (3) the monomer mixture . \n(c-d) Photographs of the actual container alone and with the microfluidic device.  Scale bar is 2 cm.  \n \n  \n 30  \nFigure S 6. Droplet -based optical shutter controlled by magnetic fields. Intensity of light transmitted \nacross the droplet array when the shutter is subjected to consecutive ON/OFF cycles. The ON state is \ntriggered  when the magnetic field is rotated within the xy -plane, whereas the OFF state is achieved \nwhen the applied field rotates within the yz -plane. The experiment data depicts the effect of the \nmagnetic flux density  B at a fixed frequency of (a) 1 Hz and (b) 5 Hz.  \n"    },    {        "title": "1201.5442v1.Dynamics_of_Magnetized_Vortex_Tubes_in_the_Solar_Chromosphere.pdf",        "content": "arXiv:1201.5442v1  [astro-ph.SR]  26 Jan 2012Dynamics of Magnetized Vortex Tubes\nin the Solar Chromosphere\nI. N. Kitiashvili1,2, A. G. Kosovichev1, N. N. Mansour3, A. A. Wray3\n1Stanford University, Stanford, CA 94305, USA\n2Kazan Federal University, Kazan, 420008, Russia\n3NASA Ames Research Center, Moffett Field, Mountain View, CA 9 4040, USA\nABSTRACT\nWe use 3D radiative MHD simulations to investigate the formation and d y-\nnamics of small-scale (less than 0.5 Mm in diameter) vortex tubes spon taneously\ngenerated by turbulent convection in quiet-Sun regions with initially w eak mean\nmagnetic fields. The results show that the vortex tubes penetrat e into the chro-\nmosphere and substantially affect the structure and dynamics of t he solar at-\nmosphere. The vortex tubes are mostly concentrated in intergra nular lanes and\nare characterized by strong (near sonic) downflows and swirling mo tions that\ncapture and twist magnetic field lines, forming magnetic flux tubes th at expand\nwith height and which attain magnetic field strengths ranging from 20 0 G in the\nchromosphere to more than 1 kG in the photosphere. We investigat e in detail the\nphysical properties of these vortex tubes, including thermodyna mic properties,\nflow dynamics, and kinetic and current helicities, and conclude that m agnetized\nvortex tubes provide an important path for energy and momentum transfer from\nthe convection zone into the chromosphere.\nSubject headings: Sun: photosphere, chromosphere, surfacemagnetism, magnet ic\ntopology\n1. Introduction\nInterest in vortex tube dynamics of the quiet Sun was recently initia ted by the de-\ntection of ubiquitous small-scale swirling motions in the photosphere ( Wang et al. 1995;\n1e-mail: irinasun@stanford.edu– 2 –\nP¨ otzi & Brandt 2005; Bonet et al. 2008, 2010; Balmaceda et al. 20 10; Steiner et al. 2010)\nandthechromosphere(Wedemeyer-B¨ ohm & Rouppe van der Voor t 2009)withhigh-resolution\nsolar telescopes. Previous to this discovery, vortex tubes on the Sun were predicted by theo-\nreticalmodels(e.g.,Stenflo 1975)andnumerical simulations(e.g.,Br andenburg et al. 1996;\nStein & Nordlund 2000), giving a clear illustration of the turbulent nat ure of solar convec-\ntion. Both observations and numerical simulations show concentra tions of vortex tubes in\nthe intergranular lanes. According to recent radiative hydrodyna mic simulations, vortical\nmotions can be also form inside granules (Kitiashvili et al. 2012). Thes e simulations have\nalso shown that vortex tube formation in the near-surface layers can be caused by two basic\nmechanisms associated with: 1) small-scale convective instability dev eloping inside granules,\nand 2) the Kelvin-Helmholtz instability of shearing flows.\nThe convective instability leads to formation of a vortex sheet and it s subsequent over-\nturning during a localized upflow (plume) or splitting of a granule. The p rocess of the\nvortex sheet overturning, which results in a vortex tube, is often accompanied by a grad-\nual migration of the vortex tube into an intergranular lane (see Fig. 2 in Kitiashvili et al.\n2012). Shearing flows that lead to the development of the Kelvin-He lmholtz instability can\nbe present in both granules and intergranular lanes. However, in th e intergranular lanes the\nshearing flows arestronger andcanlead toa series ofvortices (re sembling theKarmanvortex\nstreet). Also, converging downflows in the intergranular lanes mak e the vortex tubes more\nstable, with characteristic lifetimes up to 40 min, whereas inside gran ules the lifetime is less\nthan 10 min. These processes can explain why the observed vortex tubes are predominantly\nconcentrated in the intergranular lanes.\nNumerical simulations also show connections between vortex tube d ynamics and vari-\nous other solar phenomena, such as the hydromagnetic dynamo (B randenburg et al. 1996),\nspontaneous organization of emerged magnetic field into self-maint ained pore-like structures\n(Kitiashvili et al.2010), excitationofacousticwavesinthequietSun (Kitiashvili et al.2011),\nandothers. InthisLetter, wepresent newnumerical simulations t hatdemonstrateimportant\nlinks between the turbulent subsurface layers and the solar atmos phere though the dynamics\nof penetrating vortex tubes.\n2. Computational setup\nNumerical simulations of the quiet Sun are performed by using a 3D ra diative MHD\ncode (‘SolarBox’) developed at the NASA/Ames Research Center an d the Stanford Cen-\nter for Turbulence Research by Alan Wray and his colleagues (Jacou tot et al. 2008) for\nmodeling the outer part of the solar convection zone and lower atmo sphere in a carte-– 3 –\nsian geometry. The code was developed for realistic-type numerica l simulations of the Sun\npioneered by Nordlund & Stein (2001) and uses a tabular real-gas eq uation of stat. Ra-\ndiative energy transfer is calculated with a 3D multi-spectral-bin met hod between fluid\nelements, assuming local thermodynamic equilibrium and using the OPA L opacity tables\n(Rogers et al. 1996). Initialization is done from a standard model of the solar interior\n(Christensen-Dalsgaard et al. 1996).\nThe physical description of the dynamical properties of solar conv ection was improved\nthrough the implementation of subgrid-scale turbulence models, wh ich effectively increase\nthe Reynolds number and allow better resolution of essential turbu lent scales. This ap-\nproach, based on Large-Eddy Simulation (LES) models of subgrid tu rbulence, has demon-\nstrated good agreement of numerically modeled acoustic wave excit ation with observations\n(Jacoutot et al. 2008) and has helped improve understanding of wa ve excitation mecha-\nnisms (Kitiashvili et al. 2011), formation of magnetic structures (K itiashvili et al. 2010),\nand Evershed flows in sunspots (Kitiashvili et al. 2009). The simulatio ns in this paper were\nobtained using a Smagorinsky eddy-viscosity model (Smagorinsky 1 963) in which the com-\npressible Reynolds stresses were calculated in the form (Moin et al. 1 991; Jacoutot et al.\n2008):τij=−2CS△2|S|(Si,j−uk,kδij/3)+ 2CC△2|S|2δij/3, where the Smagorinsky coeffi-\ncientsCS=CC= 0.001,Sijis the large-scale stress tensor, and △ ≡(dx×dy×dz)1/3with\ndx,dyanddzbeing the grid-cell dimensions.\nIn the current study, the simulation results were obtained for a co mputational domain\nof 6.4×6.4×6.2 Mm3, including a 1 Mm high layer of the atmosphere, with a grid spacing\nofdx=dy= 12.5 km and dz= 10 km. The lateral boundary conditions are periodic. The\ntop boundary is open to mass, momentum, and energy transfers a nd also to radiative flux.\nThe bottom boundary is open for radiation and flows, and simulates e nergy input from the\ninterior of the Sun. We focus mostly on a case with an initially uniform ve rtical magnetic\nfield,Bz= 10 G, representing quiet-Sun conditions (far from sunspots and active regions).\n3. Formation of vortex tubes by turbulent convection\nVortex tubes are formed by turbulent convection in near-surfac e layers of the convective\nzone (e.g. Stein & Nordlund 2000; Kitiashvili et al. 2012). The vortex tubes represent\ncompact low-density structures up to 0.5 Mm in diameter and with high -speed swirling\nmotion reaching up to 12 km/s. The vortex cores are characterize d by strong downflows\n(up to 8 km/s) and lower temperature. Large vortex tubes can ex tend deeper than 300 km\nbelow the surface.– 4 –\nOur previous simulations (Kitiashvili et al. 2012) revealed two basic me chanisms of\nvortex tube formation: one due to a granular instability (vortex sh eet overturning) and\nanother due to the Kelvin-Helmholtz instability in shearing flows. Vort ex tubes can form in\nintergranular lanes and in granules but are mostly concentrated in t he intergranular lanes\n(Fig. 1a, b). These physical mechanisms of vortex tube formation are purely hydrodynamic,\nbut in the real Sun vortices areexpected to strongly interact with ubiquitous magnetic fields.\nHowever, neither simulations nor observations have shown a clear c orrelation between vortex\nmotions and magnetic field concentrations, that is, not every vort ex is accompanied by a\nstrong magnetic field concentration. This fact has also been shown in simulations using a\nshallow domain (1.4 Mm in total height; Shelyag et al. 2011; Moll et al. 20 11).\nIn weak magnetic field regions, magnetic patches follow convective m otions. Concen-\ntration and magnification of magnetic field by swirling motions can stab ilize the vortex tube\nstructure and decrease the influence of surrounding turbulent fl ows. In our simulation case,\nwe introduce a 10 G, initially uniform, vertical magnetic field. This field g ets quickly concen-\ntrated, mostly in intergranular lanes, and we find that the stronge st magnetic field ( ∼1 kG)\nconcentrations are often associated with vortices (Fig. 1 c, d).\n4. Dynamics and properties of vortex tubes in the chromosphe re\nA new interesting result of our simulations is the extension of turbule nt vortex tubes\nfrom the convection zone into the convectively stable atmospheric layers. Figure 2 illustrates\na snapshot of enstrophy distribution showing vortex tube struct ures (yellow isosurfaces)\nabove the photosphere (the horizontal wavy light surface shows the 6400 K near-surface\nlayer). These vortex tubes are mostly concentrated in the interg ranular lanes and often form\narc-shaped structures above the surface. Other vortices pen etrate almost vertically into the\nhigher chromospheric layers (an example of such an extended vort ex tube is indicated by the\narrow; we will consider its structure in detail below). Local upflows (red color on vertical\nslices, Figure 2) cause stretching of the vortex arcs, and nearby vortices can destroy them.\nFinally, propagating shock waves interact with the vortex tubes in t he higher chromospheric\nlayers. The overall chromospheric dynamics driven by turbulent co nvection is thus very\ncomplicated. Theeffectofvortexpenetrationintothechromosph ereismostlyhydrodynamic,\nas observed in simulations with and without magnetic field. However, t he magnetic field\ntends to be captured and concentrated in the vortex tubes, cau sing new dynamical effects.\nThe structure of the vortex tube indicated by the arrow in Figure 2 is illustrated in\nFigure 3 at different heights: 200 km, 500 km, 650 km and 800 km abov e the surface. The\ntemperaturedistribution(Fig.3, row a)showslocalheatingofthevortexcoreregion,whereas– 5 –\nin the subphotospheric layers the core vortex temperature is lowe r than in the surrounding\nplasma. For this moment of time, the swirling motions in the vortex reg ion are characterized\nmostly by highly turbulent downflows (Fig. 3 b), but some upflows are noticeable near the\nedge of the vortex tube, the size of which is expanding with height. T he current helicity,\ncalculated in Alfv´ en units as χm=1\n4πρ/vectorB·(∇×/vectorB), forms a sheet-like structure oriented along\nthe intergranular lane near the photospheric layers (Fig. 3 d). The current sheet structure\ngradually changes orientation in the higher layers and becomes more circular (Fig. 3 d). The\ncurrent helicity structure is more diffuse than the kinetic helicity, χm=/vector u·(∇×/vector u), shown in\nFigure 3c. The density distribution in the lower atmosphere is similar to the surf ace layers,\nbut the vortex tube structure becomes more complicated with heig ht, forming a ring-like\nstructure at ∼800 km above the surface (Fig. 3 e).\nFigure 4 shows the time evolution of the velocity streamlines (panels a-c), magnetic\nfield lines (panels d-f), and the ratio of gas pressure to magnetic pressure (plasma β) for\nthree moments separated by 3 min. The structure of the vortex t ube in the middle column\nis shown in more detail in Fig. 5 a. In Figure 4, the grey-yellow isosurface corresponds to a\ntemperature of 5800 K. Color patches on this surface indicate var iations of magnetic field\nstrength as indicated in the right color bar. The strongest magnet ic field concentrations\n(∼1.2 kG) are associated with the vortex tubes in the photospheric laye r; and the field\nstrength decreases to ∼200 G in the upper layers of our domain ( ∼1 Mm above the\nphotosphere).\nThe numerical simulations show the penetration and dynamics of the vortex tube into\nthe chromosphere. The vortex core contains very compact helica l downflows, and we observe\nthat the vortex pulls granular fluid upward which then reverses into the downflows (Fig. 4 a).\nThe magnetic field at this stage of vortex evolution continues to con centrate in the vicinity\nof the vortex by following the swirling turbulent motions (Fig. 4 d). These strong helical\nflows capture and twist the magnetic field lines. Also, the helical magn etic loops formed\nby vortex tubes have a tendency to move upward due to local upflo ws near the vortex\ncore. Three minutes later, the helical downflows have become more compact and stronger\n(Fig. 4b), but the vortex is affecting a larger surrounding area. We begin to see evidence of\nvortex decay when this vortex starts interacting with others by s haring with them a part\nof the downflow (Fig. 4 b). Finally, during the next three minutes, the photospheric and\nchromospheric parts of the vortex tube become disconnected bu t still continue to evolve.\nFigure 4cshows remnants of the initially strong helical flows in the atmosphere . At this\nmoment, they are still weakly helical and become captured by anoth er growing vortex. The\nmagnetic field lines also keep their helical topology and start to diffuse (Fig. 4). We show\nthe plasma parameter β≡8πp\nB2= 3 level as blue isosurfaces in Figures 4 g-i; the value β= 1 is\nreached only in a small region of the vortex core. This parameter sh ows that magnetic effects– 6 –\nplay a significant role in the photospheric layers of the vortex tube a nd that the region of\ntheir influence rapidly expands with height. At the decay stage of th e vortex tube (Fig. 4 i),\nmagnetic effects are significant only in the upper layers.\nThe relative role of kinematic and magnetic effects of the swirling motio ns is illustrated\nby the kinetic, χk, and magnetic, χm, helicities. An example of relative distribution for\nboth helicities is shown in Figure 5 a(the kinetic helicity is in blue, and the current one\nis in pink) for the vortex tube that is indicated by the arrow in Figure 2 . Blue and pink\nisosurfaces correspond to helicity values of −5000 cm/s2; the current helicity is calculated in\nAlfv´ en units and has the same dimension as the kinetic helicity. In Figu re 5a, we also plot\nthe temperature isosurface for 5800 K, which has a very compact structure of a complicated\nchiralical shape, expanding into the higher layers of the atmospher e. The distribution of the\nkinetic helicity is more compact than the current helicity, meaning tha t the swirling flows in\nthe vortex tube are more compact than the twisted magnetic field lin es.\nIn general, the dynamics of the subsurface and near-surface lay ers is dominated by tur-\nbulent convective motions, while magnetic effects are noticeable in th e small-scale magnetic\nflux concentrations in the intergranular lanes (magnetic flux tubes ). In higher atmospheric\nlayers, magnetic effects are stronger because of the fast decre ase of gas pressure, which leads\nto expansion of the magnetic flux tubes.\nTo investigate the properties of the magnetized vortex tube with h eight, we selected\na region inside the −5000 cm/s2isosurface of the current helicity. In this region we have\nplotted the mean values of temperature, vertical and horizontal velocities, magnetic field,\nand kinetic and current helicities as a function of height for different moments of time\nwith a cadence of 20 sec (Fig. 5 b-f). The values of temperature and density are shown as\nperturbations from and normalized by the mean values: ( Tvortex−T)/Tand (ρvortex−ρ)/ρ.\nThe temperature distribution shows a deficit in the convective layer s of the vortex tube.\nAbove the photosphere, the temperature in the vortex tube incr eases, and we can see heating\nin the vortex core (Fig. 5 b). Occasional temperature decreases above ∼500 km reflect the\ndynamically oscillatory behavior of the vortex tube. The density dist ribution (Fig. 5) shows\nan increase below the surface due to mass concentration around t he vortex core, which has\nsignificantly lower density. Above the surface, the mean density pe rturbation in the tube\nfirst decreases and then increases above 200 km (Fig. 5 c).\nThe vertical distribution of the mean velocity inside the vortex tube shows very different\nproperties for the horizontal and vertical components. The mea n horizontal speed (blue\ncurves, Fig 5 d) is almost constant along the vortex tube, with relatively small fluct uations in\ntime around the mean speed of ∼3.5 km/s. It is interesting that the mean horizontal speed,\naveraged over time and over the whole domain, (thick dark blue curv e) shows a decrease– 7 –\nat 400−800 km above the surface, but inside the vortex tube there is no su ch decrease.\nIn contrast to the horizontal speed, the vertical velocity compo nent (red curves, Fig, 5 d) is\nvery dynamic and is characterized by predominant downflows; howe ver, local upflows can be\ndetected inside the vortex tube.\nThe vertical component of magnetic field is significantly stronger th an the magnitude\nof the horizontal field (Fig. 5 e). Both vertical (red curves) and horizontal (blue curves)\nfields show a similar tendency to decrease in the atmospheric layers, which is reflected in\nthe expanding topology of the flux tube. At a height of about 500 km , the magnitude of\nthe vertical component of magnetic field is smaller than the horizont al component because\nthe magnetic field lines become more twisted by the vortex. The mean kinetic helicity (blue\ncurves, Fig. 5 f) is significantly greater than the mean current helicity (red curves ) because\nswirling motions in the tube are accompanied by strong downflows, wh ile the strongest\nmagnetic field is only weakly twisted.\n5. Conclusion\nThe formation and dynamics of small-scale vortex tubes play key role s in various pro-\ncesses in solar surface convection and in the solar atmosphere. Ou r radiative MHD simu-\nlations reveal vortex tubes formed by turbulent convection pene trating from the subphoto-\nsphere into the chromosphere. These vortex tubes cause signific ant qualitative changes in\natmospheric dynamics, leading to strong variations in the thermody namic structure through\nlocal heating and density variations, generating twisted magnetic fl ux tubes, and creating\nlocal twisted upflows into the chromosphere. Strong localized swirlin g motions occupy large\nareas around the vortex tubes, capturing and twisting magnetic fi eld lines from nearby\nmagnetic structures. As a result of these phenomena, magnetize d vortex tubes generated\nby turbulent convective motions provide a very important link for en ergy and momentum\nexchange between the surface layers and the chromosphere.\nThis work was partially supported by the NASA grant NNX10AC55G, t he International\nSpace Science Institute (Bern) and Nordita (Stockholm). The aut hors thank Phil Goode,\nVasyl Yurchshin, Valentyna Abramenko, and participants of the N ordita and ISSI teams for\ninteresting discussions and useful suggestions.– 8 –\nREFERENCES\nBalmaceda, L., Vargas Dom´ ınguez, S., Palacios, J., Cabello, I. & Domin go, V. 2010, A&A,\n513, L6. doi: 10.1051/0004-6361/200913584\nBonet, J. A., M´ arquez, I., S´ anchez Almeida, J., Cabello, I. & Doming o, V. 2008, ApJ, 687,\nL131.\nBonet, J. 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F. & Nordlund, ˚A. 2000, Annals of the New York Academy of Sciences, 898, 21.\ndoi: 10.1111/j.1749-6632.2000.tb06161.x\nSteiner, O., Franz, M., Bello Gonz´ alez, N., Nutto, Ch., Rezaei, R., Mar t´ ınez Pillet, V.,\nBonet Navarro, J. A., del Toro Iniesta, J. C., Domingo, V., Solanki, S . K., Kn¨ olker,\nM., Schmidt, W., Barthol, P. & Gandorfer, A. 2010, ApJ, 723, L180.\nStenflo, J. O. 1975, Solar Physics, 42, 79-105.\nWang, Y., Noyes, R. W., Tarbell T. D. & Title, A. M. 1995, ApJ, 447, 41 9.\nWedemeyer-B¨ ohm, S. & Rouppe van der Voort, L. 2009, A&A, 507 , L9.\nThis preprint was prepared with the AAS L ATEX macros v5.2.– 10 –\nFig. 1.— Snapshots of vertical velocity V zand enstrophy ω2= (∇×/vector u)2on the solar surface\nfor two simulation cases: without magnetic field (panels aandb) and with initial uniform\nweak vertical magnetic field, Bz0= 10 G (panels candd). Examples of small-scale vortex\ntubes are indicated by white and black squares.– 11 –\nFig. 2.— 3D snapshot of the top part of the computational domain sh ows the penetration of\nconvective vortex tubes (yellow isosurfaces) from the subphoto sphere into the low and mid\nchromospheric layers. The horizontal wavy surface indicates the distribution of temperature\nat 6400 K, and corresponds to a photosphere layer. The vortex t ubes (yellow isosurfaces)\nare shown for the enstrophy value of 0.0075 s−2. The vertical slices in the back illustrate\nthe vertical velocity distribution. Blue color corresponds to downfl ows, yellow-red shows\nupflows.– 12 –\nFig. 3.— Vortex tube structure in four different atmospheric layers : 200 km, 500 km, 650 km\nand 800 km above the solar surface (from left to right). Each row c orresponds to various\nquantities: a)temperature T, b) vertical velocity Vz, c) kinetic helicity χk,d) current helicity\nχm, ande) density ρ.– 13 –\nFig. 4.— Evolution of the velocity field (streamlines in panels a-c), magnetic topology\n(streamlines in panels d-e), and plasma parameter β(blue isosurface for β= 3 in panels\ng-i). Each column corresponds to simulation data 3 min apart. The grey isosurface shows\nT=5800 K; additional coloring from light yellow to orange indicates var iations of the mag-\nnetic field strength in the range from 0 to 1200 G. Coloring of the velo city streamlines in\npanelsa-ccorresponds to vertical velocities from −7 km/s (blue) to +7 km/s (red).– 14 –\nFig. 5.— a) 3D rendering of a vortex tube indicated by the arrow in Figure 2 sho ws the\nrelative distribution of the kinetic (blue isosurface) and current he licities (pink) for value\n−5000 cm/s2. The grey-yellow isosurface shows the distribution of temperatur e T=5800 K.\nPanelsb-fshow the distribution with height of mean vortex tube properties at different\nmoments of time with 20 sec cadence: b) relative temperature variations, c) relative density,\nd) horizontal speed /angb∇acketleftVh/angb∇acket∇ightand vertical velocity /angb∇acketleftVz/angb∇acket∇ight,e) horizontal /angb∇acketleftBh/angb∇acket∇ightand vertical /angb∇acketleftBz/angb∇acket∇ight\nmagnetic field strength, f) kinetic /angb∇acketleftχk/angb∇acket∇ightand magnetic /angb∇acketleftχm/angb∇acket∇ighthelicities. Thick blue solid curve\nin paneldshows the mean horizontal speed averaged over the whole domain a nd time. The\nsystematic variations at z >0.95 Mm are due to the top boundary conditions."    },    {        "title": "1607.05722v1.MHD_flow_and_heat_transfer_due_to_the_axisymmetric_stretching_of_a_sheet_with_induced_magnetic_field.pdf",        "content": "1 \n MHD flow  and heat transfer  due to the axisymmetric  stretching of a sheet   \nwith induced magnetic field  \nTarek M. A. El -Mistikawy  \nDepartmen t of Engineering  Math ematics and  Phys ics, Faculty of Engineering,  \nCairo University, Giza 12211, Egypt  \nAbstract  \nThe full  MHD equations, governing the flow due to the axisymmetric stretching of a \nsheet in the presence of a transverse magnetic field, can be cast in a self similar form. \nThis allows evaluation of the induced magnetic field and its effect on the flow and heat \ntransfer. The problem involves three parameters - the magnetic Prandtl number, the \nmagnetic interaction number, and the Prandtl number. Numerical solutions are obtained \nfor the velocity field, the magnetic field, and the temperature, at different values of th e \nmagnetic Prandtl number and the magnetic interaction number. The contributions of the \nviscous dissipation, Joule heating, and streamwise diffusion to the heat flux toward the \nsheet are assessed.  \nKey words: MHD flow , axisymmetric stretching , induced magn etic field , heat transfer , \nself similarity.  \n1 Introduction   \nFlow due to the  axisymmetric stretching of a sheet  was first introduced by Wang [1]  as a \nspecial case of the three dimensional flow due to the stretching of a flat surface . Its \npractical  applicati on is to extrusion processes and polymer and glass industries  \nThe problem allowed self similarity of the governing Navier -Stokes equations.  This \nencouraged researchers to add new features that maintained self similarity. Ariel [2] and \nHayat et al. [3] add ed partial slip. Ariel [4] added suction. Ariel [ 5,6] and Hayat et al. [ 7] \nconsidered non -Newtonian fluids. The latter article included heat transfer  with improper \ntransformation of the temperature , violating self similarity . The magnetohydrodynamic \n(MHD) flow of conducting fluids was handl ed by Ariel et al. [ 4] and Ariel [ 8]. Sahoo [ 9] \ntreated the MHD flow of second grade  fluids . The assumption of small magnetic \nReynolds number was adopted, leaving the imposed transverse magnetic field unaltered \nby inducti on. \nSelf similar formulation of flow problems is of such considerable value. Reduced to \nordinary differential equations, the problems are amenable to different analysis and \nsolution methods, and allow for evaluation and comparison of these approaches. For the \ncurrent problem, n umerical solutions adopting the shooting technique were presented by \nWang [1] and Fang [10]. Variants of the homotopy perturbation method were use d by 2 \n Ariel [2,11] and Ariel et al. [4]. Hayat and co -workers [3,7] implemented the homot opy \nanalysis method. Ackroyd’s method [12] , as well as a modification of which , was used by \nAriel [2,6,8,13]. Ariel also used residual minimization [6,8,13]. Perturbation expansions \nwere developed for small and large slip coefficient [2] and second grade f luid parameter \n[6]. Comparisons of the methods are found in [2,3,6,8].  \nIn this article, the full MHD flow equations; namely, the continuity, momentum,  energy , \nand Maxwell’s equations are shown to admit self similar transformation. Numerical \nsolutions are obtained for the velocity, induced magnetic field, and temperature. The \neffect of the induced magnetic field on the flow and heat transfer is demonstrat ed. \nTraditionally ignored heat generation and transfer processes such as viscous dissipation, \nJoule heat ing and streamwise diffusion are assessed.  \n2 Mathematical model  \nAn electrically conducting incompressible Newtonian fluid is driven by the axisymmetric \nstretching of a non-conducting  sheet . The stretching speed  along the  radial \nr -direction is \nr\n, where the stretching rate \n  is constant.  In the farfield  as the transverse coordinate \n~z\n, the fluid is essentially quiescent under pressure \np  and temperature \nT , and  is \npermeated by a stationary magnetic field in the \nz -direction of strength \nB .  \nIn the absence of an imposed and induced electric field, the equations governing this \nsteady MHD  flow are   \n0z r wruu\n (1) \nsqwsu uru\nruu p wu uuzzr\nrr r z r ) ( ) ( ) (2   \n (2) \nqqwsu wrww p ww uwzzr\nrr z z r ) ( ) ( ) (    \n (3) \n) ( qwsu qsz r \n (4) \n0z r srqq\n (5) \n2 2 2\n22\n2) ( ]) () (2[ ) () ( qwsu wu wruu TrTTk wT uTcr z z r zzr\nrr z r     \n (6) \nwhere \n),(wu are components of the velocity and \n),(sq  are components of the magnetic \nfield in the \n),(zr  directions, respectively , \np is the pressure and \nT  is the temperature . \nConstant are the fluid density \n , kinematic viscosity \n , electric conductivity \n , \nmagnetic permeability \n , specific heat \nc , and thermal conductivity \nk . 3 \n Our interest being the evaluation of the magnetic field and  its effect on the flow and heat \ntransfer, we opt to invoke the  simple surface conditions of no -slip, non -porosity, and \nfreestream temperature.  \n:0z\n \nr u , \n0w , \nTT  (7) \nThe farfield  conditions are \n:~z\n \n0~u , \npp~ , \nTT~ , \n0~q , \nBs~  (8) \nthe last two of which are consistent with the physical requirement of the farfield being \nfree from any current density.  \nThe problem admits the similarity transformations :  \n2/1)/(z\n, \n)( )(22/1 f w , \nfr u  (9) \n)]( 21[ g Bs\n, \ngr B q  2/1)(  (10) \n2 2 22\n21 2 2)]( [ 2 gr B f ff pp    \n (11) \n)]()/()()[/(2 r c TT  \n (12) \nwhere primes denote differentiation with respect to \n . Expression (11) for the pressure \nindicates radial -wise variation, due to the induced magnetic field.   \nThe problem becomes  \n] ) 21(2) 1(4[ 22 2ggP fg gP fg Pf fff fm m m \n (13a) \n0)0(f\n, \n1)0(f , \n0)(f  (13b) \n) (2 gffgP f gm\n (14a) \n0)(g\n, \n0)(g  (14b) \n2 22 21f g f fPr \n (15a) \n0)0(\n, \n0)(  (15b) \n21214 21fPfPr r \n (16a) \n0)0(\n, \n0)(  (16b) \nwhere \nmP  is the magnetic Prandtl number, \n /2B  is the magnetic \ninteraction number, and \nkc Pr /  is the Prandtl number. Note that the present choice  \nof characteristic length \n2/1)/(  and velocity \n2/1)(  renders \nmP =\nmR , the magnetic \nReynolds number.  4 \n  For practical applications, \nmP  is much smaller than unity  [14]. For negligible \nmP , the \nvelocity field is uncoupled from the magnetic field and is governed by the following \nproblem  \n0 22 f fff f \n, \n0)0(f , \n1)0(f , \n0)(f  (17) \nThis is the same pro blem formulated and solved by Ariel [ 8], under the assumption of \nnegligible \nmR . Numerical results for \n)(f , \n)(f , \n)0(f , and \n)(f  for different values \nof \n  are found therein.  \nThe corresponding problem for \ng  is \nf g\n, \n0)(g , \n0)(g  (18) \nwith the solution  \n)( )( f fg\n, \n\n\nd f f g )]( )([  (19) \nso that  \n)( )0( f g\n, \n\n\n0)]( )([)0( d f f g  (20) \nThe value of \n)(f  is obtained from the solution of the problem (1 7) for \nf .  \n3 Numerical method  \nSince a closed form solution is not possible, we seek an iterative numerical solu tion. In \nthe nth iteration, we solve, for \n)(nf , problem  (13a,b) with the right hand side  of Eq. (13a)  \nevaluated using the previous iteration solutions \n)(1nf  and \n)(1ng . Then we solve, for \n)(ng\n, Eq. ( 14a) with the known \n)(nf , together with conditions ( 14b). The iterations \ncontinue until the maximum error in \n)(f , \n)0(f , \n)0(g  and \n)0(g  becomes less than a \nprescribed tolerance \n =10−10. For the first iteration, we zero the right hand side of Eq. \n(13a) which corresponds to \n0)(0g .  \nThe numerical solution of the problems for \n)(nf  and \n)(ng  utilizes Keller’s two point, \nsecond order accurate , finite -difference scheme  [15].  A uniform step size \n  is used on \na finite domain \n0 . The value of \n  is chosen sufficiently large in order to insure \nthe asymptotic satisfaction of the farfield condition s. The non -linear terms  in the problem 5 \n for \n)(nf  are quasi -linearized, and an iterative procedure is implemented; terminating \nwhen the maximum error in \n)(nf  and \n)0(nf  becomes less than  \n. \nHaving determined \n)(f  and \n)(g , we solve the linear problem (1 5a,b) for \n)( , then \n(16a,b) for \n)( , using Keller’s sc heme .  \n4 Results  and discussion  \nThe results presented below are intended to explore the effect of the induced magnetic \nfield on the flow and heat transfer . Of interest  are the surface shear, the entrainment rate, \nthe \nr  and \nz  components of the induced magnetic field at the surface , and the constant \nand radial -wise varying constituents of the heat flux at the surface , which are  represented \nby \n)0(f , \n)(f , \n)0(g , \n)0(g , \n)0(  and \n)0( , respectively .  \nThe problem s for \n)(f  and \n)(g  involve t wo parameters, the magnetic Prandtl number \nmP\n and the magnetic interaction  number \n . The  problems for \n)(  and \n)(  involve \nthe Prandtl number \nrP , as a third parameter. All results presented below are for \nrP =0.72.  \nTable s 1 and 2  demo nstrate the effect of  varying \nmP when \n =1, and varying \n  when \nmP\n=0.1, respectively.  As \nmP  decreases, there is obvious tendency to the limiting case of \nmP\n=0. As \n  increases, the fluid motion is restrained more  and more , so that the induced \ncomponents of the magnetic field decrease. T he flow shapes as a boundary layer that \ndiminishes in size  leading to reduction in the  rate of fluid entrainment, and rise in  surface \nshear .  \nOn the right -hand -sides of Eqs. ( 15a) and (1 6a), the first terms represent Joule heating \nand streamwise heat diffusion, respectively, while the second terms represent heat \ndissipation . Table s 3 and 4  demonstrate the effect of these three processes. The predicted \nheat flux to the surface, represented by \n)0(  and \n)0( , is the additive effect of viscous \ndissipation and Joule heating. When both effects are neglected, th e thermal problem \npredicts zero flux. As expected, the larger the magnetic field the greater the contribution \nof Joule heating. As \n  increases, the contribution of the viscous dissipation to \n)0(  \nincreases, because of  the rise in \n2f . The fall then rise of \n)0(  can be  explained in view \nof the last two columns in Table 3, which dissect the viscous dissipation into its \nconstituents  due to \n2f  and \n2f . The higher the value  of \nf , the curvature  of the \nf  \nprofile, the faster the drop in the slope \nf ; hence , as \n increases, the effect of \n2f  rises \nwhile  the effect of \n2f  falls. The streamwise diffusion manifests itself through \n)0(  \nrelaying to it the Joule heating as well as the part of the viscous dissipation due to \n2f . 6 \n Profiles of the velocity an d induced magnetic field components are depicted in Fig. 1, for \nthe typical case of \nmP =0.1 and \n=1. Corresponding profiles of the temperature \nconstituents are shown in Fig.2. It is noted that the first set of profil es (in Fig. 1) reach the \nfarfield conditions much faster than the second set (in Fig. 2). As \n  increases, this \nbecomes more prominent. The farfield conditions are reached progressively faster by the \nfirst set and progressively slower b y the second set.    \n5 Conclusion   \nThe problem of the flow due to the axisymmetric  stretching of a sheet  in the presence of a \ntransverse magnetic field has been shown to admit self similarity of the full MHD  \ngoverning equations . Numerical solutions have be en obtained, revealing samples of \nwhich have been demonstrated. The following conclusions are drawn.   \nThe self similar formulation indicates radial variation in the pressure due to the induced \nmagnetic field, and in the temperature even when the surface t emperature is constant.  \nNo surface conditions on the induced magnetic field should be imposed. Rather, the \nrequirement of zero current density  in the farfield  should be honored . Note that the \nvectors of velocity \nV  (of magnitude \n)( 2 f ) and magnetic field \nB  are parallel, in \nthe farfield ; hence, the current density vector \nBVJ  vanishes.   \nAs the magnetic Prandtl number \nmP  diminishes, the problem and its solution app roach \nthose with \nmP =0. This is in accord with the conclusion of El -Mistikawy [16] in the \ncorresponding two-dimensional case.  \nThe increase of the imposed magnetic field  results in restraint of the flow, reducing the \nvelocity   and, consequently, the in duced magnetic field. Close to the surface, Joule \nheating increases , while  viscous dissipation  decreases then increases; being comprised of \ntwo parts, one rising and one falling.  \nStreamwise diffusion is important. It relays information from one constituen t of the \ntemperature to the other one.  \nFinally, it is noted that f eatures such as surface feed (suction or injection), velocity slip, \nthermal slip, and prescribed surface temperature or heat flux can be incorporated in the \nself-similar formulation.  \nRefer ences  \n[1] Wang  CY (1984) The Three Dimensional Flow Due to a Stretching Flat Surface . \nPhys . Fluids 27:1915 -1917 .  7 \n [2] Ariel  PD (2007)  Axisymmetric flow due to a stretching sheet with partial slip . \nComput . Math . Appl . 54:1169 -1183 .  \n[3] Hayat  T, Ahmad  I, Javed T (2009)  On comparison  of the solutions  for an \naxisymmetric  flow. Numer . Methods Part ial Diff. Equ. 25:1204 -1211 .  \n[4] Ariel  PD, Hayat  T, Asghar  S (2006)  Homotopy perturbation method and \naxisymmetric flow over a stretching sheet . Int. J. Nonlinear Sci . Numer . Simul . \n7:399-406.  \n[5] Ariel  PD (1992)  Computation of flow of viscoelastic fluids by parameter \ndifferentiation . Int. J. Numer . Methods Fluids 15:1295 -1312 .  \n[6] Ariel  PD (2001)  Axisymmetric flow of a second grade fluid past a stretching \nsheet . Int. J. Eng. Sci. 39:529-553.  \n[7] Hayat  T, Sajid  M (2007)  Analytic solution for axisymmetric flow and heat \ntransfer of a second grade fluid past a stretching sheet . Int. J. Heat and Mass \nTransf . 50:75-84.  \n[8] Ariel  PD (2009)  Computation of MHD flow due to moving boundaries . Int. J. \nComput . Math . 86:2165 -2180 .  \n[9] Sahoo  B (2010)  Effects of partial slip on axisymmetric flow of an electrically \nconducting viscoelastic fluid past a stretching sheet . Cent . Eur. J. Phys . 8:498-508.  \n[10] Fang  T (2007)  Flow over a stretchable disk . Phys. Fluids  19:128105 .  \n[11] Ariel  PD (2009)  Extended homotopy perturbation method and computation of \nflow past a stretching sheet . Comput . Math . Appl . 58:2402 -2409 .  \n[12] Ackroyd  JAD (1978)  A series method for the solution of laminar boundary layers \non moving surfaces . Z. Angew . Math . Phys . 29:729-741.  \n[13] Ariel  PD (2003)  Generalized  three -dimensional  flow due to a stretching  sheet . Z. \nAngew . Math . Mech . 83:844-852.  \n [14] Sears  WR, Resler EL Jr (1964)  Magneto -Aerodynamic  flow past bodies . Adv. \nAppl . Mech . 8:1-68.  \n [15] Keller  HB (1969)  Accurate difference methods for linear ordinary differential \nsystems subject to linear constraints . SIAM J . Numer . Anal . 6:8-30.  \n[16] El-Mistikawy TMA (2016) MHD flow due to a linear ly stretching sheet with \ninduced magnetic field . Acta Mech ., DOI: 10.1007/s00707 -016-1643 -0. "    },    {        "title": "2011.07736v1.Magnetic_Dynamic_Polymers_for_Modular_Assembling_and_Reconfigurable_Morphing_Architectures.pdf",        "content": " \nMagnetic  Dynamic Polymer s for Modular Assem bling and Reconfigurable \nMorphing  Architecture s \n \n \nXiao Kuang§, Shuai Wu§, Yi Jin, Qiji Z e, S. Macrae Montgomery, Liang Yu e, H. Jerry \nQi*, Ruike Zhao*  \n \nDr. X. Kuang, S.  M. Montgomery , Dr. L. Yue , Prof. H. Jerry Qi  \nThe George W. Woodruff School of Mechanical Engineering, Georgia Institute of \nTechnology, Atlant a, GA 30332, USA  \nE-mail: qih@me.gatech.edu  \n \nS. Wu,  Y. Jin, Dr. Q. Ze, Prof. R. Zhao  \nDepart ment of Mechanical and Aerospace Engineering, The Ohio State University, \nColumbus, OH, 43210, USA  \nE-mail: zhao.2885@osu.edu  \n \n§ These two authors are equal contribution first authors.  \n \nKeywords: magnetic soft material , covalent adaptive  polymer, modular assembly , \nshape morphing, reconfigurable architecture  \n  \nAbstract  \nShape morphing magnetic soft materials, composed of  magnetic particles in a soft \npolymer matrix, can transform  shape s reversibly,  remotely , and rapidly , finding diverse  \napplications in actuators, soft robot ics, and biomedical devices . To achieve on-demand \nand sophisticated shape morphing, the manufacturing  of structures with complex \ngeometry and magnetization distribution  is highly desired . Here, we report a magnetic \ndynamic polymer composite composed of hard-magneti c microparticles in a dynamic \npolymer network with thermal -responsive reversible linkages , which permit \nfunctionalities including targeted welding , magnetization reprogramming, and \nstructural reconfiguration . These functions not only provide highly desirable structural \nand material programmability and reprogrammability but also enable the \nmanufacturin g of structures with complex geometry and magnetization distribution.  \nThe targeted welding is exploited for modular assembl ing of fundamental building \nmodules with specific logics for complex actuation. The magnetization reprogramming  \nenables altering the morphing mode of the manufactured  structures. The shape \nreconfiguration under magnetic actuation  is coupled with network  plasticity to remotely \ntransform two-dime nsional tessellations  into complex three -dimensional architectures , \nproviding a new strategy of manufacturing functional soft architected  materials such as \nthree -dimensional kirigami . We anticipate that the reported magnetic dynamic polymer \nprovides  a new paradigm for the design and manufacturing of future multifunctional \nassemblies and  reconfigurable morphing architectures and devices. \n3 \n Shape -morphing  materials capab le of altering  the structural geometry upon external \nstimuli , such as heat, light , and magnetic field,  find diverse  applications in actuators ,[1, \n2] soft robots ,[3-5] flexible electronics ,[6-8] and biomedical devices .[9-11] Various \nstimuli -responsive smart materials, including shape memory polymer s,[12-15] \nhydrogel composites ,[16-18] liquid crystal elastomer s,[19, 20]  and magnet ic soft \nmaterials (MSMs) ,[21, 22]  have been implemented. In particular , MSM s, composed of \nhard-magnetic  materials in a soft polymer matrix , enable remot e, fast, and reversible \nshape morphing , and have  attract ed increasing attention in application s such as soft \nrobots  for minimally invasive surgery ,[23-25] where the actuation in confined and \nenclosed space s is required . The magnetic ally actuated shape -morphing is a result  of \nthe interactions between  the MSM’s magnetization (or ferromagnetic  polarity ) and the \napplied magnetic field . When the magnetization of embedded hard-magnetic particles \nis not aligned with the applied magnetic field , a body torque  is exerted  on the material  \nand leads to a deformation that tends to align the magnetization with the ma gnetic \nfield.[26-28] The on -demand  shape morphi ng is determined by b oth the structural  \ngeometry and the magnetization distribution . Therefore, manufacturing  of structures  \nwith intricate  geometry and magnetization distribution  is highly desirable, which \nenables  shape morphing in a programmable fashion . \n   From existing efforts, m olding with template -assisted post magnetization is very \ncommonly used to fabricate  the geometry and magnetization distribution .[29-31] To \nenhance the material and structural programmability , various advanced manufacturing  \ntechniques , including ultraviolet lithography [32, 33]  and additive manufacturing ,[34- \n4 \n 36] have recently been  developed  to create  complex shapes and magnetization \ndistribution via physical alignment of magnetic dipoles followed by chemical curing of \nthe matrix . For the above methods, the magnetization is coupled with the manufacturing  \nprocess, retarding the  manipulation of the geometry and the magnetization distribution  \nafter manufacturing . To tackle this issue , magnetic -assisted material assembl ing and \nmagnetization reprogramming approach es have been recently explored . In the \nmagnetic -assisted  material assembling, t he intrinsic dipole -dipole interactions enable \nthe self -organization of magnetic modules into larger  two-dimensional (2D) \nstructures .[37-40] Modules with hard magnet s as inclusion s are often used to enhance \nthe magnetic attraction force , however , it compromises the structure’s overall flexibility \nand homogeneity , which is not preferred if magnetic actuatio n with complex shape \nmorphing is needed .[37, 38]  In general, most existing magnetic -assisted assemblies are \nbased on physically  attaching the magnetic modules to each other, which can be easily \nbroken under external perturbation. Directly reprogramming magnetization distribution \nprovide s an alternative  approach to post -manipulat e actuation modes .[41-45] In this \ncase, remagnetiz ing the composite  using a  large  magnetic field  above the coerciv ity of \nthe magnetic material ,[41, 42]  or heating the magnetic material to above its Curie \ntemperature  to demagnetize before re-magnetiz ing it  [43, 44]  are the most widely used \nmethods. Another recently reported effort reprograms the magnetization  by physically \nrealigning the magnetic particles  confined in fusible  polymer microspheres  that are \nembedded in a deformed elastomer matrix .[45] Although  reprogramming \nmagnetization provides reconfigurable shape morphing, the material’s function can still  \n5 \n be limited by its unchangeable structur al geometry . \nIn this work , we develop  a magnetic dynamic polymer (MDP)  that enables both \nstructural and material programmability and reprogrammability with complex \ngeometry and magnetization distribution for multifunctional and reconfigurable shape \nmorphing. The MDP consists of  hard-magnetic microparticles (NdFeB) in a thermally \nreversible cross -linked dynamic polymer (DP) matrix (Figure  1a). As a proof of \nconcept , a DP matrix  bearing  thermally reversible Diels -Alder  reaction  between furan \nand maleimide  is used . The maleimide and furan groups can proceed with forward \nDiels -Alder (DA) reaction at low temperature to form adduct linkages , which can be \ncleave d via retro Diels -Alder  (rDA)  reaction upon heating .[46, 47]  The dynamically  \ncross -linked DP and MDP  can rearrange network topology by bond exchange reaction \n(BER) at mild temperature ( TBER) and go through  bond cleavage at elevated  temperature \n(TrDA), enabling the attractive  property of heat -induced reversible elastic -plastic \ntransition  after material manufacturing  (Figure 1b ). The MDP  integrate s three \nfunctional properties , including (i) welding -enhanced assembling, (ii) in -situ \nmagnetization  reprogram ming , and (iii) remotely controlled permanent shape  \nreconfiguration . We first illustrate the seamless welding of modules via forming n ew \ndynamic linkage s at the interfaces  to achieve  modular assembling  (Figure  1c). With the \ncleavage of linkages at TrDA, a small  magnetic field can chain up  dipole s along the field \ndirection for magnet ization reprogram ming , which further alter the shape morphing \nmodes under the actuation magnetic field  (Figure 1d ). Moreover, simultaneous \nmagnetic actuation and structural reconfiguration of the DP matrix  at mild temperature s  \n6 \n (TBER) allow remotely reshap ing the material  into a new stress -free architecture (Figure \n1e). Besides, t he MDP based mater ials and architectures with programmed shape and \nmagnetization distributions are capable of fast and reversib le morphing shapes under \nexternal magnetic fields  at room temperature .  \n \nFigure 1. Schemat ics of the working mechanism and functions of the magnetic \ndynamic polymer  (MDP) . a) Schematics  of the MDP  composition. NdFeB \nmicroparticles are embedded in a dynamic polymer (DP)  bearing reversible chemical \nbonds. b) Scheme of reversible elastic -plastic  transition via network topology transition \nin the Diels -Alder reaction -based DP at different temperatures. The bond exchange \nreaction between free furan and Diels -Alder adduct linkage for  network arrangement is \npredo minant at mild temperatures ( TBER), and reversible bond cleavage is favored at \nelevated  temperature ( TrDA). c) Schematic s of modular assembly and s eamless welding \nof MDP  modules  at a temperature near  TBER. d) Schematic s of magnetization  \nreprogramming by bond cleavage and dipole rotation under  a magnetic field at  TrDA. e) \nSchematic s of the magnetically guided  structural reconfiguration  of MDP by plasticity \nvia stress relaxation at TBER.  \n \n7 \n To prepare the DP matrix, a furan grafted prepolymer is first synthesized by the \nring-opening reaction between an epoxy oligomer and fururylamine. The prepolymer \nchains have an average of 13 pending furan groups , as indicated by the gel permeation \nchromatography (GPC)  measurement  (Figure S1 ). The linear prepolymer is cross -\nlinked by the bismaleimide cross -linker via DA reaction, forming thermally reversible \nadduct linkages, as evidenced  by Fourier transform infrared  (FTIR) spectroscopy \n(Figure S2 ). The maleimide to furan ratio ( r) is used to tune the mechanical property, \nnetwork cross -linking density, and thermal proper ty of DP  (Figure S3 ). The DP with r \n= 0.15 (Young’s modulus of 106 ± 9 kPa and glass transition temperature  of -35 oC) \nis selected for use in the rest of this paper  (Figure 2a ). To prepare MDP, NdFeB \nparticles with an average size of 25 μm are dispersed in the DP matrix (Figure S4 ). The  \nDP and MDP are soft stretahcale elastomers showing  break strain  over 200% . Young’s \nmodulus of MDP increases linearly from  106 kPa to 515 kPa with NdFeB \nmicroparticles concentration  varying from 0 to  20 vol% , respectively  (Figure S5). We \nchoose the NdFeB particle concentration of 15  vol% to manufacture MDP with low \nstiffness  and ideal magnetic properties for efficient magnetic actuation. After \nmanufacturing  and post magnetization , the obtained MDP shows Young’s modulus of \n400 ± 20 kPa and magnetization  of 75 kA  m-1 (Figure S6).   \n8 \n  \nFigure 2. Mechanical and thermomechanical characterization of the DP and MDP . a) \nTensile stress -strain curves of the DP and MDP. b) DMA heating curves for the DP and \nMDP  from -40 to 120  oC with the marked characteristic glass transition temperature  \n(Tg), bond exchange reaction temperature (TBER), and ret ro Diels -Alder reaction \ntemperature ( TrDA). c) Normalized relaxation modulus as a function of time in the stress \nrelaxation test of MDP at mild temperatures from 50 to 90  oC. d) DSC heating curves \nof DP and MDP from -60 to 150oC with ma rked characteristic temperature s. \n \nUnlike conventi onal chemical ly cross -linked polymer  and composites,  DP \nnetwork or covalent adaptable network polymer is a class of chemically cross -linked \npolymer with a sufficient amount of dynamic or exchangeable covalent bonds in the \npolymer network, which enables material flow and permanent shape change after \nactivatin g the dynamic bonds .[48-52] DP networks possess desirable attributes  of \nchemical cross -linking in thermosets and reprocessability in thermoplastics .[53-56] \nThe fabricated DP and MDP  here manifest heat-induced reversible  elastic -plastic \ntransition . At room temperature , DP and MDP show  excellent elasticity and low \n \n9 \n hysteresis  in the cyclic loading -unloading test (Figure S7 ), due to low Tg and stable \nchemical linkages. This is also evidenced by a rubbery plateau from 25 to 80 oC by \ndynamic mechanical ana lysis (DMA ) test (Figure 2b ). The rubbery modulus  of the \nMDP  slightly decrease s before 80  oC and sharply drops after 100 oC, owing to  the \ncleavage of the dynamic linkages . Note that the MDP show s a larger high -temperature \nmodul us and network stability than the DP due to  the potential polymer -filler \ninteraction.  Furthermore, c reep tests are conducted at different temperature s and \napplied stress to reveal temperature -sensitive mechanical properties  of the MDP . The \napparent zero shear rate viscosity ( η0) of the chemically cross -linked network is \nobtained by the creep test (Figure S8 ). The η0 of the MDP decreases from nearly 109 \nPas at room temperature  to 106 Pas at 90  oC, showing prominent temper ature -sensitive \nmechanical behavior . Aside from the temperature -dependent mechanical properties , the \nnetwork topology can also be rearrange d via dynamic b ond exchange reaction ,[57] \nwhich is evaluated by the stress relaxation  tests at various temperatures from 50 to 90  \noC. Figure 2c  shows the normalized relaxation modulus d ecreas ing with time . The \nrelaxation process is acc elerated at higher temperatures leading to prominent plasticity.  \nAccording to the Maxwell  model, t he network relaxation time  (τDP), defined by the time \nneeded to decay to 36.7 % of original modulus, decrease s from 153 min (at 50  oC) to \n46 s (at 90 oC). The underlying mechanism for temperature -sensitive elastic -plastic \ntransition upon heating  is the network breaking and rearrangement via dynamic  bond \ncleavage  and exchange , respectively . To evaluate the temperature -variant bond \nbreaking , differential scanning calorimetry (DSC) t esting  is condu cted. On the DSC  \n10 \n heating curves, a n endothermal peak  ranging from 50 oC to 120  oC is observed in both \nthe DP and the MDP due to continuous bond -breaking via rDA reaction (Figure 2d ). \nThe rDA reaction  degree as a function of temperature is semi -quantitatively evaluated  \nby the normalized  areal integration under the endothermal peak of the  DSC heating \ncurve  (Figure S9) . A gel point conversion (pgel~74%) is predicted between 80  oC-90 oC \n(see Supporting Information , dynamic polymer relaxation analysis ). The peak \ntemperature at around 100  oC suggests  the fastest bond -breakin g rate, which is denoted  \nas rDA temperature ( TrDA). It is noted that  the cleaved linkages can reform upon cooling  \nand annealing as supported by FTIR  (Figure S2c) . On the basis of the reversible elastic -\nplastic transition  of the MDP , τDP and magnetic dipole relaxation time ( τdiople) (see \nSupporting Information , magnetic dipole relaxation analysis ) can be tuned to \nmanipulate the shape and magnetization  by corporative  control  of heating temperature  \nand magnetic field for multifunctional reconfigurable morphing architectures.    \n  \n11 \n  \nFigure 3 . Magnetic -assisted modular assembling with seamless welding. a) Images  of \na long strip assembly consisting of  three MDP modules  via magnetic  attraction followed \nby near-infrared light (NIR) heating (80 oC for 5  min). b) Tensile stress -strain curves of \nthe original sample and the welded MDP processed samples under  different conditions. \nc) Effect of processing  conditions on the welding efficiency of the processed MDP. d) \nSchematic s of a square single -directional magnetization module  and a bidirectional \nmagnetization module  with five double -unit combination logics  via magnetic  attraction. \ne-h) Schematic designs, finite -element simulations , and experimental results of various \nassembled 2D planar structures  with programmed magnetization  for complex shape \nmorphing: a twist ing strip (e), a ‘ Z’-shape structure (f) , an ‘H’ -shape structure (g), and \na square annulus structure (h).  \n \nWe first  demonstrate the function of magnetic -assisted  assembling by magnetic \nattraction and seamless welding  of magnetic modules . Figure 3a  shows the  \nlongitudinal ly magnetized rectangular magnetic modules  automatically attach ing to \neach other rapidly (~  0.3 s ) with good contact. Afterward, the assembly is treated at 80 \n \n12 \n oC for 5  min either by direct heating or near-infrared ( NIR) light (Figure S10)  to be \nweld ed into an intact part. As  the bond exchange reaction between the contacting \ninterfaces generate s strong chemical bondin g, the welded assembly can sustai n large  \nstretch ing without bre aking  (Video  S1, Supporting Information ). The welding \nefficiency is quantified  by the tensile fracture strain ratio  of the welded samples to the \noriginal  one (Figure 3b ). The welding efficiency is 75% at room temperature after two \ndays due to the very slow dynamic reaction  at room temperature  (Figure 3c ). When at \n80 oC, the welding efficiency can increase  to 82 % for 5 min  treating  and 95  % for 20  \nmin.  \nWe further extend this concept to the modular assembly of stable 2D structures \nwith on -demand magnetization pattern s and geometries using different modules . We \nachieve this by using two basic simple building modules : a square with single -\ndirectional magnetization  (Module  1) and a square with bidirectional magnetization  \n(Module  2) (Figure 3d ). The magnetic attraction provides these two basic building \nmodules  with five different combination logics  (Video  S2, Supporting Information ). \nUnder certain boundary conditions , the response of these combination logics  under an \nout-of-plane magnetic field  generates  twisting, bending, twisting -bending, bending -\nfolding (same direction), and bending -folding (orthogonal direction) , respectively . \nThese logics can  be further used to achieve more intricate shape changes.  For example, \na strip with five pieces of Module  1 arranged in twisting logics  is assembled  and welded \nat 80  oC for 20  min. As shown in Figure 3e , the obtained straight strip assembly \ntransform s to a twisting shape rapidly  upon applying a magnetic field of 100 mT and  \n13 \n quickly recover s its original shape upon  the removal of the applied field  (Video  S2, \nSupporting Information ). The shape morphing of the assembled MDP is predicted by \nfinite -element analysis  (FEA) through  a user -defined  element subroutine ,[26] which  \nguide s the design of assemblies for target shape transformation under  the application of  \nmagnetic fields.  The simulation is conducted  using the experimental ly measured  \nmechanical  and magnetic  propert ies, showing good agreement with the experimental \nresult .  \nThe logics  presented above  are further used to assemble the building modules into \na rich set  of more intricate 2D magnetizat ion patterns on planar structures  for complex \nshape morphing  (Video  S2, Supporting Information ). In Figure 3f , the combination of \nbending, bending -folding  (same direction) , and bending -twisting  logics is  used to \ndesign a ‘Z’-shape structure  for large twisting mo rphing  under an applied magnetic \nfield of 12 mT , as predicted by our FEA simulation . An ‘H’-shape struct ure is assembled \nby using fifteen modules  with only bending and bending -folding (both types) logics to \ngenerat e large pop -up deformation under an external magnetic field  of 50 m T as \nillustrated in Figure 3g . We also assemble a closed shape  by using bending and \nbending -folding  (both types), display ing an undulating shape morphing under a \nmagnetic field of 100 mT (Figure 3h ). The simulation results match the experimental \nresults well , demonstrat ing that the modular assembling  strategy can be assi sted by FEA \nto design the shape -morphing architectures and their response under magnetic actuation . \nIt is noted that the magnetization of each module is retained after welding , which is  \nattributed to much longer τdiople than τDP at TBER (τdiople>>τDP > 0) (see Supporting  \n14 \n Inform ation , magnetic dipole relaxation analysis).  Our method of using the MDP  for \nmagnetic -assisted modular  assembling with welding enabl es material  integrity  for \ncomplicated shape morphing . Moreover, our assembling  strategy of using the logics  \ncan achieve an almost unlimited number of possible shape morphing designs .  \n \nFigure 4 . In-situ reprogramming  of magnetization in the MDP. a) Schematics of \nmagnetization  reprogramming of silicone encapsulated MDP array using a n NIR light \nsourc e and a photomask under a magnetic field. The magnetic particles  can be realigned \nby an external magnetic field at TrDA. b) Image of an as-fabricated 4 × 4 MDP array, an \naluminum photomask, and temperature profile after exposure  to NIR illumination. c) \nMicroscopic  snapshots of the in-situ realignment of NdFeB particle s in MDP at around \n110 oC under a 35 mT magnetic field. d ) Multiple cycles of magnetization  \nreprogramming from the initial isotropic dispersion  to chain -like structures along the \nexternal magnetic field direction . e-f) Design s, simulation s, and experiment al results of \nthe MDP array  with the initial magnetization (e), and reprogrammed magnetization  (f). \nScale bars in c-d, 500 µm. Scale bars  in b, e, f, 10 mm.  \n \nWe also achieve in-situ magnetization  reprogramming  via cleavage of dyna mic \nlinkages  at elevated temperatures. To selectively  reprogram the magnetization , \n \n15 \n photothermal heating by NIR ligh t illumination and photomask s is used ( Figure 4a ). \nUpon heating above TrDA, the viscosity is reduce d tremendously , and thus the magnetic \nparticles can move freely in the MDP, leading to dipole  realignment  along a small \nexternal magnetic field, which can be relocked after cooling.  As an example, in Figure \n4b, we fabricate a 4 × 4 MDP array encapsulated by silicone rubber  to illustrate the \nselective magnetization reprogramming  (Figure S1 1 and Video S3, Supporting \nInformation ). An aluminum foil as a photomask is used  to enable local photothermal \nheating of selectively exposed MDP cells to 110 - 120 oC. The magnetic particles in the \nheated  MDP cells physically rotate and reorient  under  a 35 mT  homogenous  magnetic \nfield provide d by a Halbach array magnet . The in-situ dipole realignment is observed \nunder digital microscopy coupled with a pair of electromagnetic coils (Figure S 12). As \nshown in Figure 4c , when the MDP viscosity reduces upon heating, the post-\nmagnetized NdFeB particles in  the form of small aggregation can rotate within 5 s  and \nthen tend to form a chain -like structure after 30 s under a 35 mT magnetic field (Video \nS3, Supporting Information ). Besides, due to the reversible elastic -plastic transition of \nMDP, the magnetization  realignment  can be repeated along external magnetic  fields \nwith different directions  (Figure 4d ). To test the efficacy  of magnetization \nreprogramming , the planar 4 × 4 MDP array is initial ly magn etized horizontally  in an \nalternating manner for each column of cells, as shown in Figure 4e . The 2D array \nmorph s into a three -dimensional (3D)  ‘W’ shape in a 44 mT out-of-plane magnetic field , \nwhich  is well captured by the  FEA simulation . Next,  a four-step reprogramming  \nprocedure is used to  reprogram the selective MDP cells with four different  \n16 \n reprogrammed magnetization directions, respectively  (Video S3, Supporting \nInformation ). As shown in Figure 4f , the targeted reprogrammed  magnetization  has the \npolarity oriented  in the diagon al of each cell pointing to the four centers . Upon applying \na 62 mT out-of-plane magnetic field , the reprogrammed MDP array  morphs into a 3D \nsurface with four dents . The experimental result match es the simulation  well, \nsuggesting  retained  magnetization  after reprogramming.  It is noted that the thermally \nreversible reaction also enable s hot recycling of MDP  (by both solvent and hot -\ncompression) (Figure S13 ). Compared with the existing magnetization reprogramming \ntechniques  for various magnetic soft composites , our MDP  offers additional advantages  \nof selective magnetizati on reprogramming under mild conditions , large remanent \nmagnetization  for efficient actuation, and sustainability.    \n17 \n \n \nFigure 5. Magnetic -assisted 3D structural reconfiguration of the MDP for complicated \nmultistable architectures . a) Schematic design  and the as -fabricated planar kirigami \nwith helical  cuts (helical design) . Black arrows  indicate magnetization . b) Magnetic \nactuation  of the planar helical structure . c) Stress releasing process of the magnetic -\nactuated helical structure  under the NIR irrad iation . d) Mechanical or magnetic \nactuation of reshaped bistable helical architecture . e-f) The force/rotation angle -\ndisplacement curves  and snapshots of the helical design in tensile test : initial planar  \nstructure (e) and reshaped 3D architecture (f) . g) Schematic design  and the as -fabricated \nplanar kirigami with concentric arc cuts (concentric arc design) . h) Magnetic actuation  \nof the planar concentric arc structure . i) Stress releasing process of the magnetic -\nactuated concentric arc  structure  under the NIR irradiation . j) Images  of the multistable \n3D kirigami architecture  with four stable states. k -l) The force -displacement curve s and \nsnapshots of the concentric arc  design in  tensile test : initial planar structure  (k) and \nreshaped 3D architecture (l) . Scale bar s for all images, 10 mm.  \n \nThe reshaping capability offered by the plasticity of the MDP can be harnessed to  \n18 \n manufacture 3D free-standing  intricate architectures with complex magnetization that \nwould otherwise b e very challenging to implement . In Figure 5a,  a planar kirigami \ndesign with four helical  cuts (or helical design) is first molded  and then post-magnetized \nin its mechanically  stretched -out configuration (Figure S14 ). The fast and reversible \nmagnetic actuation into a helical shape u nder a 75 mT out-of-plane magnetic field is \nshown in Figure 5b (Video S4, Supporting Information ). When applying  a magnetic \nfield and NIR heating  (~80 oC) simultaneously , actuated helical shape gradually \nreleases its internal stress due to the bond exchange in the polymer network ( Figure \n5c). Upon removing the magnetic field and the heating source after ~15 min , a stress -\nfree 3D helical architecture  with self-adjusted magnetization  is formed . The reshaped \nhelical architecture also comes with bistability, which can be triggered either by \nmagnetic actuation or mechanical loading (Figure 5 d and Video S4, Supporting \nInformation ). Such a 3D architecture would b e very challenging to manufacture but can \nbe easily achieved using our MDP approach. We conducted tensile tests  to compare the \nmechanical behaviors and properties of the as-fabricated 2D design  and the reshaped \n3D architecture . Both structures are loaded by stretching the central point while fixing \nthe edges, with results illustrated  in Figure 5e,  f (also see Figure S1 4 and Video S5, \nSupporting Information ). Before the reshaping process, the  2D structure is monostabl e, \nas the force  (black curve) and rotation angle (red curve) is monotonically increasing \nwith the displacement (Figure 5 e). After reshaping, the 3D helical architecture exhibits \nbistable behavior, which is precisely characterized by the force -displacement curve \nfrom the tensile test ( Figure 5f ) loaded from the initial downward pop -out stable state .  \n19 \n During loading, the architecture reaches the critical snapping point at around 14 mm \ndisplacement, and the structure quickly snaps through to the upward pop -out state at 21 \nmm displacement, where the force reduces to zero .  \nA more complicated 3D kirigami architecture based on concentric arc design with \nmultistable states is manufactured by the same  MDP stress releasing approach. Figure \n5g shows the design and the as-fabricated planar kirigami geometry comprising seven \nlayers of con centric arcs connected by hinges , which is initially magnetized in a \nmechanically stretched -out configuration  (Figure S15). Its magnetic actuation under a \n100 mT out -of-plane magnetic field is shown in Figure 5 h. After expos ing to  the NIR \nlight and an out -of-plane  magnetic field  for about 30 min (Figure 5i ), the planar \nconcentric arc structure  transforms  into a stress -free kirigami architecture with four \nstable states , as illustrated in Figure 5 j (see Video S6, Supporting Information ). The \ndifferent mechanical  behavior s before and after the reshaping  process  are evaluated by \ntensile test s with the same boundary conditions as previous  (Figure S15). Figure 5 k \nshows that t he initial planar kirigami is monostable . In contrast, as illustrated in Figure \n5l, the reshape d 3D kirigami architecture has four stable  states  and three  snapping  \npoints  (marked by orange and blue circles  for two loading paths ) (see Figure S15  and \nVideo S7, Supporting Information ). By either magnetic actuation or mechanical \nstretching  starting from the stable state I , two stable states  II and IV are captured with \ntwo snapping points identified between State I  and II, and State II  and IV, respectively . \nThis loading path is  illustrated by the black solid curve and top -row insets . Note that \nafter the architecture  snaps up  at the second snapping point, it  compress es the load cell  \n20 \n until reaching  stable state IV . Stable state III  can be obtained  from a different loading \npath when stretch ing from an intermediate compressed state,  indicated  by the black \ndashed curve and bottom -row insets  of Figure 5 l. This state  shows an overall upward \nconfiguration with the middle part pushed down. Af ter reaching the critical high -energy \npoint, the middle part snaps up , and the architecture achieves stable state I V (Video 7 , \nSupporting Information ). To our best knowledge,  this is the first time that complex 3D \narchitectures are  manufactured through simple remotely controlled shape morphing of \nplanar structures . The stress releasing of the MDP can be  a versatile approach to \nmanufacture functional free-standing 3D architectures such as 3D kirigami and origami  \ntessellations, which finds broad applications in morphing architectures and \nmetamaterials with programmable complex geometries and novel  mechanical \nproperties .[58-61] \nThe MDP’s merits are emphasized through its covalent adaptive network  and \nmagnetic -responsive feature. Comparing with previous shape programmable materials  \n(Figure S16 and Table S1) , MDP  stand s out  with outstandi ng performance including  \nuntethered,  fast, and reversible actuation , as well as excellent  material and structural \nprogrammability and reprogrammability . Besides , MDP s show prominent functional \nproperties, including welding , reshaping and recycling. Thus, the multifunctional \nreconfigurable morphing beha viors distinguish  MDP s from the existing shape \nmorphing and other magnetic soft materials [21, 62] . \nIn summary, we report a magnetic dynamic polymer composite to create structures  \nwith complex geometry and magnetization distribution for modular  assembling and  \n21 \n reconfigurable shape morphing  architectures . The dynamic polymer network \nrearrangement and the magnetic dipole relaxation are tuned by cooperative  control of \nthe temperature field and the magnetic field . Functional properties  and applications , \nincluding  seamless welding  of modul ar assembly with  targeted functional actuation , \nmagnetization reprogramming for reconfigurable actuation modes , and remote -\ncontrolled structural reconfiguration  with unusual properties , are demonstrated . The \nconcept of the magnetic dynamic polymer by  merging  covalent adaptive network \npolymer s and magnetic material s can be extended to diverse magnetic soft  material s, \nusing various stimuli -responsive dynamic  reactions  and numerous magnetic materials, \nwith tunable mechanical , rheological,  and magnetic properties . As magnetic soft \nmaterials  gain increasing attention, the magnetic dynamic polymer can influence  many \nareas beyond reconfigurable shape morphing. The unique performance of welding and \nremote -controlled  structural reconfiguration  in manufacturing , healing during service, \nand recycling at the end of service can provide a green material  and enhance \nfunction ality of magnetic soft materials. We envision the magnetic dynamic polymer \nand its derived functions offer great potentials for  next-generation multifunctional \nassemblies, reconfigurable shape morphing architectures and devices.  \n \nExperimental Section  \nFuran grafte d prepolymer synthesis : 50 g Poly(ethylene glycol) diglycidyl ether and \n9.8 g furfurylamine in the stoichiometric ratio were mixed in 15  g dimethylformamide \n(DMF)，and 0.374  g of 2, 6-Di-tert-butyl -4-methylphenol  was dissolved to the above  \n22 \n solution in a round bottle flask. After degassing under vacuum for 5 min , the mixture \nwas sea led for the reaction by stirring at 80  oC for 20 h and then 110  oC for another 4 h \nbefore cooling down. The obtained viscous dark yellow solution was stored at room \ntemperature before use. For prepolymer characterization, the solution was dr ied in a \nvacuum  oven at 70 oC for over one day . All reagents were purchased from Sigma -\nAldrich (St. Louis, MO , USA ) without further purification.  \nDynamic polymer and magnetic dynamic polymer preparation : Dynamic polymer (DP) \nwas prepared by cross -linking the furan grafted prepolymer with bismaleimide cross -\nlinker  (Sigma -Aldrich , St. Louis, MO , USA ). Briefly, linear prepolymer  solution \n(containing 20  wt% of solvent) and bismaleimide with various maleimide/furan ratios \n(r) were manually mixed at room temperature. To manufacture  magnetic dynamic \npolymer (MDP ), 15 vol% NdFeB microparticles (134.0  wt% to the polymer matrix) \nwith an average size of 25  µm (Magnequench , Singapore ) were added into the \nprepolymer and cross -linker mixture (with r = 0.15) followed by manually mixing to \nobtain  a homogeneous blend. The resin mixture and the composite resin were poured \non a polytetrafluoroethylene ( Teflon® PTFE , McMaster -Carr, Elmhurst, IL , USA ) film. \nThe curing was conducted with pr ecuring at 50 oC for around 60 min to form a soft gel \nand then po st-treated in a vacuum oven at 6 0 oC for one day  to remove most of the \nsolvent. To obtain films with controlled thickness, t he samples  were hot compressed at \n110 oC with spacers and PTFE film separator. The  samples  were cooled down and stored \nat room temperature for at least one day before characterization. The materials were \npost-magnetized with an initial configuration under impulse magnetic fields (1.5 T)  \n23 \n generated by an in -house built impulse magnetizer.   \nCharacterization. The uniaxial tensile test s, dynamic thermomechanical measurements, \nand stress relaxation test s were performed on a dynamic mechanical analysis (DMA) \ntester (DMA 800, TA Instruments, New Castl e, DE, USA ) using rectangular samples \n(dimension: about 2 5 × 4 × 0.9 mm). In the uniaxial tension tests, the strain rate was \n0.5 min-1 and three specimens were tested for each type  of the sample s for report ing the \naverage resu lts. The Young’s  modulus was calculated  by the secant modulus at 0.5% \nstrain. DMA test was conducted using  a constant oscillation amplitude of 30 μm, a \nfrequency of 1 Hz, and a force track of 125 %. The temperature was ramped from -60 \noC to 120 oC at a heating rate of 3 oC min-1. In the stress -relaxation experiments, the \nsamples were equilibrated at a  predetermined temperature for 10  min, and then a \nconstant strain of 2 % was applied to monitor the evolution of  stress as a function of \ntime.  Differential Scanning Calorimetry Testing (DSC) was measured on a Q200 DSC  \n(TA Instruments, New Castl e, DE, USA) using T -zero aluminum pans under a nitrogen \npurge. The testing temperature ranges from −80 to  150 °C with a heating and cooling \nrate of 5 oC min-1. The temperature field of samples during heating was captured using \na Seek Thermal  CompactPRO  thermal image camera ( Tyrian Systems, Inc. , Santa \nBarbara, CA , USA ) \nFinite -element analysis.  The shape actuation of MDP architectures and array  under \nexternal magnetic fields were simulated using a user -defined element subroutine \nimplemented in the finite -element analysis software ABAQUS 2019 (Dassault System, \nDassault System, Providence, RI, USA). The input parameters were used: Young’s  \n24 \n modulus E = 400 kPa for MDP and E=130  kPa for silicone rubber, the bulk modulus \nK = 1,000 E (approximate incompressibility), the magnetization of the MDP Mr = 75 \nkA m−1 (15 vol % of magne tic particles), and the uniform external magnetic field.    \n \nAcknowledgments  \nX. K. and S. W. contribute equally to this work.  R.Z., Q.Z., S.W., and Y .J. acknowledge \nsupport from NSF Career Award CMMI -1943070 and NSF Award CMMI -1939543.  \nH.J.Q. , X.K, S.M  and L.Y. acknowledge the support of AFOSR grants (FA9550 -16-1-\n0169 and FA -20-1-0306; Dr. B. -L. “Les” Lee, Program Manager), the gift funds from \nHP, Inc. and Northrop Grumman Corporation.  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Recent proposals using structured laser bea ms\nhave demonstrated the possibility to produce regions where intense oscillating magnetic fields are isolated from the\nelectric field. In these conditions, we show that technologi cally feasible Tesla-scale circularly polarized high-fre quency\nmagnetic fields induce purely precessional nonlinear magne tization dynamics. This fundamental result not only opens\nan avenue in the study of laser-induced ultrafast magnetiza tion dynamics, but also sustains technological implicatio ns\nas a route to promote all-optical non-thermal magnetizatio n dynamics both at shorter timescales—towards the sub-\nfemtosecond regime— and at THz frequencies.\nKeywords: Ultrafast dynamics, Non-linear dynamics, Chiral behavior\n1. Introduction\nThe pioneering work on ultrafast demagnetization in Ni[1]\npaved the way towards a large number of theoretical and\nexperimental studies on magnetization dynamics at the\nfemtosecond (fs) time scales induced by ultrashort laser\npulses[2–25]. In these studies the dynamics is mediated\nprimarily by the electric field (E-field), which can excite\nnon-equilibrium states[5–9], demagnetize the sample,[1,14–21]\ngenerate localized charge currents[24,25], or induce the\ninverse Faraday effect[22,23]. While most of the techniques\nare mediated mainly by the E-field, other techniques, such as\nthe excitation of phononic modes[26], have recently provided\nroutes for non-thermal magnetization manipulation.\nAn appealing alternative to induce coherent magnetization\ndynamics consists on the use of magnetic fields (B-field).\nThe role of the B-field in ultrafast magnetization dynamics\nhas been extensively studied, specially in the regime of\nlinear response to THz fields[27–32]. At this picosecond\ntime scale, few Tesla (T) are required to introduce small\ndeflections from the equilibrium magnetization direction,\nwhile tens of T are needed for achieving complete switch-\nCorrespondence to: Email: luis.stsj@usal.esing. Higher driving frequencies, that could break into\nthe femtosecond timescale, would require very high B-\nfield amplitudes.Although intense magnetic fields can be\nachieved, for example, using plasmonic antennas[33], in such\nregime, the associated E-field would potentially demagneti ze\nthe sample[34]or even damage it. Besides, although sub-\nstantial advances have been made towards the generation of\nelectromagnetic fields in the range of THz ( 0.1to30 THz ),\ntheir intensity is still small as compared to the infrared\ncase[35–37].\nIn this work we introduce an appealing alternative to drive\nmagnetization dynamics at the sub-picosecond timescale,\nby using isolated ultrafast intense B-fields. Recent devel-\nopments in structured laser sources have demonstrated the\npossibility to spatially decouple the B-field from the E-\nfield of an ultrafast laser pulse. For instance, azimuthally -\npolarized laser beams present a longitudinal B-field at the\nbeam axis, where the E-field is zero[38]. Depending on\nthe laser beam parameters, the contrast between the B-field\nand E-field can be adjusted, so to design a local region in\nwhich the B-field can be considered to be isolated[39]. In\nsuch region, the stochastic processes driven by the E-field\ncould be avoided, and the coherent precession induced by\n12 L. S´ anchez-Tejerina et al.\nthe B-field can be exploited. Indeed, azimuthally-polarize d\nlaser beams have been shown to induce isolated mili-Tesla\nstatic B-fields[40], with applications in nanoscale magnetic\nexcitations and photoinduced force microscopy[41,42]. More\nrecently, ultrafast time-resolved magnetic circular dich roism\nhas been proposed[43]. In addition, theoretical propos-\nals[39,44]and experiments[45,46]have raised the possibility\nto generate isolated Tesla-scale fs magnetic fields by the\ninduction of large oscillating currents through azimuthal ly\npolarized fs laser beams.\nOur theoretical study unveils the non-linear, chiral, pre-\ncessional magnetization response of a standard ferromagne t\nto a Tesla scale circularly polarized ultrafast magnetic fie ld\nwhose polarization plane contains the initial equilibrium\nmagnetization. First, we show in section 2 the feasibility\nto use state-of-the-art structured laser beams to create a\nmacroscopic region in which such B-fields are found to\nbe isolated from the E-field by particle-in-cell (PIC) sim-\nulations. Then, we present our micromagnetic ( µMag)\nsimulations for moderate fields in section 3 showing the\npresence of measurable magnetization dynamics in CoFeB\nwhen a circularly polarized 10ps B-field pulse of 10 T\nand central frequency 30 THz is applied. Additionally,\nwe compare the dynamics triggered by a B-field with lin-\near polarization, circular polarization with the polariza tion\nplane perpendicular to the equilibrium magnetization, and\ncircular polarization with the polarization plane paralle l to\nthe equilibrium magnetization. Measurable magnetization\ndynamics are found in the later case. In section 4, we provide\nfor a complete analytical model to describe such dynamics,\nand compare it with full µMag simulations. This model\nallows us to predict the complete magnetization switching\nby using 1 ps, 275 T ,60 THz , B-field pulses, verifed by\nfullµMag simulations. Finally, section 5 summarizes the\nmain conclusions of the work and gives some perspectives\non possible implications in the field.\n2. Spatially isolated circularly polarized B-fields out of\nstructured laser beams\nIn order to study the interaction of an isolated, circularly\npolarized B-field with a standard ferromagnet (CoFeB), we\nconsider a B-field, B, oscillating in the xzplane (see Fig.\n1(a)) given by\nB(t) =b(t)eiωt+b∗(t)e−iωt(1)\nb(t) =B0\n2F(t)/parenleftig\ncosθ0ˆux+sinθ0eiφ0ˆuz/parenrightig\n, (2)\nwhereωis the central angular frequency, ( ω= 2πf),B0is\nthe amplitude, and θ0andφ0define the relative amplitude\nand phase between the xandzcomponents, respectively.\nF(t)is the field envelope, given by F(t) = sin2(πt/Tp)\nfor0≤t≤Tp, withTp= 3/8tpits full duration, tpbeing\nthe full-width-at-half-maximum (FWHM) pulse duration in\nFigure 1. a) Sketch of the system under consideration. A circularly\npolarized magnetic field illuminates a magnetic sample whos e dimensions\nare smaller than the region for which the E-field can be consid ered\nnegligible. This field can trigger ultrafast magnetization dynamics. b)\nTwo crossed azimuthally polarized beams of 30 THz and peak in tensity\n2.1×1013W/cm2define a spatial region of ≃100nm in which the E-field\nis lower than 100 MV /m, as depicted in panel. In such region, a constant\nB-field of amplitude 10.5 T and central frequency 30 THz is fou nd.\nintensity. A right-handed—RCP— (left-handed—LCP—)\ncircularly polarized B-field in the xzplane corresponds to\nφ0=π/2(φ0=−π/2) andθ0=π/4, while a linearly\npolarized B-field corresponds to φ0= 0orπ.\nIn our simulations, we do not include any E-field coupling,\nas the B-field is assumed to be isolated. Such assumption\nis valid for CoFeB in spatial regions where the E-field is\nlower than 100 MV/m, for which the demagnetization has\nbeen predicted to be less than 7%[14,48]. The conditions\nfor which an intense circularly polarized B-field can be\nfound spatially isolated from the E-field can be obtained\nby using two crossed azimuthally polarized laser beams, as\nsketched in Fig. 1(b). We have performed PIC simulations\nusing the OSIRIS 3D PIC code[49–51], in order to show\nhow such isolated B-fields can be achieved with the state-\nof-the-art ultrafast laser technology. We have considered\ntwo orthogonal azimuthally polarized laser beams with wais t\nw0= 3.125λ= 31.25µm, a central wavelength of λ= 10\nµm (30 THz), and E-field amplitude of 12.5 GV /m (peak\nintensity of 2.1 ×1013W/cm2) at their radius of maximum\nintensity, w0/√\n2. The temporal envelope is modeled as\na sin2function of 88.8 fs FWHM. Due to computational\nlimitations the temporal envelope is much shorter than thos e\nconsidered in the µMag simulations presented in this work,\nwhich lies in the ps regime. However, we do not foresee\nany deviation in the results presented if longer pulses with\nsimilar amplitudes are considered.\nIn Fig. 1(b) we also show the spatial distribution of the\nB-field (color background) and the E-field (contour lines) at\noverlapping region. We have highlighted the region in which\nthe E-field is lower than 100 MV/m, and thus the E-field\ncan be neglected against the B-field. Thus, we can define aMagnetization dynamics driven by structured lasers. 3\nregion of radius ≃100nm in which the B-field exhibits a\nconstant amplitude of 10.5 T and the E-field is maintained\nbelow100MV/m. Though the use of additional currents,\nlike in Refs.[39,44], could enhance the B-field amplitude, our\nsimulations demonstrate that moderately intense laser bea ms\ncan already reach the B-field amplitudes required to observe\nthe non-linear magnetization dynamics described below.\n3. Nonlinear magnetization response to ultrafast B-\nfields\nThe interaction between the oscillating B-field and the mag-\nnetization is given by the Landau-Lifshitz-Gilbert (LLG)\nequation[29,52]\n/parenleftig\n1+α2/parenrightigdm\ndt=−γm×Beff−αm×(m×Beff)(3)\nwhere mis the normalized magnetization where both spatial\nand temporal dependencies are implicitly assumed, αis\nthe Gilbert damping parameter, and Beffis the effective\nmagnetic field. We have performed µMag simulations using\nthe well-known software MuMax3[53]to solve the LLG\nequation. The system under study is sketched in Fig. 1(a),\nwhere we consider a circular nanodot with 1 nm thickness\nand64 nm diameter discretized into 1 nm cubic cells.\nThe material parameters correspond to CoFeB grown over\na heavy metal layer: inhomogeneous exchange parameter\nA= 19 pJ/m, saturation magnetization MS= 1 MA/m,\nperpendicular uniaxial anisotropy (i.e. the anisotropy fie ld is\ndirected along the zdirection) parameter Ku= 800 kJ /m3,\nDzyaloshinkii-Moriya interaction (DMI) D= 1.8 mJ/m2\nand Gilbert damping α= 0.015.\nIn Fig. 2(a) we show the in-plane magnetization dynamics\n(perpendicular to the equilibrium configuration, mz= 1)\ninduced by RCP and LCP B-fields lying in the xzplane.\nNote that the equilibrium magnetization lies in the polariz a-\ntion plane. In both cases, B0= 10 T ,f= 30 THz , and\ntp= 10 ps . We can observe a magnetization precession\naround the zaxis triggered by a non-linear chiral response to\nthe B-field. While the RCP B-field induces a measurable\nnegative xcomponent, the LCP leads to a positive one.\nAfter the pulse, the precession dynamics is dominated by the\nanisotropy field, and the system starts to precess around the\nz-axis. Note that the broad trace is due to the subsequent\nmagnetization oscillations during the interaction with th e\npulse.\nThe non-linear mechanism underlying such behavior can\nbe understood as follows (see bottom part of Fig. 2(a)).\nAt an initial time t= 0 , in which m(black arrow) lies\nin the polarization plane of the circularly polarized B-fiel d\n(red arrow), being perpendicular to it, a transverse torque\nτ(green arrow) drives mout-of-plane from this initial\nposition. During the next quarter-period, τdecreases and\nrotates, inducing a precession of maround its initial axis.\nFor the second quarter-period, τincreases again keeping its\nFigure 2. Micromagnetic simulation results of the temporal evolutio n (color\ncode) of the magnetization components ( mx,my) of CoFeB excited by B-\nfields with different polarization states. a) RCP (yellowish color scale) a nd\nLCP (greenish color scale) B-fields ( B0= 10 T ,f= 30 THz,tp= 10\nps). The RCP (LCP) B-field induces a measurable negative (pos itive)mx\ncomponent. In both cases the anisotropy field induces a prece ssion of\nmaround the equilibrium configuration. The bottom part sketc hes the\nmechanism during a B-field period of constant amplitude. The B-field\n(red), magnetization (black) and torque (green) vector rep resentations at\nfour different times reveal the magnetization dynamics mechanism ov er one\nperiod. b) Linear polarization along x (yellowish trace) or y (greenish trace).\nc) Circular polarization perpendicular to the equilibrium magnetization with\nRCP (yellowish trace) and LCP (greenish trace) helicities.\nrotation but, at t=T/2, it reverses its rotation direction,\nthus sweeping only half of the plane perpendicular to m.\nAs a result, along a whole period, the torque component\nperpendicular to the polarization plane averages to zero,\nwhile a residual contribution along the intersection of the\npolarization plane and the plane perpendicular to mremains.\nWith long lasting multicycle laser pulses it is then possibl e\nto accumulate the small torque along the polar coordinate on\nthe polarization plane, θ, so as to promote the system to a\ntargeted non-equilibrium state. This is reflected in Fig. 2( a),\nwhere the magnetization components mxandmyare non-\nzero at the end of the pulse, and therefore the magnetization\nis not aligned along the anisotropy direction, z. A more\ndetailed scheme of the non-linear mechanism is displayed\nin Supplementary Videos 1 and 2 for both, RCP and LCP\nB-fields, revealing the chiral nature of the reported e ffect.\nTo highlight the importance of the polarization state and\norientation to get the non-linear response, Figs. 2(b) and\n2(c) depict the temporal evolution of the magnetization com -\nponents (mx,my) obtained from full micromagnetic simula-\ntions for a linearly polarized B-field, being perpendicular to4 L. S´ anchez-Tejerina et al.\nthe equilibrium magnetization, and for a circularly polari zed\nB-field (either RCP or LCP), where the polarization plane is\nperpendicular to the equilibrium magnetization. Whereas i n\nboth cases a small magnetization deflection is observed, the\nnet torque exerted by the field on the magnetization over a\nperiod is null, and the magnetization recovers its equilibr ium\nstate after the B-field pulse. In addition, at frequencies la rger\nthan few tens of THz the response is not enough to promote\nsignificant change on the magnetization even for a B-field as\nhigh asB= 10 T . Consequently, in the cases presented\nin Figs. 2(b) and 2(c), the response is completely linear and\nthe magnetization comes back to the initial configuration at\nthe end of the B-field pulse. Nonetheless, for a circularly\npolarized B-field (either RCP or LCP) with the equilibrium\nmagnetization lying in the polarization plane, the non-lin ear\nchiral phenomenon described above triggers the magnetiza-\ntion out of equilibrium as shown Fig. 2(a). This dragging,\nbeing a non-linear e ffect, is sensitive to the B-field envelope\nand does not cancel out at the end of the pulse.\n4. Analytical model\nTo give insight into the nonlinear mechanism introduced in\nprevious section and sketched in Fig. 2(a), we derive an\napproximated analytical model. The exchange field is not\nincluded in the model because we assume that the sample\nremains uniformly magnetized. Besides, we neglect the\nanisotropy and DMI fields—which are small if compared\nto the external one— and the damping term. Similar\nassumptions has been proven reasonable at this time scale\nin previous studies[29]. With these approximations in Eq.\n(3), the magnetization dynamics out of the polarization pla ne\nreads as\ndmy\ndtuy=−γ′m/bardbl×B, (4)\nm/bardblbeing the magnetization in the polarization plane and\nγ′=γ//parenleftig\n1+α2/parenrightig\n. Considering the initial magnetization in\nthezdirection, myat any time tis given by\nmy(t)uy=−γ′/integraldisplayt\n0m/bardbl(τ)×Bdτ. (5)\nThe cartesian components of the magnetization can be de-\ncomposed at each point in its Fourier components,\nmj(t) =/summationdisplay\nqmj\nq(t)eiqωtj={x,y,z}.(6)\nUsing Eqs. (5) and (6) in the simplified LLG equation, we\nobtain\n/summationdisplay\nqdm/bardbl\nq(t)\ndt+iqωm/bardbl\nq(t)eiqωt=+γ′2/summationdisplay\nq/integraldisplayt\n0m/bardbl\nq−1(τ)×b(τ)eiqωτdτ×B(t)+\n+γ′2/summationdisplay\nq/integraldisplayt\n0m/bardbl\nq+1(τ)×b∗(τ)eiqωτdτ×B(t).(7)\nAssuming that the magnetization components in the polar-\nization plane, m/bardbl\nq±1, and the B-field envelope, b(t), evolve\nslowly, considering b(0) = 0 , and selecting only the slowly\nvarying terms ( q= 0), Eq. (7) transforms into\ndm/bardbl\n0(t)\ndt=−2iγ′2\nωm/bardbl\n0(t)×(b(t)×b∗(t)). (8)\nIt is well known that the e ffective field dependence on\nthe magnetization can lead to non-linear e ffects[48,54,55].\nHowever, it must be noticed that, di fferently from those\ncases, the described e ffect is non-linear on the external field,\nnot on the effective field. Moreover, it is proportional to the\ngyromagnetic ratio and the inverse of the frequency, being\nequivalent to a drift magnetic field /vectorBdgiven by,\nBd=γ′\n2ωsinφ0(Bx×Bz). (9)\nUsing this definition, Eq. (8) describes the slowly varying\nLLG dynamics in terms of the drift field, Bd. Eqs. (8) and\n(9) constitute the main contribution of the present work, as\nthey reveal a second-order dependency of the magnetization\ndynamics with the external B-field. From Eq. (9) we can\nalready infer that Bdis maximal for circular polarization,\ndecreases with the ellipticity, and is zero for a linearly\npolarized B-field ( φ0= 0 orπ). Note also the chiral\nnature of the presented mechanism, as the direction of Bdis\nhelicity dependent. Finally, we stress the purely precessi onal\nnature of Bd—being linear with the gyromagnetic ratio—and\nits inverse proportionality with the driving frequency. It is\nworth noting that a small misalignment of the azimuthally\npolarized laser beams would convert the circularly polariz ed\nmagnetic field into elliptically polarized magnetic field,\nand/or would introduce a small angle between the initial\nmagnetization and the polarization plane. Nonetheless,\nit is possible to decompose the total magnetic field in a\ncircularly polarized magnetic field in the xzplane and a\nlinearly polarized magnetic field in the ydirection. However,\nthis later component would not a ffect the slow dynamics\npresented here.\nWe now analyze the dependency of the magnetization\ndynamics on the B-field, both with the analytical model rep-\nresented by Eq. (8), and the full micromagnetic simulations ,\nwhere all the interactions on the e ffective field, as well as\nthe damping, are included. To highlight the accuracy of our\nmodel based on the equivalent drift field, we compare the\ntotal rotation of the magnetization from our simulations wi thMagnetization dynamics driven by structured lasers. 5\nthe magnetization rotation induced by the drift B-field, Bd,\nwhich can be computed as\n∆θ=γ′/bracketleftiggγ′\n2ωsinφ0(BxBz)/bracketrightigg\ntp. (10)\nFig. 3, presents the induced magnetization rotation as\nderived from the analytical model (solid lines) and the\nmicromagnetic simulations (dots). The excellent agreemen t\nallows us to validate our model and demonstrate the reported\nnon-linear chiral e ffect. First, Fig. 3(a) shows the total\nrotation of the magnetization as a function of the polarizat ion\nstate (characterized by φ0) of an external B-field of tp= 3\nps, for amplitudes of 60 T (blue) , 100 T (red) and 140 T\n(black). Our simulations confirm no rotation for a linearly\npolarized B-field, and a maximum rotation for circular\npolarization. The chiral character of the phenomenon is als o\nevidenced.\nFig. 3(b) depicts the inverse dependency of the magneti-\nzation rotation with the B-field frequency. This frequency\nscaling suggests that the non-linear induced rotation is pa r-\nticularly relevant for external B-fields at THz frequencies .\nHowever, note that the linear dynamics (with the external\nfield) would also contribute at those frequencies. Fig.\n3(c) shows the second-order scaling of the magnetization\ndynamics with the external B-field amplitude for central\nfrequencies of 250 THz (blue), 100 THz (red) and 50 THz\n(black). As expected, the total rotation increases with\nthe B-field amplitude, being already measurable at tens\nof T. Finally, Fig. 3(d) depicts the total rotation of the\nmagnetization for a B-field pulse of frequency 50 THz as a\nfunction of the pulse duration, tp. This latter result confirms\nthat the non-linear chiral e ffect presented in this work is\ncumulative in time, as predicted from Eq. (10).\nOne of the most appealing opportunities of this non-linear\neffect is the possibility to achieve non-thermal ultrafast all -\noptical switching driven solely by an external circularly\npolarized B-field. Based on the dependencies presented in\nFig. 3, we show in Fig. 4 two di fferent micromagnetic\nsimulation results in which switching is achieved through\nthe use of a RCP B-field pulse. The B-field envelopes of\neach case are represented in dashed-red lines, whereas the\nmagnetization components mx,my,mzare represented in\nblue, yellow and black, respectively. The first case makes\nuse of a short, 1 ps, 60 THz, 275 T B-field pulse, whereas the\nsecond case uses a 10 ps B-field pulse of 60 T and 30 THz. In\nboth cases the mzcomponent reverses its direction along the\ncourse of the pulse, showing that complete switching at the f s\nor ps timescale can be achieved, depending on the strength,\npulse duration and frequency of the B-field.\n5. Discussion\nOur results unveil a non-linear chiral magnetic e ffect\ndriven by ultrafast circularly (or elliptically) polarize d\nB-field pulses, lying in the plane containing the initialFigure 3. Analysis of the nonlinear e ffect dependencies Total\nmagnetization rotation as a function of (A)the polarization state of the\nB-field (characterized by φ0, and using θ0=π/4), and (B)the inverse\nof the frequency of a circularly polarized B-field. In both, (A)and(B)\nthree different B-field amplitudes (60 T blue, 100 T red and 140 T black)\noscillating at f= 50 THz are represented. (C)Total magnetization\nrotation as a function of the circularly polarized B-field am plitude, with\nthree different central frequencies ( f= 50 THz blue,f= 100 THz\nred andf= 250 THz black). In (A),(B)and(C)the B-field pulse\nduration is tp= 3 ps .(D)Total magnetization rotation as a function of\nthe circularly polarized B-field pulse duration, tp, with three different B-\nfield amplitudes ( 60 T blue,100 T red and140 T black) and a central\nfrequency of f= 50 THz . In all panels symbols indicate results from\nmicromagnetic simulations while lines correspond to Eq. (1 0).\n-1 0 0.5 1\n 0  2.5-0.5\n 5  7.5  10  12.5  15  17.5  20  22.5  25\nFigure 4. Micromagnetic simulation results of the temporal evolution\nof the magnetization components ( mxblue,myyellow,mzblack) of\nCoFeB excited by a RCP B-field. The normalized B-field envelope is\nshown in dashed-red. While a B-field of B0= 60 T ,f= 30 THz, and\ntp= 10 ps shows switching at the ps timescale, a B-field of B0= 275 T ,\nf= 60 THz, and tp= 1 ps achieves it at the femtosecond timescale.6 L. S´ anchez-Tejerina et al.\nmagnetization. This purely precessional e ffect is quadratic\nin the external B-field, and proportional to the inverse of\nthe frequency, being equivalent to a drift field that depends\nlinearly on the gyromagnetic ratio. This non-linearity is\nproved to be essential at this time scale, since a linear\nresponse would follow adiabatically the magnetic field and,\nconsequently, would restore the magnetization to its initi al\nstate after pulse is gone. Conversely, the reported drift fie ld\nplays a significant role in the magnetization dynamics drive n\nby moderately intense circularly-polarized B-fields —tens\nof Tesla at the ps timescale, while hundreds of Tesla at the\nfs timescale. Although we have studied the magnetization\ndynamics in CoFeB, this e ffect is a general feature of the\nLLG equation, thus being present in all ferromagnets, but\nalso in ferrimagnets and antiferromagnets. Besides, this\nrectification effect may be exploited to generate THz electric\ncurrents via the inverse spin Hall e ffect, that would emit\nelectromagnetic THz radiation[37]when illuminated with\ninfrared light.\nIn addition, it should be stressed that, even when the E-\nfield is non-negligible, the reported non-linear mechanism\non the B-field may play a role, so a complete study of the\nultrafast magnetization dynamics would require taking int o\naccount this effect. We note that recent works pointed out the\nneed of including nutation in the dynamical equation of the\nmagnetization[52,56,57]. This term could also lead to second-\norder effects. Thus, our work serves as a first step towards\nthe investigation of higher-order phenomena induced by\nmagnetic inertia, potentially leading to even shorter time -\nscale magnetization switching.\nFinally, our work demonstrates that the recently developed\nscenario of spatially isolated fs B-fields[39,44–46]opens the\npath to the ultrafast manipulation of magnetization dynami cs\nby purely precessional e ffects, avoiding thermal e ffects due\nto the E-field or magnetization damping. Although the\nspatial decoupling of the intense B-field from the E-field\nusing fs structured pulses is technologically challenging ,\nit is granted by the rapid development of intense ultrafast\nlaser sources, from the infrared (800 nm, 375 THz), to the\nmid-infrared (4 µm - 40µm, 75 - 7 THz)[58–60]. Thinking\nforward, we believe that our work paves the way towards\ninduced all-optical magnetization dynamics at even shorte r\ntimescales, towards the sub-femtosecond regime. 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In this \nwork, we systematically investigate  the magnetization dynamics in p-SAFs  combining time-\nresolved magneto -optical Kerr effect (TR -MOKE)  measurements with theoretical modeling . \nThese model analyses, based on a Landau -Lifshitz -Gilbert approach incorporating exchange \ncoupling , provide detail s about the  magnetization dynamic characteristics  including  the amplitude s, \ndirections, and phases of the precession  of p-SAFs under varying magnetic fields . These model -\npredicted characteristics are in excellent quantitative agreement with TR-MOKE measurements \non an asymmetric p -SAF. We further reveal the damping mechanisms of two precession modes  \nco-existing in the p -SAF and successfully identify individual contributions from different sources , \ninclud ing Gilbert damping of each ferromagnetic layer , spin  pumping, and inhomogeneous \nbroadening . Such a comprehensive understanding of magnetization dynam ics in p -SAFs, obtained \n                                                 \n*Author s to whom correspondence should be addressed : huan1746@umn.edu  and wang4940@umn.edu  2 \n by integrating high -fidelity TR -MOKE measurements and theoretical modeling, can guide  the \ndesign of p-SAF-based architectures for spintronic  applications . \n \nKEYWORDS:  Synthetic antiferromagnets; Perpendicular magnetic anisotropy; Magnetization \nDynamics; Time -resolved magneto -optical Kerr effect; Spintronics   3 \n 1 INTRODUCTION  \nSynthetic antiferromagnet ic (SAF) structures have attracted considerable  interest for \napplications in spin mem ory and logic devices  because of their unique magnetic  configuration s [1-\n3]. The SAF structures are composed of two ferromagnetic (FM) layers anti -parallelly coupled \nthrough a non -magnetic (NM) spacer, offer ing great  flexibilit ies for the  manipulat ion of magnetic \nconfigurations  through  external stimuli  (e.g., electric -field and spin-orbit torque , SOT) . This \npermit s the design of new architecture s for  spintronic applications , such as magnetic tunnel \njunct ion (MTJ), SOT  devices, domain wall devices, sky rmion devices, among others  [4-7]. The \nSAF structures  possess many advantages for such applications , including  fast switching speeds \n(potentially in the THz regimes),  low off set fields, small switching current s (and thus low energy \nconsumption) , high thermal stability,  excellent resilience  to perturbations from external magnetic  \nfields, and large turnabilit y of magnetic properties  [3,8-16].  \nA comprehensive study  of the magnetization dynami cs of SAF structures  can facilitate the \nunderstanding of the switching behavior of spintronic  devices , and ultimately guide the design of \nnovel device architectures . Different from a single FM free layer, magnetization dynamics of the \nSAF structures involv es two modes of precession , namely high -frequency (HF) and low -frequency  \n(LF) modes,  that result from the hybridization of magnetizations  precession  in the two FM layers . \nThe relative phase and precession amplitude in two FM layers can significantly affect the spin-\npumping enhancement of magnetic damping  [17], and thus play an important role in determining \nthe magnetization dynamic behaviors in SAFs. Heretofore, the exchange -coupling strength and \nmagnetic damping constant of SAFs have been studied by ferromagnetic resonance (F MR)  [18-\n21] and optical metrolog y [22-25]. Most FMR -based experimental studies were limited to SAFs \nwith in -plane magnetic anisotropy (IM A). For device applications, perpendicular magnetic 4 \n anisotropy (PMA) gives  better scalability  [3,26] . Therefore , the characteristics of magnetization \ndynamics of perpendicular SAF (p-SAF) structures  are of much valu e to investigat e. In addition, \nprior  studies mainly focus ed on the mutual  spin pumping between two FM layers  [22,27,28] . A \nmore thorough understanding of the contribution s from various sources, including inhomogeneous \nbroadening  [29], remains elusive . \nIn this paper, we report  a comprehensive study of the magnetization dynamics of p -SAFs  by \nintegrating high -fidelity experiments and theoretical modeling to detail the characteristic \nparameters. These parameters describe the amplitude, phase, and direction of magnetization \nprecess ion of both the HF and LF modes for the two exchange -coupled FM layers in a p -SAF.  We \nconduct all -optical time -resolved magneto -optical Kerr effect (TR -MOKE) measurements  [30-33] \non an asymmetric p -SAF structure with two different FM layers. The field-dependent  amplitude \nand phase of TR -MOKE signal s can be well captured by our theoretical  model, which in turn \nprovid es comprehensive physical insights into the magnetization dynamics of p -SAF structures.  \nMost importantly, we show that inhomogeneous broadening  plays a  critical  role in determining \nthe effective damping of both HF and LF modes, especially at low fields.  We demonstrate the \nquantification of contributions from inhomogeneous broadening  and mutual spin pumping  (i.e., \nthe exchange of angular momentum between two FM layers via pumped spin currents ) [21] to the \neffect ive damping, enabl ing accurate determination of the Gilbert damping for individual FM \nlayers. Results of this work  are beneficial  for designing  p-SAF-based architectures  in spintronic  \napplication s. Additionally, this work also serves as a successful example demonstrating that  TR-\nMOKE, as an all -optical met rology, is a powerful tool to capture the magnetization dynamics and \nreveal the rich physics of complex structures that involve multilayer coupling . \n 5 \n 2 METHODOLOTY  \n2.1 Sample preparation and characterization  \nOne SAF  structure  was deposited onto thermally oxidize d silicon  wafers with a 300 -nm SiO 2 \nlayer by magnetron sputtering at room  temperature (RT) in a six -target ultra -high vacuum (UHV) \nShamrock sputtering system. The  base pressure is below  5×10−8 Torr. The stacking structure of \nthe SAF is:  [Si/SiO 2]sub/[Ta(5)/Pd(3)] seed/[Co(0.4)/Pd(0.7)/Co(0.4)] FM1/[Ru(0.6)/Ta(0.3)] NM/ \nCoFeB(1) FM2/[MgO(2)/Ta(3)] capping . The numbers in parentheses denote the layer thicknesses in \nnanometers. After deposition, the sample was annealed at 250 ℃ for 20 minutes by a rapid -\nthermal-annealing process.  The two FM layers are CoFeB and Co/Pd/Co layers, separated by a \nRu/Ta spacer, forming an asymmetric p -SAF structure ( i.e., two FM layers having different \nmagnetic properties). The M-Hext loops were characterized by a physical propert y measurement \nsystem (PPMS) with a vibrating -sample magnetometer (VSM) module. The resulting M-Hext loops \nare displayed in Fig. 1(a). Under low out -of-plane fields ( Hext < 500 Oe), the total magnetic \nmoments in two FM layers of the SAF stack perfectly cancel out each other: M1d1 = M2d2 with Mi \nand di being the magnetization and thickness of each FM layer ( i = 1 for the top CoFeB layer and \ni = 2 for the bottom Co/Pd/Co laye r). The spin-flipping field ( Hf  ≈ 500 Oe ) in the out -of-plane \nloop indicates the bilinear interlayer -exchange -coupling (IEC) J1 between the two FM layers : J1 = \n−HfMs,1d1 ≈ −0.062 erg cm-2 [34]. The values of Ms,1, Ms,2, d1, and d2 can be found in Table SI of \nthe Supplemental Material (SM)  [35]. \n \n2.2 Theoretical foundation  of magnetization dynamics for a p -SAF structure  \nThe magnetic free energy per unit area for a p -SAF structure with uniaxial PMA can be \nexpressed as [36]: 6 \n 𝐹=−𝐽1(𝐦1⋅𝐦2)−𝐽2(𝐦1⋅𝐦2)2\n+∑2\n𝑖=1𝑑𝑖𝑀s,𝑖[−1\n2𝐻k,eff,𝑖(𝐧⋅𝐦𝑖)2−𝐦𝑖⋅𝐇ext] (1) \nwhere J1 and J2 are the strength of the bilinear and biquadratic IEC. mi = Mi / Ms,i are the normalized \nmagnetization vectors for individual FM layers ( i = 1, 2). di, Ms,i, and Hk,eff, i denote, respectively, \nthe thickness, saturation magnetization, and the effective anisotropy field of the i-th layer. n is a \nunit vector indicating the sur face normal direction of the film. For the convenience of derivation \nand discussion, the direction of mi is represented in the spherical coordinates by the polar angle θi \nand the azimuthal angle φi, as shown in Fig. 1 (b). \nThe equilibrium direction of magne tization in each layer (𝜃0,𝑖,𝜑0,𝑖) under a given Hext is \nobtained by minimizing F in the (𝜃1,𝜑1,𝜃2,𝜑2) space. The magnetization precession is governed \nby the Landau -Lifshitz -Gilbert  (LLG)  equation considering the mutual spin pumping between two \nFM layers [27,37 -40]: \n𝑑𝐌𝑖\n𝑑𝑡=−𝛾𝑖𝐌𝑖×𝐇eff,𝑖+(𝛼0,𝑖+𝛼sp,𝑖𝑖)\n𝑀s,𝑖𝐌𝒊×𝑑𝐌𝒊\n𝑑𝑡−𝛼sp,𝑖𝑗\n𝑀s,𝑖𝐌𝒊×(𝐦𝐣×𝑑𝐦𝒋\n𝑑𝑡)×𝐌𝒊 (2) \nOn the right -hand side of Eq. (2), the first term describes the precession with the effective field \nHeff,i in each layer, given by the partial derivative of the total free energy in the M space via 𝐇eff,𝑖=\n−∇𝐌𝑖𝐹. The second term represents the relaxation induced by Gilbert damping ( α) of the i-th layer, \nwhich includes the intrinsic ( 𝛼0,𝑖) and spin -pumping -enhanced ( 𝛼sp,𝑖𝑖) damping. For TR -MOKE \nmeasurements, 𝛼0,𝑖 and 𝛼sp,𝑖𝑖 are indistinguishable. Hence, we def ine 𝛼𝑖=𝛼0,𝑖+𝛼sp,𝑖𝑖 to include \nboth terms. The last term in Eq. (2) considers the influence of pumped spin currents from the layer \nj on the magn etization dynamics of the layer  i.  7 \n The time evolution of Mi can be obtained by solving the linearized Eq. (2). Details are provided \nin Note 1  of the SM  [35]. The solutions to Eq. (2) in spherical coordinates are:  \n[𝜃1(𝑡)\n𝜑1(𝑡)\n𝜃2(𝑡)\n𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+[Δ𝜃1(𝑡)\nΔ𝜑1(𝑡)\nΔ𝜃2(𝑡)\nΔ𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+\n[    𝐶𝜃,1HF\n𝐶𝜑,1HF\n𝐶𝜃,2HF\n𝐶𝜑,2HF]    \nexp(𝑖𝜔HF𝑡)+\n[    𝐶𝜃,1LF\n𝐶𝜑,1LF\n𝐶𝜃,2LF\n𝐶𝜑,2LF]    \nexp(𝑖𝜔LF𝑡) (3) \nwith Δ𝜃𝑖 and Δ𝜑𝑖 representing the deviation angles of magnetization from its equilibrium direction \nalong the polar and azimuthal directions . The last two terms are the linear combination of two \neigen -solutions, denoted by  superscripts HF (high -frequency mode) and LF (low -frequency mode). \nω is the complex angular frequencies of two modes, with the real and imaginary parts representing \nthe precession angular frequency ( 𝑓/2𝜋) and relaxation rate (1/ τ), respectively. For each mode, \nthe complex prefactor vector [𝐶𝜃,1,𝐶𝜑,1,𝐶𝜃,2,𝐶𝜑,2]𝑇 contains detailed information about the \nmagnetization dynamics. As illustrated in Fig. 1 (c), the moduli, |𝐶𝜃,𝑖| and |𝐶𝜑,𝑖| correspond to the \nhalf cone angles of t he precession in layer i along the polar and azimuthal directions for a given \nmode immediately after laser heating, as shown by Δ𝜃 and Δ𝜑 in Figs. 1 (b-c). The phase \ndifference between Δ𝜃𝑖 and Δ𝜑𝑖, defined as Arg(Δ𝜃𝑖/Δ𝜑𝑖)=Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖) with Arg  \nrepresenting the argument of complex numbers, determines the direction of precession. If Δ𝜃𝑖 \nadvances Δ𝜑𝑖 by 90°, meaning Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=90°, the precession is counter -clockwise (CCW) \nin the θ-φ space  (from a view against Mi).Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=−90°, on the contrary, suggests \nclockwise (CW) precession [ Fig. 1 (d)]. Further, the argument of 𝐶𝜃,2/𝐶𝜃,1 provides the relative \nphase in two FM layers. Arg(𝐶𝜃,2/𝐶𝜃,1)=0° corresponds to the precession motions in two FM \nlayers that are in -phase (IP) in terms of θ for a given mode. While the out -of-phase (OOP) \nprecession in terms of θ is represented by Arg(𝐶𝜃,2/𝐶𝜃,1)=180°  [Fig. 1 (e)]. Given the precession 8 \n direction in each  layer and the phase difference between the two FM layers in terms of θ, the phase \ndifference in terms of φ can be automatically determined.   \n \n \nFIG. 1 (a) Magnetic hysteresis ( M-Hext) loops of the p -SAF stack. The magnetization is n ormalized \nto the saturation magnetization ( M/Ms). (b) Schematic illustration of the half cone angles (Δ θ and \nΔφ) and precession direction of magnetization. The precession direction is defined from a view \nagainst the equilibrium direction ( 0, φ0) of M. The representative precession direction in the \nschematic is counterclockwise (CCW). (c) The relation between precession half cone angles and \nthe prefactors. (d) The relation between precession direction and the prefactors. (e) The relative \nphase between two FM layers for different prefactor values.  \n \nAs for the effective damping 𝛼eff=1/2𝜋𝑓𝜏, in addition to the intrinsic damping ( α0,i) and the \nspin-pumping contribution ( αsp,ii and αsp,ji) considered in Eq. (2), inhomogeneities can also bring \nsubstantial damping enhancement [32,33,41,42] . Here, we m odel the total relaxation rate as \nfollows:  \n9 \n 1\n𝜏Φ=−Im(𝜔Φ)+1\n𝜏inhomoΦ (4) \nThe superscript Φ = HF or LF, representing either the high -frequency or low -frequency precession \nmodes. 𝜔Φ includes both the intrinsic and spin -pumping contributions. The inhomogeneous \nbroadening is calculated as:  \n1\n𝜏inhomoΦ=∑1\n𝜋|𝜕𝑓Φ\n𝜕𝐻k,eff,𝑖|\n𝑖Δ𝐻k,eff,𝑖+∑1\n𝜋|𝜕𝑓Φ\n𝜕𝐽𝑖|\n𝑖Δ𝐽𝑖 (5) \nwhere the first summation represents the contrib ution from the spatial variation of the effective \nanisotropy field of individual FM layers (Δ Hk,eff, i). The second summation denotes the contribution \nfrom the spatial fluctuations of the bilinear and biquadratic IEC (Δ J1 and Δ J2). According to \nSlonczewski’s “thickness fluctuations” theory, Δ J1 generates J2 [43,44] . Therefore,  the fact that  J2 \n= 0 for our sample suggests that  ΔJ1 is sufficiently small, allowing us to neglect  the inhomogeneous \nbroadening from th e fluctuations of both the bilinear and biquadratic IEC  in the following analyses . \n \n2.3 Detection of magnetization dynamics  \nThe magnetization dynamics of the p -SAF sample is detected by TR -MOKE, which is \nultrafast -laser -based metrology utilizing a pump -probe configuration. In TR -MOKE, pump laser \npulses interact with the sample, initiating magnetization dynamics in magnetic layers via inducing \nultrafast thermal demagnetization.  The laser -induced heating brings a rapid decrease to  the \nmagnetic anisotropy  fields  and IEC [45,46] , which changes 𝜃0,𝑖, 𝜑0,𝑖 and initiates the precession.  \nThe magnetizati on dynamics due to pump excitation is detected by a probe beam through the \nmagneto -optical Kerr effect. In our setup, the incident probe beam is normal to the sample surface \n(polar MOKE); therefore, the Kerr rotation angle ( 𝜃K) of the reflected probe beam  is proportional \nto the z component of the magnetization [47]. More details about the experimental setup can be 10 \n found in Refs. [30,32] . For p -SAF, TR -MOKE signals contain two oscillating frequencies that \ncorrespond to the HF and LF  modes  (𝑓HF>𝑓LF). The signals  are proportional to the change in \n𝜃K and can be analyzed as follows:  \nΔ𝜃K(𝑡)=𝐴+𝐵𝑒−𝑡/𝜏T+𝐶HFcos(2𝜋𝑓HF𝑡+𝛽HF)𝑒−𝑡/𝜏HF+𝐶LFcos(2𝜋𝑓LF𝑡+𝛽LF)𝑒−𝑡/𝜏LF (6) \nwhere the exponential term  𝐵𝑒−𝑡/𝜏T is related to the thermal background  with 𝜏T being the time \nscale of heat dissipation . The rest two terms on the right -hand side are the precession terms  with \nC, f, β, and τ denoting , respectively, the amplitude, frequency, phase, and relaxation time of the \nHF and LF  modes.  \nAfter excluding the thermal background from TR -MOKE signals, the precession is modeled \nwith the initial conditions of step -function de creases in 𝐻k,eff,𝑖 and 𝐽𝑖, following the ultrafast laser \nexcitation [48]. This is a reasonable  approximation since the precession period (~15 -100 ps for \nHext > 5 kOe) is much longer than the time scales of the laser excitation (~1.5 ps) and subsequent \nrelaxations among electrons, magnons, and lattice (~ 1 -2 ps) [49], but much shorter than the time \nscale of heat dissipation -governed recovery (~400 ps). With these  initial conditions , the prefactors \nin Eq. ( 3) can be determined  (see m ore details in Note 1  of the SM  [35]).  \nFor our SAF structure, 𝜃K detected by the probe beam contain s weighted contributions from \nboth the top and  bottom FM layers:  \n𝜃K(𝑡)\n𝜃K,s=𝑤cos𝜃1(𝑡)+(1−𝑤)cos𝜃2(𝑡) (7) \nwhere 𝜃K,s represents the Kerr rotation angle when the SAF s tack is saturated along the positive \nout-of-plane ( z) direction. w is the weighting factor, considering the different contributions to the \ntotal MOKE signals from two FM layers. w can be obtained from static MOKE measurements [50], \nwhich gives 𝑤= 0.457 (see more details in Note 2  of the SM  [35]). 11 \n  \n3 RESULTS AND DISCUSSION  \n3.1 Field -dependent p recession frequencies and equilibrium magnetization directions  \nTR-MOKE signals measured at varying  Hext are depicted in Fig. 2 (a). The external field is \ntilted 15 ° away from in-plane [θH = 75°, as defined by Fig. 2 (c)] to achieve larger amplitdues of  \nTR-MOKE signals [51]. The signals can be fitted to Eq.  (6) to extract the LF and HF precession \nmodes. The field -dependen t precession frequenc ies of both modes  are summarized in Fig. 2 (b). \nFor simplicity, when analyzing precession frequencies, magnetic damping  and mutual spin \npumping  are neglected due to its insignificant impacts on  precession frequencies.  By comparing \nthe experimental data and the prediction of ωHF/2π and ωLF/2π based on E q. (3), the effective \nanisotropy fields and the IEC strength  are fitted as Hk,eff, 1 = 1.23 ± 0.28  kOe, Hk,eff, 2 = 6.18 ± 0.13  \nkOe, J1 = −0.050 ± 0.020  erg cm−2, and J2 = 0. All parameters and their determination methods are \nsummarized in Table SI of the SM  [35]. The fitted J1 is close to that obtained from the M-Hext \nloops (~−0.062 erg cm-2). The inset of Fig 2 (b) shows the zoom ed-in view of field -dependent \nprecession frequencies around Hext = 8 kOe, where a n anti -crossing feature is observed: a  narrow \ngap (~2  GHz) open s in the frequency dispersion curves of the HF and LF modes owing to the weak \nIEC between two FM layers. Without a ny IEC, the precession frequencies of two FM layers would \ncross at Hext = 8 kOe, as indicated by the green dashed line and blue dashed line in the figure. We \nrefer to t hese two sets of crossing frequencies as the single -layer natural frequencies of two FM \nlayers (FM 1 and FM 2) in the following discussions .  12 \n  \nFIG. 2 (a) TR -MOKE signals under varying Hext when θH = 75° [as defined in panel (c)]. Circles \nare the experimental data and black lines are the fitting curves based on Eq. (6). (b) The precession \nfrequencies of the HF and LF modes as functions of Hext. Circles are experimental data and solid \nlines are fitting curves. The  inset highlights the zoomed -in view of the field -dependent frequencies \naround 8 kOe, where the green dashed line and blue dashed line are the single -layer (SL) \nprecession frequencies of FM 1 and FM 2 without interlayer exchange coupling. (c) Schematic \nillustration of the definition of the equilibrium polar angles ( θ0,1 and θ0,2), and the direction of the \nexternal magnetic field ( θH). The illustration is equivalent to Fig. 1(b) due to symmetry. (d) θ0,1 \nand θ0,2 as functions of Hext. The dash -dotted line plots the difference between the two equilibrium \npolar angles.  \n \n13 \n Based on the fitted stack properties ( Hk,eff,1, Hk,eff,2, J1, and J2), the equilibrium magnetization \ndirections in the two layers can be calculated. For SAFs with weak IEC compared with uniaxial \nPMA, the azimuthal angles of the magnetization in two FM layers are always the same as that of \nthe external field at equilibrium status. Therefore, two polar angles will be sufficient to describe \nthe equilibrium magnetization con figuration. Figure 2(c) illustrates the definition of the \nequilibrium polar angles of two FM layers ( θ0,1, θ0,2) and the external field ( θH). The values of θ0,1, \nθ0,2, and the difference between these two polar angles as functions of Hext are shown in Fig. 2(d). \nWhen Hext is low (<  1.6 kOe), magnetic anisotropy and antiferromagnetic coupling are dominant \nand |θ0,1 − θ0,2| is larger than 90 °. As Hext increases, both θ0,1 and θ0,2 approach θH. When Hext is \nhigh (> 15 kOe), the Zeeman energy becomes dominant and both M1 and M2 are almost aligned \nwith Hext. \n \n3.2 Cone angle, direction, and phase of magnetization  precession revealed by modeling  \nBesides the equilibrium configuration, using sample properties extracted from Fig. 2 (b) as \ninput parameters, the LLG -based modeling (described in section 2.2) also provide s information o n \nthe cone angle, direction, and phase of magnetization precession for each mode ( Fig. 1 ). The \ndiscussion in this section  is limited to the case  without damping an d mutual spin pumping . They \nwill be  considered  in Note 4 of the SM  [35], sections 3.3, and 3.4. The calculation results are \nshown in  Fig. 3 , which are categorized into three regions. At high external fields ( Hext > 1.6 kOe, \nregions 2 and 3), both FM layers precess CCW [ Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=90°], and the polar angles of \nmagnetization in two layers are  in-phase [Arg(𝐶𝜃,2/𝐶𝜃,1)= 0°] for the HF mode and  out-of-phase \n[Arg(𝐶𝜃,2/𝐶𝜃,1) = 180° ] for the LF mode. This is the reason for the HF mode (LF mode) also \nbeing called the acoustic mode ( optical mode) in the literature  [23]. The criterion to differentiate 14 \n region 2 from region 3 is the FM layer that dominat es a given precessional mode  (i.e., the layer \nwith larger precession cone angles) . In region 2  (1.6 kOe < Hext < 8 kOe) , the HF mode is \ndominated by FM 2 because FM 2 has larger cone angles than FM 1. This is reasonable since the \nhigher precession frequency is closer to the natural frequency of FM 2 [see Fig. 2(b)] in region 2. \nSimilarly, in region 3, the HF mode is dominated by FM 1 with larger precession cone angles.  \nWhen Hext is low (region 1), the angle between two magnetizations is larger than 90° [ Fig. 2 (d)] \nowing to the  more  dominan t AF-exchange -coupling  energy as compared with the Zeeman energy . \nIn this region, magnetization dynamics exhibits some unique features.  Firstly, CW [ Arg(𝐶𝜃,𝑖/\n𝐶𝜑,𝑖)=−90°] precession emerges: for each mode, the dominant layer precesses CCW (FM 2 for \nthe HF mode and FM 1 for the  LF mode) and the subservient layer precesses CW (FM 1 for the HF \nmode and FM 2 for the LF mode). This is because the effective field for the subservient layer [ e.g., \nHeff,1 for the HF mode, see Eq.  (2)] precesses CW owing to the CCW precession of the dominant \nlayer when |𝜃0,1−𝜃0,2|>90° [Fig. 2(d)] . In other words, a low Hext that makes |𝜃0,1−𝜃0,2|>\n90° is a necessary condition for the CW precession.  However, it is not a sufficient condition. In \ngeneral, certain degrees  of symmetry breaking ( Hk,eff,1 ≠ Hk,eff,2 or the field is tilted away from the \ndirection normal to the easy axis ) are also needed to generate CW precession.  For example, for \nsymmetric a ntiferromagnets ( Hk,eff,1 = Hk,eff,2) under fields perpendicular to the easy axis, CW \nprecession does not appear even at low fields (Fig. 2(a) in Ref. [52]). See Note 5 of the SM  [35] \nfor more details.  Secondly, as shown in  Fig. 3 , the precession motions in two FM layers are always \nin-phase for both HF and LF modes; thus, there is no longer a clear differentiation between \n“acoustic mode” and “optical mo de”. Instead, the two modes can be differentiated as “right -handed” \nand “left -handed” based on the chirality [53]. Here, we define the chirality with respect to a \nreference direction taken as the projection of Hext or M2 (magnetization direction of the layer with 15 \n a higher Hk,eff) on the easy axis [ -z direction in Fig. 3 (c)]. Lastly, the shape of the precession cone \nalso varies in different regions. Δ θi and Δφi are almost the same for both modes in region 3, \nindic ating the precession trajectories are nearly circular. While in regions 1 and 2, Δ θi and Δφi are \nnot always equal, suggesting the precession trajectories may have high ellipticities.  \n \n \nFIG. 3 The calculated  half cone angle, direction, and phase of magnetization precession for (a) the \nHF mode and (b) the LF mode. In the top row, four curves represent the polar and azimuthal  half \ncone angles of precession in two FM layers. All  half cone angles are normalized with r espect to \nΔθ1. The middle row shows the value of Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖) under different Hext. A value of 90°  (−90°) \nrepresents CCW (CW) precession. The bottom row is the phase difference of the polar angles in \ntwo layers. A value of 0° (180°) corresponds to the polar angles of the magnetization in two layers \nare IP (OOP ) during precession.  Dashed lines correspon d to the reference case where damping is \nzero in both layers. (c) Schematic illustrations of the cone angle, direction, and phase of \n16 \n magnetization precession for the HF and LF modes in different regions, and their corresponding \ncharacteristics regarding ch irality and phase difference.  \n \n3.3 Amplitude and phase of TR -MOKE signals  \nActual magnetization dynamics is resolvable  as a linear combination of the two eigenmodes  \n(the HF and  the LF modes ). By taking into account the initial conditions (i.e., laser excitation , see \nNote 1  of the SM  [35]), we can determine the amplitude and phase of the two modes in TR -MOKE \nsignals . Figure 4 (a) summarizes the amplitudes of both HF and LF modes [CHF and CLF in Eq.  (3)] \nunder different  Hext. Noted that  the y-axis represents Kerr angle ( θK) instead of the cone angle of \nprecession.  The LF mode  has a local minimum near 8 kOe, where the two FM layers have similar \nprecession cone angles but opposite phase s for the LF mode [ Fig. 3 (b)]. The amplitude s of both \nmodes decrease with Hext in the high -field region. This is similar to the single -layer case, where \nthe amplitudes of TR -MOKE signals decrease with Hext because the decrease  in Hk,eff induced by \nlaser heating is not able to significantly  alternate  the equilibrium magnetization direction when the \nZeeman energy dominates  [51]. The LF mode also has an amplitude peak at low fields ( Hext < 3 \nkOe), where the dominant layer of FM 1 changes its equilibrium direction dramatically with Hext \n(from ~75° to 170°) as shown in Fig. 2(d).  \nTo directly compare the amplitudes of TR -MOKE signals and the LLG -based calculations , the \nweighting factor w and the initial conditions are needed. The initial conditions are determined by \n𝐻k,eff,1′,𝐻k,eff,2′, and 𝐽1′, representing the instantan eous effective anisotropy fields and IEC strength \nupon laser heating. These instantaneous properties are different from their corresponding room -\ntemperature values ( Hk,eff,1, Hk,eff,2, and J1). The accurate determination of𝐻k,eff,1′,𝐻k,eff,2′, and 𝐽1′ \ndemands the modeling of the laser heating process as well as the temperature dependence of stack \nproperties, which are challenging. Here, we treat these three variables as adjustable parameters and 17 \n determine their values by fitting the field -dependent amp litudes of TR -MOKE signals , which \nyields 𝐻k,eff,1′𝐻k,eff,1⁄=0.90±0.01, 𝐻k,eff,2′𝐻k,eff,2⁄=0.95±0.01, and 𝐽1′𝐽1⁄=0.83±0.01. \nIt is apparent that the field dependence of TR -MOKE signal amplitude is in excellent agreement \nwith the theoretical modeling , as s hown in Fig. 4 (a). \nFigure 4 (b) shows the calculated  half polar cone angles for each mode in each FM layer. In \nTR-MOKE signals, the optical mode (the LF mode in regions 2 and 3) tends to be partially \ncanceled out  because the two layers precess out -of-phase.  Therefore, compared with Fig. 4 (a), the \ninformation in Fig. 4 (b) better reflects the actual intensity of both modes in FM 1 and FM 2. In Fig. \n4(b), the precession cone angles of both modes in FM 1 (Δ𝜃1HF,Δ𝜃1LF) have local maxima at the \nanti-crossing field (Hext ≈ 8 kOe). On the contrary, Δ𝜃2LF and Δ𝜃2HF of FM 2 have their maxima \neither above or below  the anti-crossing field. This is because FM 2 has larger precession amplitudes \n(cone angles) than FM 1 at the anti-crossing field if there is no IEC [the dotted  lines of FM 1 (SL) \nand FM 2 (SL) in Fig. 4 (b)]. With IEC, FM 2 with larger cone angles can drive the precession motion \nin FM 1 significantly near the anti-crossing field, where IEC is effective. Subsequently, the \nprecession amplitudes of FM 1 exhibit local maxima as its cone angle peaks at the anti-crossing  \nfield [solid lines in Fig. 4(b)]. Also, compared with the uncoupled case [FM 1 (SL) in Fig. 4(b)], \nFM 1 in the SAF structure has a much larger cone angle at the boundary between regions 1 and  2 \n(Hext ≈ 1.6 kOe).  This corresponds to the case where FM 1 fast switch ing is driven by Hext, as shown \nin Fig. 2( d). The energy valley of FM 1 created by IEC and uniaxial anisotropy is canceled out by \nHext. As a result, any perturbation in Hk,eff,1 or IEC can induce a large change in 𝜃1. \nBesides amplitude, the phase of TR -MOKE signals [ HF and LF in Eq. (6)] also provides \nimportant information about the magnetization dynamics in SAF  [Fig. 4 (c)]. In Fig. 4 (c), the phase \nof the HF mode stays constant around π. However, the LF mode goes through a π-phase shift at 18 \n the transition from region 2 to region 3. Th is phase shift can be explained by the change of the \ndominant layer from region 2 to region 3 for the LF mode [ Fig. 3(c)]. As illustrated in Fig. 4 (d), \nthe LF  mode (optical mode in regions 2 and 3) has opposite phases in FM 1 (~0°) and FM 2 (~180°). \nConsidering the two FM layers have comparable optical contributions to TR -MOKE signals ( w ≈ \n0.5), TR -MOKE signals will reflect the phase of the dominant layer for each mode. In region 3, \nFM 2 has larger p recession cone angles than FM 1 for the LF  mode ; therefore, LF TR-MOKE signals \nhave the same phase as FM 2 (~180°). However, in region 2, the dominant layer shifts from FM 2 to \nFM 1 for the LF mode. Hence, the phase of LF TR-MOKE signals also change s by ~180° t o be \nconsistent with the phase of FM 1 (~0°). As for the HF mode, since the two layers always have  \nalmost  the same phase ( ~180°), the change of the dominant layer does not cause a shift in the phase \nof TR -MOKE signals.   \nBy comparing Fig. 4 (d) and Fig. 3 (a-b), one can notice that the phase difference between two \nFM layers could deviate from 0° or 180° when damping  and mutual spin pumping  is considered \n[Fig. 4(d)]. The deviation of phase allows energy  to be transferred from one FM layer to the other \nduring precession via exchange coupling  [54]. In our sample system, FM 2 has a higher damping \nconstant ( 𝛼1= 0.020 and 𝛼2=0.060); therefore, the net transfer of energy is from FM 1 to FM 2. \nMore details can be found in Note 4 of the SM  [35], which shows the phase of TR -MOKE signals \nis affected by Gilbert damping in both layers and the mutual spin pumping . By fitting the phase \n[Fig. 4(c)] and the damping [ Fig. 5(a) ] of TR -MOKE signals simultaneously, we obtained 𝛼sp,12 \n= 0.010 ± 0.004, 𝛼sp,21=0.007−0.007+0.009, 1= 0.020 ± 0.002, and 2 = 0.060 ± 0.008. Nonreciprocal \nspin pumping damping ( 𝛼sp,12≠𝛼sp,21) has been reported in asymmetric FM 1/NM/FM 2 trilayers \nand attributed to the different spin -mixing conductance ( 𝑔𝑖↑↓) at the two FM/NM interfaces [27], \nfollowing 𝛼sp,𝑖𝑗=𝑔𝑖𝜇B𝑔𝑗↑↓/(8𝜋𝑀s,𝑖𝑑𝑖), with 𝑔𝑖 the 𝑔-factor of the i-th layer and 𝜇B the Bohr 19 \n magneton [55]. The above equation neglects the spin -flip scattering in NM and assumes that the \nspin accumulation in the NM spacer equally flows back to FM 1 and FM 2  [37]. However, the \nuncertainties of our 𝛼sp,𝑖𝑗 are too high to justify the nonreciprocity of 𝛼sp,𝑖𝑗 (see Note 3 of the SM  \n[35] for detailed uncertainty analyses). In fact, if the spin backflow to FM i is proportional  to 𝑔𝑖↑↓, \nthen 𝛼sp,𝑖𝑗=𝑔𝑖𝜇B𝑔𝑖↑↓𝑔𝑗↑↓/[4𝜋𝑀s,𝑖𝑑𝑖(𝑔𝑖↑↓+𝑔𝑗↑↓)] (Eq. 1.14 in Ref. [56]). In this case, the different \nspin-mixing conductance at two FM/NM interfaces ( 𝑔1↑↓≠𝑔2↑↓) will not lead to nonreciprocal \n𝛼sp,𝑖𝑗. Although differences in 𝑔𝑖 and magnetic moment per area ( 𝑀s,i𝑑𝑖) can potentially  lead to \nnonreciprocal 𝛼sp,𝑖𝑗, the values of 𝑔𝑖 and 𝑀s,i𝑑𝑖 for the two FM layers are expected to be similar \n(the net magnetization of SAF is zero without external fields). Therefore, nearly reciprocal 𝛼sp,𝑖𝑗 \nare plausible for our SAF stack. Assu ming 𝑔𝑖↑↓ values are similar at the two FM/NM interfaces \n(𝑔1↑↓≈𝑔2↑↓=𝑔↑↓), this yields 𝑔↑↓ =8𝜋𝑀s,𝑖𝑑𝑖𝛼sp,𝑖𝑗/(𝑔𝑖𝜇B) = 1.2 ~ 1.7 × 1015 cm−2. 𝑔↑↓ can also \nbe estimated from the free electron density per spin ( n) in the NM layer: 𝑔↑↓ ≈ 1.2𝑛2/3 [57]. With \nn = 5.2 × 1028 m−3 for Ru [58] (the value of n is similar for Ta [59]), 𝑔↑↓ is estimated to be 1.7  × 1015 \ncm−2, the same order as the 𝑔↑↓ value from TR -MOKE measurements, which justifies the 𝛼sp,𝑖𝑗 \nvalues derived from TR -MOKE are within a reasonable range. The values of 𝛼1 and 𝛼2 will be \ndiscussed in section 3.4.  \n 20 \n  \nFIG. 4 (a) Amplitudes of TR -MOKE signals a s functions of Hext. The circles and curves represent \nexperimental data and modeling fitting , respectively. (b) The calculated precession  half cone \nangles at different Hext. Red curves and black curves represent the cone angles of the HF  mode and \nthe LF mode in FM 1 (solid lines) and FM 2 (dash ed lines). Dotted lines are the precession cone \nangles of single -layer (SL) FM 1 and FM 2 without IEC. (c) Phases of TR -MOKE signals at  varying  \nHext. Circles and curves are experimental data and modeling fitting (𝛼sp,12=0.010, 𝛼sp,21=\n0.007, 𝛼1=0.020, 𝛼2=0.060). (d) Simulated precession phase of the HF mode (red curves) and \nthe LF mode (black curves) in FM 1 (solid lines) and FM 2 (dash ed lines).  \n \n3.4 Magnetic damping of the HF and LF precession modes  \nIn addition to the amplitude and phase of TR -MOKE signals for the p -SAF stack, the model \nanalyses also provide a better understanding of magnetic damping. Figure 5 (a) shows the effective \ndamping constant ( 𝛼eff=1/2𝜋𝑓𝜏) measured at different Hext (symbols), in comparison with \n21 \n model ing fitting (solid lines). The general Hext dependence of αeff can be well captured by the \nmodel. The fitted Gilbert damping, 1= 0.020 ± 0.002 and 2 = 0.060 ± 0.008 are close to the \nGilbert damping of Ta/CoFeB(1 nm)/MgO thin films (~0.017) [41,60]  and Co/Pd multilayers with \na similar tCo/tPd ratio (~0.085)  [61]. Other fitted parameters are  Δ𝐻k,eff,1=0.26±0.02 kOe, \nΔ𝐻k,eff,2= 1.42±0.18 kOe, 𝛼12sp=0.010±0.004  𝛼21sp=0.007−0.007+0.009. Δ𝐽1 and Δ𝐽2 are set to be \nzero, as explained in Sec. 2.2. More details regarding the values and determination methods of all \nparameters involved in our data reduction are provided in Note 3 of the SM  [35]. Dashed lines \nshow the calculated 𝛼eff without inhomogeneous broadening. At high Hext, the difference between \nthe solid lines  and dashed lines approaches zero because the inhomogeneous broadening is \nsuppressed. At low Hext, the solid lines are  significantly higher than the dashed lines , indicating \nsubstantial inhomogeneous broadening contributions .  \nThe effective  damping shows interesting features  near the anti-crossing field. As shown in \nFig. 5(b), due to the effective coupling  between two FM layers near the anti-crossing field, the \nhybridization of precession in two FM layers leads to a mix of damping with contributions from \nboth layers. The effective damping  of the FM 1-dominant mode  reaches a maximum within the \nanti-crossing region ( 7  Hext  10 kOe) and is higher than the single -layer (SL)  FM 1 case. \nSimilarly, the hybridized HF and LF  modes at 8.5 kOe exhibit  a lower 𝛼eff (~0.073) compared to \nthe SL FM 2 case. eff consists of contributions from Gilbert damping ( 𝛼𝑖), mutual spin pumping \n(𝛼sp,𝑖𝑗, 𝑖≠𝑗), and inhomogeneous broadening ( Δ𝐻k,eff,𝑖 and Δ𝐽𝑖). To better understand the mixing \ndamping behavior, Fig. 5 (c) shows eff after excluding the inhomogeneous contribution ( 𝛼effinhomo). \nCompared to the SL layer c ase (green and blue dashed lines), the HF and LF modes (red and black \ndashed lines) clearly suggest that IEC effectively mixes the damping in two layers around the anti -\ncrossing field. Without the IEC, precession in FM 2 with a higher damping relaxes faster  than that 22 \n in FM 1. However, the IEC provides a channel to transfer energy from FM 1 to FM 2, such that the \ntwo layers have the same precession relaxation rate for a given mode. Near the anti -crossing field, \ntwo layers have comparable precession cone angles; therefore, the damping values of the \nhybridized modes are roughly the average of two FM layers. In addition to the static IEC, dynamic \nspin pumping can also modify the damping of individual modes. The black and red solid lines \nrepresent the cases with mutu al spin pumping ( 𝛼sp,12 = 0.01 and 𝛼sp,21 = 0.007). Generally, in \nregions 2 & 3, mutual spin pumping reduces the damping of the HF mode and increases the \ndamping of the LF mode because the HF (LF) mode is near in -phase (out -of-phase). Overall, the \nstatic  IEC still plays the essential role for the damping mix near the anti -crossing field.  \n \n \nFIG. 5 (a) Effective damping constant under varying Hext. Circles are experimental data. Solid \nlines are fitting curves based on Eqs. (4 -5). Dashed lines denote eff after the removal of \ninhomogeneous -broadening contribution. (b) A zoomed -in figure of panel (a) between 5 kOe and \n15 kOe. Blue and green circles are measured effective damping of the mode dominated by FM 1 \nand FM 2, respectively. Blu e and green dashed lines are the 𝛼eff of FM 1 and FM 2 single layer \nwithout IEC. (c) Effective damping after excluding the inhomogeneous contribution as a function \nof Hext. The HF mode (red curves) and the LF mode (black curves) are represented by solid (o r \ndashed) curves when the mutual spin pumping terms ( 𝛼sp,12 and 𝛼sp,21) are considered (or \nexcluded). The d ashed green and blue lines are the SL cases for FM 1 and FM 2, respectively.  \n \n \n \n23 \n 4 CONCLUSION  \nWe systematically investigate d the magnetization dynamics excited by ultrafast laser pulses in \nan asymmetric p -SAF sample both theoretically and experimentally. We obtained d etailed \ninformation regarding magnetization dynamics, including the cone angles, directions, and phases \nof spin precession in each layer under different Hext. In particular, the dynamic features in the low -\nfield region  (region 1)  exhibiting  CW precession, were  revealed. The r esonance between the \nprecession of two FM layers occurs at the boundary between regions 2 an d 3, where an anti -\ncrossing feature is present in the frequency vs. Hext profile . The dominant FM layer for a given \nprecession mode also switches from region 2 to region 3. The amplitude and phase of TR -MOKE \nsignals are well captured by theoretical modeling . Importantly , we successfully quantified the \nindividual contributions from various sources to the effective damping , which enables the \ndetermination of Gilbert damping for both FM layers.  At low Hext, the contribution of \ninhomogeneous broadening to the effective damping is significant. Near the anti-crossing field, \nthe effective damping of  two coupled  modes contains substantial contributions from both FM \nlayers owing to the strong hybridization  via IEC . Although the analyses were made for an \nasymme tric SAF sample, this approach can be directly  applied to study magnetization dynamics \nand magnetic properties of  general complex material systems with  coupled multilayers , and thus  \nbenefits the design and optimization of spintronic materials via structural engineering.  \n \nAcknowledgements  \nThis work is  primarily  supported by the National Science Foundation ( NSF, CBET - 2226579). \nD.L.Z gratefully acknowledges the funding support from  the ERI program (FRANC) “Advanced \nMTJs for computation in and near ra ndom access memory” by DARPA, and ASCENT, one of six 24 \n centers in JUMP (a Semiconductor Research Corporation program, sponsored by MARCO and \nDARPA).  J.P.W  and X.J.W also appreciate the partial support from the UMN MRSEC Seed \nprogram (NSF, DMR -2011401 ). D.B.H . would like to thank the support from the UMN 2022 -2023 \nDoctoral Dissertation Fellowship. 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Magn. 48, 3288 (2012).  \n 1 \n Supplement al Material  for \nMagnetization Dynamics in Synthetic Antiferromagnets  with Perpendicular \nMagnetic Anisotropy   \n \nDingbin Huang1,*, Delin Zhang2, Yun Kim1, Jian -Ping Wang2, and Xiaojia Wang1,* \n1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA  \n2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN \n55455, USA  \n \nSupplement al Note 1: Analyses of the magnetization precession in each  Ferromagnetic  (FM) \nlayer  \nFor the convenience of derivation, mi is represented in  the spherical coordinate s with the polar \nangle θi and the azimuthal angle φi, as shown in Fig. 1(b): \n𝐦𝑖=(sin𝜃𝑖cos𝜑𝑖,sin𝜃𝑖sin𝜑𝑖,cos𝜃𝑖) (S1) \nAccordingly, t he expressi on of  Eq. ( 2) in the spherical coordinate s is:  \n{        𝜃̇1=−𝛾1\n𝑑1𝑀s,1sin𝜃1∂𝐹\n∂𝜑1−𝛼1sin𝜃1𝜑̇1+𝛼sp,12sin𝜃2cos(𝜃2−𝜃1)𝜑̇2\n𝜑̇1=𝛾1\n𝑑1𝑀s,1sin𝜃1∂𝐹\n∂𝜃1+𝛼1\nsin𝜃1𝜃̇1−𝛼sp,12\nsin𝜃1𝜃̇2\n𝜃̇2=−𝛾2\n𝑑2𝑀s,2sin𝜃2∂𝐹\n∂𝜑2−𝛼2sin𝜃2𝜑̇2+𝛼sp,21sin𝜃1cos(𝜃1−𝜃2)𝜑̇1\n𝜑̇2=𝛾2\n𝑑2𝑀s,2sin𝜃2∂𝐹\n∂𝜃2+𝛼2\nsin𝜃2𝜃̇2−𝛼sp,21\nsin𝜃2𝜃̇1 (S2) \n \n*Author s to whom correspondence should be addressed : huan1746@umn.edu  and wang4940@umn.edu  2 \n where, a dot over variables  represents  a derivative with respect to time. When Mi  precesses  around \nits equilibrium direction:  \n{𝜃𝑖=𝜃0,𝑖+Δ𝜃𝑖\n𝜑𝑖=𝜑0,𝑖+Δ𝜑𝑖 (S3) \nwith \ni  and \ni  representing the deviation angles of  Mi from  its equilibrium direction  along the \npolar and azimuthal directions.  Assuming the deviation is small, under the first -order \napproximation, the first -order partial derivative of F in Eq. (S2) can be expanded as:  \n{    ∂𝐹\n∂𝜃𝑖≈∂2𝐹\n∂𝜃𝑖2Δ𝜃𝑖+∂2𝐹\n∂𝜑𝑖∂𝜃𝑖Δ𝜑𝑖+∂2𝐹\n∂𝜃𝑗∂𝜃𝑖Δ𝜃𝑗+∂2𝐹\n∂𝜑𝑗∂𝜃𝑖Δ𝜑𝑗\n∂𝐹\n∂𝜑𝑖≈∂2𝐹\n∂𝜃𝑖𝜕𝜑𝑖Δ𝜃𝑖+∂2𝐹\n∂𝜑𝑖2Δ𝜑𝑖+∂2𝐹\n∂𝜃𝑗∂𝜑𝑖Δ𝜃𝑗+∂2𝐹\n∂𝜑𝑗∂𝜑𝑖Δ𝜑𝑗 (S4) \nBy substituting Eq. ( S4), Equation ( S2) is linearized  as [1]: \n[   Δ𝜃̇1\nΔ𝜑̇1\nΔ𝜃̇2\nΔ𝜑̇2]   \n=𝐊[Δ𝜃1\nΔ𝜑1\nΔ𝜃2\nΔ𝜑2] (S5) \nwhere, K is a 4×4 matrix, con sisting of  the properties  of individual FM layers  and the second -\norder derivatives of F in terms of 𝜃1,𝜑1,𝜃2,and𝜑2. Equation (S5) has four eigen -solutions, in the \nform of 𝐶exp(𝑖𝜔𝑡), corresponding to four precession frequencies:  ±𝜔HF and ±𝜔LF. A pair of \neigen -solutions with the same absolute precession frequency are physically equivalent. Therefore, \nonly two eigen -solutions need to be considered:  \n{Δ𝜃𝑖=𝐶𝜃,𝑖HFexp(𝑖𝜔HF𝑡)\nΔ𝜑𝑖=𝐶𝜑,𝑖HFexp(𝑖𝜔HF𝑡) and {Δ𝜃𝑖=𝐶𝜃,𝑖LFexp(𝑖𝜔LF𝑡)\nΔ𝜑𝑖=𝐶𝜑,𝑖LFexp(𝑖𝜔LF𝑡) (S6) \nAfter r earrange ment , the full solutions in the spherical coordinates are expressed as below (also \nEq. (3) in the main paper).  3 \n [𝜃1(𝑡)\n𝜑1(𝑡)\n𝜃2(𝑡)\n𝜑2(𝑡)]=\n[   𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]   \n+[Δ𝜃1(𝑡)\nΔ𝜑1(𝑡)\nΔ𝜃2(𝑡)\nΔ𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+\n[    𝐶𝜃,1HF\n𝐶𝜑,1HF\n𝐶𝜃,2HF\n𝐶𝜑,2HF]    \nexp(𝑖𝜔HF𝑡)+\n[    𝐶𝜃,1LF\n𝐶𝜑,1LF\n𝐶𝜃,2LF\n𝐶𝜑,2LF]    \nexp(𝑖𝜔LF𝑡)    (S7) \nThe prefactors of these  eigen -solutions  provide information about magnetization dynamics of both \nthe HF and LF modes. Directly from solving Eq. (S2), one can obtain the relative ratios of these \nprefactors , which are [𝐶𝜑1HF/𝐶𝜃1HF,𝐶𝜃2HF/𝐶𝜃1HF,𝐶𝜑2HF/𝐶𝜃1HF] and [𝐶𝜑1LF /𝐶𝜃1LF ,𝐶𝜃2LF /𝐶𝜃1LF ,𝐶𝜑2LF /𝐶𝜃1LF ]. \nThese ratios provide  precession information of each mode, as presented in Fig. 3.   \nObtaining the absolute values of [𝐶𝜃,1,𝐶𝜑,1,𝐶𝜃,2,𝐶𝜑,2]𝑇 for each mode requires the initial \nconditions of precession , which i s necessary for fitting the actual precession amplitudes in TR -\nMOKE signals . In TR -MOKE measurements, magnetization precession is initiated by laser \nheating, which reduces the magnetic anisotropy of each  FM layer and the interlayer exchange \ncoupling streng th between two FM layers  [2]. Considering the laser heating process is ultrafast \ncompared with magnetization precession while the following cooling due to heat dissipation is \nmuch slower than magnetization dynamics, we approximately model the temporal profiles of \neffective anisotropy fields and exchange coupling as step functions.  Owing to the sudden change \nin magnetic properties  induced by laser heating , magnetization in each layer will establish a new \nequilibrium direction (𝜃0,𝑖′,𝜑0,𝑖′). In other words, M i deviates from its new eq uilibrium direction \nby Δ𝜃𝑖=𝜃0,𝑖−𝜃0,𝑖′, Δ𝜑𝑖=𝜑0,𝑖−𝜑0,𝑖′. Substituting 𝑡=0 to Eq. ( S7), one can get the initial \nconditions for magnetization dynamics:  \nΔ𝜃𝑖(𝑡=0)=𝐶𝜃,𝑖HF+𝐶𝜃,𝑖LF=𝜃0,𝑖−𝜃0,𝑖′  \nΔ𝜑𝑖(𝑡=0)=𝐶𝜑,𝑖HF+𝐶𝜑,𝑖LF=𝜑0,𝑖−𝜑0,𝑖′=0 (S8) \nOnce the initial conditions are set, the absolute values of all prefactors can be obtained . \n 4 \n Supplementa l Note 2: Estimation of each layer’s contribution to total TR -MOKE signals  \nThe contribution from each FM layer is estimated by static MOKE measurement. According \nto Ref. [3], the resu lt from this method matches well with that from the optical calculation. The \nsample is perpendicularly saturated before the static MOKE measurement. Then the out -of-plane \nM-Hext loop ( Fig. S1) is measured by static MOKE. As shown in the figure, two different \nantiferromagnetic (AF) configurations have different normalized MOKE signals, indicating the \ndifferent contribution s to the total signals by two layers.  The weighting factor is calculated by:   \n−𝑤+(1−𝑤)=0.085 (S9) \nwhich gives 𝑤=0.457. Considering the relatively small layer thicknesses [FM 1: CoFeB(1), spacer: \nRu(0.6)/Ta(0.3), and FM 2: Co(0.4)/Pd(0.7)/Co(0.4)], it is reasonable that FM 1 and FM 2 make \ncomparable contributions to the total TR -MOKE signals ( i.e., w ≈ 0.5). \n \nFIG. S1 Static MOKE hysteresis loop. Magnetic fields are applied along the out -of-plane direction.  \n \n \n \n5 \n Supplemental Note 3: Summary of the parameters and uncertainties for data reduction  \nGiven that a number of variables are involved in the analysis, TABLE SI summarizes the major \nvariables discussed in the manuscript, along with their values and determinatio n methods.  \nTABLE SI. Summary of the values and determination methods of parameters used in the data \nreduction. The reported uncertainties are one -sigma uncertainties from the mathematical model \nfitting to the TR -MOKE measurement data.  \nParameters  Values  Determination Methods  \nHf ~500 Oe  VSM  \nMs,1 1240 emu cm−3 VSM  \nMs,2  827 emu  cm−3 VSM  \nd1 1 nm  Sample structure  \nd2 1.5 nm  Sample structure  \nHk,eff,1  1.23 ± 0.28 kOe  Fitted from f vs. Hext [Fig. 2(b)]  \nHk,eff,2 6.18 ± 0.13 kOe  Fitted from f vs. Hext [Fig. 2(b)]  \nγ1 17.79 ± 0.04  \nrad ns−1 kOe−1 Fitted from f vs. Hext [Fig. 2(b)]  \nγ2 17.85 ± 0.04  \nrad ns−1 kOe−1 Fitted from f vs. Hext [Fig. 2(b)]  \nJ1 −0.050 ± 0.020  \nerg cm−2 Fitted from f vs. Hext [Fig. 2(b)]  \nJ2 0 Fitted from f vs. Hext [Fig. 2(b)]  \nw 0.457  Static MOKE  \n𝐻k,eff,1′/𝐻k,eff,1 0.90 ± 0.01  Fitted from Amp vs. Hext [Fig. 4(a)]  \n𝐻k,eff,2′/𝐻k,eff,2 0.95 ± 0.01  Fitted from Amp vs. Hext [Fig. 4(a)]  \n𝐽1′/𝐽1 0.83 ± 0.01  Fitted from Amp vs. Hext [Fig. 4(a)]  \n𝛼1 0.020 ± 0.002  Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \n𝛼2 0.060 ± 0.008  Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \nΔ𝐻k,eff,1 0.26 ± 0.02 kOe  Fitted from eff vs. Hext [Fig. 5(a)]  \nΔ𝐻k,eff,2 1.42 ± 0.18 kOe  Fitted from eff vs. Hext [Fig. 5(a)]  \n𝛼sp,12 0.010 ± 0.004  Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \n𝛼sp,21 0.007−0.007+0.009 Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \n 6 \n Supplemental Note  4: Impacts of 𝜶𝟏, 𝜶𝟐, and mutual spin pumping on the phase  \nWithout damping, the phase difference in the precession polar angles of two FM layers \n[Arg(𝐶𝜃2/𝐶𝜃1)] is always 0° or 180°, as shown in Fig. 3 of the main article. However, this does not \nnecessarily hold if either the damping or mutual spin pumping is considered. The changes in the \nphase difference due to damping are depicted in Fig. S2. When 1 = 2, the phase difference \nbetween two layers stays at 0° or 180° [ Fig. S2(a)], identical to the lossless case ( 1 = 2 = 0) in \nFig. 3. As a result, the initial phase of TR -MOKE signals ( ) also stays at 0° or 180° [ Fig. S2(b)]. \nHowever, when 𝛼1≠𝛼2, Arg(𝐶𝜃2/𝐶𝜃1) deviates from 0° or 180° especially at high fields ( Hext > \n5 kOe) [ Fig. S2(c,e)]. The layer with a higher damping [FM 1 in (c) or FM 2 in (e)] tends to have a \nmore advanced phase at high fields (regions 2 and 3). For example, in Fig. S2(e), 0° < Arg( 𝐶𝜃2/𝐶𝜃1) \n< 180° for both HF and LF modes in regions 2 and 3. The deviation from the perfect in -phase (0°) \nor out -of-phase (180°) condition allows the IEC to transfer energy from the low -damping layer to \nthe high -damping layer, such that the precession in  both layers can damp at the same rate [4]. As \na result, the initial phase of the TR -MOKE signals also changes, which opens a negative or positive  \ngap at high fields (> 10 kOe) for both modes, as shown in Fig. S2(d,f). This enables us to determine \nthe difference between 1 and 2 by analyzing the in itial phase of TR -MOKE signals.  7 \n  \nFIG. S 2 Impact of 𝛼1 and 𝛼2 on the phase without mutual spin pumping. (a,c,e) The phase \ndifference between the polar angles in two layers for HF and LF modes. (b,d,f) The calculated \ninitial phase of TR -MOKE signals for each mode with 1 = 2 = 0.02 (a,b), 1 = 0.06 and 2 = \n0.02 (c,d), and 1 = 0.02 and 2 = 0.06 (e,f). The mutual spin pumping is set as 𝛼sp,12=𝛼sp,21 = \n0 for all three cases. The rest of the parameters used in this calculation can be found in TABLE SI.  \n \nThe impact of mutual spin pumping on the precession phase is illustrated in Fig. S3, where \nthree different cases of either the one -way (𝛼sp,12 or 𝛼sp,21) or two -way (both 𝛼sp,12 and 𝛼sp,21) \nspin pumping are considered. A reference case without the consideration of mutual spin pumping \n(1 = 0.02, 2 = 0.06, and 𝛼sp,12= 𝛼sp,21 = 0) is also plotted (dashed curves) for the ease of \ncomparison. In general, it can be seen that mutual spin pumping could also change the phase \ndifference in the precession polar angles of two layers, and thus the initial phase o f TR -MOKE \nsignals noticeably. This can be explained by the damping modification resulting from spin \npumping. In regions 2 and 3, Eq. (2) can be approximately rearranged as:  \n \n8 \n 𝑑𝐦𝑖\n𝑑𝑡≈−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+𝛼𝑖𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡−𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2)𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡\n≈−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+[𝛼𝑖−𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2)]𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡\n=−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+𝛼̅𝑖𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡 (S10) \nwhere 𝐶𝑗/𝐶𝑖 represents the ratio of the cone angles in the j-th FM layer to the i-th FM layer. 𝐶𝑗/𝐶𝑖 \nis positive for the in -phase mode and negative for the out -of-phase mode.  θ0,1 and θ0,2 are the \nequilibrium polar angle s of M1 and M2, as defined in Fig. 2(c).  Therefore, the mutual spin -pumping \nterm either enhances or reduces the damping depending on the mode. 𝛼̅𝑖 = 𝛼𝑖−\n𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2) represents the effective Gilbert damping in the i-th FM layer after \nconsidering the mutual spin -pumping effect. This modification to damping is more significant \nwhen the i-th layer is subservient with a smaller cone angle ( e.g., FM 2 for the HF mode in region \n3), while the j-th layer is dominant with a mu ch larger precession cone angle ( e.g., FM 1 for the LF \nmode in region 3), leading to a large ratio of |𝐶𝑗/𝐶𝑖|. \nIn Fig. S 3(a), only the spin current injected from FM 1 to FM 2 is considered. According to the \nabove analysis, 𝛼sp,21 can only bring noticeable modifications to the damping of FM 2 when FM 1 \nis the dominant layer. Based on Fig. 3 in the main article, the LF mode in region 2 and HF mode \nin region 3 satisfy this condition (FM 1 dominant and FM 2 subservient). As shown in Fig. S 3(a), \nthe phase difference  noticeably deviates from the reference case without mutual spin pumping \n(dashed curves) in region 2 for the LF mode (black curves) and in region 3 for the HF mode (red \ncurves). For the LF mode in region 2, the precession motions in two layers are nearly o ut-of-phase \n(negative C1/C2); therefore, the spin pumping from FM 1 enhances the damping in FM 2. Since 1 \n(0.02) is less than 2 (0.06), the spin pumping from FM 1 to FM 2 further increases |  𝛼̅1 − 𝛼̅2| \nbetween the two layers. Consequently, the phase difference shifts further away from 180°. While 9 \n for the HF mode in region 3, 𝛼sp,21 reduces the damping of FM 2 because C1/C2 is positive resulting \nfrom the near in -phase feature of this mode. Hence, |  𝛼̅1 − 𝛼̅2| becomes smaller and the phase \ndifference gets closer to 0°. In Fig. S 3(c), only 𝛼sp,12 is considered, which requires FM 2 as the \ndominant layer (the HF mode in region 2 and LF mode in region 3) for noticeable changes in |  𝛼̅1 \n− 𝛼̅2|. For the HF mode  in region 2, spin pumping from FM 2 reduces 𝛼̅1 given that the precession \nmotions in two layers are nearly in phase (positive C2/C1). Therefore, |  𝛼̅1 − 𝛼̅2| increases and the \nphase difference in Fig. S 3(c) shifts further away from 0° in region 2. However,  for the LF mode \nin regions 3, the nearly out -of-phase precession in two FM layers (negative C1/C2) increases 𝛼̅1 \nand reduces |  𝛼̅1 − 𝛼̅2|. As a result, the phase difference in Fig. S 3(c) shifts toward 180°. When \nboth 𝛼sp,12 and 𝛼sp,21 are considered [ Fig. S 3(e)], a combined effect is expected for the phase \ndifference with noticeable changes for both the HF and LF modes in regions 2 and 3.  \nThe impacts of mutual spin pumping on the phase difference between the HF and LF modes \nare reflected by the initial phase of TR -MOKE signals [  in Fig. S 3(b,d,f)]. Compared with the \nreference case without mutual spin pumping (dashed curves), the introduction of mutual spin \npumping tends to change the gap in  between the two modes. As shown in Fig. S3(e,f), the values \nof two mutual -spin-pumping induced damping terms are chosen as 𝛼sp,12 = 0.013 and 𝛼sp,21 = \n0.004, such that the  gap of the initial phase of TR -MOKE signals is closed at high fields \n(region  3). Therefore, the initial phase of TR -MOKE signals provides certain measurement \nsensitivities to 𝛼sp,12 and 𝛼sp,21, which enables us to extract the values of 𝛼sp,𝑖𝑗 from \nmeasurement fitting. Here, we acknowledge that the measurement sensitivity to 𝛼sp,𝑖𝑗 from TR -\nMOKE is limited, which subsequently leads to relatively large error bars for 𝛼sp,𝑖𝑗 (see Table SI).  \n 10 \n  \nFIG. S3 Impact of mutual spin pumping on the phase with fixed damping values of 1 = 0.02 and \n2 = 0.06. (a,c,e) The phase difference between the polar angles in two layers for HF and LF modes. \n(b,d,f) The calculated initial phase of TR -MOKE signals ( ) for each mode with 𝛼sp,12= 0 and \n𝛼sp,21 = 0.01 (a,b), 𝛼sp,12 = 0.01 and 𝛼sp,21 = 0 (c,d), and  𝛼sp,12 = 0.013 and 𝛼sp,21 = 0.004 (e,f). \nFor the third case (e,f), the values of mutual spin pumping are chosen to close the  gap in panel \n(f) for Hext > 15 kOe. The rest of the parameters used in this calculation can be found in TABLE \nSI. Dashed lines  represent the reference case without mutual spin pumping ( 1 = 0.02, 2 = 0.06, \nand 𝛼sp,12= 𝛼sp,21 = 0).  \n \nSupplemental Note 5: Region diagram s for p -SAFs with different degrees of asymmetries  \nFigure S4 shows the region diagrams for p -SAFs with different degrees of asymmetries , \nrepresented by the difference of Hk,eff in two FM layers. Hk,eff,1 = Hk,eff,2 corresponds to the \nsymmetric case (lowest asymmetry), as shown by Fig. S4(c). While  the SAF  in Fig. S4(a) has the \nhighest asymmetry: Hk,eff,1 = 2 kOe, Hk,eff,2 = 6 kOe. Figure S4 clearly shows that |𝜃0,1−𝜃0,2|>\n90° is a necessary but not sufficient condition for region 1 (CW precession). Because regions 2 or \n3 also appear to the left of the red cu rve (where |𝜃0,1−𝜃0,2|>90°), especially when θH is close \nto 90° and Hk,eff,1 is close to Hk,eff,2. \n11 \n  \nFIG. S4 Region diagrams of p -SAFs with different  degrees of  asymmetries: Hk,eff,1 = 2 kOe, Hk,eff,2 \n= 6 kOe (a), Hk,eff,1 = 4 kOe, Hk,eff,2 = 6 kOe (b), Hk,eff,1 = 6 kOe, Hk,eff,2 = 6 kOe (c). The blue \nbackground represents region 1. The green background covers regions 2 and 3. The red curve \nshows the conditions where |𝜃0,1−𝜃0,2|=90°. |𝜃0,1−𝜃0,2|>90° to the left of the red curve. 𝛼1, \n𝛼2, 𝛼sp,12, and 𝛼sp,21 are set as zero. 𝛾1=𝛾2=17.8 rad ns−1 kOe−1. Values of the rest parameters are \nthe same as those in Table SI.  \n \nReferences  \n[1] Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Angular dependence of ferromagnetic \nresonance in exchange -coupled Co/Ru/Co trilayer structures, Phys. Rev. B 50, 6094 (1994).  \n[2] W. Wang, P. Li, C. Cao, F. Liu, R. Tang, G. Chai, and C. Jiang, Temperature dependence \nof interlayer exchange coupling and Gilbert damping in synthetic antiferromagnetic \ntrilayers investigated using broadband ferromagnetic resonance, Appl. Phys. Lett. 113, \n042401 (2018).  \n[3] G. Malinowski, F. Dalla Longa, J. H. H. Rietjens, P. V. Paluskar, R. Huijink, H. J. M. \nSwagten, and B. Koopmans, Control of speed and efficiency of ultrafast demagnetization \nby direct transfer of spin angular momentum, Nat. Phys. 4, 855 (2008).  \n[4] D. H. Zanette, Energy exchange between coupled mechanical oscil lators: linear regimes, J. \nPhys. Commun. 2, 095015 (2018).  \n \n"    },    {        "title": "2006.11710v1.Rotational_properties_of_annulus_dusty_plasma_in_a_strong_magnetic_field.pdf",        "content": "Rotational properties of annulus dusty plasma in a strong magnetic \feld\nMangilal Choudhary,1,a)Roman Bergert,2Sandra Moritz,2Slobodan Mitic,2and Markus H. Thoma2\n1)Institute of Advanced Research, The University for Innovation, Koba, Gandhinagar, 382426,\nIndia\n2)I. Physikalisches Institut, Justus{Liebig Universitt Giessen, Henrich{Bu\u000b{Ring 16, D 35392 Giessen,\nGermany\nThe collective dynamics of annulus dusty plasma formed between a co-centric conducting (non-conducting)\ndisk and ring con\fguration is studied in a strongly magnetized radio-frequency (rf) discharge. A supercon-\nducting electromagnet is used to introduce a homogeneous magnetic \feld to the dusty plasma medium. In\nabsence of the magnetic \feld, dust grains exhibit thermal motion around their equilibrium position. The dust\ngrains start to rotate in anticlockwise direction with increasing magnetic \feld (B >0.02 T), and the constant\nvalue of the angular frequency at various strengths of magnetic \feld con\frms the rigid body rotation. The\nangular frequency of dust grains linearly increases up to a threshold magnetic \feld (B >0.6 T) and after that\nits value remains nearly constant in a certain range of magnetic \feld. Further increase in magnetic \feld (B\n>1 T) lowers the angular frequency. Low value of angular frequency is expected by reducing the width of\nannulus dusty plasma or the input rf power. The azimuthal ion drag force due to the magnetic \feld is assumed\nto be the energy source which drives the rotational motion. The resultant radial electric \feld in the presence\nof magnetic \feld determines the direction of rotation. The variation of \roating (plasma) potential across the\nannular region at given magnetic \feld explains the rotational properties of the annulus dusty plasma in the\npresence of magnetic \feld.\nI. INTRODUCTION\nDusty plasma, which is an admixture of electrons, ions\nand negatively charged micron sized solid particles ex-\nhibits solid- as well as liquid-like characteristics, depend-\ning on coulombic interactions among charged particles1.\nDue to such unique features, dusty plasma is consid-\nered as part of soft condensed matter similar to granu-\nlar material and colloidal suspension2{5. The negatively\ncharged dust grains have a large inertia; therefore, dusty\nplasma responses to a very low frequency \feld which al-\nlows to study solid state phenomena, \ruid dynamics, tur-\nbulences, etc. at kinetic level using optical observation\ntechnique1,6{8.\nIn laboratory experiments, dusty plasma dynamics de-\npends on the forces acting on the dust grains9{11. The\ndust charging process as well as forces acting on the\ndust grain are associated to the ambient plasma back-\nground. Collective dynamics of dusty plasma is expected\nto change with altering the background plasma. It is\nknown that input power and pressure are major exter-\nnal variables to alter plasma dynamics. Instead of these\nvariables, an external magnetic \feld can also modify the\nbackground plasma12. Therefore, magnetic \feld is con-\nsidered as an external parameter (variable) to study the\ncollective dynamics of dusty plasma. Hence, a wide spec-\ntrum of theoretical as well as experimental works have\nbeen performed to study the role of an external magnetic\n\feld on dusty plasma13.\nIn the presence of an axial magnetic \feld, strongly cou-\npled dust particles con\fned over a cathode in a DC dis-\ncharge exhibit the rotational motion with a constant an-\na)Electronic mail: jaiijichoudhary@gmail.comgular velocity in a plane and velocity shear in the vertical\ndirection14. The role of a longitudinal magnetic \feld to\na small planar dust cluster con\fned in a striation head of\nthe positive column of a DC discharge has been reported\nby many researchers15{18. In such con\fguration, they\nobserved the inversion of dust structure rotation above\na threshold magnetic \feld at given discharge conditions.\nKonopka et al.19investigated the role of a vertical mag-\nnetic \feld to a 2-dimensional dust crystal con\fned by a\nconducting ring on the sheath of a radiofrequency (rf)\ndischarge and observed two types of rotations named as\nrigid-body rotation and sheared rotation. Cheung et al.20\ninvestigated the characteristics of a rotating dust cluster\ncontaining 1 up to 12 dust grains in a rf discharge under\nthe in\ruence of an external magnetic \feld. This work\ncon\frms the role of dust particles number in a dust clus-\nter to determine the rotational characteristics in presence\nof a magnetic \feld. The role of gas pressure in determin-\ning the rotational properties of a dust cluster has been\nstudied by Huang et al.21. They reported di\u000berent rota-\ntional speeds for upper and lower dust layers as well as\nreversal of rotation direction at a threshold pressure.\nRecently, Dzlieva et al.22investigated the in\ruence of\na strong magnetic \feld (B \u00181 T) to a dust structure in\na DC discharge. They observed the inversion of rotation\nat lower magnetic \feld strength and a slow increase of\nthe angular frequency after B >0.1 T. Karasev et al.23\nreported a rotating dust cluster and shell structure in the\npresence of a strong magnetic \feld. Apart from DC dis-\ncharges, some interesting features of 3D dusty plasmas,\nas damping of dust-acoustic waves24and counter-rotating\ndust torus25have been reported in recent experiments by\nChoudhary et al. in rf discharge. Melzer et al.26investi-\ngated the e\u000bect of a strong magnetic \feld (B >5 T) to\na dust cluster con\fned by a ring shaped electrode in a\nrf discharge. They observed the increase of rotation fre-arXiv:2006.11710v1  [physics.plasm-ph]  21 Jun 20202\nquency of the two-dimensional (2D) dust cluster with in-\ncreasing magnetic \feld. The rotation of 2D dust clusters\ncon\fned by a radial electric \feld either in DC discharge\nor rf discharge in the presence of a longitudinal magnetic\n\feld is a result of the E\u0002Bdrift of ions in the azimuthal\ndirection of a cylindrical geometry19,27,28.\nIn above mentioned studies, dust grains (in rf dis-\ncharge) are con\fned by a radial electric \feld of an ad-\nditional ring on the lower electrode or the ring-like struc-\nture of the lower electrode. Dust grains form a cluster\ndue to the radial electric \feld of the ring edge and ex-\nhibit rotational motion in presence of an external mag-\nnetic \feld19. Few work has been performed to study\nthe collective dynamics of an annulus dusty plasma in\na magnetized rf discharge. Bandyopadhyay et al.29ob-\nserved the sheared or opposite dusty plasma \row in the\nannular region of co-centric conducting rings in the pres-\nence of a magnetic \feld. They claimed an increase of\nangular frequency of opposite \rowing dust grains with\nincreasing magnetic \feld. It is di\u000ecult to get answers to\nmany open questions that arise after these \frst experi-\nmental observations. Is it possible to induce shear \row\nin an annulus dusty plasma at any discharge condition in\nthe presence of a magnetic \feld? Does an annulus dusty\nplasma remain stable at a strong magnetic \feld? Is it\npossible to induce rigid rotational motion in an annu-\nlus dusty plasma in the presence of an external magnetic\n\feld? How does the angular frequency depend on the ex-\nternal magnetic \feld?. There are many other questions\nregarding the \row characteristics of the annulus dusty\nplasma at a strong magnetic \feld. To get the answers\nto some of these questions, experiments were performed\nin a strongly magnetized rf discharge where dust grains\nwere con\fned between the co-centric (conducting or non-\nconducting) disk-and-ring-con\fguration.\nSection II deals with the detailed description of the ex-\nperimental set-up and the plasma and dusty plasma pro-\nduction. The dynamics of annuls dusty plasma produced\nbetween the co-centric (conducting and non-conducting)\ndisk-and-ring-con\fguration is discussed in Section III.\nThe origin of rotational motion is discussed in Section IV.\nA brief summary of the work along with concluding re-\nmarks is provided in Section V.\nII. EXPERIMENTAL SETUP\nThe present set of experiments is carried out in a vac-\nuum chamber, which is placed at the center of a supercon-\nducting electromagnet ( Bmax\u00184 T) to introduce a ho-\nmogeneous external magnetic \feld to the dusty plasma.\nThe details of the magnetized dusty plasma device avail-\nable at Justus-Liebig University Giessen is provided in\nRef.12. A schematic diagram of the experimental setup is\npresented in Fig. 1(a). Before starting the experiments, a\ndisk (conducting or non-conducting) of diameter Ddisk=\n15 mm and a ring with an inner diameter of Din\nring30 mm,\nouter diameter of Dout\nring= 50 mm and thickness of 2 mmare placed on the lower electrode. The centres of the disk\nand the ring coincide with the center of the lower elec-\ntrode to make an annular region. Aluminium and Te\ron\nare used as conducting and non-conducting materials, re-\nspectively. After placing the co-centric disk and ring on\nthe lower electrode, the vacuum chamber is evacuated\nto base pressure p <10\u00002Pa using a pumping system\nconsisting of a rotary and a turbo molecular pump. The\nexperiments are performed with argon gas and the pres-\nsure inside the chamber is controlled by using a mass \row\ncontroller (MFC) and gate valve controller. At a given\ngas pressure, plasma is ignited between an aluminium\nelectrode of 65 mm diameter (lower electrode) and an in-\ndium tin oxide-coated ((ITO-coated) electrode of 65 mm\ndiameter (upper electrode) using a 13.56 MHz rf gener-\nator with a matching network. Between the electrodes\nthere is a gap of 30 mm. A dust dispenser, which is in-\nstalled at one of the side ports of the vacuum chamber,\nis used for injecting the Melamine Formaldehyde (MF)\nparticles with a diameter of D'6.28\u0016m into the plasma\nvolume. The MF particles acquire negative charges and\nare con\fned in the annular region of the co-centric disk\nand ring. To investigate the \row dynamics or rotational\nproperties of the annulus dusty plasma in presence of the\nmagnetic \feld, a red laser sheet and a CMOS camera are\nused to illuminate the particles and to capture the scat-\ntered light coming from dust grains, respectively. The\nCMOS camera is installed to observe the dust dynamics\nthrough the transparent upper electrode in the horizon-\ntal plane (X{Y plane) at a frame rate of 60 fps and with\na resolution of 2048 \u00022048 pixels. The stored images are\nlater analyzed with help of ImageJ30software and MAT-\nLAB based open-access code, called openPIV31. A full\nview of the annulus dusty plasma formed between the co-\ncentric aluminium disk and ring in the horizontal (X{Y)\nplane is shown in Fig. 1(b).\nIII. ANNULUS DUSTY PLASMA DYNAMICS IN THE\nPRESENCE OF A MAGNETIC FIELD\nIt has been reported experimentally that 3-dimensional\ndusty plasma exhibits the vortex motion instead of az-\nimuthal rotation in the presence of a strong magnetic\n\feld25. However, a two-dimensional dust cluster always\nrotates in the direction of the magnetic \feld-induced ion\n\row1926. To investigate the dynamics of a nearly 2-\ndimensional annulus dusty plasma (con\fned between the\nco-centric disk and ring) in presence of a strong mag-\nnetic \feld, two types of combinations of disk and ring\nare used. In the \frst set of experiments, a combination of\nco-centric non-conducting (Te\ron) disk and ring is used\nto create the annulus dusty plasma. In another set of\nexperiments, annulus dusty plasma is produced between\nthe co-centric conducting (Aluminum) disk and ring at\ngiven discharge conditions. The detailed study of the an-\nnulus dusty plasma in presence of a strong magnetic \feld\nis presented in the subsections III A and III B.3\n \n(9) (1) \n(2) \n(3) (7) \nB \nY Z \nX    \n  \n(6) (4) (8) (5) (a)\n Y \n \nX (b) \n \nr (b) \nY E E \nr = 0 mm r = 18 \nmm  \nmm  I II III \nX \nFIG. 1. (a) Schematic diagram of the experimental setup to study the magnetized dusty plasma (1) with vacuum chamber, (2)\nupper ITO-coated transparent electrode, (3) lower aluminum electrode, (4) RF power generator, (5) aluminum (Te\ron) disk\nand ring, (6) con\fned dust particles, (7) CMOS camera, (8) mirror, and (9) red laser with a cylindrical lens. The blue error\nindicates the direction of external magnetic \feld. (b) The experimental view of the annulus dusty plasma formed between the\nco-centric aluminum disk ( Ddisk= 15 mm) and ring ( Din\nring= 30 mm). In this study we used both Cartesian and Cylindrical\nCoordinates to understand the rotational motion. The experimental region is divided into three di\u000berent regions; disk edge\nregion (I), central or dusty plasma region (II) and ring edge region (III). The direction of electric \feld in the annular region is\nrepresented by a yellow arrow.\nFIG. 2. PIV images of the rotational motion of dust grains in a 2D (X{Y) plane at various strengths of magnetic \felds. Dust\ngrains are con\fned between the Te\ron disk ( Ddisk= 15 mm) and ring ( Din\nring= 30 mm) at electrode voltages Vup= 60 V and\nVdown = 50 V and argon pressure, p = 30 Pa. The PIV images are corresponding to B = 0.2 T, B = 0.7 T and B = 1.1 T,\nrespectively. The central blue color region includes the disk, as well as a dust free region (see Fig. 1(b)) which can be identi\fed\nusing the scale of plots.\nA. Dusty plasma con\fned between the co-centric\nnon-conducting disk and ring\nThe potential well created between the disk and the\nring is a result of modi\fcation of the potential distribu-\ntion in rf sheath of the lower powered electrode. After\ninjecting the dust grains into plasma, they acquire nega-\ntive charges and get con\fned in the potential well. Gas\npressure p and electrodes voltage (or rf power) are ad-\njusted to make a 2-dimensional (3 to 5 layers in vertical\nplane) annulus dusty plasma at B = 0 T. The annulus\nwidth of the dust grain medium can be changed by in-\njecting less or more particles into the plasma for a givendischarge condition. The dust density ( nd) is expected\nto increase with increasing the annulus width of dusty\nplasma. In the \frst set of experiments, dusty plasma is\nproduced at p = 30 Pa and voltages of the upper elec-\ntrode,Vup= 60 V and of the lower electrode, Vdown =\n50 V. The annulus width of dusty plasma at B = 0 T is\nestimated to be around 5 mm. As the magnetic \feld is\napplied to the dusty plasma, at \frst the con\fning poten-\ntial well gets modi\fed at low magnetic \feld strengths,\nresulting in a slight change in position of dust grains.\nTherefore, the tracking of the rotational motion at low\nmagnetic \feld (B <0.02 T) is di\u000ecult. The anticlock-\nwise rotational motion of annuals dust grain medium is4\n78 9 1011121314150.10.20.30.40.5 \nB = 0.1 T \nB = 0.3T \nB = 0.6 T \nB = 0.8 T \nB = 0.9 T \nB = 1.1 T \nB = 1.3 TΩ [rad/s]r\n [mm](a)\n91 01 11 21 30.000.070.140.210.280.35 \nB = 0.1 T \nB = 0.2 T \nB = 0.4 T \nB = 0.6 T \nB = 0.8 T \nB = 1 TΩ [rad/s]r\n [mm](b)\n8 9 1011121314150.000.040.080.120.16\n B = 0.1 T\n B = 0.3 T\n B = 0.6 T\n B = 0.8 T\n B = 1 T\n B = 1.1 T\nΩ [rad/s]\nr [mm](c)\nFIG. 3. The radial variation of angular frequency (\n) of rotating particles at various strengths of the magnetic \feld. (a)\nTe\ron disk ( Ddisk= 15 mm) and ring ( Din\nring= 30 mm), Vup= 60 V and Vdown = 50 V, p = 30 Pa and annulus width of dusty\nplasma \u00185 mm (at B = 0 T). (b) Te\ron disk ( Ddisk= 15 mm) and ring ( Din\nring= 30 mm), Vup= 60 V and Vdown = 50 V, p\n= 30 Pa and annulus width of dusty plasma \u00182.5 mm. (c) Te\ron disk ( Ddisk= 15 mm) and ring ( Din\nring= 30 mm), Vup=\n50 V and Vdown = 50 V, p = 30 Pa and annulus width of dusty plasma \u00184 mm.\npredicted above B >0.02 T. It is noticed that the an-\nnulus width of dusty plasma medium starts to increase\nabove a threshold value of magnetic \feld strength of B\n>0.6 T. AT B = 1.3 T, an estimated width of dusty\nplasma is\u00186.5 mm. This variation in annulus width\nof dusty plasma is expected due to the modi\fcation in\nthe con\fning potential well at strong magnetic \feld. The\ndust density at this annulus width comes out to be \u00183-5\n\u0002104cm\u00003. For a detailed study of rotational motion\n(e.g. direction, angular frequency) at various strengths\nof magnetic \feld, the still images are further analyzed.\nIn Fig 2, PIV images of rotating dust grains in horizon-\ntal plane (X{Y plane) at di\u000berent strengths of magnetic\n\feld are displayed. The PIV images are constructed us-ing an adaptive 2-pass algorithm (a 64 \u000264, 50% over-\nlap followed by a 32 \u000232, 50% overlap analysis). The\ncontour maps (Fig. 2) of the average magnitude of veloc-\nity are constructed after averaging the velocity vectors\nof consecutive 50 frames. In the color map of the PIV\nimages, the direction of velocity vectors represents the\ndirection of rotating particles in the annular region. The\nmagnitude of the average velocity of rotating particles is\nrepresented by color bars. It is clear from Fig. 2 that\nparticles rotate in anticlockwise direction at given mag-\nnetic \feld. Magnitude of the azimuthal velocity ( v\u001e) of\nrotating particles increases from the disk edge region (r\n\u00188 mm) to the ring edge region (r \u001814.5 mm), which\nindicates a velocity gradient. For getting the angular fre-5\n0.00.20.40.60.81.01.21.40.00.10.20.30.4\nΩ [rad/s]\nB [T]\nFIG. 4. The angular frequency variation against magnetic\n\feld corresponds to the discharge conditions of Fig. 3 \u0004Vup\n= 60 V, Vdown = 50 V and annulus width of dusty plasma \u0018\n5 mm, \u000fVup= 60 V, Vdown = 50 V, and annulus width \u00182.5\nmm,NVup= 50 V and Vdown = 50 V and annulus width \u0018\n4 mm.\nquency of rotating particles, !=v\u001e=r, the PIV images\nat di\u000berent magnetic \felds are further analysed. An av-\nerage angular frequency (\n) of the rotating particles be-\ntween the co-centric disk and ring is shown in Fig. 3(a).\nThe angular frequency of rotating particles is found to be\nnearly constant within the errors at di\u000berent strengths of\nmagnetic \feld, B = 0.1 T to 1.3 T. The nearly constant\nvalue of \n represents the rigid body rotation of annulus\ndusty plasma in the presence of a magnetic \feld.\nDoes the rotational properties of annulus dusty plasma\ndepend on its width? To see the e\u000bect of annulus width\nof the dust grain medium on the rotational motion, a\nsecond set of experiments is carried out for dusty plasma\nwith a lower annulus width \u00182.5 mm for the same disk-\nand-ring-con\fguration. The discharge conditions (p = 30\nPa,Vup= 60 V and Vdown = 50 V) are kept similar to\nthe earlier experiments (Fig. 3(a)). In this case, we esti-\nmate thend\u00181-2\u0002104cm\u00003at B = 0 T. The average\nangular frequency of rotating particles from disk edge (r\n\u00189 mm) to ring edge (r \u001813 mm) at di\u000berent mag-\nnetic \feld strengths is presented in Fig. 3(b). The dusty\nplasma width increment from \u00182.5 mm to\u00184 mm after\nincreasing the magnetic \feld above a threshold value (B\n>0.6 T) is also noticed. The variation of \n is observed to\nbe nearly constant at given magnetic \feld strengths (B\n= 0.1 T to 1 T). The constant value of \n represents the\nrigid body rotation of dust grain medium. In comparison\nof \n at given B for large annulus width (Fig. 3(a)) and\nsmall annulus width (Fig. 3(b)) dusty plasma conforms\nthe increase of \n with increasing the annulus width of\ndusty plasma (or dust density) at same discharge con-\nditions. However, the \row (or rotational) characteristics\n(e.g. direction, rigid rotation) of di\u000berent annulus widthsdusty plasmas remain same at given magnetic \feld.\nAnother set of experiments is performed to observe the\ne\u000bect of electrode voltage (rf input power) on the rota-\ntional properties of annulus dusty plasma in presence of\na magnetic \feld. Fig. 3(c) shows the variation of angular\nfrequency of particles at pressure p = 30 Pa and peak-\nto-peak voltages of 50 V in the presence of a magnetic\n\feld. It should be noted that the con\fning potential well\nbetween disk and ring gets modi\fed if Vupis lowered.\nTherefore, the annulus width of dusty plasma ( \u00184 mm)\nis always smaller than that of high values of Vup. \n has\na lower value near the disk edge (r \u00189 mm to 10 mm)\nin the low magnetic \feld regime (B <0.8 T). However,\nnearly constant \n is observed within an error range of\n<15 %. At a distance far away from the disk edge (r\n>10 mm), \n remains almost constant for all values of\nmagnetic \feld strength. Thus, annulus dusty plasma ex-\nhibits rigid rotational motion in anticlockwise direction\nfor a given value of magnetic \feld strength (B <1.1 T).\nHigher values of \n are expected at large peak-to-peak\nelectrode voltages (or input rf power) in the presence of\na magnetic \feld as seen in Fig. 3(a) and Fig. 3(b).\nIt is also important to plot the average angular\nfrequency against the strength of the external magnetic\n\feld. In Fig. 4, the average values of averaged \n(Fig. 3(a) to Fig. 3(c)) is plotted against magnetic \feld\nstrength. In this \fgure, we see a linear increase of angu-\nlar frequency up to a threshold magnetic \feld strength\nB>0.6 T and after that it remains nearly unchanged or\nsaturated in a certain range of magnetic \feld strength\n(0.6 T<B<1 T). Further increase in magnetic \feld\nstrength (B >1 T) slightly lowers the \n-value. The\nrate of increase of \n below a threshold magnetic \feld\nstrongly depends on the electrodes' peak-to-peak voltage\nfor a given pressure. The saturation value of \n shifts\nto a lower value if the dust width is decreased or input\npower is lowered.\nB. Dusty plasma con\fned between the co-centric\nconducting disk and ring\nOne of our previous studies reports the in\ruence of\nring materials (conducting and non-conducting) on the\ndepth of the con\fning potential well in the presence of a\nmagnetic \feld25. This is probably due to the di\u000berence\nof surface potentials of the materials in an rf discharge.\nA dip potential well is expected between the conducting\nco-centric disk and ring. At this point, it is not easy to\ndescribe the dust grain dynamics in an annular region of\nthe conducting disk and ring in the presence of a strong\nmagnetic \feld. Therefore, a set of experiments is carried\nout using the aluminium disk-and-ring-con\fguration in\nthe place of the Te\ron con\fguration (III A). It is ob-\nserved in experiments that plasma between ring and disk\nis unstable at high rf power (or large voltage on elec-6\nFIG. 5. PIV images of the rotational motion of dust grains in the X{Y plane at various strengths of magnetic \feld. Particles\nare con\fned between the aluminum disk ( Ddisk= 15 mm) and ring ( Din\nring= 30 mm) at electrode voltages Vup= 55 V and\nVdown = 55 V and argon pressure, p = 30 Pa. The inner blue color region includes the disk as well a dust free region (see\nFig. 1(b)).\nFIG. 6. PIV images of the rotational motion of dust grains in the X{Y plane at various strengths of magnetic \feld. Dust\ngrains are con\fned between the aluminum disk ( Ddisk= 5 mm) and ring ( Din\nring= 30 mm) at electrodes voltage Vup= 55 V\nandVdown = 55 V and argon pressure, p = 30 Pa. The inner blue color region includes the disk as well a dust free region that\ncan be separated using the plots scale.\ntrodes) and high magnetic \feld (B >0.4 T). Therefore,\na study is conducted at low peak-to-peak electrodes volt-\nage of 55 V ( Vup= 55 V and Vdown = 55 V). At this\ndischarge conditions, two separate experiments with an\naluminium disk of diameter ( Ddisk) 15 mm and 5 mm are\nperformed. The aim of using a small diameter (5 mm)\ndisk is to demonstrate the role of the magnetic \feld on\nthe \row characteristics of a greater annulus width of the\ndusty plasma. The annulus width of dusty plasma for a\n15 mm diameter disk is \u00184 mm and for a 5 mm disk is\n\u00189 mm at B = 0 T. For getting more details about ro-\ntational properties of dusty plasma, PIV analysis of still\nimages is performed as discussed in Sec. III A.\nFig. 5 shows the PIV images of rotating particles be-\ntween the co-centric disk and ring ( Ddisk= 15 mm,Din\nring= 30 mm) at various strengths of magnetic \feld. The ve-\nlocity distribution con\frms the radial velocity gradient\nfrom the disk edge region (r \u00188 mm) to ring edge region\n(r\u001812 mm). The direction of rotation is anticlockwise\nfor all values of magnetic \feld strength (B <1 T) at P =\n30 Pa and peak-to-peak electrodes voltage of 55 V. The\nvelocity distribution of rotating particles between the co-\ncentric disk and ring ( Ddisk= 5 mm,Din\nring= 30 mm) is\npresented in Fig. 6. We observe the radial velocity gra-\ndient as well as uni-directional (anticlockwise) rotational\nmotion similar to Fig. 5 at various strengths of magnetic\n\feld.\nThe variation of angular frequency of dust grains at\ndi\u000berent magnetic \feld strengths (correspond to Fig. 5\nand Fig. 6) is depicted in Fig. 7(a) and Fig. 7(b), respec-7\n89 1 01 11 21 30.00.10.20.30.40.5Ω [rad/s]r\n [mm] B = 0.2 T \nB = 0.4 T \nB = 0.6 T \nB = 0.7 T \nB = 0.9 T(a)\n4567891011120.00.10.20.3\n \n \nΩ [rad/s]\nr [mm]\n B = 0.1 T\n B = 0.3 T\n B = 0.5 T\n B = 0.7 T(b)\n9 10 11 120.000.040.080.12\nΩ [rad/s]\nr [mm]\n B = 0.3 T\n B = 0.5 T\n B = 0.6 T\n B = 0.7 T(c)\nFIG. 7. The radial variation of angular frequency (\n) of rotating particles at various strengths of magnetic \feld is shown. (a)\nAluminum disk ( Ddisk= 15 mm) and ring ( Din\nring= 30 mm), Vup= 55 V and Vdown = 55 V, p = 30 Pa and annulus width of\ndusty plasma \u00184 mm at B = 0 T. (b) Te\ron disk ( Ddisk= 5 mm) and ring ( Din\nring= 30 mm), Vup= 55 V and Vdown = 55\nV, p = 30 Pa and annulus width \u00189 mm at B = 0 T. (c) Te\ron disk ( Ddisk= 15 mm) and ring ( Din\nring= 30 mm), Vup= 45\nV and Vdown = 45 V, and p = 30 Pa.\ntively. The nearly constant values of \n from disk edge\nto ring edge region demonstrate the rigid body rotation\nof dust grain medium for a given magnetic \feld. To ob-\nserve the e\u000bect of the electrodes voltage on annulus dusty\nplasma, the experiments are performed at low peak-to-\npeak electrodes voltage of 45 V ( Vup= 45 V and Vdown =\n45 V) and p = 30 Pa in the presence of a magnetic \feld.\nThe variation of \n from disk edge to ring edge region at\nvarious strengths of magnetic \feld is shown in Fig. 7(c).\nAt lower electrodes voltage (or low rf power), dust grains\nalso rotate in anticlockwise direction with a constant an-\ngular frequency. A lower value of \n is observed in case of\nlow rf power or low Vpp. The direction of rotation of an-\nnulus dusty plasma is found to be anticlockwise at various\nsets of pressure (p = 30 to 50 Pa) and electrodes volt-ages (40 to 75 V) for such disk-and-ring-con\fgurations.\nIt should also be noted that the reversal of rotation is not\nobserved at di\u000berent discharge conditions in the presence\nof a magnetic \feld.\nThe average value of \n at a given magnetic \feld\nstrength (Fig. 7) is used to get the plots of \n against\nmagnetic \feld strength. Fig. 8 represents the angular\nfrequency variation of rotating particles with magnetic\n\feld. It is observed that \n increases almost linearly with\nincreasing magnetic \feld up to a threshold value of mag-\nnetic \feld strength (B >0.6 T). Above the threshold\nmagnetic \feld strength, a slight change in \n with increas-\ning magnetic \feld is noticed. The change rate of \n below\nthe threshold magnetic \feld depends on the rf power (or\nplasma density). The threshold value of magnetic \feld8\n0.2 0.4 0.6 0.8 1.00.10.20.30.4\n \n \nΩ [rad/s]\nB [T]\nFIG. 8. The angular frequency plots against magnetic \feld\ncorrespond to discharge conditions of Fig. 7 \u0004Vup= 55 V,\nVdown = 50 V and annulus width of dusty plasma \u00184 mm,\n\u000fVup= 55 V, Vdown = 55 V, and annulus width \u00189 mm,N\nVup= 45 V and Vdown = 45 V and annulus width \u00183 mm.\nstrength is found to be almost the same for di\u000berent an-\nnular widths dusty plasmas at same discharge conditions\nbut it shifts to higher or lower values with changing the\nelectrodes voltage ( VupandVdown).\nIV. DISCUSSION\nIn an rf discharge, massive dust particles are sus-\npended in the strong electric \feld of the sheath region to\nbalance the gravitational force and simultaneously con-\n\fned by a radial electric \feld created by an additional\nring on the lower powered electrode10,11. In this study,\ndust grains are con\fned in a potential well created by\na co-centric conducting (non-conducting) disk-and-ring-\ncon\fgurations. The grains are con\fned in the annular\nregion of a disk and ring, where the radial electric \felds\nare in opposite directions. The radial electric \feld points\ninward and outward due to the disk and ring, respec-\ntively (see Fig. 1(b)). Before turning on the magnet (B\n= 0 T), dust grains exhibit the thermal motion around\ntheir equilibrium position. At low magnetic \feld strength\n(B<0.02 T), \frst the con\fning potential well gets modi-\n\fed and with increasing the magnetic \feld (B >0.02 T)\ndust grains start to rotate in anticlockwise direction .\nIn unmagnetized plasma, ions \row ( ~ vi=\u0016i~Er) radi-\nally in the direction of electric \feld ( ~Er). In the presence\nof a magnetic \feld, the radial ion \row gets deviated due\nto the Lorentz force (q ( ~ vi\u0002~B)). Due to the Lorentz or\n~Er\u0002~Bforce on ions, they also have a velocity compo-\nnent in the azimuthal direction. The azimuthal drifted\nions drag the dust grains in the direction of \row and set\nthem into rotational motion in the presence of a mag-\nnetic \feld19,27,32. The direction of rotation of dust grainmedium is determined by ~E\u0002~Bdrift of ions. Since the\nradial electric \felds are in opposite directions due to the\ndisk and ring, an opposite or shear dust \row (clockwise\ndue to the disk and anticlockwise due the ring) is ex-\npected in the presence of a magnetic \feld. However, only\nanticlockwise rotational motion of annulus dusty plasma\nis observed at various strengths of magnetic \feld at given\ndischarge condition. We observe the rotational inversion\nwith changing the direction of magnetic \feld in our ex-\nperiments. It con\frms the role of azimuthal ion drag or\n~E\u0002~Bdrifted motion of ions to rotate the annulus dust\ngrain medium in presence of the magnetic \feld.\nIt is known that the radial electric \feld strength can be\nestimated by measuring the plasma potential ( Vp) across\nthe radial distance from disk edge region (I) to ring edge\nregion (III). Since the annular region is \u00188 mm, the use\nof an emissive probe to measure plasma potential is a\nchallenging task at strong magnetic \feld strength. It is\ndi\u000ecult to measure the small variation in Vpacross this\nradial distance. It is also known that Vpcan be estimated\nby using the measured value of the \roating potential ( Vf)\nfor a given value of Te33. In the present experiments, Te\nvariation across the radial distance (r \u00186 mm to 16 mm)\nis expected to be negligible?. Therefore, the radial vari-\nation ofVfis assumed to be the Vpvariation at a given\nmagnetic \feld strength. A cylindrical probe of length\n\u00182 mm and radius 0.125 mm is used to measure radial\nvariation of Vf12from disk edge region (r \u00184mm) to ring\nedge region (r\u001818 mm).\nIt should be noted that plasma parameters are\nnot strongly a\u000bected by the dust grains if Havnes\nparameter34Ph=Zdnd=nihas a lower value, i.e,\nPh<<1. Here,ndis the dust density, niis the ion\ndensity and Zdis the dust charge number. In such dusty\nplasma (Ph<1), ambient plasma parameters without\ndust grains can be used to understand the dynamics of\ndust grain medium. In our experiments, we estimate\nPh<1 fornd\u00181\u0002103to 5\u0002103cm\u00003,ni\u00186\n\u0002108cm\u00003to 9\u0002108cm\u00003, andZd\u00181\u0002104. Hence,\nVfmeasurements are carried out in ambient plasma\n(without dust grains) to understand the rotational\nmotion of annulus dusty plasma.\nFig.9 shows the radial variation of Vfat Z\u00180.7\ncm for (a) non-conducting (Te\ron) and (b) conducting\n(aluminium) disk and ring ( Ddisk= 15 mm,Din\nring= 30\nmm) con\fgurations at given discharge conditions in the\npresence of a magnetic \feld. For a clearer understanding\nofVfvariation, the plots are divided into three regions\nas presented in Fig. 1(b). The dust grains are con\fned\nin the central region. Fig. 8 represents clearly that Vf\nhas a larger gradient in the ring edge region (I) than in\nthe disk edge region (III) for a given magnetic \feld. The\ngradient in Vfde\fnes the strength of the radial electric\n\feld in that region. Therefore, a stronger radial electric\n\feld is expected in the ring edge region (III) than in\nthe disk edge region (I) at a given magnetic \feld. The\nresultant electric \feld ( ER) in the central region (II) is\nthe superposition of both opposite radial electric \felds.9\n2 4 6 810121416182078910\nFloating potential [V]\nr [mm]\n B = 0.2 T\n B = 0.3 T\n B = 0.5 T\n B = 0.7 T\n B = 0.9 T(a)\nCentral region Disk edge region Ring edge region\nErEr\nER\n2 4 6 81012141618678910\nFloating potential [V]\nr [mm]\n B = 0.2 T\n B = 0.3 T\n B = 0.4 T\n B = 0.5 T\n B = 0.7 T\n B = 0.8 T(b)\nCentral region Ring edge region Disk edge region\nErEr\nER\nFIG. 9. Floating potential variation across the radial dis-\ntance at various strengths of magnetic \feld. (a) Te\ron disk\n(Ddisk= 15 mm) and ring ( Din\nring= 30 mm), Vup= 60 V and\nVdown = 50 V, and p = 30 Pa. (b) Aluminum disk ( Ddisk= 15\nmm) and ring ( Din\nring= 30 mm, Vup= 55 V and Vdown = 55\nV, and p = 30 Pa. The green arrows represent the direction\nof radial electric \feld (E).\nTherefore, the annulus dust cluster is expected to rotate\nin accordance to the direction of the resultant electric\n\feld (~ER). Thus, the direction of rotation is observed\nas anticlockwise at various strengths of magnetic \feld.\nThe strength of azimuthal ion drag force determines the\nmagnitude of \n at a given magnetic \feld and increases\nwith increasing magnetic \feld. Above the threshold\nmagnetic \feld strength (B >0.6 T), the Vfgradient\nis found to be strong near the edge of the ring (see\nFig. 9), resulting in a low resultant electric \feld in the\ncentral region. We also observe a slight expansion of\ndust cluster width (slightly weak coupling) above a\nthreshold magnetic \feld strength. These two factors,\nresultant electric \feld and coupling strength among\nthe dust grains, determine the magnitude of angular\nfrequency (lower value) at strong magnetic \feld strength.V. SUMMARY\nThe rotational properties of an annulus dust grain\nmedium con\fned in a potential well created between a\nco-centric conducting (or non-conducting) disk and ring\nare explored in a strongly magnetized rf discharge. A su-\nperconducting electromagnet is used to introduce a uni-\nform magnetic \feld to the dusty plasma. The plasma is\nignited between a lower aluminum electrode and an up-\nper ITO-coated transparent electrode using a 13.56 MHz\nrf generator with the matching network for a given ar-\ngon pressure. The dynamics of the annulus dust grain\nmedium at various strengths of magnetic \feld are anal-\nysed using PIV Technique.\nIn an unmagnetized plasma (B = 0 T), the con\fned\ndust particles exhibit a thermal motion around their equi-\nlibrium position. At \frst, the con\fning potential well\ngets modi\fed while the magnetic \feld is applied. There-\nfore, rotational motion is not recorded at low magnetic\n\feld (B<0.2 T). Above a magnetic \feld strength of B\n= 0.02 T, the dust grains start to rotate in anticlock-\nwise direction. The frequency of rotational motion in\nthe annular region is found to be nearly constant (or\nrigid rotation) for the conducting, as well as for the\nnon-conducting con\fguration at a given magnetic \feld\nstrength. The angular frequency \frst increases linearly\nwith magnetic \feld strength up to a threshold value (B\n>0.6 T), and after that \n remains nearly unchanged in\nthe magnetic \feld range (0.6 T <B<1 T). Above B\n= 1 T, a slight reduction in \n is observed in the non-\nconducting con\fguration. The rate of change of \n with\nthe magnetic \feld strength depends on the annulus width\nof the dusty plasma (or the dust density) and the input rf\npower (or plasma density) at a given argon pressure. The\nsaturation value of \n shifts to a lower value as the dust\nwidth or rf power is reduced. The annulus dusty plasma\nof di\u000berent widths exhibit an anticlockwise rigid rota-\ntional motion at di\u000berent discharge conditions. At simi-\nlar discharge conditions, a higher rotational frequency is\nexpected for the conducting disk-and-ring-con\fguration\nthan for the non-conducting case (observed in experi-\nments). The uni-directional (or anticlockwise) rotation\nat di\u000berent electrodes voltages between the electrodes\nand for di\u000berent pressures in the presence of a magnetic\n\feld were observed. There were no discharge parameters\nfound to observe the opposite dust \row (or shear \row)\nin the annular region in the presence of a magnetic \feld.\nThe dust rotation in presence of a magnetic \feld is\na result of the azimuthal ion drag force. The direction\nof rotation of dust grains con\fned by opposite radial\nelectric \felds depends on the direction of resultant\nradial electric \feld ( ER). The direction of rotation and\nthe angular frequency variation with magnetic \feld are\nexplained based on the magnitude of the radial potential\ngradient (or electric \feld). Since the radial potential10\ngradient (or Er) is found to be di\u000berent in both disk\nedge region (I) and ring edge region (III), the resultant\nelectric \feld ( ER) is expected to drive the rotational\nmotion through ~ER\u0002~Bdrift of ions. Thus, anticlockwise\nrotational motion of annulus dusty plasma is observed\nat various strengths of magnetic \feld. The challenges\nof using probe diagnostics restrict us to explain the\nrotational motion qualitatively. Therefore, numerical\nsimulations are required to understand the dynamics\nof an annulus dusty plasma in a strong magnetic \feld.\nIn future, numerical simulation of such dusty plasma\nexperiments are planned.\nVI. ACKNOWLEDGEMENT\nThis work is supported by the Deutsche Forschungsge-\nmeinschaft (DFG). The authors want to thank Dr. M.\nKretschmer for his experimental assistance.\n1G. E. Mor\fll and A. V. Ivlev, \\Complex plasmas: An interdisci-\nplinary research \feld,\" Rev. Mod. Phys. 81, 1353{1404 (2009).\n2H. M. JAEGER and S. R. NAGEL, \\Physics of the granular\nstate,\" Science 255, 1523{1531 (1992).\n3H. M. Jaeger, S. R. Nagel, and R. P. Behringer, \\Granular solids,\nliquids, and gases,\" Rev. Mod. Phys. 68, 1259{1273 (1996).\n4W. C. K. Poon, \\The physics of a model colloid polymer mix-\nture,\" Journal of Physics: Condensed Matter 14, R859{R880\n(2002).\n5A. Stradner, H. Sedgwick, F. Cardinaux, W. C. K. Poon, S. U.\nEgelhaaf, and P. 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The\nspin Coulomb drag an intrinsic friction mechanism which ope rates whenever the average velocities\nof up-spin and down-spin electrons differ.\nIntroduction: The spatial-temporal dynamics of mag-\nnetization /vectorM(/vector r,t) in a ferromagnetic system is described\nby the phenomenological Landau-Lifshitz-Gilbert (LLG)\nequation [1]. Along with this, there is also an inverse\nprocess when the spatial-temporal change in magneti-\nzation induces additional electromagnetic fields acting\non conduction electrons. The spin-driving force (SDF)\ncaused by these fields was first predicted by Berger [2]\nand has been recently formulated by generalizing Fara-\nday’s law [3]. These fields are non-conservative and spin-\ndependent. Explicit expressions for induced spin electro-\nmagnetic fields can be represented as [4–6]:\nE↑↓\ni=±¯h\n2e/vector m·(∂t/vector m×∂i/vector m),\nB↑↓\ni=∓¯h\n2eǫijk/vector m·(∂j/vector m×∂k/vector m), (1)\nHere/vector m=/vectorM/|/vectorM|isaunitvectordirectedalongthemag-\nnetization /vectorM(/vector r,t). Arrows ( ↑,↓) characterize the elec-\ntron spin directions.\nThe fields in (1) generate a spin-dependent Lorentz\nforce/vectorF↑↓\ns=e(/vectorEs+/vector v×/vectorB↑↓\ns). The manifestation of a\nspin-driving force is a universal phenomenon in magnetic\nmetals and can be understood through the gauge field\ntheory, the equation of motion, and Berry’s spin phase\n[7, 8]. A spin electric field was experimentally observed\nin the case of a moving domain wall [9].\nUnlike conventional electromagnetic fields, spin elec-\ntric and magnetic fields /vectorEs,and/vectorBsdramatically af-\nfect the kinetics of conduction electrons. In the field\n/vectorEs, electrons with spins ↑, or↓drift in opposite direc-\ntions; thereby inducing a spin-polarized current, /vectorjs=\n(n↑−n↓)e/vector vs.Heree, and/vector vs,are the charge of an elec-\ntron and the drift velocity of the carriers due to the spin\nelectric field Es, respectively. When emerging, the spin\nmagnetic field associated with the Berry phase produces\nthe spin Hall effect that is detected as an anomalous Hall\neffect[10]. TheLorentzforcecausedbythespinmagnetic\nfield deflects free charge carriers in opposite directions,\ndepending on the orientation of the spin ( ↑,↓), thereby\ninducing a Hall voltage and a spin current if the electron\nconcentrations n↑/negationslash=n↓.Here special attention should be drawn to one more\ncharacteristic feature of spin electromagnetic fields. As\ncan be seen, the electric and magnetic components of the\nspin fields act as a separator, selecting free charge carri-\ners in the spin direction ( ↑,↓). Consequently, the system\nofconductionelectronscanbe treatedasaset oftwospin\nsubsystems,eachofwhichischaracterizedbyitsownspin\ndirection. Since the electric field governs the drift veloc-\nity of charge carriers, it is obvious that the drift velocity\nof each of the subsystems is spin-dependent. It can be\nassumed that the spin-dependent fields create conditions\nunder which the manifestations of effects associated with\ntheCoulombinteractionoftwospinsubsystemsofcharge\ncarriers become possible.\nElectron-electron (e-e) interactions play a critical role\nin various phenomena, ranging from high-temperature\nsuperconductivity, the fractional quantum Hall effect, a\nWigner crystal, a Mott transition, etc. However, both\nthe magnitude and the influence of this interaction on\nthe kinetic properties of crystals are difficult to measure.\nOne of the methods that proved to be effective in mea-\nsuring scatteringrates directly due to the Coulomb inter-\naction is the Coulomb drag effect of charge carriers. The\nessence of the effect is to arise a response as a generated\nvoltage or electric current in a conductive system when\ncurrentpassesthroughanotherconductivefilm separated\nfrom the first one by a dielectric layer [11] (see literature\nthere). The effect is underlain by the interlayer Coulomb\ninteraction of conduction electrons separated by a dielec-\ntric.\nThe drag effect of charge carriers due to the Coulomb\ninteraction exhibits itself also in a system of spin-\npolarized charge carriers. Consider the conductivity of\ncarriers with different spins within a two-channel model\n(conductivity along two channels with different spin ori-\nentations ( ↑,↓) in each channel). Let the drift veloc-\nitiesv↑, v↓of carriers in each of the spin subsystems\nbe different. If other electron scattering sources lack,\nthe Coulomb interaction preservesthe system’s total mo-\nmentum and redistributes it between charge carriers in\ndifferent ( ↑) or (↓) channels. In this case, faster carri-\ners drag slow ones through transferring a part of their\nmomentum to implement the spin Coulomb drag effect2\n(SCD) [12]. Then, it is obvious that the charging current\nje=e(n↑+n↓)ve(veisthedriftvelocityofelectrons)will\nbeunchangedagainstthespincurrent js= (n↑v↑−n↓v↓).\nWhen the drift velocities of carriers in the spin channels\nare equalized, v↑=v↓, the spin current is js= 0, if\nn↑=n↓.\nIt should be underscored that for the spin Coulomb\ndrag effect to be observed, the spin system of conduc-\ntion electrons divides into two subsystems not physically\nbut through only the spin orientation ( ↑,↓). Such a rep-\nresentation of the electron system for the manifestation\nof the SCD effect is fundamental. Some of the possible\nscenarios for such a representation are considered in [12].\nNote that the concomitant spin electric and magnetic\nfields produced in the case of a spatial-temporal inhomo-\ngeneousmagneticstructurenaturallylead tospin separa-\ntion of charge carriers to provide the condition necessary\nfor SCD to arise as the difference in the drift velocities\nof electrons in spin channels.\nThe quantitative measure of SCD is the transresistiv-\nityρ↑↓; it is defined as the ratio between the electro-\nchemical potential gradient ζ↑for electrons with spin ↑\nand the electric current j↓,(j↓= 0).This definition is\nsimilar to that of the Coulomb drag effect in two spa-\ntially separated layers. The results of calculating ρ↑↓for\na three-dimensionalelectrongas in the random-phaseap-\nproximation are presented in [11, 13].\nNext, we demonstrate that the separation of charge\ncarriers by their spin state, associated with the concomi-\ntant spin electric and magnetic fields, when taking into\naccount the Coulomb interaction between charge carri-\ners, leads to the effect of spin Coulomb drag of charge\ncarriers.\nThe spin Coulomb drag effect can be calculated apply-\ning the Kubo method, Green’s functions, or the Boltz-\nmann kinetic equation. However, it is easier to under-\nstand SCD at the phenomenological level.\nSo, let us look into an electric current flowing through\na non-equilibrium inhomogeneous magnetic system; the\nlatter’s magnetization /vectorM(/vector r,t) has a spatial-temporal in-\nhomogeneity. The Hamiltonian of the system at hand\nis:\nH=He+HeE+Hsd+Hee+Hev+Hv,\nwhereHe=Hk+Hsis the Hamiltonian of free conduc-\ntionelectrons, HeEisinteractionwithanexternalelectric\nfield/vector e0.Hsdis exchange interaction between free and lo-\ncalized electrons. Heeis Coulomb interaction between\nconduction electrons. Hevdescribes the interaction with\nscattering centers. Hvis the lattice Hamiltonian.\nIt is assumed that upon performing the Hamiltonian’s\ncanonical transformation that diagonalizes the exchange\ninteraction, we arrive at the realization of concomitant\nspin electric Esand magnetic Bsfields. As a conse-\nquence, the divisionofthe systemofconductionelectrons\ninto two spin subsystems ( ↑,↓) takes place.To start with, we explore the SCD effect purely phe-\nnomenologically. For this, we resort to the equation of\nmotionm(d/vector υ/dt) =/summationtext\ni/vectorFi. The drift velocities of elec-\ntrons/vector υ↑, /vector υ↓are formed by forces acting on electrons,\ninduced by electric fields and the Coulomb interaction\nbetween electrons:\n/vector υ↑↓=<(/vectorF/vectorE∗+/vectorFc)τ\nm>\nHere/vectorF/vectorE∗+/vectorFcare the forces acting on the electrons from\nthe electric E∗=/vectorE0+Esfields and are also due to the\nCoulombinteraction /vectorFc. Timeτcharacterizesdissipative\nprocesses.\nFor simplicity, we restrict ourselves to investigation of\na spin subsystem of conduction electrons (for example,\nwith spin α=↑,↓). Letυαbe the speed of the mass-\ncenter,nαis the number ofsuch chargecarriers. We write\ndown an equation of motion for the electrons of the spin\nsubsystem with spin ( α=↑,↓) as follows:\n˙pα=−enα/vectorE∗+Fα,−α\nc+ ˙pα\nev, (2)\nwhere/vectorE∗=/vectorE0+/vectorEα./vectorE0,/vectorEαare the external and spin\nelectric fields. The first summand on the right side of\nthe equation involves the interaction of electrons with\nelectric fields. The second summand is responsible for\nthe Coulomb interaction between electrons with differ-\nent spin orientations (different spin subsystems). At the\nsame time, it is obviously that F↑↓\nc=−F↓↑\nc. Preserv-\ning the total momentum of the electronic system, the\nCoulomb interaction of electrons redistributes it between\nthe spin ( ↑,↓) subsystems of electrons (see figure). Then\n|n↑/vector vc|=|n↓/vector vc|, where /vector vcis the momentum lost (ac-\nquired) in a single electron-scattering act. The last term\n˙p↑\nev= (i¯h)−1[p↑,Hev] as relaxational one can be written\nin the relaxation time approximation.\nBeforeproceedingto the microscopicdescription ofthe\nSCD effect, consider it phenomenologically. For this, we\nwrite down expressionsfor the total charge jeand spin js\ncurrents, believing that each of the forces acting on the\nconduction electrons contributes to forming the electron\ndrift velocity. Besides, the momentum is assumed to be\ntransferred from the subsystem ↑to the subsystem ↓due\nto the Coulomb interaction. An expressionfor the charge\ncurrent is given as\nje=e[n↑(/vector υ0+/vector υs−/vector υc)+n↓(/vector υ0−/vector υs+/vector υc)],\nwhere/vector υ0,/vector υs,/vector υcare the components of the drift velocity\nof electrons, due to the action of the electric fields E0,Es\nand Coulomb interaction\nIf the charge current is spin-polarized ( n↑/negationslash=n↓, n=\nn↑+n↓), then\n/vectorje=e[n/vector υ0+(n↑−n↓)(/vector υs−/vector υc)] =en/vector υ0,\ne.g. the drag effect contributes to the spin-polarized cur-\nrent until /vector υs/negationslash=/vector υc. When /vector υs=/vector υc, the contribution to3\nthe spin-polarized current induced by the spin drag effect\nvanishes and /vectorje=e[(n↑+n↓)]/vector υ0.\nConsider an expression for the spin current. We have\n/vectorjs=e[n↑(/vector υ0+/vector υs−/vector υc)−n↓(/vector υ0−/vector υs+/vector υc)].\nor\n/vectorjs=e[(n↑−n↓)/vector υ0+n(/vector υs−/vector υc).\nAs can be seen, at n↑=n↓, the spin current is nonzeroas\nlong as there is a possibility of pumping the momentum\nbetween spin subsystems and /vectorjs= 0 when /vector υs=/vector υc. If\nn↑/negationslash=n↓, the SCD effect in the spin current manifests\nitself until pumping the electron momentum between the\nspin subsystems is possible to happen. When /vector υs=/vector υc,\ntheeffectoftheSCDturnstozeroand /vectorjs=e[(n↑−n↓)/vector υ0.\nUsing the mass-center approximation, the expression\nthat determines the Coulomb part of the force /vectorFccan be\nwritten as [12]\n/vectorFc≡/vectorF↑↓=−dmn↑(n↓/n)(/vector υ↑−/vector υ↓),(3)\nwheren=n↑+n↓. The coefficient dcontrols the\nspin drag. An explicit expression for the coefficient\ndcomes from the expression for the resistance matrix\n/vectorE↑=/summationtext\n↓ρ↑↓/vectorj↓. Then\nd=ne2\nmρ↑↓. (4)\nIn the Borninteractionapproximation,the expressionfor\ntransresistivity ρ↑↓can be deduced by various methods,\nfor example, such as the formalism of the Green’s func-\ntion method, the method of projection operators [14], or\nwithin the Boltzmann kinetic equation [16]. The micro-\nscopic theory of calculating the quantity ρ↑↓is similar\nto calculating the conventional Coulomb resistance be-\ntween parallel two-dimensional layers of an electron or\nhole gas [11]. However, it differs in some important as-\npects. Firstly, chargecarrierswithoppositespinsinteract\nwith the same set of scattering centers. Secondly, the en-\nergy spectrum of chargecarriersbecomes spin-dependent\nas a result of the canonical transformation of the original\nHamiltonian.\nIn keeping with [14, 15], we write down a general ex-\npression for ρ↑↓:\nρ↑↓=π¯h2\n4n↑n↓m2/summationdisplay\n/vector p′,/vector p,/vector q∞/integraldisplay\n−∞dωq2|V(q)|2f′\n1(ε/vector p,↑)\n×(f2(ε/vector p′,↓)−f2(ε/vector p′+/vector q,↓))/braceleftbiggf1(ε/vector p−/vector q,↑)\ne∆−1−1−f1(ε/vector p−/vector q,↑)\n1−e−∆]/bracerightbigg\nδ(ε/vector p−/vector q,↑−ε/vector p,↑+¯hω)δ(ε/vector p′+/vector q,↓−ε/vector p′,↓−¯hω).(5)\nHere ∆ = ¯ hω/kBT,kBis the Boltzmann constant, Vq=\n4π2/q2ǫisFouriercomponentoftheinteractionpotential,\nǫis the dielectric constant of material, fi(ε/vector p,σ),(i= 1,2)are the Fermi-Diracdistribution functions for conduction\nelectrons.\nGoingoverfromthesummationovervectors /vector p, /vector p′toin-\ntegrationandevaluating the integrals, we arriveat an ex-\npression for the degenerate statistics of conduction elec-\ntrons and ¯ hω≪kBT:\nρ↑↓=β\n6πn↑n↓e2V/summationdisplay\n/vector qq2∞/integraldisplay\n0dω×\n|V(q)|2χ′′\n0↑(q,ω)χ′′\n0↓(q,−ω)\n|ǫ(q,ω)|2sinh2(βω/2),(6)\nwhereβ= 1/kBT.Vis the volume of the system.\nχ0α(q,ω) is the noninteracting spin resolved density-\ndensity responsefunction, and ε(q,ω) = 1−Vqχ0↑(q,ω)−\nVqχ0↓(qω) is the random-phase approximation dielectric\nfunction.\nFurther calculations of the transresistivity ρ↑↓, as well\nas its various approximations, accounting for dissipation\nprocesses and possible methods for detecting the spin\nCoulomb drag effect face no difficulties (see, for example,\n[13, 17]).\nFinally, we note that the presence of a spin-dependent\nmagnetic field B↑↓acting differently on charged parti-\ncles with spin ( ↑,↓) (shifts them in opposite directions),\nalso affects the formation of drift velocities of charge car-\nriers, and, consequently, the charge and spin currents,\ndepending on the direction of their spin. Thus, the spin-\ndependent magnetic field will also manifest itself in the\nspin Coulomb drag effect.\nThe research was carried out within the state assign-\nment of Ministry of Science and Higher Education of the\nRussian Federation (theme ”Spin” No. 122021000036-3).\n[1] Y.Tserkovnyak A.Brataas,G.E.W Bauer,\nPhys.Rev.Lett. 88, 117601-4 (2002).\n[2] L.Berger,Phys. Rev. B. 33, 1572 (1986).\n[3] S. E.Barnes and S.Maekawa, Phys. Rev. Lett. 98,\n246601-4 (2007).\n[4] Y. Yamane, J. Ieda, J. Ohe, et all, J. Appl. Phys 109,\n07C735-3 (2011).\n[5] J. Ohe, S.Maekawa, J. Appl.Phys 105, 07C706-3 (2009).\n[6] G. Tatara, Physica E: Low-dimensional Systems and\nNanostructures 106, 208 (2019).\n[7] M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984)\n[8] A. Stern, Phys. Rev. Lett. 68, 1022 (1992).\n[9] S. A. Yang, G. S. D. Beach, C. Knutson, et all, Phys.\nRev. Lett. 102, 067201-4 (2009).\n[10] N.Nagaosa, Rev. Mod. Phys. 82, 1539-1593 (2010) .\n[11] A. G. Rojo,J. Phys. Condens. Matter 11, R31-R52\n(1999).\n[12] I. D’Amico, G. Vignale, Europhys. Lett. 55, 566 (2001).\n[13] D’Amico and G. Vignale, Phys. Rev. B 62, 4853 (2000)\n[14] I.I. Lyapilin, H.M. Bikkin, Fiz TverdTela 45, 339 (2003).4\n[15] Antti-Pekka Jauho, H. Smith, Phys. Rev. B 47, 4420\n(1993)\n[16] D’Amico and G. Vignale, Phys. Rev. B 65, 085109-12\n(2002). Phys. Rev. B 33, 1572 (1986).[17] B. N. Narozhny, A. Levchenko, Rev. Mod Phys. 88,\n025003-55 (2016)."    },    {        "title": "2309.12776v1.Electric_control_of_optically_induced_magnetization_dynamics_in_a_van_der_Waals_ferromagnetic_semiconductor.pdf",        "content": "Electric control of optically-induced magnetization dynamics in a van der\nWaals ferromagnetic semiconductor\nFreddie Hendriks,1Rafael R. Rojas-Lopez,1, 2Bert Koopmans,3and Marcos H. D. Guimar˜ aes1\n1)Zernike Institute for Advanced Materials, University of Groningen, The Netherlands\n2)Departamento de F´ ısica, Universidade Federal de Minas Gerais, Brazil\n3)Department of Applied Physics, Eindhoven University of Technology, The Netherlands\n(Dated: 22 September 2023)\nElectric control of magnetization dynamics in two-dimensional (2D) magnetic materials is an essential step\nfor the development of novel spintronic nanodevices. Electrostatic gating has been shown to greatly affect\nthe static magnetic properties of some van der Waals magnets, but the control over their magnetization\ndynamics is still largely unexplored. Here we show that the optically-induced magnetization dynamics in the\nvan der Waals ferromagnet Cr 2Ge2Te6can be effectively controlled by electrostatic gates, with a one order of\nmagnitude change in the precession amplitude and over 10% change in the internal effective field. In contrast\nto the purely thermally-induced mechanisms previously reported for 2D magnets, we find that coherent\nopto-magnetic phenomena play a major role in the excitation of magnetization dynamics in Cr 2Ge2Te6.\nOur work sets the first steps towards electric control over the magnetization dynamics in 2D ferromagnetic\nsemiconductors, demonstrating their potential for applications in ultrafast opto-magnonic devices.\nKeywords: magnetization dynamics, two-dimensional magnets, electric control, magneto-optics, van der Waals\nmaterials\nI. INTRODUCTION\nEver since the experimental confirmation of mag-\nnetism in two-dimensional (2D) van der Waals (vdW)\nmaterials1,2, researchers have tried to understand their\nfundamentals and to utilise their unique properties for\nnew technologies, such as novel spintronic devices for in-\nformation storage and processing3–6. The use of magne-\ntization dynamics is particularly interesting since it pro-\nvides an energy efficient route to transfer and process\ninformation7–11. A key challenge in this field, named\nmagnonics, is the effective control over the magnetiza-\ntion and its dynamics using electrostatic means, allowing\nfor energy efficient, on-chip, reconfigurable magnonic cir-\ncuit elements12–14. For conventional (tree-dimensional)\nsystems this control has been shown to be very promis-\ning to reduce the energy barriers for writing magnetic\nbits using spin-orbit torques15,16. Nonetheless, the ef-\nfect is still relatively modest17–20. In contrast, 2D mag-\nnetic semiconductors provide an ideal platform for elec-\ntric manipulation of magnetization. Their low density\nof states and high surface-to-volume ratio allow for an\neffective control over the magnetic parameters in these\nsystems, such as the magnetic anisotropy and saturation\nmagnetization21–25. Additionally, 2D magnetic semicon-\nductors offer a bridge to another exciting field: the com-\nbination of optics and magnetism. These materials have\nshown to possess strong light-matter interaction and high\nmagneto-optic coefficients which strength can be further\ntuned by the use of vdW heterostructures6,26–31. These\nproperties make 2D magnetic semiconductors ideal for\nthe merger of two emerging fields: magnonics and pho-\ntonics.\nMost works on the electric control of magnetiza-\ntion in vdW magnets have focused on their magneto-static properties, such as the magnetic anisotropy, sat-\nuration magnetization and Curie temperature, in both\nmetallic24,25,32–35and semiconducting21–25materials. In\ncontrast, their magnetization dynamics have only re-\ncently started to receive more attention, and studied us-\ning microwave driven magnetic resonance36–42, or time-\ndependent magneto-optic techniques43–51. The latter\nwere used on antiferromagnetic bilayer CrI 3to show\nthat its magnetic resonance frequency can be electrically\ntuned by tens of GHz52. Nonetheless, the electric con-\ntrol over the optical excitation of magnetization and its\nsubsequent dynamics in 2D ferromagnets remains to be\nexplored.\nHere we show that the magnetization dynamics of the\nvdW semiconductor Cr 2Ge2Te6(CGT) can be efficiently\ncontrolled by electrostatic gating. Using ultrafast (fs)\nlaser pulses we bring the magnetization out of equilib-\nrium and study its dynamics with high temporal reso-\nlution through the magneto-optic Faraday effect. Us-\ning both top and bottom electrostatic gates, we inde-\npendently control the gate-induced change in the charge\ncarrier density (∆ n) and the electric displacement field\n(∆D) in the CGT, and show that both have drastic ef-\nfects on the optically-induced oscillation amplitudes and\na more modest effect on its frequency. Finally, we observe\na strong asymmetric behavior on the magnetization os-\ncillation amplitudes with respect to a reversal of the ex-\nternal magnetic field, which is also strongly affected by\nboth ∆ nand ∆ D. This asymmetry can be explained\nby a strong influence of coherent opto-magnetic phenom-\nena, such as the inverse Cotton-Mouton effect and photo-\ninduced magnetic anisotropy, on the excitation of the\nmagnetization dynamics.arXiv:2309.12776v1  [cond-mat.mes-hall]  22 Sep 20232\nΔta d\nc bHext\nprobe\npump\nGr\nGrhBN\nhBNCr2Ge2T e6Vt\nVb\nFused quartz0.0 0.5 1.0 1.5\nΔt (ns)012TRFE signal (arb.units)\nΔn (10  cm )13−2\n-1.8\n0.0\n1.2\nMHeff Hint\n1\n2\n3e\n0 5 10 15\nFrequency (GHz)012TRFE RMS power (a.u.)\nΔn (10  cm )13−2\n-1.8\n0.0\n1.20.1 T\n0.25 T\n0.40 TΔD = 0\nFIG. 1. Magnetization dynamics in a CGT based heterostructure. a , Illustration of time-resolved Faraday ellipticity\nmeasurements, combined with an optical micrograph of the sample (the scale bar is 10 µm). The CGT flake is outlined in\nblue. b, Schematic of the layers comprising the sample, including electrical connections for gating. c, Process of laser-induced\nmagnetization precession (see main text). d, Time-resolved Faraday ellipticity traces at µ0Hext= 100 mT for three different\nvalues of ∆ nwith ∆ D= 0. A vertical offset was added for clarity. e, RMS power of the frequency spectrum of the oscillations\nin the data shown in d. Different transparencies indicate different values of Hext.\nII. DEVICE STRUCTURE AND MEASUREMENT\nTECHNIQUES\nOur sample consists of a CGT flake, encapsulated in\nhexagonal boron nitride (hBN), with thin graphite layers\nas top gate, back gate, and contact electrodes, as depicted\nin Fig. 1a and b (see Methods for more details). The\nmeasurements were performed at low temperatures (10\nK), with the sample mounted at 50 degrees with respect\nto the magnetic field axis for transmission measurements.\nThe laser light is parallel to the magnetic field axis.\nWe use the time-resolved magneto-optic Faraday effect\nto monitor the magnetization dynamics in our system us-\ning a single-color pump-probe setup similar to the one\ndescribed in53,54(more information in Methods). The\nprocess of optical excitation of magnetization dynam-\nics in van der Waals magnets has been previously re-\nported as purely thermal43–47,52, similar to many studies\non conventional metallic thin-films55–57. Here we find\nstrong evidence that coherent opto-magnetic phenomena\nalso play an important role in the excitation of the mag-\nnetization dynamics. The detailed microscopic descrip-\ntion of how the magnetization dynamics is induced isdescribed later in the article, but in short, the excitation\nof the magnetization dynamics can be described as fol-\nlows (Fig. 1c): In equilibrium (1), the magnetization M\npoints along the total effective magnetic field Heff, which\nis the sum of the external field ( Hext), and the internal\neffective field ( Hint) caused by the magnetocrystalline\nanisotropy ( Ku) and shape anisotropy. For CGT, Hint\npoints out-of-plane2,45,58, meaning that Kudominates\nover the shape anisotropy. The linearly polarized pump\npulse interacts with the sample (2), reducing the magne-\ntization and changing the magnetocrystalline anisotropy\nthrough the mechanisms mentioned above, which causes\nMto cant away from equilibrium. Since MandHeffare\nnot parallel anymore, this results in a precession of M\naround Heff, while they both recover to their equilibrium\nvalue as the sample cools.\nIII. GATE CONTROL OF MAGNETIZATION\nDYNAMICS\nThe dual-gate geometry of our device allows for the\nindependent control of both the charge carrier density3\nand the perpendicular electric field. The dependence of\n∆nand ∆ Don the top and back gate voltages – Vtand\nVb, respectively – is derived in the Methods. The change\nin the Fermi level induced by ∆ nis expected to affect\nthe magnetic anisotropy of CGT due to the different Cr\nd-orbitals composition of the electronic bands59. The ef-\nfect of ∆ Dis, however, more subtle. The inversion sym-\nmetry breaking caused by ∆ Dcan allow for an energy\nshift of the (initially degenerate) electronic bands, po-\ntentially also modulating the magnetization parameters.\nAdditionally, the perpendicular electric field can induce\na non-uniform distribution of charge carriers along the\nthickness of the CGT flake, leading to ∆ n-induced local\nchanges in the magnetization parameters.\nTypical results from the time-resolved Faraday ellip-\nticity (TRFE) measurements for different values of ∆ n,\nwith ∆ D= 0, are shown in Fig. 1d. For ∆ t <0 the sig-\nnal is constant, since the magnetization is at its steady\nstate value. All traces show a sharp increase at ∆ t= 0,\nindicating a fast laser-induced dynamics. For ∆ t >0, the\nTRFE traces show oscillations, indicating a precession of\nthe magnetization induced by the pump pulse.\nWe observe that the magnetization dynamics strongly\ndepends on ∆ n, with the amplitude, frequency and start-\ning phase of the oscillations in the TRFE signal all being\naffected. The most striking observation is that the ampli-\ntude of the TRFE signal increases by more than a factor\nof seven when ∆ nis changed from 1 .2×1013cm−2to\n−1.8×1013cm−2. The observations of modulation of\nboth the amplitude and starting phase of the oscillations\nhint at a change in the pump excitation process. The\nchange in oscillation frequency due to ∆ nis better visi-\nble in the Fourier transform of the signals, shown in Fig.\n1e (see Methods for details on the Fourier transform).\nThis analysis clearly shows that both the frequency and\namplitude of the magnetization precession are tuned by\n∆n. All these observations point to an effective control\nof the (dynamic) magnetic properties of CGT by electro-\nstatic gating.\nThe origin of the electric control of the magnetization\ndynamics can be further understood by analyzing the\nprecession frequency at various magnetic fields and val-\nues of ∆ n(Fig. 2a). For magnetic fields below 250 mT\nwe observe a significant shift of the frequency (4 – 10%)\nby changing the charge carrier density. This is clearly\nvisible in the inset of Fig. 2a, which shows a close-up\nof the data up to 150 mT. The change in precession fre-\nquency for different values of ∆ nstrongly points towards\na modulation of the magnetization parameters of CGT\nas a function of the Fermi level, controlled by ∆ n.\nA quantitative analysis of the oscillation frequency ( f)\nas a function of Hextcan be used to extract the magne-\ntization dynamics parameters of the device. Our data is\nwell described by the ferromagnetic resonance mode ob-\ntained from the Landau-Lifshitz-Gilbert (LLG) equation\nwith negligible damping60:\nf=gµBµ0\n2πℏ/radicalig\n|Heff|/parenleftbig\n|Heff| −Hintsin2(θM)/parenrightbig\n,(1)where gis the Land´ e g-factor, µBthe Bohr magneton,\nHeff=Hext+Hintcos(θM)ˆz, with Hint= 2Ku/(µ0Ms)−\nMs,Msthe saturation magnetization, and θMthe angle\nbetween Mand the sample normal (z-direction). The\nangle θMis calculated by minimizing the magnetic en-\nergy in the presence of an external field, perpendicular\nmagnetic anisotropy, and shape anisotropy45. We ob-\ntain the g-factor and Hextof the CGT by fitting the f\nversus Hextdata (e.g. the data presented in Fig. 2a)\nusing Eq. (1), as explained in the Methods. This yields\ng≈1.89 with no clear dependence on ∆ nor ∆D, which is\nin agreement with (albeit slightly lower than) the values\nreported for CGT40,42,45(see Supplementary Sections 4\nand 5 for more details). We also find no clear dependence\nof the precession damping time ( τosc) on ∆ nor ∆D. The\nintrinsic Gilbert damping we obtain form our measure-\nments is about 6 ×10−3(see Supplementary Section 6),\nin line with values found in literature41,45.\nThe internal effective field shows a clear dependence on\nboth VtandVb, as shown in Fig. 2c, with values similar\nto the ones found in other studies45. Upon comparing\nFig. 2c to 2b, one notices that the gate dependence of\nHintis very similar to that of the precession frequency\natµ0Hext= 100 mT. This suggests that the gate de-\npendence of the precession frequency is caused by the\ngate-induced change in Hint. From the dependence of\nHintonVtandVb, we extract its behavior as a function\nof ∆nand ∆ D, shown in Fig. 2d and e. We observe\nthat Hintdecreases with both increasing ∆ nand ∆ D.\nThe dependence of Hinton ∆ nis consistent with the-\noretical calculations59, showing that Ku, and therefore\nHint, is reduced upon increasing the electron density in\nthe same order of what we achieve in our sample. The\n∆ndependence of Hintis also consistent with the depen-\ndence of the coercive field obtained from static measure-\nments (see Supplementary Section 3), providing further\nevidence that the change in fis driven by a change in\nKu.\nNow we draw our attention to the large modulation of\nthe oscillations in the TRFE measurements with varying\ngate voltage, as shown in Fig. 1d and 1e. Here we at-\ntribute this change in magneto-optical signal amplitude\nto an actual increase in amplitude of the magnetization\nprecession (increase in the precession angle) and not to\nan increase in the strength of the Faraday effect. This\nis supported by our observation that the time-resolved\nmeasurements for different combinations of gate voltages\nare not simply scaled – i.e. the amplitude of the oscilla-\ntions and their (ultrafast demagnetization) background\nscale differently. A detailed discussion can be found in\nSupplementary Section 7.\nFigure 3a clearly shows that the magnetization preces-\nsion amplitude is mostly affected by ∆ n, and to a much\nlesser extend by ∆ D. The precession amplitude versus\nHextfor various values of ∆ nis presented in Fig. 3c.\nWe note that for |µ0Hext|<50 mT the magnetization is\nnot completely saturated (see Supplementary Section 3),\nwhich can lead to multi-domain formation61and a devi-4\n−6 0 6−606μH0int (mT)a b\n4.63 5.03\ncd\ne\n118 132\n−6 0 6−606Frequency (GHz)\n0.0 0.2 0.4 0.6\nμ0Hext (T)02468101214Frequency (GHz)\nΔn (10 cm )13−2\n1.20.0-1.8\n0.10 0.15456\n−5 0 5\n(V  + )/2b tV120125130μH0int (mT)\n-0.56\n0.00\n0.561 0 −1Δn (10 cm )13−2\n−5 0 5\n(V  - )/2b tV120125130μH0int (mT)\nΔn (10 cm )13−2\n-0.62\n0.00\n0.62−1 0 1Vt (V)\nVb (V)Vt (V)\nVb (V)ΔD/ε0 (V/nm)\nΔD/ε0 (V/nm)ΔD\nΔnΔD = 0\nFIG. 2. Gate-dependence of precession frequency and internal effective field. a , Frequency of oscillations as a\nfunction of external magnetic field, for different values of ∆ n. The circles are the frequencies extracted from the TRFE data\nfor ∆ t >26 ps. Solid lines are best fits of Eq (1). Inset : Close-up of the data for low fields, showing the frequency shift due to\ngating. The error bars are smaller than the markers. b, Frequency of the oscillations in the TRFE signal at µ0Hext= 100 mT\nfor various values of top ( Vt) and back gate voltages ( Vb). The black and gray arrows indicate, respectively, the directions of\nconstant ∆ Dand varying ∆ n, and of constant ∆ nand varying ∆ D. For other values of Hextsee Supplementary Fig. S10. c,\nInternal effective field as a function of VtandVb.d,e, The dependence of the internal effective field on ∆ nfor fixed ∆ D(d)\nand on ∆ Dfor fixed ∆ n(e), with solid lines to guide the eye. The traces are taken along the dotted lines indicated in c.\nation from the general trend. We find that not only the\nprecession amplitude for a given Hextis strongly mod-\nulated by ∆ n, but its decaying trend with Hextis also\nstrongly affected. Additionally, we observe another in-\nteresting effect: the amplitude shows an asymmetry in\nthe sign of the applied magnetic field, which is also de-\npendent on ∆ n. This latter is unexpected, especially\nsince the observed precession frequency is symmetric in\nHext(see Supplementary Section 9). A similar precession\nfrequency for opposite magnetic fields indicates that the\nmagnetocrystalline anisotropy and the saturation magne-\ntization are independent of the sign of Hext. Therefore,\nwe conclude that the origin of the modulation of the pre-\ncession amplitude is related to the excitation mechanism\nof the magnetization precession (see Supplementary Sec-\ntion 10 for the complete discussion).\nTo get further insight into the microscopic mechanisms\ninvolved in the optical excitation of magnetization dy-\nnamics, we analyze the magnetic field dependence of the\nstarting phase ( ϕ0) of the precessions for different values\nofVtandVb(Fig. 3b). Unlike the amplitude, we find thatϕ0depends on both ∆ nand ∆ D. As can be seen in Fig.\n3d, the behavior of ϕ0withHextis also modulated by ∆ n.\nFor a purely thermal excitation of the magnetization dy-\nnamics one would expect ϕ0(−Hext) =π+ϕ0(Hext) in\nour geometry. Nonetheless, we observe that ϕ0for pos-\nitive and negative magnetic fields differ by less than π.\nMoreover, ∆ nseems to also affect the trend on how ϕ0\napproaches the values at high magnetic fields. Combined\nwith the observed asymmetry of the precession ampli-\ntude, our data strongly suggests that the optical exci-\ntation of the magnetization dynamics is not dominated\nby a thermal excitation (∆K mechanism) as previously\nreported for other van der Waals magnets43–47,52.\nIV. OPTO-MAGNETIC EFFECTS\nCoherent opto-magnetic mechanisms provide possible\nalternatives for the optical excitation of magnetization\ndynamics in CGT. Here we find that our data can be ex-\nplained by two of these mechanisms that are compatible5\n1.02� 1.65��0 (rad)\nAmplitude (arb. units)\n0.07 0.51a\nbc\nde\n1\n2\n−6 0 6−606\n−6 0 6−606\n−0.5 0.0 0.50.00.51.0Amplitude (arb. units)Δn (10 cm )13−2\n1.20.0-1.8\n−0.5 0.0 0.50�2�ϕ0 (rad)\nΔn (10 cm )13−2\n1.20.0-1.8Vt (V) Vt (V)Vb (V)\nVb (V)\nHICME\nxyz\nHICME\nHextHext\nM\nM\nHeffMHeff\nMΔD = 0\nΔD = 0Δn\nμ0Hext (T)\nμ0Hext (T)\nFIG. 3. Gate dependence of magnetization precession amplitude and phase. a ,b, Gate dependence of the amplitude\n(a) and starting phase ( b) of the oscillations in the TRFE measurements at µ0Hext= 100 mT. For other values of Hextsee\nSupplementary Figs. S8 and S9. c,d, External magnetic field dependence of the amplitude ( c) and starting phase ( d) of the\noscillations for different values of ∆ nat ∆D= 0. The values are extracted from the TRFE data for ∆ t >26 ps. e, Schematics\nof the inverse Cotton-Mouton effect for opposite directions of Hext. The magnetization direction is depicted by a red arrow,\nthe external magnetic field in blue, the effective magnetic field induced by the ICME and the effective field are shown in cyan.\nThexz-plane is highlighted by the shaded region.\nwith a linearly-polarized pump pulse, the inverse Cotton-\nMouton effect (ICME)62–64and photo-induced magnetic\nanisotropy (PIMA)65, in addition to the conventional\n(thermal) ∆K mechanism66. The ICME, which could be\ndescribed by impulsive stimulated Raman scattering, re-\nlies on the generation of an effective magnetic field upon\ninteraction with linearly polarized light in a magnetized\nmedium62–64,67,68. This effective magnetic field is propor-\ntional to both the light intensity and magnetization. For\npulsed laser excitation, the ICME generates a strong im-\npulsive change in Hextthat results in a fast rotation of the\nmagnetization. Therefore, this effect can cause the am-\nplitude of the precession to be asymmetric in Hext64,68.\nFigure 3e illustrates how the ICME could result in an\nasymmetric magnetic field dependence of the amplitude.\nFor simplicity we only consider the y-component of the\ngenerated effective magnetic field, since this component\nis responsible for the asymmetry. (1) A sample with per-\npendicular magnetic anisotropy is subject to an external\nmagnetic field Hext(−Hext) in the xz-plane, pointing in\nthe positive (negative) direction of both axes. In equilib-\nrium, the magnetization points along the total effectivefield, as indicated by the light gray arrow. During laser\npulse excitation, the ICME results in an effective mag-\nnetic field along the y-axis, rotating the magnetization\neither towards the z-axis or the x-axis, depending on the\nsign of Hext. Additionally, the ultrafast demagnetization\nprocess leads to a reduction of the magnetization. (2)\nAfter the laser pulse, the magnetization precesses around\nthe total effective field that is comprised of HextandHint.\nDepending on the sign of Hext, the ICME has either ro-\ntated Mtowards or away from Heff, resulting in different\nprecession amplitudes.\nThe second coherent mechanism for laser-induced mag-\nnetization dynamics is PIMA, which leads to a step-like\nchange in Heffdue to pulsed laser excitation64. This\nmechanism has been reported to arise from an optical\nexcitation of nonequivalent lattice sites (e.g. dopants\nand impurities), which effectively redistributes the ions\nand hence changes the magnetic anisotropy69–71. Unlike\nthe ICME, the PIMA mechanism is not expected to lead\nto an asymmetry of the magnetization precession am-\nplitude of upon a reversal of Hext, because it is present\nfor times much longer than the period of precession and6\ntherefore acts as a constant change of the effective mag-\nnetic field65,70–72.\nAll three discussed mechanisms for inducing magne-\ntization precession – ICME, PIMA and the ∆K mech-\nanism – are affected by electrostatic gating. The opto-\nmagnetic effects can be affected through a change in e.g.\nthe polarization-dependent refractive index and the oc-\ncupation of charge states of ions and impurities. Ad-\nditionally, the ∆K mechanism can be affected by the\nchanges in charge relaxation pathways through, for exam-\nple, electron-electron and electron-phonon interactions.\nWe find that the combination of the above mechanisms\ncan describe quantitatively the starting phase and qual-\nitatively the amplitude of the observed magnetic field\ndependence shown in Fig. 3 (see Supplementary Sec-\ntion 11). The balance between these three mechanism\naffects the magnetic field dependence of the amplitude\nand the starting phase, increases or decreases the asym-\nmetry in the induced precession amplitude, and changes\nthe steepness of the starting phase versus magnetic field\ngraph. Therefore, since our data shows a change in these\nproperties, we conclude that the relative strength of the\nmechanisms for excitation of magnetization precession\nare effectively controlled by electrostatic gating.\nV. CONCLUSIONS\nWe envision that the electric control over the optically-\ninduced magnetization precession amplitude demon-\nstrated here can be applied to devices which make use\nof spin wave interference for signal processing12–14. This\nshould lead to an efficient electric control over the mixing\nof spin waves, leading to an easier on-chip implementa-\ntion of combined magnonic and photonic circuits. Even\nthough the control over the precession frequency shown\nhere is still modest ( ≈10%), we believe it can be further\nenhanced by the use of more effective electrostatic dop-\ning, such as using high- κdielectrics or ionic-liquid gating\nwhich is capable of achieving over one order of magni-\ntude higher changes in carrier densities than the ones\nreported here21,24,25,32,73. We note that due to the non-\nmonotonic behavior of the magnetic anisotropy energy\nwith changes in charge carrier density, one might expect\nmore drastic changes on Hintfor larger changes in ∆ n.\nThis control over the magnetic anisotropy can then be\nused for the electrostatic guiding and confinement of spin\nwaves, leading to an expansion of the field of quantum\nmagnonics. Finally, the presence of coherent optical ex-\ncitation of magnetization dynamics we observed in CGT\nshould also lead to a more energy-efficient optical control\nof magnetization74. Therefore, the electric control over\nmagnetization dynamics in CGT shown here provides the\nfirst steps towards the implementation of vdW ferromag-\nnets in magneto-photonic devices that make use of spin\nwaves to transport and process information.VI. METHODS\nA. Sample fabrication\nThe thin hBN and graphite flakes are exfoliated from\nbulk crystals (HQ graphene) on an oxidized silicon wafer\n(285 nm oxide thickness). The CGT flakes are exfoli-\nated in the same way in an inert (nitrogen gas) envi-\nronment glove box with less than 0.5 ppm oxygen and\nwater to prevent degradation. The flakes are selected\nusing optical contrast and stacked using a polycarbon-\nate/polydimethylsiloxane stamp by a dry transfer van\nder Waals assembly technique75. First an hBN flake (21\nnm thick) is picked up, followed by the CGT flake. Next,\na thin graphite flake is picked up to make electrical con-\ntact with a corner of the CGT, and extends beyond the\npicked-up hBN flake. After this, a second hBN flake\n(20 nm thick) is picked up and a thin graphite flake to\nfunction as the back gate electrode. This stack is then\ntransferred to an optically transparent fused quartz sub-\nstrate finally a thin graphite flake is transferred on top\nthe stack to function as the top gate electrode. The de-\nvice is then contacted by Ti/Au (5/50 nm) electrodes\nfabricated using conventional electron-beam lithography\nand thin metallic film deposition techniques.\nB. Measurement setup\nAll measurements are done at 10 K under low-pressure\n(20 mbar) Helium gas. The sample is mounted at an\nangle, such that the sample normal makes an angle of\n50 degrees with the external magnetic field and the laser\npropagation direction.\nThe∼200 fs long laser pulses are generated by a mode-\nlocked Ti:Sapphire oscillator (Spectra-Physics MaiTai),\nat a repetition rate of 80 MHz. After a power dump,\nthe pulses are split in an intense pump and a weaker\nprobe pulse by a non-polarizing beam splitter. The pump\nbeam goes through a mechanical delay stage, allowing us\nto modify the time-delay between pump and probe by a\nchange in the optical path length. To allow for a double-\nmodulation detection53,54, the pump beam goes through\nan optical chopper working at 2173 Hz. The polarization\nof the pump is set to be horizontal (p-polarized with re-\nspect to the sample), to allow us to block the pump beam\nthrough a polarization filter at the detection stage. The\ninitially linearly polarized probe pulse goes through a\nphotoelastic modulator (PEM) which modulates the po-\nlarization of the light at 50 kHz. A non-polarizing beam\nsplitter is used to merge the pump and probe beams on\nparallel paths, with a small separation between them.\nFrom here, they are focused onto the sample by an as-\npheric cold lens with a numerical aperture of 0.55. The\nprobe spot size (Full Width at Half Maximum) is ∼1.8\nµm and the pump spot size is ∼3.4µm, both elongated\nby a factor of 1 /sin(50◦) because the laser hits the sam-\nple at 50◦with respect to the sample normal. The fluence7\nof the pump and probe pulses are Fpump = 25 µJ/cm2\nandFprobe = 5.7 µJ/cm2, respectively. The transmitted\nlight is collimated by an identical lens on the opposite\nside of the sample and leaves the cryostat. The pump\nbeam is blocked and the probe beam is sent to a detection\nstage consisting of a quarter wave plate, a polarization\nfilter, and an amplified photodetector. The quarter wave\nplate and the polarization filter are adjusted until they\ncompensate for the change in polarization caused by the\noptical components between the PEM and the detection\nstage, ensuring that our signals are purely due to the ro-\ntation or ellipticity of the probe polarization induced by\nour samples. The first and second harmonic of the signal\n(50 or 100 kHz) obtained at the photodetector are then\nproportional to the change in ellipticity and rotation due\nto the Faraday effect of the sample. For static magneto-\noptic Faraday effect measurements we have blocked the\npump beam before reaching the sample.\nC. Calculating ∆nand∆Dfrom the gate voltages\nThe gate-induced change in charge carrier density\n(∆n) and displacement field (∆ D) at the CGT are cal-\nculated from the applied gate voltages using a parallel\nplate capacitor model. The displacement field gener-\nated by the top ( Dt) and back ( Db) gates is given by\nDi=εhBNEi=1\n2σfree,i, where idenotes torb,εhBN=\n3.8ε0is the hBN dielectric constant76with ε0the vac-\nuum permittivity, and σfreethe free charge per unit area.\nThe applied top and back gate voltages are related to\nσfreebyVi=−/integraltext\nDi/εdz. This equation, combined with\nthe condition of charge neutrality, gives the following 3\nrelations:\nVt/dt=σt−σCGT−σb\n2εhBN,\nVb/db=σb−σCGT−σt\n2εhBN,\n0 =σt+σb+σCGT,\nwhere dt,bdenotes the thickness of the top (21 nm) and\nbottom (20 nm) hBN flakes, and σithe free charge per\nunit area in the top gate ( t), back gate ( b), and the CGT\nflake. Solving this set of equations yields:\nσt=εhBNVt/dt,\nσb=εhBNVb/db,\n∆n=σCGT/e=−εhBN\ne/parenleftbiggVt\ndt+Vb\ndb/parenrightbigg\n,\nwhere eis the positive elementary charge. Note that for\npositive gate voltages, a negative charge carrier density\nis induced in the CGT. For the gate-induced change in\nthe displacement field at the CGT layer, we get:\n∆D= (σb−σt)/2 =−εhBN(Vt/dt−Vb/db)/2Filling in the values for the thickness of the hBN flakes\nand dielectric constant of hBN gives ∆ D/ε 0and ∆ nat\nthe CGT:\n∆n=−(1.00Vt+ 1.05Vb)×1012V−1cm−2\nD/ε 0=−(0.090Vt−0.095Vb) nm−1.\nThroughout the main text we use ∆ D/ε 0instead of ∆ D\nfor easier comparison of our values of the gate-induced\nchange in the displacement field with values mentioned\nin other works. Note that we use the conversion factor ε0\nand not the permittivity of CGT. Therefore, the values\nfor the ∆ Dthat we report are the equivalent electric field\nvalues in vacuum , not in CGT.\nD. Windowed Fourier transform\nThe RMS power spectra of the TRFE oscillations\nshown in Fig. 1e are calculated from the TRFE measure-\nments using a windowed Fourier transform. The type of\nwindow used for this calculation if the Hamming window,\nwhich extends from ∆ t= 0 up to the last data point.\nThe RMS power spectrum ( PRMS(f)) of the TRFE os-\ncillations is calculated as\nPRMS(f) =/parenleftigg/summationdisplay\n∆t>0[WHam(∆t)y(∆t) sin(2 πf∆t)]2+\n[WHam(∆t)y(∆t) cos(2 πf∆t)]2/parenrightigg1/2\n,\nwhere WHamis the Hamming window, y the data points\nof the TRFE measurements, and fthe frequency.\nE. Determining the g-factor and Hint\nThe Land´ e g-factor and Hintcan be extracted by fit-\nting the magnetic field dependence of the precession fre-\nquencies with Eq. (1). The values of gandHintwe\nobtained from the fit were, in most cases, strongly cor-\nrelated. Therefore, we first determined gby fitting the\ndata for µ0Hext≥125 mT, since gis most sensitive to\nthe slope at high fields. This yields g= 1.89 ±0.01.\nI we further allow for an additional uncertainty in the\nmounting angle of the sample, the g-factor can change\nby∼0.1. Then we determine Hintby fitting Eq. (1)\nfor all remaining measurements fixing g= 1.89. We note\nthat the values for Hintdo depend on the exact value of\ng, but the modulation due to electrostatic gating is not\naffected, as is shown in Supplementary Section 4.\nF. Extracting the magnetization precession parameters\nfrom the TRFE measurements\nWe extract the amplitude, frequency, and starting\nphase of the oscillations in the TRFE measurements by8\nfitting the data for ∆ t >26 ps with the phenomenological\nformula45,60\ny=y0+ae−∆t/τosccos (2 πf∆t−ϕ0)\n+Ale−∆t/τl+Ase−∆t/τs. (2)\nThis formula describes a phase shifted sinusoid on top\na double exponential background. The background cap-\ntures the demagnetization and remagnetization of the\nCGT, while the sinusoid describes the magnetization pre-\ncession.\nVII. DATA AVAILABILITY\nThe raw data and the data underlying the figures in the\nmain text are publicly available through the data reposi-\ntory Zenodo at https://doi.org/10.5281/zenodo.8321758.\nVIII. ACKNOWLEDGMENTS\nWe thank Bart J. van Wees for critically reading\nthe manuscript and providing valuable feedback, and\nwe thank J. G. Holstein, H. Adema, H. de Vries,\nA. Joshua and F. H. van der Velde for their techni-\ncal support. This work was supported by the Dutch\nResearch Council (NWO) through grants STU.019.014\nand OCENW.XL21.XL21.058, the Zernike Institute for\nAdvanced Materials, the research program “Materials\nfor the Quantum Age” (QuMat, registration number\n024.005.006), which is part of the Gravitation program\nfinanced by the Dutch Ministry of Education, Culture\nand Science (OCW), and the European Union (ERC, 2D-\nOPTOSPIN, 101076932). Views and opinions expressed\nare however those of the author(s) only and do not nec-\nessarily reflect those of the European Union or the Eu-\nropean Research Council. Neither the European Union\nnor the granting authority can be held responsible for\nthem. The device fabrication and nanocharacterization\nwere performed using Zernike NanoLabNL facilities.\nIX. AUTHOR INFORMATION\nM.H.D.G. conceived and supervised the research. F.H.\ndesigned and fabricated the samples, performed the mea-\nsurements, analyzed the data, and calculated the effect of\ncoherent excitations on the magnetization precession un-\nder M.H.D.G. supervision. F.H. and R.R.R.L built and\ntested the measurement setup. F.H., M.H.D.G, and B.K\ndiscussed the data and provided the interpretation of the\nresults. F.H. and M.H.D.G co-wrote the manuscript with\ninput from all authors.X. ETHICS DECLARATION\nA. Competing interests\nThe authors declare no competing interests.\n1B. Huang, G. Clark, E. Navarro-Moratalla, D. R. 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ATOMIC FORCE MICROSCOPY\nThe thicknesses of the flakes of the sample were measured using atomic force microscopy\n(AFM). The height map is shown in Fig. S1a. The thickness of the hexagonal boron nitride\n(hBN) flakes are determined from the line trace along the blue line, shown in Fig. S1b,\nyielding 20 nm for the bottom and 21 nm for the top flake respectively. The Cr 2Ge2Te6\n(CGT) thickness was extracted from the line profile along the green line, shown in Fig. S1c,\nyielding 10 nm.\n0 2 4 6 8\nx (  m)μ010203040height (nm)\n0 2 4 6 8\nx (  m)μ05101520height (nm)\na b\nc20 nm21 nm\n10 nm\nFIG. S1. Atomic force microscopy scan. a , AFM height image. Height profiles taken along\nthe blue and green line are used to extract the thickness of the top and bottom hBN, and of\nthe thickness of the CGT respectively. b, Height profile taken along the blue line to extract the\nthickness of the top (21 nm) and bottom (20 nm) hBN flakes. c, Height profile taken along the\ngreen line to extract the thickness of the CGT flake (10 nm).arXiv:2309.12776v1  [cond-mat.mes-hall]  22 Sep 20232\nS2. FINDING OPTIMAL PARAMETERS FOR THE FARADAY EFFECT\nThe sensitivity of the Faraday effect on changes in the magnetization depends on the\nwavelength. Additionally, the change in rotation and ellipticity of the light caused by the\nFaraday effect are generally different. To find the optimal wavelength and polarization\nmode, we probe the magnetization using the Faraday effect while sweeping the magnetic\nfield. The change in polarization when the magnetic field reverses the magnetization direc-\ntion of the CGT indicates the sensitivity of the Faraday effect. We determine this sensitivity\nfor various values of the wavelength, ranging from 830 nm to 940 nm, measuring change\nin rotation and ellipticity simultaneously. We found that the ellipticity at 870 nm had\nthe highest sensitivity, and therefore we use this wavelength and polarization mode for the\ntime-resolved Faraday ellipticity (TRFE) measurements.\nS3. GATE VOLTAGE DEPENDENCE OF MAGNETIZATION CURVES MEASURED BY\nSTATIC FARADAY ELLIPTICITY\nWe measured the gate dependence of the magnetization curves of CGT by means of the\nFaraday effect using a pulsed laser with a fluence of 5.7 µJ/cm2at a wavelength of 870 nm.\nWe measured the change in ellipticity of the light, as described in the Methods. The gate\ndependence of the magnetization curves for various values of ∆ nand ∆ Dare presented in\nFig. S2.\n−0.06 −0.04 −0.02 0.00 0.02 0.04 0.06−3−2−10123Faraday ellipticity (arb. units)\n−0.06 −0.04 −0.02 0.00 0.02 0.04 0.06−3−2−10123Faraday ellipticity (arb. units)\na\nΔn (10 cm )13−2\n1.2\n0.0\n-1.21.1\n0.0\n-1.1b\nΔD/�� (V/nm)\nΔn = 0 ΔD = 0\nμ0Hext (T) μ0Hext (T)\nFIG. S2. Gate dependence of the magnetization curves measured by static Faraday\nellipticity. a , Effect of ∆ Don the magnetization curves for ∆ n= 0. b, Effect of ∆ non the\nmagnetization curves for ∆ D= 0.\nIn addition, we measured the magnetization curves at 920 nm using the polar magneto-\noptic Kerr effect (MOKE) in a different experimental setup as a function of ∆ nat ∆D= 0.\nFrom this data we extracted how the coercive field changes with ∆ n, which is shown in Fig.\nS3, to check if it is similar to the ∆ ndependence of Hint. We extract the coercive field by\nindividually fitting the trace and retrace of the magnetization curves with the formulas\nytrace=/braceleftigg\nAtanh/parenleftig\nx−x0,t\nw/parenrightig\n+ax+b,ifx > x s,t\nA+ax+b, otherwise(S1)\nyretrace =/braceleftigg\nAtanh/parenleftig\nx−x0,r\nw/parenrightig\n+ax+b,ifx < x s,r\nA+ax+b, otherwise(S2)\nwhere Ais half of the difference in MOKE ellipticity for large positive and negative fields,\nx0the horizontal shift, wthe width of the tanh (which is proportional to the saturation3\nfield), and xsthe position of the sharp switch close to zero field that indicates the transition\nfrom a single domain to multi domain state1. The parameters aandbare respectively the\nslope and offset of the background. The coercive field is generally not fitted well. To get\na better estimate for the coercive field, we subtract the background, and separately fit\na straight line, with equal slope, through the trace and retrace where the signal is close\nto zero and approximately linear (we used the signal value <0.002). The coercive field\nis then calculated as half of the horizontal separation of these two lines. The graph of\nµ0Hcversus ∆ npresented in Fig. S3 is very similar to the graph of µ0Hintversus ∆ nat\n∆D= 0 presented in Fig. 2d in the main text. This means that the ∆ ndependence of Hint\nobtained from TRFE measurements of the magnetization precession is consistent with the\n∆ndependence of Hcobtained from static MOKE mearements.\n−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0\nΔ (10  cm )n13−20.30.40.50.6μH0 c (mT)\n−30 −20 −10 0 10 20 30\nμH0 ext (mT)−1.0−0.50.00.51.0εK (arb. units)\nHcΔD = 0a b\nΔn\nΔD= 0\n= 0\nFIG. S3. Coercive field from static MOKE measurements a , Magnetization curve measured\nby static MOKE at ∆ n= ∆D= 0. The coercive field is half of the horizontal shift of the trace\nand retrace near zero. b, Effect of ∆ non the coercive field at ∆ D= 0.\nS4. θHDEPENDENCE OF gAND Hint\nWe extract the g-factor and the internal effective field ( Hint) from the external magnetic\nfield ( Hext) dependence of the magnetization precession frequency ( f), as explained in the\nmain text and the Methods. To obtain gandHint, we need to fix the value of the angle\nθHbetween Hextand the sample normal. Fig. S4 shows the gate dependence of gandHint\nfor different values of θH, ranging from 45 degrees to 60 degrees. Panels a, d, g, and j show\nthat for θH= 45◦and 50◦the data is fitted best. Since 50◦is more likely based on how\nwe mounted the sample in the cryostat, we have chosen θH= 50◦for our analysis, despite\nthe observation that θH= 60◦results in a g-factor that is more in line with values reported\nin literature. From the results presented in Fig. S4, we can conclude that, apart from an\noffset of the mean value, the gate dependence of gandHinthardly changes with θn, even if\nwe choose θHsuch that we obtain a gfactor that agrees with literature values. Therefore,\nthe conclusions in the main on the gate dependence of the Hintdo not depend on the exact\nvalue of θH, as long as these values are reasonable.4\n−6 0 6\nVb (V)−606Vt (V)μHθ0int H  (mT), =  50deg\n120125130\n−6 0 6\nVb (V)−606Vt (V)μHθ0int H  (mT), =  55deg\n125130135\n−6 0 6\nVb (V)−606Vt (V)μHθ0int H  (mT), =  60deg\n125130135\n−6 0 6\nVb (V)−606Vt (V)μHθ0int H  (mT), =  45deg\n120125\n−6 0 6\nVb (V)−606Vt (V)gθ, =  45deg H\n1.801.811.821.831.84\n−6 0 6\nVb (V)−606Vt (V)gθ, =  50deg H\n1.871.881.89\n−6 0 6\nVb (V)−606Vt (V)gθ, =  55deg H\n1.941.951.961.97\n−6 0 6\nVb (V)−606Vt (V)gθ, =  60deg H\n2.022.042.062.080.0 0.2 0.4 0.60.02.55.07.510.012.515.0Frequency (GHz)\nθH= 45 deg\nΔn (10 cm )13−2\n1.2\n-0.0\n-1.8\n0.0 0.2 0.4 0.60.02.55.07.510.012.515.0Frequency (GHz)\nθH= 50 deg\nΔn (10 cm )13−2\n1.2\n-0.0\n-1.8\n0.0 0.2 0.4 0.60.02.55.07.510.012.515.0Frequency (GHz)\nθH= 55 deg\nΔn (10 cm )13−2\n1.2\n-0.0\n-1.8\n0.0 0.2 0.4 0.60.02.55.07.510.012.515.0Frequency (GHz)\nθH= 60 deg\nΔn (10 cm )13−2\n1.2\n-0.0\n-1.8a b c\nd e f\ng h i\nj k lμ0Hext (T)\nμ0Hext (T)\nμ0Hext (T)μ0Hext (T)\nFIG. S4. Effect of the sample mounting angle on gand Hint.Fit of precession frequency\nversus Hext(a) data to obtain g(b) and Hint(cfor)θH= 45 deg. Results for θH= 50,55 and 60\ndegrees are presented in respectively d-f,g-handj-l.\nS5. PUMP POWER DEPENDENCE OF PRECESSION FREQUENCY\nWe found that the frequency of the magnetization precession depends on the fluence of\nthe pump pulse. For the data presented in the main text, the pump fluence was measured\nto be 25 µJ/cm2. Fig. S5 shows how the TRFE signal is affected by different values of\nthe pump fluence. The frequency for each of these measurements is extracted by fitting the\ndata with Eq. (2) of the Methods, and is plotted in Fig. S6. There it can be seen than\nthe magnetization precession frequency decreases with pump fluence for all the extreme\ngate voltage combinations. The precise rate at which the frequency depends on the pump\npower depends on the applied gate voltages. This can be due the photo-induced magnetic\nanisotropy effect (PIMA) since an increase in pump fluence would result on an increase5\non the increase (or decrease) of the population of the dopant or impurity states. Since\nthe PIMA effect depends on the population of these states, an increase in the pump fluence\nshould result on an increase on the effective field generated by it. The change in the observe\nfrequency should then depend on the specific sign (with respect to the magnetic anisotropy\nand applied magnetic field) of the PIMA. Additionally, a decrease in the saturation magne-\ntization ( Ms) and magnetocrystalline anisotropy caused by heating by the laser could offer\nanother possible explanation. This does affect the values of the Land´ e g-factor and the\neffective internal field ( Hext). Since we used the same pump fluence for all measurements,\nit does not affect the general trends we observe for the gate voltage dependence of the\nmagnetization precession frequency, starting phase, and amplitude shown in this work.\n0.0 0.5 1.0 1.5\nΔ (ns)t123TRFE signal (arb.units)\nV Vt b = 0V,  = 0V0.0 0.5 1.0 1.5\nΔ (ns)t123TRFE signal (arb.units)\nV Vt b = 6V,  = 6V\n0.0 0.5 1.0 1.5\nΔ (ns)t123TRFE signal (arb.units)\nV Vt b = 6V,  = -6V\n0.0 0.5 1.0 1.5\nΔ (ns)t123TRFE signal (arb.units)\nV Vt b = -6V,  = 6V0.0 0.5 1.0 1.5\nΔ (ns)t123TRFE signal (arb.units)\nV Vt b = -6V,  = -6Va b\nc d\ne26\n123858F (�J/cm2)\n26\n123858F (�J/cm2)\n26\n123858F (�J/cm2)\n26\n123858F (�J/cm2)\n26\n1258F (�J/cm2)\nFIG. S5. Pump fluence dependence of the TRFE measurements. Dependence of the TRFE\nsignal on the pump fluence, measured at µ0Hext= 100 mT, for various values of V tand V b. The\nsolid lines indicate the best fit of Eq. (2) in the Methods, which is used to extract the magnetization\nprecession frequency.6\n0 10 20 30 40 50 60\nPump funce ( J/cm )μ24.64.85.05.25.4Frequency (GHz)\n-6,\n6,\n-6,\n6,\n0,-6\n6\n6\n-6\n0VtVb\nFIG. S6. Pump fluence dependence of the precession frequency. Dependence of the\nprecession frequency on the pump fluence, measured at µ0Hext= 100 mT, extracted from Fig. S5.\nAll the extreme gate voltage combinations used for the gate voltage scans of the magnetization\ndynamics (Figs. 2a, 3a, and 3b of the main text) are shown.\nS6. DAMPING OF MAGNETIZATION PRECESSION\nThe damping time of the magnetization precession ( τosc) is an important parameter to\ndescribe the magnetization dynamics. As mentioned in the main text, the precession fre-\nquency we measured is well described by the Landau-Lifshitz-Gilbert (LLG) equation in the\nlimit of low damping. In the LLG equation, the effective damping is included through the\nphenomenological dimensionless parameter αeff. This damping describes how the magne-\ntization relaxes to its equilibrium state after excitation. The effective damping parameter\nαeffcan be decomposed into a frequency-independent intrinsic damping α, and a frequency-\ndependent extrinsic damping αext, resulting in αeff=α+αext. For our measurements we\nassume that the main source of extrinsic damping is spatial fluctuations in Hint. This con-\ntributes to the damping via inhomogeneous broadening of the precession frequency, and is\nquantified by αext=/radicalbig\n2 ln(2) |dω/dH int|∆Hint//bracketleftbiggµBµ0\nℏ(2|Hext| −Hintsin2(θM))/bracketrightbig2. Here,\ngis the Land´ e g-factor, µBthe Bohr magneton, µ0the vacuum permeability, Hextthe\nexternal magnetic field, θMthe angle between the magnetization and the sample normal, ω\nthe angular frequency of the precession, and Hint= 2Ku/(µ0Ms)−Mswith Kuthe uniaxial\nmagnetic anisotropy energy and Msthe saturation magnetization. Note that in3|Heff|is\nexpressed as Hextcos(θ−θM) +Hintcos2(θM). The angle θMis calculated by minimizing\nthe magnetic energy in the presence of an external field, perpendicular magnetic anisotropy,\nand shape anisotropy2. The derivative |dω/dH int|is calculated numerically using Eq. (1) of\nthe main text. For large external magnetic fields, ∆ Hintbecomes less relevant, and αeff≈α.\nThe expression for the magnetization precession damping time ( τosc) obtained from the\nferromagnetic resonance (FMR) mode of the LLG equation for small values of αeffis given\nby2,3\nτosc= 2ℏ//bracketleftbig\ngµBµ0αeff/parenleftbig\n2|Hext| −Hintsin2(θM)/parenrightbig/bracketrightbig\n. (S3)\nTo obtain the damping parameters αand ∆ Hint, this equation is fitted to the values of\nτoscextracted from the TRFE measurements at various values of ∆ nwith ∆ D= 0, which\nis presented in Fig. S7. We used the values g= 1.886 and µ0Hint= 134, 125 and 120\nmT for respectively ∆ n= 1.2, 0 and −1.8×1013cm−2, which are obtained from fitting the\nprecession frequency versus Hextdata using Eq. (1) of the main text. The fit results for\nthe damping parameters are summarized in Tab. I. As can be seen in Fig. S7, the data for7\n∆n=−1.8×1013cm−2is fitted well by Eq. (S3). The data for ∆ n= 1.2×1013cm−2is\nfitted reasonably well, but for ∆ n= 0 the fit is quite poor, even when taking the errors in\nτoscinto account.\nA bad fit likely results from the large spread of the measurements points at high external\nmagnetic fields, and (when present) from the peak in τoscaround Hext= 200 mT. This\npeak, which starts around Hint≈125 mT, consistently shows up for other small values\nof ∆nand ∆ Das well. A possible explanation for this is that there are more factors\naffecting the extrinsic damping than ∆ Hintalone. The large spread and errors in τoscfor\n∆n= 0 and 1 .2×1013cm−2are caused by a combination of two factors. The first factor\nis the low precession amplitude, which results in a lower signal to noise ratio for all fitting\nparameters related to the amplitude of the precession. This can be improved by inducing\nprecession with a larger amplitude (e.g. higher pump fluence), or by reducing the noise\nin the system (e.g. higher probe fluence, or better blocking of pump before the detection\nstage). The second factor is the relatively short maximum pump-probe delay (∆ tmax) that\nwe can achieve in our experimental setup, which is about 1.5 ns. For a good estimate of the\nmagnetization precession damping time, ∆ tmax≫τosc. In our case however, ∆ tmax≈τosc.\nUsing all three fit results, we obtain an intrinsic Gilbert damping of α≈6×10−3, and a\nspread in the internal effective field of ∆ Hext≈9 mT. Both of these values are in line with\nvalues found in literature for thin CGT flakes2,4.\n0.0 0.2 0.4 0.6\nμH0 ext (T)0.00.51.01.52.02.5\nΔn (10 cm )13−2\n1.20.0-1.8ΔD = 0τosc (ns)\nFIG. S7. Gate dependence of precession decay time. Gate dependence of the relation\nbetween decay time of the magnetization precession and Hext. The circles indicate decay times\nextracted from the TRFE measurements. The solid lines correspond to the best fit of Eq. (S3)\n∆n(1013cm−2)α(10−3)µ0∆Hint(mT)\n-1.8 6.5±0.3 9 .0±0.2\n0 12±3 5 ±1\n1.2 2±1 12 .6±0.7\nTABLE I. Damping parameters. Fit results of the damping of the magnetization precession for\nthree values of ∆ nwith ∆ D= 0. The error in these values is the statistical error obtained from\nthe least squares fitting procedure.8\nS7. DETAILED DISCUSSION ON OTHER POSSIBLE MECHANISMS OF THE TRFE\nAMPLITUDE MODULATION CAUSED BY ELECTROSTATIC GATING\nThe change in amplitude of the oscillations in the TRFE signal with gating can have\nmultiple causes. The most obvious one is a change in the magnetization precession ampli-\ntude. It could however also be caused by a change in the strength of the Faraday effect.\nIn this case, the TRFE curves for different values of the gate voltages (measured at the\nsame external magnetic field) would only differ by scale factor, meaning that the curves can\nbe mapped onto each other by simply scaling them along the vertical axis. This is clearly\nnot happening in the measurements shown in Fig. 1a in the main text and Fig. S12, and\ntherefore the change in amplitude is not caused by a pure change in the strength of the\nFaraday effect. Gate dependent magnetization curve measurements performed in the same\nsetup, shown in Fig. S2, confirm that the strength of the Faraday effect is not significantly\nchanged by electrostatic gating.\nAnother possibility would be that the equilibrium angle of Mchanges due to a change\nin the induced charge carrier density (∆ n). This then would change the projection of the\nprecession plane onto the propagation direction of the laser, and could change the TRFE\namplitude without changing the magnetization precession amplitude. The canting would\nbe due to a change in the effective internal field ( Hint). However, our measurements show\nthat the change in Heffis less than 15%, which is not enough to explain the large increase\nin amplitude of the TRFE oscillations.\nS8. COMPLETE DATA SETS FOR THE GATE VOLTAGE DEPENDENCE OF THE\nMAGNETIZATION PRECESSION PARAMETERS\nThis section displays the complete data sets of the gate voltage dependence of the mag-\nnetization precession amplitude ( A), starting phase ( ϕ0), and frequency ( f).9\nVb (V)\n−6 0 6\nVb (V)−606�0Hext = 50 mT\n0.95.0Vt (V)\n−6 0 6−606�0Hext = 75 mT\n3.119.5Vt (V)�0Hext = 100 mT\n−6 0 6−606\n0.75.1\nVb (V)Vt (V)\n−6 0 6−606\n0.54.9�0Hext = 125 mT\nVb (V)Vt (V)\n−6 0 6−606\n0.44.4�0Hext = 150 mT\nVb (V)Vt (V)\n−6 0 6−606\n0.23.3�0Hext = 250 mT\nVb (V)Vt (V)\n−6 0 6−606\n0.12.5�0Hext = 350 mT\nVb (V)Vt (V)\n−6 0 6−606\n0.01.8�0Hext = 500 mT\nVb (V)Vt (V)a b c\nd e f\ng hMagnetization precession amplitude (arb. units)\nFIG. S8. Gate voltage dependence of the magnetization precession amplitude. a-h ,\nThe complete data set of the gate voltage dependence of the magnetization precession amplitude\nextracted from the TRFE measurements, for all values of the magnetic fields used during the\nmeasurements. The color scale has a different range for each plot. The scale bar limits indicate\nthe minimum and maximum amplitude, all in the same arbitrary units, for µ0Hextequal to 50 mT\n(a), 75 mT ( b), 100 mT ( c), 125 mT ( d), 150 mT ( e), 250 mT ( f), 350 mT ( g), and 500 mT ( h).10\nMagnetization precession starting phase (rad)\n−6 0 6−606\n1.0�1.7�\nVb (V)Vt (V)�0Hext = 50 mT\n−6 0 6−606\n1.0�1.7�\nVb (V)Vt (V)�0Hext = 75 mT\n−6 0 6−606\n1.0�1.7�\nVb (V)Vt (V)�0Hext = 100 mT\n−6 0 6−606\n1.0�1.7�\nVb (V)Vt (V)�0Hext = 125 mT\n−6 0 6−606\n1.0�1.8�\nVb (V)Vt (V)�0Hext = 150 mT\n−6 0 6−606\n0.9�1.8�\nVb (V)Vt (V)�0Hext = 250 mT\n−6 0 6−606\n0.9�1.8�\nVb (V)Vt (V)�0Hext = 350 mT\n−6 0 6−606\n0.7�1.9�\nVb (V)Vt (V)�0Hext = 500 mTa b c\nd e f\ng h\nFIG. S9. Gate voltage dependence of the magnetization precession starting phase. a-h ,\nThe complete data set of the gate voltage dependence of the magnetization precession starting\nphase extracted from the TRFE measurements, for all values of the magnetic fields used during\nthe measurements. The color scale has a different range for each plot. The limits indicate the\nminimum and maximum phase, in radians, for µ0Hextequal to 50 mT ( a), 75 mT ( b), 100 mT ( c),\n125 mT ( d), 150 mT ( e), 250 mT ( f), 350 mT ( g), and 500 mT ( h).11\n−6 0 6−606\n3.53.9\nVb (V)�0Hext = 50 mTVt (V)\n−6 0 6−606\n4.04.4\nVb (V)�0Hext = 75 mTVt (V)\n−6 0 6−606\n4.65.0�0Hext = 100 mT\nVb (V)Vt (V)\n−6 0 6−606\n5.25.5�0Hext = 125 mT\nVb (V)Vt (V)\n−6 0 6−606\n5.86.3�0Hext = 150 mT\nVb (V)Vt (V)\n−6 0 6−606\n7.98.2�0Hext = 250 mT\nVb (V)Vt (V)\n−6 0 6−606\n10.310.5�0Hext = 350 mT\nVb (V)Vt (V)\n−6 0 6−606\n14.014.2�0Hext = 500 mT\nVb (V)Vt (V)a b c\nd e f\ng hMagnetization precession frequency (GHz)\nFIG. S10. Gate voltage dependence of the magnetization precession frequency for all\nmagnetic fields. a-h , The complete data set of the gate voltage dependence of the magnetization\nprecession frequency extracted from the TRFE measurements, for all values of the magnetic fields\nused during the measurements. The color scale has a different range for each plot. The limits\nindicate the minimum and maximum frequency, in GHz, for µ0Hextequal to 50 mT ( a), 75 mT\n(b), 100 mT ( c), 125 mT ( d), 150 mT ( e), 250 mT ( f), 350 mT ( g), and 500 mT ( h).12\nS9. MAGNETIZATION PRECESSION FREQUENCY FOR NEGATIVE MAGNETIC FIELDS\nThe measured magnetization precession frequency is found to be independent of the\nsign of Hext, (Fig. S11). There are two outliers for ∆ n= 0 cm−2(at -0.75 T and -0.5\nT), and one for ∆ n= 1.2×10−13cm−2(at -0.75 T). The precession amplitude in the\nTRFE measurements was too small for these measurements to obtain a good fit. Fig.\nS11b shows the difference in precession frequency for positive and negative fields, f(Hext)−\nf(−Hext). The absolute difference is smaller than 0.2 GHz, and typically smaller than\n0.1 GHz (disregarding the outliers). Fig. S11c shows the relative difference in precession\nfrequency for positive and negative fields, 2[ f(Hext)−f(−Hext)]/[f(Hext) +f(−Hext)].\nDisregarding the outliers, the relative difference is smaller than 2%, and typically smaller\nthan 1%. We point out that especially for ∆ n=−1.8×1013cm−2, for which the precession\namplitude is very large, the frequency difference is very small (0.02 GHz, and 0.2% for fields\nabove 125 mT). The TRFE traces at µ0Hext=±100 mT are shown in Fig. S12. From\nthese measurements it is already clear that reversing Hextdoes not affect the frequency of\nthe oscillations. These measurements also show again the arguments presented in the main\ntext that the magnetization precession is not purely caused by the ∆ Kmechanism: both\nthe amplitude and starting phase of the oscillations are different for positive and negative\nfield. Furthermore, these measurements also show again the argument presented in the\nmain text and Supplementary Section S7 that the gate dependence of the TRFE oscillation\namplitude is not (purely) caused by a gate-induced change in the strength of the Faraday\neffect: the background on top of which the oscillations in the TRFE measurements are,\nscales differently with ∆ nthan the amplitude of the oscillation.\n−0.2−0.10.00.10.2Frequency diff. (GHz)\nΔ(10  cm )n13−2\n1.2\n0.0\n-1.8\n05101520Frequency (GHz)\nΔ(10  cm )n13−2\n1.2\n0.0\n-1.8\n−0.02−0.010.000.010.02Rel. frequency diff\nΔ(10  cm )n13−2\n1.2\n0.0\n-1.8a b c\n0.0 0.2 0.4 0.6 0.8\nμH0ext  (T)−0.5 0.0 0.5\nμH0ext  (T)0.0 0.2 0.4 0.6 0.8\nμH0ext  (T)\nFIG. S11. Hextsymmetry of the magnetization precession frequency .a, Magnetization\nprecession frequency for positive and negative external magnetic fields. b, Difference in precession\nfrequency for positive and negative external magnetic fields. c, Relative difference in precession\nfrequency for positive and negative external magnetic fields. All data presented has ∆ D= 0.13\n−0.4−0.20.00.20.4\n−0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50−0.6−0.4−0.20.00.20.4\n−1.0−0.50.00.51.0\nΔt (ns)+100 mT\n−100 mT\n+100 mT\n−100 mT\n+100 mT\n−100 mTΔn (1013cm−2) = −1.8 \n0.0\n1.2TRFE (arb. units)ΔD = 0\nFIG. S12. TRFE measurements for positive and negative external magnetic fields .\nTime-resolved Faraday ellipticity measurements at positive and negative external magnetic field\n(µ0Hext=±100 mT) and ∆ D= 0, for various values of ∆ n. The offset at ∆ t <0 is set to zero for\nclarity. The difference in oscillation amplitude and starting phase for positive and negative Hextis\nclearly visible. Note the different scale for the ∆ n=−1.8×1013cm−2data.\nS10. DETAILED DISCUSSION ON THE CONTRIBUTION OF NON-LINEAR\nMAGNETO-OPTIC EFFECTS TO THE HextASYMMETRY OF THE TRFE OSCILLATIONS.\nOur measurements show that the amplitude of the TRFE oscillations is not symmetric\nin the external magnetic field. This could be explained by a change in amplitude of the\nmagnetization precession, as discussed in the main text. Another possibility is that the\namplitude of the magnetization precession is still symmetric in Hext, but the change in\npolarization of the probe by magneto-optic effects in the CGT is not. This could happen if\nthe probe is not only affected by the Faraday effect, which is linear in the magnetization,\nbut also by higher order magneto-optic (MO) effects, such as the Cotton-Mouton effect.\nThe change in polarization of the probe ( ψ) due to the MO effects of the sample depends\non the sample magnetization as ψ=aiMi+bijMiMj, where the parameters aiandbij\nare independent of the magnetization, bij=bji, and i, j=x, y, z . The Faraday effect is\ndescribed by the first term, and the quadratic MO effect, e.g. Cotton-Mouton effect, by the\nsecond term. In the time-resolved Faraday ellipticity measurements, we observe variations\ninψcaused by small variations in the magnetization around the equilibrium magnetization\nM0. This is expressed as:\n∆ψ=ai∆Mi+ 2bijM0,i∆Mj+bij∆Mi∆Mj. (S4)\nWe will now discuss how the quadratic MO effect could affect our TRFE measurements.\nThe last term in Eq. (S4) is quadratic in ∆ M, and would result in a double frequency\ncomponent in the TRFE oscillations. As this is not observed, this term is assumed to be\nnegligible in our measurements, meaning that bijand therefore the second order MO effect\nis negligibly small, or that ∆ Miis much smaller than M0,i\nThe second term in Eq. (S4) is linear in ∆ M, and could be of the same order of magnitude14\nas the first term for arbitrarily small values of ∆ Mi. If this is the case, i.e. if bijM0,iis of\nthe order of ai, this term will result in the oscillation amplitude of ψbeing not symmetric\ninHext, even for a thermal induced precession where the oscillation amplitude of ∆ Mis\nsymmetric in Hext. This is because upon reversing Hext,M0,ichanges sign, changing ψfrom\n(aj+bijM0,j)∆Mjto (aj−bijM0,j)∆Mj. Therefore, a quadratic MO effect could result in\nthe observed asymmetry in the amplitude of the TRFE oscillations while the amplitude of\nthe magnetization precession is still symmetric in Hext.\nHowever, even though quadratic MO effects could explain that the TRFE oscillation am-\nplitude is asymmetric in Hextfor a purely thermal excitation of the magnetization preces-\nsion, they cannot explain all of our observations. For example, the pump fluence dependence\nof the TRFE measurements presented in Fig. S5e shows a jump in the signal at ∆ t= 0 of\nwhich the sign depends on the pump fluence. For convenience, the data is presented again\nin Fig. S13a, with a close-up of the interesting part in Fig. S13b. The data shows that\nfor high pump fluences, the TRFE signal increases sharply right after ∆ t= 0, indicating\na fast decrease in, or canting of, the magnetization. For the lowest pump fluence however,\nthe signal sharply decreases right after ∆ t= 0. The argument for why a purely thermal\nexcitation cannot explain this, as detailed below.\na b\n0.0 0.5 1.0 1.5 2.0\nΔ (ns)t0.51.01.52.02.53.0TRFE signal (arb.units)\n−0.2 −0.1 0.0 0.1 0.2 0.3 0.4\nΔ (ns)t0.500.600.700.800.90TRFE signal (arb.units)\n26\n123858F (�J/cm2)\nFIG. S13. Supporting evidence for coherent optical excitation of the magnetization\nprecession. a , Pump fluence dependence of the TRFE measurements for Vt=−6V,Vb= 6V\n(D= 1.1 V/nm, ∆ n= 0) and µ0Hext= 100 mT. The solid line is the best fit of the data, using\nEq. (2) of the Methods for ∆ t >0, and a constant for ∆ t <0. The dashed lines indicate the\ndouble exponential background of the oscillation b, Close-up of the data for a pump fluence of 12\nµJ/cm2, indicated by the black rectangle in a.\nFor a purely thermal demagnetization, the magnetization and the magnetocrystalline\nanisotropy decrease right after excitation5. Canting of the magnetization only happens if\nthe magnetocrystalline anisotropy briefly changes on a time scale shorter than the precession\nperiod. This is then accompanied by a a starting phase that is close to ±π/2, i.e. if\nthe oscillations start as a sine instead of a cosine (in our experimental geometry)6. Our\nmeasurements however show a clear cosine-like start of the oscillations. Therefore the\nmagnetization only decreases during the heating by the laser. Assuming the direction of\nthe magnetization, Mi/|M|, has not changed during after the excitation, the TRFE signal\ncan be expressed in terms of the magnitude of the magnetization. Doing this in Eq. (S4)\nyields:\nψ=aiMi+bijMiMj=A|M|+B|M|2(S5)\n∆ψ=A∆|M|+ 2B|M0|∆|M|= (A+ 2B|M0|) ∆M. (S6)\nwhere A=aiMi/|M|andB=bijMiMj/|M|2are constant. The sign of ∆ ψis thus\nindependent of ∆ Mand therefore independent of the fluence of the pump. Hence the\nchange from a sharp increase to a sharp decrease by changing the power of the pump\ncannot be explained by a pure thermal excitation process.15\nA coherent excitation on the other hand, caused by e.g. the inverse Cotton-Mouton\neffect, does have the ability to create an effective magnetic field to cant the magnetization\non sub-picosecond timescales7. The combination of thermal excitation and coherent laser\nexcitation could explain the observed behavior of the TRFE with pump power. How this\ncombination can explain the asymmetry in the amplitude and phase with external magnetic\nfield is explained in Supplementary Section S11 below.\nS11. MODEL FOR OPTICAL EXCITATION OF MAGNETIZATION DYNAMICS\nIn this section we develop a model for the excitation of the magnetization dynamics,\nincluding coherent optical excitations, and thermal effects. Using this model, we obtain the\namplitude and starting phase of the magnetization precession.\nThe coherent effects we consider are the inverse Cotton-Mouton effect (ICME) and photo-\ninduced magnetic anisotropy (PIMA). The effective magnetic field generated by these effects\nis described by8:\nHeff,i=GijklMiEkEl (S7)\nwhere Gijklis a rank 4 polar tensor that is symmetric under exchanging the first or last\npair of indices, Mithe magnetization, Eithe electric field of the light, and i, j, k, l =x, y, z .\nNote that the two effects are described by two different tensors. The crystal symmetry of\nthe CGT (space group R¯39) restricts the number of independent elements of these tensors to\n12, and forces some to be zero, as indicated in Tab. II. In this table, Gijis an abbreviation\nofGklmn, where i and j take the values 1 to 6, determined by klandmnrespectively. The\nvalues for i(j) = 1, 2 and 3 correspond to kl(mn) =xx,yyandzz, while the values 4, 5\nand 6 correspond to kl(mn) consisting of the combination of yandz,xandz, and xand\nyrespectively. The 4 boxed elements are responsible for generating an effective magnetic\nfield in the ydirection when both the magnetization and the polarization of the pump are\nin the xzplane. These elements are needed to explain the the observed asymmetry of the\namplitude in the TRFE oscillations for positive and negative values of Hext.\njGij1 2 3 4 5 6\n1G11 G12G13G14 G15 G16\n2G12 G11G13−G14−G15 −G16\n3G31 G31G33 0 0 0\n4G41−G410G44−G54 −G51\n5G51−G510G45 G44 G41i\n6−G16G16 0−G15G14(G11−G12)/2\nTABLE II. Symmetry-adapted form of G ijkl.This table, calculated using Ref.10, shows the\nrelation between the different elements of Gijklimposed by the space group R¯3 and the invariance\nunder interchanging iandj, and kandl. using The 12 independent elements are indicated in\nblack, the dependent ones in gray. The elements that are zero are highlighted in red. The boxed\nelements are responsible for generating an effective magnetic field in the ydirection when both the\nmagnetization and the polarization of the pump are in the xzplane. Conversion between Gklmn\nandGijis explained in the text.\nTo determine effect of the ICME and photo-induced magnetic anisotropy on the the\nprecession, we use the following simplified model. Light travels in the direction /hatwidek=\n(sinθk,0,cosθk), where θkthe angle between /hatwidekand the z-axis. The orientation of the\naxes is indicated in Fig. 3e of the main text. The polarization of the pump is linear, and\nin the xzplane. Its electric field vector is described by E=E0(cosθk,0,sinθk), where E0\nis the amplitude of the electric field. In equilibrium, the magnetization lies in the xzplane16\ntoo:M=Ms(sinθM,0,cosθM). Plugging the above expression in Eq. (S7) reveals that the\ncomponents of effective magnetic field induced by these coherent optical excitations depends\nsinusoidally on θMand on 2 θk, where the amplitude and phase are in general different for\nthe different components of Heff,i.\nWhen the pump pulse hits the sample, we assume that the magnetization is firstly affected\nby the ICME. This effect can be described by a delta pulse in the effective magnetic field,\nresulting in an instantaneous rotation of the magnetization7,8,11. Afterwards, the effective\nmagnetic field is changed instantaneously to a new value due to both the PIMA and a\nthermal induced change in the magnetocrystalline anisotropy. The former changes the\neffective magnetic field according to Eq. (S7). The latter decreases the magnetocrystalline\nanisotropy of the sample, resulting in a decrease of the effective internal field along the\nsample normal5. From this point on, the magnetization starts to precess in a circular orbit\naround the effective magnetic field, as described by the Landau-Lifshitz-Gilbert equation.\nThe time dependence of Mfor negligible damping is conveniently given by Rodrigues’\nrotation formula:\nM(t) =M0cosωt+/parenleftig\n/hatwideHeff×M0/parenrightig\nsinωt+/hatwideHeff/parenleftig\n/hatwideHeff·M0/parenrightig\n(1−cosωt) (S8)\nwhich rotates the vector M0around a unit vector /hatwideHeffwith angular frequency ω.\nThe Faraday effect is assumed to be only sensitive to changes in the magnetization along\nthe propagation direction of the laser, /hatwidek. The amplitude and phase of the precession as\nmeasured by Faraday ellipticity are therefore obtained by projecting the motion of the\nmagnetization on the propagation direction of the laser. This is achieved by taking the\ninner product of M(t) and/hatwidek. The resulting expression has a term with time dependence\ncos(ωt), and a term with time dependence sin( ωt). By combining the two into a single phase\nshifted cosine, the amplitude and starting phase of the TRFE oscillations are obtained. The\namplitude angle ( θA) of the precession, which is the angle between the magnetization and\nthe effective magnetic field, is calculated too.\nFor the calculations, we use the experimental parameters µ0Hint= 125 mT and θH= 50\ndegrees. We set the small thermal induced change of the effective field µ0∆Hthermal , caused\nby a reduction of the magnetocrystalline anisotropy and the magnetization, to -1 mT. A\ngood qualitative agreement with the TRFE oscillation amplitude, and a good quantitative\nagreement with the starting phase phase, shown in Fig. 3c,d of the main text, are obtained if\nthe ICME and the PIMA depend on the orientation of the magnetization as sin ( θM−50◦).\nThe ICME then rotates Malong a fixed direction ( /hatwideHICME ), and the PIMA generates an\neffective field HPIMA along a fixed axis, of which the orientations are independent of the\nvalue of Hext. Furthermore, for the angle over which the ICME rotates M(θICME ), we\ntook 0 .4◦sin(θM−50◦) along the vector /hatwideHICME = (−0.19,0.96,0.19), and µ0∆HPIMA =\n0.4 sin( θM−50◦) mT along the direction (0 .19,0.96,−0.19).\nThe resulting precession angle, and the TRFE oscillation amplitude and starting phase\nare shown in Fig. S14. The values are only plotted for external field magnitudes larger than\n75 mT, since at lower values the magnetization is not fully saturated after laser excitation.\nThe external magnetic field dependence of both θAand the Faraday ellipticity amplitude\nare not symmetric in Hext, and have an inflection point around µ0Hext=±200 mT. This\nagrees with the measured amplitude for ∆ n=−1.8×1013cm−2and 1 .2×1013cm−2.\nA possible reason for the discrepancy for ∆ n= 0 is given below. The rate at which the\nmeasured amplitude decreases for large fields, and the asymmetry in this rate, is much\nbetter captured by θAthan by the calculated Faraday ellipticity amplitude. This could\nmean that the calculation of the TRFE signal from the magnetization precession is too\nsimplistic in our model model.\nThe calculated starting phase ( ϕ0) also captures the features of the measured starting\nphase. For positive fields, ϕ0is mostly constant, at a value slightly below π, and starts\nto increase as the field approaches zero. At negative fields, the phase increases as the\nmagnitude of the field grows, saturating to a value of about 1 .7π. This is in quantitative17\nagreement with our measurements for ∆ n=−1.8×1013cm−2and 1 .2×1013cm−2, and\nin qualitative agreement for ∆ n= 0.\nThe behavior for ∆ n= 0 is very different from to the other two. The amplitude is mostly\nsymmetric in Hextand the relative change with Hextfor low field is much faster. Also, the\nphase appears to be shifted to more positive value of Hext. A possible explanation for this\nis that the thermally induced change in Heffdoes not happen instantaneously, but on a\ntime scale close to the period of the magnetization precession for low fields. in that case,\nthe amplitude and phase can both be greatly affected if the two time scales are close. The\nstrengths of the coherent excitations can also be different than for ∆ n= 1.2 or−1.8×1013\ncm−2.\n−0.5 0.0 0.5\nμH0ext  (T)012345\n−0.5 0.0 0.5\nμH0ext  (T)0.00.20.40.60.81.01.21e−3\n−0.5 0.0 0.5\nμH0ext  (T)�2� ϕ0 (rad)\na b cθA (mrad)\nTRFE osc. ampl. (arb. units)\nFIG. S14. Model for the effect of coherent excitations on the magnetization precession.\na, The angle between the magnetization and the effective magnetic field versus Hext.b, the\namplitude of the percession versus Hext, as measured by Faraday ellipticity, c, The starting phase\nof the precession versus Hext.\n1M. Lohmann, T. Su, B. Niu, Y. Hou, M. Alghamdi, M. Aldosary, W. Xing, J. Zhong, S. Jia, W. Han,\nR. Wu, Y.-T. Cui, and J. Shi, “Probing Magnetism in Insulating Cr2Ge2Te6 by Induced Anomalous Hall\nEffect in Pt,” Nano Letters 19, 2397–2403 (2019).\n2T. Zhang, Y. Chen, Y. Li, Z. Guo, Z. Wang, Z. Han, W. He, and J. Zhang, “Laser-induced magnetization\ndynamics in a van der Waals ferromagnetic Cr2Ge2Te6 nanoflake,” Applied Physics Letters 116, 223103\n(2020).\n3S. Mizukami, E. P. Sajitha, D. Watanabe, F. Wu, T. Miyazaki, H. Naganuma, M. Oogane, and Y. Ando,\n“Gilbert damping in perpendicularly magnetized Pt/Co/Pt films investigated by all-optical pump-probe\ntechnique,” Applied Physics Letters 96(2010).\n4C. W. Zollitsch, S. Khan, V. T. T. Nam, I. A. Verzhbitskiy, D. Sagkovits, J. O’Sullivan, O. W. Kennedy,\nM. Strungaru, E. J. G. Santos, J. J. L. Morton, G. Eda, and H. Kurebayashi, “Probing spin dynamics\nof ultra-thin van der Waals magnets via photon-magnon coupling,” Nature Communications 14, 2619\n(2023).\n5M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W. J. de Jonge, and B. Koopmans,\n“All-optical probe of coherent spin waves,” Physical Review Letters 88, 227201 (2002).\n6F. Dalla Longa, Laser-induced magnetization dynamics , Ph.D. thesis, Eindhoven University of Technology\n(2008).\n7A. M. Kalashnikova, A. V. Kimel, R. V. Pisarev, V. N. Gridnev, P. A. Usachev, A. Kirilyuk, and T. Ras-\ning, “Impulsive excitation of coherent magnons and phonons by subpicosecond laser pulses in the weak\nferromagnet FeBO3,” Physical Review B 78, 104301 (2008).\n8I. Yoshimine, T. Satoh, R. Iida, A. Stupakiewicz, A. Maziewski, and T. Shimura, “Phase-controllable\nspin wave generation in iron garnet by linearly polarized light pulses,” Journal of Applied Physics 116,\n043907 (2014).\n9V. Carteaux, D. Brunet, G. Ouvrard, and G. Andre, “Crystallographic, magnetic and electronic structures\nof a new layered ferromagnetic compound Cr2Ge2Te6,” Journal of Physics: Condensed Matter 7, 69–87\n(1995).\n10S. V. Gallego, J. Etxebarria, L. Elcoro, E. S. Tasci, and J. M. Perez-Mato, “Automatic calculation of\nsymmetry-adapted tensors in magnetic and non-magnetic materials: a new tool of the Bilbao Crystallo-\ngraphic Server,” Acta Crystallographica Section A Foundations and Advances 75, 438–447 (2019).\n11L. Q. Shen, L. F. Zhou, J. Y. Shi, M. Tang, Z. Zheng, D. Wu, S. M. Zhou, L. Y. Chen, and H. B. Zhao,\n“Dominant role of inverse Cotton-Mouton effect in ultrafast stimulation of magnetization precession in\nundoped yttrium iron garnet films by 400-nm laser pulses,” Physical Review B 97, 224430 (2018)."    },    {        "title": "2308.12616v2.Coercivity_Mechanisms_of_Single_Molecule_Magnets.pdf",        "content": "Coercivity Mechanisms of Single-Molecule Magnets\nLei Gu,1,∗Qiancheng Luo,2Guoping Zhao,1,†Yan-Zhen Zheng,2,‡and Ruqian Wu3,§\n1College of Physics and Electronic Engineering,\nSichuan Normal University, Chengdu 610101, China\n2Frontier Institute of Science and Technology (FIST),\nXi’an Jiaotong University, Xi’an 710054, Shaanxi, China\n3Department of Physics and Astronomy,\nUniversity of California, Irvine, California 92697, USA\n1arXiv:2308.12616v2  [cond-mat.mtrl-sci]  7 Sep 2023Abstract\nMagnetic hysteresis has become a crucial aspect for characterizing single-molecule magnets, but\nthe comprehension of the coercivity mechanism is still a challenge. By using analytical derivation\nand quantum dynamical simulations, we reveal fundamental rules that govern magnetic relaxation\nof single molecule magnets under the influence of external magnetic fields, which in turn dictates the\nhysteresis behavior. Specifically, we find that energy level crossing induced by magnetic fields can\ndrastically increase the relaxation rate and set a coercivity limit. The activation of optical-phonon-\nmediated quantum tunneling accelerates the relaxation and largely determines the coercivity. Intra-\nmolecular exchange interaction in multi-ion compounds may enhance the coercivity by suppressing\nkey relaxation processes. Unpaired bonding electrons in mixed-valence complexes bear a pre-spin-\nflip process, which may facilitate magnetization reversal. Underlying these properties are magnetic\nrelaxation processes modulated by the interplay of magnetic fields, phonon spectrum and spin state\nconfiguration, which also proposes a fresh perspective for the nearly centurial coercive paradox.\nIntroduction. The field of magnetism continues to grapple with long-standing topics\nof coercivity mechanisms, especially the paradox of notably lower coercivity compared to\nthe theoretical predictions [1–9]. This discrepancy is usually referred to as Brown’s coer-\ncive paradox since it was brought up in the 1940s [1]. While diverse mechanisms such as\nimperfection [3], inter-grain interactions [4], boundary effects [5], and nonlocal exchange\ninteraction [6] have been proposed to bridge the gap between theory and experiment, the\nparadox has not been satisfactorily resolved, and the underlying physics is still a subject of\nongoing debate. The paradox is especially prevalent when considering the magnetic hystere-\nsis measurements of single-molecule magnets (SMMs) [10–28]. The coercivity mechanism in\nSMMs essentially links to how magnetic fields impact the relaxation processes of individ-\nual spins, which manifests as non-trivial imprints in the magnetic hysteresis behavior. As\ndemagnetization at finite temperature and magnetic fields can be generally considered as\nmagnetic relaxation [29, 30], investigation into these magneto-modulation effects in SMMs\ncould also illuminate coercivity mechanisms in other magnetic materials.\nSMMs have attracted great interests in recent decades for potential applications in quan-\ntum information technologies [31–33]. Many efforts have been paid to the slow magnetic\nrelaxation of SMM [34–45], as the ability to maintain magnetization underpins their func-\n2tionality. Recently, high temperature magnetic hysteresis in dysprosocenium [10–12] and\nultrahard magnetism in mixed-valence dilanthanide complexes [13] have marked advance-\nments toward practical applications. These developments necessitate an in-depth investiga-\ntion into magnetic relaxation of SMMs under the influence of strong magnetic fields. Despite\nvarious mechanisms of the magnetic relaxation of SMMs being available [21, 39, 46–49], a\ntheoretical description of magnetization evolution in strong magnetic fields has not been\nestablished. Fundamentally, it is still unclear what the coercivity of SMMs actually entails.\nAs the inter-molecular exchange interactions are negligible, SMMs are paramagnetic\nmolecular crystals of magnetic complexes. Their magnetization either aligns to an external\nmagnetic field or decays in the absence of a field. As a result, magnetic hysteresis be-\ncomes obvious only when the magnetic relaxation is slow compared to the sweep rate of the\nscanning magnetic field. Especially, the width of hysteresis curves depends on the rate of\nmagnetization reversal under an opposing magnetic field. While specific shape of a hysteresis\nloop may varies with the experimental setup, understanding coercivity of SMMs essentially\ninvolves elucidating how the magnetization evolves under different magnetic fields.\nIn classical approaches, the magnetization of a system at zero temperature can be de-\ntermined by minimizing the energy, with the thermal effects incorporated via empirical\nthermal activation [7] and stochastic dynamics [50, 51]. The coercivity can be then derived\nfrom behavior of the magnetization in a magnetic field. Given that the magnetization of\neach molecule in a SMM evolves rather independently, its coercivity at zero temperature\ncan be studied through a simple classical model for a single molecule. However, as magnetic\nstates of SMMs are quantized and the transition rates among spin states depends on their\nwavefunctions, a classical model is unsuitable for describing their hysteresis behavior. In\nthis study, we carry out quantum dynamical simulation of magnetic relaxation processes\nunder generic settings. Our results account for a broad range of experimental observations,\nand the effects of more specific features can be deduced based on the mechanisms that are\nelucidated.\nCritical strengths of external magnetic fields. In both model analysis and numerical\nsimulation conducted in this work, we set the initial states to the saturation magnetization\nalong the −zdirection, i.e., classical magnetic moments Sz=−Sor the quantum counter-\npart|Sz=−S⟩, where Sdenotes the magnitude of the moments. Instead of plotting the\nhysteresis curve, which depends on sweep rate of the scanning field, we address the more\n3general problem of how magnetization evolves under an external magnetic field, particularly\none in the opposite direction (+ z). As an example, we set S= 5/2 in this work. However,\nthe results can readily be adapted to other scenarios.\nWe first consider the single-ion compounds, in which each magnetic molecule contains a\nsingle metallic atom wrapped within organic groups. The magnetic moment, characterized\nby an uniaxial anisotropy, is described by the Hamiltonian Hs=−DS2\nz. According to the\nclassical approach, a magnetic field is needed to overcome the anisotropy barrier between the\ntwo ground states ( Sz=±S) to cause a flip in the magnetization. Once the critical value is\nreached, an abrupt reversal in magnetization occurs due to the elevated magnetic potential\nenergy. The coercivity is given by Hc= 2DS/µ Bg[52] with µBthe Bohr magneton and g the\ng-factor. Assuming moderate values S= 5/2 and D= 2 meV, the coercive field is Hc= 82.3\nT, which is an order higher than what is typically observed experimentally. It is apparent\nthat there must be acceleration mechanism at play to facilitate fast magnetic relaxation at\nlower magnetic field, beyond the standard understanding for the swift magnetization reversal\naround the coercive field.\nOur computation suggests that the magnetic field can induce variation of relaxation rate\nby more than ten orders at a fixed temperature (such as 5 K in this case), suggesting\nsubstantial magneto-modulation effects. We find that the rate maximum in Fig. 1(a) is\nrelated to level crossing induced by the magnetic field [Fig. 1(c)]. In our setting the magnetic\nfield points to the + zdirection, so the energies of states with |Sz<0⟩increase with magnetic\nfield, and the energies of states with |Sz>0⟩decrease. The peak region of Fig. 1(a) suggests\nthat the relaxation rate skyrockets when the energies of |Sz=−S⟩and|Sz=S−1⟩get\nclose and peaks at the level crossing point. The soaring relaxation rate implies that the\nmagnetization would reverse at an enormous speed. Therefore, it sets an effective limit on\nthe coercivity, which approximates to H′\nc≈D/µ Bg.\nThe skyrocketing increase of the relaxation rate can be interpreted using quantum per-\nturbation theory. In addition to the uniaxial magnetic anisotropy, SMMs also exhibit slight\ntransverse magnetic anisotropy, which at the lowest order takes the form E(S2\nx−S2\ny). Since\nthe transverse anisotropy does not commute with Sz, it leads to mixture of eigenstates of Sz\ni.e.,|Sz=m⟩withmdifferent integer values in the range [ −S, S]. For E≪D, the state mix-\ning is perturbative, and an eigenstate of the combined Hamiltonian Hs=−DS2\nz+E(S2\nx−S2\ny)\nconsists of predominantly one of the |Sz=m⟩and small proportions of |Sz=n⟩withn̸=m.\n4-3 -2 -1 0 1 2 3024681012Energy (meV)\n-3 -2 -1 0 1 2 30369121518Energy (meV)\nopE/g90/g39 /g32ℏ(2 1)E D S/g39 /g124 /g160 5 10 15 2010-410-21001021041061081010Relaxation rate (s-1)\n-6 -4 -2 0 2 4 6-3-2-10123\nH (T)zS(a)\n(b) (c) (T)HFIG. 1. (a) The relaxation rate varies by more than ten orders under different magnetic fields.\nThe fast magnetic reversal in the hysteresis loop is due to the opening up of the optical-phonon-\nmediated direct process between the ground states when the energy splitting matches energy of the\nsignificant optical phonons, as sketched in (b). (c) When energies of the circled states get close,\nthe relaxation rate skyrockets and reaches the maximum at the level crossing point, corresponding\nto the peak region in (a).\nThe mixing weights initially scale as ( E/D )|m−n|and increase when the energy difference\n|Em−En|is reduced by magnetic fields. In particular, two states may mix equally by a\n50−50 proportion, if their energies degenerate, regardless of the weakness of the transverse\nanisotropy. The most influential is the mixing of |Sz=S−1⟩into|Sz=−S⟩, since it\nbridges direct tunneling between the ground states.\nTaking the rotational spin-phonon coupling [53–56] that reads Hsp∝P\nα=x,y,z[Hs, Sα]\nfor instance, the action of Hsdoes not cause overlap among the eigenstates of itself (i.e.,\nspin eigenstates). It is the action of SxandSythat contributes to a considerable transition\n5product ⟨s2|Hsp|s1⟩, where |s1⟩and|s2⟩are two spin eigenstates. As Hspis first order in\nSxandSy, it imbricates two states |Sz=m⟩and|Sz=n⟩only when |m−n| ≤1. The\nmixing weight of |Sz=S−1⟩in the elevated ground state built on |Sz=−S⟩is decisive\nfor the transition between the two ground states, because in this case the weight times the\nterm⟨Sz=S−1|Hsp|Sz=S⟩approximates to the transition product. Other forms of\nspin-phonon coupling may cause state overlap with a broader span, i.e., |m−n| ≤∆Szwith\n∆Sz>1. The rate skyrocketing persists, since ⟨Sz=S−1|Hsp|Sz=S⟩is still a major\ncomponent of the transition product.\nBeing a result grounded in sound theory, the skyrocketing relaxation rate should lead\nto noticeable experimental signatures. Their absence in previous observations may be at-\ntributed to the use of scanning fields, where the demagnetization at lower field obscures\nthe effect. Moreover, since the rate skyrocketing is triggered by the same condition with\nresonant tunneling [57], it might partially contribute to previous observations of resonant\ntunneling of multi-ion SMMs [58–61], especially when the temperature is not close to zero.\nWe should point out that the rate skyrocketing has nothing to do with the resonant tun-\nneling between the degenerate states [57]. Rather, it characterizes fast phonon-mediated\ntransition between the ground states (one is lifted), and the excited state built on |S−1⟩\nitself is not an influential intermediate states.\nHowever, we note that H′\nc=D/µ Bgis still noticeably too high compared to what is\ntypically observed in experiments. For example, H′\nc= 17.3 T when D= 2 meV. Alterna-\ntively, the relaxation rate leap around 1 .7 T is consistent with typical observations. We find\nthat this critical value corresponds to activation of the direct process between the ground\nstates mediated by optical phonons. To open up this relaxation channel, the energy dif-\nference between the two states should match the energy of optical phonons [Fig. 1(b)]. As\nthe Bose-Einstein distribution implies that the availability of a phonon decays exponentially\nas its energy increases, the low energy optical phonons with effective spin-phonon coupling\ndominate the process. The energy match implies µBgH[S−(−S)]≈¯hωop, where ωopde-\nnotes frequency of the significant optical phonons. Thus, the critical field strength is given\nbyH′′\nc≈¯hωop/2SµBg.H′′\nc≈1.7 T results from our setting ωop= 1 meV and S= 5/2.\nTo more comprehensively explain our proposed coercive mechanism, let us revisit an\nopinion advocated in previous work [45, 62–64]: the optical phonons dominate the magnetic\nrelaxation of SMMs, while the acoustic phonons play a minor role [65]. This is ascribed to\n6the weak coupling between acoustic phonons and the magnetic state of a molecule, which\nin turn is because the molecules in SMMs are mechanically rather independent, due to\nthe strong intra-molecular and weak inter-molecular interactions. On the other hand, the\nRaman processes are second order effects and in general weaker than direct processes. There-\nfore, once activated by relatively strong magnetic field, the optical-phonon-meditate direct\ntransition between the ground states shall dominate the magnetic relaxation and largely\ndetermine the coercivity. In other words, the coercivity of SMMs is a manifestation of this\nmagneto-phononic modulation effect.\nExperimental evidences. In our derivation and simulations, we assume a magnetic field\naligning with the easy axis. For polycrystals of compounds with an anisotropic g-factor, the\nactivation of the rapid reversal proceeds gradually and includes more molecules as the field\nstrength increases. For instance, dysprosium complexes usually have a g-factor biased to the\nmagnetic easy axis, and the two transverse components are close to zero. In this case, H′′\nc\nshould correspond to the projection onto the magnetic easy axis, as sketched in the right\ninset of Fig. 2. The larger is the angle between the easy axis and the magnetic filed, the\nfield should be stronger to open the gap (∆ E) in Fig. 1(b). Therefore, in polycrystalline\nsamples, H′′\ncis the critical field strength at which the acceleration of magnetization reversal\nstarts, and the g-factor anisotropy smoothes the overall reversal and enhances the coercivity,\nwhich can be seen by comparing the inset of Fig. 1(a) and the experimental hysteresis loop\nin Fig. 2.\nThe lowest phonon energy and the axial g-factor of the investigated compound are calcu-\nlated as ¯ hω≈1.2 meV and g= 1.3, respectively [52]. With S= 15/2, we have H′′\nc≈0.8 T,\nwhich well agrees with the experimental value (cf. Fig. 2). The proposed acceleration mech-\nanism is also supported by other works in that acceleration of the relaxation at similar field\nstrength is prevalent in the magnetic hysteresis measurements of SMMs [10–28]. Moreover,\ngiven that the needed data are available, we find that the consistency is quantitative. For ex-\nample, in Ref. [11] the lowest phonon energy is about twice the value here, and the observed\nH′′\ncis doubled accordingly.\nWhen computing the relaxation rate, we neglect the nuclear-spin driven quantum tunnel-\ning [47–49], which may cause drop of magnetization around H= 0 T. When the tunneling\nis strong, it diminishes the magnetization to nearly zero, and the coercivity is very weak.\nIn this case, the optical-phonon-mediated direct process does not take effect in the stage of\n7 (T)H ( )BM/g80\n-4 -2 0 2 4-6-4-20246 2 K\n 5 K\nH\n\"cHMagnetic easy axisFIG. 2. Measured magnetic hysteresis loops for the [Dy(1-AdO) 2(py) 5]BPh 4complex. As the\ng-factor is completely biased to the magnetic easy axis, the projection of the magnetic field onto\nthe easy axis is actually responsible for the Zeeman effect. When the magnitude of the projection\nreaches H′′\nc, rapid magnetic relaxation of the corresponding molecule is activated. In a polycrystal\nthe activation starts at H=H′′\nc(marked by the dashed line) and proceeds to involve molecules\nwhose easy axes do not align with H.\nmagnetization reversal but results in accelerated magnetization toward the saturation (see\ne.g. [24–26]).\nRoles of intra-molecular exchange interaction. We focus on a simple setting that the\nmolecule contains two magnetic ions with ferromagnetic interaction. To have a controlled\ncomparison with the single ion case, we assume that the spin-lattice dynamics of the two\nions are independent. Namely, Hspis summation of the respective coupling terms. With this\nsetting, the transition products are null when both S1zandS2ztake different values. For\nexample, ⟨S1z=−S, S 2z=−S|Hsp|S1z=S, S 2z=S⟩= 0, as the summation form implies\nthatHsponly acts on one subspace or the other. Transition products ⟨S1z=−S, S 2z=\nS|Hsp|S1z=S, S 2z=S⟩and⟨S1z=S, S 2z=−S|Hsp|S1z=S, S 2z=S⟩are the counterpart\nof⟨Sz=−S|Hsp|Sz=S⟩in the single ion setting. These transitions constitute significant\nelements of the transition matrix, and states like |S1z=S, S 2z=−S⟩can be considered as\nsignificant intermediate states for the overall magnetic relaxation.\n8-6 -4 -2 0 2 4 605101520Energy (meV)\n-6 -4 -2 0 2 4 605101520Energy (meV)\n-6 -4 -2 0 2 4 605101520Energy (meV)0 5 1010-2100102104Relaxation Rate (s-1) 0.0   meV \n 0.05 meV  \n 0.1   meV\n 0.4   meV\n0 5 1010-2100102104Relaxation Rate (s-1) 0.0   meV  \n 0.05 meV\n 0.1   meV\n 0.4   meVIsotropic\nexchangeIsing\nexchange\nIncreasing field strength(a) (b)\n(c) (d) (e)\nopE/g90/g39 /g33ℏopE/g90/g39 /g33ℏ\nopE/g90/g39 /g31ℏ (T)H  (T)HFIG. 3. (a) Intra-molecular exchange interaction can slow down the relaxation by elevating\nthe intermediate states. When the energy elevation opens up the optical-phonon-mediated direct\nprocess, as sketched in (c), significant relaxation enhancement may result. (d) Then, a magnetic\nfield can close the direct process by narrowing the energy gap and results in the drop of the\nrelaxation rate. (e) Further strengthened fields reopen the direct process and causes the leap of\nthe relaxation rate. (b)The Ising exchange has similar effects as the isotropic exchange. The low\nrelaxation rate indicated by the blue curve implies that strong Ising exchange can harness the\nbenefits and largely avoid the harmful effects.\nThe exchange interaction plays two competitive roles. On one hand, it lifts the significant\nintermediate states, which reduces their thermal accessibility and generally slows down the\nrelaxation. On the other hand, when exchange terms such as JxS1xS2xandJyS1yS2yare\npresent, they yield extra state mixing among eigenstates of S1zandS2z, besides that from\nthe transverse magnetic anisotropy, since the exchange terms do not commute with S1z\nandS2z. The state mixing may further facilitate relaxation and compromise the coercivity.\nAccordingly, Ising exchange that has negligible JxandJyis beneficial. As shown Fig. 3(b),\nthe benefit is obvious when the exchange is strong.\n9By a mechanism similar to that illustrated in Fig. 1(b), elevation of the significant inter-\nmediate states shoots up the relaxation rate, when the energy difference with the ground\nstates matches the energy of the significant optical phonons [Fig. 3(c)], which activates the\noptical-phonon-mediated direct process. Then, performing a magnetic field and tuning its\nstrength can reduce the energy difference, break the energy match [Fig. 3(d)] and turn off\nthe rapid relaxation, resulting in drop of relaxation rate. A regime of low relaxation rate\nfollows, until the direct process is opened up again by further strengthened fields [Fig. 3(e)].\nWhen the exchange is strong, although the undesirable direct process is always on before a\nhigh field turns it off, the relaxation is kept relatively slow, because the high-lifted significant\nintermediate states imply low availability of the associated phonons.\nWe note that this setting is barely explored in previous experiments, since the anions\nbridging the exchange usually had unpaired electrons that lead to direct exchange interaction\nwith the metallic ions [14–18, 20]. Those are mixed-valence compounds investigated in the\nfollowing. In the setting of double ionic moments, one way to enhance the coercivity is to\nmildly lifted the significant intermediate states with a relatively weak field. The optimal\nenergy gap with the ground states is ∆ E<∼¯hωop, i.e., smaller than ¯ hωopbut close to it. Then,\nthe optical-phonon-mediated direct process is initially off and activated when reaching the\nconfiguration in Fig. 3(e) with H′′\nc≈¯hωop/Sµ Bg. The critical field strength is doubled\ncompared to the single ion setting. We can also enhance the coercivity by highly elevating\nthe significant intermediate states with strong Ising exchange [blue curve in Fig. 3(b)], which\ncould result in a very strong coercivity.\nMagnetization reversal in mixed-valence compounds. In regard to the molecules studied\nin Ref. [13] we consider a structure where two ionic magnetic moments are coupled to a\nbonding electron through ferromagnetic exchange interaction [66]. As there is no exchange\nbetween the ionic moments, the significant intermediate states consist mainly of opposite\nionic moments are not lifted effectively, so the consequential reduction of relaxation rate is\nweak. On the other hand, the state mixing due to the isotropic exchange lead to sizable\nrelaxation rate increase [52]. In contrast, Ising exchange can maintain low relaxation rate and\nenhance H′′\ncby elevating the intermediate state [52]. This explains why magnetic hysteresis\nwas usually observed in mixed-valence compounds with Ising exchange [14–18, 20]. The\nremarkable coercivity enhancement in Ref. [13] should arise from other factors such as the\ndoubled magnetic anisotropy.\n10HWeakexchange Strongexchange\n0 1 2 3 4 53035404550556065Coercive field (T)\nExchange (meV)\n-6 -4 -2 0 2 4 60510152025Energy (meV)\n-6 -4 -2 0 2 4 60510152025Energy (meV)(a) (b)\n(c) (d)FIG. 4. (a) Schematics of magnetization reversal processes in the cases of weak and strong\nexchange, respectively. (b) By the classical approach the transition between the two phases yields\na turning point of coercivity. (d) The quantum dynamical simulation suggests that strong exchange\ndoes suppress the intermediate process representing the pre-spin-flip, which takes effect in the weak\nexchange case, as indicated by the left arrow in (c).\nThe magnetic moment of the binding electron induces a peculiar magnetization reversal\nprocess. In the classical picture, when the exchange is weak, one can expect that the spin of\nthe bonding electron would be flipped by a relatively weak magnetic field before the two ionic\nmagnetic moments are reversed by a stronger field [Fig. 4(a)]. This is because the reversal\nof the ionic magnetic moments requires overcoming the anisotropy barrier, but the electron\nspin is stabilized solely by the exchange interaction. Consequentially, the pre-spin-flip of the\nbonding electron would facilitate the overall magnetization reversal, since the ferromagnetic\nexchange implies that the flipped electron spin counters part of the anisotropy barrier. When\nthe exchange is strong enough, the pre-spin-flip is suppressed, and the magnetic moments are\nreversed in whole. Fig. 4(b) presents coercivity by the classical approach, which manifests\nthe transition point between these two phases.\nInterestingly, this classical picture has quantum correspondence in terms of magnetic\n11relaxation. As shown in Fig. 4(c) and Fig. 4(d), the key lies in the ground state built on\n|Sz=−2S−1/2⟩(the first from the left that has been lifted by the magnetic field), and\nthe excited state mainly composed of |Sz=−2S+ 1/2⟩(the second from the left). Here,\nSzis the total zcomponent, Sz=S1z+sz+S1z. Inspection of the wavefunctions indicates\nthat the difference in Szdoes arise from spin flip of the bonding electron [52]. When the\nexchange is weak, the energy gap between these two states is small, magnetic field could\nefficiently reduce the gap and even make the energy of |Sz=−2S−1/2⟩higher [Fig. 4(c)].\nIn this case, our simulations [52] shows that the system fleets from |Sz=−2S−1/2⟩to\n|Sz=−2S+ 1/2⟩in the beginning stage of the relaxation [Fig. 4(c)], representing flip of\nthe electron spin. When the exchange becomes stronger, |Sz=−2S+ 1/2⟩is less accessible\nfrom|Sz=−2S−1/2⟩because of the larger energy gap. Then, the relaxation does not\nundergo this intermediate state [Fig. 4(d)].\nConclusions and outlook. We have showed that the magnetic field can substantially mod-\nulate the magnetic relaxation of SMMs. Because of the prominent role of optical phonons,\nactivation and suppression of related relaxation pathways due to energy match and mis-\nmatch should be one of our focuses for understanding and tuning the coercivity. Especially,\nin SSMs with slow magnetic relaxation, the tuning on of direct process mediated by optical\nphonons is the reason for the accelerated relaxation that determines the measured coerciv-\nity. The level-crossing-induced relaxation rate skyrocketing is theoretically sound and can be\nobserved with proper experimental setup. While it can be easily obscured in measurement\nwith scanning fields and does not actually concern the coercivity of SMM with strong zero\nfield splitting, it offers a means of ultra-fast magnetization reversal of SMMs.\nA classical view [1, 67] of demagnetization at macroscopic scale is coherent evolution of\nmagnetic moments where each needs to overcome the magnetic anisotropy barrier. A core\nlesson from our results is that spin-phonon relaxation involves shortcut channels that can\nbe significantly enhanced by magnetic fields. 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Lunghi, Toward exact predictions of spin-phonon relaxation times: An ab initio im-\nplementation of open quantum systems theory, Science Advances 8, eabn7880 (2022),\nhttps://www.science.org/doi/pdf/10.1126/sciadv.abn7880.\n[73] D. Reta, J. G. C. Kragskow, and N. F. Chilton, Ab initio prediction of high-temperature mag-\nnetic relaxation rates in single-molecule magnets, Journal of the American Chemical Society\n143, 5943 (2021), pMID: 33822599, https://doi.org/10.1021/jacs.1c01410.\n[74] J. G. C. Kragskow, A. Mattioni, J. K. Staab, D. Reta, J. M. Skelton, and N. F. Chilton,\nSpin–phonon coupling and magnetic relaxation in single-molecule magnets, Chem. Soc. Rev.\n52, 4567 (2023).\n[75] D. Gatteschi, R. Sessoli, and J. Villain, Molecular Nanomagnets , Mesoscopic Physics and\nNanotechnology (Oxford University Press, Oxford, 2011).\n[76] L. Landau and E. Lifˇ sic, Quantum Mechanics: Non-Relativistic Theory , 3rd ed., Course of\ntheoretical physics (Elsevier Science, Oxford, 1991).\n[77] A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions , Oxford\n19Classic Texts in the Physical Sciences (Oxford University Press, Oxford, 2012).\n[78] L. F. Chibotaru and L. Ungur, Ab initio calculation of anisotropic magnetic properties of\ncomplexes. I. Unique definition of pseudospin Hamiltonians and their derivation, The Journal\nof Chemical Physics 137, 064112 (2012).\n20"    },    {        "title": "2307.01907v1.Phonon_induced_magnetization_dynamics_in_Co_doped_iron_garnets.pdf",        "content": "1 \n Phonon -induced magnetization dynamics in Co-doped iron garnet s \n \nA. Frej1,a), C.S. Davies2, A. Kirilyuk2, and A. Stupakiewicz1 \n \n1Faculty of Physics, University of Bialystok, Ciolkowskiego 1L, 15 -245 Bialystok, Poland  \n2FELIX Laboratory, Radboud University, Toernooiveld 7, 6525 ED Nijmegen, The Netherlands  \n \nAbstract.  The developing field of strain -induced magnetization dynamics  offers  a promising  \npath to ward  efficiently  controlling  spins and phase transitions . Understanding the underlying \nmechanisms is crucial in finding the optimal parameters supporting the phonon ic switching of \nmagnetization . Here, we present an experimental and numerical study  of time -resolved \nmagnetization dynamics  driven  by the resonant exci tation of  an optical  phonon mode in iron \ngarnets . Upon pumping the latter with  an infrared pulse obtained from  a free-electron laser , we \nobserve  spatially -varying magnetization precession , with its phase depending on the direction \nof an external magnetic fiel d. Our micromagnetic simulations  effectively  describe the  \nmagnetization precession and switching  in terms of laser -induced changes in the crystal’s  \nmagneto -elastic energy.  \n \nThe excitation of the crystal lattice  with laser irradiatio n in the infrared spectral range  open s \na plethora of possibilities for  achieving  control over magnetic order. The resonant excitation of  \nhigh-frequency optical  phonon modes  can provide a direct and selective  ability to manipulate  \nmagneti zation . It has been  shown , for example, that such driving  of phonons can result in an \neffective magnetic field and the creation of a magnon .1 The wavelength dependence of  laser -\ninduced demagnetization of an iron garnet by intense THz pulses proves the importance of \nfrequency -matching.2 Similarly, ultrafast coherent magnetic phase transitions in DyFeO 3 can \nbe driven by the resonant excitation of phonons.3 Laser -induced strain can briefly change the \ncrystal structure and thus modify  the magnetization ,4,5 as exemplified  by the recently -reported  \npossibility of  using mid -infrared optical pulses to  manipulate  the domain structure of the \nantiferromagnet nickel oxide .6 It is also possible for spin waves to hybridize with acoustic \n                                                 \na) Author to whom correspondence should be addressed:  a.frej@uwb.edu.pl  2 \n waves, forming magnon -polarons.7,8 Constantly emerging new materials, which allow one  to \ncontrol spins and magnetic ordering through the activation of the crystal ’s lattice vibrations, \nhas prompted  intensive research to  focus on thoroughly  studying  light-induced phonon -\nmagnon interactions.  \nRecently , we demonstrate d the resonant ultrafast phononic switching of magnetization in \nCo-doped yttrium  iron garnet (YIG:Co) using pulses of light within  the spectral range  20-\n40 THz.9 The r ich spectrum of  phonon  modes characteristic of  yttrium iron garnets  in this \nfrequency range10,11 offers a broad platform for such investigation s. Studies  of the excitation \nand decay times of these  phonon modes provide valuable guidance in identifying which phonon \nmodes could effectively support  the manipulation  of magnetization .12 While the phononic \nswitching of magnetization clearly depends on the use of optical pulses with frequencies \nmatching the characteristic frequencies of longitudinal  optical phonons, the resulting  switching \nwas only measured  in Ref.  9 across equilibrium time scales, using magneto -optical microscopy \nto record the final state of the addressed magnetization.  Tracking the magnetization precession \nactually involved in the switching process would provide  insight into  the phonon -magnetic \nreversal mechanisms  and possibly reveal the speed of the switching process.  \nIn this Letter , we  present experimental measurements and micromagnetic numerical \nsimulations study ing time-resolved magnetization dynamics driven by the excitation of optical  \nphonon s. Our pump -probe measurements , utilizing single terahertz  optical pulses delivered by \na free -electron laser,  allow us to experimentally track  the magnetization precession within an \niron garnet thin film.  Complimentary micromagnetic simulations are used to predict and \nunder stand the observed effects. Our results directly reveal the ultrafast magnetization \ndynamics that can be triggered by driving optical phonons at resonance.  \nThe experimental studies of the phonon -induced magnetization dynamics  were performed  \nusing  a 7.5-μm-thick iron garnet thin film  with composition Y 2CaFe 3.9Co0.1GeO 12. The sample 3 \n was grown on a gadolinium gallium garnet (GGG) substrate . At room temperature, the \nconstants of cubic  and uniaxial anisotropy are  K1 ≈ -1 kJ/m3 and K u ≈ -0.1 kJ/m3, \ncorresponding to anisotropy fields of about 290  mT and 28  mT, respectively.  The four ea sy \nmagnetization axes  are close to <111> -type orientations .13,14 The saturation magnetization of \nYIG:Co at room temperature was 7 kA/m, and the Gilbert damping coefficient was α  ≈ 0.2.15 \nTo conduct  micromagnetic simulations , we used the Object -Oriented Micro Magnetic \nFramework (OOMMF)  software .16 The sample volume was (120  × 120 × 1) μm3 with the size \nof a single cell set to (1  × 1 × 1) μm3. The uniaxial magnetic anisotropy field parameter was \nartificially increased to Ku = -600 kJ/m3 in order to keep the dynamics of the magnetization \nfixed within the sample’s plane. This simplification effectively confined the problem to two \ndimensions , with two easy axes  along [110] or [ -1-10]. Additionally, t he strong damping \nobserved in the system posed a challenge during our simulation. With ultrashort excitation \npulses (~1  ps), the simulation requires intolerably short time steps while producing oscillati ons \nthat decay too rapidly. It is important to note that in the experiment, the level of damping \nexperienced by the magnetization depends on the cone angle of precession, but such time -\nvarying damping cannot yet be incorporated in micromagnetic simulations . To therefore \novercome these difficulties, we used a realistic damping constant set to 0.2 but increased the \npulse duration to 60  ps. In turn, this led to appropriately scaled properties of the considered \nsystem (saturation magnetization parameter MS = 20.4×105 A/m and the cubic \nmagnetocrystalline anisotropy field strength K1 = 500 kJ/m3). This substantial ly-stronger  \nsaturation magnetization allowed us to maintain the properties of our garnet in the simulation \ndespite the modification of the anisotropy c onstants. Moreover, we note that in simulation \neffective anisotropy cubic and uniaxial fields via the relation H = 2K/M S are about 490  mT and \n588 mT, respectively . The alteration of one parameter , therefore,  necessitates modification of  \nthe other to obtain similar material properties in terms of acting magnetic fields. The las er-4 \n induced strain was introduced with the extension developed by Y ahagi et al.,17 described in \nterms of  a magneto -elastic field pulse  with details given in Ref. 9. In short, we assume that the \nanharmonic interaction of the phonon modes shifts the in -plane coordinates with an amplitude \nproportional to the Gaussian profile of the pumping laser. This results in a macroscopic \ndistribution of strains that is equivalent to  the scenario of a non -uniformly heated object (e.g. a \ncylinder) with an axially -symmetric distribution of temperature. This gives rise to a magneto -\nelastic field containing spatial derivatives of the strain.9 \nFigure 1 presents the  spatially -resolved  results of the micromagnetic  calculations,  \nsimulati ng the magnetization reversal driven by a s train pulse . The top panel contains  images  \nof the calculated magnetic pattern s (see Fig. 1a,b) showing the evolution of magnetization  at \ndifferent times Δt after the arrival of the strain pulse . To provide a point of comparison, the \nmagneto -optical image in Fig. 1c  shows  the quadrupol ar magnetic pattern  that one obtains after  \nexperimentally  exposing YIG :Co to a laser pulse of wavelength  14 μm.9 At Δt = 0 – defined as \nthe time at which  the intensity of the  strain  pulse is maximum – the magnetization starts to \nprecess  in two opposite  directions  (black and white areas  with the in-plane component of \nmagnetization along [ -110] and [1-10] directions in YIG:Co ). After precession, at Δ t > 1 ns, \nthe magnetic domains are formed and are stable  due to  the non-zero coercive field of YIG:Co . \nThe formation of two contrasting magnetic domain pairs  is a result of laser -induced strain and  \nthe initial magnetization state.9 The color ed lines in the images are extracted from  the cross -\nsections of the magnetization indicat ed in Fig. 1d. The solid black line shows the beginning of \nthe domain -formation process. One can observe  that the magnetization at the center of the \nirradiated area  remains unchanged, while  the spatial division of the magnetic pattern can \nalready be discerned . Integrat ion of  half of the solid black line  results in  obtain ing half of the \nGaussian distribution (not shown here). The strongest strain appear s in the part of the Gaussian \nbeam’s profile with the highest gradient, i.e., the steepe st slope on its side. The r ed and blue 5 \n lines  in Fig.  1d are profiles of the switched magnetic domains obtained from simulation and \nexperiment, respectively. The maxima of the black line fit very well within the  experimentally -\nobserved  switched region s. In the middle of the experimental image (Fig. 1c) and its profile in \nFig. 1d  (blue line with circles ), the demagnetization pattern can be seen , caused by  the locally -\nexces sive laser fluence. However, due to the low strain across the central part of the Gaussian \nbeam, a uniform magnetic domain is not created  here. The green line  in Fig.  1d is the profile \nobtained at Δt = 0 but with the strength of the  magneto -elastic field just b elow the switching \nthreshold. A lower intensity results in the strain not being strong enough to create a domain.  \nThe signal ’s amplitude strongly depends on spatial  localization , and the quadrupole symmetry \nis also visible. As can be seen again, the peaks , corresponding to the region with the strong est \nstrain, are  confined to the area  within the switched magnetic domains.  \n \nFIG. 1. Simulation of the switching regime. (a)-(b) I mages  of the calculated distribution of \nmagnetization taken during and after  the strain -induced switching;  (c) Magneto -optical image  of the \nmagnetization in YIG :Co taken  after excitation by a laser pulse of wavelength  14 µm. (d) Normalized \nprofile of magnetization taken  from  the cross -sections marked in the  images  shown  in panel s (a)-(c). \nThe r ed and blue profile s correspond to the s witched domains  from the simulation and experiment , \nrespectively. The b lack line is a profile  for switching power  at the time of strain  pulse maximum  \nintensity  Δt = 0. The green line was obtained at the same time Δt = 0 but with a magneto -elastic field \n6 \n with strength  below the switching threshold . The dashed black line marks the switching threshold.  The \narrows in (b) mark the magnetization directions.  \n \nIn Fig. 2, we present the  calculated results showing how the magnetization reversal depends \non the  amplitude of the magneto -elastic field  pulse . Fig. 2a  shows images of the magnetic \npattern for different magneto -elastic field values (Hme1–Hme6). The red and blue colors \ncorrespond to magnetic domains with opposite magnetization, as marked with arrows ( see also \nin Fig. 1b).  The dashed  line in the graph  below  marks the threshold level  below which  magnetic \nswitching is not achieved . The size of the switched magnetic domains increases linearly with \nthe magneto -elastic field  (red line).9 In the simulations, we do not observe any  saturation  of the \nswitched area , i.e., the domain grow s continuously in size  with increasing field value (see Fig. \n2b). In the experiment,  in contrast,  a continu ed increase of the laser fluence result s in \nirrevocable damag e to the sample  due to heating .9 \n \nFIG. 2. (a) Calculated spatial distribution of the switched magnetization with increasing magneto -elastic \nfield value  (Hme1–Hme6 correspond to 485, 520, 557, 595, 635, and 674 mT, respectively) and (b) the \n7 \n magneto -elastic field dependence of the normalized switched a rea. The t hreshold power  is marked with \nthe black dashed line , while t he red line is a linear fit. The normalized switched area is calculated as th e \nratio of the single switched domain area to the area of the field distribution . The arrows mark the \nmagnetization directions.  \n \nAfter analy zing the  switched magnetic pattern  across equilibrium timescales , we proceed \nto study how strain affects the  magnetization dynamics  across sub -nanosecond timescales . In \nFig. 3a, we present the stabilized distribution of magnetization following strain -induced \nexcitation . The right  and left panels were obtained with the magnetization initially uniformly \noriented a long the  [110] and [ -1-10] direction s (marked +M and –M, respectively) . Such a \nconfiguration  resembles  the visualization of  a monodomain magnetic state in YIG:Co, \nachieved  by applying an external magnetic field . We analyzed the calculations for three probe  \nspot positions relative to the switched magnetization area for  Hme > Hme1 (Fig. 3a). The \ncalculated  precession  of magnetization at position “1”, obtained for  the opposite initial \nmagnetization orientation  with low magneto -elastic field value  < Hme1 (i.e., below the \nswitching threshold) , is shown in Fig . 3b. Depending on the initial state, the magnetization \nvector starts to precess with different  phase . Similarly, we observed a change of phase \nprecession when  probing at position “3” (not shown here), whereas we observe  no change in \nmagnetization  at position  “2”. The spatial separation  between  the two opposite ly-oriented \nmagnetic  domains ( at positions “1” and “3”) results in the simultaneous emergence of \nprecession  with opposite phase s that exactly  compensate for each other. When  pumping with \nmagneto -elastic field amplitude  greater than the previously -identified threshold > Hme1 (see \nFig. 2b), we observe  a permanent switching  of the M [110] component at probe positions “1” and \n“3” (blue and red line s in Fig. 3c, respectively ). The precession after 20  ps is significantly \ndamped  since it is aligned mostly along the adjacent  axis of anisotropy  (see inset).  There is no \nswitching at position “2” due to the relation between the directions of the strain and the initial \nmagnetization vector.  8 \n  \nFIG. 3. Micromagnetic s imulation of strain -induced magnetization dynamics . (a) The magnetization \nstate after magneto -elastic field  pulse for opposite initial states M[110]/M[-1-10] (+M/–M) with different \nlocations of the probe spot. (b) Precession of  the magnetization component M[-110] for the two different \nstarting magnetic states as indicated . (c) Magnetization switching for different  spatial localization of  \nthe probing (positions “1”-“3”) with initial magnetization +M. \n \nComplimentary e xperimental measurement s of the phonon -induced magnetization \ndynamics  were obtained using  the setup presented in Fig . 4a. The magnetization dynamics w ere \nmeasured using  an ultrafast two-color pump -probe technique  at the FELIX facility in the \nNetherlands .18 In order to e xcite at resonance  a longitudinal optical  phonon mode in YIG:Co , \nwe used  a single 1-ps-long optical  pulse with central  wavelength λ  = 14 μm delivered by the \n9 \n pulse -sliced free -electron laser .19 The fluence of this pump pulse  was set below the threshold  \nrequired to achieve perm anent switching of magnetization .9 The probe was provided by a \nsynchronized fiber -based oscillator (Orange High Power, Menlo), which delivers 150 -fs-long \noptical pulses with a central wavelength  of 520 nm. The linearly -polarized probe is delayed in \ntime with respect to the pump by our use of a retroreflector mounted on a motorized translation \nstage. Upon transmitting through the YIG :Co sample, the probe’s polarization is rotated due to \nthe magneto -optical  Faraday effect , providing sensitivity to the out-of-plane component of \nmagnetization  MZ at the focused probe’s position.  This polarization rotation is  measured  using \na polarizing beam -splitter and  two balanced photodetect ors. By chang ing the spatial position  \nof the  tightly -focused  probe pulse  in relation to th at of the  pump  pulse , we are able  to measure  \nmagnetization dynamics across  different part s of the quadrupolar domain  pattern.  The position \nof the probe pulse relative  to the quadrupolar magnetic domains was obtained using in-situ \nstatic magneto -optical microscopy (similar to that used in Ref.  9). Our m easurements were \nperformed in the presence of a constant  external magnetic field  applied at a direction tilted  \nabout 18 away from  the sample’s  [110] crystal axis , with amplitude H = ±15 mT. \nFigure 4b presents  the magnetization dynamics  obtained for two opposite field directions \nat the probe ’s position “1” (analogous to that shown  in Fig. 3a).  Depending on the initial  in-\nplane direction  of magnetization , the opposite phase of the magnetization precession  is \nobserved. Similar  behavior is seen for simulation s with different initial magnetization \ndirection s (see Fig. 3b) , which confirms the magnetic nature of the observed signal.  The step -\nlike precession observed experimentally  across the first  sixty picoseconds  may be as signed to \nthe strain -induced effect of magneto -elastic energy modification , while  the induced magnetic \nanisotropy has a characteristic rise -time of ~20 -30 ps.20 We fitted the experimental data with a \ndamped sine function (solid lines) ,15 omitting the initial step , showing  that the magnetization \nprecession has a frequency of 2.34 GHz . This frequency is similar for laser -induced photo -10 \n magnetic precession in YIG:Co  with the laser pump wavelength of 1.3 μm.21 The period of the \nmagnetization precession  observed experimentally  is much longer than that seen in the  \nmicromagnetic  simulations. However,  we must emphasize that  the numerical calculations were \nperformed for the in-plane orientation of magnetization due to the strong “easy -plane ” type of  \nanisotropy . The period of magnetization precession can be tuned by modifying the values of \nthe magnetization saturation MS and anisotropy constants K1 and Ku. It is important to note that \nsuch an operation influences also the switched magnetization pattern , necessitating  proper \nbalancing of the se material  parameters  as well as the strength and duration  of the  pulsed  \nmagneto -elastic field. We also studied experimentall y the dynamics for  three different \npositions of the probe beam (according to Fig. 3a) , with the  results  shown  in Figure 4c. \nDepending on the position, the signal is positive , negative , or in-between, but the phase remains \nunchanged  throughout . Our detectio n of the out -of-plane component of magnetization does not \nprovide information about the direction of the precession  along [110] and [1 -10] axes of \nYIG:Co.  Due to that, the phase does not differ for the different probe locations.  However, the \nmeasurements o btained at p ositions “1” and “3” also contain the step -like offset  with opposite  \nphase s. This result  suggest s a different direction of  the magneto -elastic anisotropy term.9 11 \n  \nFIG. 4. The time-resolved Faraday rotation with a single infrared  pump pulse of wavelength  14 μm. \n(a) Scheme of the experimental setup. (b) Magnetization precession  measured with  opposite \norientation s of the external magnetic field  H and (c) precession measured by focusing the  probe pulse \nat three different position s as indicated  (see Fig.  3a). The s olid color ed lines in  panels  (b) and (c) are \ndamped sine function fittings.  \n \nIn conclusion, we  have used micromagnetic simulations and experimental pump -probe \nmeasurements to study  magnetization dynamics driven by strongly -coupled optical phonons \nthat are excited at resonance. This excitation triggers a displacement of the equilibrium atomic \npositions,  which is equivalent to macroscopic  crystallographic strains. The obtained result s \nreveal  a strong dependence  of the magnetization pr ecession  on both the probe’s position and \n12 \n the external magnetic field. The initial step-like offset confirms th at the precession mechanism \narises from the  strain -induced magnetic anisotropy  through magneto -elastic energy term . The \nresults  presented he re reveal the dynamical process of phononic manipulation of \nmagnetization, suggesting that the process of phononic switching takes place across a timescale \nshorter than 100  ps. \nWork  was supported by the Foundation for Polish Science (POIR.04.04. 00-00-413C/17 -00). \nThe authors thank all technical staff at FELIX for providing technical support.  \nThe authors have no conflicts to disclose.  \nDATA AVAILABILITY . 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B, 81, 214440 (2010).  \n21A. Frej, I. Razdolski, A. Maziewski and A. Stupakiewicz, \"Nonlinear subswitching regime \nof magnetization dy namics in photomagnetic garnets \", Phys. Rev. B, 107, 134405 (2023).  "    },    {        "title": "1804.10102v1.Orbital_quantum_magnetism_in_spin_dynamics_of_strongly_interacting_magnetic_lanthanide_atoms.pdf",        "content": "Orbital quantum magnetism in spin dynamics of strongly interacting magnetic\nlanthanide atoms\nMing Li,1Eite Tiesinga,2and Svetlana Kotochigova1\n1Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA\n2Joint Quantum Institute and Center for Quantum Information and Computer Science,\nNational Institute of Standards and Technology and the University of Maryland, Maryland 20899, USA\nLaser cooled lanthanide atoms are ideal candidates with which to study strong and unconventional\nquantum magnetism with exotic phases. Here, we use state-of-the-art closed-coupling simulations\nto model quantum magnetism for pairs of ultracold spin-6 erbium lanthanide atoms placed in a\ndeep optical lattice. In contrast to the widely used single-channel Hubbard model description of\natoms and molecules in an optical lattice, we focus on the single-site multi-channel spin evolution\ndue to spin-dependent contact, anisotropic van der Waals, and dipolar forces. This has allowed us\nto identify the leading mechanism, orbital anisotropy , that governs molecular spin dynamics among\nerbium atoms. The large magnetic moment and combined orbital angular momentum of the 4f-\nshell electrons are responsible for these strong anisotropic interactions and unconventional quantum\nmagnetism. Multi-channel simulations of magnetic Cr atoms under similar trapping conditions\nshow that their spin-evolution is controlled by spin-dependent contact interactions that are distinct\nin nature from the orbital anisotropy in Er. The role of an external magnetic field and the aspect\nratio of the lattice site on spin dynamics is also investigated.\nPACS numbers: 31.10.+z, 34.50.-s, 03.65.Nk, 03.75.Mn, 37.10.Jk\nMagnetic moments of atoms and molecules originate\nfrom their electrons intrinsic spin as well as their or-\nbital angular momentum. In solids the orbital compo-\nnent of the magnetic dipole moment revolutionized spin-\ntronics research and led to a novel branch of electron-\nics, orbitronics [1]. The orbital magnetic moment is\nlarge in materials containing partially filled inner shell\natoms. The distinquishing feature of such atoms is the\nextremely large anisotropy in their interactions. This or-\nbital anisotropy is the crucial element in various scientific\napplications and magnetic technology [2–4].\nInspired by the role of orbital anisotropy in magnetic\nsolids and to deepen our knowledge of quantum mag-\nnetism, we study orbital anisotropy at the elementary\nlevel by capturing the behavior of strongly-interacting\nmagnetic lanthanide atoms in an optical lattice site. In\nparticular, we simulate the time-dependent multi-channel\nspin-exchange dynamics of pairs of interacting erbium\natoms in sites of a deep three-dimensional optical lattice\nwith negligible atomic tunneling between lattice sites.\nTheoretical spin models have a long established role as\nuseful tools for understanding interactions in magnetic\nmaterials. Recently, researchers demonstrated that laser-\ncooled ultracold atoms and molecules in optical lattices\nare a nearly perfect realization of these models with con-\ntrol of the local spin state and spin-coupling strength [5–\n15]. However, most of the microscopic implementation\nof spin models has focussed on atoms with zero orbital\nangular momentum and often based on a single-state\n(Fermi- or Bose-) Hubbard model description [16, 17].\nThe limitation of simplified Hubbard models essen-\ntially holds true for optical lattices filled with magnetic\nlanthanide atoms that have open electronic 4f-shells and\npossess a large unquenched orbital angular momentum.\nThese atoms are now emerging as a promising platformfor the investigation of high-spin quantum magnetism.\nTheoretical simulations of ultracold lanthanide atoms in\nan optical lattice is a difficult task that requires multi-\nchannel analyses of interactions and correlations.\nAlthough quantum many-body effects have been ob-\nserved and studied with quantum gases of lanthanide\natoms [15, 18–21], the on-lattice-site spin dynamics of\nhighly magnetic atoms remains not fully understood as\nit requires a precise knowledge of the two-body interac-\ntions.\nWe, first, focus on single-site multi-channel spin evo-\nlution due to spin-dependent contact, anisotropic van-\nder-Waals, and dipolar forces. Then, we go beyond the\none-site model in order to account for coupling between\nour atom pair and other (doubly-)occupied lattice sites\nand find a slow damping of the local spin oscillations. In\nour Mott insulator regime, the single-site quasi-molecule\nis surrounded by atom pairs in neighboring sites. This\nonly permits the magnetic dipole-dipole coupling be-\ntween the atom pairs justified by studies [8, 22] show-\ning that for short-range interactions between atoms in\nneighboring lattice sites is orders of magnitude weaker\nthan the dipole-dipole interaction.\nDipole-dipole interaction induced quantum magnetism\nwas pioneered in experiments of Laburthe-Tolra’s group\n[23]. The authors used magnetic chromium atoms with\ntheir three units of angular momenta and demonstrated\nthe most-dramatic spin oscillations observed to date.\nThese oscillations resulted from the strong coupling be-\ntween the spins, both from isotropic spin-dependent\non-site contact interactions and between-site magnetic\ndipole-dipole interactions.\nWe have shown here that the orbital-induced molecu-\nlar anisotropy, absent in alkali-metal and chromium col-\nlisions, is much stronger in interactions of lanthanidearXiv:1804.10102v1  [physics.atom-ph]  26 Apr 20182\natoms than that of the magnetic dipole-dipole interaction\nat interatomic separations smaller than 200 a0, wherea0\nis the Bohr radius. At shorter separations the strength\nof this anisotropy for Dy and Er atoms is about 10% that\nof the spin-independent isotropic interaction.\nFollowing Refs. [24–26], we prepare ground-state spin-6\nerbium atom pairs in the energetically-lowest vibrational\nstate of a lattice site and Zeeman sublevel |ja,ma/angbracketright=\n|6,−6/angbracketright, wherejaandmaare the atomic angular momen-\ntum and its projection along an applied magnetic field,\nrespectively. To initiate spin dynamics we transfer each\natom to Zeeman sublevel |6,−5/angbracketrightwith a short rf pulse.\nWe single out this pair state among the many internal\nhyperfine states, due to its ability to collisionally evolve\ninto the|6,−6/angbracketright+|6,−4/angbracketrightmolecular state. We then use\nthe spin-oscillation frequency of the atomic populations\nto extract the energy splitting between |6,−5/angbracketright+|6,−5/angbracketright\nand|6,−6/angbracketright+|6,−4/angbracketright. This magnetic-field dependent split-\nting is, unlike for alkali-metal and chromium atoms, due\nto strong orbital anisotropy in interactions between the\natoms.\nDetails of the Hamiltonian of an Er atom pair in a lat-\ntice site modeled by a cylindrically-symmetric harmonic\npotential, the computational tools, analysis of the eigen-\npairs, and state preparation by rf pulse can be found\nin the Supplemental Material. For later use we define\nthe “isotropic” trap frequency ωviaω2= (ω2\nz+ 2ω2\nρ)/3,\nwhereωzandωρare the axial and radial harmonic\nfrequencies, respectively. Eigenstates |i,M/angbracketrightwithB-\ndependent energy Ei,Mare uniquely labeled by projec-\ntionMof the total molecular angular momentum and\ninteger index i.\nIn our model, the molecular interactions U, i.e. oper-\nators that depend on separation r, contain a dominant\nangular-momentum-independent potential with strength\nV0(r), an isotropic spin-spin interaction with strength\nVjj(r)∝1/r6that entangles the spins of the atoms,\nand anisotropic terms that entangle spin and orbital\ndegrees of freedom. The latter terms are separated\ninto a magnetic dipole-dipole and orbital contribution\nwith strengths Vdip(r)∝1/r3andVorb(r)∝1/r6, re-\nspectively. These interactions play a similar role as\nspin-orbit couplings induced by synthetic magnetic fields\n[27, 28]. The short- and long-range shapes of of Uare\ndetermined from a combination of experimental mea-\nsurements and ab initio electronic-structure calculations\n[25, 29]. Our results, unless otherwise noted, are based\non anUthat reproduces the experimentally-determined\nscattering length of 137 a0for the168Er|6,−6/angbracketrightZeeman\nlevel near zero magnetic field [26]. Consequences of the\nuncertainties in the potentials will be discussed later on.\nFigures 1a), b), and d) show our predicted |6,−5/angbracketrightpop-\nulation as a function of time after the short rf pulse for\nthree characteristic, but small B. The population is seen\nto oscillate. Those in panels a) and b) are single sinu-\nsoids with a frequency Ω( B) and reflect the fact that only\ntwo|i,M =−10/angbracketrightare populated by the pulse. We find\nthat these states to good approximation are labeled by\n00.510\n0.50\n0.020.04 0.06 0.08 0.1 Time (ms)\n00.500.02 0.04 B (mT)\n00.020.040.060.080.10.12Ω (MHz)0.02\n0.024 B (mT)\n0.70.750.8E/h (MHz)Population of |6,-5〉 a)b)\nc)d)\ne) FIG. 1. Spin population dynamics of168Er in an isotropic\ntrap withω/(2π) = 0.4 MHz. Panels a) and b) show sinu-\nsoidal time traces of the population in |6,−5/angbracketrightforB= 0.1µT\nand 0.03 mT, respectively. Panel c) shows the spin-oscillation\nfrequency as a function of Bwith blocked out field regions\n(gray bands) where the oscillations are not sinusoidal. The\ntwo open circles correspond to the field values shown in pan-\nels a) and b). A complex oscillation pattern for B= 0.0206\nmT, located in one of the banded regions, is shown in panel\nd) . Panel e) shows the corresponding avoided crossing be-\ntween the three populated energy levels. We assume a slow\ndamping rate of γ= 1.2·104s−1. Its origin is discussed in\nthe text.\ndiabatic states|/lscript;j,m/angbracketright/angbracketright=|s; 10,−10/angbracketright/angbracketrightand|s; 12,−10/angbracketright/angbracketright,\nwhere thescorresponds to /lscript= 0,s-wave partial wave\nscattering and /vectorjis the sum of the two atomic angular\nmomenta (mis its projection). Figure 1c) shows that\nthe frequency Ω is a sharply decreasing function of B.\nIn the banded regions of Fig. 1c) avoided crossings be-\ntween three states occur and the spin oscillations have\nmultiple periodicities. An example from near B= 0.02\nmT is shown in panel d) together with a blowup of the\nrelevant populated energy levels in panel e). The third\nstate away from this avoided crossing is characterized\nby|d; 12,−12/angbracketright/angbracketrightwith an energy that decreases with B\nand a large d-wave character. Near the avoided cross-\ning eigenstates are superpositions of the three diabatic\nstates|k/angbracketright/angbracketright=|s; 10,−10/angbracketright/angbracketright,|s; 12,−10/angbracketright/angbracketrightand|d; 12,−12/angbracketright/angbracketright.\nSee Supplemental Material for more details. Neither the\nB-dependence of Ω( B) nor the presence of avoided cross-\nings can be observed in atomic chromium and both are\nsolely the consequence of the anisotropic dispersive inter-\nactions in magnetic lanthanides and form two important\nresults of this article.\nIn an experiment thousands of simultaneous spin-\noscillations occur, one in each site of a D-dimensional op-\ntical lattice. Atom pairs in different lattices sites are cou-\npled by magnetic dipole-dipole interactions. This leads\nto dephasing of the intra-site spin oscillation. Here, we\nestimate the timescale involved. The inter-site dipolar3\nstrength for a typical lattice period δλbetween 250 nm\nand 500 nm is an order of magnitude smaller than the en-\nergy spacings between the local |s;j,mj/angbracketright/angbracketrightand|d;j,mj/angbracketright/angbracketright\nstates. In principle, this dipole-dipole interaction can\nchange di-atomic projection quantum numbers jandmj.\nWe, however, focus on couplings that are superelastic and\nignore exchange processes. That is, transitions that leave\njand the sum of the Zeeman energies unchanged, i.e.\n|s;jpmp/angbracketright/angbracketrightp|s;jqmq/angbracketright/angbracketrightq←→|s;jpmp+ 1/angbracketright/angbracketrightp|s;jqmq−1/angbracketright/angbracketrightq\netc., where the subscripts pandqon the kets indicate dif-\nferent lattice sites. For Nunit cells and focusing on states\n|s;j,mj/angbracketright/angbracketrightpof a single jthis leads to≈(2j+ 1)Nspin\nconfigurations when N/greatermuch2j+ 1 and with nonuniformly-\ndistributed eigenenergies that span an energy interval of\norder 2D×µ0/(4π)×(gµBj)2/δλ3≡∆ accounting for\nnearest-neighbor coupling only. Here, gis the atomic g-\nfactor,µBis the Bohr magneton, and µ0is the magnetic\nconstant. For Er 2, ∆/h=D×320 Hz with j= 12 and\nδλ= 250 nm and Planck constant h. The value ∆ is a\nlower bound for the energy span.\nWe then simulate the intra-site spin-evolution in the\npresence of these super-elastic dipolar processes with\ndi-atoms in other sites with a master equation for the\nreduced density matrix ρkk/prime(t) with up to three di-\natomic basis states |k/angbracketright/angbracketrightand Lindblad operators Lk=√γk|k/angbracketright/angbracketright/angbracketleft/angbracketleftk|[30] that damp coherences but not popula-\ntions at rate γk= 2πηk∆/hwith dimensionless parame-\ntersηkof the order unity. Figures 1a), b), and d) show\nthis dephasing assuming γk=γ= 1.2·104s−1for allk\n(i. e.ηk= 2 andD= 3). Fort→∞ the overlap with\nthe initial state approaches |cj=10|4+|cj=12|4≈0.50 for\npanels a) and b) and to a value that depends on the pre-\ncise mixing of the three diabatic states for panel d). The\ncjare defined in the Supplemental Material.\nThe short-range shapes of the Er interaction potentials\nhave significant uncertainties even when taking into ac-\ncount of the 137 a0scattering length of the |6,−6/angbracketrightstate.\nThis modifies the expected spin-oscillation frequencies.\nWe characterize the distribution of the spin-oscillation\nfrequency by changing the depth of the isotropic and\nspin-independent V0(r) over a small range, such that its\nnumber of s-wave bound states changes by one, while\nkeeping its long-range dispersion coefficient fixed. The\nnominal number of bound states is 72, much larger than\none, and based on quantum-defect theory [31] we can as-\nsume that each depth within this range is equally likely.\nFigure 2a) shows the distribution of oscillation frequen-\ncies Ω =|Ei,M−Ei/prime,M|/hatB= 0.1µT away from\navoided crossings assuming an isotropic harmonic trap.\nHere, the eigenstates are |i,M =−10/angbracketright ≈ |s; 10,−10/angbracketright/angbracketright\nand|i/prime,M=−10/angbracketright≈|s; 12,−10/angbracketright/angbracketright. (Few of the realiza-\ntions show evidence of mixing with other states.) The\ndistribution is broad and peaked at zero frequency. Its\nmean frequency is ≈0.5/planckover2pi1ωand should be compared to\nthe 2 /planckover2pi1ωspacing between the harmonic levels.\nWe have also computed the distribution for Hamilto-\nnians, where one or more parts of Uhave been turned\noff. In particular, Fig. 2b) shows the distribution for the\n0.00.10.20\n0.1 0.2 0.3 0.4 0.5 0.6 Ω\n (MHz)0.00.40.80.00.10.20\n0.1 0.2 0.3 0.4 0.5 0.6 Ω\n (MHz)00.40.8Probability distributiona) Fullc) Only U\njjb) Only Uorbd) Only U\ndipFIG. 2. Probability distributions of the M=−10 spin-\noscillation frequency Ω showing the role of anisotropic inter-\nactions in168Er. Data is for an isotropic trap with ω/2π= 0.4\nMHz andB= 0.1µT. Panels a), b), c), and d) show distribu-\ntions for the full interaction potential U,U→V0(r)+Vorb(/vector r),\nU→V0(r) +Vjj(r), andU→V0(r) +Vdip(/vector r), respectively.\nIn each panel the mean spin-oscillation frequency is indicated\nby a vertical dashed line.\ncase where Uis replaced by V0(r) and the term propor-\ntional toVorb(/vector r), while Figs. 2c) and d) show that for\nU→V0(r) +Vjj(r) andU→V0(r) +Vdip(/vector r), respec-\ntively. The distribution in panel b) is about as broad\nas that in panel a) indicating that the anisotropic dis-\npersion ofUorb(/vector r) is the most important factor in deter-\nmining the distribution for the full U, even though the\nprecise distribution has noticeably changed. In panel b)\nsmaller and larger splittings are now suppressed relative\nto those near the mean. In panel c) with only isotropic\nspin-spin interactions and panel d) with only the mag-\nnetic dipole-dipole interaction coupling spins and orbital\nangular momenta the distributions are highly localized.\nOne of the most-studied ultracold magnetic atom is\nchromium with its magnetic moment of 6 µBand spin\ns= 3 [23, 32–36]. This moment is only slightly smaller\nthan that of Er, ≈7µB. The dipolar parameter /epsilon1dd=\nadd/afor these atoms, however, is very different. Here,\ndipolar length add= (1/3)×2µ(gµBj)2µ0/(4π/planckover2pi12),ais a\nrelevants-wave scattering length, and µis the reduced\nmass. In fact, /epsilon1ddis 0.16 for52Cr [34] and near two\nfor168Er [15] for the|6,−6/angbracketrightZeeman level due to the\nlarge difference in their mass. Another distinction is their\norbital electronic structure. Chromium has a7S3ground\nstate and no orbital anisotropy, whereas erbium has a\n3H6ground state and a large orbital anisotropy.\nQuantum magnetism of pairs of Cr atoms in a lattice\nsite was investigated in Refs. [23, 36]. Cr atoms were\nprepared in the|s,ms/angbracketright=|3,−3/angbracketrightstate and spin dynamics\nwas initiated by transfer into the |3,−2/angbracketrightstate. Then\nthe quasi-molecular state |3,−2/angbracketright+|3,−2/angbracketrightevolves into\n|3,−3/angbracketright+|3,−1/angbracketrightstate and back. The energy difference\nbetween these states is due to molecular interactions.\nThe potential operator Ufor Cr 2only contains\nisotropic interactions except for the magnetic dipole-\ndipole interaction. These isotropic potentials can be4\n00.03 0.06 0.09 Time (ms)\n00.51|3,-2> state0\n0.03 0.06 0.09 Time (ms)\n00.51|6,-5> state78 9 1011 12 R (units of a\n0)-0.8-0.6-0.4-0.202\n4 6 8 10 R (units of a\n0)-15-10-50-0.78\n-0.75-0.72V/(hc) (103 cm-1)b) Er\n2a) Cr2V/(hc) (103 cm-1)c)\nd) \nFIG. 3. Ground-state adiabatic potentials of Cr 2and Er 2\nand a comparison of their spin evolution. Panel a) shows the\nseven2S+1Σ+BO potentials of Cr 2as functions of separation\nrcalculated in Ref. [37]. Panel b) shows the forty-nine gerade\npotentials of Er 2with Ω = 0 ,..., 12. Panels c) and d) show\nthe population evolution of the |3,−2/angbracketrightand|6,−5/angbracketrightstates of\nCr and Er atoms at B= 0.1µT, respectively. The solid and\ndashed curves in panel d) are for the full interaction potential\nand a potential that only includes the isotropic interactions,\nrespectively. Damping is due to dephasing from dipolar in-\nteractions with atoms in neighboring lattice sites. The lattice\ngeometry is the same for both atomic species and as in Fig. 1.\nrepresented as a sum of tensor operators, Uiso(r) =\nV0(r) +V1(r)/vector sa·/vector sb+V2(r)T2(/vector sa,/vector sa)·T2(/vector sb,/vector sb) +···for\natomsaandb, whereV0(r) has an attractive well and\na dispersion potential with C6= 733Eha6\n0forr→∞\n[33] andV1(r) is the exchange potential proportional\ntorγe−κrforr→∞ . The non-negligible V2(r) is the\nstrength of a second spin-dependent contribution. It and\nany other additional operator also decrease exponentially\nwithr. The rank-2 tensor T2(/vector sa,/vector sb) is constructed from\nangular momenta /vector saand/vector sb.\nFor Cr 2the tensor description of Uis equivalent to a\ndescription in terms of Born-Oppenheimer (BO) or adi-\nabatic potentials, labeled2S+1Σ+\ng/uwith/vectorS=/vector sa+/vector sband\nS= 0,1,···,6 (Even and odd spin Scorresponds to ger-\nade(g) and ungerade (u) symmetry, respectively.) They\nwere computed in Ref. [37] and reproduced in Fig. 3a).\nThe complex relationship between the BO potentials in-\ndicates that Vq>0(r) are on the order of Vq/(hc) = 103\ncm−1forr<6a0. Figure 3b) shows the equivalent graph\nfor two Er atoms as obtained by diagonalizing our Uat\neachrwith/vector raligned along the internuclear axis. The\npotentials have depths D/(hc) between 750 cm−1and\n790 cm−1at the equilibrium separation, where the split-\ntings are mainly due to the anisotropic interaction pro-\nportional to Vorb(r). Visually, the adiabatic potentials\nof the two species are very distinct. The crucial physical\ndistinction, however, lies in the origin of the splittings\nbetween the potentials, i.e. isotropic versus anisotropic\ninteractions.\nWe find it convenient to simulate the spin dynamics\nof Cr by replacing the Vq(r) by delta-function or contactpotentials 4 π(/planckover2pi12/2µ)Aqδ(/vector r)∂/∂r forq= 0,1 and 2, with\nfitted lengthsAqsuch that the contact model reproduces\nthe measured scattering lengths of the BO potentials.\nWe usedA0= 60.6a0,A1= 6.73a0andA2=−0.243a0\nleading to the measured scattering lengths of −7(20)a0,\n58(6)a0, and 112(14) a0for theS= 2, 4, and 6 potentials,\nrespectively [33]. (The numbers in parenthesis are the\nquoted uncertainties. The scattering length for the1Σ+\ng\npotential is not known. The Aqwill change once this\nscattering length is determined.)\nFigures 3c) and d) compare the spin oscillations for a\npair of Cr and Er atoms for the same lattice geometry and\nB= 0.1µT, respectively. The eigen energies of a Cr pair\nin an isotropic harmonic trap interacting via the delta\nfunction potentials are found with the help of the non-\nperturbative analytical solutions obtained in Ref. [38] .\nThe spin evolution for Cr is solely due to the isotropic\nspin-spin interactions proportional to lengths A1andA2\nand are independent of B. Two time traces for Er are\nshown corresponding to the full Uand one where only\nthe isotropic interactions are included. We see that the\noscillation period is slower when the anisotropic interac-\ntions are excluded and the behavior is much more alike\nto that of a Cr pair. The curves in panels c) and d)\nalso include an estimate of dephasing due to atom pairs\nin neighboring lattice sites again using a lattice spacing\nof 250 nm. Dephasing of a chromium-pair is only 25%\nsmaller than that of an erbium-pair due to the slightly-\nsmaller magnetic moment of Cr.\nWe have presented simulations that give valuable in-\nsight into the interactions of lanthanide-based quantum\nsystems with periodic arrays or lattices containing atom\npairs in each lattice site. As our atoms carry a spin that\nis much larger than that of spin-1/2 electrons in mag-\nnetic solid-state materials, this might result in a wealth\nof novel quantum magnetic phases. The ultimate goal\nof any Atomic, Molecular and Optical implementation\nof quantum magnetism is to develop a controllable en-\nvironment in which to simulate materials with new and\nadvanced functionality.\nIn spite of the successes of previous analyses of quan-\ntum simulations with (magnetic) atoms in optical lattices\n[39], simplified representations with atoms as point parti-\ncles and point dipoles can not always be applied to mag-\nnetic lanthanide atoms. Important information about the\nelectron orbital structure within the constituent atoms is\nlost.\nIn the current study the orbital anisotropy of magnetic-\nlanthanide electron configurations is properly treated in\nthe interactions between Er atoms in optical lattice sites\nand is used to describe and predict spin-oscillation dy-\nnamics. We illuminated the role of orbital anisotropy in\nthis spin dynamics as well as studied the interplay be-\ntween the molecular anisotropy and the geometry of the\nlattice site potential. The interactions lift the energy-\ndegeneracies of different spin orientations, which, in turn,\nleads to spin oscillations.5\nI. ACKNOWLEDGMENTS\nWork at Temple University is supported by the\nAFOSR Grant No. FA9550-14-1-0321, the ARO-MURI Grants No. W911NF-14-1-0378, ARO Grant No.\nW911NF-17-1-0563, and the NSF Grant No. PHY-\n1619788. The work at the JQI is supported by NSF\nGrant No. PHY-1506343.\n[1] D. Go, J.-P. Hanke, P. M. Buhl, F. Freimuth,\nG. 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Rev. 112, 5012 (2012).Orbital Quantum Magnetism in Spin Dynamics of Strongly Interacting Magnetic\nLanthanide Atoms: Supplemental Material\nMing Li,1Eite Tiesinga,2and Svetlana Kotochigova1\n1Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA\n2Joint Quantum Institute and Center for Quantum Information and Computer Science,\nNational Institute of Standards and Technology and the University of Maryland, Gaithersburg, Maryland 20899, USA\nRelative Motion of two bosonic lanthanide atoms\nA single site of an optical lattice potential is well ap-\nproximated by a cylindrically-symmetric harmonic trap.\nThe Hamiltonian for the relative motion of two ground-\nstate bosonic lanthanide atoms is then Hrel=H0+U,\nwhere\nH0=/vector p2/(2µ) +µ(ω2\nρρ2+ω2\nzz2)/2 +HZ\nwith relative momentum /vector p, relative coordinate /vector r=\n(r,θ,ϕ ) = (ρ,θ,z ) between the atoms in spherical and\ncylindrical coordinates, respectively, and reduced mass\nµ. Moreover, ωρandωzare trapping frequencies and\nHZ=gµB(/vector a+/vector b)·/vectorBis the Zeeman Hamiltonian with\natomic g-factor gand Bohr magneton µB. Zeeman states\n|ja,ma/angbracketright|jb,mb/angbracketrightare eigen states of HZ, where/vector αis the to-\ntal atomic angular momentum of atom α=aorband\nmαis its projection along /vectorB.\nWe have assumed that the harmonic lattice-site po-\ntential is the same for all atomic states and a /vectorBfield\ndirected along the axial or zdirection of the trap. Small\ntensor light shifts, proportional to ( ma)2[?], induced by\nthe lattice lasers, have been omitted. Here and through-\nout, we use dimensionless angular momenta, i.e. /vector //planckover2pi1is\nimplied when we write /vector with /planckover2pi1=h/2πand Planck con-\nstanth. For angular momentum algebra we follow Ref. [ ?\n].\nThe Hamiltonian term Udescribes the molecular in-\nteractions. It contains an isotropic contribution Uiso(r)\nthat only depends on ras well as an anisotropic con-\ntributionUaniso(/vector r) that also depends on the orientation\nof the internuclear axis. In fact, we have Uiso(r) =\nV0(r) +Vjj(r)/vector a·/vector b+···, where for r→∞ the angular-\nmomentum independent V0(r)→−C6/r6with disper-\nsion coefficient C6= 1723Eha6\n0for Er 2[? ? ]. For\nsmallrthe potential V0(r) has a repulsive wall and an\nattractive well with depth De/(hc)≈790 cm−1. We\nuseVjj(r) =−cjj\n6/r6for allrwith dispersion coeffi-\ncientcjj\n6=−0.1718Eha6\n0. Furthermore, Uaniso(/vector r) =\nUorb(/vector r) +Udip(/vector r) with\nUorb(/vector r) =Vorb(r)/summationdisplay\ni=a,b3(ˆr·/vector i)(ˆr·/vector i)−/vector i·/vector i√\n6+···,\nand\nUdip(/vector r) =−µ0(gµB)2\n4π3(ˆr·/vector a)(ˆr·/vector b)−/vector a·/vector b\nr3is the magnetic dipole-dipole interaction. We use\nVorb(r) =−corb\n6/r6for allrwithcorb\n6=−1.904Eha6\n0.\nFinally,Ehis the Hartree energy, cis the speed of light\nin vacuum, and µ0is the magnetic constant.\nThe coefficient C6is the largest dispersion coeffi-\ncient by far, making the van-der-Waals length R6=\n4/radicalbig\n2µC6//planckover2pi12the natural length scale for the dispersive in-\nteractions. The contribution to Uorb(/vector r) with strength\nVorb(r) is the strongest anisotropic orbital interaction.\nIt couples the angular momentum of each atom to the\nrotation of the molecule. For future reference and follow-\ning the convention in the literature the natural length\nscale for the magnetic dipole-dipole interaction is add=\n(1/3)×2µC3//planckover2pi12with coefficient C3=µ0(gµBj)2/(4π),\nwherej= 6 andg= 1.16381 for Er.\nThe Hamiltonian Hrelcommutes with Jzand only cou-\nples even or odd values of /vector/lscript, the relative orbital angular\nmomentum or partial wave. Here, Jzis thezprojection\nof the total angular momentum /vectorJ=/vector/lscript+/vector with/vector =/vector a+/vector b.\nForB= 0Hrelalso commutes with J2. Eigenstates\n|i,M/angbracketrightofHrelwith energy Ei,Mare labeled by projection\nquantum number Mand indexi. These eigen pairs have\nbeen computed in the spin basis\n|(jajb)j/lscript;JM/angbracketright≡/summationdisplay\nmjm/angbracketleftj/lscriptmjm|JM/angbracketright|(jajb)jmj/angbracketrightY/lscriptm(θ,ϕ)\nwith|(jajb)jmj/angbracketright=/summationtext\nmamb/angbracketleftjajbmamb|jmj/angbracketright|jama/angbracketright|jbmb/angbracketright,\nspherical harmonic function Y/lscriptm(θ,ϕ), and\n/angbracketleftj1j2m1m2|jm/angbracketrightare Clebsch-Gordan coefficients. For our\nbosonic system only basis states with even /lscript+jexist. We\nuse a discrete variable representation (DVR) [ ? ? ] to\nrepresent the radial coordinate r. The largest rvalue is\na few times the largest of the harmonic oscillator lengths/radicalbig\n/planckover2pi1/(µωρ,z) and for typical traps R6/lessmuch/radicalbig\n/planckover2pi1/(µωρ,z). We\nfurther characterize eigen states by computing overlap\namplitudes with those at different Bfield values. In\nparticular, overlaps with B= 0 eigenstates give us the\napproximate Jvalue. The expectation value of operators\nj2and/lscript2are also computed.\nEigenstates in an isotropic lattice site\nFigure 1 shows even- /lscript, above-threshold168Er2energy\nlevels in an isotropic harmonic trap and M=−12,−11,\nor−10 as functions of Bup to 0.05 mT. The energies\nof two harmonic oscillator levels are also shown. ThearXiv:1804.10102v1  [physics.atom-ph]  26 Apr 20182\n0.60.81.01.21.40.6\n0.81.01.21.4E/h (MHz)0\n0.01 0.02 0.03 0.04 0.05 B (mT)\n0.60.8M=-10M=-11\nM=-12\n|s; j=12〉〉 |s; j=12〉〉 |s; j=10\n〉〉 |s; j=12\n〉〉 a)b)\nc)\n|d; j=12〉〉 \nFIG. 1. Near-threshold energy levels of the relative motion of\ntwo harmonically trapped and interacting168Er atoms with\nM=−10,−11, and−12 in panel a), b), and c), respectively.\nThe trap is isotropic with ω/(2π) = 0.4 MHz. In the three\npanels the zero of energy corresponds to the Zeeman energy\nof two atoms at rest with ma+mb=−10,−11, and−12,\nrespectively. Dashed red lines correspond to the first and\nsecond harmonic-oscillator levels with energies (3 /2)/planckover2pi1ωand\n(7/2)/planckover2pi1ω, respectively. Blue curves indicate the energy levels\nthat are involved in the spin-changing oscillations. Some of\nthe eigenstates have been labeled by their dominant /lscriptandj\ncontribution. Orange arrows (not to scale) and a red rf pulse\nindicate single-color two-photon rf transitions that initiate the\nspin oscillations starting from two atoms in the |6,−6/angbracketrightstate.\nenergetically-lowest is an /lscript= 0 ors-wave state with en-\nergy (3/2)/planckover2pi1ω; the second is degenerate with one s- and\nmultiple/lscript= 2,d-wave states and energy (7 /2)/planckover2pi1ω. In\neach panel one or two eigenstates with energies that run\nnearly parallel with these oscillator energies exist. In\nfact, their energy, just above (3 /2)/planckover2pi1ω, indicates a repul-\nsive effective atom-atom interaction [ ? ?]. For our weak\nBfields and away from avoided crossings their wavefunc-\ntions are well described by or are correlated to a single\nJ= 10 or 12 zero- Beigenstate. Further, they have an\ns-wave dominated spatial function, j≈J, andmj≈M.\nThese eigenstates will be labeled by |s;jmj/angbracketright/angbracketrightaway from\navoided crossings.\nA single bound state with E < 0 whenB= 0 and a\nnegative magnetic moment can be inferred in each panel\nof Fig. 1. It avoids with several trap states when B >\n0.015 mT and has mixed g- andi-wave character away\nfrom avoided crossings. In free space this bound state\nwould induce a narrow Feshbach resonance near B= 6\nµT (not shown).\nFigure 1 also shows eigenstates with energies close to\nE= (7/2)/planckover2pi1ωwhenB= 0. ForB > 2µT and away\nfrom avoided crossings, these states have d-wave char-\nacter, are well described by a single j,mjpair, and\nhave a magnetic moment, −dEi,M/dB, that is an in-\nteger multiple of gµB.D-wave states with a positive\nmagnetic moment in the figure have avoided crossingswiths-wave states|s;jmj/angbracketright/angbracketrightand play an important role\nin our analysis of spin oscillations. We focus on the three-\nstate avoided crossing in panel a) near B= 0.020 mT,\nwhere the corresponding d-wave state has j= 12 and\nmj=−12 and will be labeled by |d;jmj/angbracketright/angbracketright. Close to the\navoided crossings the three eigenstates of Hrelare super-\npositions of the B-independent|s;jmj/angbracketright/angbracketrightand|d;jmj/angbracketright/angbracketright\nstates. The mixing coefficients follow from the overlap\namplitudes with eigenstates well away from the avoided\ncrossing. In other words |i,M =−10/angbracketright=/summationtext\nkUi,k(B)|k/angbracketright/angbracketright\nwithk={s; 10,−10},{s; 12,−10}, and{d; 12,−12}and\nU(B) is aB-dependent 3×3 unitary matrix.\nFinally, Fig. 1 shows the rf pulse that initiates spin\noscillations. We choose a non-zero Bfield and start in\n|s; 12−12/angbracketright/angbracketrightwithM=−12 in panel c). After the pulse,\nvia near-resonant intermediate states with M=−11, the\natom pair is in a superposition of two or three eigenstates\nwithM=−10. The precise superposition depends on ex-\nperimental details such as carrier frequency, polarization\nand pulse shape. We, however, can use the following\nobservations. The initial |s; 12−12/angbracketright/angbracketrightstate can also be\nexpressed as the uncoupled product state |6,−6/angbracketright|6,−6/angbracketright\nindependent of B. As rf photons only induce transitions\nin atoms (and not couple to /vector/lscriptof the atom pair), the ab-\nsorption of one photon by each atom leads to the product\ns-wave state|6,−5/angbracketright|6,−5/angbracketright, neglecting changes to the spa-\ntial wavefunction of the atoms. To prevent population in\natomic Zeeman levels with ma,b>−5 we follow Ref. [ ?]\nand assume that light shifts induced by optical photons\nbriefly break the resonant condition to such states.\nThes-wave|6,−5/angbracketright|6,−5/angbracketrightstate then evolves under\nthe molecular Hamiltonian. It is therefore conve-\nnient to express this state in terms of M=−10 eigen-\nstates. First, by coupling the atomic spins to /vector we note\n|6,−5/angbracketright|6,−5/angbracketright→c10|s; 10,−10/angbracketright/angbracketright+c12|s; 12,−10/angbracketright/angbracketright, where\ncj=/angbracketleft66−5−5|j−10/angbracketrightand each|k/angbracketright/angbracketrightis a superposition\nof three eigenstates |i,M =−10/angbracketrightas given by the inverse\nofU(B). After free evolution for time twe measure the\npopulation of remaining in the initial state.\nAnisotropic harmonic traps\nThe lattice laser beams along the independent spatial\ndirections do not need to have the same intensity and\nthus the potential for a lattice site can be anisotropic.\nThis gives us a means to extract further information\nabout the anisotropic molecular interaction potentials.\nA first such experiment for Er provided evidence of the\norientational dependence due to the intra-site magnetic\ndipole-dipole interactions on the particle-hole excitation\nfrequency in the doubly-occupied Mott state [ ?].\nFigure 2 shows the near-threshold energy M=−10\nlevels in an anisotropic harmonic trap as a function of\ntrap aspect ratio ωz/ωρat fixedω/(2π) = 0.4 MHz for\ntwo magnetic field strengths. Dashed lines correspond to\nnon-interacting levels, including only the harmonic trap\nand Zeeman energies. For ωz/ωρ→0 and∞the trap3\n0.11 10 0.60.91.21.50.1\n1 10 Aspect ratio, \nω z/ωρa)E/h (MHz)b)\nFIG. 2. Near-threshold energy levels of168Er2withM=−10\nin an anisotropic harmonic trap as a function of trap aspect\nratioωz/ωρat fixedω/(2π) = 0.4 MHz for B= 0.1µT and\n0.05 mT in panels a) and b), respectively. The red dashed\nlines correspond to non-interacting levels. Blue lines are rele-\nvant for spin-oscillation experiments. The zero of energy in a\npanel is that of two free atoms at rest with ma+mb=−10\nat the corresponding Bfield.is quasi-1D and quasi-2D, respectively. Energy levels in-\nvolved in spin-oscillations are highlighted in blue and for\nfiniteBtheir avoided crossings with other states can be\nstudied."    },    {        "title": "2007.14687v1.Impact_of_magnetic_dopants_on_magnetic_and_topological_phases_in_magnetic_topological_insulators.pdf",        "content": "arXiv:2007.14687v1  [cond-mat.str-el]  29 Jul 2020Impact of magnetic dopants on magnetic and topological phas es\nin magnetic topological insulators\nThanh-Mai Thi Tran and Duc-Anh Le\nFaculty of Physics, Hanoi National University of Education , Hanoi, Vietnam\nTuan-Minh Pham, Kim-Thanh Thi Nguyen, and Minh-Tien Tran\nGraduate University of Science and Technology, Vietnam Aca demy of Science and Technology, Hanoi, Vietnam and\nInstitute of Physics, Vietnam Academy of Science and Techno logy, Hanoi, Vietnam\nAtopological insulator dopedwith randommagnetic impurit iesis studied. Thesystemis modelled\nby the Kane-Mele model with a random spin exchange between co nduction electrons and magnetic\ndopants. The dynamical mean field theory for disordered syst ems is used to investigate the electron\ndynamics. The magnetic long-range order and the topologica l invariant are calculated within the\nmeanfield theory. Theyreveal arich phasediagram, where diff erentmagnetic long-range orders such\nas antiferromagnetic or ferromagnetic one can exist in the m etallic or insulating phases, depending\non electron and magnetic impurity fillings. It is found that i nsulator only occurs at electron half\nfilling, quarter filling and when electron filling is equal to m agnetic impurity filling. However,\nnon-trivial topology is observed only in half-filling antif erromagnetic insulator and quarter-filling\nferromagnetic insulator. At electron half filling, the spin Hall conductance is quantized and it\nis robust against magnetic doping, while at electron quarte r filling, magnetic dopants drive the\nferromagnetic topological insulator to ferromagnetic met al. The quantum anomalous Hall effect is\nobserved only at electron quarter filling and dense magnetic doping.\nI. INTRODUCTION\nMagnetic topological quantum materials are a rela-\ntively new type ofmaterials, where topologicallynontriv-\nial electron properties coexist with magnetic ordering1–5.\nThese novel materials include magnetic topological in-\nsulators (MTIs), magnetic Weyl semimetals, magnetic\nDirac semimetals... Experiments observed a remarkable\nquantization of the anomalous Hall conductance in a\nnumber of materials, for instance (Bi,Sb) 2(Se,Te) 3doped\nwith magnetic impurities3,4. In these materials the ori-\ngin of the quantum anomalous Hall (QAH) effect relies\non the spin-orbital coupling (SOC) and magnetism6–11.\nUpon magnetic impurity doping, the spin exchange (SE)\nbetween the conduction electrons and magnetic moments\ninducesaspontaneousmagnetizationatlowtemperature.\nThe macroscopic magnetization could act on conduction\nelectrons as the magnetic field in the anomalous Hall ef-\nfect. The SOC keeps the topologically nontrivial band\nstructure in the magnetic state. Without the topolog-\nically nontrivial band structure, the magnetic impurity\ndoping alone could not cause the QAH effect, for in-\nstance, in the dilute magnetic semiconductors (DMSs),\nthe magnetic dopants also induce a magnetic ordering,\nbut the anomalous Hall conductance is not quantized,\nbecause the band structure of the DMSs is topologically\ntrivial12.\nAlthough the QAH effect emerges as a result of the in-\nterplay between the SOC and magnetism, its occurrence\nalso depends on the filling level of conduction electrons\nand the concentration of magnetic dopants7–11. The\ndopings of conduction electrons and magnetic impuri-\nties can drive both the topological and magnetic phase\ntransitions7–11. In particular, first principle calculationsshowed successivetopologicaland magnetic phase transi-\ntions from quantum spin Hall (QSH) state to QAH state,\nand then to ferromagnetic state, when the concentra-\ntion of magnetic impurities increases9. In these phase\ntransitions magnetic dopants alter the SOC and the SE\nstrengths, and the successive topological and magnetic\nphases are established as a result of the interplay be-\ntween the SOC and the SE only9. Moreover, when mag-\nnetic impurities are doped, disorder and inhomogeneity\nare inevitably introduced. As a consequence, the SE be-\ntween conduction electrons and magnetic impurities is a\nrandom variable. Disorder of magnetic dopants can also\ninduce random deviations of the magnetic moments from\nthe macroscopic magnetization. Therefore, the induced\nmagnetic ordering at low temperature depends on the\nconcentration and the distribution of magnetic dopants.\nIn many materials such as the DMSs or the colossalmag-\nnetoresistant materials, the doping of magnetic impuri-\nties is crucially important in determining the electronic\nand magnetic properties13,14. While the impact of elec-\ntron and magnetic dopings on the magnetic and topolog-\nical properties of the MTIs was experimentally studied,\nit has received less theoretical attention.\nIn this work, we study the impact of magnetic dopants\non the magnetic and topological phases existing in the\nMTIs. We will construct a minimal model for MTIs. It\nshould include at least two terms: the SOC which is re-\nsponsible for the topologically nontrivial band structure\nand the SE between the magnetic dopants and conduc-\ntion electrons that could induce a magnetic ordering at\nlow temperature. The QAH effect emerges as a result of\nthe interplay between the SOC and the SE3,4,11. In con-\ntrast to the first principle calculations, where the SOC\nand the SE are altered by magnetic dopants9, in the min-\nimal model the SOC and the SE are fixed upon magnetic2\ndoping. This allows us to study the direct impact of\nmagnetic dopants on the the magnetic and topological\nproperties. In the previous study, a theoretical model\nfor the MTIs was proposed11. This model is based on a\ncombination of the Kane-Mele model15and the SE be-\ntween conduction electrons and magnetic impurities. It\nalso looks like the double-exchange (DE) model with a\nSOC14,16. The DE model is a minimal model proposed\nfor different magnetic materials such as DMSs12, colossal\nmagnetoresistancematerials14,16. We foundthatthe pro-\nposed model exhibits various magnetic insulating states,\nwhich occur at electron half and quarter (or three quar-\nters) fillings, and they are topological insulator at appro-\npriate values of the SE11. However, the previous studies\nassumed that the magnetic impurities are present at ev-\nery lattice site3,4,11. The doping of magnetic impurities\naway from the full filling and the disorder effect intro-\nduced by magnetic dopants were not previously consid-\nered. In this work, we study the impact of magnetic\ndoping on the magnetic and topological properties, tak-\ning into account disorder and inhomogeneity introduced\nby magnetic dopants. The dynamical mean field the-\nory (DMFT) for disordered systems is used to study the\nproposed model17–21. Originally, the DMFT was intro-\nduced in order to correctly treat local electron correla-\ntions in infinite dimensional systems22. It has widely\nbeen used to study strong electron correlations23. Espe-\ncially, the DMFT has successfully treated the SE in the\nDE-based models24–31. By adopting the DMFT for dis-\norderedsystems, wecalculateboth the spontaneousmag-\nnetization and the topological invariant self-consistently.\nThey reveal rich phase diagrams, depending on electron\nand magnetic dopings. It is found that the insulating\nstate only occurs at electron (hole) half, quarter fillings,\nand at electron filling equaled to the concentration of\nmagnetic dopants. However, the insulating state is topo-\nlogically nontrivial only at electron (hole) half and quar-\nter fillings. At electron half filling, the QSH effect is\nobserved and it is robust against the magnetic impurity\ndoping, while at electron quarter filling, the magnetic\ndoping away from full magnetic filling suppresses the ob-\nserved QAH effect. These findings reveal that magnetic\ndopants impact differently on the topological properties\nof the MTIs depending on electron filling.\nThepresentpaperisorganizedasfollows. InSec. II we\ndescribe the minimum model for MTIs and the DMFT\nfor treating the SE and disorder introduced by magnetic\ndopants. The numerical results are presented in Sec. III.\nFinally, Sec. IV is the conclusion of the present work.\nII. MODEL AND DYNAMICAL MEAN FIELD\nTHEORY\nWe consider a topological insulator doped with mag-\nnetic impurities. For the sake of simplicity, the\ntopological insulator is modelled by the Kane-Mele\nHamiltonian15. The Kane-Mele model consists of anearest-neighbor hopping and an intrinsic SOC. In addi-\ntion, magnetic impurities are randomly distributed over\nthe lattice. They are locally coupled with conduction\nelectronsviaaSE.TheHamiltoniandescribingthemodel\nreads\nH=−t/summationdisplay\n/angbracketlefti,j/angbracketright,σc†\niσcjσ+iλ/summationdisplay\n/angbracketleft/angbracketlefti,j/angbracketright/angbracketright,s,s′νijc†\nisσz\nss′cjs′\n−/summationdisplay\ni,ss′JiSic†\nisσss′cis′, (1)\nwherec†\niσ(ciσ) is the creation (annihilation) operator\nfor electron with spin σat siteiof a honeycomb lat-\ntice./angbracketlefti,j/angbracketrightand/angbracketleft/angbracketlefti,j/angbracketright/angbracketrightdenote the nearest-neighbor and\nnext-nearest-neighbor lattice sites, respectively. tis the\nhopping parameter for the nearest-neighbor sites, and λ\nis the strength of the intrinsic SOC. The sign νij=±1\ndepends on the hopping direction, as shown in Fig. 1. σ\nare the Pauli matrices. The honeycomb lattice is chosen,\nsincetheSOCinthislatticeinducesatopologicalinsulat-\ning state15.Siis spin of magnetic impurity at lattice site\ni. We also treat it classically, as widely used in the stud-\nies of materials doped with magnetic impurities14,24–31.\nIndeed, the magnetic moment of magnetic dopants is of-\nten big, for instance doped Mn ions in topological insu-\nlator Bi 2−xMnxTe3have the magnetic moment ∼4µB32.\nThis consideration excludes any possibility of the Kondo\neffect33–36. In fact, no any signature of the Kondo effect\nwas observedin the MTIs. Jiis the strength of SE at lat-\ntice sitei. We consider only the substitutional doping of\nmagnetic impurities, and avoid any interstitial one. In-\ndeed, first principle calculations show the substitutional\ndoping is energetically more favorable than the intersti-\ntial one37. In contrast to the previous study3,4,11, in this\nstudy magnetic impurities are randomly doped, and the\nSE is valid only on the lattice sites, where magnetic im-\npurities are located. We consider a binary distribution of\nmagnetic dopants\nP(Ji) = (1−x)δ(Ji)+xδ(Ji−J), (2)\nwherexis the concentration of magnetic dopants. Basi-\ncally, only xfraction of lattice sites has the local SE be-\ntween conduction electrons and magnetic dopants. The\nparameter xcan also be interpreted as a disorder mea-\nsurement of magnetic impurities. However, both x= 0\n/c110/c61/c49 /c110/c61/c45/c49A B A B\nFIG. 1: (Color online) The sign structure νijof the SOC term\nin the honeycomb lattice.3\nandx= 1 correspond to the non-disordered cases. When\nthe magnetic impurities are absent ( x= 0), the pro-\nposed model returns to the Kane-Mele model15. The\nSOC causes a band gap at half filling, and the insulating\nstate has an integer spin Chern number15. This yields\nthe QSH effect. In the opposite limit, x= 1, magnetic\nimpurities are present at every lattice site. Hamiltonian\nin Eq. (1) essentially describes the interplay between\nthe SOC and the SE11. It exhibits a coexistence of the\nQSH effect and antiferromagnetism at electron half fill-\ning and a ferromagnetic topological insulator at electron\n(hole) quarterfilling11. Between these two limiting cases,\n0< x <1, magnetic dopants are randomly distributed,\nand they may drive the magnetic and topological phase\ntransitions. For classical impurity spin, the sign of the\nSE is irrelevant. Without loss of generality, we adopt the\nferromagnetic sign J >0.\nWe divide the honeycomb lattice into two penetrating\nsublattices AandB, as shown in Fig. 1. Then we denote\naIσ(bIσ) being the annihilation operator of electron at\nunit cell Ifor the sublattice A(B). We introduce a four-\ndimensional spinor\nΨI=\naI↑\nbI↑\naI↓\nbI↓\n.\nFor a fixed configuration of magnetic impurities, we in-\ntroduce the Green function\nGIJ(iωn,Ji) =−β/integraldisplay\n0dτ e−iωnτ/angbracketleftTΨI(τ)Ψ†\nJ/angbracketright,(3)\nwhereωnis the Matsubara frequency and β= 1/Tis the\ninverse temperature. Magnetic impurity disorder breaks\nthe lattice translation invariance. However, the lattice\ntranslation invariance of the Green function is restored\nwhen Green-function averaging over the magnetic impu-\nrity distribution is made. We obtain the averaged Green\nfunction in the momentum space\nG(k,iωn) =/summationdisplay\nI,Je−ik·(RI−RJ)GIJ(iωn,Ji),\nwhere the bar denotes the average over the magnetic im-\npurity distribution. The averaged Green function obeys\nthe Dyson equation\nG(k,iωn) = [z−H0(k)−Σ(k,iωn)]−1,\nwhereΣ(k,iωn) is the self energy, and H0(k) is the non-\ninteracting and non-disordered Bloch Hamiltonian. The\nBloch Hamiltonian reads\nH0(k) =/parenleftbigg\nh↑(k) 0\n0h↓(k)/parenrightbigg\n, (4)\nwhere\nhσ(k) =/parenleftbigg\nσλξk−tγk\n−tγ∗\nk−σλξk/parenrightbigg\n,andγk=/summationtext\nδeik·rδ,ξk=i/summationtext\nηνηeik·rη. Hereδandη\ndenote the nearest-neighbor and next-nearest-neighbor\nsitesofagivensiteinthehoneycomblattice, respectively.\nThe self energy Σ(k,iωn) includes all effects of interac-\ntion and disorder in an average manner. It renormalizes\nthe dynamics of noninteracting and non-disordered con-\nduction electrons.\nWe calculate the electron Green function by means of\nthe DMFT. Here we will use the arithmetic average ver-\nsion of the DMFT for disordered systems17–21. It basi-\ncally is equivalent to the coherent potential approxima-\ntion (CPA)23. There is also a geometric average version\nof the DMFT that is usually called the typical medium\ntheory17–21. The typical medium theory appropriately\ndescribes the Anderson localization in disordered sys-\ntems. In this work we focus on the effect of magnetic\ndopants on the magnetic and topological phases ofMTIs,\nwhere the Anderson localization is perhaps absent6–10.\nApparently, the Anderson localization is induced by non-\nmagnetic diagonal disorder or by off-diagonal disorder of\nconduction electron hopping38,39. Such disorders are ab-\nsent in the proposed Hamiltonian in Eq. (1). Within\nthe DMFT, the self energy depends only on frequency\nΣ(k,iωn)→Σ(iωn). The DMFT neglects nonlocal cor-\nrelations at finite dimensions. In the honeycomb lat-\ntice, the DMFT overestimates the critical value of the\nsemimetal-insulator transition, but it is still capable to\ndetect the insulating or magnetic states40–43. Due to the\nlocal nature, the DMFT does not mix the different spin\nand sublattice sectorsof the self energy, therefore Σ(iωn)\nis a 4×4 diagonal matrix. The self energy obeys the\nDyson equation\nGaσ(iωn) =Gaσ(iωn)+Gaσ(iωn)Σaσ(iωn)Gaσ(iωn),(5)\nwhereais the sublattice notation ( a=A,B), and\nGaσ(iωn) =/summationtext\nkGaσ(k,iωn)/Nis the local averaged\nGreenfunction ( Nisthe numberofsublattice sites). The\nGreen function Gaσ(iωn) actually represents the effective\ndynamical mean field of conduction electrons. Within\nthe DMFT, the self energy is determined from an effec-\ntive single-site action, where Gaσ(iωn) serves as the bare\nnoninteracting Green function. The action of the effec-\ntive single site of sublattice awith a fixed SE Jais\nSa(Ja) =−/summationdisplay\nsβ/integraldisplay\n0β/integraldisplay\n0dτdτ′Ψ†\nas(τ)G−1\nas(τ−τ′)Ψas(τ′)\n−/summationdisplay\nαss′β/integraldisplay\n0dτJaSα(τ)Ψ†\nas(τ)σα\nss′Ψas′(τ).(6)\nFor classical impurity spin S, this effective single-site ac-\ntion can exactly be solved11. Indeed, it is basically the\naction of an one-particle problem. Therefore, the DMFT\nis actually the CPA. After solving the effective single-site\nproblem, weobtainthelocalGreenfunction Gaσ(iωn,Ja)\nfor a fixed SE Ja. The averaged local Green function can4\nbe calculated by\nGaσ(iωn) =/integraldisplay\ndJaP(Ja)Gaσ(iωn,Ja)\n= (1−x)Gaσ(iωn,Ja= 0)+xGaσ(iωn,Ja=J).(7)\nThen, the self energy is determined by the Dyson equa-\ntion(5)again. Sofar,wehaveobtainedtheselfconsistent\nequations of the DMFT. They can be solved by simple\niterations. After solving the DMFT equations, we obtain\nthe self energy and the averaged Green function. The\nspontaneous magnetizations of sublattice AandBare\ndefined as\nmA=1\n2N/summationdisplay\nI,σσ/angbracketlefta†\nIσaIσ/angbracketright,\nmB=1\n2N/summationdisplay\nI,σσ/angbracketleftb†\nIσbIσ/angbracketright,\nwhereσ=±1. When mA=±mB/negationslash= 0 the ground state\nis ferromagnetic or antiferromagnetic, respectively. Due\ntothelocalnature,theDMFTdoesnotmixdifferentspin\nsectors of the Green function, hence the magnetization is\nnot non-coplanar. The topological property can be de-\ntermined through the disorder-average transport44or by\nthetopologicalBottindex45,46. TheBottindexisdefined\nin the real space with a realized configuration of random\nmagnetic impurities. However, calculating the Bott in-\ndex requires extensive numerical calculations. Instead of\ncalculating the Bott index, here we will use the disorder-\naverage approach proposed in Ref. 44. Within this ap-\nproachtheselfenergyofthedisorder-averageGreenfunc-\ntion renormalizes the noninteracting and non-disordered\nBloch Hamiltonian. Therefore, the renormalized Bloch\nHamiltonian Heff(k) =H0(k) +Σ(i0) =−[G(k,i0)]−1\ndetermines the topological invariant, like in the non-\ndisordered interacting case47. The topological invariant\nis determined by\nCν=1\n2π/integraldisplay\nd2kFν\nxy, (8)\nwhereFν\nij=∂iAν\nj−∂jAν\ni,Aν\ni=i/angbracketleftkν|∂ki|kν/angbracketright, and|kν/angbracketright\nis the orthonormalized eigenstate of matrix Heff(k), cor-\nresponding to the eigenvalue Eν(k). This topological\ninvariant is actually the Chern number of the effective\nHamiltonian, where its renormalization is given by disor-\nder and interaction in the mean field approximation. For\nweak disorder the disorder-average approach gives con-\nsistent results with the Bott index approach48. In fact,\nthe disorder-average approach has widely been used in\ndetermining the ground state topology48–52. In numeri-\ncal calculations one can use the efficient method of dis-\ncretization of the Brillouin zone to calculate the Chern\nnumber in Eq. (8)53./s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s48/s49/s50\n/s32/s74/s83/s61/s52/s32 /s32/s48/s49/s50\n/s74/s83/s61/s50\n/s32/s48/s49/s50/s110 /s32/s32\n/s109\n/s65/s32/s32/s32/s32/s32/s32/s32\n/s109\n/s66/s32/s32/s32/s32/s32 /s32/s32\n/s74/s83/s61/s48/s46/s53\nFIG.2: (Color online)The electron filling nandthesublattice\nmagnetization mA,mBvia the chemical potential µfor dif-\nferent values of the SE at magnetic doping x= 0.3 and SOC\nλ= 0.5. For guiding the eye electron fillings n= 0.3,1.0,1.7\nare indicated by the horizontal dotted lines.\nIII. NUMERICAL RESULTS\nWe numerically solve the DMFT equations by itera-\ntion for a given magnetic doping x. The numerical cal-\nculations are performed at fixed fictitious temperature\nT= 0.01, which serves as the cell size of the Matsub-\nara frequency mesh. The emergence of magnetism and\ntopology occurs in insulator, therefore we focus on de-\ntecting the insulating state. It is detected by a plateau\nin the curve n(µ), the dependence of electron filling on\nthe chemical potential11. Actually, the plateau reflects\nthe band gap as well as the vanishing of the charge com-\npressibility. They are the signals of the insulating stabil-\nity.\nFirst, we consider the case of dilute magnetic doping\n(x <0.5). In Fig. 2 we plot the dependence of electron\nfillingnand the sublattice magnetizations mA,mBon\nthe chemical potential µfor a small value x. It shows\nthe plateau appearance at fillings n= 1,n=xand\nn= 2−xwith appropriate values of the SE. At elec-\ntron half filling n= 1, the system is transformed from\nan insulating state to a metallic state when the SE in-\ncreases. When the SE vanishes ( J= 0), the SOC opens\na band gap in the electron structure15. A weak SE does\nnot change the insulating state, however, it reduces the\nband gap. As a consequence, at an appropriate value of5\nthe SE, the band gap closes, and ground state becomes\nmetallic. At the same time, the SE also drives a mag-\nnetic phase transition14,27. At electron half filling, it can\ninduce an antiferromagnetic (AF) long range order with\nmA=−mB/negationslash= 0 at low temperature14,27. In a mean\nfield picture, the AF magnetization can act on conduc-\ntion electrons like a staggered magnetic field, and this\nfield reduces the gap opened by the SOC. Indeed, when\na staggered magnetic field is present, the energy spectra\nof conduction electrons become\nE(k) =±/radicalbig\nt2|γk|2+(λξk−h)2, (9)\nwherehis the strength of the staggered magnetic field.\nActually, in the mean field approximation h∼J. At the\ncorners of the Brillouin zone K= 2π(1/3,±1/3√\n3),γk\nvanishes, whereas ξkremains finite. When h=λξK, the\ngap closes. However, in contrast to the non-disordered\nmagnetic case ( x= 1), at finite magnetic doping ( x <1)\nstrong SE does not open the band gap again, as can be\nseen in Fig. 2. At strong SE, instead of antiferromag-\nnetic insulator (AFI), antiferromagnetic metal (AFM) is\nestablished. This is an effect of magnetic impurity dop-\ning. Upon the magnetic doping, some lattice sites are\nfree of the magnetic impurity occupation. Therefore, at\nthese sites conduction electrons are also free of the SE\ncoupling. As a consequence, these conduction electrons\ngive a contribution to the electrical conductivity. How-\never, this effect occurs only for strong SE, which aligns\nelectron spins and magnetic moments in order to opti-\nmize the electron kinetic energy16. As can be seen in\nFig. 2, when the SE is strong, additional plateaus appear\nin the curve n(µ) atn=xandn= 2−x. Actually,\nn=xandn= 2−xare equivalent due to the particle-\nhole symmetry. In this case, the concentrations of con-\nduction electrons (holes) and of magnetic impurities are\nthe same. As we will see later, depending on magnetic\ndopingxand the SE strength, the ground state at n=x\n(orn= 2−x) may become magnetic. In particular,\nat strong SE and dense magnetic doping ( x>∼0.8), the\nground state at n=xandn= 2−xis AFI. In the\nlimitx→1, these AFI states at n=xandn= 2−x\nmerge into the single AFI at electron half filling n= 1.\nAs a consequence, at filling n=x= 1, AFI occurs again\nwhen the SE is strong. This can also be interpreted that\nthe AFI at half filling in the full magnetic case ( x= 1)\nis actually split into two AFI states in the electron and\nhole domains upon doping of magnetic impurities.\nIn dense magnetic doping ( x>∼0.8), additional\nplateaus in the curve of n(µ) are observed at n= 0.5 and\nn= 1.5 and strong SE, as can be seen in Fig. 3. Elec-\ntron fillings n= 0.5 andn= 1.5are equivalent due to the\nparticle-holesymmetry. Atelectronquarterfillingthe in-\nsulating state is ferromagnetic because mA=mB/negationslash= 0.\nThis ferromagnetic insulator (FI) is also established in\nthe non-disordered magnetic case ( x= 1)11. Figure 3\nalso shows discontinuities of the electron filling nand\nthe sublattice magnetizations mA,mBat certain values\nof the chemical potential. At these values of the chem-/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s48/s49/s50\n/s32/s74/s83/s61/s52/s32 /s32/s48/s49/s50\n/s74/s83/s61/s50\n/s32/s48/s49/s50/s110 /s32/s32\n/s109\n/s65/s32/s32/s32/s32/s32/s32/s32\n/s109\n/s66/s32/s32/s32/s32/s32 /s32/s32\n/s74/s83/s61/s48/s46/s53\nFIG. 3: (Color online) The electron filling nand the sub-\nlattice magnetization mA,mBvia the chemical potential µ\nfor different values of the SE at magnetic doping x= 0.9\nand SOC λ= 0.5. For guiding the eye electron fillings\nn= 0.5,0.9,1,1.1,1.5 are indicated by the horizontal dotted\nlines.\nical potential, the electron filling is uncertain, and ac-\ntually the ground state is spontaneously separated into\ntwo phases with the electron fillings corresponding to the\nextremesofthe discontinuityinthecurve n(µ). Thiscon-\nstitutes a phase separation14,25. The phase separation is\nnot a disorder effect, because it also occurs in the non-\ndisordered magnetic case x= 111. It occurs at the phase\nboundary between different symmetry phases, such as\nthe magnetic and paramagnetic states. In the magnetic\nstate, the electron ground-state energy is optimized by\naligning electron spins and magnetic moments through\nthe SE coupling, while in the paramagnetic state elec-\ntron spins are not aligned with the magnetic moments,\nand the optimization of the ground-state energy via the\nSE coupling is not operative14,25. As a result of the com-\npetition of these two phases, a magnetic pattern is en-\nergetically formed at the phase boundary. The phase\nseparation often occurs in the DE model upon electron\ndoping14,25.\nSofar,wehaveobservedtheinsulatingstateatelectron\n(hole) fillings n= 1,n= 0.5, andn=x. However,\nthe insulating state at quarter filling ( n= 0.5) occurs\nonly when the doping of magnetic impurities closes to\nx= 1. In the case of dilute magnetic doping, it is absent.\nElectron fillings n= 1 and n= 0.5, where insulator6\nis stable, reflect the number of occupied bands in the\nproposedmodel. Athalffilling n= 1, theinsulatingstate\noccurswhentwolowestbandsarefully occupied, whereas\natquarterfilling n= 0.5,the fulloccupationofthelowest\nband yields the insulating state. For other models, where\nthe number ofenergybands is larger, the filling condition\nfor the insulating stability may be changed55.\nA. Half filling n= 1\nIn Fig. 4, we plot the phase diagram for a fixed SOC\nat electron half filling n= 1. It shows that the insulating\nstate exists regardless of magnetic impurity doping when\nSE is weak. At weak SE the paramagnetic insulator (PI)\nis established. We have also calculated the Chern num-\nber defined in Eq. (8). It turns out this PI is topological\nwith the spin Chern number C= 1. Actually, it adiabat-\nically connects to the Z2topologicalinsulator in the non-\ninteracting and non-disordered case x= 015. The topo-\nlogical invariant is robust against the SE coupling until\nthe SE closes the band gap. On the other hand, the SE\ncouplingalignselectronspinswiththemagneticmoments\nin order to optimize the ground state energy14,16. When\nthe SE strength is larger a certain value, AF ordering is\nestablished at low temperature. At electron half filling\nthe ground state is insulating, hence there are no medi-\nated itinerant electrons that can generate the magnetic\nlong-range order by the DE mechanism14,16. However,\nthe spontaneous magnetization in the insulating states\ncan occur due to the direct coupling between the mag-\nnetic moments and electron spins through the van Vleck\nmechanism6. We find that the AFI at half filling is also a\ntopological insulator with the spin Chern number C= 1.\nActually, the SE coupling drives only the magnetic phase\ntransition from paramagnetic to antiferromagnetic state.\nAcross this phase transition the topological invariant is\nnot changed, because the band gap is still not closed by\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s50/s52/s54/s56/s49/s48/s74/s83\n/s120/s80/s84/s73/s65/s70/s84/s73/s65/s70/s77\nFIG. 4: Phase diagram at electron half filling n= 1 (λ=\n0.5). Abbreviations PTI, AFTI, AFM denote paramagnetic\ntopological insulator, antiferromagnetic topological in sulator,\nand antiferromagnetic metal, respectively.the SE. The magnetic phase transition is quite similar to\nthe one in the non-disordered magnetic case ( x= 1)11.\nWith further increase of the SE coupling, the band gap\nis closed and the ground state is AFM, except for x= 1,\nwherethe groundstateisAFI. Aswe havepreviouslydis-\ncussed, the AFI at strong SE in the non-disordered mag-\nnetic case ( x= 1) adiabatically connects to the merged\ninsulatingstateat equalfilling n=xandn= 2−x, when\nx→1. Therefore, the ground state at magnetic dopings\nx <1andx= 1hasdifferent origins. Figure4alsoshows\nthat the non-trivial topology of the AF ground state at\nelectron half filling is robust against magnetic doping.\nThe topological invariant remains the same regardless of\nmagnetic doping x. This indicates that the QSH effect\nis protected even in the presence of magnetic dopant dis-\norder as long as the band gap is still open. Some MTI\nmaterials doped with magnetic impurities favor the AF\nstate, for instance, first-principle calculations show an\nAF state in Bi 2Se3doped with Fe ions37. However, it is\nstill challenge to find the coexistence of the QSH effect\nand AF ordering in MTIs.\nB. Quarter filling n= 0.5\nIn Fig. 5 weplot the phasediagramatelectronquarter\nfillingn= 0.5. The insulating state exists only at strong\nSE coupling and large values of magnetic impurity dop-\ning (x>∼0.8). At small values of x, only metallic state\nexists. The insulating state is ferromagnetic, since strong\nSE coupling energetically favors the parallel alignment of\nelectron spins like in the DE mechanism14,25,27. In the\nferromagneticinsulator(FI), only the lowestband is fully\noccupied, and three other bands are empty. It turns out\n/s48/s46/s55 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48/s48/s49/s50/s51/s52\n/s32/s80/s77\n/s32/s70/s77\n/s32/s70/s84/s73/s32/s32/s74/s83\n/s120\nFIG. 5: (Color online) Phase diagram at electron quarter fill -\ningn= 0.5 (λ= 0.5). Abbreviations PM, FM, FTI denote\nparamagnetic metal, ferromagnetic metal, and ferromagnet ic\ntopological insulator, respectively.7\nthat the FI is topological since the Chern number calcu-\nlated by Eq. (8) gives C= 1 for the lowest band. This\nyieldstheQAHeffect. Firstprinciplecalculationsforreal\nmaterial Bi 2Se3doped with Cr ions also reveal the QAH\neffect6,7. The phase diagram plotted in Fig. 5 also shows\nthat doping of magnetic impurities can drive the topo-\nlogical FI to non-topological ferromagnetic metal (FM).\nHowever, this topological phase transition is actually an\ninsulator-metal transition. At the phase boundary, the\ngap closes. However, a further decrease of magnetic dop-\ning does not open the gap again, because electron filling\nis fixedn= 0.5 and the chemical potential lies within an\nenergy band. The phase transition at electron quarter\nfilling is quite different in comparison with the magnetic\ntopological phase transition at electron half filling. At\nelectron half filling the topological invariant remains the\nsame across the magnetic phase transition, whereas at\nelectron quarter filling, the spontaneous ferromagnetic\nmagnetization is maintained across the insulator-metal\ntransition, and the non-trivial topological invariant ap-\npears in the insulating side only. Doping of magnetic im-\npurities away from full filling suppresses the gap, hence\nsimultaneously destroys the topological invariant. The\nanomalous Hall effect was also suggested to exist in con-\nduction ferromagnets, however it cannot be quantized in\nmetals54.\nFigure 5 also shows a magnetic topological phase tran-\nsition driven by SE at fixed magnetic doping. When\nthe SE is weak, the ground state is paramagnetic metal\n(PM) although the SOC is present. Actually, the SOC\nopens a band gap only at electron half filling. Therefore\nat quarter filling, the SOC does not affect the metal-\nlic properties. Both the metal-insulator and the mag-\nnetic transitions are driven solely by the SE. However,\nthe SOC causes non-trivial topological invariants of two\nlowest bands. One lowest band has the Chern number\nC= 1, and the other one has C=−1. Since the two\nlowest bands have opposite spins, the QSH effect occurs\nat electron half filling. When the two lowest bands are\nseparated by a gap, the ground state is also insulator\nat electron quarter filling. Since its topological invari-\nant is integer, the QAH effect occurs. The separation of\ntwo lowest bands at electron quarter filling also indicates\nthe fully ferromagnetic state. This can be achieved by\nstrong SE11. Therefore the QAH effect occurs only at\nthe FI state. However, the SE coupling separates two\nlowest bands only at dense magnetic doping. At dilute\nmagnetic doping, the SE is valid only at a small number\nof lattice sites, and in an average manner, it cannot open\na band gap at electron quarter filling. In real MTI ma-\nterials, the QAH effect was observed at certain range of\nmagnetic impurity concentration6–10.\nC. Equal filling n=x\nIn this filling case, the concentration of electrons\n(holes) is equalto the concentrationofmagneticdopants./s48/s46/s49 /s48/s46/s51 /s48/s46/s53 /s48/s46/s55 /s48/s46/s57/s48/s49/s50/s51/s52/s53\n/s32/s80/s77\n/s32/s80/s73\n/s32/s70/s77\n/s32/s65/s70/s73/s32/s32/s74/s83\nFIG. 6: (Color online) Phase diagram at equal filling n=\nx= 0.3. Abbreviations PM, PI, FM, AFI denote param-\nagnetic metal, paramagnetic insulator, ferromagnetic met al,\nand antiferromagnetic insulator, respectively. All insul ating\nphases are topologically trivial.\nThe extreme case n=x= 1 is non-disorder and was pre-\nviously studied11. In Fig. 6 we plot the phase diagram\nat a fixed equal filling n=x <1. It exhibits differ-\nent magnetic states depending on the SOC and the SE.\nAs we have previously discussed, the SOC opens a band\ngap only at electron half filling n= 1 regardless of the\nSOC strength. When filling n=x <1, the valence\nband is partially occupied, therefore the ground state\nis metal. Weak SE does not change this paramagnetic\nmetal (PM). However, the SE polarizes electron spins\nand shifts the energy bands of opposite spins in opposite\ndirections. This effect of the SE looks like the one of an\nexternal magnetic field. Actually, in a mean field approx-\nimation, the SE can be treated as a magnetic field. As a\nconsequence, depending on the relationbetween the SOC\nand the SE, the ground state may become FM as can be\nseen in Fig. 6 (see also Fig. 2). This phase transition\nis similar to the one obtained in the interplay between\nthe SOC and external magnetic field55. With further in-\ncreasing SE, a band gap can be opened by the SE, and\nthe ground state becomes paramagnetic insulator (PI).\nActually, Fig. 2 also shows when the SE increases, the\nferromagnetic state occurs not at a fixed electron filling.\nIt moves toward the domain of lower electron filling as\nthe SE increases. Therefore, when the magnetic doping\nx=nis fixed, the FM state only occurs in a finite range\nof the SE. When the SE is strong enough, the ground\nstate is AFI. Indeed, upon magnetic doping, the AFI at\nelectron half filling is split into two AFIs at fillings n=x\nandn= 2−x. In Fig. 7 one can also see the impact\nof magnetic doping on the magnetic states at equal fill-\ningn=x. The FM state exists only in a finite range\nofx, because the band shift due to the SE lowers the\nenergy band of one spin component, and hence it can8\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s49/s50/s51/s52/s53\n/s32/s80/s77\n/s32/s80/s73\n/s32/s70/s77\n/s32/s65/s70/s77\n/s32/s65/s70/s73/s32/s32/s74/s83\n/s120\nFIG. 7: (Color online) Phase diagram at equal filling n=x\n(λ= 0.5). Abbreviations PM, PI, FM, AFM, AFI denote\nparamagnetic metal, paramagnetic insulator, ferromagnet ic\nmetal, antiferromagnetic metal, and antiferromagnetic in sula-\ntor, respectively. All insulating phases are topologicall y triv-\nial.\nmaintain the FM state only at certain electron filling n.\nSincen=x, asxvaries, the electron filling nvaries too.\nTherefore, the phases presented in Fig. 7 have varying\nelectron filling, from almost empty filling to almost half\nfilling. The insulating state only exists when the SE is\nstrong enough. A strong SE aligns spins of conduction\nelectrons and magnetic moments of impurities. Since the\nnumbersofconductionelectronsandofmagneticdopants\nare the same, there are no free conduction electrons. As\na consequence, the insulating state is established. In the\ndomain of dilute magnetic doping, the insulator is para-\nmagnetic, while in the opposite domain, when the mag-\nnetic doping is dense, it is antiferromagnetic. This yields\na magnetic phase transition driven by magnetic dopants.\nIn the case of dense magnetic doping, the ground-state\nenergy is optimized when the AF state is formed like in\nthe limit case n=x= 1. However, in the dilute doping\ncase, the aligning orientation of electron spins at each\nlattice site is random. Therefore the macroscopic mag-\nnetization vanishes and the PI is established. We want to\nemphasize that the magnetic phase transtion driven by\nmagnetic dopants occurs not at a fixed electron filling n,\nbut at the constraint n=x. In the insulating states at\nn=x, the Chern number calculated by Eq. (8) vanishes.\nAlthough magnetic dopants can maintain the insulating\nstates at equal filling n=x, and they can drive the mag-\nnetic phase transition from PI to AFI, neither QAH nor\nQSH effect occurs. Nevertheless, the phase diagram at\nequal filling n=xshows rich phase diagrams. Despite\nthe SOCdoes notcauseanytopologicallynontrivialinsu-\nlator atn=x, its interplay with magnetic dopants gives\nrise to rich magnetic phases.IV. CONCLUSION\nWe have studied the impact of magnetic dopants on\nthe magneticand topologicalphaseswhichcould occurin\nMTIs. When magnetic impurities are doped into MTIs,\nthey are coupled with conduction electrons via the SE,\nand simultaneously introduce disorder and inhomogene-\nity. The interplay between the random SE and the SOC\ncauses rich magnetic and topological phases in MTIs.\nHowever, non-trivial topology of the insulating ground\nstate exists only at electron half and quarter fillings. At\nelectron half filling the AFI is stable between the PI and\nAFM, when the SE strength increases. It exhibits the\nQSH effect that is robust against the magnetic impu-\nrity doping. However, disorder and inhomogeneity which\nare introduced by magnetic dopants induce the AFM at\nstrong SE, while in the non-disordered case, the AFI is\ninstead established. Actually, the AFI at electron half\nfilling is split into two AFIs in the electron and hole do-\nmains upon magnetic doping. Although the AFI is topo-\nlogically nontrivial at electron half filling, its split AFI\nstates upon magnetic doping are topologically trivial.\nAt electron quarter filling, the QAH effect could occur\nat the strong SE and dense magnetic doping. However,\nthe magnetic doping also drives the FI to the FM, when\nit decreases, and therefore it completely suppresses the\nQAHeffectatitsappropriatevalue. Thesefindingsreveal\nthat magnetic dopants impact differently on the topolog-\nical properties of the MTIs, depending on electron filling.\nAt electron half filling the topological invariant is robust\nagainst magnetic dopants, while at electron quarter fill-\ning it is suppressed by magnetic doping. In addition to\ntheelectronhalfandquarterfillings, wealsoobservedthe\ninsulating ground states at equal fillings (i.e., the concen-\ntration of electrons (holes) is equal to the concentration\nof magnetic dopants). However, the insulating states\nare topologically trivial. In comparison with the non-\ndisordered case, the phase diagram becomes very rich.\nDisorder and inhomogeneity cause different magnetic or-\nderings in both insulating and metallic states.\nDespite the explicit presence of magnetic impurities,\nthe proposed model is also appropriate for intrinsic\nMTIs, where instead of magnetic impurities, the d-\nband correlated electrons establish magnetic long-range\nordering56,57. The intrinsic MTIs were recently discov-\nered and have attracted intensive attention56,57. Actu-\nally, in the intrinsic MTIs only the spin degree of free-\ndom of the d-band correlated electrons is relevant for es-\ntablishing magnetism, and the charge degree of freedom\ncan be discarded. The SE between the d-band corre-\nlated electrons and conduction electrons may interplay\nwith the SOC of conduction electrons and this emerges\nthe topologically nontrivial magnetic ground state11. 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Kiss,3Bertalan Gy¨ orgy\nSzigeti,2Vladimir Tsurkan,4,5Alois Loidl,4and Istv´ an K´ ezsm´ arki1,2,4\n1MTA-BME Lend¨ ulet Magneto-optical Spectroscopy Research Group, 1111 Budapest, Hungary\n2Department of Physics, Budapest University of Technology a nd Economics 1111 Budapest, Hungary\n3Department of Experimental Solid State Physics, Institute for Solid State Physics and Optics,\nWigner-MTA Research Centre for Physics, 1121 Budapest, Hun gary\n4Experimental Physics V, Center for Electronic Correlation s and Magnetism,\nUniversity of Augsburg, 86135 Augsburg, Germany\n5Institute of Applied Physics, Academy of Sciences of Moldov a, MD 2028, Chisinau, Republica Moldova\n(Dated: April 3, 2017)\nWe report on the slow magnetization dynamics observed upon t he magnetic phase transitions\nof GaV 4S8, a multiferroic compound featuring a long-ranged cycloida l magnetic order and a N´ eel-\ntype skyrmion lattice in a relatively broad temperature ran ge below its Curie temperature. The\nfundamental difference between GaV 4S8and the chiral helimagnets, wherein the skyrmion phase\nwas first discovered, lies within the polar symmetry of GaV 4S8, promoting a cycloidal instead of\na helical magnetic order and rendering the magnetic phase di agram essentially different from that\nin the cubic helimagnets. Our ac magnetic susceptibility st udy reveals slow relaxation dynamics\nat the field-driven phase transitions between the cycloidal , skyrmion lattice and field-polarized\nstates. At each phase boundary, the characteristic relaxat ion times were found to exhibit a strong\ntemperature dependence, starting from the minute range at l ow temperatures, decreasing to the\nmicro- to millisecond range at higher temperatures.\nI. INTRODUCTION\nMagnetic skyrmions are topologically non-trivial,\nwhirling spin structures, which can form 2-dimensional\ncrystals, so-called skyrmion lattices1,2. The emergence\nof the skyrmion lattice phase was first identified in the\nclosevicinity ofthe paramagnetic-helicalphaseboundary\nof cubic chiral helimagnets, known as B20 compounds\nwith a P2 13 space group3–7. Cu2OSeO3, belonging to\nthe same space group with a different crystal structure8\nwas the first insulating material demonstrated to host\nskyrmions9,10, with a magnetoelectric character11–14.\nSince their experimental discovery, skyrmions have at-\ntracted much attention owing to their potential ap-\nplication as magnetic bits in high-capacity and low-\nconsumption memory devices15–19.\nGaV4S8, a member of the lacunar spinel family, char-\nacterized by the space group F ¯43m, is the first known\nexample of skyrmion-host materials with non-chiral but\npolar crystal structure20. GaV 4S8is a semiconductor\nwith a non-centrosymmetric cubic crystal structure (T d)\nat room temperature. The compound undergoes a coop-\nerative Jahn-Teller distortion at T S=42K through the\nstretching of the lattice along any of the four cubic\nbody diagonals reducing the crystal symmetry to po-\nlar rhombohedral21–24. As a result a sizable ferroelec-\ntric polarization develops along the C 3vrhombohedral\naxis25,26. Depolarization field is reduced by the forma-\ntion of submicron-sized structural domains of the four\npossible rhombohedrally distorted variants with the dif-\nferent/angbracketleft111/angbracketright-type rhombohedral axes21, assembling into\nan alternating lamellar domain structure27.\nLong-range magnetic ordering arises and therebythe compound becomes a type-I multiferroic at\nTC=13K25,26,28,29. The interplay of the symmetric ex-\nchangeinteractionandtheantisymmetricDzyaloshinskii-\nMoriya (DM) exchange interaction gives rise to a long-\nwavelength spin ordering. However, in contrast to the\nB20compounds featuringa helicalspin orderin zerofield\nand longitudinal conical spin structure in finite magnetic\nfields, in GaV 4S8the different pattern of DM vectors,\ndictated by the polar C 3vsymmetry, leads to a cycloidal\nspin order (Cyc) and precludes the emergence of the lon-\ngitudinalconicalstructureinfinitefields20. Thecycloidal\nnature of the magnetic modulations has been confirmed\nexperimentally by polarized SANS measurements30.\nIn low magnetic fields a N´ eel-type skyrmion lattice\n(SkL) develops. Due to the lack of the longitudinal con-\nical state, being the main competitor of the SkL phase\nin the B20 compounds, the N´ eel-type SkL is stable over\na broad temperature region20. On the other hand, the\neasy-axisanisotropyin the rhombohedraldomains31pro-\nmotestheferromagneticordering(FM) ofthespinsinthe\nground state, suppressing the modulated phases below\nT=5K20.\nThe modulation vectors of the Cyc and SkL states are\nconfined to the rhombohedral plane, irrespective of the\ndirection of the applied magnetic field20. Furthermore,\nthe skyrmions were demonstrated to carry magnetoelec-\ntric polarization25,29, offering new possibilities of their\nmanipulation with external electric fields.\nMeasurement of static magnetization or magnetic sus-\nceptibility is a primary methodology to reveal the phase\ntransitions between the modulated magnetic states in\nnon-centrosymmetric magnets9,20,32–34. Beyond the de-\ntermination of the phase diagram via dc susceptibil-2\nity measurements, recently, several works have been de-\nvoted to analyzing the dynamic response of the modu-\nlated spin structures via ac susceptibility measurements.\nSimilar features have been identified in the ac suscep-\ntibility in many compounds of the B20 family as well\nas in CuO 2SeO332,35–39. Generally, the real component\nof the ac susceptibility follows well the static suscep-\ntibility, however in the vicinity of the magnetic phase\nboundaries, it deviatesfrom the static value and becomes\nstrongly frequency dependent, accompanied by a finite\nimaginary component of the susceptibility. In chiral heli-\nmagnets,thisfeatureisgenerallyinterpretedasthesigna-\nture of first-order transitions between the different mag-\nnetic phases, involving slow relaxation processes of large\ncorrelated magnetic volumes, such as the reorientation\nof the long-wavelength spin-spirals or the nucleation of\nskyrmioniccores35,36,38,39. Theanalysisofthefrequency-\ndependence of the ac susceptibility in all cases revealed a\nbroad distribution of relaxation times with macroscopic\nvalues at the phase boundaries36,38,39.\nHere, we report a systematic study of the ac sus-\nceptibility in GaV 4S8, a compound essentially different\nfrom the cubic helimagnets in terms of crystal sym-\nmetries and magnetic phase diagram20. The analy-\nsis of the susceptibility measurements reveals a dra-\nmatic increase of the relaxation times in the vicinity of\nthe magnetic phase boundaries, with a strong temper-\nature dependence, which is similar to findings in other\nsystems36,38,39. In magnetic fields close to the critical\nvalues, the average relaxation times are much shorter\nthan 1ms at the high-temperature end of each phase\nboundaryand rise well abovethe minute scaleat the low-\ntemperature limits. This strongtemperature dependence\ninvokes a thermally activated behavior of the relaxation\nprocesses characterized by energy barriers in the order of\n1000K.\nII. RELAXATION MODEL\nThe modulated magnetic structures in skyrmion host\ncompounds are generally characterized by a long correla-\ntion length, forming coherent magnetic regions with di-\nmensions of hundreds of nanometers3,20. The ac suscep-\ntibility of correlated spin-structures consisting of clusters\nand/or domains of various volumes is generally described\nby the Cole-Cole relaxation model. This phenomeno-\nlogical model has been effectively applied to various\nsystems40comprising large magnetic volumes, such as\nspin glasses41, superparamagnetic nanoparticles42, and\nmore recently it has been proposed for the description of\nthe phase transitionsbetweenmodulated magnetic states\nin chiral helimagnets, Fe 1−xCoxSi38and Cu 2OSeO336,39.\nThe dynamic response of the magnetic system in the\nCole-Cole model is formulated as an extension of the\nDebye-relaxation by introducing a distribution of relax-\nation times, while keeping the exponential time depen-\ndence of the relaxation40,43:χω=χ∞+(χ0−χ∞)1\n1+(iωτc)1−α,(1)\nwhereτcrepresents the central value of the relaxation\ntimes and the αparameter is connected to the width of\ntheir distribution. The adiabatic susceptibility, χ∞orig-\ninates from the almost immediate response of the spins,\nas compared to the time scale of the studied relaxation\nprocesses, hence, it is a purely real quantity. The static\nlimit of the ac susceptibility is denoted as χ0. The distri-\nbution of the relaxation times, g(ln( τ)), is expressed by\ntheτcandαparameters as follows43:\ng(lnτ) =1\n2π/parenleftbiggsinαπ\ncosh[(1−α)ln(τ/τc)−cosαπ]/parenrightbigg\n.(2)\nThe distribution is symmetric on the logarithmic scale\nwith the central value of τc. Zero value of αrepresents\na single Debye-relaxation process, while values close to\nunityleadtoaninfinitely broaddistributionofrelaxation\ntimes.\nOwing to the phase sensitivity of the ac-susceptibility\nmeasurements, both the real and imaginary components\nof the susceptibility, χ′andχ′′, can be recovered. The\nfrequency dependence of the two components, expressed\nfrom Eq.1, reads as:\nχ′=χ∞+(χ0−χ∞)ωτα−1\nc+sin/parenleftbigαπ\n2/parenrightbig\nωτα−1c+ωτ1−αc+2sin/parenleftbigαπ\n2/parenrightbig(3)\nχ′′= (χ0−χ∞)cos/parenleftbigαπ\n2/parenrightbig\nωτα−1c+ωτ1−αc+2sin/parenleftbigαπ\n2/parenrightbig.(4)\nIII. METHODS\nA. Sample synthesis and characterization\nSingle crystalline GaV 4S8samples were grown by\nchemical vapour transport method using iodine as trans-\nport agent. The high crystalline quality of the sam-\nples has been confirmed by X-ray diffraction. A cuboid-\nshaped sample with a mass of 23.4mg was selected for\nthe ac susceptibility measurements.\nB. Static and ac susceptibility measurements\nA 5T Quantum Design MPMS SQUID magnetometer\nwas used for the static and ac susceptibility measure-\nments. Both the static magnetic field and the ac modu-\nlation field were normal to the (111) plane of the GaV 4S8\ncrystal and the longitudinal magnetic moment was mea-\nsured in a phase sensitive manner. The field dependence3\nof the ac susceptibility was measured in the 0-80mT\nrange with a modulation amplitude of µ0Hω=0.3mT.\nIn order to probe the dynamics of the magnetically or-\ndered spin system in the low-frequency regime, the drive\nfrequency of the modulating coil was varied between\nf=0.1Hz and 1kHz. The in-phase and out-of-phase com-\nponents of the oscillating magnetization were measured\nand normalized by the drive amplitude, µ0Hωto obtain\nthe real and imaginary parts of the ac-susceptibility, re-\nspectively. The dc magnetization was measured in a\nsubsequent run after the ac susceptibility measurements.\nThe static susceptibility, ∂m/∂H, was obtained from the\nmeasured dc magnetization curves by numeric derivation\nusingthecentraldifferencemethod. Thetypicalduration\nof the static measurements was ≈100s per data point,\nwhich involved the ramping of the magnetic field and\nperforming the measurement by the reciprocating sam-\nple option (RSO).\nIV. RESULTS AND DISCUSSION\nThemagnetic-fielddependenceofthestaticandacsus-\nceptibility was recorded at various temperatures within\nthe magnetically ordered phases of GaV 4S8, i.e. between\nT=6.5K and T=12K with magnetic fields applied par-\nallel to the crystallographic /angbracketleft111/angbracketright-axis. The static sus-\nceptibility and the real part of the ac susceptibility are\nshown in Fig. 1 (a). The imaginary component of the ac\nsusceptibility is presented in Panel (b) in the same figure.\nA. Magnetic phase diagram established by the\nstatic susceptibility\nThe field-driven phase transitions are associated\nwith peaks in the field dependence of the static\nsusceptibility20,32. The susceptibility measurements in\nFig.1 reveal the same magnetic phase diagram as estab-\nlished by K´ ezsm´ arki et al.in earlier static magnetiza-\ntion measurements20. The peaks in the dc susceptibil-\nity separate three magnetic phases in the temperature –\nmagnetic field plane identified as the Cyc, SkL and FP\nstates. The three states adjoin at the triple point located\nat T=9.5K and µ0H=32mT. Despite the coexistence of\nstructural domains in the compound, within the stud-\nied field range, phase transitions are only observed in\nthe structural variant wherein the rhombohedral axis is\nparallel to the field20. Remarkably, the susceptibility is\nenhanced in the SkL phase by almost a factor of 2 as\ncompared to the Cyc phase, implying a larger suscepti-\nbility of the former phase along the rhombohedral axis.\nIn the FP state the susceptibility is nearly zero.B. Slow relaxation phenomena at the magnetic\nphase boundaries\nIn the following, we discuss the ac susceptibility mea-\nsurements, indicating slow dynamic processes occurring\nin the modulated magnetic phases of GaV 4S8.\nIt is clearly seen in Figs. 1 (a) and (b) that away from\nthe phase transitions, the ac susceptibility is frequency-\nindependent and purely real, with values identical to\nthe static susceptibility, i.e. they correspond to χ∞.\nIn these regions, the characteristic relaxation times are\nmuch shorter than 1ms, the time period of the highest-\nfrequency modulation in our experiments.\nOn the other hand, as approachingthe magnetic phase\nboundaries, the real component of the susceptibility falls\nbehind the static values, exhibiting a strong frequency\ndependence. This effect becomes the most prominent be-\nlow T=9.5K, i.e. for the Cyc-FP phase transition. More-\nover,neareach phasetransition, apeak is seen the imagi-\nnary component of the ac susceptibility, as a signature of\ndissipative processes occurring at extremely low frequen-\ncies, which also accounts for the frequency dependence\nof both components of the ac susceptibility. Such be-\nhavior was also reported in cubic helimagnets of the B20\nfamily33,35,38,44and in Cu 2OSeO336,39near the magnetic\nphase transitions.\nAsignificantdifferenceisthatinthecubichelimagnets,\nthe spiral magnetic order undergoes multiple phase tran-\nsitions. In the absence ofa magnetic field, a multidomain\nhelical state is realized with propagation vectors selected\nby the cubic anisotropy in these materials9,10,45–47. As\nthe magnetic field is increased and the weak anisotropy\nis overcome by the Zeeman energy, the domains redis-\ntribute upon a first-order phase transition48. Finally,\nthe wave vectors of the modulations flip towards the\nmagnetic field, establishing a mono-domain longitudi-\nnal conical structure. Ac susceptibility measurements in\nFeGe44, MnSi32,48, Cu2OSeO336,39and Fe 1−xCoxSi38in-\ndicate that these transitions occur on macroscopic time\nscales, attributed to the rearrangementof large magnetic\nspirals.\nIn GaV 4S8, onthe otherhand, notraceofslowdynam-\nics is seen in magnetic fields away from the Cyc-FP, SkL-\nFP and Cyc-SkL phase boundaries, even though multiple\ndynamic processes are expected within the Cyc as well\nas the SkL phase. According to theoretical considera-\ntions generic to spin-spirals in external magnetic field49\nand SANS measurements in particular in GaV 4S820, be-\nsides the anharmonic deformation of the cycloids and\nskyrmions induced by the field, the cycloidal wavelength\nand the skyrmion lattice constant increase substantially\nwith increasing magnetic fields. The absence of dissipa-\ntion within the Cycand SkL phasessuggeststhat this ex-\npansion takes place on much faster time-scales than 1ms\nin both phases. Furthermore, as opposed to the cubic\nhelimagnets, there is no sign of an abrupt redistribution\nof the cycloidal wave vectors induced by the field.\nOur susceptibility data indicate slow dynamics only4\n(a)\nCyc FPSkLPM\n11.25K\n7K8K9K\n8.5K9.5K10K10.5K10.75K11K12K\n11.5K\n6.5K\n0H(mT)051015202530\n0 10 20 30 40 50 60 70f=0.1Hz\nf=1Hz\nf=10Hz\nf=45Hz\nf=110Hz\nf=500Hz\nf=1000Hzdc, m/HCycFPSkLPM\n0 10 20 30 40 50 60 706.5K11.25K\n7K8K9K\n8.5K9.5K10K10.5K10.75K11K12K\n11.5K\n050100150m/H,   (cm3/mol) '(b)\nf=0.1Hz\nf=1Hz\nf=10Hz\nf=45Hz\nf=110Hz\nf=500Hz\nf=1000Hz\n0H(mT)\n  (cm3/mol) ''\nFIG. 1. (Color online) Static and ac susceptibility measure d in GaV 4S8, plotted against the external magnetic field. Panel\n(a) presents the static susceptibility, ∂m/∂H , plotted in gray, along with the real component of the ac susc eptibility, χ′(H).\nPanel (b) displays the imaginary component of the ac suscept ibility,χ′′(H). Different colors represent data measured at various\nac frequencies in the f=0.1 Hz-1 kHz range. Measured data are shifted proportionally with the sample temperature, which is\nindicated on the left side of each curve. The continuous line s connecting the dots are guides to the eye. The magnetic phas es\nseparated by the susceptibility peaks in the B-T plane are in dicated in the graphs. The paramagnetic state above T C=13 K is\ndenoted as PM.\nnear the phase boundaries between the Cyc-FP, SkL-\nFP and Cyc-SkL phases. At the two boundaries of the\nSkL phase, the first-order nucleation processes of the\nskyrmions may account for the long time scales. As\nopposed to the transition from the longitudinal coni-\ncal phase to the field polarized state in the B20s and\nCu2OSeO3, dissipative processes occur at low frequen-\ncies at the Cyc-FP transition in GaV 4S8. This suggests\nthe first-order character of the transition, further sup-\nported by recent neutron scattering experiments30. The\nunderlying dynamics may be related to the unwinding\nof the 2πdomain walls of the anharmonic spin-cycloids\npolarized by the external field.\nC. Temperature and magnetic-field dependence of\nthe relaxation processes\nThe frequencydependence ofthe peaks in both suscep-\ntibility components [Fig. 1 (a) and (b)] shows a strong\nvariation with the temperature. Concerning the Cyc-FP\nphase boundary, at T=7K the dissipation is the largest\nfor the smallest modulation frequency, whereasat T=9Kit becomes almost frequency independent. Above the\ntriple point between T=10K and T=11K, a reversal can\nbe seen in the hierarchy of the frequencies in the dis-\nsipation peaks at both the Cyc-SkL and SkL-FP phase\nboundaries, indicating that the characteristic frequencies\noftherelaxationprocessespassthroughthemeasurement\nwindow. Additionally, the peaks in the imaginary part\nof the susceptibility are shifted in magnetic field with the\nchange of the drive frequency, which is most prominent\nat T=10.5K. In order to systematically investigate the\nbehaviorofthe relaxationas the function ofthe tempera-\nture and magnetic field, the frequency dependence of the\nreal and imaginary components of the susceptibility was\nanalyzed at all measured (H,T) points in the vicinity of\nthe magnetic phase boundaries.\nFigure 2 presents the frequency dependence of χ′and\nχ′′in various magnetic fields. Three representative tem-\nperatures are selected above the triple point, where the\npeak in the imaginary part of the susceptibility passes\novertheexperimentalwindow, indicatingthattheinverse\nrelaxation times go through the range of the measure-\nment frequencies. Figure 2 only shows data measured in\nrepresentative magnetic field regions near the Cyc-SkL5\nand SkL-FP phase transitions. The frequency depen-\ndence of the complex susceptibility can be fitted well by\nthe Cole-Cole relaxation model (Eqs. 3 and 4) using the\nsame set of parameters for the real and imaginary com-\nponents. The shifting of the peak in χ′′towards lower\nfrequencies with decreasing temperature is well traced\nby the fitted curves for both the Cyc-SkL and the SkL-\nFP transitions implying an overall slowing down of the\nrelaxation.\nUsing the τcandαparametersretrievedfromthe Cole-\nCole fits, the distribution of the relaxation times, g(lnτ)\nwas calculated for each (H,T) point, according to Eq. 2.\nFigures 3 (a), (b) and (c) display the calculated distribu-\ntionsoftherelaxationtimesfortheCyc-SkL,SkL-FPand\nCyc-FP phase transitions, respectively. For each transi-\ntion, the characteristic time scales fall below τ <<1ms\nat the high-temperature end of the phase boundary, ex-\nhibiting a further dramatic increase towards lower tem-\nperatures,reachingvalues >>10satthelow-temperature\npart of the phase boundaries. Similar tendencies have\nbeen identified in Cu 2OSeO336,39.\nFigure 4 presents the temperature dependence of the\nfitted relaxation times averaged over the range of mag-\nnetic fields near the phase transitions as log( τav) =/summationtext\nilog(τ(Hi)). The sum runs over the values of relax-\nation times, τ(Hi), which are determined by fitting at\neach field, Hi, where the susceptibility shows observ-\nable frequency dependence in the vicinity of the phase\nboundaries. The rapid drop in the relaxation times with\nincreasing temperatures is clearly seen for each phase\nboundary. Thediscontinuousjumpintherelaxationtime\nat the triple point marks an abrupt change in the relax-\nation processes between the Cyc-FP and the Cyc-SkL\nphase.\nThe exponential character of the temperature de-\npendence of the relaxation times suggests a thermally\nactivated behavior related to an energy barrier, ∆ E,\nseparating the two thermodynamically stable magnetic\nphases. The energy barriers for the Cyc-SkL and SkL-\nFPphasetransitionsweredetermined bylinearfits tothe\nArrhenius-plots, i.e. ln( τav) against 1 /T, as presented in\nthe inset of Fig. 4. The fitted values yield average acti-\nvation energies of 1293K and 1137K at the Cyc-SkL and\nthe SkL-FP boundaries, respectively. These large values\nunderline the stability of the modulated phases extend-\ning over large volumes, i.e. fluctuations of sizable regions\ninstead of individual spins. Since the susceptibility at\nthe Cyc-FP boundary could not be accurately fitted (as\ndiscussed later), the relaxation times for this transition\nhave not been analyzed quantitatively.\nApparently, the characteristic relaxation times are\nstrongly affected also by the magnetic field in case of\nthe SkL-FP transition [Fig. 3 (b)], whereas such promi-\nnent field dependence is not found at the other two phase\ntransitions [Figs. 3 (a) and (c)]. In the former case, the\nrelaxation substantially accelerates with increasing mag-\nnetic field, as approaching the FP state. This may be ex-\nplained by a phase coexistence of the SkL and FP states10-310-1100101102103105(cm3/mol)\n0102030405060\n24mT26mT28mT30mT32mT44mT46mT48mT52mT\n50mT54mT56mTT=10.75K'(c)\n10-310-1100101102103105024681012141618\n24mT26mT28mT30mT32mT44mT46mT48mT52mT\n50mT54mT56mTT=10.75K(cm3/mol) ''(d)\n10-310-1100101102103105(cm3/mol)\n010203040506070\n24mT26mT28mT30mT32mT40mT42mT44mT48mT\n46mT50mT52mT54mTT=10.5K'(e)\n10-310-110010110210310505101520\n24mT26mT28mT30mT32mT40mT42mT44mT48mT\n46mT50mT52mT54mTT=10.5K(cm3/mol)''(f)10-310-1100101102103105(cm3/mol)\n0102030405060\nT=11K'(a)\n22mT24mT26mT28mT30mT46mT48mT50mT54mT\n52mT56mT58mT\n10-310-1100101102103105024681012141618\nT=11K(cm3/mol) ''(b)\n22mT24mT26mT28mT30mT46mT48mT50mT54mT\n52mT56mT58mT\nCyc-SkL SkL-FPCyc-SkL SkL-FP\nCyc-SkL SkL-FPCyc-SkL SkL-FP\nCyc-SkL SkL-FPCyc-SkL SkL-FP\nFrequency, f(Hz) Frequency, f(Hz)\nFIG. 2. (Color online) Frequency dependence of the real (lef t\ncolumn) and imaginary components (right column) of the sus-\nceptibility in various magnetic fields above the temperatur e\nof the triple point, at T=11K (top), T=10.75K (middle) and\nT=10.5K (bottom row). Measured values are shifted with re-\nspect to the magnetic field. The color coding of the measured\nvalues represents the different ac frequencies in accordanc e\nwith Fig. 1. Solid lines are fitted curves according to Eqs.\n3 and 4 with the same set of parameters for the real and\nimaginary parts of the susceptibility. The blue and red bars\nnext to the right axes represent the range of magnetic fields\ncorresponding to the Cyc-SkL and SkL-FP phase transitions,\nrespectively.6\n(a)\nCyc-SkL\n9.51010.51111.512T(K)\n10\n- \u0000 \u0001\u0000 \u0001\n\u0001203040\n(s)0H(mT)(b)\nSkL-FP\n9.51010.51111.512T(K)\n10-10\n1010100\n20304050\n(s)0H(mT)(c)\nCyc-FP\n6.577.588.59T(K)\n10-10\n1010100\n102030\n(s)0H(mT)\nFIG. 3. (Color online) Distribution of relaxation times, g(ln (τ)), plotted as a function of the temperature and magnetic fiel d.\nThe distributions were calculated employing Eq. 2 with the τcandαparameters obtained from the fits to the frequency\ndependence of the complex susceptibility. Panels (a),(b) a nd (c) present the relaxation times in the ranges of magnetic fields\ncorresponding to the Cyc-SkL, SkL-FP and Cyc-FP transition s, respectively. The distribution curves are shifted propo rtionally\nwith the temperature along the z-axis, which is also indicat ed in the right side of the graphs. The curves are colored acco rding\nto a color map representing decreasing temperatures rangin g from T=12 K (red) to T=6.5 K (blue). The average relaxation\ntime shows a dramatic decrease with decreasing temperature s along all the three phase boundaries, with values well abov e 10 s\nat their low-temperature limits.\nwith adecreasingtypicalsize ofpersistingSkL islandsto-\nwards larger magnetic fields, exhibiting a faster response\nto the ac modulation.\nIn contrast to the other two phase boundaries, the\nCole-Cole model fails to fit the frequency dependence of\nthe complex susceptibility for the Cyc-FP transition, as\ndemonstrated in Fig. 5 for two selected temperatures be-\nlow the triple point. Even though the real and the imagi-\nnary components can be fitted separatelywith two differ-\nent sets of parameters (see dashed gray curves in Fig. 5),\nthe resulting parameters convey no physical meaning, as\nthe Kramers-Kronig relation does not hold between the\ntwo components of the response function. The large dif-\nference between the static susceptibility values and the\nreal part of the ac susceptibility measured even at the\nlowest frequency of f=0.1Hz, as seen in 1 (a), suggests\nthat dynamic processes exist with characteristic relax-\nation times far beyond 10s.\nThe Cole-Cole model assumes a symmetric distribu-\ntion of relaxation times on the logarithmic scale43, which\nmaynotapplyformorecomplexprocessesinvolvedinthe\nmagnetic phase transitions in GaV 4S8. A generalization\nof the Cole-Cole function was provided by Havriliak and\nNegami50allowing for an asymmetric distribution of re-\nlaxation times51. Applying the Havriliak-Negami model\nto our data, however, yielded the same parameters as the\nCole-Cole fits returning the same symmetric distributionof relaxation times, hence did not improve the fit.\nOnly a few recent studies made an attempt to quanti-\ntativelydescribe the relaxationprocessesat the magnetic\nphase boundaries in cubic skyrmion host compounds,\neachwithin theframeworkoftheCole-Colemodel36,38,39.\nHowever, in most of these studies the real and imaginary\ncomponents of the ac susceptibility were handled sepa-\nrately, which may lead to unphysical parameters, as seen\nfor the Cyc-FP transition in GaV 4S8(Fig. 5). Qian\net al.correlated the Cole-Cole fits to the real and the\nimaginary parts of the susceptibility in Cu 2OSeO3, find-\ning good agreement in case of the conical-to-skyrmion\nand skyrmion-to-conical transitions, whereas a discrep-\nancy was reported at the helical-to-conical transition.\nThe authors attributed this difference to additional re-\nlaxation processes present at extremely low frequencies.\nBannenberg et al.38also identified a low-frequency con-\ntribution to the dissipation in Fe 1−xCoxSi that could not\nbe described by the Cole-Cole model both at the conical-\nto-skyrmion and the skyrmion-to-conical transitions.\nV. CONCLUSION\nIn this study, we investigatedthe dynamicsofthe field-\ndriven phase transitions between the magnetic states in\nGaV4S8via ac susceptibility measurements. The mag-7\n/s54 /s55 /s56 /s57 /s49/s48 /s49/s49 /s49/s50/s49/s48/s45/s54/s49/s48/s45/s51/s49/s48/s48/s49/s48/s51/s49/s48/s54\n/s48/s46/s48/s57/s48 /s48/s46/s48/s57/s53 /s48/s46/s49/s48/s48/s45/s50/s48/s45/s49/s48/s48/s49/s48/s32/s67/s121 /s99/s45/s70/s80\n/s32/s67/s121 /s99/s45/s83/s107/s76\n/s32/s83/s107/s76/s45/s70/s80/s32\n/s84 /s101/s109 /s112/s101/s114/s97/s116/s117/s114/s101/s44/s32/s84 /s40/s75/s41/s65/s118/s101/s114/s97/s103/s101/s32/s114/s101/s108/s97/s120/s97/s116/s105/s111/s110/s32/s116/s105/s109/s101/s44/s32\n/s97/s118/s40/s115/s41\n/s69/s61/s49/s50/s57/s51/s75\n/s32/s32/s108/s110/s40\n/s97/s118/s41\n/s49/s47/s84/s69/s61/s49/s49/s51/s55/s75\nFIG. 4. (Color online) Temperature dependence of the log-\narithmic average of the relaxation times obtained from the\nCole-Cole fits, where the averaging was performed over the\nfitted values in the magnetic field region close to the phase\ntransitions. The green, red and blue circles correspond to t he\naverage relaxation times at the Cyc-FP, Cyc-SkL and SkL-FP\ntransitions. The lines connecting the data points are guide s to\nthe eye. The dashed horizontal lines represent the measure-\nment window defined by the inverse of the highest (1 kHz)\nand lowest (0.1 Hz) ac frequencies. The inset presents ln τav\nas the function of 1 /Talong the Cyc-SkL and SkL-FP phase\nboundaries. Relaxation time values close to the experiment al\nwindow are plotted, as indicated by the black dotted frame.\nLinear fits to the data (solid black lines) yield the average\nactivation energies of 1293 K and 1137 K for the Cyc-SkL and\nSkL-FP transitions, respectively.\nnetic response related to the continuous deformations of\nthecycloidalstructureandtheskyrmionlatticeoccurring\ninside the phases, such as the field-induced anharmonic-\nity and transverse distortions, are accompanied with an\ninstantaneous response of the spin system on the fre-\nquency scale well abovethe kHz range. The emergenceof\nextremely slow dynamics was demonstrated at the phase\nboundaries between the cycloidal, skyrmion lattice and\nfield-polarized states.\nSimilar frequency-dependent susceptibility and peaks\nin the imaginary (dissipative) part of the susceptibility\ncharacterize the phase boundaries between modulated\nmagnetic states in all cubic helimagnets32. However, in\ncontrasttothelackoflow-frequencydissipativeprocesses\nneartheconical-FPboundaryinhelimagnets,inGaV 4S8,\nthe Cyc-FP transition is characterized by extremely slow\ndynamicalprocesses,possiblyassociatedtothediscontin-\nuous transition into the FP state through the unwinding\nof the 2πdomain walls.\nMagnetization dynamics at the Cyc-SkL and SkL-FP(d)(a) (b)\n(c)(cm3/mol) '\n(cm3/mol) ''(cm3/mol) '\n(cm3/mol)''10-310-110010110210310501020304050\n22mT24mT26mT28mT30mT32mT34mT36mT40mT\n38mTT=9K ''' Fitted   , \nFitted    '\n10-310-11001011021031050246810121416\n22mT24mT26mT28mT30mT32mT34mT36mT40mT\n38mTT=9K''' Fitted   , \nFitted    ''\n10-310-110010110210310501020304050\n14mT16mT18mT20mT22mT24mT26mT28mT32mT\n30mTT=8K\nFrequency, f (Hz)''' Fitted   , \nFitted    '\n10-310-11001011021031050246810121416\n14mT16mT18mT20mT22mT24mT26mT28mT32mT\n30mTT=8K\nFrequency, f (Hz)''' Fitted   , \nFitted    ''\nCyc-FP Cyc-FPCyc-FP Cyc-FP\nFIG. 5. (Color online) Frequency dependence of the real (lef t\ncolumn) and imaginary components (right column) of the sus-\nceptibility in various magnetic fields below the temperatur e of\nthe triple point at T=9K (top), T=8K (bottom row). Mea-\nsured values are shifted with respect to the magnetic field.\nThe color coding of the measured values represents the differ -\nent ac frequencies in accordance with Fig. 1. Solid lines are\nfitted curves according to Eqs. 3 and 4 with the same set of\nparameters for the real and imaginary parts of the suscepti-\nbility. Gray dashed lines are separate fits to the real [(a) an d\n(c)] and the imaginary components [(b) and (d)] of the sus-\nceptibility. The green bars next to the right axes emphasize\nthat the range of magnetic fields at the given temperatures\ncorrespond to the Cyc-FP phase transition.\nphasetransitionsinGaV 4S8iswelldescribedbytheCole-\nCole relaxation model. However, discrepancies found at\nthe Cyc-FPtransitionindicate the presenceofmorecom-\nplex dynamics that cannot be described by a distribution\nof Debye-relaxation processes.\nEach field-driven phase transition in GaV 4S8shows\nsimilar behavior as a function of the temperature: at\nthe high-temperature end of the phase boundaries the\ncharacteristic relaxation times are much shorter than\n1ms, whereas at the lower-temperature part of the phase\nboundaries, the dynamics is drastically slowed down,\ncharacterized by relaxation times well above the minute\nscale. Similar temperature dependence of the relaxation8\ntimes has been reported for Cu 2OSeO336,39and may be\na common feature for all bulk skyrmion host materi-\nals. The broad distribution of relaxation times and their\nstrong dependence on the temperature, implying large\nactivation energies, are likely the results of the collec-\ntive relaxation of large magnetic structures of various\nvolumes.\nThe extremely slow relaxation times at the low-\ntemperature end of the phase boundaries may be ex-\nploited for the creation of non-equilibrium skyrmionic\nclusters in the FP state with a long lifetime, whereas\na relatively fast switching can be realized between themodulated states at higher temperatures.\nVI. ACKNOWLEDGEMENT\nThe authors thank I. ˇZivkovi´ c and H. Rønnow for en-\nlighting discussions. This work was supported by the\nDeutsche Forschungsgemeinschaft through the Transre-\ngional Collaborative Research Center TRR 80 and by\nthe Hungarian Research Funds OTKA K 108918, OTKA\nPD 111756 and Bolyai 00565/14/11.\n∗butykai@dept.phy.bme.hu\n1A. Bogdanov and D. Yablonskii, Zh. Eksp. Teor. Fiz 95,\n182 (1989).\n2N. Nagaosa and Y. Tokura, Nat. 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Maegawa\nGraduate School of Human and Environmental Studies, Kyoto U niversity, Kyoto 606-8501, Japan\n(Dated: November 10, 2018)\nWe show that the dynamic magnetic susceptibility and the sup erparamagnetic blocking tempera-\nture of an Fe 8single molecule magnet oscillate as a function of the magnet ic fieldHxapplied along\nits hard magnetic axis. These oscillations are associated w ith quantum interferences, tuned by Hx,\nbetween different spin tunneling paths linking two excited m agnetic states. The oscillation period\nis determined by the quantum mixing between the ground S= 10 and excited multiplets. These\nexperiments enable us to quantify such mixing. We find that th e weight of excited multiplets in the\nmagnetic ground state of Fe 8amounts to approximately 11 .6 %.\nPACS numbers:\nHigh-spin molecular clusters [1, 2] display superpara-\nmagnetic behavior, very much as magnetic nanoparticles\ntypically do. Below a time- (or frequency-)dependent\nblocking temperature Tb, the linear magnetic response\n”freezes” [3, 4] and magnetization shows hysteresis [5].\nThe slow magnetic relaxation of these single-molecule\nmagnets (SMMs) arises from anisotropy energy barri-\ners separating spin-up and spin-down states. Because\nof their small size, the magnetic response shows also ev-\nidences for quantum phenomena, such as resonant spin\ntunneling [3, 6–8]. In the case of molecules that, like Fe 8\n(cf Fig. 1A and [4]), have a biaxial magnetic anisotropy,\ntunneling between any pair of quasi-degenerate spin\nstates±mcan proceed via two equivalent trajectories,\nwhich, asillustratedinFig. 1B,crossthehardanisotropy\nplane close to the medium anisotropy axis. A magnetic\nfield along the hard axis changes the phases of these tun-\nneling paths, leading to either constructiveor destructive\ninterferences. This phenomenon is known as Berry phase\ninterference [9, 10].\nExperimental evidences for the ensuing oscillation of\nthe quantum tunnel splitting ∆ m, shown in Fig. 1C,\nwere first observed in Fe 8[11, 12] and then in some other\nSMMs [13–19] by means of Landau-Zener magnetization\nrelaxation experiments. Interference patterns measured\non Fe8at very low temperatures, which correspond to\ntunneling via the ground state doublet m=±10, are\nreproduced by the following spin Hamiltonian\nH=−DS2\nz+E(S2\nx−S2\ny)+C/parenleftbig\nS4\n++S4\n−/parenrightbig\n−gµB− →S·− →H(1)\nthat applies to the lowest lying spin multiplet, with\nS= 10, and where D/kB= 0.294 K,E/kB= 0.046 K,\nHx Easy axis, \nz \nMedium  \naxis, y A \nB C \nHard  \naxis, x 0.0 0.5 1.0 1.5 2.0 10 -10 10 -8 10 -6 10 -4 10 -2 10 0\n  D/kB (K) \nm0Hx (T) + m \n- m \ny \nx z \nFIG. 1: A: Molecular structure of the\n[(C6H15N3)6Fe8O2(OH)12] molecular magnet, briefly re-\nferred to as Fe 8. B: Two equivalent tunneling paths linking\nstates with m= +Sandm=−S. A magnetic field along\nthe hard axis (denoted by x) shifts the relative phases of\nthese trajectories, thus leading to constructive and destr uc-\ntive interferences. C: Dependence of the quantum tunnel\nsplittings ∆ monHxcalculated with Eq. (1) for states ±m,\nwithm= 10 (bottom curve) to m= 1 top curve).\nC/kB=−2.9×10−5K are magnetic anisotropy param-\neters, and g= 2. The sizeable fourth-order parameter C\nreflects not only the intrinsic anisotropy but, mainly, it\nparameterizes quantum mixing of the S= 10 with ex-\ncited multiplets ( S-mixing) and how it influences quan-\ntum tunneling via the ground state [20].\nIn the present paper, we study the influence of Berry\nphase interference on the ac magnetic susceptibility χ\nandTbof Fe8, that is, on those quantities that character-\nize the standard SMM (or superparamagnetic) behavior.\nClose to Tb, magnetic relaxation is dominated by tunnel-\ning near the top of the anisotropy energy barrier, thus\nalso near excited multiplets with S/negationslash= 10. In this way, we2\naim also to use the interference pattern to gain quanti-\ntative information on the degree of S-mixing in Fe 8.\nThe sample employed in these experiments was a\n3×2×1 mm3single crystal of Fe 8. Each molecule\nhas a net spin S= 10 and a strong uniaxial magnetic\nanisotropy. Equation (1) defines x,yandzas the\nhard, medium and easy magnetization axes. In the tri-\nclinic structure of Fe 8,x,y, andzaxes are common to\nall molecules [21]. The complex magnetic susceptibility\nχ=χ′(T,ω)−iχ′′(T,ω) was measured between 90 mK\nand 7 K, and in the frequency range 3 Hz ≤ω/2π≤20\nkHz, using a purpose built ac susceptometer thermally\nanchored to the mixing chamber of a3He-4He dilution\nrefrigerator. A dc magnetic field− →Hwas applied with\na 9 T×1 T×1 T superconducting vector magnet, which\nenables rotating− →Hwith an accuracy better than 0 .001◦.\nThe magneticeasyaxis zwasparalleltothe acexcitation\nmagnetic field of amplitude hac= 0.01 Oe. The sample\nwas completely covered by a non-magnetic epoxy to pre-\nvent it from moving under the action of the applied mag-\nnetic field. The alignment of− →Hperpendicular ( ±0.05◦)\ntozand close ( ±5◦) to the hard xaxis was done at low\ntemperatures ( T= 2 K), using the strong dependence\nofχ′(T,ω) on the magnetic field orientation (see [22] for\nfurther details). These data were scaled with measure-\nments performed, for T/greaterorequalslant1.8 K, using a commercial\nSQUID magnetometer and a physical measurement plat-\nform equipped with ac susceptibility options.\nThe zerofield ( H= 0) ac susceptibility χ′andχ′′com-\nponents of Fe 8show the typical SMM behavior (Fig. 2).\nMaxima of χ′′measured at different frequencies define\nTband occur when τ≃1/ω, whereτis the magnetic\nrelaxation time. As Fig. 2 shows, τapproximately fol-\nlows Arrhenius’ law τ≃τ0exp(U/kBT), where Uis the\nactivation energy and τ0is an attempt time. However, a\ncloser inspection reveals that the slope of the Arrhenius\nplot increases gradually with temperature, from a low- T\nvalueU/kB≃22 K to more than 32 K. While the former\nUvalue agrees with tunneling taking place via m=±5\nstates, the latter is close to the maximum energy ( ≃32.5\nK) of the S= 10 multiplet.\nFigure 3 shows χ′(T) andχ′′(T) data measured at\nω/2π= 333 Hz and under three different transverse\nmagnetic fields. By increasing µ0Hxfrom 0 to 0 .19 T,\nthe superparamagnetic blocking shifts towards higher T.\nThereafter, Tbdecreases again with further increasing\nµ0Hxto 0.27 T. The right-hand panel of Fig. 3 shows\nthatTboscillates as a function of Hx. This behavior con-\ntrasts sharply with the rapid and monotonic decrease of\nTbthat is observed when− →His parallel to the medium\naxisy(see the inset of Fig. 3 and [22, 23]).\nOscillations of Tblead also to oscillations of the dy-\nnamical susceptibility. Figure 4A shows χ′vsHxdata\nmeasured at ω/2π= 333 Hz and T= 2.6≃Tb(Hx=\n0) K. Under such conditions, small shifts of Tbre-/s48 /s50 /s52 /s54/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48\n/s49/s48/s45/s54/s49/s48/s45/s53/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49\n/s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54/s50/s48/s50/s53/s51/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s49/s48/s50/s48/s51/s48/s52/s48/s32/s49/s51/s51/s32/s72/s122\n/s32/s51/s51/s51/s32/s72/s122\n/s32/s54/s51/s51/s32/s72/s122\n/s32/s49/s51/s51/s51/s32/s72/s122\n/s32/s53/s51/s51/s51/s32/s72/s122\n/s32/s49/s51/s51/s51/s51/s32/s72/s122\n/s32/s32/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s84 /s40/s75/s41/s40/s65/s41\n/s40/s66/s41/s40/s68/s41\n/s32/s32/s40/s115/s41/s40/s67/s41\n/s109/s32 /s61/s32/s177/s51\n/s109/s32 /s61/s32/s177/s52\n/s32/s32/s85 /s47/s107\n/s66/s40/s75/s41\n/s49/s47 /s84 /s32/s40/s75/s45/s49\n/s41/s109/s32 /s61/s32/s177/s53/s32/s49/s46/s57/s48/s32/s75/s32 /s32/s50/s32/s75/s32/s32/s32/s32/s32/s32/s32 /s32/s50/s46/s50/s48/s32/s75/s32\n/s32/s50/s46/s51/s53/s32/s75/s32 /s32/s50/s46/s53/s48/s32/s75/s32/s32 /s32/s50/s46/s54/s48/s32/s75\n/s32/s50/s46/s55/s48/s32/s75/s32 /s32/s50/s46/s56/s53/s32/s75/s32/s32 /s32/s51/s46/s48/s48/s32/s75/s32\n/s32/s51/s46/s50/s48/s32/s75/s32 /s32/s51/s46/s53/s48/s32/s75/s32/s32 /s32/s51/s46/s56/s48/s32/s75\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s52/s46/s50/s48/s32/s75\n/s32/s32/s39/s39/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s47/s50 /s32/s40/s72/s122/s41\nFIG. 2: Ac susceptibility of an Fe 8single crystal measured\nat several frequencies. A: temperature dependence of the\nreal (solid symbols) and imaginary (open symbols) compo-\nnents. B: frequency dependence of the imaginary component.\nLines are least-square fits with Cole-Cole function χ′′(ω,T) =\n∆χ(ωτ)βsin(βπ/2)/[1 + (ωτ)2β+ 2(ωτ)βcos(βπ/2)], where\n∆χ≃χT, the equilibrium susceptibility, and β≃0.92. C:\nArrhenius plot of the relaxation time τextracted from these\nfits. The solid line is a least-squares linear fit of data mea-\nsured below 2 .6 K. D: Effective activation energy Ufor the\nmagnetic relaxation process, obtained from the slope of the\nArrhenius plot. The horizontal lines show magnetic energy\nlevels derived from the giant spin Hamiltonian (1).\nsult in large changes of χ′, thus allowing us to moni-\ntor these changes very precisely. χ′shows three min-\nima, at µ0Ha≃0.20(1) T, µ0Hb≃0.56(1) T, and\nµ0Hc≃0.90(1) T, with an approximate periodicity\nµ0∆Hx≡µ0[2Ha+(Hb−Ha)+(Hc−Hb)]/3≃0.37\nT. Again, this behavior contrasts with the abrupt in-\ncrease towards equilibrium that is observed when /vectorHis\napplied along y[22]. From the susceptibility we esti-\nmate also τ≃[r/(sinβπ/2−rcosβπ/2)](1/β)/ω, where\nr=χ′′/χ′andβ≃0.92 was determined from Cole-Cole\nfits performed at Hx= 0 (see Fig. 2B). Figure 4B shows\nthat, as one could anticipate, τalso oscillates with Hx.\nData of Figs. 3 and 4 strongly suggest that the oscilla-\ntion period ∆ Hxremains approximately constant in the\ntemperature range between 2 and 3 K coveredby present\nexperiments.\nThe oscillations can be qualitatively accounted for by\nrecallingthedependenceofthequantumtunnelsplittings\n∆monHx(Fig. 1). At zero field and close to T= 2.6\nK, magnetic relaxation takes place mainly via thermally\nactivated m=±4 states. By increasing Hx, ∆4gets pe-\nriodically quenched and therefore tunneling is inhibited.\nThis leads to an increase of τand thus also of Tb, as it\nis observed experimentally.\nHowever, the giant spin model Eq.(1) is unable to pro-\nvide aquantitative description of the interference pat-\ntern. The magnetic relaxation time and the frequency3\n/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s50/s46/s53/s50/s46/s54/s50/s46/s55\n/s48 /s49 /s50/s48/s49/s50/s51/s32\n/s48/s72\n/s120 /s32/s61/s32/s48\n/s32\n/s48/s72\n/s120 /s32/s61/s32/s48/s46/s49/s57/s32/s84\n/s32\n/s48/s72\n/s120 /s32/s61/s32/s48/s46/s50/s55/s32/s84\n/s34\n/s84 /s40/s75/s41/s32/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41/s39\n/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116/s115\n/s32/s71/s105/s97/s110/s116/s32/s115/s112/s105/s110\n/s32/s83 /s45/s109/s105/s120/s105/s110/s103\n/s48/s72\n/s120/s40/s84/s41/s32/s84\n/s98/s40/s75/s41/s72 /s32/s97/s108/s111/s110 /s103/s32 /s121\n/s48/s72 /s40/s84/s41/s32/s84\n/s98/s40/s75/s41/s72 /s32/s97/s108/s111/s110 /s103/s32 /s120\nFIG. 3: Left: χ′andχ′′susceptibility components of Fe 8\nmeasured at ω/2π= 333 Hz and for three different Hxval-\nues. Right: blocking temperature Tbas a function of Hx\n(hard axis) and Hy(medium axis) for the same frequency.\nDotted and solid lines are theoretical predictions followi ng\nfrom, respectively, the giant spin model [Eq. (1)] and the\ntwo-spin model [Eq. (2)], which includes S-mixing effects,\nforφ= 4 deg. The inset compares blocking temperatures\nmeasured with /vectorHalong the hard ( x) and medium ( y) axes.\ndependent-susceptibility have been calculated by solving\na Pauli master equation for the populations of all energy\nlevels of (1), following the model described in [24]. For\nsimplicity, we have simulated the effect that dipolar in-\nteractions between different Fe 8clusters have on the spin\ntunneling probabilities [24, 25] by introducing an effec-\ntive bias field µ0Hdz= 31 mT. This value is the width\nof the distribution of dipolar fields in a magnetically un-\npolarized Fe 8crystal [26, 27]. The results are compared\nto experimental data in Figs. 3 and 4. Although they\nshow oscillations, the theoretical period µ0∆Hx= 0.28\nT is about 20 % smaller than the experimental one. It is\nimportant to emphasize that the discrepancy cannot be\nascribedto a small uncertaintyin angle φ. All theoretical\ncurves with distinct oscillations (for φ/lessorsimilar10 deg.) show\n∆Hxsmaller than observed.\nThe discrepancy originates instead from the fact that\nµ0∆Hxobtained from Eq. (1) strongly decreases with\nm, from 0 .4 T for m=±10 to 0.25 T for m=±2.\nThis dependence arises from the presence of a strong\nfourth-order anisotropy term. The same effect occurs\nfor higher-order terms. The giant spin approximation,\nwhich neglects all excited multiplets, is therefore unable\nto simultaneously account for the quantum interference\npatterns observed at low and high- T.\nThese results call for a more complete description,\nable to explicitly incorporate the effects of S-mixing. A\nschematic diagram of the magnetic structure of the Fe 8\nmolecular core[28, 29] is shown in Fig. 5. The centraldi-\namond(or”butterfly”)ofspins1 −4stronglycoupleanti-\nferromagnetically to give a net spin S1−4≃0. Couplings\nvia the butterfly generate effective interactions between\nthe remaining spins 5 −8. The resulting spin configu-/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48\n/s48 /s49 /s50 /s51/s48/s46/s50/s56/s48/s46/s51/s50/s48/s46/s51/s54/s48/s46/s52/s48/s48/s46/s52/s52\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s49/s50/s51\n/s67/s65\n/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116/s115/s32 /s84 /s32/s61/s32/s50/s46/s54/s32/s75/s32\n/s32/s71/s105/s97/s110/s116/s32/s115/s112/s105/s110/s32/s109/s111/s100/s101/s108/s44/s32 /s32/s61/s32/s52/s32/s100/s101/s103\n/s32/s83/s112/s105/s110/s45/s109/s105/s120/s105/s110/s103/s44/s32 /s32/s61/s32/s52/s32/s100/s101/s103\n/s32/s32\n/s32\n/s48/s72\n/s120/s32/s40/s84/s41/s39/s40/s101/s109/s117/s47/s109/s111/s108/s32/s79/s101/s41\n/s66\n/s32/s32/s48/s72\n/s120/s40/s84/s41\n/s84 /s40/s75/s41/s32/s50/s46/s50/s53/s32/s75\n/s32/s50/s46/s52/s32/s75\n/s32/s50/s46/s53/s32/s75\n/s32/s50/s46/s54/s32/s75\n/s32/s50/s46/s55/s53/s32/s75\n/s32/s51/s32/s75\n/s32/s32/s47 /s40/s72\n/s120/s61/s48/s41\n/s48/s72\n/s120/s40/s84/s41\nFIG. 4: A: χ′of Fe8vsHxmeasured at T= 2.6 K and ω/2π\n= 333 Hz. B: transverse field dependence of the magnetic\nrelaxation time τdetermined at different temperatures. C:\ntemperature dependence of the quantum oscillation periods\nestimated from Landau-Zener relaxation measurements ( ⋆)\n[11],χ′vsHxatT= 2.6 K (•), andχ′vsTat different\nHx(∗). Dotted and solid lines are theoretical predictions, for\nφ= 4 deg., that follow from the giant spin model [Eq. (1)]\nand the two-spin model [Eq. (2)], respectively.\nration, with a ground state S= 10 and a first excited\nS= 9 multiplet lying about δE9,10/kB= 44 K above it\n[28–31], follows from the fact that |J4|>|J3|(all cou-\nplings are antiferromagnetic). In addition to symmetric\nexchange interactions, one has to consider also single-ion\nmagnetic anisotropies, dipolar interactions and antisym-\nmetric Dzyaloshinskii-Moriya (DM) interactions, which\nmix states of different S[20]. Concerning the latter, al-\nthoughit isin principlepossibleto makeDM interactions\nirrelevant on a specific bond by a gauge transformation,\ncompatibility conditions must be satisfied to extend this\nover the molecule. To be precise, the product of the\ngauge transformations associated to each bond along any\nclosed exchange path in the molecule should be equal to\n(−1)nI(I≡identity, ninteger) [32]. The low symmetry\nof Fe8ensures that no such global similarity transforma-\ntion should exist, mainly because the DM interactions\nbetween the individual spins are certainly non uniform.\nBased on the above considerations, we reduce the full\nspin Hamiltonian of 8 Fe3+ions to a simpler and compu-\ntationally more affordable one, involving only two spins\nSI= 5 and SII= 5, defined in Fig. 5. This approxima-4\ny \nx z 5 \n7 6 8 \n1 \nJ1 J4 J3 \nJ2 \n3 4 S1-4 = 0 SI = 5  SII  = 5  \n2 \nFIG. 5: Scheme ofexchange pathways connecting Fe3+ions in\nthe Fe 8core. Approximate values of the exchange constants\nare [29]J1/kB=−36 K,J2/kB=−201 K,J3/kB=−26 K,\nandJ4/kB=−59 K.\ntion can be justified by the fact that exchange interac-\ntions in Fe 8as well as in other SMMs have been treated\nby iteratively coupling spins in pairs [28, 33]. The two-\nspin Hamiltonian reads as follows\nH=/summationdisplay\ni=I,IIHanis,i−/summationdisplay\ni=I,IIgiµB− →Si·− →H\n−J− →SI− →SII−− →SIˆA− →SII+− − →dI,II− →SI×− →SII(2)\nwhereHanis,i=−DiS2\ni,z+Ei(S2\ni,x−S2\ni,y)+Ci/parenleftbig\nS4\ni,++S4\ni,−/parenrightbig\naccounts for the magnetic anisotropy of each spin, J >\n0 is an effective isotropic exchange constant, ˆAis an\nanisotropic coupling tensor, and− − →dI,IIis a DM inter-\naction vector. We set DI/kB=DII/kB= 0.625 K,\nEI/kB=EII/kB= 8.94×10−2K andCI/kB=CII/kB=\n−5.7×10−5K, which give, for the S= 10 multiplet, pa-\nrameters D= 0.47DI,E= 0.47EIandC= 0.128CIthat\nagree with those determined from EPR experiments [29].\nWithin this model, dominant symmetric exchange in-\nteractions(mainly J2andJ4)contributetotheformation\nof the two giant spins /vectorSIand/vectorSII, which are then coupled\nby a weaker effective symmetric exchange. We have set\nJ/kB= 3.52 K to fit the energy gap δE9,10between the\nS= 9 and the S= 10 multiplets. The last two terms\nin Eq. (2) parameterize the effects that dipolar and DM\ninteractions, consideredas perturbations, haveon the en-\nergies of the subspace defined by different /vectorSIand/vectorSIIori-\nentations. Dipolar interactions between spins (5 ,7) and\n(6,8) give predominantly rise to a term −AxxSI,xSII,x,\nwithAxx/kB≃2.8×10−2K. This term hardly has any\nnoticeable influence on the period of quantum oscilla-\ntions and, furthermore, it tends to reduce∆Hx. Because\nof the close to planar molecular structure and its pseudo\nC2symmetry, the same considerations apply to terms\narising from dipolar interactions between any of these\nspins and those forming the central butterfly. In orderto account for the observations, S-mixing must therefore\npredominantly arise from antisymmetric exchange inter-\nactions.\nDM interactions between individual Fe3+spins are\ngenerally of the order of ∆ g/g≈0.01 times the symmet-\nric interactions [34–36], thus about 0 .3−2 K in the case\nof Fe8, see Fig. 5. The oscillation periods of ∆ mvsHx\ndepend on the magnitude and orientation of− − →dI,II: they\nincreasewithincreasing dI,IIwhen− − →dI,IIpointsalong xbut\ndecrease when− − →dI,IIis alongy. We have therefore set this\nvectoralong xand varied dI,IIto fit the experimental sus-\nceptibility oscillations. As with the giant spin model, the\ndynamical susceptibility has been calculated by solving\na Pauli master equation for the energy level populations\nof Eq.(2). The best agreement, shown in Figs. 3 and 4,\nis found for dI,II/kB= 1.28±0.05 K, which is compat-\nible with the above estimates. The model accounts for\nthe oscillations observed in the ac susceptibility and re-\nlated quantities ( Tbandτ) measured between 2 K and 3\nK. In addition, it describes well the overall temperature\ndependence of ∆ Hxbetween very low temperatures and\n3 K (Fig. 4C). Finally, it predicts a ground state tun-\nnel splitting ∆ 10/kB= 10−7K, which agrees with that\ndetermined from Landau-Zener experiments [11, 12].\nIt can be concluded that Eq. (2), despite its relative\nsimplicity, agrees not only with available spectroscopic\nand magnetic data, but also provides a much more ac-\ncurate description of the spin dynamics in Fe 8than the\ngiant spin model Eq. (1). It also enables one to quantify\nthe degree of S-mixing. The ground state of Eq.(2) con-\ntains 88.4 % ofS= 10 states, 10 .9 % ofS= 9 states and\n0.7 % of states from other multiplets.\nSummarizing, we have shown that the ac linear mag-\nnetic response and the superparamagnetic blocking of\nmolecular nanomagnets are governed by quantum inter-\nferences, which can be tuned by an external magnetic\nfield. Furthermore, the period of oscillations depends\non the nature of the spin states involved in the tunnel-\ning processes, i.e. on whether they are pure ground- S\nstates or quantum superpositions with states from other\nmultiplets. These results confirm that an accurate de-\nscription of quantum phenomena in single molecule mag-\nnets should take into account quantum mixing between\nground and excited multiplets. They also illustrate the\nsensitivity of interference phenomena to small changes in\nthe wave function describing a physical system. 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Lett. 109, 067205 (2012)."    },    {        "title": "1706.05177v1.Observation_of_Various_and_Spontaneous_Magnetic_Skyrmionic_Bubbles_at_Room_Temperature_in_a_Frustrated_Kagome_Magnet_with_Uniaxial_Magnetic_Anisotropy.pdf",        "content": "  Submitted to  \n1 \n \nDOI: 10.1002/((please add manuscript number))  \nArticle type: Communication  \n \nObservation of  Various  and Spontaneous Magnetic Skyrmionic Bubbles at Room -\nTemperature in a Frustrated Kagome Magnet with Uniaxial Magnetic Anisotropy  \n \nZhipeng Hou*, Weijun Ren*,Bei Ding*, Guizhou Xu, Yue Wang, Bing Yang, Qiang Zhang, \nYing Zhang,  Enke Liu,  Feng Xu, Wenhong Wang, Guangheng Wu, Xi -xiang Zhang, Baogen \nShen, Zhidong Zhang  \n Dr. Z. P. Hou, B. Ding, Y. Wang, Dr. Y. Zhang, Dr. E. K. Liu, Prof. W. H. Wang, Prof. G. H. \nWu, Prof. B. G. Shen  \nBeijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China  \nE-mail: wenhong.wang@iphy.ac.cn (W. H. Wang)  \nDr.W. J. Ren, Dr. B. Yang, Prof. Z. D. Zhang  \nShenyang Materials Science National Laboratory, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China  \nE-mail:wjren@imr.ac.cn  (W. J . Ren)  \nDr. G. Z. Xu , Prof. F. Xu  \nSchool of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China  \nDr. Qiang Zhang, Prof. X. X. Zhang King Abdullah University of Science and Technology (KAUST), Physical Science and \nEngineering  (PSE) , Thuwal 23955- 6900, Saudi Arabia  \n [*] Z.P.H, W.J.R, and B.D contributed equally to this work.  Keywords:  skyrmion ic bubbles , topological spin textures , kagome  magnet , Fe\n3Sn2, spintronic \ndevices  \n \n \nThe quest for materials hosting topologically protected nanometric spin textures, so- called \nmagnetic skyrmions or  magnetic skyrmionic  bubbles, continues to be fuelled by the promise \nof novel devices .[1-5] The skyrmionic spin textures  have  been mostly  observed in non-\ncentrosymmetr ic crystals , such as  the cubic chiral magnets MnSi,[6-9] (Mn,Fe)Ge,[10,11]  \nFeCoSi ,[12] Cu2OSeO 3,[13-15] and also  the polar magnet  GaV 4S8,[16] where Dzyaloshinskii -\nMoriya interaction (DMI)  is active . A number of intriguing electromagnetic phenomena, \nincluding the topological Hall effect,[7] skyrmion magnetic resonance,[17] thermally induced \nratchet motion,[18] and effective magnetic monopoles,[19] have been demonstrated  to be closely \nrelated to the topologically nontrivial spin texture of skyrmions. T hese novel topological \nproperties , together with nanoscale dimensions, a stable particle -like feature, and an ultralow \nthreshold for current -driven motion, make magnetic skyrmions  fundamentally promising for \napplications  in next -generation high -density and low-dissipation memory devices .[1-5]   Submitted to  \n2 \n \nHowever, although a very recent  publication by Tokunage et al .[20]  reported the observation \nof skyrmion lattices above room temperature  (RT) in β–type Cu -Zn-Mn alloys, skyrmions in \nthe bulk chiral magnets  have been mostly observed below RT .[6-16] Moreover , the spin texture \nof skyrmions  that were stabilized by DMI  in chiral  magnets  is quite limited.  The s kyrmion s \nwith variable spin textures  may be more attractive for further  technological applications  \nbecause they can adapt to various external stimuli act ing as information carriers in  spintronic \ndevices . Therefore,  one particularly important current research direction aims at the discovery \nof new materials that host  skyrmions  with variable  spin textures  at room temperature.  \nIn addition to the non- centrosymmetric chiral magnets  in which the magnetic  skyrmions  are \nstabled  by DMI , the centrosymmetric materials with uniaxial magnetic anisotropy (UMA)  are \nanother family of materials that can host skyrmions.[21-26] In these materials, the competition \nbetween the magnetic dipole interaction and uniaxial easy axis anisotropy is the key force  in \nthe formation of skyrmion s.[21-26] The skyrmions in centrosymmetric materials  are \ntopologically equivalent to those  in the chi ral magnets[21-26] but posse ss two degrees of \nfreedom, i.e. helicity and vorticity .[5] Since their internal  degree of freedom  is similar to that  \nin the topologically trivial magnetic bubbles , they are also called skyrmionic bubbles .[24] The \nmost interesting physics  in skyrmionic bubbles  is that the vorticity of the ir spin textures  varies  \nwith the internal structure of the Bloch lines (BLs) , resulting  in a variety of spin textures .[22-26] \nFor instance,  a new type of spin texture formed by two skyrmions  with opposite spin vorticity \n(the topological number  equals 2), called  a biskyrmion, has been experimentally discovered in \nthe centrosymmetric  tetragonal magnetite La1-xSrxMnO 3(x=0.315)[23] and hexagonal (Mn 1-\nxNix)65Ga35(x=0.5)[25] at temperatures  around 60 K and 300K, respect ively. More recently , Yu \net al.[24] have found a variety of spin textures  in magnetic skyrmionic bubbles in orthorhombic \nmagnetite  La1-xSrxMnO 3(x=0.175)  at 100K. Thus, the  multifarious topological nature  of \nskyrmionic bubbles offers us an opportunity  to manipulate  their topological spin textures  \nthrough external stimuli.  \nRecently, the domain structure s in the  frustrated magnet with tunable UMA  were studied \nnumerically by Leonov et al [27]. They sh owed that the UMA strongly affects spin ordering. \nThey also predicted that different spin structures, including isolated magnetic skyrmions, may \ncoexist in a frustrated magnet,  and that the isolated magnetic skyrmion s also possess \nadditional degrees of freedom  (spin vorticity  and helicity ), similar to that in the magnetic \nskyrmionic bubbles . Their predictions are not only extremely interesting, but also point to \nfurther investigat ion of  the variable topological spin textures  in the frustrated magnets . \nFurthermore,  Pereiro  et al .[28] have theoretically  shown that Heisenberg and DMI   Submitted to  \n3 \n \ninteractions in kagome  magnets can overcome  the thermal fluctuation  and stabilize  the \nskyrmions at relatively high temperature s, even at room temperature . Based on the theoretical \ninvestigations above, we revisit the frustrated magnet s with kagome lattice to search for the \npossible skyrmionic bubbles that are able to stabilize  at room temperature.   \nOne of the promising materials is Fe 3Sn2, suggested  by Pereiro et al.,[28] which  has a \nlayered rhombohedral structure with alternate stacking of  the Sn layer  and the Fe -Sn bilayers  \nalong the c -axis, as shown in Figure 1 a. The Fe atoms form bilayers of offset kagome \nnetworks, with Sn atoms  throughout the kagome layers as well as between the kagome \nbilayers. Very importantly, this material is a non- collinear frustrated ferromagnet with a high \nCurie temperature T c of 640K, and shows a spin reorientation that  the easy  axis rotates \ngradually from the  c-axis to the ab- plane as the temperature decreases.[29-32] Recently, a large \nanomalous Hall effect was  observed in this material, which is strongly related to the frustrated \nkagome bilayer of Fe atoms.[33,34] In this communication, we report that  magnetic skyrmionic  \nbubbles with various spin textures  can indeed be realized in  the single crystals of Fe 3Sn2 at \nroom temperature . The emergence of skyrmionic bubbles and the magnetization dynamics \nassociated with the transition of different bubbles via the field -driven motion of the Bloch \nlines are  revealed by in -situ Lorentz transmission electron microscopy  (LTEM) , and further  \nsupported by the  micromagnetic simulations and magnetic transport measurements. The se \nresults demonstrate that Fe 3Sn2 facilitates a unique magnetic control of topological spin \ntextures at room temperature, making it a promising candidate for further  skyrmion -based \nspintronic devices.  \nHigh -quality single crystals of Fe 3Sn2 were synthesized by a Sn -flux method, as describ ed \nin the Methods section. These crystals are layered and exhibit mirror -like hexagonal faces \nwith a small thickness  (see Figure S1 , SI). By using single -crystal X -ray diffraction (SXRD), \nthe crystal parameters were identified as a = b = 5.3074Å  and c = 19.7011Å , with respect to \nthe rhombohedral unit cell (space group R -3m), agreeing well with the previous studies.[32-34] \nHaving determined the unit lattice parameters and orientation matrix, we then found that the  \nhexagonal face was  normal to [001] with the (100), (010), and (110) faces around (see Figure \nS2, SI). In addition, as shown in Figure S3 , both the  temperature -dependent  in-plane and out -\nof-plane magnetization curves measured on the bulk crystal indicate that the ferromagneti c \ntransition temperature  Tc is about 640K . \nFigure 1b  shows the temperature -dependent UMA coefficient  (Ku), which was estimated \nby the approximation of K u=HkMs/2, where M s is the saturation magnetization  and Hk is the \nanisotropy field  defined as the critical field above which the difference in magnetization   Submitted to  \n4 \n \nbetween the two magnetic field directions (H//ab and H// c) becomes smaller than 2% (see \nFigure S4 , SI). One can notice that the value  of Ku  increases  monotonically  with the decreas e \nof temperature. Simultaneously, the Fe moments gradually rotate from the c -axis towards the \nab-plane (see the inset of Figure  1b), demonstrating a gradual transformation from the \nmagnetically easy axis (uniaxial magnetic anisotropy)  into a  magnetically eas y plane with \ndecreasing the temperature. The most important message conveyed to us by Figure 1b  is that \nthe magnetic domain configuration in Fe 3Sn2 may vary over  a very wide temper ature range of \n80-423K , because the domain structure is strongly  affected  by the magnetic anisotropy. \nTherefore, Fe 3Sn2 should be a good platform for us to explore the correlation between the spin \ntexture and K u in a very wide temper ature range.  \nWe then imaged the magnetic domain structure s using LTEM under zero magnetic field in \nthe temperature range of 300K ~130K, as shown in Figure 1c- f. The corresponding selected-\narea electron diffraction (SAED) patterns suggest that the sample is normal to the [001] \ndirection (see the inset of Figure 1b ). Nanosized stripe domains with an average periodicity \nof ~150nm  were clearly observed . Notably, the value of periodicity  is comparable to that in \nthe bulk (Mn 1-xNix)65Ga35(x=0.5) ,[25] but nearly two times larger than that  in the La 1-\nxSrxMnO 3(x=0.175).[24] The sharp contrast between the dark stripe domains and the bright \nwalls  suggests that the doma ins possess out -of-plane magnetizations and are separated by \nBloch domain walls. With the decrease of temperature, the stripes ’ periodicity l widened \nwhile the d omain wall thickness D  remained  almost a constant (see Figure S5 , SI). \nInterestingly, we found that when the temper ature was lower than 130K, the stripe domains  \ndisappeared and vortex domains formed, indicating that the spin starts to lie into the ab- plane \nbelow 130K . The critical temperature of the LTEM sample co incides with that  of the bulk \nsample , but with a slight deviation .[32-34] This feature suggests that  the spin texture of domains \nin the bulk and LTEM samples show little differences. Th at can be attributed to the fact that \nthe magnetic anisotropy in Fe 3Sn2 is high enough to over come the spin rearrangement effect \nresulting from the increase of demagnetizing energy in the thin LTEM sample. To understand \nthe physics behind this domain structure t ransformation, we performed numerical simulations \nwith estimated parameters of exchange constant ( A) and anisotropy energy  (Ku⊥) associated \nwith the perpendicular component of the anisotropy field (see Methods). As shown in Fig ure \n1g, the stripe domain gradually transformed into a vortex domain with decreasing  Ku⊥, \nagreeing well with the LTEM images. This feature suggests that the domain morphology  in \nFe3Sn2 is mainly governed by the anisotropy perpendicular  to the ab-plane , which is \nconsistent with the simulated results based on a frustrated magnet .[27] It is well known that  the   Submitted to  \n5 \n \nmagnetic domain structure also depends greatly on an ext ernal magnetic field . Therefore , we \nhave simulated the domain structure under different magnetic fields that are perpendicular to \nthe ab-plane  (as shown in Figure 1h). It is interesting  to note that the stripe domains \ngradually transformed into bubbles with increasing the external field. Mo re strikingly, isolated \nmagnetic skyrmions formed when the magnetic field increased to 400 mT.  \nTo experimentally observe the domain structure variation with a n external  magnetic field \npredicted by the simulation, we imaged the domain structure under different magnetic fields \nat room temper ature using LTEM. Figure 2 a-d show s the over-focused  LTEM images under \ndifferent out -of-plane magnetic fields (the corresponding zero- field image is shown in Figure \nS6, SI). The gradual transformation from a  stripe domain structure into magnetic skyrmionic  \nbubbles is clearly observed as the magnetic field increases from 0 to 860mT. In Figure 2a,  we \nshow a snapshot of the transformation process under a magnetic field  of 300mT . One can \nnotice that  the stripe  domains, dumbbell -shaped domains and magnetic bubbles coexist in the \nimage . We actually observed that, during the evolution of the domain structures, the stripe  \ndomains  gradually broke into the dumbbell -like domains first before evolving into the \nmagnetic bubbles. When the magnetic field increased above 800m T, all the stripe s and \ndumbbell -like domains completely transformed into magnetic bubbles. One should note that \nwe o nly changed the external magnetic field and kept all other conditions constant in Fig. 2a-\nd. Therefore, the changes in domain structure can be ent irely ascribed to the exter nal field \neffect.  A closer analysis  of the entire transformation process reveals that the spin texture of \nthe magnetic bubbles changes dramatically  with increasing the magnetic field, as shown in \nFigure  2e-h (the bubbles with different spin textures are notated by different numbers). The \nstruct ural evolution of the magnetic bubbles with increasing external magnetic field should be \nclosely related to the change of topology in  the bubble s, as previously observed.[24,26]  \nTo characterize the topological spin textures of the magnetic bubbles, a transport -of-\nintensity equation (TIE) was employed to analyze the over - and under -focused LTEM images. \nFigure 2i -l display s the spin textures of the bubble domains shown in Figure 2e- h. The white \narrows show the directions of the in- plane  magnetic inductions , while the black regions \nrepresent the domains with out -of-plane magnetic inductions . Bubble “1”, composed of a pair \nof open Bloch lines (BLs), is characteristic of the domain structure com monly observed in \nferromagnetic compounds with uniaxial magnetic anisotropy. In this bubble, the topological \nnumber N is determined to be 0, because the magnetizations of the BLs are nonconvergent. \nFurther increase of the magnetic field induced the formation of bubble “2”, which has two arc-shaped walls with opposite helicity, separated by two BLs. Similar to that of  bubble  “1”,   Submitted to  \n6 \n \nthe spin texture  of bubble “2” is  also not convergent, leading to the same topological number, \nN=0. H owever,  when the  magnetic fi eld increased above 800mT, the topological number of \nbubbles “3” and “4” transforms from 0 to 1, being equal to that of skyrmions. As shown in \nFigure 2k , bubble “3” has a pendulum structure with two BLs, in which the spin textures \nbecome conver gent. Compared with the recent  results obtained in La 1-xSrxMnO 3(x=0.175) by \nYu et al,[24] bubble “3” can also be regarded as a specific type of rarely observed skyrmionic \nbubble. Bubble “4” is the most orthodox  skyrmionic bubble , possessing the same domain \nstructure as the skyrmions observed in chiral magnets. One important feature of this bubble is \nthat the thickness of the Bloch wall is comparable to the radius of the bubble, leading to a small core region. Following previous reports,\n[24] we understand that bubble “3” transformed \ninto bubble “4” through the motion of BLs driven by the magnetic field. Therefore, although \nthe magnetic textures in bubbles “3” and “4” are strikingly different, they are homeomorphi c \nin topology. To understand the above transformations of spin texture in more detail, we \nsuccessfully recorded the transformation process from the topologically nontrivial magnetic \nbubbles to the topologically protected skyrmions in a Fe 3Sn2 (001) thin- plate using in -situ \nLTEM  (see Supplementa ry Movie , SI). Figure 3 a-f presents several  snapshots of the \ntransformation process, selected from a  movie taken by LTEM at 300 K and under different \nout-of-plane magnetic fields. We observed a  pair of BLs move  along the bubble wall and \neventually die out  within 8.3 seconds , as the field increased from 840mT to 850mT.  These \nresults demonstrate clear ly that isolated  magnetic  skyrmions can be realized  in the frustrated  \nFe3Sn2 magnet through BL motion by tuning the external magneti c field, even at room \ntemperature.  \nIn addition to the observation of various spin textures  of skyrmionic  bubbles during BL \nmotion, we further found that the magnetic  domain configurations after turning  off the \nexternal fields depend strongly on the strength of the external fields. If the thin- plate s ample is \nfirst magnetized to saturation  (i.e. the sample is in  single -domain state ), then the domain will \nreturn to the multi- stripe  state after turning off the external magnetic field ( see Figure S7 , SI). \nHowever, if the sample  is magnetized to an  intermediate state  (for example, at a field of \n700mT  as shown in Figure S8 , SI), the domain structure evolves differently  after turning off \nthe external magnetic field.  Figure 4 a shows the under -focused LTEM image taken at 300K  \nafter turning off the external field of  700mT by which the sample was magnetized to an \nintermediate state. The coexistence of different types of magnetic domains, e.g. magnetic \nbubbles, stripes and dumbbell -like domains is clearly observed. After careful analysis of the \nimage using TIE, the detailed spin texture of the domain enclosed by the  square  in Figure  4a   Submitted to  \n7 \n \nis shown in Figure 4e. Unexpectedly, the magnetic bubbles possess three concentric rings . It \nis particularly interesting that the winding directions of  the inner and outer rings are opposite \nto that of  the middle ring, indicating that the helicity reverses inside the bubbles. These  \nspontaneous bubbles  can be considered skyrmions, analogous to those  observed in  BaFe 12-x-\n0.05ScxMg 0.05O19.[22] \nTo explore the evolution of the domain structure with temper ature, the domain structures  in \nthe same region as shown in Figure 4a  were also imaged at several lower temperatures  \nwithout changing the external  field ( Figure 4b -d). These results help us to understand how \nthe domain structure evolve s under the influence of different magnetic energies, i.e. magnetic \nanisotropy, exchange energy , and static magnetic energy.  As the temperature decreases , the \nsize of the  spontaneous bubbles change slightly  and the stripes gradually transform into \nskyrmions. Consequently, the number of spontaneous bubble s increases  with decreasing \ntemperature. The maximum density of the bubbles appear s at 250K. The spontaneous bubbles \nvanish as the temperature further decrease below 100K , due to the transformation from \nuniaxial, out -of-plane anisotropy into in -plane anisotropy ( Figure  1b). The corresponding \nspin textures of the skyrmions  marked in Figure 4b -d were  also extracted by using  TIE \nanalysis, as shown in Figure 4f -h. It was found that the size of the innermost ring  increases \nwith decreasing the temperature. Micromagnetic simulations are currently ongoing  in order to \nexplore the interplay  among the different magnetic parameters behind t hese features.  \nThe formation of  magnetic skyrmionic  bubbles and isolated skyrmion spin textures  in the \nbulk Fe 3Sn2 single crystals were further studied and confirmed by magnetic and magneto -\ntransport measurements, similar to previous studies of other skyrmion -based \nmaterials .[7,20,25,35] It should be noted here that t he sample  for magnetic and magnetotransport \nmeasurements is from the same crystal s used for LTEM  observations . In Figure  5a, we  show \nthe dependence of magnetoresistance (MR) on the magnetic field ( H) that is applied normal  to \nthe ab-plane in the temperature range of 100 -400K. The inset shows the details of the MR -H \ncurve at 3 00K,  in which two broad peaks are clearly seen at about 200 mT and 800mT , \nrespectively. Based on the LTEM resul ts shown in Figure  2 and our analysis, it can be  \nconclude d that the peak at 200mT  (Ha) reflects the starting point of the transformation from \nstripes to magnetic bubbles, whereas the peak  at 800mT ( Hm) represents the starting point of \nthe transformation from  magnetic bubbles to ins ulated skyrmions. When the magnetic field \nincreases above 900mT ( Hc), the sample reaches the magnetically saturated state (see the M -H \ncurves in Figure  S4) in which  the spins are aligned along the field direction , and \nconsequent ly all the skyrmions  die out . A close inspection of the MR -H curves in the range of   Submitted to  \n8 \n \n400-100K reveals that all three critical fields shift monotonically with temperature as \nindicated by the dotted lines. As shown in Figure 5b , we can also identify the three critical \nfields in the field -dependent AC-susceptibility, though the critical fields are slightly lower \nthan those observed in the MR results and show slightly weaker  temper ature dependence. \nBased on the LTEM, MR, and AC -susceptibility results, we can roughly create a magnetic \nphase diagram as depicted in Figure  5c. Based on the phase diagram, we predict that the \ntopological spin texture states in Fe 3Sn2 may extend to a much higher temperature, perhaps up \nto Curie temperature (~640K). However, due to the technical limitations of our current \nmeasurements, we could not perform the experiments at a temperature higher than 400K. It is \nwell known that  stable skyrmion ic states at hig h temperatures are critical for technical \napplications in magnetic storage and spintronic s devices. Therefore, the observation of a \nskyrmion state in Fe 3Sn2, not only in a very wide temper ature range but also at a high \ntemper ature, strongly suggest s that Fe 3Sn2 is a very promising material for both sky rmion \nphysics and potential technical applications of magnetic skyrmions.  \nThe ongoing  and future studies will include electrically prob ing various exciting \nphenomena  in this material, such as the skyrm ion Hall effect ,[36, 37] and quantiz ing the \ntransport of magnetic skyrmions,[38,39] similar to the study of emergent electrodynamics of \nskyrmions in bulk chiral materials .[40] \n \nExperimental Section  \nSample Preparation : Single crystals of Fe 3Sn2 were synthesized by using the Sn -flux method \nwith a molar ratio of Fe : Sn = 1:19. The starting materials were mixed together and placed in \nan aluminum  crucible with higher melting temperatures at  the bottom. This process was \nperformed  in a glove box fill ed with a rgon gas. To avoid the influence of volatilization of Sn \nat high temperatures, the whole assembly was first sealed inside a tantalum (Ta) tub e under \nproper Ar pressure. The Ta tube was then sealed in a quartz tube filled with 2 mbar Ar \npressure. The crystal growth was carried out in a furnace by heating the tube from room \ntemperature up to 1150 ℃ over a period of 15  h, holding at this temperature for 72 h, cooling \nto 910℃ over  6 h, and subsequently cooling to 800℃ at a rate of 1.5 K/h. The excess S n flux \nwas removed by spinning the tube in a centrifuge at 800℃. After the centrifugation process, \nmost of the flux contamination was removed from the surfaces of crystals  and the remaining \nflux was polished. \nMagnetic and Transport Measurements: The magnet ic moment was measured by using a \nQuantum Design physical properties measurement system (PPMS)  between 10K and 400K,   Submitted to  \n9 \n \nwhereas the magnetic moment above 400K was measured by using a vibrating sample \nmagnetometer  (VSM).  To measure the (magneto -) transport properties, several single crystals \nwere milled into a bar  shape with a typical size of about 0.6 × 0.4 × 0.05 mm3. Both \nlongitudinal and Hall resistivity we re measured using a standard four -probe method on a \nQuantum Design PPMS. The field dependence of the Hall resistivity was obtained after \nsubtracting the longitudinal resistivity component. \nLTEM Measurements: The thin plates for Lorentz TEM observations were cut from bulk \nsingle -crystalline samples and thinned  by mechanical polishing and argon- ion milling. The \nmagnetic domain contrast was observed by using Tecnai F20 in the Lorentz TEM mode  and a \nJEOL -dedicated Lorentz TEM, both equipped with liquid- nitro gen, low-temperature holders \n(≈100 K) to study the temperature dependence of the magnetic domains. The magnetic \nstructures were  imaged directly in the electron microscope. To determine the spin helicity of \nthe skyr mions, three sets of images  with under -, over -, and just (or zero) focal lengths were \nrecorded by a charge- coupled device (CCD) camera, and then the high -resolution in-plane \nmagnetic induction distribution mapping was obtained by QPt software, based on the \ntransport of the intensity equation (TIE) . The inversion of magnetic contrast can be seen by \ncomparing  the under - and over -focused images. The colors and arrows depict the magnitude \nand orientation of the in- plane  magnetic induction . The objective lens was turned off when the \nsample holder was inserted, and the perpendicular magnetic field was applied to the stripe  \ndomains by increasing the objective lens gradually in  very small increments . The specimens \nfor the TEM observations were prepared by polishing, dimpling, and subsequently ion milling.  \nThe crystalline orientation of the crystals was determined by selected -area electron diffraction  \n(SAED).  \nMicromagnetic  Simulations : Micromagnetic  simulations were carried out with three -\ndimensional object oriented micromagnetic  frame work (OOMMF) code,  based on the LLG \nfunction.[41] Slab geometries of dimensions 2000 nm×  2000 nm × 100 nm were used, with \nrectangle mesh of  size 10nm×10nm×10nm. We used a damping constant α=1 to ensure quick \nrelaxation to the equilibrium state. The material parameters were chosen according to the \nexperimental values of Fe 3Sn2, where the saturation magnetization M s = 5.66×105 A/m at \nroom temperature, and the uniaxial magnetocrystalline anisotropy constant K u=1.8×105 J/m3. \nAs the magnetic moment aligned obliquely along the c -axis, we defined K u⊥as the anisotropy \nenergy associated with the perpendicular component of the anisotropic field, i.e. 𝐾𝐾u⊥=\n�1\n2�𝐻𝐻k⊥𝑀𝑀s. The exchange constant A  was estimated to be 1.4×10-11 A/m by  𝐷𝐷=π�𝐴𝐴𝐾𝐾u⁄, \nwhere D is the domain wall width obtained from the LTEM results. These three parameters   Submitted to  \n10 \n \nvary with temperature, hence we investigated the dependence of domain morphology  on \nexchange energy ( A) and anisotropy energy as s hown in Figure 1g  by fixing the M s. The \nequilibrium states are all obtained by fully relaxing the randomly distributed magnetization.  \nThe simulations on the field -dependent domain structures at 300K (as shown in Figure 1h ) \nwere conducted  at zero temperature (no stochastic field) but the values o f parameters \ncorrespond to 300K, because the change tendency of magnetic do main  alters little by \ntemperature.  \nSupporting Information  \nSupporting Information is available from the Wiley Online Library or from the author.  \n \nAcknowledgements  \nThe authors thank  Jie Cui and Dr. Yuan Yao  for discussions and thei r help in L TEM \nexperiments . This work is supported by the National Natural Science Foundation of China \n(Grant Nos. 11474343, 11574374, 11604148, 51471183, 51590880, 51331006 and 5161192), \nKing Abdullah University of Science and Technology (KAUST) Office of Sponsored \nResearch (OSR) under Award No: CRF -2015- 2549- CRG4, China Postdoctoral Science \nFoundation NO. Y6BK011M51, a project of the Chinese Academy of Sciences  with grant \nnumber KJZD -EW-M05 -3, and Strategic Priority Research Program B of the Chinese \nAcademy of Sciences under the grant No. XDB07010300. \n \nReferences:  \n[1] T.H.R. Skyrme, Nucl. Phys . 1962, 31, 556. \n[2]K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A. Thiaville,T. Ono, Nat. \nMater. 2007, 6, 269. \n[3] R. Hertel, C. M. Schneider, Phys. Rev. Lett. 2006, 97, 177202. \n[4] A. Fert, V. Cros, J. Sampaio, Nat. Nanotechnol. 2013, 8, 152. [5]N. Nagaosa, Y.Tokura, Nat. Nanotechnology . 2013, 8, 899. \n[6] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, P. \nBoni, Science 2009, 323, 915. \n[7] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Niklowitz, P. Boni,Phys. Rev. \nLett. 2009, 102, 186602. \n[8] C. Pappas, E. Lelivre -Berna, P. Falus, P.M. Bentley, E. Moskvin, S. Grigoriev, P. Fouquet, \nB. Farago, Phys. Rev. Lett . 2009, 102, 197202.   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Loidl, Nat. \nMater . 2015, 14, 1116. \n[17] Y. Onose, Y. Okamura, S. Seki, S. Ishiwata, Y . Tokura ,Phys. Rev. Lett . 2012, 109, \n037603. \n[18] M. Mochizuki, X.Z. Yu, S. Seki, N. Kanazawa, W. Koshibae, J. Zang, M. Mostovoy, Y. \nTokura, N. Nagaosa,Nat. M ater. 2014, 13, 241. \n[19] P. Milde, et al., Science2013,340, 1076.  \n[20]Y. Tokunaga, X.Z.Yu, J.S.White, H.M. Rǿnnow, D. Morkawa, Y. Taguchi, Y. Tokura, \nNat. Commun . 2015, 6, 7638. \n[21]Y.S. Lin, J. Grundy,E. A. Giess, Appl. Phys. Lett . 1973, 23, 485. \n[22]X. Z. Yu, M. Mostovoy, Y. Tokunaga, W. Z. Zhang, K. Kimoto,Y. Matsui, Y. Kaneko, N. \nNagaosa, Y. Tokura, Proc. Natl. Acad. Sci.USA 2012, 109, 8856. \n[23] X.Z.Yu, Y. Tokunaga1, Y. Kaneko1, W.Z. Zhang, K. Kimoto, Y. Matsui, Y. Taguchi , Y. \nTokura1,Nat. Commun. 2014, 5, 3198. [24] X.Z.Yu, Y. Tokunaga, Y. Taguchi, Y. Tokura, Adv. Mater. 2017, 29, 1603958. \n[25] W. H. Wang, Y. Zhang, G. Xu, L. Peng, B. Ding, Y. Wang, Z. Hou,X. Zhang, X. Li, E. \nLiu, S. Wang, J. Cai, F. Wang, J. Li, F. Hu, G. Wu,B. Shen, X. Zhang, Adv. Mater. 2016, 28, \n6887. [26]C. Phatak,  O. Heinonen, M . D. Graef,  A. P-. Long , Nano Lett . 2016, 16, 4141. \n[27]A. O. Leonov, M. Mostovoy, Nat. Commun. 2015, 6, 8275.   Submitted to  \n12 \n \n[28]M. Pereiro, D. Yudin, H. Chico, C. Etz, O. Eriksson, A. Bergman, Nat. Commun. 2014, 5, \n4815. \n[29] B. Malaman, D. Fruchart, G. L. Caër, J. Phys. F. 1978, 8, 2389. \n[30] G. L. Caër, B. Malaman, B. Roques, J. Phys. F. 1978,8, 323. \n[31]G. L. Caër, B.Malaman, L. Häggstr öm, and T. Ericsson, J. Phys.F.1979 , 9, 1905. \n[32] L. A. Fenner, A. A. Dee, and A. S. Wills, J. Phys.: Condens. Matter . 2009, 21, 452202. \n[33] T. Kida, L. A. Fenner, A. A. Dee, I. Terasaki, M. Hagiwara, A. S. Wills, J. Phys.: \nCondens. Matter. 2011, 23, 112205. [34] Q. Wang, S. S. Sun, X. Zhang, F. Pang, H. C. Lei, Phys. Rev. B. 2016, 94, 075135. \n[35] H. F. Du, J.P. DeGrave, F. Xue, D. Liang, W. Ning, J.Y.Yang, M.L. Tian, Y.H. Zhang, S. \nJin, Nano Lett . 2014,14, 2026. \n[36] T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Waner, C. Frznz, C. Pfleiderer, K. Everschor, \nM. Garst, A. Rosch, Nat. Phys . 2012, 8, 301. \n[37]W. J. Jiang, X. C. Zhang, G. Q. Yu, W. Zhang, X. Wang, M. B. Jungfleisch, J. E. Pearson, \nX. M. Cheng, O. Heinonen, K . L. Wang, Y . Zhou, A . Hoffmann , S. G. E. te  Velthuis , Nat. \nPhys . 2017, 13, 162. \n[38] K. Litzius, I. Lemesh, B. Kr üger, P. Bassirian, L. Caretta, K. Richter, F. B üttner, K. Sato, \nO. A. Tretiakov, J. F örster, R. M. Reeve, M. Weigand, L. Bykova, H. Stoll, G. Schütz, G. S. \nD. Beach, M. Kl äui, Nat. Phys . 2017, 13, 170. \n[39] S. Z.Lin, C. Reichhardt, C. D. Batista,  A. Saxena,  Phys. Rev. Lett.  2013, 110, 207202. \n[40] C.Reichhardt, D.Ray,   C. J. O.Reichhardt, Phys. Rev. B . 2015, 91, 104426. \n[41]M. J. Donahue , D. G. Porter, NISTIR 1999 6376;http://math.nist.gov/oommf . \n    Submitted to  \n13 \n \n \n \n  \n \n  \n \n  \n \n  \n \n  \n \n \nFigure 1 . Structure, magnetic properties, and micromagnetic stimulations of Fe\n3Sn2. a) The \ncrystal structure of Fe 3Sn2 (up), a top view of the  kagome layer of Fe atoms (down, left ) and a \npossible spin (arrows)  configuration of the Fe atoms (down, right ). b) The temperature \ndependence of anisotropy constant ( Ku) in the temperature range 10- 400K. The insets (from \nright to left) show the schematic of the angle between the magnetic easy axis and the c-axis at \n300, 150, and 6K, respectively. c-f) The representative images of the domain structures of \nsame area in a Fe 3Sn2 thin-plate taken by Lorentz tr ansmission electron microscopy (LTEM) \nwith an electron beam perpendicular to the ab- plane of Fe 3Sn2 when the sample temper ature \nwas lowered from 300 K to 130K in  zero external magnetic field. The inset  of (c) shows the \ncorresponding selected -area electron diffraction (SAED) pattern. g)  The plan view  of \nequilibrium states under different ratios of  exchange constant A  and perpendicular component \nfor the magnetocrystalline  magnitude  𝐾𝐾u⊥for a fixed  Ms=5.7×105 A/m and thickness (100nm) \nin zero external magnetic field. The magnetization along the z -axis (m z) is represented by \nregions in red (+m z) and blue ( -mz), whereas the in -plane magnetization (m x, m y) is \nrepresented by the white regions. h ) The simulated  field-dependent domain morphology under \nseveral magnetic fields that capture the domain evolution from stripes to type -Ⅱbubbles and \nskyrmionic bubbles.  \n  Submitted to  \n14 \n \n \n \n \n \n  \n \n  \n \n \n \n \n \n \nFigure 2. Magnetic field dependence of the magnetic domain morphology imaged using \nLTEM at  300K . a-d) The over-focused LTEM images under different out -of-plane magnetic \nfields at 300K. The regions in the white boxes show the different types of magnetic bubble \ndomains. e-h) Enlarged LTEM images of the  white boxes in (a -d) showing the  magnetic \nbubble domains . i-l) Corresponding  spin textures for the bubble domains shown in (e -h), \nextracted from the analysis usin g TIE. Colors (the inset  of (i) shows the color wheel) and \nwhite arrows represent the direction of in -plane magnetic induction, respecti vely, whereas the \ndark color represents the magnetic induction along the out -of-plane direction. ( e) and (f) \ndisplay the type -Ⅱ bubbles, and (k) and  (l) show different types of skyrmionic bubbles.  \n \n \n \n \n \n \n \n \n \n \n \n  Submitted to  \n15 \n \n \n             \n \n \n \n \n \n \nFigure 3.  Evolution of the  magnetic bubble  through Bloch line (BL) motion, induced by the \nmagnetic field. a- f) Series of LTEM images of bubble “3” observed at different times and \nfields applied along the c -axis. The field was increased  slowly for 8 second s from 840 to \n850mT. \n  \n  Submitted to  \n16 \n \n \n \nFigure 4.  LTEM images of the magnetic domain structures taken at different temper atures  \nafter turning off  the external magnetic field applied along c -axis. The field of 700mT is not \nstrong enough to saturate the sample magnetically.  a-d) Stripe domains and skyrmionic \nbubbles coexist in the sample  over the temper ature range of 300K to 170K, after turning off \nthe magnetic field. The regions enclosed in the squares are the skyrmionic bubbles. e -h) The \ncorresponding magnetization textures obtained from the TIE analysis for the bubble domains \nin the squares in (a -d). The inset  of (e) shows the color wheel. The spontaneous skyrmionic \nbubbles with triple -ring structures and random helicities are observe d.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  Submitted to  \n17 \n \n \n \n \n  \n \n \n \n  \nFigure  5. Field dependence of magnetoresistance (MR), AC -susceptibility and the magnetic \nphase diagrams of Fe\n3Sn2. a) The magnetic field ( H) dependence of MR obtained in the \ntemperature range of 100K -400K, with magnetic fields  applied normal to the ab- plane. The \ninset shows the details of the MR -H curve measured at 3 00K.  b)The magnetic field \ndependence of the real part of AC -susceptibility χ in the temperature range of 100K -400K. c) \nMagnetic phase diagram of bulk sample in the magnetic field versus the temperature plane, as \ndeduced from the temperature dependence of MR and χ curves. The three fully filled circles  \nindicate the experimental data obtained from the in situ LTEM observations. The error bars \nwere added based on the results measured on three different samples.  \n \n \n(b) (c) (a)   Submitted to  \n18 \n \nVarious  and spontaneous magnetic  skyrmionic bubbles are experimental ly observed for \nthe first time, at room temperature in a frustrated  kagome magnet Fe 3Sn2 with unixial \nmagnetic anisotropy. The magnetization dynamics were investigated using in-situ Lorentz \ntransmission electron microscopy,  revealing that the transformation between different \nmagnetic bubbles and domains are via the motion of Bloch lines driven by applied external \nmagnetic field. The results demonstrate that Fe 3Sn2 facilitates a unique magnetic control of \ntopological spin textures  at room temperature , making it a promising candidate for further  \nskyrmion -based spintronic devices.  \n \nKeywords: Skyrmio nic bubbles; Topological spin textures; Kagome magnet ; Fe 3Sn2; \nSpintronic devices  \n \n \nZhipeng Hou*, Weijin Ren *,Bei Ding*, Guizhou Xu, Yue Wang, Bing Yang, Qiang Zhang, \nYing Zhang, Enke Liu, Feng Xu, Wenhong Wang, Guangheng Wu, Xi -xiang Zhang, Baogen \nShen, Zhidong Zhang  \n \nObservation of Multiple  and Spontaneous Skyrmionic Magnetic Bubbles at Room \nTemperature in a Frustrated Kagome Magnet with Uniaxial Magnetic Anisotropy  \n \n \nTOC  Figure  \n \n \n \n  \n \n \n \n  \n \n  \n \n \n \n \n \n \n \n \n \n \n. \n \n \nFrustrated kagome lattice @ Fe3Sn2 b c \na \na b \nVarious  magnetic bubbles @ RT    Submitted to  \n19 \n \nSupplementary Information to  \nObservation of  Various  and Spontaneous Magnetic Skyrmionic Bubbles at Room -\ntemperature in a Frustrated Kagome Magnet with Uniaxial Magnetic Anisotropy \n \nZhipeng Hou*, Weijin Ren *, Bei Ding *, Guizhou Xu, Yue Wang, Bing Yang, Qiang Zhang, \nYing Zhang, Enke Liu, Feng Xu, Wenhong Wang, Guangheng Wu, Xi -xiang Zhang, Baogen \nShen, Zhidong Zhang  \n \nSingle -crystal X -ray diffraction (SXRD) was performed on the  crystal shown in Figure  S1 \nwith a Bruker APEX II diffractometer using Mo K -alpha radiation (lambda = 0.71073 A) at \nroom temperature. Exposure time was 10 seconds with a detector distance of 60 mm. Unit cell \nrefinement and data integration were performed with Bruke r APEX3 software. A total of 180 \nframes were collected over a total exposure time of 2.5 hours. The crystal lattice parameters were established to be a = b = 5.3074Å, c = 19.7011Å with respect to the rhombohedral unit \ncell (space group R -3m), agreeing well  with the previous studies. As shown in Figure  S2, to \nascertain  the crystal orientations, the sample was mounted on a holder and Bruker APEX II \nsoftware was used to indicate the face normal of the crystal , after the unit cell and orientation \nmatrix were determined. One can notice that the hexagonal face is normal to [001] with the \n(100), (010), and (110) faces around.  \n \n             \n \nFigureS1 . X-ray diffraction pattern of a Fe\n3Sn2 single crystal along the perpendicular \ndirection of  hexagonal surface, which indicates that the hexagonal surface is parallel to the \nab-plane and perpendicular to the c -axis. Inset: The typical photograph of Fe 3Sn2 single \ncrystal placed on a millimeter grid. The crystal is 0.3 mm × 0. 3 mm × 0. 2 mm in size and \npossesses hexagonal  mirror -like surfaces . \n \n \n  Submitted to  \n20 \n \n \n \n \n \n \n \n \nFigure S2. (a) Single -crystal X -ray diffraction processi ng image of the (00l ) plane in the \nreciprocal lattice of Fe 3Sn2 obtained on the crystal mentioned above. No diffuse scattering is \nseen and all the resolved spots fit the crystal lattice structure established for Fe 3Sn2. (b) \nVarious crystal planes and their corresponding normal directions of Fe 3Sn2 single crystal. The \nyellow striped region is the glue to af fix the crystal to the holder . \n \n \n   \n \n \n   \n \n \n \n \n \n \n \n \nFigure S3. Temperature dependence of magnetization with the field -cooling (FC) model in an \nexternal  magnetic field of 500Oe between 5K and 700K. As is shown in Fig ure S3, the Curie \ntemperature T\nc is established to be 660K, which is similar  to previous reports. When the \ntemperature falls below Tc, the magnetization  for both magnetic fields ( H//c and H//ab)  first \ndecreases and then starts to increase at 420K , reaching a maximum at 80K. When the \ntemperature decreases below 80K, the slight decr ease in magnetization  can be attributed  to the \nentrance of the spin glass state (SGS).    \n \n \n  Submitted to  \n21 \n \n \n  \n \n    \n \n   \n \n \nFigure S4. a) The magnetic  field dependence of magnetization in fields parallel to the c -axis \n(black line and  black symbol ) and ab- plane (red  line and  red symbol ) in the temperature range \nof 400K -6K. b)  The temperature dependence of the saturation magnetization  M\ns. \n \n \n \nFigureS 5. a) The temperature dependence of domain wall thickness D . The error bar denotes \nthe deviation of three individual width measurements. One can notice that the value of D  is \nnearly independent from  the change in  temperature. The exchange stiffness constant A  can be \nestablished by using the equation , 𝐴𝐴=𝐷𝐷𝐾𝐾𝑢𝑢2\n𝜋𝜋2.b) The temperature  dependence of A. By \ndecreasing the temperature, the value of A  increases correspondingly .  \n(a) (b)   Submitted to  \n22 \n \n \nFigureS 6. a-b) The over- and under -focused LTEM images under  zero magnetic field at \n300K. c-f) Corresponding under -focused TEM ima ges for Figure 2 (a -d). The boxed regions  \ncorrespond to the  magnetic bubbles  shown in  Figure 2 (a -d). \n \n \n \nFigure S7. The under -focused LTEM  images after a saturated magnetization. When the \nsample is magnetized to a saturated state, the domain reverts to the stripe.  \n \n  \n \n \n  Submitted to  \n23 \n \n \n \n \nFigure S8. The corresponding  over-focused  (a, b, c, d)  and under -focused (e, f, g, h)  LTEM  \nafter an unsaturated magnetization in a different region from that in Figure 3. If the sample is \nmagnetized to an intermediate state, then the  skyrmionic bubbles with concentric rings  appear \nafter the magnetic field decreases to zero.  \n \n \n \n"    },    {        "title": "1209.1280v1.Field_driven_femtosecond_magnetization_dynamics_induced_by_ultrastrong_coupling_to_THz_transients.pdf",        "content": "Title:  \nField-driven femtosecond magnetization dynamics induced by ultra-\nstrong coupling to THz transients \n  \nAuthors:  \nC. Ruchert1, C. Vicario1, F. Ardana-Lamas1,4, P.M. Derlet2, B. Tudu3, J. Luning3 and C.P. Hauri1,4 \n \nAffiliations: \n1Paul Scherrer Institute, SwissFEL, 5232 Villigen PSI, Switzerland \n2Paul Scherrer Institute, Condensed Matter Th eory Group, 5232 Villigen PSI, Switzerland \n3Université Pierre et Marie Curie, LC PMR, UMR CNRS 7614, 75005 Paris, France \n4Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland \n \nAbstract: \nControlling ultrafast magnetization dynamics  by a femtosecond laser is attracting \ninterest both in fundamental science and i ndustry because of the potential to achieve \nmagnetic domain switching at ever advan ced speed.  Here we report experiments \nillustrating the ultrastrong and fully coh erent light-matter coupling of a high-field \nsingle-cycle THz transient to the magnetizat ion vector in a ferromagnetic thin film. \nWe could visualize magnetization dynamics  which occur on a timescale of the THz \nlaser cycle and two orders of magnitude fa ster than the natural precession response \nof electrons to an external magnetic fiel d, given by the Larmor frequency. We show \nthat for one particular scattering geomet ry the strong coherent optical coupling can \nbe described within the framework of a renormalized Landau Lifshitz equation. In \naddition to fundamentally new insights to ultrafast magnetiz ation dynamics the \ncoherent interaction allows for retrievin g the complex time-frequency magnetic \nproperties and points out new opportuniti es in data storage technology towards \nsignificantly higher storage speed.   \nMain text: \nFemtosecond coherent manipulation of magnetic  moments at ever advanced speed is of \nparamount interest to future data storage and processing t echnology in order to meet the \ngrowing demand of increased data rates and access speed. Today’s widely used mass \nstorage devices, based on magnetic  domain flipping by the giant \nmagnetoresistance(GMR) effect ( 1,2), are limited to access times in the nanosecond \nrange. The quest towards faster manipulation of magnetization requires a stimulus which \nis capable of controlling the magnetization ve ctor more rapidly. Following those lines, \nfemtosecond pulses has been used since 1996 as a precursor to overcome GMR speed \nlimits by inducing faster demagnetization. Lase r-induced ultrafast demagnetization has its \norigin in the heat deposition which causes a paramagnetic phase transition upon crossing \nthe Curie/Neel temperature( 3-5). The ultrafast heat-induced  demagnetization, however, is \nincoherent and followed by a long-standing th ermal recovery phase (up to nanoseconds) \nwhich hinders femtosecond re-access. Due to th e incoherent nature of the near-infrared \nstimulus (heat), physical in sight and visualization into  the ultrafast optomagnetic \ncoupling and the magnetization dynamics occurri ng at the very onset of the laser-matter \ninteraction has been inaccessible which antic ipated a complete understanding of the \noptomagnetic dynamics. For next generation storage technology, significantly higher \nimpact is expected if heat deposition on th e storage media is omitted and magnetization is \ncontrolled directly with the stimulus’ field. First attempts to circumvent heat-induced demagnetization have been undertaken ( 6-9) but purely field-driven ultrafast dynamics of \nmagnetization remained inaccessible.  In this Letter we report on the control and visualization of femtosecond coherent \nmagnetization dynamics in a ferromagnetic coba lt thin film. An ultr astrong optomagnetic \ncoupling is established by use of an intense single-cycle THz pulse with a stable absolute phase ( 10,11 ). We give an experimental proof  that the THz-induced magnetization \ndynamics are coherently governed by the absolute phase and amplitude of the initiating magnetic field rather than its intensity enve lope and heat. Our stud ies unveil unexpected \nrich and ultrafast field-driven magnetizati on dynamics which are not governed by the \n(slow) Larmor precession fre quency, but by the ultr afast THz stimulus field oscillations. \nIn fact, we show for the first time that the field dynamics of the stimulus are directly \nimprinted on the magnetization dynamics and th at the magnetisation dynamics vanish as \nsoon as the laser field vanishes. The ultras trong coherent optomagnetic interaction gives \nfor the first time the possibility to retrie ve the complex magnetic impulse response in \nboth the time and frequency domain. The experi mental data for one particular scattering \ngeometry (\next THz B B\n ) is approximately described by an overdamped Landau-Lifshitz \nformalism thereby accounting for effects caused by the str ong co-propagating electric \nfield. The results represent the dawn of a new era towards purely optical, coherent \nmagnetic domain switching on the time scale of the stimulus’ in the complete absence of heat.   As a model system for our investigations , a 10 nm ferromagnetic cobalt thin film \nat room temperature (290 K) is used, which due to the thin-film shape anisotropy exhibits \nan in-plane magnetization. Pr ior to THz excita tion, the cobalt macroscopic magnetic \nmoment is well aligned by an external field \nextB\nand the magnetization is saturated (Fig. \n1). The THz field (blue) with a variable linear polarization is used to initiate optical field-induced coherent magnetization dynamics in the thin film which are detected by the time-\nresolved magneto-optical Kerr effect (MOKE). As a probe, a sub-50 fs near infrared \npulse (=800 nm) is employed which is inherent ly synchronized to the THz pump pulse. \nThe octave-spanning THz pulse carries 1.5 optical cycles with a 0.3 Tesla field \namplitude, centered at 2.1 THz. Most important for our investigations is the fact that the \nTHz magnetic field )(tB\n is phase-locked to its intensity envelope, )(2tB\n, as is the co-\npropagating electric field ) (tE\n. This turns out to be a prer equisite for the observation of \nsub-cycle magnetization dynamics. \nUnder approximately equilibrium conditi ons, the interaction between a magnetic \nfield B\nand the magnetization M\nis well described by the Landau-Lifschitz (LL) \nequation,   B M MMB MdtMd  \n . The first term BM\n leads to simple precession \nof the magnetic moment M\n around the external field B\n, via the Zeeman torque with the \ngyromagnetic constant . This motion is the well-known Larmor precession with the \nfrequency GHzL10 for a 0.3 T magnetic field. The s econd term leads to a damping of \nthe precession tending to orient the magnetizat ion towards the external field, with the \ndamping factor  an empirical parameter that embodi es diverse dissipative processes \ninvolving lattice and electronic degrees of fr eedom. While this picture holds well for a \nconstant or slowly varying external magnetic field, the ultrafast magnetization dynamics \ntriggered by our intense single-cycle THz stim ulus are expected to  look fundamentally \ndifferent, in particular owi ng to the strong co-propagatin g electric field component \n(THzEmax1 MV/cm).  These expectations are corroborated in fi gure 2(a). The results show the observed \nmagnetic dynamics represented by the MOKE signa l (red line) during the interaction time \nof the THz magnetic field (black line) with the sample. The THz magnetic field which is \northogonally polarized to the al igned magnetic moment (positi on “A” in Fig 1) induces a \nstrong and instant torq ue to the macroscopic magnetic moment. It turns out that the \nmagnetization dynamics are not governed by the Larmor precession frequency caused by \nthe Zeeman interaction term BM\n . In contrast the magnetic response follows almost \ninstantaneously the much faster oscillat ions of the strong femtosecond THz pulse \nfeaturing frequency components in th e multi-THz range. After a delay of ≈50 fs the \nmagnetization moment M\n pursues a de- and re-magnetization cycle similar to the \ndriving THz magnetic field os cillations. The strong couplin g of the THz pulse to the \nmagnetization vector is reflected by the la rge magnetic Kerr rotation of up to 200 mrad \nmeasured by MOKE. The MOKE frequency spectrum and its corresponding spectral \nphase depicted in figure 2b (red line) manife sts the occurrence of strong THz constituents \nwith frequencies comparable to the THz st imulus spectrum (black line). The magnetic \nresponse is thus fully dominated by the fre quency components of the THz magnetic field \n(2.1 THz and 0.5 THz) while a contribution of the Larmor precession frequency is not \nobserved. Remarkably the strong phase-loc ked stimulus completely dominates the \nmagnetic dynamics in a reproducible and coherent  manner. This fact is reflected in the \nalmost identical spectral phase of both th e THz stimulus and the MOKE response (Fig \n2b). The presented experimental results clearly manifest that the magnetization M\n \nfollows the initiating magnetic field ) (tB\nrather than its intensity envelope )(2tB\n on a \ntime scale which has not been expected to be that fast ( 12). Our findings clearly corroborate that Terahertz laser pulses with a st able absolute phase offer an entirely new \nexperimental avenue for the exploration and control of coherent sub-cycle magnetization \ndynamics. \nThe ultrafast magnetization dynamics become  observable thanks to the absence of \nheating and ionizing effects th roughout the interaction. In fact, a Terahertz photon carries \na photon energy which is almost three orders of  magnitude below that of a near-infrared \nphoton (0.004 eV versus 1.6 eV). This avoids  heating of the elect ronic sub-system and \nenables the discovery of unexpect ed rich coherent motions of  the magnetization, such as \nsub-cycle de- and re-magnetization dynamics dur ing the very first optical cycles of the \nTHz pulse interaction. These dynamics are for the first time driven coherently by the \nphase properties of the intense THz field. \nIn depth physical insights on the observe d dynamics can be gained by further \ninterrogating the LL equation. Even though LL ignores by defau lt the strong electric field \ncontribution of the THz pulse the equation is  found to mimic the experimentally observed \nmagnetization dynamics surprisingly well when a large damping term  and modified \nHext is introduced. Unde r those conditions the calculated dynamics of M\n follow the \nstimulus field oscillations and reproduce a la rge part of the measured characteristic \ntemporal dynamics (Fig. 2c). As experimentally observed the calculated magnetic moment (green) pursues the stimulus’ field osci llations (grey dashed) with a few tens of \nfemtosecond retardation and perfectly reorie nts in the external field after the THz \ninteraction terminated. A comparison with the experimental data gives evidence that the \nZeeman-driven precession is insignificant and dominated by strong damping dynamics \nwhose damping parameter is three orders of ma gnitude larger than that normally expected for equilibrium Co. While the over-damped LL limit represents well the overall \nmagnetization dynamics (green) with regards to  the experimental MOKE signal (red), the \nsubsequent developing phase lag observed in  the experiment is less well reproduced \nindicating some non-linear memory effects as  a consequence of the strong electric and \nmagnetic field interaction (13).  \nThe measured magnetization dynamics ) (tM\nshown in Fig 2 indicates that the \nphase properties of the magnetization are fully determined by the phase-locked THz \nmagnetic field. We call this coherent coupli ng. While conventional he ating of the sample \nabove T C results in a decrease of the macroscopic magnetization magnitude M\n, the \ncoherent coupling mechanism postulated here is  supposed to alter ex clusively the angular \norientation of M\n, but not its net magnitude M\n. To corroborate this assumption an \nadditional MOKE measurement was performed under identical conditions as shown in \nFig. 2a but with an oppositely  polarized THz pulse (Fig.1 po sition “B”). This corresponds \nto a  shift of the temporal phase. With this in verted coherent stimul us field (Fig. 3(a)) \nthe measured magnetic dynamics unveil a MOKE  signal which is equivalently inverted. \nThe sum of the corresponding two MOKE signa ls leads to a full annihilation of the \nmagnetization dynamics while the difference si gnal give rise to a two-times enhanced \nsignal (Fig. 3(b)), as expected from a cohe rent interaction. We emphasize that the \nannihilated (“zero”) signal in  (b) is a direct experimental proof for the absence of an \nincoherent heat stimulus. It is achieved onl y when the full vectorial (i.e. angular and \nmagnitude) information of M is governed coherently by the THz field leading to a \nvectorial sum of M\nwhich is zero. In contrast, an analog experiment with oppositely polarized heat stimuli would yi eld a sum of the two MOKE signa ls different to zero since \nthe magnitude M\n of the magnetization undergoes a ch ange via the paramagnetic phase \ntransition caused by heating. In addition th e high electronic temperature smears out the \norientation of the previously well aligned magnetic moment and introduces incoherent \nmotion to the magnetic dynamics. Those two co ntributions would resu lt in a vector sum \nof the corresponding MOKE signals unequal to  zero. A direct control on the angular \norientation of M\nwithout changing its magnitude is thus impossible for a heat-based \nstimulus, and hinders coherent contro l on magnetization dynamics. We therefore \nconclude that our results represent for the first time unambiguously a coherent and \nultrafast coupling of the strong THz magnetic field to the magnetization moment with the \ncomplete absence of heat.   \nIt is this coherent nature of interaction which gives for the first time access to the \ncomplex magnetic response (i.e. complete information on phase and amplitude), both in the time and frequency domain. As is known fr om signal processing (Fig. 4a) the impulse \nresponse, (t), of a system is determined by the measured input, I(t), and output, (t), \nand can be reconstructed in  the time domain by means of deconvolution of I and  or \nlikewise by inverse Fourier transformati on of the spectral response function ) (\n~ . In our \nexperiment the complex response function )(~/)(~)(~  I has been reconstructed \nfollowing the procedure laid out in Fig 4a  by transforming the measured complex \ntemporal THz field I(t) and MOKE signal (t) to the frequency domain. The spectral \namplitude response )( and the corresponding spectral phase calculated from the \nexperimental data shown in Fig 2(b), are presented in Fig 4b and c (red curve). The spectral response, far from being flat, unveils the dominant frequencies of the cobalt thin \nfilm in response to illumi nation by our broadband THz s timulus. The magnetic system \nunveils a strong response in the lower frequenc y part (<1 THz) which fades out towards \nhigher frequencies. For sake of comple teness the corresponding temporal impulse \nresponse of our ferromagnetic sample is shown as an inset in Figure 4(b).  \nOf particular interest is  the narrow-band frequency response at 1 THz disclosed \nby our time and frequency resolved measurements . Its origin is allocat ed to the strong co-\npropagating terahertz electri c field (1 MV/cm) which is supposed to influence the \nmagnetic response measurably. In order to corroborate this statement we performed an additional MOKE measurement. This time the magnetic field polarization was set parallel \nto the external field (\nextBB\n|| ) in order to disable magne tic coupling according to LL \n(position “C”, Fig. 1). The MOKE frequenc y spectrum recorded for this configuration \n(Fig. 4(c), black) is surprisingly still presen t yet significantly different to what is \nobserved for the Zeeman active (i.e. ext THz B B\n ) configuration  “A” (4c, red dotted line). In \nthe latter the frequency components around 1 THz are barely present, nor are higher \nfrequencies (> 2.5 THz), which hint at a different coupling mechanism for the two \nconfigurations. It is, however, remarkable th at for both field confi gurations the frequency \nimpulse response carries the narrow-band signature  at 1 THz with an associated jump in \nthe spectral phase (Fig 4b).  \nWe propose that the observed magnetic response could be associated to the \ncurrent induced by the strong electric fiel d according to Ampere circuital law. The \ncorresponding response functions  give evidence that this coupling is present in both \nconfiguration B\n||extB\n and extBB\n , though dominated by the THz magnetic field induced dynamics for the latter. While the origin of this phenomenon requires further \nstudies our investigations unve il that the strong co-propaga ting electric field has an \nobservable effect on the magnetization dynami cs. Since the standard LL equation does \nnot consider strong co-propa gating electric fields the above mentioned phenomenon \ncannot be described adequately. A more genera l theory is needed to give a complete \npicture of the magneto-electric dynamics in  this strong-field electro-magnetic regime \nwhich includes a nonlinear integro-differentia l approach for LLG coupled to the Maxwell \nequations, as developed for magneto-electric materials ( 13). Our numerical investigations \n(Fig. 2c) show, however, that for the case of ext THz B B\n , the complex nonlinear \nformalism proposed in ref. 14 can be transpos ed into an effectiv e linear LL by a simple \nrenormalization which can describe reas onably well the magnetization dynamics \nobserved in our thin film. \nIn former studies there has been eviden ce that heat-induced demagnetization with \nnear-infrared lasers (h 1.6 eV) alters the dielectric te nsor significantly and thus the \nMOKE signal due to exc itation of electrons ( 4). In those experiment s the illumination of \nmagnetic thin films give rise to hot electrons (typically ≥1000 K) and thus dichroic \nbleaching ( 14,15) which obviates a detailed response of  the MOKE signal within the first \ntens of femtoseconds. The use of a THz pum p laser prevents such deceptive MOKE \nsignals since the creation of hot electrons by THz is virtua lly excluded and so are state-\nblocking transitions. Our calculation shows that the THz interaction with the electronic system gives rise to a temperature increase of the electrons as low as a few tens of Kelvin \nabove room temperature thanks to the low THz photon energy (see \nsupplementary \ninformation) . The reported MOKE signal reflects therefore the unbiased and coherent magnetic dynamics in the cobalt thin film. Fina lly we would like to mention that the total \namount of angular moment carried by the THz photons is approximately three orders of \nmagnitudes larger than for a conventional 800 nm stimulus with equal pulse energies. \nThis, and the absence of heat during the inte raction makes us believe that intense THz \npulses are an important new t ool to investigate the magneti zation dynamics and to shed \nlight onto the not yet fully understood inte raction mechanism of angular momentum \nbetween light and matter during magnetic switching.  \nIn conclusion, the new aspects of our i nvestigations involv e the observation of \ncoherent sub-cycle femtosecond magnetizatio n dynamics in ferromagnetic thin films \ncontrolled by a strong single-cycle THz stimul us. In the absence of  heat injection the \nmagnetization vector is shown to be contro lled coherently by the complex field of the \nTHz laser, determined by its amplitude and phase. Previously inaccessible sub-cycle \nmagnetization dynamics are therefore visualiz ed and are shown to be governed by the \nfrequency components of the TH z pump rather than the natural response frequency of the \nelectrons (Larmor frequency). The coherent, ph ase-sensitive coupling of the THz field to \nthe magnetization allows for the retrieval of the complex impulse response of the \nmagnetic thin film, both in the time and fre quency domain. The presented concept of a \nphase-stabilized non-ionizing stimulus opens the door towards coherent magnetic domain \nswitching in absence of heat towards the yet unknown speed li mit of magnetization \ndynamics.  References:  \n1. M.N. Baibich, J.M. Roto, A. Fert, F. Nguy en Van Dau, F. Petroff, P. Eitenne, G. \nCreuzet, A. Friederich and J. Chazelas „G iant magnetoresistance of (001)Fe/(001)Cr \nmagnetic superlattices,” Phys. Rev. Lett. 61, 2472-2475 (1988). \n 2.  G. Binasch, P. Grunberg, F. Saurenbach  and W. Zinn „Enhanced magnetoresistance in \nlayered magnetic structures with antiferroma gnetic interlayer exchange,“ Phys. Rev. B \n39, 4828-4830 (1989).   3. E. Beaurepaire , J.C. Merle, A. Daunoi s, J.Y. Bigot, “Ultrafast spin dynamics in \nferromagnetic nickel,” Phys.Rev. Lett. 76, 4250-4253 (1996).  4.  B. Koopmans, M. van Kampen, J.T. K ohlhepp, W.J.M. de Jonge, „ultrafast magneto-\noptics in nickel:magnetics or optics, “ Phys. Rev. Lett. 85, 844-847 (2000). \n 5.  A. Kirilyuk, A. Kimel, T. Rasing, „Ultr afast optical manipulati on of magnetic order,“ \nRev. Mod. Phys. 82, 2731-2784 (2010).  6.  J.Y. Bigot, M. Vomir, E. Beaurepaire,  “Coherent ultrafast magnetism induced by \nfemtosecond laser pulses,” Nature Physics 5, 515-520 (2009).  7.  T. Kampfrath, A. Sell, G. Klatt, A. Pa shkin, S. Maehrlein, T. Dekorsky, M. Wolf, M. \nFiebig, A. Leitenstofer, R. Huber, „Coherent terahertz control of antiferromagnetic spin \nwaves,“ Nature Photonics, 5, 31-34 (2011).  \n8.  M. Foerst, R.I. Tobey, S. Wall, H. Br omberger, V. Khanna, A.L. Cavalieri, Y.D. \nChuang, W.S. Lee, R. Moore, W.F. Schlotter, J.J. Turner, O. Krupin, M. Trigo, H. Zheng, \nJ.F. Mitchell, S.S. Dhesi, J.P. Hill, A. Ca valleri, “Driving magnetic order in a manganite \nby ultrafast lattice excitation,” Phys. Rev. B 84, 2411041-2411045 (2011). \n 9.  A.V. Kimel, A. Kirilyuk, P.A. Usachev , R.V. Pisarev, A.M. Balbashov, T. Rasing, \n“Ultrafast non-thermal control of magnetization by instan taneous photomagnetic pulses,” \nNature 435, 655-657 (2005).  10.  C.P. Hauri, C. Ruchert, C. Vicario, F. Ardana, “Strong-field single-cycle THz pulses generated in an organi c crystal,” Appl. Phys . Lett. 99, 161116-161118 (2011). \n 11. H.Hirori, A. Doi, F. Blanchard and K. Tanaka, “Single-cycle terahertz pulses with \namplitudes exceeding 1 MV/cm generated by optical rectification in LiNbO\n3,” Appl. \nPhys. Lett. 98, 091106-091108 (2011).  12.  I. Tudosa, C. Stamm, A.B. Kashuba, F. Ki ng, H.C. Siegmann, J. Stoehr, G. Ju, B. Lu, \nD. Weller, “The ultimate speed of magnetic  switching in granular  recording media,” \nNature 428, 831-833 (2004).  13. J. Baker-Jarvis, P. Kabos, “Dynamic constitutive relations for polarization and \nmagnetization ,” Phys. Rev. E 64, 0561271-05612714 (2001).  14.  P.M. Oppeneer, A. Liebsch, “Ultrafast demagnetization in nickel:theory of magneto-\noptics for non-equilibrium electron distributio ns,” Journal of Physics:Cond. Matt. 16, \n5519-5530 (2004).  15. B. Koopmans, M. van Kampten, W.J.M. de Jonge, “Experimental access to femtosecond spin dynamics,” J. of Phys ics:Cond. Matter 15, 723-736 (2003). \n 16. E. Beaurepaire, J.-C. Merle, A. Daunoi s, J.Y. Bigot, “Ultrafast spin dynamics in \nferromagnetic nickel,” Phys. Rev. Lett 76, 4250 (1995)  17.  Handbook of Chemistry and Physics, online version www.hbcpnetbase.com   18.  L. D. Landau and E. M. Lifshitz, \"Theory of the dispersion of magnetic permeability in ferromagnetic bodies\", Phys . Z. Sowietunion 8, 153 (1935) \n   \nAcknowledgments: \nThis work was carried out at the Paul Sc herrer Institute and was supported by the SNF \ngrant PP00P2_128493 and SwissFEL. We acknowledge  M. Paraliev for valuable support. \nB.T. is supported by the ERASMUS Mundus  program. J.L. acknowledges the DYNAVO \nproject for financial support for upgrading the magnetron sputtering.    \nCorrespondence:  \nCorrespondence and requests for materi als should be addressed to C.P.H. \n(christoph.hauri@psi.ch ;christoph.hauri@epfl.ch) \n \n \nAuthor contributions: C.P.H. and J.L. conceived the experiment. C. P.H. coordinated the project and wrote the \nmanuscript. C.R.,C.V.,J.L., F.A. and C.P.H. ca rried out the experiments. B.T. fabricated \nand characterized the samples. F.A. performe d calculation on hot electron excitation. \nP.M.D. performed numerical investigations of the magnetization dynamics. All authors \ncontributed to data analysis.       \n \nFig. 1 . Experimental scheme with time-resolv ed magneto-optical Kerr effect (MOKE) \nthat allows for the measurement of ultraf ast moment dynamics on the Co sample surface \nkept at room temperature. The strong 0.3 T single-cycle THz magnetic field (blue) is \nlinearly polarized and carries an absolute phase  which is constant for consecutive shots. \nThe polarization can be rotated by 90 degree (position C) and inverted (position B), \ncorresponding to a  phase shift. The co-pr opagating electric field is not shown. The THz \npump pulse hits the sample under 20 degree off normal inci dence. The collinear 50-fs \nprobe pulse originates from the same laser sy stem driving the THz source and allows for \njitter free time-resolved measurement of magnetization dynamics by MOKE. The sample \nis placed in an external DC magnetic field ( 1.0extB\nT) which can be inverted after a \npump-probe cycle (by inverting the magnet cu rrent) in order to improve the signal-to-\nnoise of the Kerr rotation.   \n \n \n \n \nFig. 2 . Magnetization dynamics initiated by the strong THz transient. (a) Femtosecond \nmagnetization dynamics represented by the MOKE signal (red) and initiated by the \nphase-locked single-cycle THz magnetic field (black). The magnetization follows almost \ninstantaneously the THz field oscillations and its dynamics  is governed by frequencies \nmuch higher than the Larmor contribution (0.01 THz). The corresponding spectra and \nspectral phases are shown in (b) for th e THz pulse (black) and the magnetization \ndynamics (red). Interrogating Landau-Lifshi tz (LL) equation provides insight into \nultrafast dynamics shown in (c). A renormali zed, strongly damped LL (green) mimics the \noverall temporal magnetization dynamics surprisingly well when compared to the \nexperimental MOKE results (red, from (a)) even though LL ignores by default the co-\npropagating electric field.    \n \nFig. 3 . Investigation of coherence properties of  the initiated magnetization dynamics. (a) \nThe fundamental THz stimulus linearly polariz ed in direction “A” and the identical, but \ninverted ( -phase shifted) copy “B” used to initiate magnetization dynamics under \notherwise equal conditions. (b) The calculate d difference and sum signal of the measured \nMOKE traces show a two times amplification and complete annihilation, respectively. These findings are an unambiguou s proof for the absence of an incoherent heat-based \nstimulus. The absence of an incoherent heat  stimulus allows for the first time a fully \ncoherent steering of the magnetization vector thanks to the coherent coupling of the THz \nfield to the magnetization moment. \n \n \nFig. 4 . The complex impulse response of the c obalt thin film. (a) Fourier transformation \nof the measured THz field I(t) and MOKE signal (t) gives access to the complex \nfunctions ) (~I  and  ) (~  which allow direct calculat ion of the co mplex response \nfunction ) (~  shown in (b) with spectral amplitude )(  and spectral phase ) (  \n(both in red). The tem poral response function ) (t  is gained by inverse Fourier \ntransformation and shown in the inset of (b).  The frequency response function exhibits a \ndominant broadband frequency response cen tered around 0.4 THz and a resonance \nfrequency at 1.05 THz, which is attributed  to the strong co-pr opagating THz electric \nfield. This assumption is corroborated by the observed MOKE signal measurement with the THz B field parallel to the initial magnetization (c, black) which is significantly different to the Zeeman configuration (r ed dotted). The complex magnetic response \nfunction for this configurati on (b, black line) consists onl y of the narrowband resonant \nmode at 1.05 THz. Since no direct coupling of the magnetic THz field component to the \nmagnetization is present the observed magnetic re sponse is attributed to the strong pulsed \nco-propagating electric field.       Supplementary Materials: \n \nMethods: \n Laser-based THz source and pump-probe setup\n \nThe single-cycle THz pulses are generated by op tical rectification in the organic crystal \nDAST (4-N,N-dimethylamino-4’-N‘-methyl stil bazolium tosylate).  The output of a \nTi:sapphire amplifier system delivering up to 20 mJ, 50 fs pulses at 100 Hz is used to \npump a whitelight-seeded optical parametric am plifier (OPA) system with three amplifier \nstages. The OPA delivers 45fs, 2.5 mJ pulses at a wavelength of 1.5 µm which are used \nto initiate optical re ctification in the nonlin ear organic crystal. Fr equency conversion with \nan efficiency of 2% is achieved in a collimated pump geometry and the collinearly \ngenerated THz is redirected to the cobalt sa mple after separated of the residual pump \nlight. The broadband, single-cycle THz pulses are isolated from the infrared pump laser \nby a few mm thick teflon filter. No residual infrared (1.5 m) is detected at the sample \nposition. The THz source offers intrinsically carrier envelope phase-stable (CEP) pulses \nwith a high energy stability of <1%. The tem poral and spectral characteristics of the THz \nfield are reconstructed by el ectro-optical sampling in a 100 m thick Gallium Phosphide \n(GaP) crystal. The broadband THz spectrum s upports the formation of close to transform-\nlimited single-cycle pulses at a central fre quency of 2 THz. The radiation is linearly \npolarized and the direction of the polarizati on can be continuously varied by rotating the \ninput polarization and correspond ingly the OR crystal. Gold  coated mirrors are used \nthroughout the setup for THz transport. Time -resolved magneto-optical Kerr effect \n(MOKE) experiments are performed by splittin g the probe beam from the fundamental \nlaser (Ti:Sapphire) prior to th e optical parametric amplifier which is used for pumping the THz source. Both beams (THz and near IR ) are thus intrinsically  synchronized and hit \nthe sample collinear at close to normal incidence ( ≈20 deg from the normal). The near IR \nprobe spot size (150 m at 1/e2) is chosen to be significantly  smaller than the THz spot \nsize (800 m at 1/e2) to ensure homogeneous Terahertz fields for the MOKE \nmeasurements. The absorbed THz pump fluence on the sample is 0.8 mJ/cm2 and thus \nconsiderably lower compared to recent ne ar IR pump experime nts applying 10-100 \nmJ/cm2 (9). The MOKE system has been identified  to be free from artifacts which could \npotentially arise from the infrared (1.5 m) pump. The MOKE signal disappeared \ncompletely when the THz production was term inated by detuning the DAST from optical \nrectification. The Kerr rotation is calculated  from the polarization rotation of the optical \nprobe (probe=800 nm). \n \nSample preparation  \nThe 10 nm thick cobalt sample is covered wi th a 3 nm Al layer and evaporated onto a \nSi/Pd base substrate covered with 2 nm of Palladium. The aluminum layer on the top is \nvirtually transparent to the THz and the probe pulse. Its hcp crystalline structure exhibits \nan in-plane isotropy with the c-axis normal to the surface. \n  Electron temperature during interaction  \nThe total energy of the electron, spin and the lattice will increase during interaction with \nthe laser. Following the model in ref ( 16) we assume the existence of three thermalized \nreservoirs with temperature T e, Ts, Tl that exchange energy. Th e evolution of the system \nin time is given by the followi ng three differential equations: \n) ( ) ( )() ( ) ( )()( ) ( ) ( )(\ns l sl e l ell\nl ll s sl e s ess\ns ss e es l e ele\ne e\nTTG TTGdtdTTCTTG TTGdtdTTCtP TTG TTGdtdTTC\n\n \nwith C e, C s, C l the electronic, magnetic, lattice c ontribution to the specific heat, \nrespectively and G el, Ges, Gsl the electron-lattice, electron-sp in, and spin-lattice interaction \nconstants and P(t) the instan taneous laser power. For comp arison the evolution of the \nthree corresponding temperatures  are plotted for two diffe rent wavelengths, namely \nnIR=0.8 m and THz=150 m assuming equal fluence on the sample (0.8 mJ/cm2). The \nresults plotted in Fig 5 indicate that the elec tronic temperature in the case of illumination \nwith nIR is significantly higher than for a THz pump pulse. For THz radiation the \ntemperature increase in the electronic system  is approximately 50 K and thus very small. \nThe equivalent electronic temperature for a near IR pump pulse exceeds 1100 K which \nleads to the state-blocking and dichroic  bleaching effects described in ref ( 4). \nFor sake of completeness we add the calculated penetration depth L given by \nk L4/ with k the imaginary part of the complex refractive index iknn'. For the \nprobe beam (k=4.8, =800nm) a penetration depth of 13 nm is calculated while for the THz pump L is 28 nm (k=416, =150 m), both calculated for normal incidence \ngeometry. The k values are taken from ( 17) and extrapolated to the THz range. \n   \n  \n \nFig. 5 . Evolution of electronic, lattice and sp in temperatures after illumination of a \nnearIR pulse (a) and a THz pulse (b ) with same fluence (0.8 mJ/cm2). The maximum \nelectron temperature increase reached by THz induction is 350 K, thus only slightly \nabove room temperature (RT). For nIR illumi nation the maximum electron temperature is \n1200 K, a factor of 4 above RT. Different to experime nts with near infrared stimuli \n(=800 nm), the THz pump pulse does not signif icantly alter the electron temperature and \nprovides thus an unbiased in sight into magnetization dynamics by means of the time-\nresolved magneto-optical Kerr effect.  Landau-Lifschitz-Gilbert Equation\n \n The Landau-Lifshitz equation (\n18) is given by \n \n eff eff BM MMBMdtMd  \n\n0 , \n \nwhere M\nis the magnetization vector with magnitude 0M, effB\nis an external magnetic \nfield and the gyromagnetic ratio (T-1s-1). For the present simulations effB\n consists of the \nsum of the rapidly varying THz pulse  and the static external field ( 1.0extB\nT). In the \nabove equation, effB\n  represents the instantaneous Larmor precession frequency and  \nis a dimensionless empirical damping paramete r that sets the relaxation time-scale via \n1\neffB\n . For Co under equilibrium conditions, 014.0~ . Use of these parameter \nvalues would lead to a magnetization dynamics dominated by precession around, and \nrelaxation to, the static extern al field at a time-scale that is in the nano-second range. To \nproduce the femtosecond time evolution entailed  in fig. 2c, and approximate agreement \nwith experiment for the case of the THz ma gnetic component being perpendicular to the \nstatic external field, 50~ and 2extB\nT, the latter of which is aligned along the MOKE \nprobe direction. In this over-damped re gime the magnetization then approximately \nfollows the THz signal.  \n     "    },    {        "title": "2305.00093v1.Persistent_dynamic_magnetic_state_in_artificial_honeycomb_spin_ice.pdf",        "content": "Persistent dynamic magnetic state in artificial honeycomb spin ice\nJ. Guo1,†, P. Ghosh1,†, D. Hill1,†, Y. Chen2, L. Stingaciu3, P. Zolnierczuk3, C. A. Ullrich1,∗, and D. K. Singh1,∗\n1Department of Physics and Astronomy, University of Missouri, Columbia, MO\n2Suzhou Institute of Nano-Tech and Nano-Bionics, Chinese Academy of Sciences, China and\n3Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA\nTopological magnetic charges, arising due to the non-vanishing magnetic flux on spin\nice vertices, serve as the origin of magnetic monopoles that traverse the underlying\nlattice effortlessly. Unlike spin ice materials of atomic origin, the dynamic state in arti-\nficial honeycomb spin ice is conventionally described in terms of finite size domain wall\nkinetics that require magnetic field or current application. Contrary to this common\nunderstanding, here we show that thermally tunable artificial permalloy honeycomb\nlattice manifests a perpetual dynamic state due to self-propelled magnetic charge defect\nrelaxation in the absence of any external tuning agent. Quantitative investigation of\nmagnetic charge defect dynamics using neutron spin echo spectroscopy reveals sub-ns\nrelaxation times that are comparable to monopole’s relaxation in bulk spin ices. Most\nimportantly, the kinetic process remains unabated at low temperature where thermal\nfluctuation is negligible. This suggests that dynamic phenomena in honeycomb spin ice\nare mediated by quasi-particle type entities, also confirmed by quantum Monte-Carlo\nsimulations that replicate the kinetic behavior. Our research unveils a new ‘macro-\nscopic’ magnetic particle that shares many known traits of quantum particles, namely\nmagnetic monopole and magnon.\nMagnetic structure and dynamics in a magnetic ma-\nterial are governed by the underlying energetics due to\nthe interaction of magnetic moments with external and\ninternal magnetic fields1,2. Unlike its classical coun-\nterpart, a quantum magnet manifests a persistent dy-\nnamic tendency associated with spin or magnetic mo-\nment relaxation.2,3A magnetic moment can be thought\nof as a pair of ‘+’ and ‘ −’ quasi-particles, called magnetic\ncharges, coupled via magnetic Coulomb interactions4.\nThis concept plays a dominant role in the temperature-\ndependent evolution of magnetic and physical properties\nin spin ice compounds5–7. A direct demonstration of\nthis effect is most discernible in two-dimensional artifi-\ncial kagome ice on a honeycomb motif, where the pos-\nsible local moment configurations (2-in and 1-out, 1-in\nand 2-out, and all-in or all-out) impart low ( Q) and\nhigh (3Q) multiplicity magnetic charges on respective\nvertices8–11. Unequivocal evidence of magnetic charges\nhas been obtained using magnetic force microscopy in\nhoneycomb lattices made of permalloy (Ni 0.81Fe0.19) and\ncobalt magnets12–15. However, they are not known to be\ndynamic in the absence of external tuning agents (mag-\nnetic, thermal, or current),8,9,15,16which is in strong con-\ntrast to the well-established dynamic properties of mag-\nnetic charges in analogous spin ice compounds that ex-\nhibit persistent or continuous relaxation at low enough\ntemperature.6,17,18Magnetic charges relax by emitting or\nabsorbing net charge defects, qm, of magnitude 2 Q, also\ntermed effective monopoles.2,4,8,12,19\nA key question is whether magnetic charge in artifi-\n†These authors contributed equally to the research.\n∗email: ullrichc@missouri.edu, singhdk@missouri.educial spin ice can be described by the same quantum me-\nchanical treatment as in spin ice, where the Hamilto-\nnian representing magnetic charge interaction is defined\nby Pauli matrices.3,4. A direct comparison of magnetic\ncharge relaxation times in spin ice and artificial magnetic\nhoneycomb lattices can shed light on this fundamental\nproblem. From the Coulombic physics perspective, a\nmagnetic charge at the center of an artificial kagome lat-\ntice can be mapped onto the net magnetic flux along the\nbond direction on the parent honeycomb motif.20,21The\ncorrespondence entails the quantum treatment of mag-\nnetic charge interactions, which has been used previously\nto deduce thermodynamic and magnetic properties of\nthe statistical ensemble20. A charge defect’s motion be-\ntween vertices would flip the microscopic moment in the\nnanoscopic honeycomb element, manifesting a magnonic\npattern, thereby altering the moment direction and the\nnet charge on a given vertex, see Fig. 1a-c. The charge\ndefect or magnonic charge, qm, cannot pass through a\nhigh integer charge (3 Qor−3Q), which serves as a road-\nblock. At the same time, ±3Qcharges have a high energy\ncost, which makes them unstable and triggers the dy-\nnamic process. A thermally tunable honeycomb lattice,\nmade of ultra-small ( ∼11 nm) nanoscopic permalloy ele-\nment (see Fig. 1d),22can host such dynamic event with-\nout the application of any external stimulus. It is known\nto depict a highly disordered ground state, comprised of\nboth±Qand±3Qcharges, at low temperature.23The\nultra-small element size imparts thermally tunable ener-\ngetics due to the modest inter-elemental magnetic dipo-\nlar interaction energy of ∼45 K. We use neutron spin\necho (NSE) measurements to obtain direct evidence of\nself-propelled relaxation of magnonic charge qm, as it\nprovides accurate quantitative estimation of relaxation\nproperties across a broad dynamic range.24Furthermore,arXiv:2305.00093v1  [cond-mat.str-el]  28 Apr 20232\nFIG. 1: Self-propelled magnetic charge defect relaxation in an artificial honeycomb lattice made of ultra-small\npermalloy elements .a, Schematic illustration of magnetic charge relaxation processes between honeycomb vertices. The\ncharge defect ( qm) dynamics between nearest neighboring vertices corresponds to the reciprocal wave vector q∼0.058˚A−1.\nMagnetic charge relaxation occurs simultaneously in all three elements attached to a vertex. b, The dynamic state due to qm\nkinetics in a honeycomb element resembles magnon dynamics pattern. c, Time-dependent micromagnetic simulations depict\nmagnetization pattern in a honeycomb element at two different instances, t= 0 andt= 363 ps, of qmpropagation. d, Atomic\nforce micrograph of artificial honeycomb lattice, with typical element size of ∼11 nm (length) ×4 nm (width) ×6 nm (thickness).\nNSE spectroscopy is carried out on two stacks of permal-\nloy honeycomb samples with varying thicknesses of 6 nm\nand 8.5 nm that render a comprehensive outlook.\nIn Fig. 2a we show the color plot of NSE data obtained\non a 6 nm thick honeycomb lattice with net neutron spin\npolarization along + Zdirection. No meaningful spectral\nweight is detected in the spin down neutron polarization\n(see Fig. S2-S3 in SM),25which confirms the magnetic\nnature of the signal. To ensure that the observed signal\nis magnetic in origin, NSE spectrometer is modified by\nremoving the flipper before the sample (see Methods for\ndetail). The figure exhibits very bright localized scat-\ntering pixels that are identified with q= 0.058˚A−1and\n0.029 ˚A−1. Theseqvalues correspond to the typical re-\nlaxation length associated with the distance between the\nnearest and the next-nearest vertices, 2 π/land 2π/2l, re-\nspectively, with l= 11 nm. This immediately suggests\nthe quantized longitudinal nature of magnetic charge de-\nfect dynamics in artificial honeycomb lattices. NSE spec-\ntroscopy is a quasi-elastic measurement technique where\nthe relaxation of magnetic specimen is decoded by mea-\nsuring relative change in scattered neutron’s polarization\nvia the change in the phase current at a given Fourier\ntime (related to neutron precession). The localized ex-\ncitations exhibit pronounced sinusoidal spin echo. Fig.\n2b,c show the echo profiles obtained on honeycomb sam-\nples at room temperature at 0.02 ns Fourier time where\na strong signal-to-background ratio is detected.\nTypically, the echo intensity from a single wavelength λ\nfollows a cosine function, given by I(φ) =Acos(φλ)+B,\nwhere the phase φis directly related to the phase current\ndJ. However, if the neutron has a wavelength span of\nλavg±dλ, then the signals from all different wavelengths\nadd up (see SM). In this case, the total intensity follows\nthe following functional relation,26\nI(φ) =Acos(φλavg)sin(φdλ)\nφdλ+B. (1)\nAs the Fourier time associated to neutron precession in-\ncreases, the error bar tends to become larger. Fig. 2d\nshows the spin echo at 0.26 ns Fourier time (see Fig. S4 inSM for other Fourier times). This trend hints at the fast\nrelaxation of the magnetic charge quasi-particles, which\nis more prominent at a sub-ns time scale.\nThe sinusoidal nature of spin echo is quite evident at\nother temperatures as well. In Fig. 2e,f, we show the\nplot of characteristic spin echo profiles at low tempera-\nture in both thicknesses at q= 0.058˚A−1. Similar echo\nprofiles are observed at q= 0.029˚A−1. The strong spin\necho at low temperature suggests the persistence of sig-\nnificant charge dynamics in the nanoscopic system. The\norigin of charge relaxation at low temperature of T= 4\nK is unlikely to be thermal in nature considering that\nit costs|EQ- E−Q|(∼76 K) to create a charge defect.\nUnlike other nanostructured magnets where the domain\nwall motion, typically ascribed to the dynamic property,\nis detectable in applied magnetic field or current only, the\nabsence of any external tuning parameter at low temper-\nature suggests the non-trivial nature of magnonic charge\nqmrelaxation. In total, measurements were performed at\nsix temperatures of T= 4 K, 20 K, 50 K, 100 K, 200 K\nand 300 K (see Fig. S5 in SM). The quantitative estima-\ntion of relaxation times at different temperatures sheds\nlight on the dynamic state of artificial honeycomb lattice.\nThe estimation of magnetic charge relaxation time is\nachieved by analyzing the intermediate scattering func-\ntionS(q,t)/S(q,0) at different spin echo Fourier times.\nWe show the plot of normalized intensity as a function of\nneutron Fourier time at q= 0.058˚A−1in the 6 nm thick\nsample in Fig. 3a. The normalization is achieved by\ndividing the observed oscillation amplitude by the maxi-\nmum measurable amplitude (see SM), which is a common\npractice in magnetic systems with unsettling fluctuation\nto the lowest measurement temperature26. The nor-\nmalized intensity reduces to the background level above\nthe spin echo Fourier time of ∼0.5 ns, conforming to\nthe most general signature of relaxation process in NSE\nmeasurements26,27. The quantitative determination of\nmagnetic charge relaxation time involves exponential fit-\nting of the NSE scattering intensity as S(q,t)/S(q,0) =\nCe−t/τm, whereCis a constant28. The exponential func-\ntion provides a good description of the relaxation mech-3\nPhase current (A) Phase current (A)\na\nec db\nfT=300K\nT=4K T=4KT=300K\nT=300Kh=6nm\nh=6nmh=6nm h=8.5nm\nh=8.5nmt=0.02nst=0.02ns\nt=0.02ns t=0.02nst=0.26ns\nFIG. 2: Neutron spin echo spectroscopy revealing dy-\nnamic properties in artificial honeycomb lattice with-\nout any external tuning parameter. a , Color map of NSE\ndata on 6 nm thick honeycomb lattice with net neutron spin\npolarization (+ Z)−(−Z). Bright localized scattering pixels\nare identified with q= 0.058˚A−1andq= 0.029˚A−1(only\npartially visible near x= 0 due to geometrical limit of the\ninstrument). NSE data is obtained on a large parallel stack\nof 125 (117) honeycomb samples of 6 nm (8.5 nm) thickness.\nb,c, Spin echo profile at T= 300 K in 6 nm and 8.5 nm thick\nsamples, respectively, at t= 0.02 ns neutron Fourier time.\nStrong signal-to-background ratio is detected in the NSE sig-\nnal.d, The echo becomes weaker at higher Fourier time ( t=\n0.26 ns), indicating the dominance of charge defect qmrelax-\nation at lower time scale. e,f, Spin echo profile at T= 4 K in\n6 nm and 8.5 nm thick samples, respectively, at t= 0.02 ns\nneutron Fourier time. Self-propelled charge dynamics prevails\nat low temperature.\nanism of charge qm.\nFour important conclusions can be drawn from the es-\ntimated value of the relaxation time: first, the obtained\nvalue ofτm= 20 ps atT= 300 K suggests a highly active\nkinetic state in our permalloy honeycomb lattice, typi-\ncally not observed in nanostructured materials where the\ndomain wall motion is the primary mechanism behind the\ndynamic property29. Moreover, magnetic field or current\napplication is necessary to trigger domain wall dynamics,\nwhich is not the case here: the dynamics state of charge\nqmis entirely self-propelled. Second, the relaxation rate\nofqmin honeycomb lattice is somewhat thickness inde-\npendent. In Fig. 3b, we show S(q,t)/S(q,0) as a function\nof neutron Fourier time in the 8.5 nm thick lattice. The\nestimatedτm= 49(10) ps is comparable to that found\nin the 6 nm thick lattice (within 2 σ). Importantly, we\ndo not observe a drastic difference between the charge\ndynamics in the thinner and the thicker lattice, which isa strong indication for its quasi-particle characteristic.\nThird, the charge defect qmrelaxes at the same high\nrate at low Tdespite the absence of thermal fluctua-\ntion. As shown in Fig. 3c, the estimated relaxation time\nas a function of temperature suggests the persistence of\ncharge dynamics at lower temperatures. Given the fact\nthat the charge dynamics prevails in the absence of any\nexternal stimuli, the observed T-independence is quite\nsignificant. The conclusion is that the system lives in a\npersistent kinetic state due to the magnetic charge defect\nmotion between parent lattice vertices, which is strong\nevidence for the quantum nature of magnetic charge\nquasi-particles.\nLastly, theqmrelaxation rate in the 6 nm thick honey-\ncomb lattice is comparable to that found in the bulk spin\nice material,∼5 ps.18,27This suggests a correspondence\nbetween the dynamic behavior of magnetic charge de-\nfects and the monopolar dynamics in the atomistic spin\nice. To our knowledge, this is the first observation of its\nkind, bridging the gap between the bulk spin ice mate-\nrial and the nanostructured spin ice system. Thus, the\nquasi-particle characteristic of the monopole can be suit-\nably invoked in the artificial lattice. These conclusions\nare further supported by theoretical analysis, which we\nnow briefly summarize.\nIn order to confirm the quantum nature of magnetic\ncharge defect quasi-particles, we modeled the system\nboth fully quantum mechanically and fully classically.\nMotivated by the simplistic behavior of magnetic charge\nrelaxation, unchanged by temperature or system size, we\nsuspect the system exhibits rapid renormalization group\nconvergence to a minimal effective model of behavior, and\nas such we assume the system should be modeled effec-\ntively by a Hamiltonian with minimal features. To this\nend, for the quantum modeling we treat the magnetic\nmonopoles at each vertex as a pseudospin 3/2 due to the\nquartet nature of these sites. These pseudospins are cou-\npled minimally via nearest neighbor flip flop terms, i.e.\nHnn=/summationtext\n/angbracketleftij/angbracketrightJhS+\niS−\njwhereS+\niis the spin 3/2 repre-\nsentation of an SU(2) raising operator for the ith site\nand the sum is over nearest neighbors. We calculated\nthe average relaxation time as a function of tempera-\nture for two cases, a 4-site model solved exactly, and a\n200 site model with periodic boundary conditions time\nevolved via dynamic Monte Carlo.30, Both cases repro-\nduce the temperature independence of relaxation time for\nnegligible barrier height compared to the large exchange\nconstant, see Fig. 4a (see further detail in the SM).\nThe main result of the quantum Monte-Carlo simula-\ntion is that a magnetic charge defect qmtravels through\nthe lattice element effortlessly as a result of negligible\nenergy barrier. A similar observation was made in the\ncase of magnetic monopole dynamics in atomic spin ice.5\nIf the barrier height were comparable to the exchange\nconstantJ, then the relaxation time would increase sig-\nnificantly at low temperature; however, this is not seen\nin the experimental data. Similarly, if the kinetic be-\nhavior ofqmwere classical in origin, then the simulation4\na b2.0\n1.0\n0.5\n0.2\n2.0\n1.0\n0.4h=6nm\nT=4K\nh=6nmT=300K T=300K\nh=6nm h=8.5nm\nc\n0.0 0.4 0.8\nm=49ps m=20ps\nm=19ps\nFIG. 3: Temperature dependence of magnetic charge\ndefect’s relaxation time. a,b ,S(q,t)/S(q,0) versus neu-\ntron Fourier time at T= 300 K for the 6 nm sample ( a) and\nthe 8.5 nm sample ( b).The magnetic charge relaxation time\nτmis obtained from an exponential fit of the data. c,τmver-\nsusTin the 6 nm sample. Charge defect qmrelaxes atτ∼\n20 ps timescale at T= 4 K without any external stimuli The\nplot also shows the temperature independence of τm. Inset:\nsame as ( a), but forT= 4 K.\nwould predict a monotonic temperature dependence of\nthe relaxation time above the barrier height.\nFor the classical modeling, we have treated each joint of\nthe lattice as a classical spin obeying the Landau-Lifshitz-\nGilbert equation with a stochastic field modeling temper-\nature dependence31. In this case the magnetic charge re-\nlaxation time was found to be temperature independent\nonly up to a threshold temperature Tc, above which the\nrelaxation time decreases monotonically with increasing\ntemperature. This threshold occurs when the root-mean-\nsquare of the thermal field /vectorBthis comparable in mag-\nnitude to the effective field term associated with shape\nanisotropy /vectorBan. With an estimated effective anisotropy\n(mostly shape anisotropy) of |K1|∼1×104J/m3, we\nestimate the threshold temperature to be on the order of\nTc∼20 K, see Fig. 4b. This is inconsistent with the ex-\nperimental observations, leading us to conclude that the\ncharge defect qmdynamics has indeed quantum nature.\nThis is the first time that a comprehensive study of\nmagnetic charge relaxation processes in two-dimensional\nartificial permalloy honeycomb lattice reveals kinetic\nevents with∼ps temporal quantization. Previous efforts\nin determining the charge dynamics in artificial spin ice\nsystems focused on the usage of ferromagnetic resonance\nspectroscopy and optical measurements such as Brillouin\nlight scattering and optical cavity techniques32,33. How-\never, these techniques have intrinsic temporal thresholds\nof 1 ns or higher, which limits the scope of sub-ns inves-\nFIG. 4: Quantum mechanical nature of qmrelaxation\nprocess. a,b , Temperature dependence of τmextracted from\nquantum ( a) and classical ( b) numerical simulations. Only\nthe quantum Monte-Carlo simulations with negligible barrier\nheight, compared to exchange constant, results in tempera-\nture independence of τm, as found in experimental data. It\nsuggests effortless relaxation of qm, similar to quantum entity.\ntigation of charge dynamics. Additionally, the samples\nused for dynamic study were created via electron-beam\nlithography, which results in large element size. In most\ncases, samples created in this way are athermal,9thus\nsignificantly affecting the spatially quantized relaxation\nprocess between the vertices of the parent lattice. Our\nhoneycomb sample with small inter-elemental energy, uti-\nlized in this study, provided an archetypal platform to\nextract the charge relaxation properties in the absence\nof thermal fluctuation. The ultra-small element size im-\nparts a thermally tunable characteristic to the lattice.\nIt is also relevant to mention that the macroscopic size\nhoneycomb specimen, used in this study, tends to develop\nstructural domains that vary in size from 400 nm to few\nµm. However, the NSE study of charge dynamics only\ninvolves the nearest and the next nearest neighboring ver-\ntices. Therefore, structural domains are not expected to\naffect the outcome. The NSE technique, with a dynamic\nrange extending to sub-ns timescales, is the most suit-\nable experimental probe to study such delicate dynamic\nproperty.\nThe synergistic experimental and theoretical investiga-\ntions have not only revealed the sub-20 ps relaxation time5\nassociated to magnonic charge dynamics but also eluci-\ndated its quantum mechanical characteristics. The thick-\nness independence of the relaxation time τgives further\nsupport to the quasi-particle nature of the charge qm.\nDespite the classical characteristic of magnetic charge,\ndirectly related to magnetic moment Malong the hon-\neycomb element length ( L) viaq=M/L , the observed\nquantum mechanical nature of the charge defect dy-\nnamics is unprecedented. Unlike magnetic monopole or\nmagnon particles in bulk magnetic materials of atomic\norigin,qmis expected to possess finite ’macroscopic’ size.\nHowever, the kinetic behavior of qmmanifests similarities\nwith the known properties of magnetic monopoles, which\nis a strong indication of the particle-type character.\nThe new finding reported here can be utilized for the\nexploration of ’magnetricity’ in artificial kagome ice, orig-\ninally envisaged in the spin ice magnet.6Additionally, the\nquantum nature of magnonic charge can spur spintronic\napplication via the indirect coupling with electric charge\ncarriers.34In recent times, experimental efforts are made\nto develop a new spintronic venue in artificial spin ice\nsystem.35A dynamic system of thermally tunable mag-\nnetic honeycomb lattice renders a new research platform\nin this regard.\nMethods\nSample Fabrication\nArtificial honeycomb lattices were created using hi-\nerarchical top-down nanofabrication involving diblock\ncopolymer templates [see details in Supplementary Ma-\nterial (SM)25]. An atomic force micrograph of the hon-eycomb lattice sample is shown in Fig. 1d.\nNeutron Spin Echo Measurements\nWe performed NSE measurements on permalloy hon-\neycomb lattice samples using an ultrahigh resolution (2\nneV) neutron spectrometer SNS-NSE at beam line BL-\n15 of the Spallation Neutron Source, Oak Ridge National\nLaboratory. Time-of-flight experiments were performed\nusing a neutron wavelength range of 3.5 – 6.5 ˚A and\nneutron Fourier times between 0.06 and 1 ns. NSE is\na quasi-elastic technique where the relaxation of a mag-\nnetic specimen is decoded by measuring the relative po-\nlarization change of the scattered neutron via the change\nin the phase current at a given Fourier time (related to\nneutron precession). The experiment was carried out in\na modified instrumental configuration of the NSE spec-\ntrometer where magnetism in the sample is used as a π\nflipper to apply 180oneutron spin inversion, instead of\nutilizing a flipper before the sample. Additional magnetic\ncoils were installed to enable the XYZ neutron polariza-\ntion analysis, see the schematic in Fig. S1 in SM. Such\nmodifications ensure that the detected signal is magnetic\nin origin. NSE measurements were performed on two dif-\nferent parallel stacks of 125 and 117 samples of 20 ×20\nmm surface size and 6 nm and 8.5 nm thicknesses, respec-\ntively, to obtain good signal-to-background ratio. The\nsample stacks were loaded in the custom-made sample\ncell, which was mounted in a close cycle refrigerator with\n4 K base temperature. NSE data was collected for 8 hrs\non the average at each temperature and neutron Fourier\ntime.\n1Coey, J. M. D. Magnetism and Magnetic Materials . (Cam-\nbridge University Press, 2012).\n2Gardner, J., Gingras, M. & Greedan, J. Magnetic py-\nrochlore oxides. Rev. Mod. Phys. 82, 53 (2010).\n3Tomasello B., Castelnovo C., Moessner R. & Quintanilla\nJ. Correlated quantum tunneling of monopoles in spin ice.\nPhys. Rev. Lett. 123, 067204 (2019).\n4Castelnovo C., Moessner R. & Sondhi S. L. Magnetic\nmonopoles in spin ice. Nature 451, 42 (2008).\n5Bramwell S. T. et al. Measurement of the charge and cur-\nrent of magnetic monopoles in spin ice. Nature 461, 956\n(2009).\n6Bramwell S. Magnetricity near the speed of light. Nat.\nPhys. 8, 703 (2012).\n7Chern G. W., Maiti S., Fernandes R. M. & W¨ olfle P. Elec-\ntronic transport in the Coulomb phase of the pyrochlore\nspin ice. Phys. Rev. Lett. 110, 146602 (2013).\n8Skjærvo S. H., Marrows C. H., Stamps R. L. & Heyderman\nL. J. Advances in artificial spin ice. Nat. Rev. Phys. 12, 13\n(2020).\n9Nisoli C., Moessner R. & Schiffer P. Colloquium: Artificial\nspin ice: Designing and imaging magnetic frustration. Rev.\nMod. Phys. 85, 1473 (2013).\n10Rougemaille N. et al. Artificial kagome arrays of nanomag-\nnets: a frozen dipolar spin ice. Phys. Rev. Lett. 106, 057209\n(2011).11Parakkat V., Macauley G., Stamps R. L. & Krishnan K.\nConfigurable artificial spin ice with site-specific local mag-\nnetic fields. Phys. Rev. Lett. 126, 017203 (2021).\n12Mengotti E. et al. Real-space observation of emergent mag-\nnetic monopoles and associated Dirac strings in artificial\nkagome spin ice. Nat. Phys. 7, 68 (2011).\n13Tanaka M., Saitoh E., Miyajima H., Yamaoka T. & Iye\nY. Magnetic interactions in a ferromagnetic honeycomb\nnanoscale network. Phys. Rev. B 73, 052411 (2006).\n14Ladak S., Read D. E., Perkins G. K., Cohen L. F. &\nBranford W. R. Direct observation of magnetic monopole\ndefects in an artificial spin-ice system. Nat. Phys. 6, 359\n(2010).\n15Shen Y. et al. Dynamics of artificial spin ice: a continuous\nhoneycomb network. New J. Phys. 14, 035022 (2012).\n16Farhan A., Derlet P. M., Anghinolfi L., Kleibert A. & Hey-\nderman L. J. Magnetic charge and moment dynamics in\nartificial kagome spin ice. Phys. Rev. B 96, 064409 (2017).\n17Jaubert J. & Holdsworth P. Magnetic monopole dynamics\nin spin ice. J. Phys.: Cond. Matt. 23, 164222 (2011).\n18Wang Y. et al. Monopolar and dipolar relaxation in spin\nice Ho2Ti2O7. Sci. Adv. 7, eabg0908 (2021).\n19Zeissler K., Chadha M., Lovell E., Cohen L. F. & Branford\nW. R. Low temperature and high field regimes of connected\nkagome artificial spin ice: the role of domain wall topology.\nSci. Rep. 6, 30218 (2016).6\n20Chern G. W., Mellado P. & Tchernyshyov O. Two-stage\nordering of spins in dipolar spin ice on the kagome lattice.\nPhys. Rev. Lett. 106, 207202 (2011).\n21Moller G. & Moessner R. Magnetic multipole analysis of\nkagome and artificial spin-ice dipolar arrays. Phys. Rev. B\n80, 140409 (2009).\n22Summers B. et al. Temperature-dependent magnetism in\nartificial honeycomb lattice of connected elements. Phys.\nRev. B 97, 014401 (2018).\n23Yumnam G., Chen Y., Guo J., Keum J., Lauter V. &\nSingh D. K. Quantum disordered state of magnetic charges\nin nanoengineered honeycomb lattice. Adv. Sci 8, 2004103\n(2021).\n24Neutron spin echo spectroscopy: Basics, trends and appli-\ncations. (Springer Germany, 2003).\n25See Supplemental Material at http://... for additional ex-\nperimental and theoretical details regarding artificial hon-\neycomb lattice nanofabrication, NSE measurements, data\nanalysis, and modeling.\n26Zolnierczuk P. A. et al. Efficient data extraction from neu-\ntron time-of-flight spin-echo raw data. J. Appl. Crystal. 52,\n1022 (2019).\n27Ehlers G. et al. Dynamics of diluted Ho spin ice\nHo2−xYxTi2O7studied by neutron spin echo spectroscopy\nand ac susceptibility. Phys. Rev. B 73, 174429 (2006).\n28Ehlers G. et al. Evidence for two distinct spin relaxation\nmechanisms in ‘hot’ spin ice Ho 2Ti2O7.J. Phys.: Condens.\nMatter. 16, S635 (2004).\n29Zhang S. & Li Z. Roles of nonequilibrium conduction\nelectrons on the magnetization dynamics of ferromagnets.\nPhys. Rev. Lett. 93, 127204 (2004).\n30Adler S. B., Smith J. W. & Reimer J. A. Dynamic Monte\nCarlo simulation of spin-lattice relaxation of quadrupolar\nnuclei in solids. Oxygen-17 in yttria-doped ceria. J. Chem.\nPhys. 98, 7613 (1993).\n31Leliaert J. et al. Adaptively time stepping the stochastic\nLandau-Lifshitz-Gilbert equation at nonzero temperature:\nImplementation and validation in MuMax3. AIP Adv. 7,\n125010 (2017).\n32Jungfleisch M. B. et al. Dynamic response of an artificial\nsquare spin ice. Phys. Rev. B 93, 100401 (2016).\n33Chen X. M. et al. Spontaneous magnetic superdomain wall\nfluctuations in an artificial antiferromagnet. Phys. Rev.\nLett.123, 197202 (2019).\n34Chen Y. et al. Field and current control of the electrical\nconductivity of an artificial 2D honeycomb lattice. Adv.\nMater. 31, 1808298 (2019).\n35Kaffash M. T., Lendinez S. & Jungfleisch M. B. Nano-\nmagnonics with artificial spin ice. Phys. Lett. A 402,\n127364 (2021).\nAcknowledgements\nWe thank Valeria Lauter and George Yumnam for\nhelping us understand the magnetic charge distribution\non honeycomb vertices, and Antonio Faraone for helpful\ndiscussion on the use of NSE technique in probing mag-\nnetic materials. This work was supported by the U.S.\nDepartment of Energy, Office of Science, Basic Energy\nSciences under Awards No. DE-SC0014461 (DKS) and\nDE-SC0019109 (CAU). This work utilized the facilities\nsupported by the Office of Basic Energy Sciences, US\nDepartment of Energy.\nAuthors ContributionsD.K.S and C.A.U. jointly led the research. D.K.S. en-\nvisaged the research idea and supervised every aspect of\nthe experimental research. C.A.U supervised theoretical\nresearch. Samples were synthesized by J.G and P.G. Neu-\ntron spin echo measurements were carried out by J.G.,\nY.C., L.S., P.Z. Analysis was carried out by J.G., L.S.,\nP.Z. and D.K.S. Theoretical calculations were carried out\nby D.H. and C.A.U. The paper was written by D.K.S. and\nC.A.U. with input of all co-authors.\nAdditional Information\nSupplementary Information is available for this paper.\nAuthors declare no competing financial interests. Corre-\nspondence and requests for materials should be addressed\nto D.K.S.Supplemental material for:\nPersistent dynamic magnetic state in artificial honeycomb spin ice\nJ. Guo1,†, P. Ghosh1,†, D. Hill1,†, Y. Chen2, L. Stingaciu3, P. Zolnierczuk3, C. A. Ullrich1,∗, and D. K. Singh1,∗\n1Department of Physics and Astronomy, University of Missouri, Columbia, MO\n2Suzhou Institute of Nano-Tech and Nano-Bionics, Chinese Academy of Sciences, China\n3Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA\n∗email: singhdk@missouri.edu, ullrichca@missouri.edu and\n†These authors contribute equally to this work\nI. EXPERIMENTAL DETAILS\nA. Nanofabrication of the artificial honeycomb lattice of permalloy\nThe fabrication of the artificial honeycomb lattice involves the synthesis of a porous hexagonal template on top\nof a silicon substrate, calibrated reactive ion etching to transfer the hexagonal pattern to the underlying silicon\nsubstrate and deposition of permalloy on top of the uniformly rotating substrate in a near-parallel configuration ( ∼\n3◦) to achieve the 2D character of the system. The porous hexagonal template fabrication process utilizes diblock\ncopolymer polystyrene (PS)-b-poly-4-vinly pyridine (P4VP) of molecular weight 29K Dalton and volume fractions\n70% PS and 30% P4VP, which can self-assemble into hexagonal cylindrical structure of P4VP in the matrix of PS\nunder the right condition. A PS-P4VP copolymer solution of mass fraction 0.6% in toluene was spin-coated on a\npolished silicon wafer at around 2300 rpm for 30 seconds, followed by solvent vapor annealing at ∼25◦C for 12 hours.\nA mixture of toluene/THF (20/80, volume fraction) was used for the solvent vapor annealing. This process results\nin the self-assembly of P4VP cylinders in a hexagonal pattern in a PS matrix. Submerging the sample in ethanol for\n20 minutes releases the P4VP cylinders from the PS matrix, leaving a hexagonal porous template with an average\nhole center-to-center distance around 31 nm. The diblock template is used as a mask to transfer the topographical\npattern to the underlying silicon substrate using reactive ion etching with CF 4gas. The top surface of the etched\nsilicon substrate resembles a honeycomb lattice pattern, which is exploited to create a magnetic honeycomb lattice by\ndepositing permalloy (Ni 0.81Fe0.19) in a near-parallel configuration using E-beam physical vapor deposition. Samples\nwere uniformed rotated to achieve uniformity of the deposition. This allowed evaporated permalloy to coat the top\nsurface of the etched silicon substrate only and producing magnetic honeycomb lattice with a typical element size of\nabout 11 nm (length) ×4 nm (width) and controllable thickness. A typical atomic force micrograph of the honeycomb\nlattice is shown in Fig. 1d. Samples of thickness 6 nm and 8.5 nm were used in this work.\nB. Time-of-flight Neutron Spin Echo instrument\nThe Neutron Spin Echo (NSE) measurements were conducted on an ultrahigh resolution (2 neV corresponding to ∼\n300 ns Fourier time) neutron spectrometer at beam line BL–15 of the Spallation Neutron Source, Oak Ridge National\nLaboratory. A 3 ˚A wavelength bandwidth of neutrons (3.5–6.5 ˚A) was used in the experiment, and the scattered\nneutrons were collected with a 30 cm ×30 cm position-sensitive3He detector 3.9 m away from the sample. The\ndetector has 32×32 X, Y pixels laterally to keep track of the scattering angles of the detected neutrons. It also has\n42 time of flight (ToF) channels (tbins) that label the time frame of the detected neutron which encodes the neutron’s\nwavelength. Thus, the data collected by the detector is a 32 ×32×42 array. By carefully grouping the X, Y pixels\nand tbins, one can extract echo signals at different Fourier times t[see Eq. (S2) below for the definition] and qvalues\nfrom a single measurement. In this experiment, echo signals at multiple temperatures and nominal Fourier times (the\nFourier time associated with the neutron of the maximum wavelength) were measured.\nUnlike in conventional NSE, where a πflipper is installed before or after the sample to flip the neutron’s polarization,\na magnetic sample itself does a πflip of the neutron’s polarization due to the nature of the interactions between\nmagnetic moments and neutron spins. In this experiment, the πflipper was removed and additional magnetic coils\nwere installed for three-dimensional polarization analysis. To obtain good signal-to-background ratio, a stack of 125\n(117) samples of about 20 ×20 mm2was loaded in a custom-made aluminum sample container. The sample container\nwas inserted into a close cycle refrigerator with a base temperature of 4 K. A schematic diagram of the NSE instrument\nis presented in Fig. S1.arXiv:2305.00093v1  [cond-mat.str-el]  28 Apr 20232\nFIG. S1: Design of modified NSE instrument, utilized for the magnetic charge relaxation experiment. Inset shows the stack\nof samples sealed in an aluminum can, which serves as a πflipper. Additional magnetic coils are installed to enable the XYZ\npolarization analysis.\nC. Analysis of the Neutron Spin Echo signal\nNSE experiments probe the intermediate scattering function of the sample studied, given by\nS(q,t) =/integraldisplay\ncos(ωt)S(q,ω)dω, (S1)\nwhereS(Q,ω) is the scattering function of the sample and tis the Fourier time, defined as\nt=Jλ3γnm2\nn\n2πh2, (S2)\nwithγn,mnandλbeing the neutron’s gyromagnetic ratio, mass and wavelength, and hdenotes Planck’s constant.\nJ=/integraltext\n|B|dlis the magnetic field integral along the neutron’s path through the precession coil1. The field integrals are\ndesigned to be the same in both precession coils before and after the sample such that the precession phase acquired\nby the neutron in the first coil will be exactly recovered at the end of the second coil, given that the neutron does not\nchange its velocity while interacting with the sample. For quasi-elastic scattering, the neutron gains a net phase at the\nend of the second coil before it reaches the analyzer. By systematically stepping through an additional phase current\n(convert to an additional field integral) in either the first or the second coil, a cosine modulation of the sample’s\nintermediate scattering function is realized, which is the typical raw data, echo signal, one would analyze in an NSE\nexperiment.\nThe echo signal intensity for a given neutron wavelength λhas a cosine function shape given by\nI(φ) =Acos(φλ) +B, (S3)\nwhereφ=dJγnmn/handdJis the phase asymmetry between the two precession coils introduced via the scanning\nphase current. When signals from neutrons of a certain wavelength span λavg±dλare added up, the total intensity\nresembles a cosine function modulated by an envelope function\nI(φ) =Acos(φλavg)sin(φdλ)\nφdλ+B. (S4)3\n5\nZ+ Z-\nY Y\n15\n10\n5x10-5\nFIG. S1: Color detector image of the 6nm honeycomb lattice at room temperature with neutron’s polarization\nZ+ and Z −. The color scale on the right shows the detected neutron intensity normalized by the proton charge.\nWith the same color scale, while no meaningful spectral weight is detected in the Z −polarization, in the Z+\npolarization case, clear spectral weight is observed in the detector region corresponding to q∼0.058 ˚A−1and\nq∼0.029 ˚A−1. The intense signal at the left edge of the detector is the direct beam as the instrument hits its\ngeometrical constraint limit. Fig. 2a is obtained by subtracting the Z −intensities from the Z+ intensities.\nX pixels X pixels\nFIG. S2: The qdistribution and echo signals of the 6 nm honeycomb lattice measured at T = 4 K on the detector.\nLeft panel shows the flux-weighted qdistribution across the whole detector in ToF tbin 17, corresponding to a\nFourier time of 0.02 ns, the numbers in the individual pixels denote the qvalue in the unit of 0.001 ˚A−1. Right\npanel shows the echo signals in ToF tbin 17 on the detector. Every 2 pixels along X and Y directions are grouped\nfor the clearness of the demonstration. The intense direct beam signals on the left edge of the detector are not\nshown. This image together with the color detector image in Fig. 2a guided the grouping of detector X, Y pixels\nin data analysis.\nFIG. S2: Color detector image of the 6nm honeycomb lattice at room temperature with neutron’s polarization Z+ and Z −.\nThe color scale on the right shows the detected neutron intensity normalized by the proton charge. With the same color scale,\nwhile no meaningful spectral weight is detected in the Z −polarization, in the Z+ polarization case, clear spectral weight is\nobserved in the detector region corresponding to q∼0.058˚A−1andq∼0.029˚A−1. The intense signal at the left edge of the\ndetector is the direct beam as the instrument hits its geometrical constraint limit. Fig. 2a in the main text is obtained by\nsubtracting the Z −intensities from the Z+ intensities.\n5\nZ+ Z-\nY Y\n15\n10\n5x10-5\nFIG. S1: Color detector image of the 6nm honeycomb lattice at room temperature with neutron’s polarization\nZ+ and Z −. The color scale on the right shows the detected neutron intensity normalized by the proton charge.\nWith the same color scale, while no meaningful spectral weight is detected in the Z −polarization, in the Z+\npolarization case, clear spectral weight is observed in the detector region corresponding to q∼0.058 ˚A−1and\nq∼0.029 ˚A−1. The intense signal at the left edge of the detector is the direct beam as the instrument hits its\ngeometrical constraint limit. Fig. 2a is obtained by subtracting the Z −intensities from the Z+ intensities.\nX pixels X pixels\nFIG. S2: The qdistribution and echo signals of the 6 nm honeycomb lattice measured at T = 4 K on the detector.\nLeft panel shows the flux-weighted qdistribution across the whole detector in ToF tbin 17, corresponding to a\nFourier time of 0.02 ns, the numbers in the individual pixels denote the qvalue in the unit of 0.001 ˚A−1. Right\npanel shows the echo signals in ToF tbin 17 on the detector. Every 2 pixels along X and Y directions are grouped\nfor the clearness of the demonstration. The intense direct beam signals on the left edge of the detector are not\nshown. This image together with the color detector image in Fig. 2a guided the grouping of detector X, Y pixels\nin data analysis.\nFIG. S3: The qdistribution and echo signals of the 6 nm honeycomb lattice measured at T = 4 K on the detector. Left panel\nshows the flux-weighted qdistribution across the whole detector in ToF tbin 17, corresponding to a Fourier time of 0.02 ns, the\nnumbers in the individual pixels denote the qvalue in the unit of 0.001 ˚A−1. Right panel shows the echo signals in ToF tbin\n17 on the detector. Every 2 pixels along X and Y directions are grouped for the clearness of the demonstration. The intense\ndirect beam signals on the left edge of the detector are not shown. This image together with the color detector image in Fig.\n2a in the main text guided the grouping of detector X, Y pixels in data analysis.4\nD. Neutron Spin Echo raw data treatment\nThe echo intensities were obtained by adding up signals in X, Y detector pixels and then summing over different\nToF tbins while correcting the phase shifts among the tbins before the summation process. Grouping of the X, Y\ndetector pixels were guided by the false color detector image (Fig. 2a in the main text) as well as the detector pixel\nimage (Fig. S3) to center around the intense scattering signal. The echo profiles of the 8.5 nm honeycomb lattice\nshown in Fig. 2c,f of the main text were obtained by summing over detector pixels over the range of X = 8 to 10, Y\n= 10 to 21 in ToF tbin = 4, with a Fourier time ∼0.02 ns. In order to make direct comparisons between the 6 nm\nhoneycomb lattice and the 8.5 nm honeycomb lattice, echo profiles of the 6 nm honeycomb lattice (Fig. 2b,d,e of the\nmain text) with the same Fourier time were obtained by summing over detector pixels over the range of X = 7 to 11,\nY = 7 to 24 in ToF tbin = 17.\nFor the calculation of the intermediate scattering function (Fig. 3a,b of the main text), ToF tbins = 10 to 19, 20\nto 29, 30 to 39 were explored. The detector pixel groupings used in the calculation of the intermediate scattering\nfunctions at each temperature are identical to that used for the echo profiles. Note that a slightly larger detector area\nis selected for the data analysis of the 6 nm honeycomb lattice. This is because the detected scattering signal from\nthe 6 nm honeycomb lattice is broader than that from the 8.5 nm honeycomb lattice possibly due to a larger variation\nof the honeycomb element length scale.\nE. Calculation of the intermediate scattering function\nThe goal of every NSE experiment is to obtain the intermediate scattering function S(q,t)/S(q,0) as a function of\nthe Fourier time t, which can be calculated from the fitted echo amplitude. For non-magnetic samples,\nS(q,t)\nS(q,0)=2A\nU−D(S5)\nwhereAdenotes the fitted amplitude of the echo signal, UandDdenote the spin up (non- πflipping) and spin down\n(πflipping) measurement of direct scattering without precession. U−Dmeasures the maximum obtainable echo\namplitude. For magnetic samples, half of the magnetic scattering intensity M/2 is used for normalization2. Thus 6\nadditional polarization measurements along x,yandzdirections were performed at each temperature. MandU,D\nare calculated as\nM= 2(zup−zdn)−[(xup−xdn) + (yup−ydn)], (S6)\nU=xup+yup+zup\n3,D=xdn+ydn+zdn\n3. (S7)\nThe intermediate function is then determined as\nS(q,t)\nS(q,0)=4A\nM, (S8)\nwhich is comparable with 66 A/(U−D) in this experiment. In the calculation of the intermediate scattering function\nat a given Fourier time, the same X, Y pixels and ToF channels selection is used for the determination of both the\necho amplitude and the normalization factor.\nUnlike experiments conducted with other spectrometers, the measured dynamic structure factor Sexp(q,ω) is\na convolution between the true dynamic structure factor S(q,ω) and the instrument resolution R(q,ω), that is\nSexp(q,ω) =S(q,ω)∗R(q,ω). NSE directly probes the intermediate scattering function, which is the Fourier trans-\nform of the dynamics scattering function S(q,t) =/integraltext\ncos(ωt)S(q,ω)dω, thusSexp(q,t) =S(q,t)R(q,t). Therefore,\nin an NSE experiment, the resolution function can be simply divided out to obtain the true intermediate scattering\nfunction. At SNS-NSE the resolution of the instrument and elastic scattering contribution is assessed by measuring a\nperfect elastic scattering sample, usually mounted in the same container as the sample to investigate, measured over\nthe same q-range, tau-range, and wavelength. For soft matter measurements, solid graphite and Al 2O3as well as\nTiZr are usually used depending on the scattering angles. For paramagnetic NSE measurements we use Ho 2Ti2O7,\na well-known classical spin ice material frozen below T= 20 K where it exhibits no dynamics. However, for the\nmeasurement performed in this work, it was not possible to use Ho 2Ti2O7sample as the resolution since it only\nscatters and produces reliable echoes at high scattering angles, while the magnetic honeycomb samples scatter in the\nsmall-angle regime. Another common practice in quasi-elastic techniques is to measure the same sample at T≤4 K5\n6\nT=300KT=300K T=300K\nh=6nm h=6nm h=6nmt=0.04ns t=0.37ns t=0.11ns (a) (b) (c)\nPhase current (A) Phase current (A) Phase current (A)\nFIG. S3: Echo profiles of the 6 nm honeycomb lattice at different Fourier times measured at room temperature,\nrelated to Fig. 2. The Fourier time of the echoes are labeled on the graphs. Clearly, as the Fourier time increases,\nthe error bars become large and the echoes exhibit smaller amplitudes.\nPhase current (A) Phase current (A)\n(a) (b) (c)\nT=20K\nh=6nmT=200K\nh=6nmT=100K\nh=6nm\nFIG. S4: Echo profiles of the 6 nm honeycomb lattice at Fourier time 0.02 ns, measured at intermediate tem-\nperatures. Put together with Fig. 2b, and Fig. 3a-b, the persistence of a good signal-to-background ratio from\nroom temperature to the lowest measurement temperature (4 K) demonstrates the high quality of the data.\nh=6nm\nt (ns)\n0.02.0\n1.0\n0.4\n0.2\n0.2 0.4 0.6 0.8\nFIG. S5: The normalized intermediate scattering function of the 6 nm honeycomb lattice at q∼0.058˚A−1at\nintermediate temperatures, related to Fig. 4. The relaxation time τmat each temperature are indicated on the\ngraph.\nFIG. S4: Echo profiles of the 6 nm honeycomb lattice at different Fourier times measured at room temperature, related to Fig.\nS3. The Fourier time of the echoes are labeled on the graphs. Clearly, as the Fourier time increases, the error bars become\nlarge and the echoes exhibit smaller amplitudes.\n6\nT=300KT=300K T=300K\nh=6nm h=6nm h=6nmt=0.04ns t=0.37ns t=0.11ns (a) (b) (c)\nPhase current (A) Phase current (A) Phase current (A)\nFIG. S3: Echo profiles of the 6 nm honeycomb lattice at different Fourier times measured at room temperature,\nrelated to Fig. 2. The Fourier time of the echoes are labeled on the graphs. Clearly, as the Fourier time increases,\nthe error bars become large and the echoes exhibit smaller amplitudes.\nPhase current (A) Phase current (A)\n(a) (b) (c)\nT=20K\nh=6nmT=200K\nh=6nmT=100K\nh=6nm\nFIG. S4: Echo profiles of the 6 nm honeycomb lattice at Fourier time 0.02 ns, measured at intermediate tem-\nperatures. Put together with Fig. 2b, and Fig. 3a-b, the persistence of a good signal-to-background ratio from\nroom temperature to the lowest measurement temperature (4 K) demonstrates the high quality of the data.\nh=6nm\nt (ns)\n0.02.0\n1.0\n0.4\n0.2\n0.2 0.4 0.6 0.8\nFIG. S5: The normalized intermediate scattering function of the 6 nm honeycomb lattice at q∼0.058˚A−1at\nintermediate temperatures, related to Fig. 4. The relaxation time τmat each temperature are indicated on the\ngraph.\nFIG. S5: Echo profiles of the 6 nm honeycomb lattice at Fourier time 0.02 ns, measured at intermediate temperatures. Put\ntogether with Fig. 2b,e of the main text, the persistence of a good signal-to-background ratio from room temperature to the\nlowest measurement temperature (4 K) demonstrates the high quality of the data.\nwhere presumably all dynamics freezes. However, it is well observable in our measurements that even at T= 4 K fast\ndynamics of the magnetic charges still exists, which is, in fact, one of the main findings of our research. Therefore,\nwe decided not to reduce the data by elastic resolution, since, as explained, we do not have an elastic scatter with\nfrozen dynamics within the temperature range accessible. In support of our decision is the fact that at SNS-NSE\nbelow Fourier times t<1 ns the elastic resolution is predominantly flat (linear). This means that any reduction by\nresolution will not affect the relaxation observed in the data but only the intensity scaling on y-axis. To demonstrate\nthe above we have collected two measured Ho 2Ti2O7resolutions from previous paramagnetic measurements at T= 4\nK and 10 K, for 2 different wavelength 6.5 ˚A and 8 ˚A. In Fig. S6, these experimental elastic resolution magnetic data\nare presented, together with their fits and simulated resolution in the soft-matter regime. One can easily observe the\nlinear behavior in the range of t<1 ns, which is the predominant range of our measurements. Plots of S(q,t)/S(q,0)\natq= 0.058˚A−1at intermediate temperatures are presented in Fig. S7.\nII. THEORETICAL DETAILS\nHere we present a theoretical analysis of the ultra-small artificial magnetic honeycomb lattice, with the intention\nof elucidating the dominant physics at play behind the experimentally observed temperature independence of the\nmagnetic charge relaxation time.\nAs discussed in the main text, we suspect the system exhibits rapid renormalization group (RG) convergence to\na minimal effective model. This intuition is motivated by the simple, consistent behavior of the magnetic charge\nrelaxation over a broad range of temperatures; a feature that is unchanged with at least a modest change in lattice\nthickness. By “rapid renormalization group convergence” we specifically mean that small changes to a hypothetical6\nFIG. S6: Measured paramagnetic resolution function at SNS-NSE and simulated elastic resolution function in soft matter\nregime.\nexact theoretical description of the experimental system, such as a change in an interaction term coupling strength\nor a change in system size, would result in no qualitative change to the physically observable behavior of the model,\ndue to the presence of a well isolated and strongly stable fixed point in RG space. Literature on the structurally\nequivalent Kagome spin ice model3supports the contention that the model exhibits rapid RG convergence to the\nsame high and low temperature phases, with or without the inclusion of long or short range interaction terms. As\nsuch we assume the system should be modeled effectively by a Hamiltonian with minimal features, e.g. a Hamiltonian\nthat only captures the magnetic charge excitation energy and the dynamical capacity of these charges to transfer to\nneighboring sites.\nIn the following subsections, we consider two minimal effective models, one a classical description and one a quantum\nmechanical description, and compare the excitation relaxation time for each model to the experimental results.\nA. Quantum magnetic charge model\nThe simplest model for a Kagome spin ice involves spins\n/vectorSi=σiˆei, (S9)\nrestricted to specific site dependent directions ˆ eiwhich point along three possible axes separated by 120◦. These axes\ntogether form a honeycomb lattice structure akin to the herein studied artificial honeycomb lattices. The spins of the\nKagome spin ice are coupled via a nearest-neighbor spin exchange interaction\nH1=−J1/summationdisplay\n/angbracketlefti,j/angbracketright/vectorSi·/vectorSj=J1\n2/summationdisplay\n/angbracketlefti,j/angbracketrightσiσj, (S10)\nwhere ˆei·ˆei=−1\n2has been used, and the notation /angbracketleft.../angbracketrightis used to indicate a sum over nearest neighbors. The system\nis frustrated in the case of ferromagnetic coupling, i.e. J1>0, as can be seen by noting that for this sign of J1the7\n6\nT=300KT=300K T=300K\nh=6nm h=6nm h=6nmt=0.04ns t=0.37ns t=0.11ns (a) (b) (c)\nPhase current (A) Phase current (A) Phase current (A)\nFIG. S3: Echo profiles of the 6 nm honeycomb lattice at different Fourier times measured at room temperature,\nrelated to Fig. 2. The Fourier time of the echoes are labeled on the graphs. Clearly, as the Fourier time increases,\nthe error bars become large and the echoes exhibit smaller amplitudes.\nPhase current (A) Phase current (A)\n(a) (b) (c)\nT=20K\nh=6nmT=200K\nh=6nmT=100K\nh=6nm\nFIG. S4: Echo profiles of the 6 nm honeycomb lattice at Fourier time 0.02 ns, measured at intermediate tem-\nperatures. Put together with Fig. 2b, and Fig. 3a-b, the persistence of a good signal-to-background ratio from\nroom temperature to the lowest measurement temperature (4 K) demonstrates the high quality of the data.\nh=6nm\nt (ns)\n0.02.0\n1.0\n0.4\n0.2\n0.2 0.4 0.6 0.8\nFIG. S5: The normalized intermediate scattering function of the 6 nm honeycomb lattice at q∼0.058˚A−1at\nintermediate temperatures, related to Fig. 4. The relaxation time τmat each temperature are indicated on the\ngraph.\n7\nFIG. S6: The normalized intermediate scattering function of the 8.5 nm honeycomb lattice at q∼0.06˚A−1at\nT = 4 K, related to Fig. 4.\nFIG. S7: Left: normalized intermediate scattering function of the 6 nm honeycomb lattice at q∼0.058˚A−1at intermediate\ntemperatures, related to Fig. 3 of the main text. The relaxation time τmat each temperature are indicated on the graph.\nRight: normalized intermediate scattering function of the 8.5 nm honeycomb lattice at q∼0.06˚A−1atT= 4 K, related to\nFig. 3 in the main text\nfar right side of Eq. (S10) is equivalent to an antiferromagnetic honeycomb Ising model. The Hamiltonian (S10) can\nbe simplified further by introducing the magnetic charge at honeycomb vertex α,Qα=±/summationtext\ni∈ασi. In terms of these\noperators, the Hamiltonian is equivalent to, up to a constant,\nH1=J1\n4/summationdisplay\nαQ2\nα. (S11)\nHere the possible values of the magnetic charges are Qα=±1,±3, with the±3 states corresponding to the localized\nexcitations of the model.\nThe model described by H1is essentially classical, for the same reason that the original Ising model (without a\ntransverse field) is classical, namely because the model lacks any non-commuting observables. This originates in\nthe fact that, in deriving this Hamiltonian, we have neglected the important off axis components of the spins (S9).\nIncluding these off axis terms in the exchange coupling of Eq. (S10) results in nearest neighbor spin flip-flop terms\n(corresponding to a nearest neighbor hop of Q/prime\nαs), thus restoring the quantum nature of the model. However, instead\nof simply reintroducing the spin components, we consider a slightly different starting point in hopes of constructing\na minimal quantum mechanical model in terms of the Qα’s. To this end, we note that the quartet nature of Qα\nsuggests that the simplest possible treatment of Qαas a quantum operator would be to treat it as a zcomponent of\na pseudospin 3/2 representation of SU(2). With this in mind the natural choice of non-commuting observables would\nbe the ladder operators of the same pseudospin 3/2 representation. With a basic nearest neighbor hopping term, we\narrive at the minimal quantum mechanical Hamiltonian\nHQ=J1/summationdisplay\nαq2\nα+Jh/summationdisplay\n/angbracketleftα,β/angbracketrightq+\nαq−\nβ, (S12)\nwhere we define qα=1\n2Qαin order for qαto exhibit the traditional spin 3/2 representation eigenvalues of ±1\n2and\n±3\n2, and the ladder operators q±\nαare defined by their SU(2) commutation relations\n/bracketleftBig\nqα,q±\nβ/bracketrightBig\n=±q±\nαδαβ,/bracketleftBig\nq+\nα,q−\nβ/bracketrightBig\n= 2qαδαβ.\nIn order to model excitation relaxation time using the Hamiltonian HQ, we consider two separate approximation\nschemes, a fully integrable finite size model with 4 sites, and a dynamic Monte Carlo algorithm applied to an extended\nperiodic boundary condition model with 200 sites.\nFor the first approximation, using the short range nature of the Hamiltonian, we focus on a central vertex coupled\nto three isolated neighbors via the flip flop term of Eq. (S12). The result is a 44dimensional Hilbert space system\nwhich we time evolve exactly. For the initial ensemble, the system is set at thermal equilibrium and projected into8\nonly states with a central qeigenvalue of q0= +3\n2. We calculate the central excitation probability P(q0= +3\n2) as a\nfunction of time. The resulting time dependence of the q0= +3\n2state shows a temperature independent relaxation\ntime, in agreement with experimental results. The main qualitative change as a function of temperature is that the\nlarge time asymptote of the time average of P(q0= +3\n2) increases monotonically with temperature, as one would\nexpect from ergodicity. These features can be seen in Fig. S8.\nFor the second approximate treatment of (S12), we utilize the dynamic Monte Carlo algorithm as presented in\nRef. 4. The system we consider has dimensions of 10 unit cells by 10 unit cells, with 2 sites per unit cell, for a\ntotal of 200 sites, under periodic boundary conditions. We time evolve the system according to the dynamic Monte\nCarlo algorithm through thousands of hopping processes, recording any excitations that appear in the system and\ncomputing the average lifetime of stationary excitations. In this case, it is straightforward to model the system\nwith the addition of a potential energy barrier to magnetic charge hopping. The experimental result of temperature\nindependent relaxation time is reproduced for the case of this barrier being set to zero, as shown in Fig. 4b. Given\nthe fact that exchange energy is expected to be much larger than the estimated barrier height, ∼20 K (as discussed\nbelow), the magnetic charge relaxation between neighboring vertices can be considered barrier-free.\nB. Classical spin dynamics model\nThe large number of spins per honeycomb joint in the experiment, by conventional wisdom, would suggest the\npossibility of modeling the system according to the purely classical treatment of a Landau-Lifshitz-Gilbert (LLG)\nequation of motion. Here we provide such treatment while following the same philosophy as that above of restricting\nourselves to a minimalistic model. As the starting point for the above kagome spin ice was the nearest neighbor spin\nexchange coupling, we start with the classical analogue of this term in our LLG Hamiltonian. We further posit an\nanisotropy term in order to favor spins pointing along the honeycomb joint axes described by the ˆ eiunit vectors. The\nresult is the Hamiltonian\nHLLG=−J/summationdisplay\n/angbracketlefti,j/angbracketright/vectorSi·/vectorSj+K1/summationdisplay\ni/parenleftBig\n/vectorSi·ˆei/parenrightBig2\n, (S13)\nwhere/vectorSiare normalized classical spin variables. Physically a term like K1arises in an effective field theory typically\nwith dominant contributions from shape anisotropy; however this term can be used to wrap up a variety of both\nintrinsic and emergent anisotropic effects all into a single, phenomenological parameter. As such K1’s optimal value\ncan potentially be difficult to estimate both experimentally and theoretically. However, we have used the typical value\nfor permalloy, K1∼−1×104J/m3.\nIn order to model the effects of temperature, we include a time dependent stochastic field /vectorBthin the LLG equations\nof motion, following the method of Ref. 5, which satisfies the time averaged expectation value relations\n/angbracketleft/vectorBth(t)/angbracketright= 0, (S14)\n/angbracketleftBi\nth(t)Bj\nth(t/prime)/angbracketright=2kBTg\nMsγVδtδtt/primeδij, (S15)\nwhereγis the gyromagnetic ratio, gis the dimensionless Gilbert damping parameter, Msis the saturation magneti-\nzation,Vis the spatial volume of a magnetic unit cell, and δtis the time step size of the numerical integration.\nWe numerically integrate a model consisting of three hexagons sharing three edges, with each edge corresponding to\none spin/vectorSi(t), for a total of 15 spins. In order to model the dynamics of a classical version of a Q= 3 excitation, for the\ninitial state, the three central spins are oriented outward, with the rest of the spins arranged in an energy minimizing\norientation along their respective axes. For ferromagnetic coupling, this central Q= 3 excitation state corresponds\nto an unstable fixed point of the LLG equations of motions. For sufficiently small temperatures, corresponding to\nsmall thermal fluctuations /vectorBth(t), this model predicts an identical relaxation time for a given arrangement over a\nbroad range of temperatures. However, we find that when the root mean square of /vectorBth(t) becomes comparable to\nthe anisotropy field /vectorBan∝K1, the relaxation time begins to decrease monotonically as a function of temperature, as\nshown in Fig. 4a.\nThe existence of this anisotropy dependent crossover allows us to estimate a thermalization crossover temperature\nTcfor the classical model. The values used for this estimate are δt= 10−11s,g= 0.2,Ms= 6.6×105A/m, and\nV= 264 nm3. The resulting crossover temperature estimate is on the order of Tc∼20 K, which is much too small\nto be consistent with the experimental results. The substantially better agreement of the experiment with quantum9\nFIG. S8: Exact time evolution of a four site quantum spin ice model ensemble with a central Q= 3 excitation initial state.\nThe curves show the density of the central charge over time for three different initial temperatures of the surrounding sites.\nmodeling provides evidence for our hypothesis on the robust quantum mechanical nature of the ultra-small artificial\nmagnetic honeycomb lattice.\nWe again note that an effective value of K1can be difficult to estimate due to its emergent nature and due to\nthe fact that the shape anisotropy intrinsically depends on the geometry of the material, which in this case is highly\nnontrivial. In spite of this, we believe that our theoretical model supports our conclusions of the dominant physics at\nplay.\n1 Zolnierczuk P. A. et al. Efficient data extraction from neutron time-of-flight spin-echo raw data. J. Appl. Cryst. 52, 1022\n(2019).\n2 Pappas C., Ehlers G. & Mezei F. In-Neutron scattering from magnetic materials. Neutron-spin-echo spectroscopy and mag-\nnetism . (eds. Chatterji. T) 521-542 (Elsevier, 2006).\n3 Chern G. & Tchernyshyov O. Magnetic charge and ordering in kagome spin ice. Phil. Trans. R. Soc. A 370, 5718 (2012).\n4 Adler S. B., Smith J. W. & Reimer J. A. Dynamic Monte Carlo simulation of spin-lattice relaxation of quadrupolar nuclei in\nsolids. Oxygen-17 in yttria-doped ceria. J. Chem. Phys. 98, 7613 (1993).\n5 Leliaert J. et al. Adaptively time stepping the stochastic Landau-Lifshitz-Gilbert equation at nonzero temperature: Imple-\nmentation and validation in MuMax3. AIP Adv. 7, 125010 (2017)."    },    {        "title": "2104.06733v1.Non_resonant_circles_for_strong_magnetic_fields_on_surfaces.pdf",        "content": "arXiv:2104.06733v1  [math.DS]  14 Apr 2021NON-RESONANT CIRCLES FOR STRONG MAGNETIC FIELDS ON SURFACE S\nLUCA ASSELLE AND GABRIELE BENEDETTI\nAbstract. We study non-resonant circles for strong magnetic fields on a closed, connected, oriented surface\nand show how these can be used to prove the existence of trappi ng regions and of periodic magnetic geodesics\nwith prescribedlow speed. As acorollary, there existinfini tely many periodic magnetic geodesics forevery low\nspeed in the following cases: i) the surface is not the two-sp here, ii) the magnetic field vanishes somewhere.\n1.Introduction\n1.1.The setting. LetMbe a closed, connected, oriented surface. A magnetic system on Mis a pair ( g,b),\nwheregis a Riemannian metric on Mandb:M→Ris a function, which we refer to as the magnetic field.\nA magnetic system gives rise to a second-order differential equatio n\n∇˙γ˙γ=b(γ)˙γ⊥, (1.1)\nfor curves γ:R→M. Here∇is the Levi-Civita connection of gand⊥denotes the rotation of a tangent\nvector by ninety-degree in the positive direction. Equation (1.1) is N ewton’s second law for a charged particle\non the surface Munder the effect of the Lorentz force induced by the magnetic field b[7, 6].\nSolutions of (1.1), also known as magnetic geodesics, have constan t speeds. Moreover, a curve with\nconstant speed |˙γ| ≡s >0 is a solution of (1.1) if and only if it satisfies the prescribed curvatur e equation\nκγ=b(γ)\ns, (1.2)\nwhereκγ:R→Rdenotes the geodesic curvature of γ[7, 6]. From (1.2) we see that the geometric properties\nofγwill depend on the size of the quantity b/s. For instance, if we are in the regime of weak magnetic fields,\nnamely if sis large with respect to b, the solutions of (1.1) approximate the geodesics of g.\nIn this paper, we are interested in the opposite regime of strong ma gnetic fields, namely when sis small\nwith respect to b. In particular, we will look at the existence of periodic orbits and of t rapping regions for\nthe motion when the speed slies in this regime. Periodic orbits and trapping regions play a crucial ro le for\ntheir applications to the dynamics of charged particles and have bee n the subject of a vast literature in the\nlast decades. Periodic orbits are usually detected by variational me thods (see for instance [33, 23, 17] and the\ndiscussion below for further references) and are used to study t he presence of stable and chaotic trajectories\n(see [4, 30]). Trapping regions yield confinement of particles, a phen omenon which is observed in van Allen\nbelts [19], exploited to build plasma fusion devices [28] and is mainly studie d for magnetic systems on flat\nthree-dimensional space [34, 5].\nIn order to state our main results we need two more pieces of notat ion and a definition. We denote by µ\nthe area form of gwith respect to the given orientation and by K:M→Rthe Gaussian curvature of g.\nDefinition 1.1. A magnetic system ( g,b) withb/\\e}atio\\slash≡0 is called resonant at zero speed if either\n(1) both bandKare constant, or\n(2)M=S2and one of the following two conditions holds:\n(i)bis a non-constant, nowhere vanishing function such that the Hamilt onian flow of the function\nb−2on the symplectic manifold ( S2,µ) is fully periodic, or\n(ii)bis a non-zero constant, Kis non-constant and the Hamiltonian flow of the function Kon the\nsymplectic manifold ( S2,µ) is fully periodic.\nRemark 1.2. Fully periodic Hamiltonian flows on ( S2,µ) are completely classified, up to a change of coordi-\nnates. They are all isomorphic to the Hamiltonian flow induced by a con stant multiple of the height function\n2000Mathematics Subject Classification. 37J99, 58E10.\nKey words and phrases. Magnetic systems, KAM tori, periodic orbits, trapping regi ons.\n12 L. ASSELLE AND G. BENEDETTI\nof the euclidean unit sphere in R3endowed with a constant multiple of the euclidean area form. In part icu-\nlar, the Hamiltonian function possesses exactly two critical points w hich are additionally non-degenerate. A\ncriterion that prevents a Hamiltonian flow on ( S2,µ) to be fully periodic is given in [13, Theorem 1.7]. /squaresolid\nRemark 1.3. Given a metric on S2, one can always construct a function bsatisfying (i) in Definition 1.1.\nTo our best knowledge, it is not known whether examples of metrics w hose curvature Ksatisfies (ii) exist\nor not. In Theorem 5.1 we show that there is no such metric of revolu tion. In (5.4) we give a geometric\ncharacterization of such metrics, and in (5.5) a simple necessary co ndition that their curvature must satisfy.\n/squaresolid\n1.2.The results. Our main result asserts that magnetic systems which are not reson ant at zero speed\npossess an abundance of periodic orbits and trapping regions when the speed sis small. Here is the precise\nstatement.\nTheorem 1.4. Let(g,b)be a magnetic system on a closed, connected, oriented surfac eMwithb/\\e}atio\\slash≡0.\nSuppose that (g,b)is not resonant at zero speed. Then there exists an embedded c ircleL⊂ {b/\\e}atio\\slash= 0}such that\nfor all neighborhoods UofLthere exist s∗>0and a neighborhood U′ofLcontained in Uwith the property\nthat for all s < s∗there holds:\n(1) there exist infinitely many periodic magnetic geodesics with speed scontained in Uand homotopic to\nsome multiple of the free-homotopy class of L,\n(2) every magnetic geodesic with speed sstarting in U′stays inUfor all positive and negative times.\nThe theorem is proved using a normal form for the flow Φ sonSsM:={(q,v)∈TM| |v|=s}arising from\nthe tangent lifts of magnetic geodesics with small speed s. For the vector field generating Φ s, the normal\nform was established by Arnold in [9] and used to show that the magne tic fieldb, or the Gaussian curvature\nKifbis constant, is an adiabatic invariant for strong magnetic fields [10, Example 6.29]. For the applications\nto trapping regions and periodic orbits, we need an upgrade of the n ormal form from the vector field to the\nHamiltonian structure inducing the flow Φ s. This Hamiltonian normal form was established in [20] when b\nis not constant and in [13] when bis constant. We refer to Section 4 for more details and to [18] for a r elated\nnormal form in flat three-dimensional space.\nThe circle Lin the statement of Theorem 1.4 is any connected component of a re gular level set of the\nmagnetic field b, or of the Gaussian curvature Kifbis constant, at which the first order of the Hamiltonian\nnormal form is non-resonant (see Section 2.2 for a precise definitio n). The Moser twist theorem [31], a\nmanifestation in this setting of the celebrated KAM theorem, implies t he existence of nearby Φ s-invariant\ntwo-tori inside SsMfilled with quasi-periodic motions. Considering two of these tori lying t o the left and\nto the right of the circle L, one can trap trajectories starting inside a small neighborhood of L. Moreover,\nΦsadmits a closed annulus-like surface of section in the region between the two tori. The corresponding\nfirst-return map is twist by the non-resonance condition and the e xistence of infinitely many periodic orbits\nfollows by the Poincar´ e–Birkhoff theorem [27].\nIfKandb/\\e}atio\\slash≡0 are constant, then all magnetic geodesics are periodic for ssmall and every embedded circle\ninMhas the properties stated in Theorem 1.4. Thus the only instance wh ere the existence of an embedded\ncircle with those properties is not known is case (2) in Definition 1.1. Mo reover, in many situations we can\ngive additional information on the position of such an embedded circle .\nTheorem 1.5. Let(g,b)be a magnetic system on M. Suppose that one of the following conditions is satisfied:\n(i)b/\\e}atio\\slash≡0and the set {b= 0}is non-empty;\n(ii) the function bhas a degenerate strict local maximum or minimum z0∈ {b/\\e}atio\\slash= 0};\n(iii) the function battains a strict local maximum or minimum at an embedded circ leL′⊂ {b/\\e}atio\\slash= 0}.\nThen there exists a fundamental system {Ui}i∈Nof neighborhoods of {b= 0}, respectively of z0orL′, such\nthat∂Uiis the union of embedded circles satisfying the properties o f Theorem 1.4 for all i∈N. In particular,\neach of the sets {b= 0},z0andL′is stable in the sense of Lyapunov.\nIf in (ii) and (iii) we additionally assume that z0orL′are isolated in the set of critical points of b, then\nUican be taken to be a disc around z0or a tubular neighborhood of L′.\nSame conclusions as above hold if Ksatisfies the analogous of conditions (ii) and (iii) and bis a positive\nconstant in a neighborhood of z0andL′.\nRemark 1.6. Case (ii) in Theorem 1.5 under the hypotheses that z0is isolated in the set of critical points\nofbwas treated in [13, Corollary 1.5]. Case (iii) under the hypothesis that L′is isolated in the set of criticalNON-RESONANT CIRCLES 3\npoints of bwas treated in [20, Corollary 1.3]. On the other hand, notice that [20 , Corollary 1.2] about a\nnon-degeneratelocal maximum or minimum point of bis not correctsince such a point might be resonant. /squaresolid\nRemark 1.7. The presence of a non-degenerate saddle point z1∈Mforbwithb(z1)/\\e}atio\\slash= 0 also forces\nthe existence of embedded circles as in Theorem 1.4 intersecting the four components of {b/\\e}atio\\slash=b(z1)}in an\narbitrarily small neighborhood of z1. A partial result in this direction was obtained in [13, Theorem 1.6]. As\na consequence, trajectories starting close to z1can only escape along the four branches of {b=b(z1)}. We\nrefer to Theorem 4.3 for the precise formulation of the result. A sim ilar statement holds if bis a non-zero\nconstant and z1is a non-degenerate saddle point of K. /squaresolid\nWe end this introduction discussing the problem of existence of perio dic magnetic geodesics for strong\nmagnetic fields in more detail. We start by observing that Theorem 1.4 has the following consequence.\nCorollary 1.8. Let(g,b)be a magnetic system with b/\\e}atio\\slash≡0. IfM/\\e}atio\\slash=S2orbvanishes somewhere, then there\niss∗>0such that there are infinitely many periodic magnetic geodes ics with speed sfor every s∈(0,s∗)./squaresolid\nLet us examine how Corollary 1.8 complements previous results in the lit erature. When b/\\e}atio\\slash≡0, the\nexistence of one periodic magnetic geodesic for all low speeds follows from [22] (see also [13]). If the function\nbassumes both positive and negative values, which happens for insta nce if the two-form bµis exact, then\nthe existence of infinitely many periodic magnetic geodesics for almost every speeds <˜swas proved in the\nseries of papers [1, 3, 2, 11, 12] by employing the variational chara cterization of periodic solutions to (1.1)\nas critical points of the free-period action functional. The value ˜ scan be explicitly characterized and, when\nbµis exact,1\n2˜s2coincides with the Ma˜ n´ e critical value of the universal cover. The se methods have failed\nso far to yield existence of infinitely many periodic orbits for every sm all speed because of the lack of good\ncompactness properties of the functional (namely, the lack of th e Palais–Smale condition) and the existence\nfor almost every speed is recovered by means of Struwe’s monoton icity argument [32].\nOur results upgrade the almost everywhere existence results to everylow speed, even though a priori only\non a smaller speed range since we do not have an explicit estimate on th e values∗appearing in Corollary 1.8.\nThus the existence of infinitely many periodic orbits for s <˜sremains an open problem (the existence of one\nperiodic orbit being known by [33, 21]). We shall notice that the almost everywhere existence results extend\nto the more general class of Tonelli Lagrangians, i.e. fibrewise stric tly convex and superlinear Lagrangians,\nsee [14]. In contrast, the arguments of the present paper seem t o carry over to the Tonelli setting only if the\nenergy function of the Lagrangian attains a Morse–Bott minimum at the zero section of TM.\nIfbis nowhere vanishing, the existence of infinitely many contractible pe riodic magnetic geodesics for all\nlow speeds was proven for M=T2in [24] and for Mwith genus at least 2 in [26] in the context of the Conley\nconjecture [25]. In contrast, the periodic orbits found in Corollary 1.8 are all homotopic to the iterate of some\nfree-homotopy class but in general will not be contractible. If M=S2andbis nowhere vanishing, for every\nlow speed there are either two or infinitely many periodic magnetic geo desics [15]. Examples with exactly two\nperiodic orbits at a single speed were constructed in [16]. However, it is still an open problem to determine\nif magnetic systems on S2which are resonant at zero speed have infinitely many periodic orbits at every\nlow speed or at least for a sequence of speeds converging to zero. The latter option holds for rotationally\nsymmetric magnetic systems as we observe in Corollary 5.2.\nOrganization of the paper. The next sections are organized as follows:\n•In Section 2 we discuss the existence problem of non-resonant circ les for autonomous Hamiltonian\nsystems on a surface with finite area. The main technical result of t his section is Lemma 2.9.\n•InSection3weproveTheorem3.1whichcombinestheMosertwistthe oremandthePoincar´ e–Birkhoff\ntheorem to produce trapping regions and infinitely many periodic orb its close to a non-resonantcircle\nfor small non-autonomous perturbations of autonomous Hamilton ian systems on surfaces.\n•In Section 4, we apply the abstract results of the previous section to magnetic systems using the\nnormal form of [20, 13] and prove Theorems 1.4 and 1.5.\n•In Section 5 we discuss some results about magnetic systems on S2which are resonant at zero speed.\nAcknowledgments. L.A. is partially supported by the Deutsche Forschungsgemeinscha ft under the DFG-\ngrant 380257369 (Morse theoretical methods in Hamiltonian dynam ics). G.B. is partially supported by\nthe Deutsche Forschungsgemeinschaft under Germany’s Excellen ce Strategy EXC2181/1 - 390900948 (the\nHeidelberg STRUCTURES Excellence Cluster), under the Collaborativ e Research Center SFB/TRR 191 -4 L. ASSELLE AND G. BENEDETTI\n281071066 (Symplectic Structures in Geometry, Algebra and Dyna mics) and under the Research Training\nGroup RTG 2229 - 281869850 (Asymptotic Invariants and Limits of G roups and Spaces). We are grateful to\nViktor Ginzburg and Leonardo Macarini for their comments on the p reliminary version of the manuscript.\n2.Non-resonant circles for autonomous systems on symplectic surfaces\nIn this section Nwill denote a connected surface without boundary, not necessar ily compact. Moreover,\nωwill denote a symplectic form on NandH:N→Ra proper Hamiltonian function. A set U⊂Nis\nsaid to be a neighborhood of {H= +∞}, ifHis bounded from above on N\\U. A similar property defines\nneighborhoods of {H=−∞}. We denote by XHthe associated Hamiltonian vector field determined by the\nequation ω(XH,·) =−dH.\n2.1.Orbit cylinders. Inthisfirstsubsection,weintroduceorbitcylinderswhichwillbethe mainprotagonist\nof the crucial Lemma 2.9 in the next subsection.\nDefinition 2.1. Letγ:R→Nbe a non-constant periodic orbit for H. We call the support Lofγ, namely\nL=γ(R), anembedded circle ofHat levelc:=H(γ).\nRemark 2.2. SinceHis a first integral of motion and it is proper, its embedded circles are e xactly the\nregular connected components of the level sets of H. /squaresolid\nDefinition 2.3. Anorbit cylinder (ofH) is a smooth isotopy c/mapsto→Lc⊂Nparametrized over an interval\nI⊂Rsuch that Lcis an embedded circle of Hat levelcfor every c∈I. We say that an orbit cylinder\nc/mapsto→Lcpasses through an embedded circle LofHifL=Lc0for some c0in the interior of I.\nRemark 2.4. By the implicit function theorem, for every embedded circle LofHat levelc0there exists a\nunique orbit cylinder with I= (c0−ǫ,c0+ǫ) for small ǫ >0 passing through it. /squaresolid\nDefinition 2.5. We say that an orbit cylinder c/mapsto→Lcis bigger than another orbit cylinder c/mapsto→L′\nc, if the\nformer isotopy extends the latter. We call an orbit cylinder maximal if it is maximal with respect to such\npartial order. If c/mapsto→Lcis a maximal cylinder passing through L=Lc0, we call the restriction of c/mapsto→Lcto\nI∩[c0,+∞) themaximal forward orbit cylinder starting at L=Lc0. Analogously, we call the restriction of\nc/mapsto→LctoI∩(−∞,c0] themaximal backward orbit cylinder ending at L=Lc0.\nLemma 2.6. LetLbe an embedded circle of H. Then there exists a unique maximal orbit cylinder passing\nthroughL=Lc0and it is parametrized over an open interval I= (c−,c+). Ifc+<+∞, then the closure of\n∪c∈[c0,c+)Lccontains a critical point z∞ofH. A similar statement holds when c−>−∞.\nProof.LetV⊂Nbe the open set of regular points of H. OnVwe define a flow t/mapsto→Ψtsuch that\nH◦Ψt(z) =H(z)+t,∀t∈(t−(z),t+(z)), (2.1)\nwhere(t−(z),t+(z)) isthe maximalintervalofdefinition ofthe flowat z∈V. ForinstanceΨtcan be obtained\nintegrating the vector field X=∇H/|∇H|2where the norm and the gradient are taken with respect to any\nRiemannian metric on V. Ift+(z)<+∞, then the closure of the set S={Ψt(z)|t∈[0,t+(z))}is contained\nin the set H−1([H(z),H(z) +t+(z)]) and hence is compact in NsinceHis proper. As [0 ,t+(z)) is the\nmaximal interval of definition of the trajectory t/mapsto→Ψt(z), the closure of Scannot be contained in Vand\ntherefore there exists a critical point wz∈NofHlying in it.\nLet now Lbe an embedded circle for Hat levelc0. We define\nc+:=c0+ inf\nz∈Lt+(z), c −:=c0+sup\nz∈Lt−(z) (2.2)\nandthe orbitcylinder c/mapsto→Lc:= Ψc−c0(L)forc∈(c−,c+) whichiswell-defined by (2.1) and(2.2). If c+<∞,\nthen the lower semi-continuity of t+:V→Rimplies the existence of some z∈Lsuch that c+=c0+t+(z).\nIt follows that there exists a critical point z∞ofHin the closure of ∪c∈[c0,c+)Lc. A similar argument works\nforc−>−∞. This shows that c/mapsto→Lcis a maximal orbit cylinder. Uniqueness follows from the local\nuniqueness of orbit cylinders passing through a given embedded circ le as observed in Remark 2.4. /squaresolidNON-RESONANT CIRCLES 5\n2.2.Resonant cylinders. The goal of this subsection is to define resonant cylinders and prov e their prop-\nerties in Lemma 2.9. This result will be used in the next subsection to giv e criteria for the existence of\nnon-resonant cylinders.\nDefinition 2.7. Letc/mapsto→Lc, c∈I,be an orbit cylinder. We denote by T(c) the period of the orbit of H\nsupported on Lc. We say that a circle L=Lc0withc0∈Iisnon-resonant , if\ndT\ndc(c0)/\\e}atio\\slash= 0\nandresonant otherwise. If Lcis resonant for all c∈Ithen we say that the orbit cylinder is resonant. If\nthere is a c0∈Isuch that Lc0is non-resonant then we call the whole orbit cylinder non-resonant .\nWe recall now the classical relation connecting the period function Tto the area function of sub-cylinders.\nTo this purpose consider two parameters c1,c2∈I. Ifc1≤c2the set\nUc1,c2:=/uniondisplay\nc∈(c1,c2)Lc\nis diffeomorphic to an open cylinder inside Nand we endow it with the orientationinduced by ω. Forc2≤c1,\nwe define Uc1,c2asUc2,c1with the opposite orientation. We fix c0∈Iand define the area function\nA:I→R, A(c) :=/integraldisplay\nUc0,cω.\nLemma 2.8. There holds\ndA\ndc(c1) =T(c1),∀c1∈I.\nProof.Let us consider an orientation-preserving parametrization F:I×T→Nof the orbit cylinder, so\nthatθ/mapsto→F(c,θ) parametrizes LcandH(F(c,θ)) =c. Then there exists a function a:I×T→(0,∞) such\nthat∂θF=aXHand dH(∂cF) = 1. If tis the time parameter of XHalongLc1, then dt(∂θF) =a(c1,·) and\nT(c1) =/integraldisplay\nTdt(∂θF)dθ=/integraldisplay\nTa(c1,θ)dθ.\nTo compute the area, we observe that\nω(∂cF,∂θF) =aω(∂cF,XH) =adH(∂cF) =a.\nTherefore, we have\nA(c1) =/integraldisplayc1\nc0/integraldisplay\nTω(∂cF,∂θF)dcdθ=/integraldisplayc1\nc0/parenleftBig/integraldisplay\nTa(c,θ)dθ/parenrightBig\ndc\nand by the fundamental theorem of calculus\ndA\ndc(c1) =/integraldisplay\nTa(c1,θ)dθ=T(c1). /squaresolid\nWe are ready to prove the main lemma.\nLemma 2.9. Let(N,ω)be a symplectic surface without boundary having finite sympl ectic area. Let H:\nN→Rbe a proper Hamiltonian. Let c/mapsto→Lcbe a forward maximal orbit cylinder with c∈[c0,c+)starting\nat some L=Lc0. If this orbit cylinder is resonant, then c+<+∞and there exists a non-degenerate local\nmaximum point z∞∈NofHsuch that D:=Uc0,c+∪ {z∞}is an embedded disc in N. In particular H\ninduces a Hamiltonian circle action on Dwithz∞as the only fixed point. A similar statement holds for\nresonant backward maximal cylinders.\nProof.Since the orbit cylinder c/mapsto→Lcis resonant, there exists a smallest positive τsuch that\nΦτ\nH(w) =w,∀c∈[c0,c+),∀w∈Lc. (2.3)\nSinceτis the common period of all the orbits in the cylinder, Lemma 2.8 implies\nτ(c−c0) =/integraldisplay\nUc0,cω≤/integraldisplay\nNω <+∞6 L. ASSELLE AND G. BENEDETTI\nwhich shows that c+<+∞. By Lemma 2.6, there exists a sequence cn→c+and a sequence zn∈Lcnsuch\nthatznconverges to a critical point z∞ofH. We claim that\nsup\nz∈Lcndist(z,z∞)→0,forn→ ∞. (2.4)\nLet us assume this were not the case. Since His proper, the set/uniontext\nn∈NLcnis contained in a compact set\nofN. This fact together with the negation of (2.4) implies that we can ext ract a subsequence, which we still\nlabel by n, with the following property: There exists w∞∈N\\{z∞}and a sequence wn∈Lcnsuch that\nlim\nn→∞wn=w∞.\nBy (2.3), there is a sequence τn∈[0,τ) such that wn= Φτn\nH(zn). Up to extracting a subsequence, we can\nassume that τn→τ∞∈[0,τ] asn→ ∞. By the continuous dependence of the flow Φ Hon the initial\ncondition and the fact that z∞is a rest point of Φ H, we get the contradiction\nz∞= Φτ∞\nH(z∞) = lim\nn→∞Φτn\nH(zn) = lim\nn→∞wn=w∞.\nLet us now consider a chart ϕ:U∞→R2centered at z∞. By (2.4) the circle Lcnis contained in U∞\nfornlarge enough and therefore ϕ(Lcn) is an embedded circle in R2. By the Jordan curve theorem ϕ(Lcn)\nbounds a compact region homeomorphic to a disc Dn. By (2.4), there holds\nsup\nz∈ϕ−1(Dn)dist(z,z∞)→0,forn→ ∞. (2.5)\nIndeed, the Euclidean distance in the chart is equivalent to the dista nce function on Nand therefore Dn\nis contained in a euclidean ball of radius smaller than a fixed constant t imes supz∈Lcndist(z,z∞). As a\nconsequence,\nlim\nn→∞/integraldisplay\nϕ−1(Dn)ω= 0. (2.6)\nThe next step is to prove the following claim: There exists msuch that for all n≥m, the circle Lcis\ncontained in the interior of ϕ−1(Dn) for allc∈(cn,c+). If this were not the case, then Uc0,cn⊂ϕ−1(Dn) for\nsomenthat can be chosen arbitrarily large. Therefore,/integraldisplay\nϕ−1(Dn)ω≥/integraldisplay\nUc0,cnω≥/integraldisplay\nUc0,c1ω,\nwhich contradicts (2.6) for nlarge enough and proves the claim.\nTo prove that D:={z∞} ∪Uc0,c+is a neighborhood of z∞homeomorphic to a disc is enough to show\nthatϕ−1(˚Dm) is equal to {z∞}∪Ucm,c+. By the claim, {z∞}∪Ucm,c+is contained in ϕ−1(˚Dm). Ifzwere a\npoint ofϕ−1(˚Dm) which is not in {z∞}∪Ucm,c+, thenϕ(Lcm) is not homotopic to a constant in Dm\\{ϕ(z)}.\nHowever, ϕ(Lcm) is also homotopic in Dm\\{ϕ(z)}toϕ(Lcn) for allnlarge enough and ϕ(Lcn) is homotopic\nto a constant in Dm\\{ϕ(z)}by (2.4), a contradiction. This shows that Dis an open, embedded disc in N\nand that Φ Hinduces a Hamiltonian circle action on Dwith unique fixed point z∞. By [29, Lemma 5.5.8],\nz∞is a non-degenerate local maximum of H. /squaresolid\n2.3.Non-resonant cylinders. In this subsection, we exploit Lemma 2.9 to prove the existence of no n-\nresonant cylinders in certain situations. These results will be used in the proofs of Theorems 1.4 and 1.5.\nTheorem 2.10. Let(N,ω)be a symplectic, connected surface without boundary having finite symplectic area.\nLetH:N→Rbe a non-constant proper Hamiltonian. Then the set of maxima l orbit cylinders is non-empty\nand exactly one of the following statements holds:\n(i) All maximal orbit cylinders are non-resonant.\n(ii) There is only one maximal orbit cylinder. Such a cylinde r is resonant and Nis diffeomorphic to S2. In\nthis case, Nis the union of the cylinder and the unique maximum z∞and minimum point z−∞ofH.\nThe points z∞andz−∞are non-degenerate and ΦHinduces a Hamiltonian circle action on N∼=S2\nwith{z∞,z−∞}as fixed-point set.\nProof.SinceHis proper and non-constant, there exists an embedded circle LofHby Sard’s theorem\nand hence a maximal orbit cylinder by Lemma 2.6. If there exists a res onant maximal cylinder c/mapsto→Lc,\nc∈(c−,c+), then applying Lemma 2.9 to the forward and backward maximal orb it cylinders through some\nLc0we get item (ii) since Nis connected. /squaresolidNON-RESONANT CIRCLES 7\nTheorem 2.11. Let(N,ω)be a symplectic surface without boundary having finite sympl ectic area and such\nthat no connected component of Nis diffeomorphic to the two-sphere. Let H:N→Rbe a proper Hamil-\ntonian. If His unbounded from above, then for every c0∈Rthere exists a neighborhood V⊂ {H > c 0}of\n{H= +∞}with smooth, non-empty boundary such that ∂Vis the union of non-resonant circles. A similar\nstatement holds when His unbounded from below.\nProof.SinceHis unbounded from above, there exists a regular value c≥c0ofH. SinceHis proper, the\ncomponents L(1)\nc,...,L(k)\ncof{H=c}are embedded circles. Let\nS:={i∈ {1,...,k} |the maximal forward cylinder of L(i)\ncis resonant }.\nFor alli∈S, there exists by Lemma 2.9 an embedded disc Diwith boundary L(i)\nccontained in {H≥c}. We\nconsider two cases according to whether S/\\e}atio\\slash={1,...,k}or not. If S/\\e}atio\\slash={1,...,k}, then for all i /∈S, letL(i)\nci,\nci≥c, be a non-resonant circle in the maximal forward cylinder of L(i)\nc, and let U(i)\nc,cibe the portion of the\ncylinder between L(i)\ncandL(i)\nci. The statement of the theorem follows taking\nV:={H≥c}\\/bracketleftBig/parenleftBig/uniondisplay\ni∈SDi/parenrightBig\n∪/parenleftBig/uniondisplay\ni/∈SU(i)\nc,ci/parenrightBig/bracketrightBig\n.\nIfS={1,...,k}, then let us consider a regular value c′>maxH|Difor alli= 1,...,kwhich exists since H\nis proper and unbounded from above. We denote by L(1)\nc′,...,L(k′)\nc′the connected components of {H=c′}.\nSinceNdoes not contain connected components homeomorphicto the two -sphere, the maximal orbit cylinder\nthrough L(j)\nc′contains a non-resonant circle L(j)\nc′\nj⊂ {H=c′\nj}. By the choice of c′, we have c′\nj≥cand we\ndefineS′:={j∈ {1,...,k′} |c′\nj≥c′}. In this case the statement of the theorem follows taking\nV:=/parenleftBig/uniondisplay\nj/∈S′U(j)\nc′\nj,c′/parenrightBig\n∪{H≥c′}\\/uniondisplay\nj∈S′U(j)\nc′,c′\nj. /squaresolid\nTheorem 2.12. Let(N,ω)be a symplectic surface without boundary and let H:N→Rbe a Hamiltonian\nhaving a strict local minimum at z−∈Nwhich is degenerate. For every neighborhood Uofz−there exists\na neighborhood V⊂Uofz−such that ∂Vis a union of non-resonant circles. If z−is isolated in the set of\ncritical points of H, thenVcan be taken to be an embedded disc.\nProof.We can assume without loss of generality that Nis connected. Since z−is a strict local minimum,\nwe can choose c > H(z−) such that {H≤c}∩Uis compact. By Sard’s theorem, we can suppose that cis a\nregular value of Hand letL(1)\nc,...,L(k)\ncbe the connected components of {H=c}∩U. Let\nS:={i∈ {1,...,k} |the maximal backward cylinder of L(i)\ncis resonant }.\nFor each i∈Sthere exists a closed embedded disc Di⊂ {H≤c}∩Ucontaining the maximal cylinder. Since\nz−is degenerate, z−/∈Diby Lemma 2.9. If S={1,...,k}, thenU′:= ({H≤c} ∩U)\\/uniontextk\ni=1Diis both\nopen inUand compact. Since Nis connected and z−∈U′, there holds U′=Nwhich is a contradiction\nsinceDi⊂N\\U′for alli= 1,...,k. Therefore S/\\e}atio\\slash={1,...,k}and we can construct the required Vas in\nthe proof of Theorem 2.11. If z−is also isolated as critical point then a short topological argument sh ows\nthat{H≤c}∩Ucan be taken to be diffeomorphic to a disc. /squaresolid\nTheorem 2.13. Let(N,ω)be a symplectic surface without boundary, and let H:N→Rbe a Hamiltonian\nhaving an isolated minimum at an embedded circle L′⊂N. Then for every neighborhood U′ofL′there exists\na neighborhood V⊂U′ofL′such that ∂Vis a union of non-resonant circles. If L′is also isolated in the set\nof critical points of H, thenVcan be taken to be a tubular neighborhood of L′.\nProof.The argument is identical to that of Theorem 2.12 using that L′∩Di=∅for alli∈S, where the\ndiscsDiare defined as in the proof above. /squaresolid\nTheorem 2.14. Let(N,ω)be a symplectic surface without boundary and let H:N→Rbe a proper\nHamiltonian having a non-degenerate saddle point at z1∈Nwithc:=H(z1). Consider a positive chart\n(x,y) :U→(−δ0,+δ0)2centered at z1such that H(x,y) =c+xyfor some δ0>0. For every ǫ < δ < δ 0\nthere exist numbers ǫ1,ǫ2,ǫ3,ǫ4∈(0,ǫ)such that the following four sets belong to non-resonant cir cles\nB1:=/braceleftbig\nH(x,y) =c+δǫ1, x >0, y >0/bracerightbig\n, B 2:=/braceleftbig\nH(x,y) =c−δǫ2,−x >0, y >0/bracerightbig\n,8 L. ASSELLE AND G. BENEDETTI\nB3:=/braceleftbig\nH(x,y) =c+δǫ3,−x >0,−y >0/bracerightbig\n, B 4:=/braceleftbig\nH(x,y) =c−δǫ4, x >0,−y >0/bracerightbig\n.\nProof.DefineCǫ′:={H(x,y) =c+δǫ′, x >0, y >0} ⊂Nfor some ǫ′∈(0,ǫ) such that c+δǫ′is a\nregular value of Hwhich exists by Sard’s theorem. Let Lc+δǫ′be the embedded circle containing Cǫ′. By\nLemma 2.9 the maximal backward orbit cylinder through Lc+δǫ′contains a non-resonant circle Lc+δǫ1for\nsomeǫ1∈(0,ǫ′]. In a similar manner we construct ǫ2,ǫ3,ǫ4. /squaresolid\n3.The Moser twist theorem: Trapping regions and Poincar ´e–Birkhoff orbits\nIn this section we recall how to use non-resonant circles to prove t he existence of trapping regions and\nperiodic orbits for small perturbations of an autonomous Hamiltonia n system given by a proper Hamiltonian\nH:N→Ron a symplectic surface ( N,ω) without boundary. More precisely, for natural numbers k < l\nconsider a one-parameter family of time-dependent Hamiltonian fun ctions\nHs:N×T→R, H s(z,t) :=skH(z)+slRs(z,t),∀(z,t)∈N×T, (3.1)\nwhereRs:N×T→Ris a perturbation depending smoothly on the parameter s∈[0,s1) and 2π-periodically\non the time t∈T. LetXs,tbe the time-dependent Hamiltonian vector field associated with the f unction\nz/mapsto→Hs(z,t) and consider the map ϕt0,t1,s:Ns→Nsuch that ϕt0,t1,s(z) =z(t1) wherez(t) is the unique\nsolution of\n˙z(t) =Xs,t(z(t)) (3.2)\nwithz(t0) =z. The open set Nswhereϕt0,t1,sis defined contains an arbitrary compact subset of Nwhens\nis sufficiently small since Hsdepends periodically on time.\nAn application of the Moser twist theorem and of the Poincar´ e–Birk hoff theorem yields the following\nclassical result.\nTheorem 3.1. LetL⊂Nbe a non-resonant circle for H. For all neighborhoods UofL, there exists\ns∗∈(0,s1)and a neighborhood U′ofLcontained in Uwith the property that for all s∈[0,s∗)there holds\n(1) there exist infinitely many solutions of (3.2)whose period is a multiple of 2πand whose free-homotopy\nclass is a multiple of L,\n(2) every solution of (3.2)starting in U′at some time t∈Rstays inUfor all times.\nProof.Let (r,θ)∈I×Tbe action-angle coordinates of ωin a tubular neighborhood of LwhereIis some\nopen interval. These coordinates can be chosen so that the level s ets ofrare embedded circles of Hand\nL={r=r0}for some r0∈I[8]. Let ϕs:=ϕ0,2π,s. There exists a function f:I→Rso that in these\ncoordinates we can write ϕs(r,θ) = (rs,θs), where\nrs=r+O(sl),\nθs=θ+skf(r)+O(sl).(3.3)\nThe condition of Lbeing non-resonant is equivalent to f′(r0)/\\e}atio\\slash= 0 [13]. Therefore there exists a sub-interval\n[r−,r+] containing r0in its interior such that f′(r)/\\e}atio\\slash= 0 for every r∈[r−,r+] and{(r,θ)∈[r−,r+]×T} ⊂U.\nSinceϕsis a Hamiltonian diffeomorphism, it is exact symplectic. In other words, there exists a function\nWs:I×T→Rsuch that\nϕ∗\ns(rsdθs) =rdθ+dWs.\nThuswecanapplytheMosertwisttheorem[31]at r−andr+tofindtwo ϕs-invariantclosedcurves βs,r−,βs,r+\narbitrarily close to {r=r−}, respectively {r=r+}, forssmall enough. Thus we can assume that βs,r−and\nβs,r+bound an annular region As⊂Uwhich is invariant under ϕsand contains U′:={(r,θ)∈[˜r−,˜r+]×T}\nin its interior for some ˜ r−< r0<˜r+.\nLet us show (2) in the statement of the theorem. For every t∈RletAt,s=ϕ0,t,s(As) so that there holds\nϕt,t1,s(At,s) =At1,s∀t,t1∈R.\nSinceHsdepends periodically on time and ris constant along the flow of H0, there holds U′⊂At,s⊂Ufor\nallssmall enough and for all t∈R. Thus a solution of (3.2) passing through U′stays inUfor all times.\nLet us show (1) in the statement of the theorem. Consider the res triction of ϕsto the closed annulus As.\nRecall the definition of the rotation number of a point z∈As\nρs(z) := lim\nn→+∞˜ϕn\ns(z)−z\nnNON-RESONANT CIRCLES 9\nwhere ˜ϕsdenotes a lift of ϕsto the universal cover ˜AsofAs. Ifz±∈βs,r±, then (3.3) together with the fact\nthatβs,r±is close to {r=r±}implies that\nρs,±:=ρs(z±) =sk2πf(r±)+o(sk)\nIn particular, for ssmall enough the rotation numbers ρs,−andρs,+are different and by the Poincar´ e–\nBirkhoff fixed point theorem [27, Theorem 7.1] the map ϕs|Ashas a periodic point with rotation number p/q\nfor every rational number p/qlying between ρs,−andρs,+. /squaresolid\nCombining Theorem 3.1 with Theorem 2.10 we get the existence of infinit ely many periodic orbits and\ntrapping regions for Hsif (N,ω) is a connected, symplectic surface of finite area such that either N/\\e}atio\\slash∼=S2or\nN∼=S2andHdoes not give a Hamiltonian circle action. Combining Theorem 3.1 with The orems 2.11, 2.12,\nand 2.13, we get more precise information on the location of the perio dic orbits and the trapping regions.\nTheorem 3.2. IfHis unbounded from above (resp. below), or has a degenerate st rict local minimum\n(resp. maximum) z0∈N, or has a strict local minimum (resp. maximum) at an embedded circleL′, then ar-\nbitrarily small neighborhoods of {H= +∞},z0orL′are trapping regions for the orbits of (3.2)and contain\ninfinitely many periodic orbits of (3.2)for every ssmall enough. /squaresolid\nFinally, we apply Theorem 2.14 to study the stability properties of non -degenerate saddle points of H.\nTheorem 3.3. Let(N,ω)be a symplectic surface without boundary, H:N→Ra proper Hamiltonian with\na non-degenerate saddle point z1. Under the notation of Theorem 2.14, for every ǫ < δ < δ 0define\nUδ:=/braceleftbig\n(x,y)∈(−δ,δ)2/bracerightbig\n, A+\nδ,ǫ:=/braceleftbig\n|x|=δ,|y| ≤ǫ/bracerightbig\n, A−\nδ,ǫ:=/braceleftbig\n|y|=δ,|x| ≤ǫ/bracerightbig\n.\nThen for every ǫ < δ < δ 0there exists δ′< ǫands∗>0such that for all s < s∗every orbit zof(3.2)with\nz(t0)∈Uδ′for some t0∈Rhas the following property: There exist t−∈[−∞,t0)andt+∈(t0,+∞]such\nthatz(t)∈Uδfor allt∈(t−,t+). Moreover, either t±=±∞orz(t±)∈A±\nδ,ǫ.\nProof.Givenǫandδ, Theorem 2.14 yields the sets B1,B2,B3,B4which are parts of non-resonant circles.\nApplying Theorem 3.1 to these non-resonant circles we obtain a value s∗andδ′< ǫ, and tubular neigh-\nborhoods of these circles which cannot be intersected by solutions zof (3.2) with parameter s < s∗such\nthatz(t0)∈Uδ′for some t0. LetVbe the union of these tubular neighborhoods and let V′the connected\ncomponents of Uδ\\Vcontaining Uδ′. Then\n∂V′∩∂Uδ⊂A−\nδ,ǫ∪A+\nδ,ǫ.\nThus either zstays inUδfor all times larger than t0or it must have a first intersection time larger than t0\nwith∂UδatA−\nδ,ǫ∪A+\nδ,ǫ. Notice however that Xt,spoints inside UδonA−\nδ,ǫand outside of it on A+\nδ,ǫifsis\nsmall enough. Therefore zcan only exit UδatA+\nδ,ǫand enter at A−\nδ,ǫ. /squaresolid\n4.Applications to magnetic systems via the Hamiltonian norma l form\nIn this section we see how the normal form established in [20, 13] allo ws us to apply the results of the\nprevious section to magnetic systems at low speeds and prove Theo rems 1.4 and 1.5.\nLet (g,b) be a magnetic system on a closed, connected, oriented surface M. Equation (1.1) induces a\nflow onTM, usually referred to as the magnetic flow , which preserves the set of vectors with fixed speed\nSsM:={(q,v)∈TM| |v|q=s}, where|·|is the norm associated with g. We recall now that the flow on\nTMis Hamiltonian. To this purpose let λ∈Ω1(TM) be the Hilbert form of g, namely the pull-back of the\nstandard Liouville one-form on T∗Mby means of the isomorphism TM→T∗Minduced by the metric. We\ndefine the twisted symplectic form\nω(g,b):= dλ−π∗(bµ),\nwhereπ:TM→Mis the foot-point projectionand µis the areaforminduced by gand the givenorientation,\nand consider the kinetic Hamiltonian\nHkin:TM→R, H kin(q,v) =1\n2|v|2\nq.\nIt is a classical fact that the Hamiltonian flow Φ (g,b):R×TM→TMinduced by the pair ( Hkin,ω(g,b)) is\nthe magnetic flow of the pair ( g,b). This implies that the restriction of the magnetic flow to each SsMis\neverywhere tangent to the characteristic line distribution\nker(ω(g,b)|SsM)⊂T(SsM).10 L. ASSELLE AND G. BENEDETTI\nSettingSM:=S1Mand defining the rescaling map Fs:SM→SsM,Fs(q,v) = (q,sv), we get\nωs:=F∗\nsω(g,b)|SsM=sdλ−π∗(bµ),\nwhere on the right-hand side we have also denoted by λandπ∗(bµ) the restrictions of these objects to\nSM. Therefore the map Fsidentifies integral curves of the characteristic distribution ker ωswith magnetic\ngeodesics, namely solutions of (1.1), up to time reparametrization.\nLetU⊂Mbe an open set on which bdoes not vanish and such that π:SU→Uadmits a section\nW:U→SU. Thusω:=bµis a symplectic form and ( U,bµ) is a symplectic surface without boundary\nwith finite area. Moreover the angular function t:SU→Tassociated with Wyields a trivialization\nτ:U×T→SU. Following [13, Theorem 1.1] we see that for every open set U′such that U′⊂U, there\nexists0>0 and an isotopy of embeddings Ψ s:U′×T→SU,s∈[0,s0) with Ψ 0=τ|U′×Tsuch that\nΨ∗\nsωs= d(Hsdt)−π∗(bµ),\nwhereHs:U′×T→Ris a path of functions such that\nHs(z,t) =−s2\n2b(z)+s3Rs(z,t),∀(z,t)∈U′×T (4.1)\nfor some remainder Rs:U′×T. Ifbis constant, then we can choose Ψ sso that\nHs(z,t) =−s2\n2b−s4\n(2b)3K(z)+s5˜Rs(z,t),∀(z,t)∈U′×T, (4.2)\nwhereK:M→Ris the Gaussian curvature of gand˜Rs:U′×T→Ris some remainder. A short\ncomputation shows that the curves Γ : ( t0,t1)→U′×T, Γ(t) = (z(t),t) wherezis a solution of the non-\nautonomous Hamiltonian system with Hamiltonian Hsand symplectic form ω=bµare exactly the integral\ncurves of the characteristic foliation of ker(d( Hsdt)−π∗(bµ)) = ker(Ψ∗\nsωs).\nRemark 4.1. Up to a constant factor, the leading term in (4.1) is given by the Hamilt onianb−1:U′→R.\nThe Hamiltonian flow of b−1with symplectic form bµis the same as the Hamiltonian flow of1\n2b−2with\nsymplectic form µ. Thus non-resonant circles for b−1with symplectic form bµare the same as non-resonant\ncircles for b−2with symplectic form µ. /squaresolid\nRemark 4.2. Ift/mapsto→z(t) is a periodic orbit for Hs, thenγ(t) :=π(Ψs(z(t),t)) is a periodic magnetic geodesic\nwith speed sfreely homotopic to zsinceπ◦Ψ0=π. Moreover, if U′is an open set with U′⊂Uhaving the\nproperty that every solutions of the non-autonomous Hamiltonian system of Hspassing through U′stays in\nUfor all times, then up to slightly modifying UandU′the same holds for magnetic geodesics with small\nspeedssince Ψ sis close to the trivialization Ψ 0=τ. /squaresolid\nWe are in position to apply the results of Section 3 to the trajectorie s ofHsand then transfer them to the\nsolutions of (1.1) via the maps F◦Ψs.\nProof of Theorem 1.4. Let (g,b) be a magnetic system which is non-resonant at zero speed. Let us assume\nfirst that bis not constant and define the open set N:={b/\\e}atio\\slash= 0}. The function b−1:N→Ris proper\nand the symplectic area of Nwith respect to ω=bµis finite. Then Theorem 2.10 implies that b−1has\na non-resonant circle L: This is clear if {b= 0}is non-empty (as N/\\e}atio\\slash=S2in this case) and it is true by\nDefinition 1.1 together with Remark 4.1 otherwise. By Theorem 3.1 app lied to a neighborhood Uof the circle\nL, forssmall enough the function Hsgiven in (4.1) admits infinitely many periodic orbits freely homotopic\nto some multiple of Land there is a neighborhood U′ofLsuch that every orbit of Hsstarting in U′stay in\nUfor all times. By Remark 4.2 the same applies to magnetic geodesics fo r a small enough speed s.\nIfbis a non-zero constant, then by Definition 1.1 and Theorem 2.10 the G aussian curvature K:M→R\nadmits a non-resonant circle Lwith respect to the symplectic form µ. In this case the leading term of Hs\ngivenin (4.2) is equalto Kup to constantsand wecan run the sameargumentas in the caseof a non-constant\nbusing Theorem 3.1 to finish the proof of Theorem 1.4. /squaresolid\nProof of Theorem 1.5. Using Remark 4.1 and 4.2 the result follows from Theorem 3.2 applied to Hsgiven\nin (4.1), (4.2). Indeed observe that if the set {b= 0}is non-empty, then the Hamiltonian b−1is proper and\nunbounded on the non-empty open set {b/\\e}atio\\slash= 0}. /squaresolid\nUsing the notation of Theorem 3.3, we get a result for saddle points o fborKas promised in Remark 1.7.NON-RESONANT CIRCLES 11\nTheorem 4.3. Letz1∈ {b/\\e}atio\\slash= 0}be a non-degenerate saddle point of the function −b−1. For every ǫ < δ < δ 0\nthere exists δ′ands∗such that for all s < s∗every magnetic geodesic γwith speed sandγ(t0)∈Uδ′for\nsomet0∈Rhas the following property: There exist t−∈[−∞,t0)andt+∈(t0,+∞]such that γ(t)∈Uδ\nfor allt∈(t−,t+). Moreover, either t±=±∞orγ(t±)∈A±\nδ,ǫ. A similar statement holds if bis a non-zero\nconstant and z1is a non-degenerate saddle point for −K.\nProof.Using Remark 4.1 and 4.2 the result follows from Theorem 3.3 applied to Hsgiven in (4.1), (4.2). /squaresolid\n5.The resonant case\nIn this last section, we collect some observations about magnetic sy stems (g,b) onS2that are resonant at\nzero speed for which the main results in Section 1.2 cannot be applied. Let us have a look first at the case in\nwhich (g,b) is rotationally symmetric. This means that there are spherical coo rdinates ( r,φ)∈(r−,r+)×T\non the complement of the north and south pole such that g= dr2+a(r)2dφ2andb=b(r) for some functions\na,b: (r−,r+)→R. A closer inspection of the proof of (4.1) and (4.2) given in [13, Sectio n 3] shows that Hs\nis independent of φin this case. Thus we have the formula\nXs,t=∂rHs\nab∂φ (5.1)\nfor the time-dependent Hamiltonian vector field of Hs. Therefore the level sets of the function rwithb(r)/\\e}atio\\slash= 0\nare invariant under the flow of Hsand using the isotopy Ψ swe find that magnetic geodesics with speed sare\ntrapped around each of these circles. By (5.1) and (4.1), the dyna mics ofHson such circles over an interval\nof time of length 2 πis given by a rotation of angle\nρs(r) :=−s2πb′(r)\na(r)b3(r)+O(s3).\nTherefore, if b′(r)/\\e}atio\\slash= 0, the function s/mapsto→ρs(r) is non-constant for ssmall. Thus there is a decreasingsequence\nsnconverging to 0 such that ρsn(r) is rational. By continuity ρsn(rm) is rational for infinitely many values\n(rm)m∈Nclose tor. In particular, there are infinitely many magnetic geodesics with spe edsnfor alln∈N.\nIn the complementary case in which bis a non-zero constant, we can show that the curvature Kcannot\nyield a non-constant circle action.\nTheorem 5.1. Letg= dr2+a(r)2dφ2be a metric of revolution on S2wherea: [0,R]→[0,∞)with\nnon-constant Gaussian curvature K. ThenKdoes not induce a Hamiltonian circle action on (S2,µ).\nProof.Up to rescaling gwe can assume that area g(S2) = 4π. LetA: [0,R]→Rbe the primitive of awith\nthe property that A(0) =−1 andA(R) = 1. Since r= 0,Rcorrespond to the south and north pole of S2,\nwe also have A′(0) = 0 = A′(R) andA′′(0) = 1,A′′(R) =−1.\nWe have µ=adr∧dφ= dA∧dφ. Therefore, Kinduces a non-constant Hamiltonian circle action if and\nonly if\nK=c1A+c2\nfor some constants c1,c2∈Rwithc1/\\e}atio\\slash= 0. Since K=−(A′)−1A′′′, we can integrate this equation to\n2c1A′′+(c1A+c2)2+c3= 0 (5.2)\nfor some constant c3. Using the boundary conditions on AandA′′we get\n2c1+(c2−c1)2+c3= 0,−2c1+(c2+c1)2+c3= 0.\nEliminating c3, we find 4 c1= (c2+c1)2−(c2−c1)2. Sincec1/\\e}atio\\slash= 0, this is equivalent to c2= 1. Consequently\nc3=−1−c2\n1. Substituting c2andc3in (5.2) and dividing by c1, we arrive at\n2A′′+c1A2+2A−c1= 0.\nMultiplying this equation by A′and integrating once again, we find\n(A′)2+1\n3c1A3+A2−c1A=c4.\nUsing the boundary conditions for AandA′, we get\n1\n3c1+1−c1=−1\n3c1+1+c1,\nwhich is equivalent to c1= 0, a contradiction. /squaresolid12 L. ASSELLE AND G. BENEDETTI\nCorollary 5.2. Let(g,b)be a magnetic system on S2such that gis rotationally symmetric and bis a non-\nzero constant. Then for every ssmall enough there are infinitely many periodic magnetic geo desics of speed\ns. /squaresolid\nPrompted by these results for rotationally symmetric magnetic sys tems we can formulate the following\nconjecture for arbitrary resonant magnetic systems.\nConjecture. Let(g,b)be a magnetic system on S2which is resonant at zero speed. Then there exists a\nsequence sn→0such that there are infinitely many periodic magnetic geodes ics with speed snfor alln∈N.\nFinally, we can ask what geometric conditions general magnetic syst ems (g,b) onS2must satisfy in order\nto be non-resonant at zero speed. Surely, given gone readily finds a positive, non-constant function bsuch\nthatb−2yields a Hamiltonian circle action on ( S2,µ). Thus we consider the other case in which bis constant\nandgis a metric whose Gaussian curvature Kyields a non-constant Hamiltonian circle action on ( S2,µ).\nThenKis a Morse function with exactly one minimum and one maximum and by Lem ma 2.8\nµ= dK∧dφ φ∈R/TZ, (5.3)\nwhere\nT:=areag(S2)\nmaxK−minK.\nIntegrating (5.3) on {K≤k}for an arbitrary kwe get\nk= minK+maxK−minK\nareag(S2)areag/parenleftbig\n{K≤k}/parenrightbig\n,∀k∈[minK,maxK]. 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Plasma confinement . Dover books on physics. Courier Corporation, Mineola, New York,\n2003.\n[29] D. McDuff and D. Salamon. Introduction to symplectic topology . Oxford Mathematical Monographs. The Clarendon Press,\nOxford University Press, New York, second edition, 1998.\n[30] J. A. G. Miranda. Positive topological entropy for magn etic flows on surfaces. Nonlinearity , 20(8):2007–2031, 2007.\n[31] J. Moser. On invariant curves of area-preserving mappi ngs of an annulus. Nachr. Akad. Wiss. G¨ ottingen Math.-Phys. Kl.\nII, 1962:1–20, 1962.\n[32] M. Struwe. Existence of periodic solutions of Hamilton ian systems on almost every energy surface. Bol. Soc. Brasil. Mat.\n(N.S.), 20(2):49–58, 1990.\n[33] I. A. Taimanov. Closed extremals on two-dimensional ma nifolds.Uspekhi Mat. Nauk , 47(2(284)):143–185, 223, 1992.\n[34] F. Truc. Trajectoires born´ ees d’une particule soumis e ` a un champ magn´ etique sym´ etrique lin´ eaire. Ann. Inst. H. Poincar´ e\nPhys. Th´ eor. , 64(2):127–154, 1996.\nJustus Liebig Universit ¨at Giessen, Mathematisches Institut\nArndtstrasse 2, 35392 Giessen, Germany\nEmail address :luca.asselle@math.uni-giessen.de\nUniversit ¨at Heidelberg, Mathematisches Institut,\nIm Neuenheimer Feld 205, 69120 Heidelberg, Germany\nEmail address :gbenedetti@mathi.uni-heidelberg.de"    },    {        "title": "0905.1733v2.Dynamically_Generated_Anomalous_Magnetic_Moment_in_Massless_QED.pdf",        "content": "arXiv:0905.1733v2  [hep-ph]  2 Sep 2009Dynamically Generated Anomalous Magnetic Moment in Massle ss QED\nEfrain J. Ferrer and Vivian de la Incera\nDepartment of Physics, University of Texas at El Paso, El Pas o, TX 79968, USA\nIn this paper we investigate the non-perturbative generati on of an anomalous magnetic moment\nfor massless fermions in the presence of an external magneti c field. In thecontextof massless QED in\na magnetic field, we prove that the phenomenon of magnetic cat alysis of chiral symmetry breaking,\nwhich has been associated in the literature with dynamical m ass generation, is also responsible for\nthe generation of a dynamical anomalous magnetic moment. As a consequence, the degenerate\nenergy of electrons in Landau levels higher than zero exhibi ts Zeeman splitting. We explicitly\nreport the splitting for the first Landau level and find the non -perturbative Lande g-factor and\nBohr magneton. We anticipate that a dynamically generated a nomalous magnetic moment will be a\nuniversal feature of theories with magnetic catalysis. Our findings can be important for condensed\nplanar systems as graphene, as well as for highly magnetized dense systems as those forming the\ncore of compact stars.\nPACS numbers: 11.30.Rd, 13.40.Em, 21.65.Qr, 81.05.Uw\nI. INTRODUCTION\nThe quantum mechanical description of charged fermions in a const ant magnetic field has attracted much attention\nsince the early developments of Quantum Electrodynamics (QED) [1]- [4]. The partial Lorentz symmetry breaking\nproduced by the magnetic field is reflected in the quantum mechanica l properties of the charged fermions which\nbehave as free particles along the field, but have quantized moment a [2] (characterized by a Landau level l) in the\ntransverse direction. At the tree level, the Landau levels l≥1 are degenerate with respect to the spin projections\nalong and opposite to the magnetic field. However, this degeneracy is broken by radiative corrections [3], giving rise\nto a spin-field interaction term in the effective action and thus to an a nomalous magnetic moment.\nThe theory of the electron magnetic moment has historically played a n important role in the development of QED.\nAs it is known, the electron intrinsic magnetic moment µis related to the spin vector sbyµ=gµBs, where\nµB=e/planckover2pi1/2mcis the Bohr magneton, and gis the Lande g-factor. One of the greatest triumphs of the Dirac theory\nwas the prediction of the value g= 2 for the Lande g-factor of the electron in the non-relativistic limit. However,\nthis prediction was later challenged by more refined experimental me asurements showing a value slightly larger than\n2. The solution of this puzzle came only after Schwinger calculated th e first-order radiative correction to µdue to\nthe electron-photon interactions [3]. Schwinger’s results led to an a nomalous magnetic moment with a correction to\ntheg-factor of orderg−2\n2=α\n2π,αbeing the fine-structure constant. Higher-order radiative corr ections toghave\nsubsequently led to a series in powers of α/π[5] that is in excellent agreement with the experiment.\nFor massless electrons, however, no anomalous magnetic moment c an be found through Schwinger’s perturbative\napproach. The problem is that an anomalous magnetic moment would b reak the chiral symmetry of massless QED,\nbut this symmetry is protected against any perturbatively genera ted breaking term. However, no protection exists\nagainst non-perturbative breaking of chiral symmetry. A non-pe rturbative chiral symmetry breaking mechanism is\nknown to exist in theories of massless fermions in a constant and unif orm magnetic field [6]. It is called the magnetic\ncatalysis of chiral symmetry breaking ( MCχSB ). TheMCχSB gives rise to a chiral condensate that in turns can\nproduce adynamicalfermion mass. Until recently, all the studies o fMCχSB focused on the generationofa dynamical\nmass [6]-[10], but ignored the possibility of a dynamically generated ano malous magnetic moment. In a recent letter\n[11], we reconsidered the MCχSB in massless QED and showed that in addition to the dynamical mass, th e magnetic\ncatalysis simultaneously produces a dynamically generated anomalou s magnetic moment. Our results gave rise to a\nnon-perturbative Bohr magneton proportional to the inverse of the dynamical mass, in analogy to the way in which\nthe bare Bohr magneton depends on the bare mass. In the presen t paper we develop in detail the calculations that\nled to the magnetically catalyzed anomalous magnetic moment found in Ref. [11]. We also discuss some elucidating\npoints about the infrared dynamics of fermions in the lowest Landau level (LLL) and their role in the generation of\nthe anomalous magnetic moment for fermions in higher Landau levels.\nThe plan of the paper is as follows. In Sec. 2, we present a brief histo rical account of the appearance of magnetic\nmoment contributions in relativistic and non-relativistic theories of c harged fermions, including a summary of our\nown findings for the case of massless fermions, which will be develope d in detail in the rest of the paper. In Sec. 3,\nwe calculate the fermion full propagator in massless QED in a magnetic field, considering the Dirac’s structures for\nmass, magnetic moment and wave-function renormalization term. I n Sec. 4, an infinite system of coupled Schwinger-\nDyson (SD) equations for the fermion self-energy is derived within t he ladder approximation. This system is then2\nconsistently solved in Sec. 5, where explicit solutions of the dynamica l quantities are found in the first and in the\nLLL. The dispersion relations for fermions at different LLs in the chir al-condensate phase are calculated in Sec. 6. In\nparticular,the Zeemaneffectforfermionsinthe firstLLisanalyzed andthe correspondingnon-perturbativeLande’s g-\nfactorand Bohrmagneton areidentified. In Sec. 7, we calculate th e chiralcondensateand find the connectionbetween\nthis order parameter and the rest energy of the electrons in the L LL, which is given by the sum of the dynamical\nmass and the magnetic energy due to the anomalous magnetic momen t term. This result reflects the infrared origin of\nthe condensation phenomenon and underlines the fact that both p arameters, the mass and the anomalous magnetic\nmoment, can be induced once the chiral symmetry is broken by the c ondensate. Possible applications of the outcomes\nof this paper for planar condensed matter systems as graphene a nd for dense astrophysical systems as compact stars\nor magnetars are outlined in Sec. 8. Finally, using a chiral-spin repres entation for fermions in the LLL, it is shown in\nthe Appendix that the dynamics of such fermions reduces to that o f free particles in a (1+1)-dimensional space with\nan induced rest-energy that depends non-perturbatively on the coupling constant and applied magnetic field.\nII. MAGNETIC MOMENT HISTORICAL REVIEW\nFor the sake of understanding we will briefly review in this section the role that spin-field interactions have played\nin non-relativistic and relativistic theories of spin-1\n2charged particles in a magnetic field. We discuss massive and\nmassless QED and explain why in the massless theory we cannot follow t he perturbative approach that gave rise to\nan anomalous magnetic moment in the massive situation. It will become clear throughout the paper that a non-\nperturbative method is required in the massless case because the a nomalous magnetic moment there will be closely\nconnected to the breaking of the original chiral symmetry, hence it has to be dynamically generated through some\nnon-perturbative mechanism.\nA. Non-Relativistic Case\nLet us start with the case of a non-relativistic spinless charged par ticle in the presence of a constant and uniform\nmagnetic field H. Assuming a magnetic field pointing along the x3-direction, the energy eigenvalues ESof the\nSchrodinger Hamiltonian\nHS=1\n2m(p−eA)2(1)\nare given by [2, 12]\nES=p2\n3\n2m+|eH|\nm(n+1\n2)n= 0,1,2,..., (2)\nNotice that the particle can freely move along the field direction, while in the plane perpendicular to the field its\nmotion is confined to quantum orbits labeled by the discrete numbers n. Without loss of generality, we can assume\nfrom now on that |eH|=eH.\nFor charged fermions, the Schrodinger Hamiltonian needs to include a spin-field interaction term originally intro-\nduced by Pauli as\nHP=1\n2m(p−eA)2−µ·H, (3)\nwhere\nµ=gµBs (4)\nis the particle intrinsic magnetic moment. Here µBis the Bohr magneton, sthe spin operator given in terms of the\nPauli matrices as s=σ/2, andgthe Landeg-factor. For bare electrons g= 2.\nThe corresponding energy eigenvalues are\nEP=p2\n3\n2m+eH\nm(n+1\n2)−µBσH n = 0,1,2,..., σ =±1 (5)\nHereσdenotes spin projections along (+) and opposite ( −) to the field direction. The last equation can be rewritten\nas\nEP=p2\n3\n2m+2µB(n+1\n2−σ\n2)H=p2\n3\n2m+2µBlH (6)3\nwhere\nl=n+1\n2−σ\n2, l= 0,1,2,... (7)\nThe number lis known as the Landau level number. Notice the double degeneracy of all thel/ne}ationslash= 0 due to the two spin\nprojections contributing to each of them. The LLL ( l= 0) is not degenerate because there is only one combination\nof the non-negative integer nand spin projection σthat can produce l= 0. As we will see below, the degeneracy of\nl/ne}ationslash= 0 can be broken by radiative corrections which give rise to an anoma lous magnetic moment term in the electron\nself-energy.\nB. Relativistic Case\nIn the relativistic case we should start from the Dirac equation in the presence of an external magnetic field\n(Πµγµ−m)ψ= 0 (8)\nwhere the field is introduced through the covariant derivative\nΠµ=i∂µ−eAµ (9)\nAssuming again a uniform and constant magnetic field along the x3direction, and using the gauge A2=Hx1,\nA0=A1=A3= 0, Eq. (8) can be solved for ψ. It follows from (8) that the energy eigenvalues for the relativistic\nspin-1\n2particle are found [4] to be\nER=±/radicalbigg\nm2+p2\n3+2eH(n+1\n2−σ\n2)n= 0,1,2,..., σ =±1 (10)\nwith (+) for particles and ( −) for antiparticles. In terms of the Landau levels (7) the relativistic energy becomes\nER=±/radicalBig\nm2+p2\n3+2eHl l = 0,1,2,... (11)\nClearly the double spin degeneracy of the l/ne}ationslash= 0 states is also present for relativistic fermions.\nThe non-relativistic energy can be easily recuperated by taking the limit 2eH/m2≪1 andp2\n3/m2≪1 in (11) and\nsubtracting the rest energy m,\nENR= lim\n2eH/m2,p2\n3/m2→0ER−m= lim\n2eH/m2,p2\n3/m2→0m/radicalbigg\n1+p2\n3\nm2+2eH\nm2l−m≃p2\n3\n2m+2µBlH. (12)\nSince the spin is automatically incorporated in the relativistic treatme nt, the non-relativistic limit leads directly\nto the Pauli theory (6). Even more, it naturally gives that a unit of s pin angular momentum ( σ/2) interacts with\nthe magnetic field with a coupling of 2 µB, that is, the Dirac theory automatically produces the correct g-factor of 2,\nsomething that was a puzzle at those times.\nC. Anomalous Magnetic Moment\nDespite Dirac theory’s success in predicting g= 2, this result was later challenged by experimental findings of g>2\nfor electrons/positrons. The solution to this puzzle was provided b y Schwinger. In a classical paper on the topic\n[3], Schwinger calculated the one-loop contribution to the fermion se lf-energy in a weak magnetic field that led to an\nanomalous magnetic moment\nµA= (g′−2)µBs (13)\nAccordingly, the Lande g-factor was modified as\ng′= 2(1+α\n2π) (14)\nin good agreement with the experiment.4\nSchwinger’s result contained the first order correction in α/π. Subsequent higher-order corrections to ggive rise to\na series in powers of α/π[5]. The corrections up to the eighth order has shown an agreemen t with the experimental\nvalue that is good to one part in 1012[13].\nTaking into account Schwinger’s anomalous magnetic moment, the Dir ac equation acquires an extra structure\nσµνFµν(withσµν=i\n2[γµγν]), so\n(Πµγµ−m+κµBHΣ3)ψ= 0 (15)\nwhereκ=α/2πand Σ 3=iγ1γ2is the spin operator. The corresponding relativistic-particle energ y [14] is\nE2\nl,σ= [(m2+2eHl)1/2−µBκHσ]2+p2\n3, l= 0,1,2,..., σ=±1 (16)\nHence, once the anomalous magnetic moment contribution is conside red the spin degeneracy is removed since the\nenergy (16) explicitly depends on the spin projection σ. It is timely to clarify here a mistake appearing in Ref. [15],\nwhere it was argued that a magnetic moment term ( σµνFµν) could not be present in the self-energy because it would\ngive rise to an energy that would depend on the orientation of the ma gnetic field. It is obvious from (16), that no\nmatter what the direction of the magnetic field is, the energy of the particle with spin oriented along or opposite\nto the field will not change. Notice that a 180◦rotation equally affects the magnetic field and the spin, leaving the\nmagnetic moment contribution to the energy −µ·Hunchanged.\nAs was done in Eqs. (12), we can take the non-relativistic limit of (16) to find\nENR=p2\n3\n2m+2(n+1\n2)µBH−2(1+κ)µBσ\n2H=\n=p2\n3\n2m+2(n+1\n2)µBH−g′µBσ\n2H=p2\n3\n2m+2lµBH−κµBσH, (17)\nThis result shows that the Lande g-factor depends on the fine-structure constant as pointed out in (14), and that the\nanomalous magnetic moment breaks the spin degeneracy of all LLs w ithl≥1, hence producing the following energy\nsplitting of the levels\n∆El= 2κµBH= (g′−2)µBH (18)\nThe case with strong fields, i.e. fields about Bc∼1013G, was considered in Refs.[16]-[17]. In those works, the authors\ntreated the field exactly, (i.e. without expanding in powers of H), in the fermion one-loop self energy but kept a\nperturbative treatment in the coupling constant. In that approx imation the energy splitting due to the anomalous\nmagnetic moment no longer changes linearly with H, and besides it depends on l[17].\nD. Induced Magnetic Moment for Massless Fermions\nAs previously shown, the electron magnetic moment is related throu gh the Bohr magneton µBto its charge-to-\nmass ratio. Hence, the origin of the extra contribution due to the a nomalous magnetic moment appearing in quantum\nfield theory can be understood from the fact that there the elect ron is continuously self-interacting through radiative\ninteractions. Thus, part of the electron energy, and consequen tly of the electron mass, will be transferred to the\ncreated photon cloud. Therefore, as a consequence of the decr ease of the electron’s mass, the corresponding magnetic\nmoment will be strengthened.\nOn the other hand, as we have already stressed in the Introductio n, if one is interested in exploring the appearance\nof an anomalous magnetic moment in a theory of massless fermions, a perturbative, ”a-la-Schwinger” approach is not\npossible anymore. Since an anomalous magnetic moment would break t he chiral symmetry of the massless theory, it\ncan only be generated via non-perturbative effects.\nHenceforth, we are going to explore the dynamical generation of a n anomalous magnetic moment in the context of\nmassless 4-dimensional QED in the presence of a uniform and consta nt magnetic field. As we are going to show in\nthe next sections, the formation of a chiral condensate through theMCχSB mechanism is responsible not just for\nthe dynamical generation of a mass, but also for the appearance o f a dynamical magnetic moment.\nPhysically it is easy to understand the origin of the new dynamical qua ntity. The chiral condensate carries non-zero\nmagnetic moment, since the particles forming the condensate have opposite spins and opposite charges. Therefore,\nchiral condensation will inexorably provide the quasiparticles with bo th a dynamical mass and a dynamical magnetic\nmoment. Symmetry arguments can help us also to better understa nd this phenomenon. A magnetic moment term\ndoes not break any additional symmetry that has not already been broken by a mass term. Hence, once MCχSB\noccurs, there is no reason why only one of these parameters shou ld be different from zero.5\nWe will show below that a very important consequence of the dynamic ally generated magnetic moment is a splitting\nin the electron energy spectrum that can be interpreted as a non- perturbative Zeeman effect. In the LLL, since only\nelectrons with one spin projection are allowed, there is no energy de generacy and therefore no splitting can occur.\nHowever, for electrons in higher LLs the energy of the degenerat ed spin states is splitted by the interaction of the\ninduced magnetic moment with the applied field. The corresponding en ergy splitting can be conveniently written in\nthe well known form of the Zeeman energy splitting for the two spin p rojections as\n∆E= 2/tildewideκ/tildewideµBH (19)\nwhere/tildewideκand/tildewideµBare the non-perturbative Lande g-factor and Bohr magneton respectively. For electrons in the firs t\nLL they are given by [11]\n/tildewideκ=e−2√\nπ/α,/tildewideµB=e\n2M1(20)\nWorth to highlight in the above results are the non-perturbative de pendence on the coupling constant αin the Lande\ng-factor and the Bohr magneton’s dependence on the dynamically ind uced electron mass M1.\nIII. ELECTRON FULL PROPAGATOR IN MOMENTUM SPACE\nIn this Section we are going to obtain the fermion’s full propagator in QED with massless bare fermions in the\npresence of a constant and uniform magnetic field. The full propag ator obeys the following equation\n[Πµγµ−Σ(x,x′)]G(x,x′) =δ4(x−x′) (21)\nThe structure of the the self-energy Σ( x,x′) [18]\nΣ(x,x′) = (Z/bardblΠ/bardbl\nµγµ\n/bardbl+Z⊥Π⊥\nµγµ\n⊥+M+T\n2/hatwideFµνσµν)δ4(x−x′) (22)\ncontains the wave function’s renormalization coefficients Z/bardblandZ⊥, as well as mass Mand anomalous magnetic\nmomentTterms, all of which have to be determined self-consistently as the s olutions of the SD equations of the\ntheory.\nIn (22)/hatwideFµν=Fµν/Hdenotes the normalized electromagnetic strength tensor, with Hthe field strength. The\nexternal magnetic field breaks the rotational symmetry of the th eory, hence separating between longitudinal p/bardbl·γ/bardbl=\npν/hatwideF∗\nνρ/hatwideF∗µργµ(forµ,ν= 0,3), and transverse p⊥·γ⊥=pµ/hatwideFµρ/hatwideFρνγν(forµ,ν= 1,2), modes. /hatwideF∗\nµν=1\n2HεµνρλFρλis\nthe dual of the normalized electromagnetic strength tensor /hatwideFµν.\nThe transformation to momentum space of (22) can be done by usin g the so-called Ritus’ method, originally\ndeveloped for fermions in [19] and later extended to vector fields in [2 0]. In Ritus’ approach, the transformation to\nmomentum space is carried out using the eigenfunctions El\np(x) of the asymptotic states of the charged fermions in a\nuniform magnetic field\nEl\np(x) =E+\np(x)∆(+)+E−\np(x)∆(−) (23)\nwhere\n∆(±) =I±iγ1γ2\n2, (24)\nare the spin up (+) and down ( −) projectors, and E+/−\np(x) are the corresponding eigenfunctions\nE+\np(x) =Nle−i(p0x0+p2x2+p3x3)Dl(ρ),\nE−\np(x) =Nl−1e−i(p0x0+p2x2+p3x3)Dl−1(ρ) (25)\nwith normalization constant Nl= (4πeH)1/4/√\nl!, andDl(ρ) denoting the parabolic cylinder functions of argument\nρ=√\n2eH(x1−p2/eH), and index given by the Landau level numbers l= 0,1,2,....6\nTheEl\npfunctions satisfy the orthogonality condition [15]\n/integraldisplay\nd4xEl\np(x)El′\np′(x) = (2π)4/hatwideδ(4)(p−p′)Π(l), (26)\nwithEl\np≡γ0(El\np)†γ0,\n/hatwideδ(4)(p−p′) =δll′δ(p0−p′\n0)δ(p2−p′\n2)δ(p3−p′\n3), (27)\nand\nΠ(l) = ∆(+)δl0+I(1−δl0). (28)\nThe spin structure of the Epfunctions (23) is essential to satisfy the eigenvalue equations\n(Π·γ)El\np(x) =El\np(x)(γ·p), (29)\nand\n(Z/bardblΠ/bardbl\nµγµ\n/bardbl+Z⊥Π⊥\nµγµ\n⊥)El\np(x) =El\np(x)(Z/bardblpµ\n/bardblγ/bardbl\nµ+Z⊥pµ\n⊥γ⊥\nµ), (30)\nwithpµ= (p0,0,−√\n2eHl,p3), thuspµ\n⊥= (0,0,−√\n2eHl,0) andpµ\n/bardbl= (p0,0,0,p3).\nThe relations (29)-(30) and the orthogonalitycondition (26), fac ilitate the diagonalization of the fermion self energy\nΣ(x,x′) in momentum space\nΣ(p,p′) =/integraldisplay\nd4xd4yEl\np(x)Σ(x,y)El′\np′(y) = (2π)4/hatwideδ(4)(p−p′)Π(l)/tildewideΣl(p) (31)\nwith\n/tildewideΣl(p) =Zl\n/bardblpµ\n/bardblγ/bardbl\nµ+Zl\n⊥pµ\n⊥γ⊥\nµ+MlI+iTlγ1γ2(32)\nUsing the spin projectors (24) and introducing the longitudinal and transverse projectors\nΛ±\n/bardbl=1\n2(1±γ/bardbl·p/bardbl\n|p/bardbl|),Λ±\n⊥=1\n2(1±iγ2). (33)\nthe function /tildewideΣl(p) can be rewritten in the following form,\n/tildewideΣl(p) =Zl\n/bardbl(Λ+\n/bardbl−Λ−\n/bardbl)|p/bardbl|+iZl\n⊥(Λ−\n⊥−Λ+\n⊥)|p⊥|+(Ml+Tl)∆(+)+(Ml−Tl)∆(−) (34)\nIt is clear from (31) that when the Eptransformation is correctly used (i.e. taking into account the proj ector Π(l)\nin the orthogonal condition), the separation between the LLL and the rest of the levels is automatically produced.\nConsidering l= 0 in (34), and using that p⊥(l= 0) = 0 and ∆(+)∆( −) = 0, the spinorial structure in the RHS of\nEq. (31) reduces to\nΠ(0)/tildewideΣ0(p) = [Z0\n/bardbl(Λ+\n/bardbl−Λ−\n/bardbl)|p/bardbl|+(M0+T0)∆(+)]∆(+) (35)\nWhile atl/ne}ationslash= 0, it is given by\nΠ(l/ne}ationslash= 0)/tildewideΣl(p) =/tildewideΣl(p) =Zl\n/bardbl(Λ+\n/bardbl−Λ−\n/bardbl)|p/bardbl|+iZl\n⊥(Λ−\n⊥−Λ+\n⊥)|p⊥|+(Ml+Tl)∆(+)+(Ml−Tl)∆(−) (36)\nFrom a physical point of view the results (35) and (36) are simply refl ecting the fact that there is only one spin\nprojection in the LLL , and hence the self-energy /tildewideΣ0(p) only contains the spin projector ∆(+). Since the remaining\nLL’s contain two spin projections, /tildewideΣl(p) depends on the two projectors ∆(+) and ∆( −).\nWith the help of Eqs. (26), (29)-(31), it is straightforward to sho w that the inverse of the full fermion propagator\nin momentum space is given by\nG−1\nl(p,p′) =/integraldisplay\nd4xd4yEl\np(x)[Π·γ−Σ(x,y)]El′\np′(y) = (2π)4/hatwideδ(4)(p−p′)Π(l)[p·γ−/tildewideΣl(p)] (37)7\nThe full propagator Gl(p,p′) must satisfy\n/summationdisplay/integraldisplayd4p”\n(2π)4G−1\nl”(p,p”)Gl”(p”,p′) = (2π)4/hatwideδ(4)(p−p′)Π(l) (38)\nwhere/summationtext\nl/integraltextd4p\n(2π)4=/summationtext\nldp0dp2dp3\n(2π)4. It is easy to see that (38) is indeed satisfied by\nGl(p,p′) = (2π)4/hatwideδ(4)(p−p′)Π(l)/tildewideGl(p) (39)\nwith/tildewideGl(p) formally given by\n/tildewideGl(p) =1\np·γ−/tildewideΣl(p)(40)\nTo find the explicit form of /tildewideGl(p) we have to solve\n/tildewideGl(p)/tildewideG−1\nl(p) =/tildewideG−1\nl(p)/tildewideGl(p) =I (41)\nwhere\n/tildewideG−1\nl(p) =p·γ−/tildewideΣl(p) =γ·Vl−MlI−Tliγ1γ2 (42)\nandVµ= ((1−Zl\n/bardbl)p0,0,(1−Zl\n⊥)√\n2eHl,(1−Zl\n/bardbl)p3).\nOne can show that the matrix function\n/tildewideGl(p) =AB\ndet/tildewideG−1\nl(p), (43)\nwith\nA=γ1/tildewideG−1\nl(p)γ1, B=γ5/tildewideG−1\nl(p)Aγ5 (44)\nand\ndet/tildewideG−1\nl(p) =4/radicalBig\ndet[/tildewideG−1\nl(p)AB]\n=1\n4{[Ml+(Vl\n/bardbl−Tl)+Vl\n⊥][Ml−(Vl\n/bardbl−Tl)+Vl\n⊥]\n+ [Ml+(Vl\n/bardbl−Tl)−Vl\n⊥][Ml−(Vl\n/bardbl−Tl)−Vl\n⊥]}\n× {[Ml+(Vl\n/bardbl+Tl)+Vl\n⊥][Ml−(Vl\n/bardbl+Tl)+Vl\n⊥]\n+ [Ml+(Vl\n/bardbl+Tl)−Vl\n⊥][Ml−(Vl\n/bardbl+Tl)−Vl\n⊥]}, (45)\nsatisfies the condition (41).\nUsing (43) and working in the basis of projectors (24) and (33), th e matrix function /tildewideGldefining the full fermion\npropagator (39) can be written as\n/tildewideGl(p) =Nl(T,V/bardbl)\nDl(T)∆(+)Λ+\n/bardbl+Nl(T,−V/bardbl)\nDl(−T)∆(+)Λ−\n/bardbl\n+Nl(−T,V/bardbl)\nDl(−T)∆(−)Λ+\n/bardbl+Nl(−T,−V/bardbl)\nDl(T)∆(−)Λ−\n/bardbl\n−iVl\n⊥(Λ+\n⊥−Λ−\n⊥)[∆(+)Λ+\n/bardbl+∆(−)Λ−\n/bardbl\nDl(T)+∆(+)Λ−\n/bardbl+∆(−)Λ+\n/bardbl\nDl(−T)] (46)\nwith notation\nNl(T,V/bardbl) =Tl−Ml−Vl\n/bardbl\nDl(T) = (Ml)2−(Vl\n/bardbl−Tl)2+(Vl\n⊥)2\nVl\n/bardbl= (1−Zl\n/bardbl)|p/bardbl|\nVl\n⊥= (1−Zl\n⊥)|p⊥|= (1−Zl\n⊥)√\n2eHl. (47)8\nIn the LLL, V0\n⊥= 0, thus the LLL full propagator becomes\n/tildewideG0(p) =1\nV0\n/bardbl−(M0+T0)∆(+)Λ+\n/bardbl−1\nV0\n/bardbl+(M0+T0)∆(+)Λ−\n/bardbl\n+1\nV0\n/bardbl−(M0−T0)∆(−)Λ+\n/bardbl−1\nV0\n/bardbl+(M0−T0)∆(−)Λ−\n/bardbl(48)\nIV. SCHWINGER-DYSON EQUATION FOR THE FERMION SELF-ENERGY\nTo explore the dynamical generation of a magnetic moment in massles s QED, we can start from the SD equation for\nthefermionself-energyinthepresenceofaconstantmagneticfie ld. Wewillworkinthequenched-ladderapproximation\nwhere\nΣ(x,x′) =−ie2γµG(x,x′)γνDµν(x−x′). (49)\nHere, Σ(x,x′) is the fermion self-energy operator (22), Dµν(x−x′) is the bare photon propagator, and G(x,x′) is the\nfull fermion propagator depending on the dynamically induced quant ities and the magnetic field.\nEquation (49) can be transformed to momentum space by using the Epfunctions as\n/integraldisplay\nd4xd4x′El\np(x)Σ(x,x′)El′\np′(x′) =−ie2/integraldisplay\nd4xd4x′El\np(x)γµ\n(/summationdisplay/integraldisplayd4p”\n(2π)4El”\np”(x)Π(l”)/tildewideGl”(p”)El”\np”(x′))γνEl′\np′(x′)Dµν(x−x′) (50)\nwhere\nDµν(x−x′) =/integraldisplayd4q\n(2π)4e−iq·(x−x′)\nq2−iǫ(gµν−(1−ξ)qµqν\nq2), (51)\nwithξthe gauge fixing parameter, and we used that\nG(x,x′) =/summationdisplay/integraldisplayd4p”\n(2π)4El”\np”(x)Π(l”)/tildewideGl”(p”)El”\np”(x′) (52)\nAt this point it is convenient to consider the integrals [8]\n/integraldisplay\nd4xEl\np(x)γµEl”\np”(x)e−iq·x= (2π)4δ(3)(p”+q−p)e−iq1(p2”+p2)/2eHe−bq2\n⊥/2\n×/summationdisplay\nσ,σ”1√\nn!n”!ei(n−n”)ϕJnn”(/hatwideq⊥)∆(σ)γµ∆(σ”), (53)\nand\n/integraldisplay\nd4x′El”\np”(x′)γνEl′\np′(x′)eiq·x′= (2π)4δ(3)(p”+q−p′)eiq1(p”2+p′\n2)/2eHe−bq2\n⊥/2\n×/summationdisplay\nσ′,σ”1√\nn′!n”!ei(n”−n′)ϕJn”n′(/hatwideq⊥)∆(σ”)γν∆(σ′), (54)\nwithn≡n(l,σ),n”≡n(l”,σ”),n′≡n(l′,σ′), andn”≡n(l”,σ”), defined according to\nn(l,σ) =l+σ\n2−1\n2, l= 0,1,2,..., σ =±1. (55)\nThe notation in (53) and (54) included the use of polar coordinates f or the transverse q-momentum /hatwideq⊥≡/radicalbig\n/hatwideq2\n1+/hatwideq2\n1,\nϕ≡arctan(/hatwideq2//hatwideq1); normalized quantities /hatwideQµ=Qµ/√\n2eH; the tri-delta function\nδ(3)(p”+q−p)≡δ(p0”+q0−p0)δ(p2”+q2−p2)δ(p3”+q3−p3); (56)9\nand\nJnn”(/hatwideq⊥)≡min(n.n”)/summationdisplay\nm=0n!n”!\nm!(n−m)!(n”−m)![i/hatwideq⊥]n+n”−2m. (57)\nDoing the integrals in xandx′in (50) with the help of (53) and (54), integrating in p”, and using the Feynman gauge\n(ξ= 1), one finds\n/tildewideΣl(p)Π(l)δll′=−ie2(2eH)/integraldisplayd4/hatwideq\n(2π)4/summationdisplay\nl”/summationdisplay\n[σ]ei(n−n”+n”−n′)ϕ\n√\nn!n′!n”!n”!e−bq2\n⊥\n/hatwideq2\n×Jnn”(/hatwideq⊥)Jn”n′(/hatwideq⊥)∆(σ)γµ∆(σ”)Π(l”)/tildewideGl”(p−q)∆(σ”)γµ∆(σ′), (58)\nwherep−q≡(p0−q0,0,−√\n2eHl”,p3−q3) and [σ] means summing over σ,σ′,σ”,σ”. The appearance of the Π( l)\nfactors in both sides of the equation ensures the correct countin g of only one spin projection for the fermions at the\nLLL.\nDue to the negative exponential e−bq2\n⊥the main contribution to (58) will come from the smallest values of /hatwideq⊥. This\nallows one to keep in (58) only the terms with the smallest power of /hatwideq⊥inJn”n′(/hatwideq⊥) (see Ref. [8] for details). Hence\nJnn”(/hatwideq⊥)→[max(n′,n”)]!\n|n−n”|[i/hatwideq⊥]|n−n”|→n!δnn” (59)\nand we obtain\n/tildewideΣl(p)Π(l)δll′=−ie2(2eH)/integraldisplayd4/hatwideq\n(2π)4/summationdisplay\nl”/summationdisplay\n[σ]e−bq2\n⊥\n/hatwideq2\n×δnn”δn”n′∆(σ)γµ∆(σ”)Π(l”)/tildewideGl”(p−q)∆(σ”)γµ∆(σ′), (60)\nTaking into account that\nδn,n”=δl,l”δσ,σ”+δl+σ,l”δ−σ,σ” (61)\ntogether with the relations\n∆(±)γ⊥\nµ=γ⊥\nµ∆(∓),∆(±)γ/bardbl\nµ=γ/bardbl\nµ∆(±) (62)\n∆(±)∆(±) = ∆(±),∆(±)∆(∓) = 0, γ⊥\nµγ⊥\nνγµ\n⊥= 0, (63)\nwe can do the sums in [ σ] andl” in (60) to arrive at the SD equation\n/tildewideΣl(p)Π(l) =−ie2(2eH)Π(l)/integraldisplayd4/hatwideq\n(2π)4e−bq2\n⊥\n/hatwideq2[γ/bardbl\nµ/tildewideGl(p−q)γ/bardbl\nµ\n+∆(+)γ⊥\nµ/tildewideGl+1(p−q)γ⊥\nµ∆(+)+∆( −)γ⊥\nµ/tildewideGl−1(p−q)γ⊥\nµ∆(−)] (64)\nIf the external fermion is in the LLL ( l= 0), Eq. (64) reduces to\n/tildewideΣ0(p)∆(+) = −ie2(2eH)∆(+)/integraldisplayd4/hatwideq\n(2π)4e−bq2\n⊥\n/hatwideq2[γ/bardbl\nµ/tildewideG0(p−q)γ/bardbl\nµ+γ⊥\nµ/tildewideG1(p−q)γ⊥\nµ∆(+)] (65)\nwith/tildewideΣ0(p)∆(+) given in Eq. (35). While if the external fermion is in any higher LL (l/ne}ationslash= 0), Eq. (64) becomes\n/tildewideΣ(l/negationslash=0)(p) =−ie2(2eH)/integraldisplayd4/hatwideq\n(2π)4e−bq2\n⊥\n/hatwideq2[γ/bardbl\nµ/tildewideGl(p−q)γ/bardbl\nµ+∆(+)γ⊥\nµ/tildewideGl+1(p−q)γ⊥\nµ∆(+)\n+∆(−)γ⊥\nµ/tildewideGl−1(p−q)γ⊥\nµ∆(−)] (66)\nwith/tildewideΣ(l/negationslash=0)(p) given in Eq. (36).10\nSince the equation for a given Landau level linvolves dynamical parameters that depend on l,l−1 andl+ 1,\nthe SD equations for all the LL’s form a system of infinite coupled equ ations. Fortunately, in the infrared region,\nthe leading contribution to each equation will come from the propaga tors with the lower LL’s, since the magnetic\nfield term ( ∼lB) in the denominator of the fermion propagator for l/ne}ationslash= 0 acts as a suppressing factor. Using this\napproximation, one can find a consistent solution at each level. It is a lso convenient to notice that, since the solution\nfor the first LL depends on the one for the LLL; the solution for th e second LL depends on the one for the first\nLL, and so on, the solutions for MlandTlcan all ultimately be expressed as a function of the LLL solution. This\nindicates that the physical origin of all the dynamical quantities is ac tually due to the infrared dynamics taking place\nat the LLL.\nV. INDUCED ELECTRON MASS AND ANOMALOUS MAGNETIC MOMENT\nA. Solution of the SD Equation in the LLL\nLet’s workinthe LLL, thatmeans, /tildewidep⊥= 0, andconsequentlythe LHS of(65) isgivenby(35). In the infrar edregion,\nthe leading contribution to the RHS of Eq. (65) comes from the term with no magnetic field in the denominator, that\nis, the term with the LLL propagator /tildewideG0(p−q). Hence, in the leading approximation Eq. (65) is given by\n(M0+T0)∆(+)+Z0\n/bardbl∆(+)(Λ+\n/bardbl−Λ−\n/bardbl)|p/bardbl| ≃ −ie2(2eH)∆(+)/integraldisplayd4/hatwideq\n(2π)4e−bq2\n⊥\n/hatwideq2γ/bardbl\nµ/tildewideG0(p−q)γ/bardbl\nµ (67)\nUsing (62)-(63) together with\nγ/bardbl\nµΛ±\n/bardblγµ\n/bardbl= 1, (68)\nassumingZ(0)\n/bardbl≪1, and doing a Wick’s rotation to Euclidean space, we obtain\n(M0+T0)∆(+)+Z0\n/bardbl∆(+)(Λ+\n/bardbl−Λ−\n/bardbl)|p/bardbl|=e2(2eH)\n×∆(+)/integraldisplayd4q\n(2π)4e−bq2\n⊥\n/hatwideq2(M0+T0)\n(p/bardbl−q/bardbl)2+(M0+T0)2(69)\nFrom Eq.(69) it is clear that Z0\n/bardbl= 0, which corroborates the assumption we did before. Taking the in frared limit\n(p/bardbl∼0), and assuming that M0+T0is independent of the parallel momentum, we arrive at\n1 =e2(4eH)/integraldisplayd4/hatwideq\n(2π)4e−bq2\n⊥\n/hatwideq21\n(M(0)+T(0))2+q2\n/bardbl(70)\nIfM0+T0is replaced by mdyn, Eq. (70) becomes identical to the SD equation obtained in the phen omenon of\nmagnetic catalysis of chiral symmetry breaking. Thus, the solution of (70) is given by\nM0+T0≃√\n2eHe−√π\nα (71)\nAs in [7]-[8], this solution is obtained considering that M0+T0does not depend on the parallel momentum, an\nassumption consistent within the ladder approximation [21]. As prove din [22], when the polarizationeffect is included\nin the gap equation through the improved-ladder approximation, th e solution for mdynis of the same form as (71),\nbut with the replacement/radicalbig\nπ/α→π/αlog(π/α) in the exponent. Since the inclusion of the magnetic moment in the\nLLL SD equation merely implies the replacement mdyn→M0+T0, it is clear that a similar effect will occur in the\nsolution (71). However, this effect will not qualitatively change the n ature of our findings.\nAs the fermions in the LLL have only one spin orientation it is not possib le to findM0andT0independently (see\nthat the combination M0−T0is absent from the LHS of the SD equation (67), as well as from the R HS, because the\nspin projector ∆(+) ensures that only the terms containing M0+T0in (48) contribute to the RHS of (67)). Thus,\nthe SD equation determines the induced LLL rest-energy\nE0=M0+T0. (72)\nwhich has contributions from the dynamical mass and from the magn etic energy related to the interaction between\nthe magnetic field and the dynamically induced magnetic moment.11\nB. Solution of the SD Equation in the First-LL\nThe SD equation for a fermion in the first-LL is\n/tildewideΣ1(p) =−ie2(2eH)/integraldisplayd4/hatwideq\n(2π)4e−bq2\n⊥\n/hatwideq2[γ/bardbl\nµ/tildewideG1(p−q)γ/bardbl\nµ+\n+∆(+)γ⊥\nµ/tildewideG2(p−q)γ⊥\nµ∆(+)+∆( −)γ⊥\nµ/tildewideG0(p−q)γ⊥\nµ∆(−)] (73)\nHere again the leading contribution comes from the term with no magn etic field in the denominator. Hence to find\nthe leading contribution we just need to keep the term depending on /tildewideG0(p−q) in the RHS. Using (62)-(63), together\nwith\nγ⊥\nµΛ±\n/bardblγµ\n⊥= 2Λ∓\n/bardbl, (74)\nand working as before in the infrared limit, the SD equation after Wick ’s rotation becomes\nZ1\n⊥γ2(2eH)+(M1+T1)∆(+)+(M1−T1)∆(−) =e2(4eH)∆(−)/integraldisplayd4/hatwideq\n(2π)4e−bq2\n⊥\n/hatwideq2E0\n(E0)2+q2\n/bardbl(75)\nTherefore,\nM1+T1= 0, Z1\n⊥= 0 (76)\nand\nM1−T1=e2(4eH)/integraldisplayd4/hatwideq\n(2π)4e−bq2\n⊥\n/hatwideq2E0\n(E0)2−q2\n/bardbl(77)\nTaking into account (70) in (77) we get\nM1−T1=E0(78)\nFinally, from (76) and (78), it results\nM1=−T1=1\n2E0=/radicalbig\neH/2e−√π\nα, (79)\nThe solution (79) corroborates the relevance of the LLL dynamics (bothM1andT1are determined by E0) in the\ngeneration of the dynamical mass and magnetic moment of the ferm ions in the first LL. Given that the magnitude of\nthe magnetic moment for the electrons in the first LL is determined b y the dynamically generated rest-energy of the\nelectrons in the LLL, any modification of the theory producing an inc rease ofE0will, in turn, lead to an increase in\nthe magnitude of T1. From the experience with the MCχSB phenomenon, such modifications could be for example,\nlowering the space dimension [23], introducing scalar-fermion interac tions [9, 21], or considering a non-zero bare mass\n[24].\nFor the remaining LL’s the procedure is similar. For example, the leadin g term in the second LL will be given by\nthe/tildewideG1(p−q) contribution which in turn depends on E0through the values found for M1andT1. Therefore, the\nvalues ofMlandTlfor higher LL’s will depend on E0through the found values of the previous LL’s. This fact shows\nthat the infra-red dynamics of the electrons in the LLL is the domina nt one.\nVI. DISPERSION RELATIONS AND ZEEMAN SPLITTING\nA. Fermions in the LLL\nIn the Appendix we showed that the Dirac equation for fermions in th e LLL can be written as\n[/tildewidep·/tildewideγ−E0]ψLLL= 0, (80)\nwhereψLLLis a spin-up two-component spinor. Eq. (80) coincides with that of t he free (1+1)-D Thirring model [25],\nwith the (1+1) −Dgamma matrices /tildewideγ0=σ1,/tildewideγ1=−iσ2, defined in terms of the Pauli matrices σi, and/tildewidepµ= (p0,p3).12\nThe dispersion relation of the LLL fermions obtained from (80) is\np0=±/radicalBig\np2\n3+(E0)2, (81)\nThus, the effect of a dynamical magnetic moment is irrelevant for th e LLL fermions, since it just redefines their rest\nenergy through the replacement mdyn→M0+T0. This is physically natural, since the fermions in the LLL can\nonly have one spin projection, so for them there is no spin degenera cy and hence, no possible energy splitting due to\nthe magnetic moment. We shall see below that the dynamical anomalo us magnetic moment turns out to be really\nrelevant for fermions in higher LL’s.\nB. Fermions in the First-LL\nLet us find now the dispersion relations for fermions in higher LL’s, ta king into account the dynamically induced\nquantities. Starting from the modified field equation in the presence of the magnetic field,\n[p·γ−MlI−iTlγ1γ2]ψl= 0, (82)\nwhere we neglected the coefficients Zin the terms (1 −Zl\n/bardbl) and (1 −Zl\n⊥), since it is expected that the Z′sare much\nsmaller than one; the dispersion relations are found from\ndet[p·γ−MlI−iTlγ1γ2] = [(Ml)2−(p/bardbl−Tl)2+p2\n⊥][(Ml)2−(p/bardbl+Tl)2+p2\n⊥] = 0. (83)\nyielding\np2\n0=p2\n3+[/radicalBig\n(Ml)2+2eHl±Tl]2, (84)\nand thus showing that the induced magnetic moment breaks the ene rgy degeneracy between the spin states in the\nsame LL (see the double sign in front of Tl).\nIn particular for the first-LL, and taking into account that M1/√\n2eH ,T1/√\n2eH≪1, the leading contribution to\nthe energy becomes\np2\n0≃p2\n3+2eH+(M1)2+(T1)2±2T1√\n2eH, (85)\nWorking in the infrared region ( p2\n3/2eH≪1) the expression (85) can be approximated as\np0≃ ±[√\n2eH+p2\n3\n2√\n2eH+(M1)2+(T1)2\n2√\n2eH±T1] (86)\nAs a consequence, the energy splitting for the fermions in the first LL is\n∆E=|2T1|= 2/radicalbig\neH/2e−√\nπ/α(87)\nOne can rewrite the last expression in the usual form of the Zeeman energy splitting for the two spin projections\nalready introduced in Sec. II in Eq. (19) as ∆ E=/tildewideg/tildewideµBHwith/tildewidegand/tildewideµBrepresenting the non-perturbative Lande\ng-factor and Bohr magneton respectively given by /tildewideg= 2e−2√\nπ/α,/tildewideµB=e\n2M1.\nVII. CHIRAL CONDENSATE\nNow we are going to find the relation between the chiral condensate /an}bracketle{tΨΨ/an}bracketri}ht, that is, the order parameter of the\nmagnetically catalyzed chiral symmetry breaking, and the LLL ferm ion rest-energy, which as seen before depends on\nthe sum of the LLL mass and magnetic moment energy contribution.\nWe start from the definition of the chiral condensate\n/an}bracketle{tΨΨ/an}bracketri}ht=iTr[G(x,x)/V], (88)\nwithG(x,x) the full fermion propagatorgivenin (52), and Vthe systemvolume. As discussedbefore, thephenomenon\nof magnetic catalysis of chiral symmetry breaking is physically due to the infrared dynamics of the fermions in the13\nLLL. Thus, the leading contribution to the condensate (88) comes from the LLL fermions. It is convenient to recall\nhere that the space-dependent part of the LLL fermion wave fun ction is given by [26]\nΨ(x)∼ei(x0p0+x2p2+x3p3)e−(x1−xc)2\n4l2\nB, (89)\nwherexc=p2l2\nBis the coordinate of the center of the Landau orbits and lB= 1/√\neBis the magnetic length. It is\nclear from (89) that a particle in the LLL can be localized along the x2andx3directions up to infinite, but along\nthex1direction it is confined within a magnetic length due to the Gaussian fun ction with width lB(note that for\n(89) the standard deviation from the particle position average valu e/an}bracketle{tx1/an}bracketri}htisσ=lB). This implies that in the LLL the\nvolumeVin (88) is given by V=L0L2L3lB, withLi=/integraltext+∞\n−∞dxi.\nUsing (52) in (88) and keeping only the leading LLL contribution in the s um, we obtain after integrating in x′s,\n/an}bracketle{tΨΨ/an}bracketri}ht=iL0L2L3\nVtr{/integraldisplaydp0dp2dp3\n(2π)3∆(+)/tildewideG0(p)} (90)\nwith/tildewideG0(p) given in (48), and trdenoting the remaining spinorial trace. In the above result we used the orthogonality\nof the parabolic cylinder functions\n/integraldisplay∞\n−∞dρDl(ρ)Dl′(ρ) =√\n2πl!δll′ (91)\nThe trace operation reduces the previous expression to\n/an}bracketle{tΨΨ/an}bracketri}ht=i2\nlB/integraldisplaydp0dp2dp3\n(2π)3E0\n(E0)2+|p/bardbl|2(92)\nTransforming to Euclidean space, taking polar coordinates for the parallel momenta, and using a momentum cut-off\ndefined by the magnetic scale 1 /lB, which is dominant in the infra-red region, we have\n/an}bracketle{tΨΨ/an}bracketri}ht=−1\n(2π)2lB/integraldisplay1\nlB\n−1\nlBdp2/integraldisplay1\nlB\n0dp2\n/bardblE0\n(E0)2+|p/bardbl|2(93)\nAfter integrating in the momenta we obtain\n/an}bracketle{tΨΨ/an}bracketri}ht ≃ −eH\n2π2E0ln(eH\n(E0)2) (94)\nThis result shows that the role played by the dynamical mass in previo us works on magnetic catalysis, on which the\ninduced anomalousmagneticmomentwasignored[8], is nowplayedby E0. Consideringthe magnetic-fielddependence\nofE0given in (71), we can rewrite (94) as\nE0=2π2l2\nB\nln2−/radicalbig\nπ/α/an}bracketle{tΨΨ/an}bracketri}ht (95)\nIt shows that the induced rest-energy of the electrons in the LLL is proportional to the condensate. This reflects the\nfact that the same order parameter produces the induction of th e two quantities contributing to E0: the dynamical\nmass and anomalous magnetic moment. This also confirms that the dy namical mass and magnetic moment have a\ncommon physical origin, as they are related to one single order para meter, the chiral condensate /an}bracketle{tΨΨ/an}bracketri}ht.\nThis validates the fact that since the induction of the term σµν/hatwideFµνin the electron self-energy does not produce any\nadditional symmetry breaking to that already created by the mass term, then once the chiral symmetry is broken by\nthe condensate /an}bracketle{tΨΨ/an}bracketri}ht, both dynamical parameters M0andT0are generated.\nVIII. CONCLUDING REMARKS\nIn this paper we have presented a fresh view of the phenomenon of magnetic catalysis of chiral symmetry breaking\nin massless QED. Our results show that in this phenomenon the dynam ical generation of an additional parameter:\nthe fermion anomalous magnetic moment, has to be considered in equ al footing to that already taken into account14\nin previous works on this topic: the fermion mass. The rationale for t he increase in the number of the induced\nparameters is easy to understand on symmetry arguments. The r eason is that once the chiral symmetry is broken by\nthe magnetically catalyzed condensation of fermion/anti-fermion p airs, all the physical quantities in the action that\nwere forbidden by the chiral symmetry of the bare theory can be n ow dynamically induced. Therefore, the chiral\ncondensate induces dynamical mass and anomalous magnetic momen t, since in the massless theory their generation\nwas only protected by that same symmetry. Another important ou tcome of this paper is the lifting of the spin\ndegeneracy of all the LLs different from zero. The degeneracy is e liminated due to the Zeeman splitting produced\nby the dynamical anomalous magnetic moment. This energy splitting is given in terms of a non-perturbative Lande\ng-factor and a Bohr magneton that depends on the dynamical mass .\nWe call attention to a fact that has not been emphasized in previous works on magnetic catalysis, but that is really\nessential to prove the results here obtained. We refer to the dep endence of the dynamical quantities on the Landau\nlevels. There is no reason to assume a priori that the dynamical par ameters have to be the same for all the LLs. One\ncan see that, as soon as one is interested in exploring the SD equatio ns for higher LLs. It becomes clear that there is\nno consistent solution of the infinite system of coupled equations, u nless one takes into consideration that the mass\nand the magnetic moment depend on the LLs. This is evident already f rom Eq. (75).\nWe anticipate that the dynamical generation of the magnetic momen t will be a universal feature of theories with\nmagnetic catalysis, in the same way that occurs with the generation of the dynamical mass. Moreover, we expect\nthat an anomalous magnetic moment will be also dynamically generated in theories with fermion/fermion condensates\nin the presence of a magnetic field, as long as the symmetry broken b y the condensate coincides with the one that\nwould be explicitly broken by a magnetic moment term in the action. The se considerations point to some potential\napplications of our findings in two different areas: condensed matte r and the ultra dense matter existing in the core\nof compact stars.\nIn condensed matter, the most plausible application at present cou ld be in the physics of graphene. It is known\nthat the 2-dimensional crystalline form of carbon known as graphe ne [27] has charge carriers that behave as massless\nDirac electrons. In particular, a phenomenon where the dynamically induced Zeeman effect we have found here\ncould shed some new light is the lifting of the fourfold degeneracy of t hel= 0 LL, and twofold degeneracy of the\nl= 1 LL in the recently found quantum Hall states corresponding to fi lling factors ν= 0,±1,±4 under strong\nmagnetic fields [28]. An attempt to explain the observed lifting of the L Ls degeneracy was carried out in [29] with the\nhelp of a 2+1-dimensional four-fermion-interaction model of Dirac quasiparticles with chemical potentials interpreted\nas Quantum Hall Ferromagnetism (QFH) order parameters, and dy namical masses related to the phenomenon of\nMCχSB . Nevertheless, we should call attention to two shortcomings of Re f. [29]. First, the order parameter /tildewideµswas\nintroduced in [29] as a chemical potential and interpreted as a QFH o rder parameter, implying that its physical origin\nwas considered unrelated to the phenomenon of magnetic catalysis . However, it is not hard to understand, based on\nthe results of the present paper, that /tildewideµsshould have been identified as a 2+1 dynamical anomalous magnetic mo ment,\nthus similar to the one we have found in 3+1 massless QED. In other wo rds,/tildewideµsshould have been connected to the\nMCχSB phenomenon. Second, the LLL results reported in the second pap er of [29] are not self-consistent, since\nthey were found taking into account the single (pseudo)spin contr ibution in the LLL on the RHS of the gap equation\n(A31), but ignoring it in the self-energy operator appearing in the L HS of the same equation. As a consequence,\nseparated values for the LLL parameters analogous to our M0andT0were obtained.\nThe other possible application in the realm of ultra-dense matter relie s on the phenomenon of color supercon-\nductivity. An important aspect of color superconductivity is its mag netic properties [30]-[33]. In spin-zero color\nsuperconductivity, although the color condensate has non-zero electric charge, it is neutral with respect to a modified\nelectromagnetism. This so-called rotated electromagnetism is pres ent because a linear combination of the photon\nand the eight gluon remains massless, hence giving rise, in both the 2S C and CFL phases, to a long-range ”rotated-\nelectromagnetic” field [30]. The long-range field can propagate in the color superconductor implying that there is no\nMeissner effect for the rotated component of an external magne tic field applied to the color superconductor. Even\nthough the quark-quark condensate is neutral with respect to t he rotated charge, an applied magnetic field can inter-\nact with the quarks of a pair formed by /tildewideQ-charged quarks of opposite sign and, moreover, for large magne tic fields this\ninteraction can reinforce these pairs [32]. In this sense, the ”rota ted-electromagnetism” in the color-superconductor\nhas some resemblance with the chiral condensate in a theory with ma gnetic catalysis. It is natural to expect then\nthat a dynamically magnetic moment can also be induced in a color super conductor under an applied magnetic field.\nSince, on the other hand, the Meissner instabilities that appear in so me density regions of the color superconductor\ncan be removed by the induction of a magnetic field [33], it will be interes ting to investigate what could be the role\nin this process of a dynamically induced magnetic moment.\nAcknowledgments\nThis work was supported in part by the Office of Nuclear Physics of th e Department of Energy under contract\nDE-FG02-09ER41599.15\nAPPENDIX A: THE LLL LAGRANGIAN IN THE CHIRAL-CONDENSATE PHA SE\nWe are interested in obtaining the Dirac equation in momentum space f or fermions in the LLL of the chiral-\ncondensate phase (80). With this goal in mind, we should start from the Dirac Lagrangian in the presence of a\nconstant and uniform magnetic field that includes the self-energy c orrections\nL=/integraldisplay\nd4xψ(x)(Πµγµ−Σ(x))ψ(x) (A1)\nUsing Ritus’ transformation to momentum space for the wave func tions\nψ(x) =/summationdisplay/integraldisplayd4p\n(2π)4El\np(x)ψl(p),ψ(x) =/summationdisplay/integraldisplayd4p′\n(2π)4ψl′(p′)El′\np′(x) (A2)\nand taking into account (26), (29) and (31), we obtain\nL=/summationdisplay/integraldisplayd4p\n(2π)4ψl(p)Π(l)[γµpµ−/tildewideΣl(p)]ψl(p) (A3)\nThe factor Π( l), given in Eq. (28), separates the LLL Lagrangian L0from the rest\nL=L0+∞/summationdisplay\nl=1/integraldisplaydp0dp2dp3\n(2π)4ψl(p)[γµpµ−/tildewideΣl(p)]ψl(p) (A4)\nwith\nL0=/integraldisplaydp0dp2dp3\n(2π)4ψ0(p)∆(+)[γ/bardbl·p/bardbl−/tildewideΣ0(p)]ψ0(p) (A5)\nConsidering the Dirac matrices in the chiral representation\nγ0=β=/parenleftbigg\n0−1\n−1 0/parenrightbigg\n, γi=/parenleftbigg\n0σi\n−σi0/parenrightbigg\n, γ5=/parenleftbigg\n1\n−1/parenrightbigg\n, (A6)\nwhereσiare the Pauli matrices, we can express the spin projectors (24) in terms ofσias\n∆(±) =/parenleftbigg\nσ±\nσ±/parenrightbigg\n, (A7)\nwith\nσ±=1\n2(1±σ3), (A8)\nand introduce the chiral projection operators\nR=1+γ5\n2=/parenleftbigg\n1\n0/parenrightbigg\n, L=1−γ5\n2=/parenleftbigg\n0\n1/parenrightbigg\n. (A9)\nThe projectors (A7) and (A9) satisfy the commutation relations\n[∆(±),L] = [∆(±),R] = 0 (A10)\nIntroducing now the chiral-spin representation for the Dirac spino r\nψ(+)\nR=R∆(+)ψ, ψ(−)\nR=R∆(−)ψ, ψ(+)\nL=L∆(+)ψ, ψ(−)\nL=L∆(−)ψ (A11)\nψ(+)\nR=ψL∆(+),ψ(−)\nR=ψL∆(−),ψ(+)\nL=ψR∆(+),ψ(−)\nL=ψR∆(−) (A12)16\nand using that\nψ=ψ(+)\nR+ψ(−)\nR+ψ(+)\nL+ψ(−)\nL (A13)\nwe obtain\nL0=/integraldisplaydp0dp2dp3\n(2π)4[ψ(+)\n0R(p)γ/bardbl·p/bardblψ(+)\n0R(p)−ψ(+)\n0R(p)E0ψ(+)\n0L(p)+ψ(+)\n0L(p)γ/bardbl·p/bardblψ(+)\n0L(p)−ψ(+)\n0L(p)E0ψ(+)\n0R(p)] (A14)\nwhere we used ∆(+) /tildewideΣ0(p) =E0∆(+) since Z0\n/bardbl= 0 (see Eq. (35)). From (A14) we see that the LLL only gets\ncontribution from the wave functions of the spin up-states. Intr oducingψ⊤\n0= (ψ1,ψ2,ψ3,ψ4) for the LLL four spinors\nwe can rewrite Eq. (A14) as\nL0=/integraldisplaydp0dp2dp3\n(2π)4(ψ∗\n3,ψ∗\n1)/parenleftbigg\nE0(p0+p3)\n(p0−p3)E0/parenrightbigg/parenleftbigg\nψ1\nψ3/parenrightbigg\n, (A15)\nNotice that in the LLL the fermion spinor reduces to a bispinor which c orresponds to the two chiralities of the spin\nup state in this case.\nAt this point it is more convenient to work with the (1+1)-D gamma mat rices\n/tildewideγ0=σ1=/parenleftbigg\n0 1\n1 0/parenrightbigg\n,/tildewideγ1=−iσ2=/parenleftbigg\n0−1\n1 0/parenrightbigg\n(A16)\nwhich satisfy the algebra\n/tildewideγµ/tildewideγν=gµν+ǫµν/tildewideγν, (A17)\n/tildewideγµ/tildewideγ5=−ǫµν/tildewideγν (A18)\nwith/tildewideγ5=/tildewideγ0/tildewideγ1and the (1+1)-D metric and the totally antisymmetric tensor given r espectively by\ngµν=/parenleftbigg\n1\n−1/parenrightbigg\n, ǫµν=/parenleftbigg\n0 1\n−1 0/parenrightbigg\n(A19)\nDefining the LLL bi-spinor as\nψLLL=/parenleftbigg\nψ1\nψ3/parenrightbigg\n(A20)\nand using (A16), Eq. (A15) can be rewritten in the compact form\nL0=/integraldisplaydp0dp2dp3\n(2π)4ψLLL(p) [/tildewideγ·/tildewidep−E0]ψLLL(p) (A21)\nwhere/tildewidepµ= (p0,p3). From the Lagrangian (A21) we obtain the Dirac equation for the L LL fermions\n[/tildewidep·/tildewideγ−E0]ψLLL= 0, (A22)\nThe Hamiltonian associated to L0is\nH=E0+p3/tildewideγ1(A23)\nwhich is the free (1+1)-D Thirring model Hamiltonian [25] with the rep lacement of the fermion mass mby the rest-\nenergyE0. Thus, in the chiral-condensate phase the LLL fermions behave as free particles in a reduced (1 + 1)-D\nspace with rest energy proportional to the inverse magnetic lengt hl−1\nmag=√\neH(see Eq. (71)).\n[1] E. H. Kennard, Zeits. F. Physik 44327 (1927); G. C. Darwin, Proc. Roy. Soc. 117258 (1928); I. I. Rabi Zeits. F. Physik\n49507 (1928); L. Page, Phys. Rev. 36444 (1930); M. S. Plesset, Phys. Rev. 361728 (1930).17\n[2] L. D. Landau, Zeits. F. Physik 64629 (1930).\n[3] J. Schwinger, Phys. Rev. 73, 416 (1947).\n[4] M. H. 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D 77(2008) 014021."    },    {        "title": "1309.4641v1.Dynamical_barrier_for_flux_penetration_in_a_superconducting_film_in_the_flux_flow_state.pdf",        "content": "Dynamical barrier for \rux penetration in a superconducting \flm in the \rux \row state\nJ. I. Vestg\u0017 arden,1Y. M. Galperin,1, 2and T. H. Johansen1, 3\n1Department of Physics, University of Oslo, P. O. box 1048 Blindern, 0316 Oslo, Norway\n2Io\u000be Physical Technical Institute, 26 Polytekhnicheskaya, St Petersburg 194021, Russian Federation\n3Institute for Superconducting and Electronic Materials,\nUniversity of Wollongong, North\felds Avenue, Wollongong, NSW 2522, Australia\nThe penetration of transverse magnetic \rux into a thin superconducting square \flm in the \rux\n\row state is considered by numerical simulation. Due to the \flm self-\feld, the governing equations\nare nonlinear, and in combination with the \fnite viscosity of the moving vortices, this sets up a\ndynamical barrier for \rux penetration into the sample. The corresponding magnetization loop is\nhysteric, with the peak in magnetization shifted from the zero position. The magnetic \feld in\nincreasing applied \feld is found to form a well-de\fned front of propagation. Numerical estimates\nshows that the dynamical barrier should be measurable on \flms with low volume pinning.\nPACS numbers: 74.25.Ha, 74.78.-w\nI. INTRODUCTION\nThe penetration of magnetic \rux into superconductors\nis delayed due to the presence of surface barriers, such\nas the Bean-Livingston barrier,1{4surface pinning5, and\nvarious barriers of geometric origin.6{8(The review by\nBrandt9lists 7 di\u000berent mechanisms) The barriers are\nparticularly important in thin \flms where the equilib-\nrium \feld for existence of magnetic \rux is much reduced\nfrom the bulk lower critical \feld Hc1toHc1d=2w, where\ndis thickness and wis sample width.10The presence of\nsurface barriers implies that vortices will not necessar-\nily enter the sample when it is energetically favorable for\nthem to reside in the sample center. Of particular im-\nportance in thin \flms, is the geometric barrier caused by\nthe magnetic \felds piling up near the edges, which delays\npenetration until the external \feld reaches Hc1p\nd=w.10\nNumerical simulations show that in samples without vol-\nume pinning, the magnetic \rux that overcomes the bar-\nrier tends to pile up in the sample center.11Because the\nbarrier does not prevent vortices from leaving the sample,\nthe magnetization loop is asymmetric, and the magneti-\nzation irreversible.12\nThe attention so far has mainly been paid to the static\nnature of barriers. Yet, dynamic e\u000bects might also give\nrise to barriers for \rux penetration. In order to inves-\ntigate if this is the case we consider dynamics of a su-\nperconducting \flm in transverse magnetic \feld. We as-\nsume that the \flm is su\u000eciently wide, so that the mag-\nnetic \feld can be treated as a continuum, and the spatio-\ntemporal evolution of the system can be obtained by solu-\ntion of the Maxwell-equations. In order to separate the\ndynamical barrier from other kinds of surface barriers,\nwe disregard surface pinning, and assume that Hc1= 0\nand the critical current density, jc, is zero. Then, the\nonly mechanism that gives loss in the system is the \fnite\nviscosity of the moving vortices, which gives a \rux \row\nresistivity\u001a=\u001anjHzj=Hc2, whereHc2is the upper crit-\nical \feld. The corresponding dynamical barrier towards\n\rux penetration will thus be strongly dependent on therate of change the applied \feld.\nII. MODEL\nLet us consider a thin superconducting \flm with thick-\nnessd, shaped as a square with sides 2 a\u001dd. Due to\nabsence of pinning, jc= 0, and the resistivity is solely\ngiven by the conventional \rux \row expression13\n\u001a=\u001anjHzj=Hc2; (1)\nwhere\u001anis the normal state resistivity, Hc2is the upper\ncritical \feld, and Hzis the transverse component of the\nmagnetic \feld. The magnetic \feld has two contributions,\nthe applied \feld and self-\feld of the sample,14\nHz=Ha+F\u00001\u0014k\n2F[g]\u0015\n; (2)\nwhereF, andF\u00001are forward and inverse Fourier trans-\nforms respectively, and k=q\nk2x+k2yis the wave-vector.\nThe local magnetization gis de\fned byr\u0002^zg=J, where\nJis the sheet current. The inverse of Eq. (2) and a time\nderivative gives\n_g=F\u00001\u00142\nkFh\n_Hz\u0000_Hai\u0015\n: (3)\nInside the sample, Faraday law and the material law,\nEq. (1), gives\n_Hz=r\u0001(Hzrg)\u001an=(Hc2\u00160); (4)\nwhereHzis given from Eq. (2). Outside the sample, _Hzis\ncalculated by an iterative Fourier space -real space hybrid\nmethod which ensures g= 0 in the vacuum outside the\nsample.14Eq. (3) is non-linear due to the self-\feld of the\nsample. In this respect, the situation is di\u000berent from the\nparallel geometry, where only the constant applied \feld\nenters the expression, and the corresponding equation for\nthe \rux dynamics is linear.arXiv:1309.4641v1  [cond-mat.supr-con]  18 Sep 20132\nFIG. 1. The m\u0000Hamagnetization loop. Even in absence\nof pinning the loop is hysteric due to the dynamical barrier.\nLet us rewrite the equations on dimensionless form,\nassuming that the applied \feld is ramped with constant\nratej_Haj. We de\fne a time scale and sheet current scale\nas\nt0\u0011s\n\u00160Hc2dw\n\u001anj_Haj; J 0\u0011s\n\u00160Hc2dwj_Haj\n\u001an:(5)\nThe dimensionless quantities are de\fned as ~t=t=t0,\ng=J0w,~H=H=J 0,~k=wk. Eq. (3) becomes\n@~g\n@~t=F\u00001\"\n2\n~kF\"\n@~Hz\n@~t\u00001##\n; (6)\nwhere\n@~Hz\n@~t=~r\u0001h\n~Hz~r~gi\n; (7)\nvalid inside the sample. As long as j_Hajis constant,\nthere are no free parameters in the problem. We will\nhenceforth omit the tildes in the dimensionless quantities,\nwhen reporting the results.\nA total are of size 1 :4\u00021:4 is discretized on a 512 \u0002512\ngrid. The additional vacuum at the sides of the super-\nconductor is used to implement the boundary conditions.\nIII. RESULT\nLet us now consider the evolution of the sample as\nit completes a magnetization cycle. The external \feld\ndriven with constant rate j_Haj= 1 until the maxi-\nmum \feldHa= 3, starting from zero-\feld-cooled con-\nditions. As applied \feld is changed, shielding currents\nare induced in the sample, giving it a nonzero mag-\nnetic moment m. The magnetic moment is calculated as\nm=R\ng(x;y)dxdy . Figure 1 shows the magnetic momentas a function of applied \feld. The plot contains the vir-\ngin branch and a steady state loop. As expected for a su-\nperconducting \flm, the main direction of the response is\ndiamagnetic. The absolute value of jmjreaches a peak for\nHa= 0:54 in the virgin branch and at Ha= 0:35 in the\nsteady-state loop, while it decreases at higher magnetic\n\felds. The shape of the loop is quite similar to supercon-\nductors with a \feld-dependent critical current,15except\nthat the magnetization peak is shifted from Ha= 0.16\nIn this respect the dynamical barrier is similar to other\nkinds of surface barriers.2,8\nFigure 2 shows HzandJmagnitude and stream lines\nat various applied \felds. The state at Ha= 0:5 is close\nto the peak in magnetization in the virgin branch. The\n\rux piles up close to the edges, and falls to zero on a well\nde\fned \rux front, roughly penetrating one third of the\ndistance to the sample center. The current stream lines\nare smooth, with highest density in the \rux-penetrated\nregion. The \rux distribution has some similarity with\nthe square in the critical state,17but the most striking\ndi\u000berence is the absence of dark d-lines at the diagonals.\nAtHa= 1:1, the \rux front has reached the center of\nthe sample. The edge of the sample is still white sig-\nnifying piling up of \rux there, but the \rux distribution\nat this time is much more uniform than it was earlier,\nand the current density is correspondingly much lower.\nThis is a feature caused by the short lifetime of currents\nof superconductors in the \rux \row state. The rightmost\npanels show the remanent state after the \feld has been\nincreased to max Ha= 3 and then back to Ha= 0. The\ndistributions are star-shaped, with the inner part of the\nsample has low current and contains a lot of trapped pos-\nitive \rux. The \rux is trapped due to a line with Hz= 0\ninside the sample, where the strong shielding currents\n\row with zero resistivity. The shielding from the cur-\nrents at this line prevents the trapped \rux from leaving\nthe sample.\nLet us return to the dimensional quantities to deter-\nmine how easy it is to measure the e\u000bect of the dy-\nnamic barrier. The most likely candidate for material\nare superconductors with low intrinsic \rux pinning and\nlow \frst critical \feld. One such material is MoGe thin-\n\flms.18With the values \u00160Hc2=3 T,\u001an= 2\u000110\u00006\nm,\nd= 50 nm,w= 2 mm, and driving rate \u00160_Ha= 10 T/s,\nwe getJ0= 35 A/m and t0= 4:3 ms. The characteristic\ncurrent density will thus be J0=d= 6:9 108A/m2and the\nmagnetic \feld values will be of order \u00160J0= 0:043 mT.\nIn this case the dynamical barrier will be larger than\nthe geometric barrier obtained Ref. 10, which is of order\n\u00160Hp=\u00160Hc1p\nd=w = 0:01 mT, with \u00160Hc1= 2mT.\nExperimentally it will thus be easy to distinguish the ge-\nometric barrier from the dynamical barrier due to the\nramp-rate dependency of the latter.3\nFIG. 2. (top) The magnetic \rux distribution and (bottom) current density and stream lines, at Ha= 0:5, 1:1, and 0 (Remanent\nstate).\nIV. SUMMARY\nThe penetration of magnetic \rux into superconducting\n\flms can be delayed due to a dynamical barrier caused by\nthe viscous motion of the vortices. In this work we have\nstudied this e\u000bect on a thin \flm superconductor of square\nshape using numerical simulations. The point that makes\nthe dynamics interesting is that in transverse geometry,\nthe \rux \row equations are non-linear due to the \flm self-\n\feld, contrary to parallel geometry where they are linear.\nIn small applied magnetic \feld, the \rux penetrates into\nthe sample in a orderly manner with a well-de\fned \rux\nfront, similar to the critical state, but with absence of\ncurrent discontinuity lines. When the applied \feld is\nchanged there are fronts moving where the total mag-netic \feld is zero, and shielding currents \row without\nresistivity. In particular in the remanent state, such a\nfront will prevent magnetic \rux from leaving the sample,\nso that the remanent state contains trapped \rux. The\nmagnetization loop is hysteric with the magnetization\npeak shifted from the zero position. Numerical estimates\nshows that the e\u000bect of the dynamical barrier should be\npossible to measure on thin \flms of materials with low\nvolume pinning. The e\u000bect is easily distinguished from\nother kinds of barriers due to its dependence on the rate\nof change of the applied \feld.\nACKNOWLEDGMENTS\nThis work was \fnancially supported by the Research\nCouncil of Norway.\n1C. P. Bean and J. D. Livingston, Phys. Rev. Lett. 12, 14\n(1964).\n2L. Burlachkov, M. Konczykowski, Y. Yeshurun, and\nF. Holtzberg, J. Appl. Phys. 70, 5759 (1991).\n3M. Konczykowski, L. I. Burlachkov, Y. Yeshurun, and\nF. Holtzberg, Phys. Rev. B 43, 13707 (1991).\n4A. A. F. Olsen, H. Hauglin, T.H.Johansen, P. E. Goa, and\nD.V.Shantsev, Physica C 408-410 , 537 (2004).\n5R. Flippen, T. Askew, J. Fendrich, and C. van der Beek,\nPhys. Rev. B 52, R9882 (1995).6J. R. Clem, R. P. Huebener, and D. E. Gallus, J. Low.\nTemp. Phys. 12, 449 (1973).\n7E. H. Brandt, M. V. Indenbom, and A. Forkl, EPL 22,\n735 (1993).\n8Y. Mawatari and J. R. Clem, Phys. Rev. B 68, 024505\n(2003).\n9E. H. Brandt, Rep. Prog. Phys. 58, 1465 (1995).\n10E. Zeldov, A. I. Larkin, V. B. Geshkenbein, M. Kon-\nczykowski, D. Majer, B. Khaykovich, V. M. Vinokur, and\nH. Shtrikman, Phys. Rev. Lett. 73, 1428 (1994).4\n11E. H. Brandt, Phys. Rev. B 59, 3369 (1999).\n12E. H. Brandt, Phys. Rev. B 60, 11939 (1999).\n13J. Bardeen and M. J. Stephen, Phys. Rev. 140, A1197\n(1965).\n14J. I. Vestg\u0017 arden, P. Mikheenko, Y. M. Galperin, and T. H.\nJohansen, New J. Phys. 15, 093001 (2013).15J. McDonald and J. R. Clem, Phys. Rev. B 53, 8643 (1996).\n16D. V. Shantsev, M. R. Koblischka, Y. M. Galperin, T. H.\nJohansen, L. P\u0017 ust, and M. Jirsa, Phys. Rev. Lett. 82,\n2947 (1999).\n17E. H. Brandt, Phys. Rev. B 52, 15442 (1995).\n18S. Kubo, J. Appl. Phys. 63, 2033 (1988)."    },    {        "title": "1311.3537v2.Nonequilibrium_dynamics_of_a_mixed_spin_1_2_and_spin_3_2_Ising_ferrimagnetic_system_with_a_time_dependent_oscillating_magnetic_field_source.pdf",        "content": "arXiv:1311.3537v2  [cond-mat.stat-mech]  26 Aug 2014Nonequilibrium dynamics of a mixed spin-1/2 and spin-3/2 Is ing ferrimagnetic system\nwith a time dependent oscillating magnetic field source\nErol Vatansever\nDokuz Eyl¨ ul University, Graduate School of Natural and App lied Sciences, TR-35160 Izmir, Turkey\nHamza Polat∗\nDepartment of Physics, Dokuz Eyl¨ ul University, TR-35160 I zmir, Turkey\n(Dated: June 14, 2018)\nNonequilibrium phase transition properties of a mixed Isin g ferrimagnetic model consisting of\nspin-1/2 and spin-3/2 on a square lattice under the existenc e of a time dependent oscillating mag-\nnetic field have been investigated by making use of Monte Carl o simulations with single-spin flip\nMetropolis algorithm. A complete picture of dynamic phase b oundary and magnetization profiles\nhave been illustrated and the conditions of a dynamic compen sation behavior have been discussed\nin detail. According to our simulation results, the conside red system does not point out a dynamic\ncompensation behavior, when it only includes the nearest-n eighbor interaction, single-ion anisotropy\nand an oscillating magnetic field source. As the next-neares t-neighbor interaction between the spins-\n1/2 takes into account and exceeds a characteristic value wh ich sensitively depends upon values of\nsingle-ion anisotropy and only of amplitude of external mag netic field, a dynamic compensation\nbehavior occurs in the system. Finally, it is reported that i t has not been found any evidence of\ndynamically first-order phase transition between dynamica lly ordered and disordered phases, which\nconflicts with the recently published molecular field invest igation, for a wide range of selected system\nparameters.\nI. INTRODUCTION\nThe phenomenon of ferrimagnetism is related to the counteraction of opposite magnetic moments with unequal\nmagnitudes located on different sublattices. Ferrimagnetic materia ls have, under certain conditions, a compensation\ntemperature at which the resultant magnetization vanishes below it s critical temperature1. Recently, it has been both\nexperimentally and theoretically shown that the coercive field exhibit s a rapid increase at the compensation point2,3.\nIt is obvious that such kind of point has a technological importance4,5, because at this point only a small driving\nfield is required to change the sign of the resultant magnetization. D ue to the recent developments in experimental\ntechniques, scientists begin to synthesize new classes of molecular -based magnets6–8. For instance, it has been shown\nthat the saturation magnetization, chemical analysis and infrared spectrum analysis of V(TCNE) x.y(solvent), where\nTCNE is tetracyanoethylene, are consistent with a ferrimagnet wit h spin-3/2 at the vanadium site and a spin of 1/2\nat the TCNE sites with x ∼27. In this regard, it is possible to mention that the theoretical models referring the\nmixed systems are of great importance since they are well adopted to study and to provide deeper understanding of\ncertain type of ferrimagnetism1.\nFrom the theoretical point of view, a great deal number of studies have been realized to get a clear idea about the\nmagnetic properties of mixed spin-1/2 and spin-3/2 ferrimagnetic I sing systems. In order to have a general overview\nabout it, it is beneficial to classify the studies in two categories base d on the investigation of equilibrium and nonequi-\nlibrium phase transition properties of such type of mixed spin system s. In the former group, static or equilibrium\nproperties of these type of systems have been analyzed within the several frameworks such as exact9,10, effective field\ntheory with correlations11–22, Bethe lattice23–26, exact star-triangle mapping transformation27, high temperature se-\nries expansion method29, multisublattice Green-function technique30, Oguchi approximation31as well as Monte Carlo\nsimulation32. It is underlined in the somestudies noted abovethat when the syst emincludes onlythe nearest-neighbor\ninteraction between spins and the single-ion anisotropy, the tempe rature variation of resultant magnetization does not\nexhibit a compensation behavior. In contrary to this, when the nex t-nearest neighbor interaction between spins-1/2\ntakes into account and exceeds a minimum value which depends upon t he other system parameters, the ferrimagnetic\nsystem reveals a compensation treatment which can not be observ ed in single-spin Ising systems.\nMagnetically interacting system under the influence of a magnetic fie ld varying sinusoidally in time exhibits two\nimportant striking phenomena: Nonequilibrium phase transitions and dynamic hysteresis behavior. Nowadays, these\ntypes of nonequilibrium systems are in the center of scientists beca use they have exotic, unusual and interesting\nbehaviors. For example, the universality classes of the Ising model and its variations under a time dependent driving\nfield are different from its equilibrium counterparts33–35. It is possible to emphasize that nonequilibrium phase\ntransitions originate due to a competition between time scales of the relaxation time of the system and oscillating\nperiodoftheexternalappliedfield. Forthehightemperaturesand highamplitudesoftheperiodicallyvaryingmagnetic\nfield, the simple kinetic ferromagnetic system exists in dynamically diso rdered phase where the time dependent2\nmagnetization oscillates around value of zero and is able to follow the e xternal applied magnetic field with some delay,\nwhereas it oscillates around a non-zero value which indicates a dynam ically ordered phase for low temperatures and\nsmall magnetic field amplitudes36. The physical mechanism described briefly above points out the exis tence of a\ndynamic phase transition (DPT)33,36,37.\nDPTs and hysteresis behaviors can also be observed experimentally . For example, by benefiting from surface\nmagneto-optic Kerr effect (MOKE), dynamic scaling of magnetic hys teresis in ultrathin ferromagnetic Fe/Au(001)\nfilms has been studied, and it is reported that the dispersion of hyst eresis loop area of studied system obeys to a\npower law behavior38. A comprehensive study, which includes the hysteresis loop measur ement of well-characterized\nultrathin Fe films grown on flat and stepped W(110) surfaces, has b een done by using MOKE, and prominent\nexperimental observations are reported in Ref.39. In addition to these pioneering works mentioned briefly above, to\nthe best of our knowledge, there exist a number of experimental s tudies regarding the nonequilibrium properties of\ndifferenttypesofmagneticmaterialssuchasCofilmsonaCu(001)su rface40, polycrystallineNi 80Fe20films41, epitaxial\nFe/GaAs(001) thin films42, Fe0.42Zn0.58F243, finemet thin films with composition Fe 73.5Cu1Nb3Si13.5B944, [Co/Pt]3\nmagnetic multilayers with strong perpendicular anisotropy45as well as assembly of paramagnetic colloids46. Based\nupon the detailed experimental investigations, it has been discover ed that experimental nonequilibrium dynamics of\nconsidered real magnetic systems strongly resemble the dynamic b ehavior predicted from theoretical calculations of a\nkinetic Ising model. From this point of view, it is possible to see that the re exists an impressive evidence of qualitative\nconsistency between theoretical and experimental investigation s.\nOn the other hand, in the latter group there exists a limited number o f nonequilibrium studies concerning the\ninfluences of time varying magnetic field on the mixed spin-1/2 and spin -3/2 Ising ferrimagnetic model. For instance,\nthermal and magnetic properties of a mixed Ising ferrimagnetic mod el consisting of spin-1/2 and spin-3/2 on a square\nlattice have been analyzed by making use of Glauber-type stochast ic process47. It has been reported that the studied\nsystem always exhibits a dynamic tricritical point in amplitude of exter nal applied field and temperature plane, but\nit does not show in the single-ion anisotropy and temperature plane f or low values of amplitude of field48. Following\nthe same methodology, a similar study has been done to shed some ligh t on what happens when an oscillating\nmagnetic field is applied to the mixed spin-1/2 and spin-3/2 Ising model on alternate layers of hexagonal lattice. It\nhas been found that depending on the Hamiltonian parameters, the system presents dynamic multicritical as well\nas compensation behaviors49. However, the aforementioned studies are mainly based on molecula r field theory. It\nis a well known fact that, in molecular field theory, spin fluctuations a re ignored and the obtained results do not\nhave any microscopic information details of system. From this point v iew, in order to obtain the true dynamics of a\nmixed spin-1/2 and spin-3/2 Ising ferrimagnetic system on a square lattice under the presence of a time dependent\noscillating magnetic field, we intend to use of Monte Carlo simulation tec hnique which takes into account the thermal\nfluctuations, and in this way, non-artificial results can be obtained .\nThe outline of the paper is as follows: In section II we briefly present our model. Section III is dedicated to the\nresults and discussion, and finally section IV contains our conclusion s.\nII. FORMULATION\nWe consider a two-dimensional kinetic Ising ferrimagnetic system wit h mixed spins of σ= 1/2 andS= 3/2 defined\non a square lattice, and the system is exposed to a time dependent m agnetic field source. The Hamiltonian describing\nour model is given by\nH=−J1/summationdisplay\n/angbracketleftnn/angbracketrightσA\niSB\nj−J2/summationdisplay\n/angbracketleftnnn/angbracketrightσA\niσA\nk−D/summationdisplay\nj(SB\nj)2−H(t)\n/summationdisplay\niσA\ni+/summationdisplay\njSB\nj\n (1)\nwhere the σi=±1/2, andSj=±3/2,±1/2 are the Ising spins on the sites of the sublattices A and B, respect ively.\nFirst and second sums in Eq. (1) are over the nearest- and next-n earest neighbor pairs of spins, respectively. We\nassumeJ1<0 such that the exchange interaction between nearest neighbour s is antiferromagnetic. J2is the exchange\ninteraction parameter between pairs of next-nearest neighbors of spins located on sublattice A, and Dis single ion-\nanisotropy term which affects only S= 3/2 spins located on sublattice B. The time varying sinusoidal magnetic fi eld\nis as following\nH(t) =h0sin(ωt) (2)\nhere,h0andωare amplitude and angular frequency of the external field, respec tively. The period of the oscillating\nmagnetic field is given by τ= 2π/ω.3\nThe linear dimension of the lattice is selected as L = 40 through all simula tions, and Monte Carlo simulation based\non Metropolis algorithm50is applied to the kinetic mixed Ising ferrimagnetic system on a 40 ×40 square lattice\nwith periodic boundary conditions in all directions. Configurations we re generated by selecting the sites sequentially\nthrough the lattice and making single-spin-flip attempts, which were accepted or rejected according to the Metropolis\nalgorithm. Data were generated over 50 independent samples realiz ations by running the simulations for 60000 MC\nsteps per site after discarding the first 20000 steps. This amount of transient steps is found to be sufficient for\nthermalization for the whole range of the parameter sets. Error b ars are found by using Jacknife method51. Because\nthe calculated errors are usually smaller than the sizes of the symbo ls in the obtained figures, they have not been\ngiven in this study.\nThe instantaneous values of the sublattice magnetizations MAandMB, and also the total magnetization MTat\nthe time t are defined as\nMA(t) =2\nL2/summationdisplay\ni∈AσA\ni, M B(t) =2\nL2/summationdisplay\nj∈BSB\nj, M T(t) =MA(t)+MB(t)\n2. (3)\nBy benefiting from the instantaneous magnetizations over a full pe riod of oscillating magnetic field, we obtain the\ndynamic order parameters as follows\nQA=1\nτ/contintegraldisplay\nMA(t)dt, Q B=1\nτ/contintegraldisplay\nMB(t)dt, Q t=1\nτ/contintegraldisplay\nMT(t)dt, (4)\nwhereQA,QBandQtdenote the dynamic order parameterscorrespondingto the subla tticesAandB, and the overall\nlattice, respectively. To determine the dynamic compensation temp eratureTcompfrom the computed magnetization\ndata, the intersection point of the absolute values of the dynamic s ublattice magnetizations was found using\n|QA(Tcomp)|=|QB(Tcomp)|, (5)\nsign(QA(Tcomp)) =−sign(QB(Tcomp)), (6)\nwithTcomp< Tc, whereTcis the dynamic critical temperature. We also calculate the time avera ge of the cooperative\npart of energy of the kinetic mixed Ising ferrimagnetic system over a full cycle of the magnetic field as follow52\nEcoop=−1\nL2τ/contintegraldisplay\nJ1/summationdisplay\n/angbracketleftnn/angbracketrightσA\niSB\nj+J2/summationdisplay\n/angbracketleftnnn/angbracketrightσA\niσA\nk+D/summationdisplay\nj(SB\nj)2\ndt. (7)\nThus, the specific heat of the system is defined as\nCcoop=Ecoop\ndT, (8)\nwhereTrepresents the temperature. We should mention here that DPT po ints separating the dynamically ordered\nand disordered phases are determined by benefiting from the peak s of heat capacities. We also verified that the peak\npositions of heat capacities do not significantly alter when larger Lis selected.\nIII. RESULTS AND DISCUSSION\nIn this section, we will focus our attention on the nonequilibrium dyna mics of the mixed spin-1/2 and spin-3/2 Ising\nferrimagnetic system under a time dependent magnetic field. First o f all, we will discuss the dynamic nature of the\nsystem when the system includes only the nearest-neighbor intera ction between spins, the single-ion anisotropy and\nexternal applied field. Next, we will give and argue the global dynamic phase diagrams including the both dynamic\ncritical and compensation temperatures in the case of the existen ce the next-nearest neighbor interaction between\nspins-1/2 located on sublattice A. Before we discuss the DPT featu res of the considered system, we should notice that\nthe situation of h0/|J1|= 0.0 indicates the equilibrium case, and our Monte Carlo simulation findings for this value\nofh0/|J1|are completely in accordance with the recently published work32where the equilibrium properties of the\npresent system were analyzed by following a numerical methodology of heat-bath Monte Carlo algorithm.4\nThe considered system exhibits three types of magnetic behaviors depending on the Hamiltonian parameters. These\nare dynamically ferrimagnetic ( i), ferromagnetic ( f) and paramagnetic ( p) phases, respectively. In the first type of\nphase, namely in iphase,|QA| /negationslash=|QB|, and, the time dependent sublattice magnetizations, MA(t) andMB(t) oscillate\nwith time around a non-zero value whereas they alternate around a non-zerovalue and |QA|=|QB|in the second type\nof phase, namely in fphase. In pphase which corresponds to the third type of phase, |QA|=|QB|andMA(t) and\nMB(t) oscillate around zero value, and they are delayed with respect to t he external applied magnetic field. Keeping\nin this mind, we illustrate the dynamic phase diagrams in a ( D/|J1|−kBTc/|J1|) plane with three oscillation periods\nτ= 50,100 and 200 and for some selected values of the applied field amplitude s (h0/|J1|) in Figs. 1(a)-(c). One of\nthe main findings is that DPT temperature decreases as the value of applied field amplitude increases. The physical\nmechanism underlying this observation can be much better underst ood by following a simple way: If one keeps the\nsystem in one well of a Landau type double well potential, a certain am ount of energy coming from magnetic field\nis necessary to achieve a dynamic symmetry breaking. If the amplitu de of the applied field is less than the required\namount then the system oscillates in one well. In this situation, the ma gnetization does not change its sign. In\nother words, the system oscillates around a non-zero value corre sponding to a dynamically ordered phase. As the\ntemperature increases, the height of the barrier between the tw o wells decreases. As a result of this, the less amount\nof magnetic field is necessary to push the system from one well to an other and hence the magnetization can change\nits sign for this amount of magnetic field. Consequently, the time ave raged magnetization over a full cycle of the\noscillating magnetic field becomes zero. For the relatively high oscillatio n period values, dynamic magnetizations\ncorresponding to the instantaneous sublattice order parameter s can respond to the oscillating magnetic field with\nsome delay, whereas a competition occurs between the period τof the field and the relaxation time of the system\nas the period of the external magnetic field decreases. Hence, th e dynamic magnetizations can not respond to the\nexternal magnetic field due to the increasing phase lag between the field and the time dependent magnetization. This\nmechanism makes the occurrence of the DPT difficult for the conside red system. Another important observation is\nthat an unexpected sharp dip occurs between dynamically ordered and disordered phases in the ( D/|J1|−kBTc/|J1|)\nplane with increasing value of the amplitude of the external applied fie ld, and our results show that observation such\nkind of treatment explicitly depends upon the value of τof field. Furthermore, it is necessary to state that for both\nlarge negative and positive values of single-ion anisotropies, the pha se transition points saturate a certain temperature\nregions, and they tend to shift to the lower temperature regions w ith increasing amplitude and period of the external\napplied field.\n/s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s112\n/s105/s32/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s48\n/s32/s48/s46/s53\n/s32/s61/s32/s53/s48\n/s32/s32/s107\n/s66/s84\n/s99/s47/s124/s74\n/s49/s124\n/s68/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s102\n/s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s112\n/s105\n/s102/s40/s99/s41\n/s32/s61/s32/s50/s48/s48\n/s68/s47/s124/s74\n/s49/s124/s32\n/s32\n/s32/s32\n/s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s112\n/s105\n/s102\n/s32/s32\n/s68/s47/s124/s74\n/s49/s124/s40/s98/s41\n/s32/s61/s32/s49/s48/s48\nFIG. 1: (Color online) Dynamic phase boundaries of the syste m in the ( D/|J1|−kBTc/|J1|) plane with some selected values\nof external field amplitudes h0/|J1|= 0.0 and 0.5. The curves are plotted for three values of oscillating per iod: (a) τ= 50, (b)\nτ= 100 and (c) τ= 200. The dotted lines are boundary lines between two dynami cally ordered phases.\nIn Figs. 2(a)-(b), we depict the effect of the single-ion anisotropy on the thermal variations of dynamic order\nparameters corresponding to the phase diagram illustrated in Fig. 1 (a) for value of h0/|J1|= 0.5. It is clear\nfrom the figures that the treatments of the thermal variations o f sublattices as well as total magnetizations curves\nsensitively depend upon the value of single-ion anisotropy, for selec ted values of Hamiltonian parameters. In the5\nbulk ferrimagnetism of N` eel, it is possible to classify the thermal var iation of the total magnetization curve in\ncertain categories1. According to this nomenclature, for D/|J1| ≥0, the considered system clearly points out a Q-\ntype behavior, where the magnetizations of system begin to decre ase gradually starting from their saturation values\nwith increasing thermal agitation, and then they vanish at the DPT p oint. One can easily see that, in the range\n−1< D/|J1|<0, the magnetizations tend to fall prominently from their saturatio n values, and the system undergoes\na second order DPT as temperature increases. In addition to thes e, whenD/|J1|<−1, the system exhibits a L-type\nbehavior at which the total magnetization shows a temperature ind uced maximum which definitively depends on the\nvalue of single-ion anisotropy as well as other Hamiltonian parameter s. Based on the above simulation observations,\nit is possible to make an inference that the studied system has three types of dynamic magnetic behavior. It is also\nworthy of note that even though the magnitudes of spins are differ ent from each other, both QAandQBexhibit a\nDPT at the same critical temperature, which is a result of the neare st-neighbor exchange coupling J1.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s32/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s49/s46/s48/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s48/s46/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s45/s48/s46/s50/s53\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s45/s48/s46/s53\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s45/s48/s46/s55/s53\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s45/s49/s46/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s45/s49/s46/s50/s53\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s45/s49/s46/s53\n/s124/s81\n/s65/s124/s44/s32/s124/s81\n/s66/s124/s32/s32/s124/s81\n/s116/s124\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s53\n/s32/s61/s32/s53/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s124/s81\n/s65/s124/s32\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s98/s41\n/s124/s81\n/s66/s124\nFIG. 2: (Color online) Effects of the single-ion anisotropy o n the thermal variations of order parameters |Qt|,|QA|and|QB|\nfor a combination of Hamiltonian parameters corresponding to the phase diagram depicted in Fig. 1.\nThe influences ofthe applied field amplitude h0/|J1|onthe thermalvariationsoftotal and sublatticemagnetizations\nas well as dynamic heat capacity of the system are plotted in Figs. 3( a)-(c) corresponding to the phase diagram\nconstructed in Fig. 1(a) with value of single-ion anisotropy D/|J1|= 1.0.In Fig. 3(a), total magnetization curves\nof the system are shown. As seen in this figure, magnetization curv es exhibit Q-type behavior and dynamic critical\ntemperatures decreases with increasing h0/|J1|values. On the other hand, dynamic heat capacity curves which are\ndepicted in Fig. 3(c) show a sharp peak behavior indicating the phase transition temperature. Moreover, one can\nreadily see that increasing value of the applied field amplitude gives rise to decrease the maximum of the dynamic\nheat capacity curves. We should note here that such kinds of dyna mic heat capacity behaviors have been found in\na ferrimagnetic core-shell nanoparticle composed of a spin-3 /2 ferromagnetic core which is surrounded by a spin-1\nferromagnetic shell layer under the presence of a time dependent magnetic field53,54.\nIn Fig. 4, we investigate the effect of the applied field period on the DP T features of the system. Phase diagrams\nin Fig. 4(a) are constructed for a value of the applied field amplitude h0/|J1|= 0.5.It is possible to say that DPT\npoints are depressed with increasing applied field period especially in th e high values of the single-ion anisotropy.\nThe physics behind of these findings are identical to those emphasiz ed in Fig. 1. Therefore, we will not discuss these\ninterpretations here. Instead of this, in Fig. 4(b) we will give the infl uence of applied field period on the thermal\nvariations of sublattice magnetizations for a considered value of sin gle-ion anisotropy D/|J1|= 1.0 corresponding to\nthe phase diagram illustrated in Fig. 4(a). It is found that as the the rmal agitation increases starting from zero,\nthe values of sublattice magnetizations begin to gradually decrease and the system undergoes a DPT at the critical\ntemperature which sensitively depends on value of the τ.\nAccording to our simulation results, the kinetic mixed spin-1/2 and sp in-3/2 Ising ferrimagnetic system including\nonly nearest-neighborinteraction between spins, the single-ion an isotropyand external applied field does not display a\ndynamic multicritical behavior for a wide range of Hamiltonian paramet ers used in here, in contrary to the previously6\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s72/s101/s97/s116/s32/s67/s97/s112/s97/s99/s105/s116/s121/s124/s81\n/s65/s124/s44/s32/s124/s81\n/s66/s124/s68/s47/s124/s74\n/s49/s124/s61/s49/s46/s48\n/s32/s61/s32/s53/s48/s32/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s48/s46/s53\n/s32/s32/s124/s81\n/s116/s124\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s40/s99/s41\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s32\n/s32\n/s32/s32\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s124/s81\n/s66/s124\n/s124/s81\n/s65/s124\n/s32/s32\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s98/s41\nFIG. 3: (Color online) Temperature dependencies of (a) tota l magnetization |Qt|, (b) sublattice magnetizations |QA|and|QB|,\nand (c) dynamic heat capacity for D/|J1|= 1.0 andτ= 50 with h0/|J1|= 0.0 and 0.5.\n/s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s49/s46/s50/s53/s49/s46/s53/s48/s49/s46/s55/s53\n/s32/s32 /s32/s61/s32/s53/s48\n/s32/s32/s49/s48/s48\n/s32/s32/s50/s48/s48\n/s32/s32/s107\n/s66/s84\n/s99/s47/s124/s74\n/s49/s124\n/s68/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s53\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s124/s81\n/s66/s124\n/s124/s81\n/s65/s124/s40/s98/s41\n/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s49/s46/s48\n/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s53\n/s32/s32/s124/s81\n/s65/s124/s44/s32/s124/s81\n/s66/s124\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124\nFIG. 4: (Color online) (a) Dynamic phase boundaries of the sy stem in ( D/|J1| −kBTc/|J1|) plane for h0/|J1|= 0.5 with\nτ= 50,100 and 200. (b) Effects of the external applied field period on the thermal variations of sublattice magnetizations |QA|\nand|QB|forD/|J1|= 1.0 andτ= 50,100 and 200.\npublished molecular field investigation where dynamic first order phas e transitions and dynamic tricritical points have\nbeen reported for the same model48.\nIn the following analysis, in order to shed some light on the effect of th e next-nearest neighbor interaction between\nspins-1/2 of the studied system, we give the dynamic phase bounda ries in (J2/|J1| −kBT/|J1|) plane for some\nconsidered values of applied field amplitudes with τ= 100 in Figs. 5(a-c). The phase boundaries containing both\ndynamic critical and compensation temperatures are plotted for t hree values of single-ion anisotropies D/|J1|=\n−1.0,0.0 and 1.0, respectively. At first sight, one can clearly see that the dynamic compensation temperatures do not\nemerge until J2/|J1|reaches a certain amount of value. After the aforementioned valu e ofJ2/|J1|, with an increment\ninJ2/|J1|does not lead to change the location of dynamic compensation point f or fixed values of D/|J1|andh0/|J1|.7\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s49/s50/s51/s52/s53/s54/s55\n/s112/s32/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s48\n/s32/s49/s46/s48\n/s32/s50/s46/s48\n/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s45/s49/s46/s48\n/s32/s61/s32/s49/s48/s48\n/s32/s32/s107\n/s66/s84/s47/s124/s74\n/s49/s124\n/s74\n/s50/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s105\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s49/s50/s51/s52/s53/s54/s55\n/s112\n/s105/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s49/s46/s48\n/s74\n/s50/s47/s124/s74\n/s49/s124/s40/s99/s41/s32\n/s32\n/s32/s32\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s49/s50/s51/s52/s53/s54/s55\n/s112\n/s105/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s48\n/s32/s84\n/s99\n/s32/s84\n/s99/s111/s109/s112\n/s74\n/s50/s47/s124/s74\n/s49/s124\n/s32/s32\n/s40/s98/s41\nFIG.5: (Color online)Dynamicphaseboundariesincludingb othcriticalandcompensation temperaturesin(J 2/|J1|−kBT/|J1|)\nplane for τ= 100 and with some selected values of external field amplitud es h0/|J1|= 0.0,1.0,and 2.0. The curves are plotted\nfor three values of single-ion anisotropies (a) D/|J1|=−1.0, (b)D/|J1|= 0.0 and (c) D/|J1|= 1.0.\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s32/s74\n/s50/s47/s124/s74\n/s49/s124/s32/s61/s32/s49/s48/s46/s48/s32\n/s32/s57/s46/s48\n/s32/s56/s46/s48\n/s32/s55/s46/s48\n/s72/s101/s97/s116/s32/s67/s97/s112/s97/s99/s105/s116/s121/s124/s81\n/s65/s124/s44/s32/s124/s81\n/s66/s124 /s32/s32/s124/s81\n/s116/s124\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s68/s47/s124/s74\n/s49/s124/s61/s45/s49/s46/s48\n/s104\n/s48/s47/s124/s74\n/s49/s124/s61/s49/s46/s48\n/s32/s61/s32/s49/s48/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s49/s46/s50/s53/s40/s99/s41\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s32\n/s32\n/s32/s32\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s124/s81\n/s65/s124\n/s32/s32\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s98/s41\n/s124/s81\n/s66/s124\nFIG. 6: (Color online) Influences of the next-nearest neighb or interactions on the thermal variations of (a) total magne tization\n|Qt|, (b) sublattice magnetizations |QA|and|QB|and (c) dynamic heat capacity for some selected values of Ham iltoanian\nparameters corresponding to the phase diagram illustrated in Fig. 5.\nIn contrary to this behavior, in accordance with the expectations , as the value of J2/|J1|gets bigger starting from\nzero, the much more thermal energy is necessary to reveal a DPT . On the other side, both the dynamic critical and\ncompensation temperatures strongly depend upon the selected v alues ofD/|J1|andh0/|J1|. An increase in the value\nofh0/|J1|gives rise to shift the dynamic critical and compensation points to low er temperatures and also allows the\nsystem to display a dynamic compensation behavior at the relatively lo w value of J2/|J1|. Additionally, it is possible\nto make an inference that with increasing value of single-ion anisotro py, the region where the dynamic compensation\nbehavior occurs shifts to upward for considered Hamiltonian param eters. This situation can be well understood by\ncomparing the Figs. 5(a), (b) and (c) with each other.8\nEffects of the next-nearest neighbor interactions on the therma l variations of total and sublattice magnetizations\nas well as on dynamic heat capacity of the studied system for some s elected values of D/|J1|=−1.0,h0/|J1|= 1.0\nwithτ= 100 corresponding to the phase diagram shown in Fig. 5(a) are see n in Figs. 6(a-c). We give the total\nmagnetizationcurvesforchangingvalueof J2/|J1|inFig. 6(a). Thesecurvesexplicitlyrefertheexistenceofadynamic\ncompensation behaviors, and they also exhibit a N-type magnetic be havior. As we discussed before, varying value of\nJ2/|J1|does not give rise to cause a change in value of dynamic compensation point in temperature plane. Besides, in\nordertoshowhowthe dynamiccompensation phenomenonrises, th ermalvariationsofthe absolutevaluesofsublattice\nmagnetizations are given in Fig. 6(b). It can be said that an increase in value of J2/|J1|leads to the existence of\na dynamically stronger ferromagnetic interaction between σspins. In this way, the σspins can remain ordered\nat relatively higher temperatures. When the thermal energy incre ases starting from zero, the values of sublattice\nmagnetizations begin to gradually decrease from their saturation v alues until both sublattices magnetizations are\nequail in magnitude at a certain temperatures at which dynamic comp ensation point emerges below the dynamic\ncritical transition temperature. J2/|J1|dependencies of dynamic heat capacities are presented in Fig. 6(c) . It is\nobvious from the figure that the dynamic heat capacity curves exh ibit a sharp peak at the transition temperature,\nand when the value of J2/|J1|increases, the dynamically ordered phase region gets wider, in othe r words, the location\nof the sharp peak slides to a higher value in the temperature plane. T he shape of the heat capacity curves are also\nnearly same for selected values of Hamiltonian parameters.\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s32/s104\n/s48/s47/s124/s74\n/s49/s124/s32/s61/s32/s48/s46/s48\n/s32/s49/s46/s48\n/s32/s50/s46/s48\n/s124/s81\n/s65/s124/s44/s32/s124/s81\n/s66/s124 /s32/s32/s124/s81\n/s116/s124\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s97/s41\n/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s45/s49/s46/s48\n/s74\n/s50/s47/s124/s74\n/s49/s124/s32/s61/s32/s49/s48/s46/s48\n/s32/s61/s32/s49/s48/s48\n/s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s124/s81\n/s65/s124/s32\n/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s40/s98/s41\n/s124/s81\n/s66/s124\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s49/s50/s51/s52/s53/s54/s55\n/s32/s84\n/s99\n/s32/s84\n/s99/s111/s109/s112/s32 /s32/s61/s32/s53/s48\n/s32/s49/s48/s48\n/s32/s50/s48/s48/s68/s47/s124/s74\n/s49/s124/s32/s61/s32/s45/s49/s46/s48\n/s74\n/s50/s47/s124/s74\n/s49/s124/s32/s61/s32/s49/s48/s46/s48/s107\n/s66/s84/s47/s124/s74\n/s49/s124/s32\n/s104\n/s48/s47/s124/s74\n/s49/s124/s40/s99/s41\n/s105/s112\nFIG. 7: (Color online) Effects of external applied field ampli tude on the thermal variations of order parameters |Qt|,|QA|and\n|QB|forD/|J1|=−1.0,J2/|J1|= 10.0 andτ= 100 with h0/|J1|= 0.0,1.0 and 2.0.\nAs a final investigation, we give and discuss the influences of varying value of applied field amplitude on the thermal\nvariations of total and sublattices magnetizations exhibiting dynam ic compensation as well as critical temperatures\nin Figs. 7(a-b) corresponding to the dynamic phase boundary seen in Fig. 5(a) for J2/|J1|= 10.0.It is possible\nto mention that both dynamic compensation and critical temperatu res strongly depend on value of the external\napplied field amplitude and they tend to shift to a lower region in temper ature plane, and compensation behavior\ndisappears with increasing value of h0/|J1|. In order to demonstrate the detailed magnetic behavior of syste m, we\ngive the dynamic phase boundaries in ( h0/|J1|−kBT/|J1|) planes for three values of the applied field periods such as\nτ= 50,100 and 200 for selected values of single-ion anisotropy D/|J1|=−1.0 and next-nearest neighbor interaction\nJ2/|J1|= 10.0 in Fig. 7(c). Based on the calculated phase diagrams, it can be said t hat the decreasing (increasing)\napplied field period has no effect on the dynamic behavior of compensa tion behavior whereas it affects prominently\nthe dynamic critical temperature of the system such that the dyn amically ordered phase region gets wider (narrower).9\nIV. CONCLUDING REMARKS\nIn conclusion, it has been carried out a detailed Monte Carlo investiga tion based on standard single-spin flip\nMetropolis algorithm to determine the true DPT properties of a mixed spin-1/2 and spin-3/2 Ising ferrimagnetic\nsystem under a time varying magnetic field. A complete picture of glob al dynamic phase diagrams separating the\ndynamically disordered and ordered phases has been constructed by benefiting from the peaks of thermal variations\nof dynamic heat capacities in order to have a better understanding of the physical background underlying of the\nconsidered system. The most important observations reported in the present study can be briefly summarized as\nfollows:\n•When the considered system only includes the nearest-neighbor int eraction, single-ion anisotropy and a time\ndependent sinusoidally oscillating magnetic field, it does not point out a dynamic compensation point.\n•Stationary state solutions of the system strongly depend on the s elected system parameters. As discussed in\ndetail in previous section, with increasing values of amplitude and per iod of the external applied field, the\ndynamic phase boundaries tend to shift to the lower temperature r egions in related planes, and a sharp dip\noccurs between dynamically ordered and disordered phases.\n•In contrary to the previously published investigation for the same m odel where dynamic first-order phase tran-\nsitions and tricritical points have been reported48, it has not been found any evidence of the dynamic first-order\nphase transitions in our present work. The reason is most likely the f act that the method we used completely\ntakes into account the thermal fluctuations, which allows us to obt ain non-artificial results.\n•When the next-nearest neighbor interaction between spins-1/2 is included and exceeded a characteristic value\nwhich sensitively depends on value of the single-ion anisotropy and am plitude of the external applied field, the\nsystem exhibits a dynamic compensation behavior below its critical te mperature. According to the our simula-\ntion results, the changing value of applied field period has no effect on the location of dynamic compensation\npoint.\nFinally, we should note that it is possible to improve the obtained result s by making use of a more realistic system\nsuch as Heisenberg type of Hamiltonian. From the theoretical point of view, such an interesting problem may be\nsubject of a future work in order to provide deeper understandin g of ferrimagnetic materials under a time dependent\nalternating magnetic field source.\nAcknowledgements\nThe numerical calculations reported in this paper were performed a t T¨UB˙ITAK ULAKBIM (Turkish agency), High\nPerformance and Grid Computing Center (TRUBA Resources).\n∗Electronic address: hamza.polat@deu.edu.tr\n1L. N´ eel, Ann. Phys. (Paris) 3, 137 (1948).\n2P. Hansen, J. Appl. Phys. 62, 216 (1987).\n3G.M. Buendia, and E. Machado, Phys. Rev. B 61, 14686 (2000).\n4M. Mansuripur, J. Appl. Phys. 61, 1580 (1987).\n5H.-P.D. Shieh, and M.H. Kryder, Appl. Phys. Lett. 49, 473 (1986).\n6J.M. Manriquez, G.T. Yee, R.S. McLean, A.J. Epstein, and J.S . Miller, Science 252, 1415 (1991).\n7B.G. Morin, P. Zhou, C. Hahm, and A.J. Epstein, J. Appl. Phys. 73, 5648 (1993).\n8G. Du, J. Joo, and A.J. Epstein, J. Appl. Phys. 73, 6566 (1993).\n9C. Domb, Adv. 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In agreement with a recent model, we observe that coupling dipolar\nand quadrupolar magnetic modes by an asymmetric forcing of t he flow generates field reversals.\nIn addition, we show that this mechanism strongly depends on the value of the magnetic Prandtl\nnumber.\nPACS numbers: 47.65.-d, 52.65.Kj, 91.25.Cw\nThe generation of magnetic field by the flow of an elec-\ntricallyconductingfluid, i.e., the dynamoeffect, hasbeen\nmostly studied to understand the magnetic fields of plan-\nets and stars[1]. The Earthand the Sun providethe best\ndocumented examples: they both involve a spatially co-\nherent large scale component of magnetic field with well\ncharacterized dynamics. Earth’s dipole is nearly station-\nary on time scales much larger than the ones related to\nthe flow in the liquid core, but displays random rever-\nsals. Reversals also occur for the Sun but nearly pe-\nriodically. The magnetic field changes polarity roughly\nevery 11 years. Reversals have been displayed by direct\nsimulations of the equations of magnetohydrodynamics\n(MHD) [2] or of mean field MHD [3] and have been mod-\neled using low dimensional dynamical systems [4, 5]. It\nhas been observed recently that the magnetic field gen-\nerated by a von Karman flow of liquid sodium (VKS\nexperiment) can display either periodic or random re-\nversals [6] as well as several other dynamo regimes, all\nlocated in a small parameter range [7]. The ability of all\nthese very different dynamos to reverse polarity is their\nmost striking property. This is obviously related to the\nB→ −Bsymmetry of the MHD equations, implying\nthat if a magnetic field Bis a solution, −Bis another\nsolution. However, this does not explain how these two\nsolutions can be connected as time evolves. The VKS\nexperiment has provided an interesting observation. In\nthis experiment, the flow is driven in a cylindrical con-\ntainer by two counter-rotating coaxial propellers. When\nthey rotate at roughly the same frequency, a magnetic\nfieldwithadominantdipolarcomponentalignedwiththe\naxis of rotation is generated. Time dependent magnetic\nfield with periodic or random reversals are observed only\nwhen the difference between the two rotation frequencies\nis large enough [6]. We have shown that this can be re-\nlated to broken Rπsymmetry (the rotation of an angle π\nalong any axis in the mid-plane is a symmetry when the\npropellers rotate at the same frequency) [12]. Magnetic\nmodes changed to their opposite by Rπare dipolar ones,\nwhereas quadrupolar ones are unchanged. Breaking theRπsymmetry by rotating the two propellers at different\nfrequencies generates additional coupling terms between\ndipolar and quadrupolar modes. If their dynamo thresh-\nold is close enough, a slightly broken symmetry is suffi-\ncienttoleadtoasaddlenodebifurcation,whichgenerates\na limit cycle that connects opposite polarities. Slightly\nbelowthis bifurcation threshold, ±Bstationarysolutions\nare stable but a small amount of hydrodynamic fluctua-\ntions is enough to generate random reversals. Although\nthe flow in the Earth’s core strongly differs from the one\nof the VKS experiment, a similar type of interaction be-\ntween two marginal dynamo modes can be considered\nand provides a simple explanation of several features of\npaleomagnetic records of Earth’s reversals [13].\nThe purpose of this work is to strengthen this\nphenomenological scenario by displaying reversal of a\nmagnetic dipole coupled with a quadrupolar mode in\na direct numerical simulation of MHD equations. To\nwit, we consider a flow driven in a spherical geometry\nby volumic forces that mimic the motion of two co-axial\npropellers. We observe reversals of the generated\nmagnetic field for a wide range of parameters when\nthe propellers rotate at different speeds. We show that\nthe value of the magnetic Prandtl number Pmstrongly\naffects the magnetic modes involved in the dynamics\nof reversals. Reversals that involve a coupling between\ndipole and quadrupolemodes occur for Pmsmall enough.\nFinally, we present a minimal model for the reversal\ndynamics.\nThe MHD equations are integrated in a spherical ge-\nometry for the solenoidal velocity vand magnetic B\nfields,\n∂v\n∂t+(v·∇)v=−∇π+ν∆v+f+1\nµ0ρ(B·∇)B,(1)\n∂B\n∂t=∇×(v×B)+1\nµ0σ∆B. (2)\nIn the above equations, ρis the density, µ0is the mag-\nnetic permeability and σis the electrical conductivity2\nFIG.1: Magnetic fieldlinesobtainedwithasymmetricforcin g\n(C= 1) for Rm= 300 and Pm= 1. Note that the field\ninvolves a dipolar component with its axis aligned with the\naxiszof rotation of the propellers.\nof the fluid. The forcing is f=f0F, where Fφ=\ns2sin(πsb), Fz=εsin(πsc),forz >0, and opposite\nforz <0. We use polar coordinates ( s,φ,z), normalized\nby the radius of the sphere a.Fφgenerates counter-\nrotating flows in each hemisphere, while Fzenforces a\nstrong poloidal circulation. The forcing is only applied\nin the region 0 .25a <|z|<0.65a,s < s0. In the simula-\ntions presented here, s0= 0.4,b−1= 2s0andc−1=s0.\nThis forcing has been used to model the Madison experi-\nment [8]. It is invariant by the Rπsymmetry. In order to\nbreak it, we consider in the present study a forcing of the\nformCf, whereC= 1 forz <0 but can be different from\noneforz >0. Thisdescribestwopropellersthat counter-\nrotate at different frequencies. Although performed in a\nspherical geometry, this simulation involves a mean flow\nwith a similar topology to that of the VKS experiment.\nWe solve the above system of equations using the Parody\nnumerical code [10]. This code was originally developped\nin the context of the geodynamo (spherical shell) and we\nhave modified it to make it suitable for a full sphere.\nWe use the same dimensionless numbers as in [9], the\nmagnetic Reynolds number Rm=µ0σamax(|v|), and\nthe magnetic Prandtl number Pm=νµ0σ. The kinetic\nReynolds number is then Re=Rm/Pm.\nIn a previous study with symmetric forcing ( C= 1),\nwe showed that different magnetic modes can be gener-\nated depending on Pm[9]. For large enough Pm, the\ndynamo onset corresponds to small Reand the flow is\naxisymmetric and generates first an equatorial dipole. In\ncontrast, for Pmsmall, the dynamo onset occurs when\nReis already large and the flow involves non axisymmet-\nric fluctuations. A magnetic field with a dominant axial\ndipole is observed (see figure 1). In the present study, all\nthe simulations are made for Re >300, so that an axial\ndipole is obtained for symmetric forcing.\nWe next break the symmetry Rπof the forcing to\ncheck whether time dependent magnetic fields involv-\ning reversals between both polarities are obtained as in510 15 20 25 30\ntime [resistive unit]-0.03-0.02-0.0100.010.02Magnetic field [arb.]Axial dipole\nEquatorial dipole\nQuadrupole\nFIG. 2: Time recording of the axial dipolar magnetic mode\n(inblack), the axial quadrupolarmode (in blue)andthe equa -\ntorial dipole (in red) for Rm= 300,Pm= 1 and C= 2..\nthe VKS experiment. Time recordings of some compo-\nnents of the magnetic field are displayed in figure 2 for\nRm= 300,Pm= 1 and C= 2, which means that one\nof the propellers is spinning twice as fast as the other\none. We observe that the axial dipolar component (in\nblack) randomly reverses sign. The phases with given\npolarity are an order of magnitude longer than the dura-\ntion of a reversal that corresponds to an Ohmic diffusion\ntime. The magnetic field strongly fluctuates during these\nphases because ofhydrodynamic fluctuations. It also dis-\nplays excursions or aborted reversals, i.e., the dipolar\ncomponent almost vanishes or even slightly changes sign\nbut then grows again with its direction unchanged. All\nthese features are observed in paleomagnetic records of\nEarth’s magnetic field [11] and also in the VKS experi-\nment [6]. However, these simulations also display strong\ndifferences with the VKS experiment. The equatorial\ndipole is the mode with the largest fluctuations whereas\nthe axial quadrupolar components is an order of magni-\ntude smaller than the equatorial modes. In addition, it\ndoes not seem to be coupled to the axial dipolar compo-\nnent.\nMagnetic Prandtl numbers relevant to liquid metals\nare much smaller than unity ( ∼10−5–10−6). While re-\nalistic values cannot be achieved owing to computational\nlimitations, Pmcanbedecreasedtovalueslessthanunity.\nWe now turn to simulations using Pm= 0.5, thus in-\ntroducing a distinction between the viscous and ohmic\ntimescale. The time evolution of the magnetic modes for\nRm= 165 and C= 1.5 is represented on figure 3 (left).\nIt differs significantly from the previous case ( Pm= 1).\nFirst of all, the quadrupole is now a significant part of\nthe field, andreversestogetherwith the axialdipole. The\nequatorial dipole remains compartively very weak and\nunessential to the dynamics. The high amount of fluc-3\n30 40 506070 80 90\nTime [resistive unit ]-0.02-0.0100.010.02Magnetic field [arb.]Equatorial dipole\nAxial quadrupole\nAxial dipoleRe=330\n20 40 60 80 100 120 140\nTime [resistive unit]-0.02-0.0100.010.020.03Magnetic field [arb.]Equatorial dipole\nAxial quadrupole\nAxial dipoleRe=360\nFIG. 3: Time recordings of the axial dipole (black), theaxia l quadrupole (blue) andthe equatorial dipole (red). Left: Rm= 165,\nPm= 0.5 andC= 1.5. Right: Rm= 180,Pm= 0.5 andC= 2.\ntuations observed in these signals points to the role of\nhydrodynamic fluctuations on the reversal mechanism.\nOne could be tempted to speculate that a higher degree\nof hydrodynamic fluctuations necessarily yields a larger\nreversal rate. Such is in fact not the case. A more sen-\nsible approach could be to try to relate the amount of\nfluctuations of the magnetic modes in a phase with given\npolarity, to the frequency of reversals. Increasing Rm\nfrom 165 to 180 does yield larger fluctuations as shown\nin figure 3 (right). However the reversal rate is in fact\nlowered because Cwas modified to C= 2. This clearly\nshows that the asymmetry parameter Cplays a more im-\nportant role than the fluctuations of the magnetic field.\nForPm= 0.5, reversals occur only in a restricted re-\ngion, 1.1< C <2.5, which is also a feature of the VKS\nexperiment.\nLet us now investigate the detail of a polarity rever-\nsal (figure 4). Interestingly the dipolar and quadrupolar\ncomponents do not vanish simultaneously. Instead the\ndecreaseofthedipoleisassociatedwithasuddenincrease\nof the quadrupolar component, related with a burst of\nactivity in the non axisymmetric velocity mode m= 1\n(zonal velocity) breakingthe Rπsymmetry (i.e. coupling\nthe dipolar and quadrupolar families). The quadrupole\nquickly decays as the dipole recovers with a reversed po-\nlarity. Immediatlyafterthereversalthedipolesystemati-\ncally overshootsits mean value during a polarityinterval.\nThis behavior of the magnetic modes is typical of rever-\nsals obtained with this value of Pmand is in agreement\nwith the model presented in [12, 13].\nThese direct numerical simulations illustrate the role\nof the magnetic Prandtl number in the dynamics of re-\nversals. When Pmis of order one, the magnetic per-\nturbations due to the advection of magnetic field lines\nby the velocity field, evolves with a time scale similar\nto the one of the velocity fluctuations. We thus expect32 33 34 35\ntime[resistive unit]-0.02-0.0100.010.02Magnetic field [arb. value]Axial dipole\nAxial quadrupole\nZonal flow\nFIG. 4: Time recordings of the axial dipole (black), the axia l\nquadrupole (blue) and zonal velocity (yellow) during a reve r-\nsal.Rm= 165,Pm= 0.5 andC= 1.5.\nthese two fields to be strongly coupled. Modification of\nthe magnetic field lines due to their advection by a local\nfluctuation of the flow can then trigger a reversal of the\nfield[14]. Thistypeofscenariohasbeenobservedinsome\ndirect numerical simulations, usually performed with Pm\nof order one [15]. When Pmis small, magnetic perturba-\ntions decay much faster and we expect only the largest\nscale magnetic modes to govern the dynamics. To illus-\ntrate this argument in a more quantitative way, we have\ncomputed the correlation rof the most significant mag-\nnetic and velocity modes with the axial magnetic dipole\nforRe= 330 and 0 .3< Pm<1. ForPm∼1, all\nthe modes are weakly correlated with the axial magnetic\ndipole (r <0.3). When Pmis decreased, the correlation\nof most modes decay except the one of the zonal veloc-\nity mode that slightly increases and the one of the axial4\n80000 84000 88000 92000\nTime [arb.]-3-2-10123Amplitude Q\nFIG. 5: Numerical integration of the amplitude equa-\ntions (3,4,5). Time recording of the amplitude of the\nquadrupolar mode for µ= 0.119,ν= 0.1 and Γ = 0 .9.\nquadrupole mode that strongly increases up to 0 .8.\nWe now write the simplest dynamical system that in-\nvolves the three modes that display correlation in the\nlowPmsimulations: the dipole D, the quadrupole Q,\nand the zonal velocity mode Vthat breaks the Rπsym-\nmetry. These modes transform as D→ −D,Q→Qand\nV→ −Vunder the Rπsymmetry. Keeping nonlinear\nterms up to quadratic order, we get\n˙D=µD−VQ, (3)\n˙Q=−νQ+VD, (4)\n˙V= Γ−V+QD. (5)\nA non zero value of Γ is related to a forcing that breaks\ntheRπsymmetry, i.e. propellers rotating at different\nspeeds. The dynamical system (3,4,5) with Γ = 0 oc-\ncurs in different hydrodynamic problems and has been\nanalyzed in detail [16]. The relative signs of the coeffi-\ncients of the nonlinear terms are such that the solutions\ndo not diverge when µ >0 andν <0. Their modulus\ncan be taken equal to one by appropriate scalings of the\namplitudes. The velocity mode is linearly damped and\nits coefficient can be taken equal to −1 by an appropriate\nchoice ofthe time scale. Note that similar equationswere\nobtained with a drastic truncation of the linear modes of\nMHD equations [5]. However, in that context µandν\nshould be both negative and the damping of the velocity\nmode was discarded, thus modifying the dynamics.\nThis system displays reversals of the magnetic modes\nDandQfor a wide range of parameters. A time record-\ning is shown in figure 5. The mechanism for these rever-\nsals results from the interaction of the modes DandQ\ncoupled by the broken Rπsymmetry when V/negationslash= 0. It is\nthus similar to the one described in [12] but keeping the\ndamped velocity mode into the system generates chaoticfluctuations. Thus, it is not necessary to add external\nnoise to obtain random reversals. We do not claim that\nthis minimal low order system fully describes the direct\nsimulations presented here. For instance, in the case of\nexact counter-rotation ( C= 1, i.e. Γ = 0), equations\n(3,4,5) do not have a stable stationary state with a dom-\ninant axial dipole. The different solutions obtained when\nµis increased cannot capture all the dynamo regimes of\nthe VKS experiment or of the direct simulations when\nRmis increased away from the threshold. Taking into\naccount cubic nonlinearities provides a better descrip-\ntion of the numerical results for Pm= 0.5. However, this\nthree mode system with only quadratic nonlinearities in-\nvolves the basic ingredients of the reversals observed in\nthe present numerical simulations for low enough values\nof the magnetic Prandtl number.\nWe thank F. P´ etr´ elis for useful discussions. Computa-\ntions were performed at CEMAG and IDRIS.\n[1] H. K. Moffatt, Magnetic field generation in electrically\nconducting fluids , Cambridge University Press (Cam-\nbridge, 1978); E. Dormy and A.M. Soward (Eds), Math-\nematical Aspects of Natural dynamos , CRC-press (2007).\n[2] See for instance, P. H. Roberts and G. A. Glatzmaier,\nRev. Mod. Phys. 72, 1081 (2000).\n[3] F. Stefani and G. Gerbeth, Phys. Rev. Lett. 94, 184506\n(2005); F. Stefani and al., G. A. F. D. 101, 227 (2007).\n[4] T. Rikitake, Proc. Camb. Phil. Soc. 54, 89-105 (1958);\nE. Knobloch and A. S. Landsberg, Mon. Not. R. Astron.\nSoc.278, 294 (1996); I. Melbourne, M. R. E. Proctor\nand A. M. Rucklidge, Dynamo and dynamics, a math-\nematical challenge , Eds. P. Chossat et al., pp. 363-370,\nKluwer Academic Publishers (2001); P. Hoyng and J. J.\nDuistermaat, Europhys. Lett. 68, 177 (2004).\n[5] P. Nozi` eres, Phys. Earth Planet. Int. 17, 55-74 (1978).\n[6] M. Berhanu et al., Europhys. Lett. 77, 59001 (2007).\n[7] F. Ravelet et al., Phys. Rev. Lett. 101, 074502 (2008).\n[8] R. A. Bayliss et al., Phys. Rev. E 75, 026303 (2007).\n[9] C.Gissinger, E. Dormy and S. Fauve, Phys. Rev. Lett.\n101, 144502 (2008).\n[10] E. Dormy, PhD thesis (1997); E. Dormy, P. Cardin, D.\nJault,Earth Plan. Sci. Lett. 160, 15–30 (1998); U. Chris-\ntensenet al,128, 25-34 (2001); and later collaborative\ndevelopments.\n[11] J.-P. Valet , L. Meynadier, Y. Guyodo, Nature 435, 802-\n805 (2005).\n[12] F. P´ etr´ elis and S. Fauve, J. Phys. Condens. Matter 20,\n494203 (2008).\n[13] F. P´ etr´ elis et al., Phys. Rev. Lett. 102, 144503 (2009).\n[14] E. N. Parker, Astrophys. J. 158, 815-827 (1969).\n[15] G.R. Sarson and C.A.Jones, Phys. Earth Planet. Int.\n111, 3-20 (1999); J. Wicht and P. Olson, Geochemistry,\nGeophysics, Geosystems 5, Q03GH10 (2004); J. Aubert,\nJ.AurnouandJ.Wicht, Geophys.J.Int. 172, 945(2008).\n[16] D. W. Hughes and M. R. E. Proctor, Nonlinearity 3,\n127-153 (1990)."    },    {        "title": "1609.08543v1.Sustained_Turbulence_in_Differentially_Rotating_Magnetized_Fluids_at_Low_Magnetic_Prandtl_Number.pdf",        "content": "arXiv:1609.08543v1  [astro-ph.SR]  27 Sep 2016DRAFT VERSION JUNE2, 2021\nPreprint typesetusingL ATEX styleAASTeX6v. 1.0\nSUSTAINEDTURBULENCE INDIFFERENTIALLYROTATINGMAGNETIZ EDFLUIDSATLOWMAGNETIC\nPRANDTL NUMBER\nFARRUKH NAUMAN AND MARTINE. PESSAH\nNiels Bohr International Academy, TheNiels Bohr Institute , Blegdamsvej 17, DK-2100, Copenhagen Ø,Denmark.\n(Dated: June 2,2021)\nABSTRACT\nWe show for the first time that sustained turbulence is possib le at low magnetic Prandtl number for Keplerian\nflowswithnomeanmagneticflux. Ourresultsindicatethatinc reasingtheverticaldomainsizeisequivalentto\nincreasingthedynamicalrangebetweentheenergyinjectio nscaleandthedissipativescale. Thishasimportant\nimplications for a large variety of differentially rotatin g systems with low magnetic Prandtl number such as\nprotostellardisksandlaboratoryexperiments.\nKeywords: accretion,accretiondisks— magnetohydrodynamicturbule nce— plasma,dynamotheory\n1.INTRODUCTION\nDifferentiallyrotatingflowsareubiquitousinnature,fro m\naccretion flows around young stars and compact objects to\nconvectionzones inside stars. Magnetic fields are importan t\nin many of these flows. Understanding the stability prop-\nerties of these flows can shed light into the processes driv-\ning the transport of angular momentum and energy in these\nenvironments. The Magnetorotational Instability (MRI) is a\nlinear instability that has emerged as a potential explanat ion\nfor the origin of turbulence in weakly magnetized differen-\ntially rotating flows (e.g. magnetized Taylor-Couette flows ,\naccretion disks Velikhov (1959);Chandrasekhar (1960);\nBalbus&Hawley (1991)). MRI requires a weak magnetic\nflux to work but given the paucity of direct observations of\nmagnetic field amplitudes and geometries, it is of interest\nto explore a range of cases for the magnetic flux threading\nan accretion flow, including the case of zero mean magnetic\nflux.\nThe earliest 3D simulation of ideal magnetohydrody-\nnamic (MHD) Keplerian shear flow without a mean mag-\nnetic flux by Hawleyet al. (1996) showed that the saturated\nlevelofturbulentstressesdecreaseswith resolution(see also\nPessah etal. (2007)). Later work by Fromangetal. (2007)\nused physical diffusion coefficients and showed that turbu-\nlence does not persist for magnetic Prandtl number, Pm=\nRm/Re /lessorsimilar2. Recent work on linearly stable hydrody-\nnamic shear flows (e.g., Hofet al. (2006)) suggests that tur-\nbulence lifetime is finite and increases exponentially with\nthe Reynolds number, Re.Rempelet al. (2010) explored\nthe question of turbulence lifetime in Keplerian shear flows\nwith zero net magnetic flux with a small vertical domain\nsize (Lz= 1) and found that the turbulence lifetime in-\ncreases exponentially with Rmbetween9,000and11,000\n(Re= 3,125).\nnauman@nbi.ku.dkExamples of Pm≪1systems include astrophysical sys-\ntemssuchasprotostellardisks(formoreastrophysicalexa m-\nples,seetable1of Brandenburg&Subramanian (2005))and\nlaboratoryplasmas(e.g., Sisan et al. (2004),Seilmayeret al.\n(2014)). ThePm≪1limit has been studied extensivelyin\nnon-helically forced isotropic MHD turbulence simulation s\nandtheory,withtheconsensusthatalarger Rmcritisrequired\nin this limit compared to the Pm≫1Schekochihinet al.\n(2007). Earlier work by Schekochihinet al. (2004) had\nclaimed that the small scale dynamo (growth of the mag-\nnetic field on dissipation scales) does not exist in the low\nPmlimit but later work ( Iskakovet al. 2007 ) demonstrated\ngrowthandsustenanceofmagneticfields(atahigher Rmcrit).\nIn their study, Schekochihinet al. (2004) conjectured that a\nlowPmdynamo might only be possible in the presence of\na mean field. Indeed for MRI simulations with net mag-\nneticflux,turbulencecanbesustainedinthelow Pmregime\n(Lesur&Longaretti (2007),Meheutetal. (2015)).\nThe question of whether a large domain size can have\na significant effect on flow stability has been long un-\nder discussion (e.g., Pomeau(1986),Philip& Manneville\n(2011)). The shearing box is a local approximation\nGoldreich& Lynden-Bell (1965) for differentially rotating\nflows such as accretion disks (or Taylor-Couette flows) and\nit is unclear whether it can exhibit similar behavior as the\nspatiotemporal chaotic patterns that might emerge in a real -\nistic accretionflowwithitsverylargespatialextentandve ry\nlargeRe(Cross&Hohenberg (1993)). Nevertheless, within\nthe shearing box framework, it is of interest to explore how\na small domain size (‘minimal flow unit’ Jimenez&Moin\n(1991),Rinconet al. (2007)) is differentfrom a system with\nlarger degrees of freedom as a result of larger domain size,\nresolution, ReandRm. Using an asymptotic analysis\nwithnettoroidalandverticalflux, Julien&Knobloch (2007)\nshowedthatscaleseparationbetweenthemostunstablemode\nandtheverticaldomainsizeleadstoasaturatedstateofMRI\nwheretheenergyisdominatedbytheboxscaleinthevertical\ndirection.2\nIn this Letter, we explore the question of whether sus-\ntained turbulence can be found in unstratified1incompress-\nible MHD Keplerian shear flows with zero mean magnetic\nflux with a variety of box sizes and dissipation coefficients.\nWe find that the turbulence lifetime is very sensitive to the\nverticalboxsize2andthatlargescalestructuresdevelopboth\ninthevelocityandthemagneticfields.\n2.NUMERICALSIMULATIONSAND RESULTS\nWe use the publicly available pseudospectral code\nSNOOPY3(Lesur&Longaretti (2007)). All of our sim-\nulations employ a zero net flux initial field Bini=\nB0sin(kxx)ez. The shear profile is Vsh=−Sxey, where\nS=qΩ = 1(q= 1.5for Keplerian shear) is the shear\nparameter and Ωis the angular frequency. We apply pertur-\nbations of order LSto the first few velocity modes. All of\nourresultsarereportedinunitsofsheartimes 1/S(forcom-\nparison,10,000shear times are equal to about 1,061orbits\n(2π/Ω) in our units). The dissipation coefficients are char-\nacterized by the Reynolds number, Re=SL2/νand the\nmagnetic Reynolds number, Rm=SL2/η; whereL= 1.\nFor most of our runs, Lx=L. The magnetic Prandtl num-\nberPm=Rm/Re=ν/η, is of course independent of the\nchoiceof characteristicscale. We reportourdomainsizes a s\nLx×Ly×Lz.\n2.1.FixedRe,Variable LzandPm\nIn the top panel of fig. 1, we find that as we increase Lz\nwhile keeping Lx= 1andLy= 2, evenPm <1man-\nages to sustain turbulence for several thousand shear times .\nThis is to be contrasted with earlier work of Fromangetal.\n(2007) (Lz= 1,Re= 3,125) andShiet al. (2016) (Lz=\n4,Re= 3,125),which showed that Pm≥2is requiredfor\nsustained turbulence. Shiet al. (2016) reports that Pm= 2\nsimulationsustainedturbulencefor thousandsof sheartim es\nwhile their Pm= 1run decayed after about 2,000shear\ntimes. In an extensive numericalstudy, Rempelet al. (2010)\n(usingSNOOPY) showed that turbulence is transient and the\nlifetime of turbulence increases exponentially with Rm(for\n2.88≤Pm≤3.52) withLz= 1,Re= 3,125. We do not\nconductsuchanexhaustivestudyheretodeterminethefunc-\ntionaldependenceoflifetime on LzandPm, sincea way of\nconductingsuchastudyistoevolveaflowtoafullyturbulent\nstateandthentakethatstateasinitialconditionforrunsw ith\ndifferent dissipation coefficients ( Rempeletal. 2010 ). Due\ntothespatiotemporalchaoticnatureofturbulence,itcant ake\nhundredsofrunsforeach Lzto getgoodstatistics.\n2.2.FixedLz, Variable ReandPm\nWe show lifetime of the turbulent flow as a function of\nReandRmwith fixed Lz= 4in the bottom panel of fig.\n1Davis etal. (2010)suggestthatdensitystratification duetogravitylow-\ners the critical Pmslightly butthey did notexplore the Pm <1regime.\n2We do not explore elongated azimuthal domains as Riols etal. (2015)\nalreadyexplored Ly= 20anddidnotfindevidenceforsustainedturbulence\natPm <1.\n3http://ipag.osug.fr/ ˜lesurg/snoopy.html\n1 2 345 6 78 9 10\nRm ×1034812LzRe=104\n0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Pm\n1 2 345 6 78 9 10\nRm ×10312345678910Re×103\nPm=1Lz=4\n01 2 345 6 78 9 10\nLifetime ( S−1) ×103\nFigure 1. Turbulence lifetime in units of 1,000S−1for runs with:\n(i)fixedRe= 10,000butvariable verticalboxsize LzandPm=\nRm/Re(toppanel); (ii)fixedverticalboxsize Lz= 4(1×2×4)\nbutvariable ReandRm(bottompanel). Thelifetimeofturbulence\nincreases with larger ReandRm, but it is most sensitive to Lz.\nNote that we stopped all of our simulations at 10,000shear times\nso the runs that show 10,000shear timesmight have a much larger\nlifetime. The number of zones used for these simulations are 64×\n64×(64∗Lz).\n1. The lifetime increases with Pm(andRm, the bottom\nright of the panel). More importantly, we find that at high\nenoughRe, turbulencecan last several thousandshear times\natPm <1. Combined with fig. 1, this result hints that as\nRe→ ∞andL≫1,Pmcritreachesalowasymptoticvalue\n(Fromangetal. (2007))andourworksuggeststhat Pmcrit<\n1(Schekochihinet al.2004 ).\nWalker et al. (2016) did simulationswith a domainsize of\n2×4×1at an unprecedented high resolution of 1024×\n1024×512where they fixed Re= 45,000. They found\nthatzeronetfluxsimulationimmediatelydecaysat Pm= 1.\nWe used a lower resolution but larger vertical domain size\nLzrangingfrom 1,2,4,8at fixedRe=Rm= 10,000. As\nshown in fig. 2, the2×4×4runis turbulentfor 2,000S−1\nuntil it decays while the 2×4×8runremains turbulentfor3\n0 1000 2000 3000 4000 500010-810-610-410-2100\n0 1000 2000 3000 4000 5000\nTime ( S-1)10-810-610-410-2100StressReynolds\nMaxwell2x4x1\n2x4x2\n2x4x4\n2x4x8\nFigure2. Time history for runs with the same horizontal domain\nsize (Lx= 2,Ly= 4) asWalker et al. (2016) but larger verti-\ncal domains. We see sustained turbulence even though our Re=\nRm= 10,000, much lower thanthat of Walker etal. (2016).\nFigure3. Comparison of the /angbracketleftBy/angbracketright(angled brackets represent xy\naverage) profilesfor three different boxsizes 1×2×1,2×4×2,\n4×8×4withRe=Rm= 10,000with the same aspect ratio.\nTurbulence is only sustained for significant duration for th e largest\ndomain,4×8×4.\nthe entire duration of our run (and possibly much longer),\nwhichis10,000S−1. This clearlydemonstratesthat vertical\ndomainsize is an importantparameterin transition to turbu -\nlencestudiesinthe shearingbox.2.3.Sameaspectratio( Lz/Lx=1),largerboxsize\nWe donotfindanysignificantevidenceforturbulenceina\ndomainwithsize 1×2×1(witharesolutionof 643)andshort\nlivedturbulence( ∼175sheartimes)for 2×4×2(1283). The\n4×8×4(2563) run is turbulent for nearly 5,000thousand\nshear times. The dissipation coefficients are set by Re=\nRm= 10,000,whichisindependentoftheabsoluteboxsize\nsince we scale both coefficientswith a fixed L= 1. We plot\nthe/angbracketleftBy/angbracketright(angledbracketsrepresent xyaverage)forthethree\nruns1×2×1,2×4×2and4×8×4infig.3. Acoherentfield\ndevelops in all cases but only the biggest box size manages\nto sustain turbulencefor a long time. To comparethe effects\nof increasing the domain size to the effect of increasing the\nresolution,weranasimulationat 1×2×1witharesolution\nof2563(the same number of zones used in the 4×8×4)\nand foundnosignsofturbulencewith Re=Rm= 10,000.\nThiscomparisonsuggeststhattheturbulencelifetimeismo re\nsensitive totheboxsize thantheresolution.\n2.4.Velocityfields\nWe plot the azimuthal velocity profile /angbracketleftVy/angbracketrightin fig.4for\nthreedifferentverticalboxsizes Lz= 4,8,12whilekeeping\nLx= 1,Ly= 2,Re=Rm= 10,000. The spatiotemporal\ndependence of the velocities suggest that velocities behav e\nlike:sin(κt+sinkzz), wherekz= 2πnz/Lz, and2π/κ∼\n9.42S−1is the epicyclic time, with κ2= 2Ω2(2−q). The\nvertical profile of differentvelocity componentsseems to b e\ndominated by nz= 1regardless of box size, which indi-\ncates that the gap between the energy containing scale and\nthe dissipation scale increases with the increase in the ver -\ntical domain. The corresponding azimuthal magnetic field\ncomponent profiles for these runs behave much in the same\nway as in fig. 3so they are also dominated by nz= 1ir-\nrespective of the domain size. Our results are largely con-\nsistent with the recent suggestion by Sekimotoetal. (2016),\nthat shearing box flow has the vertical box size as the outer\nscale.\n2.5.Magneticfields(dynamo)\nBothShiet al.(2016)andWalkeret al. (2016)referto‘dy-\nnamo’intheirdiscussion,withthefocusonthelargescalei n\ntheformerandsmallscaleinthelatter. Onemightaskwhich\none of these two differentmechanisms(or both) is responsi-\nbleforthegrowthandsustenanceofmagneticfieldsinazero\nnet flux MHD Keplerian flows. This distinction is difficult\nsince it is not entirely clear whether there is a separation o f\nspatialscales between the box scale and the forcing scale.\nAn argumentcan be madethat since the azimuthalmagnetic\nfieldvariesonseveralepicyclictimes(e.g. 4×8×4runhas\nacycleperiodofabout 500S−1,seefig.3)asopposedtoone\nepicyclic time ( ∼9.42S−1) for the velocity field, there is at\nleast atemporal scale separation4and hence a large scale\ndynamo( Gressel&Pessah (2015),Bhat etal. (2016)).\nThe small scale dynamo, on the other hand, refers to the\ngrowth of magnetic fields on the viscous scale ( Pm >1)\n4Wethank EricBlackman for pointing this out.4\nFigure4. Comparisonofthe /angbracketleftVy/angbracketrightprofilesforthreedifferentvertical\nboxsizes 1×2×Lz:Lz= 4,8,12withRe=Rm= 10,000. A\ncoherent velocity field ( ∼sin(κt+ sinkzz)), which is dominated\nby the vertical box scale suggests that the vertical box size has a\nverysignificantroleindeterminingthe(kinetic)‘energyc ontaining’\nscale of the system.\nor the resistive scale ( Pm <1) (Schekochihinet al. (2007))\nanditmightbethemechanismresponsibleforincreasingthe\nlifetime as Rmincreases. There is yet another possibility:\ngrowthdue to a mean velocityflow (e.g., Cattaneo& Tobias\n(2005),Mininni& Montgomery (2005)). Thismightbequite\nsignificant in the simulationswe reporthere since the veloc -\nities seems to settle into large scale periodic structures ( fig.\n4).Schekochihinet al. (2007)suggestcomparingthegrowth\nrate dependence of the magnetic field on Rmto distinguish\nbetween the small scale dynamoand the mean velocity flowdriven dynamo but since we trigger our simulations with fi-\nnite amplitude perturbations, it is unclear how to determin e\nthe growthratesandhencedistinguishbetweenthetwo.\n2.6.Resolutiontest\nVy\n0.1 1.0 10.0 100.0\nkz10−1010−810−610−4Power SpectrumLz=4 Res 64\nLz=4 Res 128\nLz=8 Res 64\nLz=8 Res 128\nLz=12 Res 64\nLz=12 Res 128\nFigure 5. Powerspectrumof /angbracketleftVy/angbracketright(kz)averagedover 251−500S−1,\nwherekz= 2πnz/LzforLz= 4,8,12at two different res-\nolutions64×64×(64∗Lz)and128×128×(128∗Lz).\nRe=Rm= 10,000for all of these runs. This plot suggests\nthat resolution does not significantly alter the azimuthal v elocity as\na function of resolution.\nBy\n0.1 1.0 10.0 100.0\nkz10−1010−810−610−410−2Power SpectrumLz=4 Res 64\nLz=4 Res 128\nLz=8 Res 64\nLz=8 Res 128\nLz=12 Res 64\nLz=12 Res 128\nFigure 6. Same as fig. 5but power spectrum of /angbracketleftBy/angbracketright(kz).\nAs a resolution check, we compare the power spectrum\ninkz(= 2πnz/Lz) of three different runs 1×2×Lzat\nLz= 4,8,12(Re=Rm= 10,000) at two different res-\nolutions64×64×(64∗Lz)and128×128×(128∗Lz)\nin figs.5and6. We find that the increase in resolution does\nnotsignificantlyaltertheturbulentpowerspectrum. There is\nsome discrepancy at low kzbut we point out that since both\nVyandByshowcyclicbehavior,thechosenrangeandinitial\npoint for temporal averagingcan have a significant effect on\nthe largestwavelengths.5\n3.DISCUSSIONAND CONCLUSIONS\nRecent numerical simulations and experiments on hydro-\ndynamic shear flows suggest that transition to turbulence\nmimicsa secondorderphasetransitionandcan bedescribed\nby the directed percolation universality class ( Shihetal.\n(2016);Lemoultetal. (2016);Sano&Tamai (2016)). The\nbasic idea is that linearly stable flows develop two kinds of\ndomains: laminar and turbulent corresponding to dead and\nactive states. It is through the interaction of these two typ es\nofdomainsthatturbulencestartstospreadandeventuallyfi lls\nthewholedomainastheReynoldsnumberisincreased. This\nmight explain why the box size playsa special role: a larger\nbox size allows for more turbulent and laminar domains to\nfit in the box and thus allows for complex pattern forma-\ntion (Cross&Hohenberg (1993)). A potential analogy for\noursimulationsisthegenerationof Bx(activestate)through\nstochasticprocessesonsmall scalesandthesubsequentgen -\nerationof By(deadstate)throughthe SBxterm(seealsothe\nrecent study by Riols etal. (2016) who describe the dynamo\nusingthe conceptof‘active’and‘slave’perturbations).\nWe have demonstrated for the first time that Keplerian\nshear MHD turbulence (without using any external forcing\nor net magnetic flux) can sustain for several thousand sheartimes atPm≤1(forRe= 10,000), if one uses domain\nsizes with Lz≥4. We find that with the ReandRmup to\n10,000thereisnoturbulenceat Pm≤1whenLz= 1,con-\nsistent with previous work. It seems that by increasing Lz,\none increases the separation between the outer scale ( ∼Lz)\nandthedissipationscale,whichisqualitativelysimilart othe\neffect of increasing Rm. But the increase in Lzat fixedRe\nandRmhas a more dramatic effect on turbulence lifetime\nthan increasing ReandRmatLz= 1. Our work clearly\nillustrates that flow stability studies should not be confine d\nto very small domains and has especially important impli-\ncations for laboratory plasmas, which can only explore the\nPm≪1regimeSisanet al. (2004),Seilmayeretal. (2014).\nWe thankShantanuAgarwal,EricBlackman,OliverGres-\nsel, Tobias Heinemann, Paul Manneville, Jiming Shi, Jim\nStone and Francois Rincon for discussions. Colin McNally\nis thanked for suggestions and help with the figures. The\ncomputations were performed on the BlueStreak cluster at\nthe CenterforIntegratedResearch Computing(CIRC) at the\nUniversityofRochester. Theresearchleadingtotheseresu lts\nhas received funding from the European Research Council\nunderthe EuropeanUnionsSeventhFrameworkProgramme\n(FP/2007-2013)underERC grantagreement306614.\nREFERENCES\nBalbus, S.A.,&Hawley, J.F.1991, ApJ,376, 214\nBhat, P.,Ebrahimi, F.,&Blackman, E.G.2016, MNRAS,462, 81 8\nBrandenburg, A.,&Subramanian, K.2005, PhR, 417, 1\nCattaneo, F.,&Tobias, S.M.2005, Physics of Fluids, 17, 127 105\nChandrasekhar, S. 1960, Proceedings of the National Academ y of Science,\n46, 253\nCross,M. C.,&Hohenberg, P.C.1993, Rev. 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The magnetic operator separates the\neffect of the laser pulse on the magnetic system from other mag netic interactions. The effective\nmagnetic operator provides the equations of motion of magne tic vectors during the excitation by\nthe laser. We show that magnetization dynamics calculated w ith these equations is equivalent to\nmagnetization dynamics calculated with the time-dependen t Schr¨ odinger equation, which takes into\naccount the interaction of an electronic system with the ele ctric field of light. Wemodel and compare\nlaser-induced precessions of magnetic sublattices of an ea sy-plane and an easy-axis antiferromagnetic\nsystems. Usingthesemodels, weshowhowtheultrafast inver seFaradayeffectinducesanetmagnetic\nmoment in antiferromagnets and demonstrate that a crystal fi eld environment and the exchange\ninteraction play essential roles for laser-induced magnet ization dynamics even during the action of\na pump pulse. Using our approach, we show that light-induced precessions can start even during\nthe action of the pump pulse with a duration several tens time s shorter than the period of induced\nprecessions and affect the position of magnetic vectors afte r the action of the pump pulse.\nPACS numbers: 75.78.Jp,75.40.Gb,78.20.Ls\nI. INTRODUCTION\nUltrafast optical control of a magnetic state of a\nmedium is a rapidly developing field of research [1, 2].\nLaser-induced magnetization dynamics can take place on\na subpicosecond time scale providing the possibility to\novercomethe time limit of several picoseconds for preces-\nsional magnetic switching. Therefore, laser manipulation\ntechniques are extremely promising for the development\nofdataoperationdeviceswhichareseveralordersofmag-\nnitude faster than that available now. However, despite\nthe importance of subpicosecond laser-induced magneti-\nzation effects for technological applications, their origin\nis poorly understood.\nThe inverseFaradayeffect (IFE) that leadstononther-\nmal induction of magnetization by circularly-polarized\nlaser pulses [3, 4] is one of magneto-optical effects, which\ncan take place on a femtosecond time scale. The ul-\ntrafast inverse Faraday effect (UIFE) is particularly im-\nportant for potential applications in magnetic recording\nand magneto-optical devices, since it enables nonthermal\ncoherent control of magnetization dynamics at a subpi-\ncosecond time scale. It avoids problems caused by ma-\nterial heating, which limits a repetition frequency due\nto a required cooling time and a recording density due\nto heat diffusion. Therefore, the demonstration of the\nUIFE, in which magnetic oscillations in canted antifer-\nromagnet DyFeO 3were induced by circularly polarizedultrashortlaserpulses [5], motivated intensivetheoretical\n[6–18] and experimental [19–23] studies of this process.\nA significant progress in development of techniques of ul-\ntrafast spin control using femtosecond laser pulses based\non the UIFE was demonstrated in recent years [22, 24–\n33]. It has recently became possible to gain atomic- and\nspin-selective real-time insight into light-induced magne-\ntization dynamics during the excitation by a pump pulse\nemploying attosecond extreme ultraviolet pulses [34–36].\nA big effort is being performed at Free-Electron Laser\nFacilities such as Linear Coherent Light Source LCLS to\nenable attosecond x-ray imaging experiments [37]. Such\nattosecond x-ray pulses can provide a detailed insight\ninto real-time laser-driven electron and spin dynamics\n[38, 39]. These advances give rise to a demand to bet-\nter understand light-induced magnetization dynamics at\nultrashort time scales.\nEquations of motion for a magnetic moment are a\nstandard tool to describe magnetization dynamics. Usu-\nally, it is straightforward to derive magnetic equations\nof motion after the action of a pump pulse taking the\nnew laser-induced magnetic state as an initial condi-\ntion. Landau-Lifschitz-Bloch equation is usually applied\nfor a macroscopic description of magnetization dynam-\nics [1, 40–42], which can be derived from a microscopic\ndescription within the Heisenberg representation. How-\never, the ability to fully control magnetization dynamics\nin a material requires not only a description of magneti-\nzation dynamics after the excitation, but also an ability2\nto calculate evolution of a magnetic moment due to the\naction of a pump pulse depending on its parameters and\nmaterial properties. An inclusion of a coupling between\na magnetic system and a pump pulse into magnetic equa-\ntions of motion is necessary for this goal.\nThephenomenologicaldescriptionoftheIFE[3,4]sug-\ngests that the action of a circularly polarized laser pulse\nshould be considered as an effective magnetic field, which\nis proportional to the pulse intensity. However, it was\nshown by us theoretically [7–9] and demonstrated in sev-\neral experimental works [19, 20] that this description of\nthe IFE is not applicable at subpicosecond time scales,\nsinceitwasdevelopedforlaserpulseswithdurationmuch\nlonger than system relaxation times. Thus, a proper de-\nscription of this process at subpicosecond time scales is\nnecessaryforfurtherachievementsin the fieldofultrafast\nmagnetization dynamics control.\nA sudden approximation can be used as an alternative\nto the phenomenological description [43–46]. Within this\napproach,the effectofalaserpulseisconsideredasanac-\ntion of an ultrashort magnetic pulse, amplitude of which\nis proportionalto light intensity multiplied by the Verdet\nconstant. However, there are two disadvantages of this\napproach. First, if the action of a laser pulse is sub-\nstituted by a magnetic field, then a new magnetic state\nafter the excitation cannot be predicted accurately and,\nhence, thisapproachcannotbeappliedforadevelopment\nof a mechanism to control magnetization dynamics. The\nsecond problem is that the condition that a laser pulse\nduration is much shorter than a system’s oscillation pe-\nriod is not valid in many cases. For instance, it is not ap-\nplicable to the description of experiments demonstrating\nlight-induced terahertz precessions[24, 26, 27, 47], when\na magnetic oscillation with a period of one or several pi-\ncoseconds is induced by a laser pulse of several hundreds\nof femtoseconds duration.\nIn this article, we derive an effective magnetic opera-\ntor expressed in terms of total angular momentum oper-\nators that accurately describes magnetization dynamics\nat a subpicosecond time scale during the action of an ul-\ntrashort laser pulse via the UIFE. The time-dependent\nfunctions entering the operator depend on parameters of\na laser pulse and the coupling of the electric field of a\nlaser pulse to the electronic system of a material. They\nare nonzero during the action of a laser pulse and zero\nafterwards. The action of the effective magnetic opera-\ntor is separated from that of other magnetic operators\ndescribing fields acting on a magnetic system apart from\nlight, such as an external magnetic field, exchange inter-\nactionetc. Thus, after the action of the laser pulse, the\naction of the effective magnetic operator turns off and\nmagnetization dynamics persist due to a deviation of a\nmagnetic vector from its ground state.\nThe equations of motion of magnetic vectors are ob-\ntainedfromthecommutatorsoftotalangularmomentum\noperators with the effective magnetic operator and other\nmagneticoperatorsdescribingfields acting ona magnetic\nsystem. This way, we calculate magnetization dynamicsduring and after the action of light within the Heisenberg\nrepresentation. We show that magnetization dynamics\nobtained in the Heisenberg picture are equivalent to that\nobtained with the time-dependent Schr¨ odinger equation,\nwhich describes the perturbation of an electronic system\nbytheelectricfieldofalaserpulse. Thefirstadvantageof\nthe Heisenberg representation over the Schr¨ odinger rep-\nresentationisthatitshowshowtheopticalprocessaffects\ncomponents of the total angularmomentum individually.\nAnd, most importantly, the Heisenberg representation\nprovides a link for the derivation of the description of\nthe UIFE at a nanoscale [48].\nIn our previous studies, we described the effect of the\nspin-orbit coupling on the IFE and how light-induced\nelectronic transitions lead to the change of a spin state\n[7, 8]. In this article, we also consider the effects of the\nZeeman, exchange and crystal-field interactions, and in-\nclude their corresponding effective magnetic Hamiltoni-\nans into the derived equations of motion. We find that\nthe exchange interaction and, especially, the crystal-field\ninteraction can considerably influence the dynamics of\nmagnetic vectors even during the action of the pump\npulse. The approach derived in this article is able to\ndescribe the interplay between the deviation of magnetic\nvectors due to the action of the pump pulse and light-\ninduced magnetic precessions launched due to this de-\nviation. As we will show, this effect can be substantial\neven if the pump-pulse duration is several tens of times\nshorter than the period of precessions.\nThe article is organizedas follows. We derive the effec-\ntive magneticoperatoroperatorandthe equationsofmo-\ntion of magnetic moment due to the action of the UIFE\nin the Section I. We apply this formalism to describe the\ndynamics of a single spin system in an external mag-\nnetic field during and after the excitation by a circularly\npolarized laser pulse in Section II. We show that an ac-\ncurate calculation of the time evolution of a magnetic\nvector during the action of the pump pulse is necessary\neven if the period of the induced oscillations is several\ntens of times longer than the laser pulse duration. We\nderive equations of motion for magnetic vectors of a sys-\ntem consisting of two antiferromagnetically coupled sub-\nlattices during and after the action of an ultrashort laser\npulse in Section III. With the help of the equations of\nmotion, we describe the mechanism of the generation of\na non-zero magnetic moment and its precessions in com-\npensated antiferromagnets due to the UIFE. It is shown\nthat acrystalfieldenvironmentand exchangeandcrystal\nfield coupling have a strong effect on laser-induced mag-\nnetization dynamics even during the action of a pump\npulse.\nII. DERIVATION OF THE EFFECTIVE\nMAGNETIC OPERATOR\nThe UIFE leads to a nonthermal change of a mag-\nnetic state of a system by a circularly-polarized laser3\npulse via the stimulated Raman scattering process [4, 7–\n9]. Thereby, the laser pulse excites electron transitions\nin the system in such a way that the initial and the fi-\nnal states belong to the same ground state manifold, but\nare energetically separated by internal or external mag-\nnetic interactions [49, 50]. The magnetic state of this\nelectronic system is described by a spinor Ψ gwith com-\nponents, which are projections of the wave function of\nthe system on the eigenstates of the total angular mo-\nmentum. Since the UIFE leads to the change of a mag-\nnetic state of the electronic system, it must be possible\nto introduce an effective magnetic operator ˆHJ, which\ndescribes this effect. ˆHJmust act on the spinor Ψ gand\nfulfil the time-dependent Schr¨ odinger equation\niΨ′\ng= [ˆHm+ˆHJ]Ψg. (1)\nHereandthroughoutthisarticle,weusetheatomicunits.\nHmis the magnetic Hamiltonian, which includes all ex-\nternal and internal magnetic interactions acting on the\ntotal angular momentum apart from ˆHJ. If the operator\nˆHJis known, one can determine the equations of motion\nof projections Jx,JyandJzof the total angular momen-\ntum due to the action of the IFE using the Heisenberg\nrepresentation as\niJ′\nα=/angbracketleftig\nΨg/vextendsingle/vextendsingle/vextendsingle/bracketleftig\nˆJα,ˆHm+ˆHJ/bracketrightig/vextendsingle/vextendsingle/vextendsingleΨg/angbracketrightig\n, (2)\nwhereJα=i/angbracketleftig\nΨg/vextendsingle/vextendsingle/vextendsingleˆJα/vextendsingle/vextendsingle/vextendsingleΨg/angbracketrightig\n,αstays forx,yandz.\nIn order to derive ˆHJ, we take into account that the\ntime evolution of spinor Ψ gobtained with the time-\ndependent Schr¨ odinger equation in Eq. (1) should be\nequivalent to the one derived from the solution of time-\ndependent Schr¨ odinger equation for the wave function\n/tildewideΨ(t) of the electronic system\ni/tildewideΨ′(t) = (ˆH0−d·E)/tildewideΨ(t). (3)\nˆH0=/summationtext\nip2\ni+ˆV, wherepiis the momentum of an elec-\ntron, the summation is over all electrons in the system.\nˆVincludes the kinetic energy of nuclei, the interaction\nenergy between electrons and nuclei, mutual Coulomb\nenergy of the electrons and nuclei, and all internal and\nexternal magnetic interactions [7–9]. dis the dipole mo-\nment of the system. Eis the electric field of a laser pulse\nwith a frequency ω0\nE=nEp(t/T−r/(cT))sin(ω0t). (4)\nThe electronic system is assumed to have the spatial ex-\ntend much smaller than the wavelength λ0=c/ω0is con-\nsidered. Also it is assumed that λ0≪cT, thus the pulse\nspatial dependence is ignored. Eis the amplitude of the\nelectricfield, p(t/T)describesthetime-dependenceofthe\namplitude of the electric field. nis a unit vector perpen-\ndicular to the light propagation direction. Throughout\nthis article, we consider the action of a left-circularly po-\nlarized laser pulse propagating in the zdirection, which\ncorresponds to n= (nx+iny)/√\n2.The solution of the Eq. (3) can be represented by an\nexpansion in functions /tildewideΨnof order (1/c)n\n/tildewideΨ(t) =ˆU(/tildewideΨ0+/tildewideΨ1(t)+/tildewideΨ2(t)+...)\n=ˆU/parenleftbigg\n/tildewideΨ0−i/integraldisplayt\n−∞dt′ˆU−1ˆVˆU/tildewideΨ0 (5)\n−/integraldisplayt\n−∞dt′ˆU−1ˆVˆU/integraldisplayt′\n−∞dt′′ˆU−1ˆVˆU/tildewideΨ0+.../parenrightigg\n,\nwhere/tildewideΨ0=/tildewideΨ(0),ˆV=−A·/summationtext\nipi/c,Ais avectorpoten-\ntial, which is relatedto the electricfield by E=−˙A/c.ˆU\nis the time evolution operator, which fulfils the equation\niˆU′=ˆH0ˆU. The IFE experiments are typically done\nat frequencies corresponding to a material transparency\nregion, where the absorption is very weak. The interme-\ndiate states can be considered as virtually excited, and\nthe contribution of the first order wave function is not\ntaken into account. The stimulated Raman scattering is\ndescribedbythesecondorderwavefunction /tildewideΨ2(t). Thus,\nthe normalized spinor Ψ g(t) is obtained by\nΨg(t) =ˆU/parenleftbig\nΨ0+Ψ2(t)/parenrightbig\n/bardblΨ0+Ψ2(t)/bardbl, (6)\nwhere Ψ 0and Ψ 2(t) are the spinors associated with the\nwave functions /tildewideΨ0and/tildewideΨ2(t).\nThe spinor Ψ 0, which describes the magnetic state of\nthe electronic system before the action of light, can be\nrepresented as\nΨ0=\nP01\nP02\n...\nP0n\n=P01/vextendsingle/vextendsingle/vextendsingleJ,Jz=J/angbracketrightig\n+P02/vextendsingle/vextendsingle/vextendsingleJ,J−1/angbracketrightig\n+...,(7)\nwhereJis the total angular magnetic momentum, n=\n2J+1andP0kistheprojectionofΨ 0onthestate |J,Jz=\nJ+1−k/angbracketrightig\n./summationtext\nk|P0k|2= 1 is the normalization condition.\nΨ2(t) is a spinor with ntime-dependent components.\nEach component of this spinor is given by a transition\namplitude ofstimulated Ramanscatteringtoa finalstate\nwith a corresponding projection of Jz(see Ref. 7–9 for\ndetails). According to the selection rules for the stimu-\nlated Raman scattering process induced by a circularly\npolarized laser pulse propagating in the zdirection, a\ntransition from a state |J,Jz/angbracketrightback only to the same state\nis allowed. This means that the k-th component of Ψ 2(t)\nis a transition amplitude Tkdescribing transition from\nan initial state |J,J+1−k/angbracketrightto a final state |J,J+1−k/angbracketright.\nDue to the presence of the spin-orbit coupling, transition\namplitudes Tkare different for each k. Thus, the spinor\nΨ2(t) is not proportional to Ψ 0and describes a magnetic\nstate, which is different from the initial one.\nWe introduce time-dependent factors Ak(t) such that\nthek-th element of the spinor Ψ 0+ Ψ2(t) is equal to4\nAk(t)P0kand satisfy Ak(0) = 1. The factors Ak(t) are\ndetermined by Eq. (5) and depend on the electric field of\nthe laser pulse, selection rules and energies of system’s\nground and excited states, but does not depend on the\ninitial state of the system. Applying Eq. (6), Ψ g(t) can\nbe represented as\nΨg=ˆU\nN(t)\nA1(t)P01\n...\nAk(t)P0k\n...\n=\nP1(t)\n...\nPk(t)\n...\n,(8)\nwhereN(t) =/bardblΨ0+Ψ2(t)/bardblis the normalization factor.\nWederivedtheoperator ˆHJprovidingΨ gfromEq.(1),\nwhich is equivalent to the solution of Eq. (6) (see Ap-\npendix A and Ref. 9 for the details). We obtain that\nˆHJ=\n...\n−γa···iPaP∗\nb(νa−νb)···\n......\niP∗\naPb(νb−νa)\n...\n,\n(9)\nwherea= 1...nis the row number and b= 1...nis\nthe column number. Namely, the diagonal elements of\nthe operator are [ ˆHJ]aa=−γa(t), and the off-diagonal\nelements are [ ˆHJ]ab=iPa(t)Pb(t)∗(νa(t)−νb(t)).Pa(t)\nandPb(t) are the components of Ψ g, and\nνa(t) = Re(Ya(t)),\nγa(t) = Im(Ya(t)), (10)\nYa(t) = [ˆUA′]a/[ˆUA]a,\nwhereAisaspinorwithelements Aa(t)andA′isaspinor\nwith elements A′\na(t). [ˆUA′]aand [ˆUA]aare thea-th com-\nponents of the spinors ˆUA′andˆUA, correspondingly.\nOne can express the operator ˆHJvia operators ˆNab+\nandˆNab−and their expectation values, where the ele-\nments of these operators are\n(ˆNab+)ab= (ˆNab+)ba= 1,whereb≥a,\n(ˆNab−)ba= (ˆNab−)∗\nab=i,whereb>a,\nˆNa=ˆNaa+, (11)\n(ˆNa)aa= 1,\n(ˆNab±)mn= 0,ifm/negationslash=a, m/negationslash=b,n/negationslash=a,n/negationslash=borb<a.For example, if J= 3/2, these operators are\nˆN12+=\n0 1 0 0\n1 0 0 0\n0 0 0 0\n0 0 0 0\n,\nˆN12−=\n0−i0 0\ni0 0 0\n0 0 0 0\n0 0 0 0\n, (12)\nˆN1=\n1 0 0 0\n0 0 0 0\n0 0 0 0\n0 0 0 0\n.\nApplying that PaP∗\nb=/angbracketleftΨg|ˆNab+−iˆNab−|Ψg/angbracketright/2 and\nP∗\naPb=/angbracketleftΨg|ˆNab++iˆNab−|Ψg/angbracketright/2, the effective magnetic\noperator ˆHJcan be represented as\nˆHJ=−n/summationdisplay\naγaˆNa (13)\n+1\n2n/summationdisplay\na,b(νa−νb)/parenleftig\n/angbracketleftˆNab−/angbracketrightˆNab+−/angbracketleftˆNab+/angbracketrightˆNab−/parenrightig\n.\nThis representation allows deriving a convenient system\nof differential equations for the variables /angbracketleftˆNab±/angbracketright, which\nareconnectedtotheexpectationvaluesofthemomentum\noperators ˆJx,ˆJyandˆJzas will be shown below.\nThe equations of motion for the expectation values of\nˆNab±operators are given by [see Eq. (2)]\n/angbracketleftˆNab±/angbracketright′=−i/angbracketleftig/bracketleftig\nˆNab±,ˆHJ+ˆHm/bracketrightig/angbracketrightig\n(14)\nThe commutators −i/angbracketleftig/bracketleftig\nˆNab±,ˆHJ/bracketrightig/angbracketrightig\nare derived in Ap-\npendix A and result in\n/angbracketleftˆNab±/angbracketright′=/parenleftigg\n−2/summationdisplay\nkνk/angbracketleftˆNk/angbracketright+νa+νb/parenrightigg\n/angbracketleftˆNab±/angbracketright\n±(γa−γb)/angbracketleftˆNab∓/angbracketright−i/angbracketleftig/bracketleftig\nˆNab±,ˆHm/bracketrightig/angbracketrightig\n.(15)\nThe matrices representing total angular momentum\noperators ˆJx,ˆJyandˆJzcan be expressed by linear\ncombinations of ˆNab±operators. For example, ˆJx=√\n3ˆN12+/2+ˆN23++√\n3ˆN34+/2,ˆJy=√\n3ˆN12−/2+ˆN23−+√\n3ˆN34−/2 andˆJz= 3ˆN1/2+ˆN2/2−ˆN3/2−3ˆN4/2 for\nJ= 3/2. This way, the time evolutions of Jx,JyandJz\ncan be obtained from the equations of motion for /angbracketleftˆNab±/angbracketright\nas will be shown with the examples presented in the Sec-\ntions III and IV.\nThe advantage of our approach derived in the Heisen-\nberg representation is that one can include the dissipa-\ntion processes using the density matrix formalism and\nthe Redfield equation [51]\niρ′= [ˆHm+ˆHJ,ρ]−iˆΓ(ρ−ρeq).(16)5\nHere,His the total magnetic Hamiltonian, Hmis the\nmagneticHamiltonianintheabsenceofthelightfieldand\nˆHJis the effective magnetic Hamiltonian due to the ac-\ntionofthe pumppulsethatwederivedinourmanuscript.\nρ=|Ψg/angbracketright/angbracketleftΨg|is the time-dependent density matrix in the\nHilbert space of the magnetic system, ρeqis the density\nmatrix at the equilibrium. ˆΓ is the relaxation superop-\nerator. For example, it can be diagonal with the matrix\nelements being relaxation rates. The time evolution of\nthe magnetic vectors is then derived using the trace of ρ\nactingon correspondingmagneticoperators,e.g. Tr[ ρJx].\nWe do not consider the dissipation in the processes de-\nscribed in the following Sections, since its treatment is\nbeyond our study.\nIII. SINGLE SPIN DYNAMICS DUE AN\nEXTERNAL MAGNETIC FIELD AND UIFE\nA. Equations of motion\nIn this Section, we describe single-spin dynamics due\nto a joint action of the UIFE and an external magnetic\nfieldB. We assume that an external magnetic field Bis\naligned in the −xdirection and the gyromagnetic ratio\nis equal to 1 (see Fig. 1). The ground state is split into\ntwo states: spin parallel, |x−/angbracketright, and antiparallel, |x+/angbracketright,\nto the magnetic field with the corresponding energies\nǫx−=ǫ1s+B/2 andǫx+=ǫ1s−B/2, whereǫ1sis the\nground state energy in the absence of the magnetic field.\nThe system is initially in the lowest energy state with\nthe spin aligned in the + xdirection, which is described\nby the spinor Ψ 0=/parenleftig\n1/√\n2\n1/√\n2/parenrightig\n. A left-circularly polarized\nultrashort laser pulse propagating in the zdirection trig-\ngers Raman transitions via a virtual excited state 2 p,\nwhich is split due to the spin orbit coupling λ(L·S) and\nZeeman interaction µ((2S+L)·B) into six states (see\nFig. 2 and Appendix B). Here, µis the gyromagnetic ra-\ntio,λis the spin-orbit coupling constant and Lis orbital\nmomentum. We assume µ=−1/2 andλ= 20 meV. The\nUIFE leads to a change of a spin state, thereby the spin\ndeviatesfromitsinitialpositionandstartsprecessingdue\nto the magnetic field.\nThe total magnetic Hamiltonian ˆHtot\nm, which deter-\nmined spin dynamics, isthe sumofthe effective magnetic\noperator ˆHJand Hamiltonian describing the Zeeman in-\nteraction with the external magnetic field\nˆHtot\nm=ˆHJ−BxˆSx. (17)\nIn the case of S= 1/2, the spin operators are linear\ncombinations of the ˆNab±operators: ˆSx=ˆN12+/2,ˆSy=\nˆN12−/2,ˆSz= (ˆN1−ˆN2)/2,ˆS2= 3(ˆN1+ˆN2)/2. Thus,\nthe operator ˆHJcan be represented as\nˆHJ=f(t)/parenleftig\nSyˆSx−SxˆSy/parenrightig\n+g(t)ˆSz+h(t)ˆS2,(18)\nFIG. 1: Spin 1/2 in an external magnetic field\nwheref(t) = 2Re(Y2(t)−Y1(t)),g(t) = Im(Y2(t)−Y1(t))\nandh(t) =−2\n3Im(Y2(t) +Y1(t)),Sα=/angbracketleftˆSα/angbracketrightforα=\nx,y,z. The function Y1(t) is determined by the transi-\ntion amplitude of the Raman transitions from the state\n|1s,Sz= +1\n2/angbracketrightback to the state |1s,Sz= +1\n2/angbracketrightandY2(t)\nis determined by the transition amplitude the Raman\ntransitions from the state |1s,Sz=−1\n2/angbracketrightto the state\n|1s,Sz=−1\n2/angbracketright[see Eq. (10)].\nThe third term of ˆHJin Eq. (18) determined by func-\ntionh(t) is proportional to the identity matrix and does\nnot influence the spin. The actionofthe first and the sec-\nond terms of ˆHJ, which are determined by the functions\nf(t) andg(t), lead to spin rotation. Both functions f(t)\nandg(t) are given by the difference between Y1(t) and\nY2(t). Thus, the larger the difference between the transi-\ntion amplitudes for spin-up and spin-down states is, the\nlarger functions f(t) andg(t) are, and the more effective\nthe spin is rotated by light [see Eq. (10)]. If transition\namplitudes for different spin components are equal, then\nY1(t) =Y2(t),ˆHJis proportional to the identity matrix,\nand no rotation of spin by light is possible.\nThe equations of motion for the spin vector can be\nderived either from the general equation (15) or from the\nrelationS′\nα=−i/angbracketleft[ˆSα,ˆHtot\nm]/angbracketrightresulting in\nS′\nx=−f(t)SxSz−g(t)Sy,\nS′\ny=−f(t)SySz+g(t)Sx+BSz,(19)\nS′\nz=−f(t)(S2\nx+S2\ny)−BSy.\nIt follows from these equations that the first term of the\neffective operator ˆHJdetermined by the function f(t)\nresults in a quadratic effect on a spin and the second\nterm determined by g(t) - in a linear one. Both terms\ndescribe rotation of a spin around some axis with a time-\ndependent frequency. The first term describes nonlinear\nrotation of spin around an axis, which is perpendicular\nto the light propagation direction and to the initial di-\nrection of spin. The second part determined by g(t) de-\nscribes rotation of the spin around the zaxis. This is\nin agreement to a conclusion that a spin is rotated only\naround the light propagation direction via the UIFE, if\na laser pulse has a frequency far from a resonance[8].\nIn this case, both functions f(t) andg(t) are relatively\nsmall since the transition amplitudes are low and, conse-\nquently, the quadratic effect is stronger suppressed than\nthe linear one.\nThe functions f(t) andg(t) can be calculated as fol-\nlows. According to Eq. (10), the functions Y1(t) and6\nFIG. 2: The energy level scheme of the single-spin sys-\ntem.\nY2(t) depend on the spinor A(t) and the time evolution\noperator ˆU. The time evolution operator, which fulfils\nthe relation iˆU′=ˆHmˆU, isˆU=e−iBˆSxt. The spinor\nA(t) =/parenleftig\n1+ψ2↑(t)\n1+ψ2↓(t)/parenrightig\nis determined by the transition am-\nplitudes of the Raman scattering process for the spin-up\nstateand spin-downstate, ψ2↑(t) andψ2↓(t), correspond-\ningly. According to Eq. (5),\nA(t) =/parenleftbigg\n1\n1/parenrightbigg\n+/parenleftbiggE\nω0/parenrightbigg2/integraldisplayt\n−∞dt′ˆU(t′)/parenleftbigg/summationtext\nj|d↑j|2Gj(t′)/summationtext\nj|d↓j|2Gj(t′)/parenrightbigg\n,\n(20)\nwhere\nGj(t′) =e−i∆ω0jt′F(t′)/integraldisplayt′\n−∞dt′′ei∆ω0jt′′e−iBt′′/2F(t′′),\nF(t) =p(t/T)cos(ω0t). (21)\nHere, the summation is over intermediate states j.\n∆ω0j=ǫ2p,j−ǫ1s,ǫ2p,jare the energies of the inter-\nmediate states. d↑jandd↓jare the dipole matrix ele-\nments of the transitions from the states |1s,Sz= 1/2/angbracketright\nand|1s,Sz=−1/2/angbracketrightto the state j. The dipole matrix\nelements are calculated in Appendix B. It is also shown\nin Appendix B that a spin state can be changed via the\nRaman scattering process only if the spin-orbit coupling\nis nonzero even if an external magnetic field is nonzero.\nAs shown in the next Subsection, the functions f(t)\nandg(t)arenonzeroduringtheactionofalaserpulseand\nsmoothly become zero at time τp, when the excitation is\nfinished. Thus, duringtheactionofalaserpulse, Eq.(19)\ndescribes the spin motion due to both laser excitation\nand the external magnetic field. The terms in Eq. (19),\nwhich describe spin motion due to the coupling to the\nelectromagnetic field, are smoothly turning off while the\nexcitation is finishing, and spin moves only due to an\nexternal magnetic field after the action of a laser pulse.\nThus, we have obtained a system of differential equations\nwith terms separately describing spin motion due to the\nZeeman interaction and due to the UIFE, which shows\nthe interplay of the interactions leading to the rotation\nof spin.0.450.4750.50.5250.55Sx\n-0.2-0.100.10.2Sy\n-0.2-0.100.10.2\n-0.2 0 0.2 0.4 0.6Sz\nTime, psB= 0 T\nB= 7 T\nB= 20 T\nFIG. 3: Time evolution of the spin-vector components\ndue to the excitation at different applied magnetic fields\nin thexdirection. The gray line represents the time\nevolution of the electric field amplitude. Laser pulse du-\nration is 117 fs.\nB. Time evolution of the spin vector\nWe calculateandcomparelaser-inducedspin dynamics\nin the presence of an external magnetic field with a mag-\nnitude of 7 T and of 20 T. Although the chosen magnetic\nfieldmagnitudesareratherhigh, theyarereasonablefora\ngoalofcomparisontotheexperimentsstudyingtheUIFE\nin the presence of an applied external magnetic field. Al-\nthough external magnetic fields with magnitudes ofup to\n0.5 T are usually applied in experiments, materials un-\nder consideration obtain gyromagnetic factor about ten\ntimes higher than that of a single electron spin (see e.\ng.Refs. 19, 46, 52, and 53). Thus, the chosen magni-\ntudes of the magnetic fields result in Larmor precession\nfrequencies comparable to the ones in experiments.\nWe consider the action of a left-circularly polar-\nized Gaussian shaped laser pulse with p(t/T) =\nexp(−t2/T2)/√\nπ3and fluence Efl≈2 mJ/cm2. We as-\nsumeT= 100fs, whichcorrespondstothepulseduration\nof 117 fs at FWHM of the intensity and the bandwidth\nof 15 meV. The laser pulse central frequency ω0is equal\nto (ǫ2p−ǫ1s), whereǫ2pis the energy of the unsplit 2 p\nstate, i.e. the energy of the state in the absence of the\nZeeman and spin-orbit interactions. The Larmor preces-\nsion periods of the magnetic fields of 7 T and 20 T are\napproximately 5 ps and 1.7 ps, correspondingly (see Ta-\nble I). The ratios of the induced precession periods to\nthe pulse duration are 40 and 15, respectively.\nThe time evolutions of the spin-vector components Sx,7\nBTLTL/Tdr\n7 T5 ps 40\n20 T1.7 ps 15\nTABLE I: Larmor periods TLand their ratios to the\nlaser pulse duration, TL/Tdr, for the chosen magnetic\nfields.\nSyandSzare shown on Fig. (3) for B= 0, 7 T and\n20 T.Sxcomponent is influenced only due to the in-\nteraction with light and its dynamics does not depend\non the magnetic field [see Eq. (19)]. But comparing the\ntime evolutions of the spin components SyandSzat dif-\nferent magnetic fields, one can see that they noticeably\ndepend on the magnitude of the magnetic field even dur-\ning the interaction with light. The values of SyandSz\nat timeτp= 200 fs, when the excitation is negligible, are\nverystronglyaffected bythe magneticfield. This demon-\nstrates that Larmor precession has a considerable impact\non laser-inducedspin dynamics alreadyduring the action\nof the laser pulse.\nA sudden approximation suggests that the laser exci-\ntation is immediate and thus the spin oscillation in an\nexternal magnetic field during the excitation can be ig-\nnored. Thus, one may assume that spin Larmor preces-\nsion starts after the excitation, namely, at time τp= 200\nfs. The spin state at τpcalculated without an applied\nmagnetic field would be taken as the initial condition for\nthe precession. Applying this assumption and compar-\ning it to the results obtained with Eq. (19), we obtain\nthat the phase disagreement between spin dynamics cal-\nculated with and without this assumption are 14◦in the\ncase ofB= 7 T (the corresponding Larmor frequency\nof 5 ps) and 47◦in the case of B= 20 T (the corre-\nsponding Larmor frequency of 1.7 ps). Thus, the sudden\napproximation does not work correctly even if the oscil-\nlation period is about fifty times larger than the pulse\nduration. This statement is confirmed by the observa-\ntion of Satoh et al.in Ref. 24 that models, which ignore\nthe time-dependency of a laser pulse, are not sufficient\nto describe the initial state of a magnetic precession.\nFig. 4 shows the 3D picture of spin-vector trajectories\nduring and after the excitation. This Figure additionally\ndemonstrates how an applied magnetic field influences\nthe spin dynamics. When the magnetic field is zero, the\nspin moves during the action of the laser pulse almost al-\nwaysinthe xyplane. Butthe spintrajectoriesduringthe\nexcitation in the presence of the external magnetic fields\nare rotated around the zaxis relatively to the trajectory\natazeromagneticfield. Furthermore, thespintrajectory\nstartstofollowthat ofthe Larmoroscillationevenduring\nthe action of the laser pulse at B= 20 T corresponding\nto the Larmor frequency of 1.7 ps (see Fig. 4f).\nFigure 4 compares the fields acting on the spin due\nto the magnetic field, B, and the interaction with light,\nf(t) andg(t) [see Eq. (19)]. We obtain that f(t) andg(t)xyz(f)(d)(b)\n-4-2024\n-0.2 -0.1 0 0.1 0.2 0.3Fields, meV\nTime, ps(e)-4-2024Fields, meV(c)-4-2024Fields, meV(a)f(t)\ng(t)\nB\nFIG. 4: Left column: fields (in energy units) acting on\nthe spin during the excitation. Right column: the corre-\nsponding time evolution of the spin vector on the Bloch\nsphere. The black arrow shows the initial alignment of\nspin. The dotted lines on the left plots show the dynam-\nics of the spin during the excitation, i. e.att <200 fs,\nthe continuous lines show the dynamics of the spin after\nthe excitation at t>200 fs. The external magnetic fields\nare (a), (b) B= 0. (c), (d) B= 7 T. (e), (f) B= 20 T.\nduring the excitation are of the same order of magnitude\nas that of the chosen magnetic fields. Although f(t) and\ng(t)dependonthemagneticfieldviatheirdependenceon\nthe operator ˆU=e−iˆSxB[see Eq. (10)], this dependence\nis negligible even at the magnetic field of 20 T. Since the\nbandwidth of the laser pulse, which is equal to 15 meV\nin the considered case, is much larger than energy split-\ntings induced by the Zeeman interaction, the transition\namplitudes are almost not affected by the Zeeman inter-\naction. Thus, the dependence of the functions f(t) and\ng(t), which are determined by the transition amplitudes,\non the magnetic field is negligible.\nWe have shown that the action of an external mag-\nnetic field can strongly affect spin dynamics and lead to\nanaccumulationofaphaseoftheLarmorprecessioneven8\nduring the excitation by light. Our simple model demon-\nstrates that an exact calculation of the time evolution of\nthe magnetic moment during the action of a laser pulse\nis necessary even if the pulse duration is several tens of\ntimes shorter than the period of an induced magnetic\nprecession.\nIV. DYNAMICS OF AN ANTIFERROMAGNET\nIn this Section, we apply the Heisenberg picture to\ndescribe magnetization dynamics of two types of antifer-\nromagnets, an easy-plane and an easy-axis antiferromag-\nnet. We demonstrate the mechanism of the excitation\nof magnetic precessions and induction of a magnetic mo-\nment by the UIFE. A study of antiferromagnet dynam-\nics is especially interesting, since magnetic resonances of\nsuch a system would not be observed, if a circularly-\npolarizedlaserpulse couldbe treated asan effectivemag-\nnetic field. If it were true, magnetic precessions would be\npossible only in a material with a net magnetic moment,\nso that the effective magnetic field can produce a torque\ntoit, whichwouldresultinmagnetizationprecession[54].\nHowever, light-induced terahertz magnetic precessions in\nantiferromagnets have been observed [24, 26].\nThe phenomenological model of Ref. 55 indeed pre-\ndicted the possibility of the UIFE in an antiferromagnet.\nHowever, there are several problems to apply this ap-\nproach to the interpretation of the experiments demon-\nstrating light-induced terahertz magnetic precessions in\nantiferromagnets. First, the model is based on an as-\nsumption that the duration of a laser pulse is much\nshorter than the period of an induced spin precession.\nThis approximationis not applicable for this experiment,\nsince the ratio of the induced precessions periods to the\npulse duration is about ten. Second, the model considers\nthe light excitation as an ultrashort magnetic pulse and\ndoes not provide the information about the dependence\nof the effect on laser-pulse and material parameters.\nThe method introduced by us, first, does not make any\nassumptions on a pulse duration and, thus, can be ap-\nplied to describe subpicosecond magnetization dynamics.\nSecond, the technique involves the analysis of material\nproperties and thus provides details about the depen-\ndence of the effect on a material structure.\nWe apply several assumptions to treat the antiferro-\nmagnetic systems. However, our technique is not re-\nstrictedtotheseassumptions, andothermagneticmodels\ncan be chosen depending on a considered magnetic sys-\ntem. The model, which we use, is an example that can\nbe adjusted to a realistic magnetic system.\nA. Antiferromagnetic systems\nWe treat the exchange interaction in the framework of\nthe Weiss mean field theory [56]. According to this the-\nory, the quantum fluctuations can be neglected, and theexchange interaction between any two atoms is consid-\nered as the Zeeman interaction of a spin of each atom\nwith a magnetic field, which is the spin average of the\nother atom. This means that the Hamiltonian ˆHex12=\nJex0ˆS1·ˆS2issubstituted by ˆHex=Jex0(ˆS1/angbracketleftS2/angbracketright+/angbracketleftS1/angbracketrightˆS2).\nWith the assumption that the exchange interaction only\nwith the next neighbor atoms is relevant, the Hamilto-\nnian acting on an atom iis expressed as\nˆHex(i)=ZJex0/angbracketleftSnn/angbracketrightˆSi, (22)\nwhereZis the number of the neighboring atoms and\n/angbracketleftSnn/angbracketrightisthe averagespin ofanext neighboratom. Apply-\ning that the magnetic moment ofan atom in the presence\nofthespin-orbitcouplingisproportionalto gJJ=L+2S,\nwheregJis the Land´ efactorand J=L+S, the exchange\ninteraction acting on atom ican be expressed as\nˆHex(i)=ZJex0(gL−1)2/angbracketleftJnn/angbracketrightˆJi. (23)\nThe approximation is valid, when the fluctuations of the\neffective magnetic field Z/angbracketleftJnn/angbracketrightare small, which is true,\nwhen each spin has many nearest neighbors.\nWe consider a system consisting of two equal sublat-\ntices coupled antiferromagnetically ( Jex0>0). Every\natom belonging to the sublattice 1 is surrounded by Z\natoms belonging to the sublattice 2 and vice versa. The\nexchange interaction acting on atoms belonging to the\nsublattices 1 and 2 can be written in the framework of\nthe Weiss mean field theory as\nˆH(1)\nex=Jex/parenleftig\nJx2ˆJx1+Jy2ˆJy1+Jz2ˆJz1/parenrightig\n,(24)\nˆH(2)\nex=Jex/parenleftig\nJx1ˆJx2+Jy1ˆJy2+Jz1ˆJz2/parenrightig\n,\nwhereJex=ZJex0(gL−1)2.\nWe treat the crystal field environment in the frame-\nwork of the crystal field theory. Two types of uniax-\nial crystal fields are considered: one with the symme-\ntry along the pulse propagation direction, z, and one\nin thexdirection, perpendicular to the pulse propaga-\ntion direction. The spin-orbit coupling is assumed to\nbe much larger than the crystal field, and the crystal\nfield Hamiltonians can be expressed via the total an-\ngular momentum operators. The crystal fields are de-\nscribed by Hamiltonians ˆHcrz= ∆z/parenleftig\n3ˆJ2\nz1,z2−ˆJ2\n1,2/parenrightig\nand\nˆHcrx= ∆x/parenleftig\n3ˆJ2\nx1,x2−ˆJ2\n1,2/parenrightig\n, correspondingly. This as-\nsumption and the treatment of the crystal field are ap-\nplicable to rare-earth-based magnetic materials [56] that\nare often used in magneto-optical experiments.\nThe magnetic Hamiltonian of the antiferromagnetic\nsystem in the case of the two types of crystal fields are\nˆHm=ˆH(1)\nex+ˆH(2)\nex+ (25)\n+∆z(x)/parenleftig\n3ˆJ2\nz1(x1)−ˆJ2\n1/parenrightig\n+∆z(x)/parenleftig\n3ˆJ2\nz2(x2)−ˆJ2\n2/parenrightig\n.\nThe alignment of the magnetic vectors in the ground\nstate are determined by the sign of the crystal field con-\nstants ∆z,x[56]. We chose the sign of both constants,9\n∆zand ∆xin such a way that in both cases the align-\nment of the magnetic vectors perpendicular to the pulse\npropagationdirection, z,isenergeticallyfavorable. Thus,\nwe choose ∆ z>0, which results in the xyplane being\nthe easy plane (the zaxis becomes energetically unfavor-\nable), and ∆ x<0, which results in the xbeing the easy\naxis.\nThus, in the ground-state, the magnetic vectors are\naligned along the xdirection in the case of the crystal\nfieldˆHcrx, and are aligned along some direction in the\nxy-plane, which we define as the xaxis, in the case of the\ncrystal field ˆHcrz. Therefore, in both cases, the energy\nof the system is the lowest, when the absolute values\nofJx1andJx2are the largest, but the vectors M1and\nM2are antiparallel. We assume that the ground state is\ncharacterized by the term J= 3/2, therefore, the spinors\nin both cases have initially the form\nΨ(1)\n0=\nc\nd\nd\nc\n,Ψ(2)\n0=\nc\n−d\nd\n−c\n, (26)\nIm(c) = Im(d) = 0, c>0, d>0.\nThese spinors correspond to Jx1=−Jx2,Jy1,y2=\nJz1,z2= 0 (see Appendix C). In the case of ˆHcrx,c=\n1/(2√\n2),d=√\n3/(2√\n2) providing the functions Ψ(1,2)\n0,\nwhich are the eigenfunctions of the operators ˆJx1,x2with\nthe corresponding expectation values Jx1= 3/2 and\nJx2=−3/2. In the case of ˆHcrz, the factors cand\nddepend on the exchange interaction and the crystal\nfield. The crystal field interaction ˆHcrzleads to a partial\nquenching of the total magnetic moment, and the ex-\npectation values of the ˆJx1,x2operators are smaller than\n±3/2 (see Appendix C and Ref. 9).\nWe assume, that the laser-induced Raman transitions\nof the atoms belonging to each sublattice involve excited\nstates characterized by a term with J= 5/2. Other ex-\ncitedstates, e.g.withJ= 3/2and1/2,areassumedtobe\nenergetically inaccessible for the applied laser pulse. It is\nalso assumed that the Hamiltonian for the excited state\nis simply ∆ ze/parenleftig\n3ˆJ2\nz1,z2−ˆJ2\n1,2/parenrightig\nor ∆xe/parenleftig\n3ˆJ2\nx1,x2−ˆJ2\n1,2/parenrightig\nand that the effect of the exchange interaction between\nthe sublattices on the excited state is negligible. Thus,\nJ= 5/2 term is splitted into three doubly degenerate\nlevels. The crystal field constants ∆ ze(xe)depend on the\norbital radius and are not necessary equal to ∆ x,z. We\ntake crystal field constants for the excited states ∆ ze= 3\nmeV and ∆ xe=−3 meV.\nIt is assumed that all atoms belonging to the same\nsublatticeareexcitedcoherentlybythelaserpulse. Thus,\nthe dynamics of all atoms belonging to a same sublattice\ncan be simulated by one system. Therefore, we describe\nthe dynamics of the antiferromagnets by considering the\ndynamics of the interacting sublattices 1 and 2.B. Time-dependent functions\nThe functions entering the operators ˆH(1)\nJandˆH(2)\nJare\ndetermined by dipole matrix elements involved in the\nlight-induced electronic transitions between electronic\nstates. Calculating the matrix elements, we take into\naccount that the character of these electronic states is\naffected by crystal-field and magnetic interactions as de-\nscribed in Appendix C1.\nThe functions ν(1,2)\naandγ(1,2)\naentering the operators\nˆH(1)\nJandˆH(2)\nJdepend on the time evolution operators\nˆU(1,2), which are defined by the equation iˆU(1,2)′=\n[ˆH(1,2)\nex+∆z(x)(3ˆJ2\nz1,2(x1,2)−ˆJ2\n1,2)]ˆU(1,2), correspondingly.\nSince the Hamiltonians ˆH(1)\nexandˆH(2)\nexare not equal, the\ntime evolution operators ˆU(1)andˆU(2)are not equal\nas well. However, the functions ν(1,2)\naandγ(1,2)\naare\nindeed equal for both systems, ν(1)\na=ν(2)\na=νaand\nγ(1)\na=γ(2)\na=γa, due to the symmetry considerations\n(see Appendix C and Ref. 9). Thus, the functions νaand\nγacan be calculated only for the sublattice 1.\nThese functions are νa= Re(Ya) andγa= Im(Ya),\nwhereYa= [ˆU(1)A′]a/[ˆU(1)A]a. Thea-th element of A\nisAa= 1−Ca/P0a, ifP0a/negationslash= 0, otherwise Aa= 0. Here,\nP0ais thea-th element of Ψ(1)\n0andCais thea-th element\nof the spinor\nC(t) =E2|d0|2/bracketleftigg/integraldisplayt\n−∞dt′F(t′)/parenleftig\nˆU(1)/parenrightig−1\n(t′)×(27)\nˆDTˆUe(t′)/integraldisplayt′\n−∞dt′F(t′′)′ˆUe(t′′)ˆDˆU(1)(t′′)/bracketrightigg\nc\nd\nd\nc\n,\nwhere the operator in the squared brackets acts on the\ninitial state vector of the sublattice 1, Ψ(1)\n0.ˆUe=\nexp[−i∆ze(xe)(3ˆJ2\nz1(x1)−ˆJ2\n1)t] is the time evolution op-\nerator related to the Hamiltonian acting on the excited\nstate andF(t) is defined in Eq. (21). See Appendix C2\nand Ref. 9 for the details of the derivation and the defi-\nnition of ˆD.\nNote that the operator ˆH(1)\nex, which determines ˆU(1),\nis time-dependent, since it depends on expectation val-\nues ofˆJx2,ˆJy2andˆJz2and, in general, this dependence\nshould be taken into account. However, we obtain that\nthe variation of ˆH(1)\nex(t) in time, |ˆH(1)\nex(t)−ˆH(1)\nex(0)|, is\nmuch smaller than the laser pulse bandwidth [9]. Ac-\ncording to our calculations, the dependence of νaandγa\non the time evolution of ˆH(1)\nex(t) is negligible, and the\nthe time evolution operator can be written as ˆU(1)(t) =\nexp[−i(ˆH(1)\nex(0)+∆z(x)(3ˆJ2\nz1,2(x1,2)−ˆJ2\n1,2))t]. The calcu-\nlation ofνaandγacan be even simplified further, if the\npump laser pulse bandwidth ∆ ωswis much larger than\nthe splitting of the ground state manifold. As shown in\nSection IIIB the dependence of νaandγaon the time\nevolution operator ˆUcan be neglected in this case.10\nAs discussed in Section IIIA, the diversity of the el-\nements of the vector Ais a necessary condition for the\nIFE. In the considered case, all four elements of Aare\ndifferent. As in the previous example, this is due to\nthe spin-orbit coupling, which leads to dipole matrix el-\nements of the Raman transitions from the states with\ndifferent projections of the total angular momentum on\nthe pulse propagation direction being different.\nC. Equations of motion\nThe dynamics of the magnetic vectors of the sublat-\nticesM1= (Jx1,Jy1,Jz1) andM2= (Jx2,Jy2,Jz2), can\nbe derived from the dynamics of the expectation values\nof the operators ˆN(1)\nab±andˆN(2)\nab±[see Eq. (12)], which act\nin the Hilbert space of spinorscorresponding to the sub-\nlattice 1 and 2, correspondingly. The dynamics of /angbracketleftˆN(1)\nab±/angbracketright\nand/angbracketleftˆN(2)\nab±/angbracketrightcan be derived from the equations of motion\n(15), which must be solved for /angbracketleftN(1)\nab±/angbracketrightand/angbracketleftN(2)\nab±/angbracketrightsimul-\ntaneously, since the systems 1 and 2 are coupled via the\nexchange interaction term. The action of a laser pulse on\nthe sublattice 1 and 2 is described by operators ˆH(1)\nJand\nˆH(2)\nJ,correspondingly. Notethatsincethemeanfieldthe-\nory is applied and quantum fluctuations are ignored, the\noperators ˆN(1)\nab±,and, consequently, the operators ˆJx1,ˆJy1\nandˆJz1commute with operators ˆH(2)\nex, ∆z/parenleftig\n3ˆJ2\nz2−J2\n2/parenrightig\n,\nˆH(2)\nJ. Correspondingly, ˆN(2)\nab±,ˆJx2,ˆJy2andˆJz2commute\nwithˆH(1)\nex, ∆z/parenleftig\n3ˆJ2\nz1−J2\n1/parenrightig\n,ˆH(1)\nJ.\nWe consider the equations of motion for the functions\nmab±(t) andlab±(t), which are defined as [cf. Eq. (C16)]\nmab±(t) =/angbracketleftˆmab±/angbracketright=papb/angbracketleftig\nˆN(1)\nab±+ˆN(2)\nab±/angbracketrightig\n,ifb>a,\nlab±(t) =/angbracketleftˆlab±/angbracketright=papb/angbracketleftig\nˆN(1)\nab±−ˆN(2)\nab±/angbracketrightig\n,ifb>a,(28)\nma(t) =maa+(t) =/angbracketleftˆma/angbracketright=/angbracketleftˆmaa+/angbracketright=/angbracketleftig\nˆN(1)\na+ˆN(2)\na/angbracketrightig\n,\nla(t) =laa+(t) =/angbracketleftˆla/angbracketright=/angbracketleftˆlaa+/angbracketright=/angbracketleftig\nˆN(1)\na−ˆN(2)\na/angbracketrightig\n,\nwhereaandbare integers between 1 and 4, p1=p4=√\n3/2 andp2=p3= 1. This way, it is convenient to\ntake into account symmetries of the antiferromagnetic\nsystems.\nAdditionally, instead of vectors M1andM2, we con-\nsider the dynamics of vectors M=M1+M2and\nL=M1−M2, which are proportional to ferromagnetic\nand antiferromagnetic vectors of the antiferromagnets.\nInstead of operators ˆJα, (αstays forx,yorz) we use\nˆMα=ˆJα1+ˆJα2andˆLα=ˆJα1−ˆJα2, whichareconnectedto ˆmab±andˆlab±via the relations\nˆMx= ˆm12++ ˆm23++ ˆm34+,\nˆMy= ˆm12−+ ˆm23−+ ˆm34−,\nˆMz=3\n2ˆm1+1\n2ˆm2−1\n2ˆm3−3\n2ˆm4,(29)\nˆLx=ˆl12++ˆl23++ˆl34+,\nˆLy=ˆl12−+ˆl23−+ˆl34−,\nˆLz=3\n2ˆl1+1\n2ˆl2−1\n2ˆl3−3\n2ˆl4.\nIn the Heisenberg picture, the equations of motion of\nthe functions mab±andlab±are given by\nim′\nab±=/angbracketleftig/bracketleftig\nˆmab±,ˆHm+ˆHJ/bracketrightig/angbracketrightig\n,\nil′\nab±=/angbracketleft[ˆlab±,ˆHm+ˆHJ]/angbracketright, (30)\nˆHJ=ˆH(1)\nJ+ˆH(2)\nJ.\nThere are sixteen functions mab±and sixteen functions\nlab±. ItisshowninAppendixCthatsixteenofthethirty-\ntwo functions always remain zero, namely, m12±(t) =\n0,m23±(t) = 0,m34±(t) = 0,m14±(t) = 0,l13±(t) =\n0l24±(t) = 0 and l1,2,3,4(t) = 0. Applying this result\ntogether with Eq. (15), we obtain\nm′\nab±=/parenleftigg\n−4/summationdisplay\nk=1νkmk+νa+νb/parenrightigg\nmab±\n±(γa−γb)mab∓−i/angbracketleftig/bracketleftig\nˆmab±,ˆHm/bracketrightig/angbracketrightig\n,(31)\nl′\nab±=/parenleftigg\n−4/summationdisplay\nk=1νkmk+νa+νb/parenrightigg\nlab±\n±(γa−γb)lab∓−i/angbracketleftig/bracketleftig\nˆlab±,ˆHm/bracketrightig/angbracketrightig\n.\nThe commutators of ˆ mab±andˆlab±with the magnetic\nHamiltonian ˆHmare given in Table II in Appendix C.\nCombining Eqs. (29) and (31), we obtain the following\nequations of motion for the vectors MandL\nMx=0, My= 0, Lz= 0,\nL′\nx=F0Lx+gLy+Fxy(l12+,l34+)+Gxy(l12−,l34−)\n−i/bracketleftig\nˆLx,ˆHm/bracketrightig\n,\nL′\ny=F0Ly−gLx+Fxy(l12−,l34−)−Gxy(l12+,l34+)\n−i/bracketleftig\nˆLy,ˆHm/bracketrightig\n, (32)\nM′\nz=F0Mz+Fz−i/bracketleftig\nˆMz,ˆHm/bracketrightig\n,11\nwhere\ng(t) =γ2(t)−γ3(t),\nF0(t) =−4/summationdisplay\naνa(t)ma(t)+ν2(t)+ν3(t),\nFxy(t)(l12±,l34±) = (ν1(t)−ν3(t))l12±(t)\n+(ν4(t)−ν2(t))l34±(t), (33)\nGxy(t)(l12±,l34±) = (γ1(t)−2γ2(t)+γ3(t))l12±(t)\n+(−γ2(t)+2γ3(t)−γ4(t))l34±(t),\nFz(t) =ν2(t)−ν3(t)\n2+(3ν1(t)−2ν2(t)−ν3(t))m1(t)\n+(ν2(t)+2ν3(t)−3ν4(t))m4(t).\nAnalogously to the system of differential equations in\nEq. (19) describing the dynamics of the single-spin sys-\ntem, Eq. (32) contains a linear term determined by the\nfunctiong(t) describing a rotation of the magnetic vec-\ntors around the zaxis. Also, analogously to the single-\nspin system, the terms determined by the function F0\ndescribe arotationaroundthe yaxis, whichis perpendic-\nular to the light propagation direction and to the initial\nalignment of the magnetic vectors. These terms are also\nnonlinear, since they depend on the variables ma. The\ntermsFxy,GxyandFzdo not appear in the equations\nfor the single-spin system.\nThe set of the first-order differential equations in\nEq. (32) is not sufficient to obtain the time evolution\nofMz,LxandLz, since apart from these variables, the\nfunctionsmab±andlab±alsoenterthesedifferentialequa-\ntions. Thus,incontrasttothesingle-spinsystem,itisnot\npossible to describe dynamics of a system with the total\nangular momentum J= 3/2 induced by the UIFE with\ndifferentialequations,whichincludeonly Jx,JyandJzto\na first order (or Mx,y,zandLx,y,zin the considered case).\nEven in the absence of the optical excitation, the dynam-\nics of the considered antiferromagneticsystems could not\nbe described by first-order differential equations, which\ninclude solely Mx,y,zandLx,y,zvariables, due to the\npresence of the crystal field. Thus, rather than apply-\ning Eq. (32), it is more convenient to solve the system of\ndifferential equations in Eq. (31) for variables lab±and\nmab±and to derive the time evolutions of vectors Mand\nLwith Eq. (29). In our case, it is sufficient to solve a sys-\ntem of fifteen first order differential equations, which in-\nvolvesfifteen variables l12±,l23±,l34±,l14±,m13±,m24±,\nm1,m2,m3. The sixteenth variable m4is derived from\nthe constrain/summationtext4\nama=/summationtext4\na/angbracketleftˆN(1)\na/angbracketright+/summationtext4\na/angbracketleftˆN(2)\na/angbracketright= 2.\nD. Time evolution of the magnetic vectors\nAccording to Eq. (32), Mx= 0,My= 0 andLz=\n0. Thus, the action of the UIFE makes the magnetic\nvectors of the sublattices M1andM2deviate from their\nequilibrium positions in such a way that their xandy\ncomponentsareoppositeand zcomponentsareequal[seeM1M2xyz\n(a)M1M2\n(b)\nM1M2\n(c)M1 M2\n(d)\nFIG. 5: (a) The initial alignment of the magnetic vec-\ntors. (b) The circular mode due to the exchange inter-\naction. The elliptical modes due to the crystal field with\nthe symmetry along (c) zaxis and (d) xaxis.\nFig. 5]. Since the magnetic vectors are deviated from\ntheir equilibrium positions, precession modes due to the\nexchange interaction and the crystal field are evoked in\nthe antiferromagnetic system as shown in Fig. 5(b)-(d).\nThe exchange interaction leads to the circular rotation\nof the magnetic vectors around the zaxis [see Fig. 5(b)].\nThe crystal field with the symmetry in the zdirection\nleadsto elliptical rotationofthe magnetic vectorsaround\nthezaxis [see Fig. 5(c)]. However, this elliptical mode\nexists only in the presence of the exchange interaction.\nThe crystal field with the symmetry in the xdirection\nleadsto elliptical rotationofthe magnetic vectorsaround\nthexaxis [see Fig. 5(d)] in such a way that their xand\nyprojections are always opposite to each other, and z\nprojections are alway equal to each other.\nWe calculated the time evolutions of the magnetic\nvectors triggered by a left-circularly polarized Gaussian\nshaped laser pulse of the duration of 117 fs, which cor-\nresponds to bandwidth of 15 meV. We assume the peak\nintensity of 2 ×1010W/cm2, the fluence 8 mJ/cm2and\nthe central frequency of 2.0 eV. Figure 6 shows the\ntime evolution of the components Lx=M1x−M2x,\nLy=M1y−M2yandMz=M1z+M2zand the cor-\nresponding 3D picture of the trajectories of the magnetic\nvectorsM1andM2atdifferentvaluesoftheexchangein-\nteraction and the crystal field interaction constants. The\nred and blue arrows on the right panel show the initial\nalignments of the magnetic vectors of the sublattices 1\nand2correspondingly. Thedottedyellow-redandyellow-\nbluelinesshowthedynamicsofthecorrespondingvectors\nduring the excitation. The continuous red and blue lines\nshow their dynamics after the excitation.\nThe dynamics of the magnetic vectors during and after\nthe action of the laser pulse depends considerably on the\nexchange and crystal field interactions. Figures 6(a)-(d)\nshow the dynamics of the magnetic vectors in the case\nof the crystal field ˆHcrzwith the symmetry along the z\naxis. In this case, vectors M1andM2move upwards\nduring the action of the pump pulse and start to precess12\n-3-2-10123\n0 1 2 3\nTime,ps-3-2-10123-3-2-10123-3-2-10123-3-2-10123\nxyz\nJex≪|∆x|xyz\nJex∼|∆x|xyz\nJex≫|∆x|xyz\nJex≫∆zxyz\nJex∼∆z\nLx\nLy\nMz(i)(g)(e)(c)(a)\n(j)(h)(f)(d)(b)\nFIG. 6: Left column: the time evolutions of Lx,Lyand\nMzdepending on the values of Jexand ∆x,z. The gray\nline represents the time evolution of the electric field am-\nplitude. Right column: the corresponding trajectory of\ntheM1(red) and M2(blue) vectors. The dotted yellow\nline represents the dynamics of the vectors during the\nexcitation. (a),(b) Jex= 3 meV, ∆ z= 2 meV. (c),(d)\nJex= 3 meV, ∆ z= 0.02 meV. (e), (f) Jex= 3 meV,\n∆x= 0.02 meV. (g), (h) Jex= 3 meV, ∆ x=−2 meV.\n(i), (j)Jex≈0 meV, ∆ x=−2 meV.around the zaxis slightly before the excitation finishes.\nTheirzprojections remain constant after the excitation,\nsince the oscillation modes due to the exchange interac-\ntion and the crystal field ˆHcrzcorrespondto constant Mz\n[cf. Figs. 5(b) and (c)]. The precession of the magnetic\nvectors involves both elliptical and circular modes, when\nJex∼∆z[cf. Figs. 6(a) and (b)]. The length of the mag-\nnetic vectors is slightly lower than 3/2 due to the partial\nquenching of the angular momentum by the crystal field\nas discussed earlier. The magnetic vectors simply circu-\nlate around the zaxis, when Jex≫∆z[cf. Figs. 6(c) and\n(d)].\nIn the case of the crystal field ˆHcrxwith the symmetry\nalong thexaxis, the dynamics of the magnetic vectors\nduring the excitation are much more dependent on the\nvalueofthecrystalfieldconstant[cf. Figs.6(e)-(j)]. After\nthe excitation, the magnetic vectors start to follow an\nelliptical trajectory around the xaxis, which is slightly\nbent to the zaxis in the case of Jex≫ |∆x|[cf. Fig. 6(e)-\n(f)]. As discussed earlier, the magnetic vectors always\nmove in such a way that their projections on the zaxis\nare equal to each other, and the xandyprojections are\nopposite.\nThe trajectories of the magnetic vectors during the ex-\ncitation in the case of Jex≫ |∆x|[cf. Figs. 6(e) and (f)]\nare quite similar to the the ones in the case of Jex∼∆z\n[cf. Figs. 6(a) and (b)] and Jex≫∆z[cf. Figs. 6(c) and\n(d)]. Magnetic vectors on Figs. 6(e) and (f) also start to\nprecess slightly before the excitation has finished. This\nmeans that the time evolutions of the magnetic vectors\nduring the excitation are not strongly influenced by the\nexchange and crystal field interactions, and approximate\npositions of the magnetic vectors after the excitation are\ndetermined mainly by the interaction with the pump\npulse. Still, it is noticeable that the values of Lyright\nafter the excitation on Figs. 6(a),(c) and (e) differ from\neach other. In all these cases, the periods of the induced\nprecessions are more than twenty times larger than the\npulse duration.\nA situation is quite different in the case of Jex∼ |∆x|\nandJex≪ |∆x|. The magnetic vectors start to pre-\ncess during the excitation at approximately half of the\npulse duration [cf. Figs. 6(g)-(j)]. The trajectories dur-\ning the excitation are strongly dependent on the value\nof the crystal field constant ∆ x. The magnetic vectors\neven move downwards before the start of the precession\non Figs. 6(i) and (j)], which is the opposite direction to\nthe ones in all other cases in Fig. 6. This means that\nthe dynamics of the magnetic vectors during the action\nof the pump pulse is strongly dependent on the exchange\ninteraction and the crystal field ˆHcrx. The period of the\ninduced precessions is about two times longer than the\npulse duration, when Jex∼ |∆x|, and about three time\nlonger than the pulse duration, when Jex≪ |∆x|.\nThus, if the pulse duration is several tens of times\nshorter than the period of laser-induced magnetic pre-\ncessions, then the trajectories of the magnetic vectors\nduring the excitation are similar even at different values13\nofthe crystal-fieldconstantsoftheantiferromagneticsys-\ntem. However,the positionsofthemagneticvectorsright\nafter the excitation still slightly differ in this case. If the\npulse duration is just several times shorter than the pe-\nriod of the laser-induced magnetic precessions, then the\ndynamics of the magnetic vectors during the excitation\ncan be absolutely divergent at different values of the ex-\nchange and crystal field constants. Thus, the dynamics\nof the magnetic vectors during the excitation are mainly\ndetermined by the action of the pump pulse in the for-\nmer case. In the latter case, the magnetization dynamics\nduring the excitationare mainly determined by the inter-\nnal magnetic interactions and the pump pulse serves as\na slight impulse prompting the dynamics. This demon-\nstrates that an accurate calculation of the magnetization\ndynamics during the action of light is necessary to pre-\ndict correct positions of the magnetic vectors right after\nthe excitation.\nV. CONCLUSIONS\nThe action of the UIFE on a magnetic system leads\nto deviation of its magnetic moment from the ground\nstate, which prompts the magnetic moment to precess\n[9]. In our article, we have shown that this process can-\nnotbe describedwithin the suddenapproximationevenif\nthe pump-pulse duration is several tens of times shorter\nthan the period of laser-induced precessions. We pro-\nvide a technique that accurately describes magnetization\ndynamics during the action of a laser pulse at a subpi-\ncosecond time scale.\nWederivedtheHeisenbergrepresentationfortheUIFE\nfrom the Schr¨ odinger picture, which describes coupling\nof light to electrons of a magnetic system. We obtained\nan operator ˆHJacting in the Hilbert space of total an-\ngular momentum with time-dependent elements, which\ndepend on laser-pulse parameters and transition ampli-\ntudes of the electronic system under the action of a laser\npulse. This way we substituted the operator −d·Eby\nthe nondiagonal effective magnetic operator ˆHJ. The\neffective magnetic operator allows to separate the mo-\ntion of a magnetic vector due to the action of light from\nthat induced by other fields acting on a magnetic sys-\ntem. Commuting the magnetic operator with total angu-\nlar momentum operators, we obtained equations of mo-\ntion for magnetic vectors of a magnetic system. During\ntheactionoflight, magneticvectorsmoveduetothejoint\naction of ˆHJand other magnetic operators acting on the\nmagnetic system. After the action of light, the elements\nofˆHJnaturally become zero.\nThe Heisenberg representation of the UIFE could\nbe implemented for macroscopic calculations of laser-\ninduced magnetization dynamics, which are a practical\ntechnique allowing to take simultaneously many differ-\nent magnetic effects into account [1, 40–42]. The effec-\ntive magnetic operator ˆHJcould be also used as a con-\nvenient tool to adjust pump-pulse properties enhancingthe UIFE, since it directly illustrates how a pump pulse\ncouples to the total angular momentum.\nWith the help of the illustrative single-spin system in\nan external magnetic field, we showed that laser-induced\nmagnetization dynamics can be strongly affected by the\nLarmorprecessioneven duringthe actionofa laserpulse.\nThe spin started to precess during the action of the laser\npulseevenwhentheLarmorperiodwasfortytimeslonger\nthan the pump-pulse duration. Thus, even in this case, it\nwas necessaryto calculate the joint action of the external\nmagnetic field and the UIFE on spin in order to obtain\nthecorrectpositionofthespinvectoraftertheexcitation.\nWe calculated magnetization dynamics induced by the\nUIFE in model antiferromagnetic systems consisting of\ntwo sublattices with opposite magnetic vectors. The\nmagnetization dynamics in these systems were described\nby a system of fifteen first-order differential equations\nwithin the Heisenberg picture. We demonstrated that\nthe action of the UIFE induced by a pump pulse prop-\nagating in the zdirection made both magnetic vectors\nbend upwards to the zaxis and rotate around it in such\na way that their zprojections were equal, and xand\nyprojections were opposite. The deviation of the mag-\nnetic vectors from their initial positions evoked a circular\nprecession mode due to the exchange interaction and an\nelliptical precession mode due to the crystal field.\nIn the case of the xy-easy-plane antiferromagnetic sys-\ntem with the exchange-interactionconstantof3meVand\nthe crystal-field constant of 2 meV, the period of the\ninduced precessions was more than twenty times longer\nthan the pump-pulse duration. The motion of the mag-\nnetic vectors was mainly determined by the UIFE and to\nsome extend affected by the crystal field and exchange\ninteractions. In the case of the z-easy-axis antiferromag-\nnetic system with the same absolute values of the ex-\nchange interaction and crystal-field constants, the period\nof the induced precessions was just two times longer than\nthe pump-pulse duration. The motion of the magnetic\nvectors was mainly determined by the crystal field and\nthe exchange interaction, and the pump pulse served just\nas a slight impulse prompting the magnetization dynam-\nics. The magnetic vectors moved even in the opposite di-\nrection to the one, in which the action of the UIFE alone\nwould make the magnetic vectors move. This example\nalso demonstrates that an accurate calculation of mag-\nnetization dynamics during the action of a pump pulse is\nnecessary even if the pump-pulse duration is several tens\noftimes shorter than the period of induced magnetic pre-\ncessions.\nWe thus observed that the character of induced mag-\nnetic precessions depends on the ratio of the pump-pulse\nduration to the period of magnetic-oscillation modes.\nThis effect could be used for control of magnetic preces-\nsions by varying the pump-pulse duration in the regime,\nwhen it is comparable with the period of magnetic os-\ncillation modes. In this regime, the motion due to the\nUIFE-driven deviation of magnetic vectors can cooper-\nate or compete with the one due to magnetic precessions14\ncaused by this deviation. For example, in the z-easy-axis\nantiferromagnetic system, the UIFE-induced deviation is\nin the upward zdirection, but the crystal-field interac-\ntion causes the precession upwards and downwards the\nzaxis. Varying the pump-pulse duration, one can apply\nthe driving during the time, when it either enhances or\ncounteracts the precession mode, which would influence\nthe magnitude and the phase of resulting precessions af-\nter the action of the pump pulse. This effect does not\nfollow from the phenomenological description of the IFE,\nin which the pump-pulse fluence alone determines the\ncharacter of magnetic precessions.\nThe developed technique to study the magnetization\ndynamics induced by the UIFE can be applied to other\ntypes of magnetic materials, not necessary antiferromag-\nnetic. The Heisenberg representation of other magneto-\noptical effects driven by Raman transitions [57–61] can\nbe derived analogously to our methodology. The pre-\nsented concept of the time-dependent effective magnetic\noperator paves the way towards the macroscopic descrip-\ntion of ultrafast laser-induced magnetization dynamics\nthat accurately takes electronic transitions induced by\nan ultrashort light pulse into account.\nACKNOWLEDGMENT\nThis work was supported by European Community’s\nSeventh Framework Programme FP7/2007-2013 under\nGrant Agreement No. 214810 (FANTOMAS). D. P.-G.\nacknowledgesthe support of the VolkswagenFoundation.\nAppendix A: Effective magnetic operator describing\nthe UIFE\n1. Derivation of the effective magnetic operator\nLet us first derive the operator for the case, when\nthere is no field except light acting on the total angu-\nlar momentum of the system, meaning that ˆU=1and\niΨ′\ng=ˆH0\nJΨ0\ng. Here,ˆH0\nJand Ψ0\ngare the operator and the\ntime-dependent spinor in this case. The relation (8) can\nbe rewritten as\nΨ0\ng=AΨ0\nN=1\nN(t)\nA1(t)eiφ1(t)P01\n...\nAk(t)eiφk(t)P0k\n...\n=\nP1(t)\n...\nPk(t)\n...\n,\n(A1)\nwherethe elementsofthe spinor AareAk=Ak(t)eiφk(t).\nWe apply the condition that the magnetic moment\nof a system is not rotated by the IFE, if it is parallel\nto the laser pulse propagation direction. The magnetic\nmoment is parallel to the quantization axis, if all ele-\nments except one of the spinor Ψ 0are zero. This meansthat, if Ψ 0=\n0\n...\nP0k\n...\n, there|P0k|= 1, then the effec-\ntive magnetic operator acts only on the k-th component\nof the state vector, so that the other magnetic com-\nponents remain zero, Ψ0\ng=1\nN(t)\n0\n...\nAk(t)eiφk(t)\n...\n. Since\n|Ψ0\ng|= 1, the spinor during and after the action of light\nis Ψ0\ng=eiφk(t)Ψ0. Thus, the diagonal elements of ˆH0\nJare\n(ˆH0\nJ)aa=−φ′\na(t), and if the magnetic moment is par-\nallel to the light propagation direction, the off-diagonal\nelements of ˆH0\nJmust become zero.\nLet us now derive the off-diagonal elements of ˆH0\nJ,\nwhich are non-zero, if the magnetic moment is not paral-\nlel to the light propagation direction. The a-th element\nofiΨ0\ng′can be expressed as\niP′\na=−φ′\naPa+i/summationdisplay\nb,b/negationslash=aPb/bracketleftbigg\nPaP∗\nb/parenleftbiggA′\na\nAa−A′\nb\nAb/parenrightbigg/bracketrightbigg\n,(A2)\nwhich follows from the relations Pa(t) =\nAa(t)eiφa(t)P0a//summationtext\nk|Pk(t)|andP0a= const. Ap-\nplying this expression with iΨ0\ng′=ˆH0\nJΨ0\ng, we obtain\nˆH0\nJ=\n...\n−γa···iPaP∗\nb(νa−νb)···\n......\niP∗\naPb(νb−νa)\n...\n,\n(A3)\nwhereνa= Re(A′\na/Aa),γa=φ′\na= Im(A′\na/Aa) forˆU=\n1.\nIf there is a field ˆHm, which acts on the magnetic\nsystem apart from light, then ˆU /negationslash=1andiΨ′\ng=/bracketleftig\nˆHJ+ˆHm/bracketrightig\nΨg. In this case the spinor is Ψ g=\nˆUAΨ0/N=ˆUΨ0\ng. Substituting ˆUΨ0\ngfor Ψg, we obtain\nthati(ˆUΨ0\ng)′=/bracketleftig\nˆHJ+ˆHm/bracketrightig\nˆUΨ0\ng. Applying that, by def-\ninition,iˆU′=ˆHmˆU, we obtain ˆHJ=ˆUˆH0\nJˆU−1. We\nderiveˆHJapplying this expression, which results in the\noperator ˆHJhaving the same form as ˆH0\nJin Eq. (A3),\nbut with different functions νaandγa:\nνa= Re(Ya), γa= Im(Ya), Ya= [ˆUA′]a/[ˆUA]a,\n(A4)\nwhere [ˆUA′]ais thea-th element of the spinor obtained\nby the action of the operator ˆUonA′, which is the time-\nderivative of the spinor A. [ˆUA]ais thea-th element of\nthespinorobtainedbytheactionoftheoperator ˆUonthe\nspinorA. Applying that PaP∗\nb= (/angbracketleftˆNab+/angbracketright+i/angbracketleftˆNab−/angbracketright)/2 =\n/angbracketleftΨg|ˆNab++iˆNab−|Ψg/angbracketright/2, the effective magnetic operator15\ncan be written as\nˆHJ=−n/summationdisplay\naγaˆNa (A5)\n+1\n2n/summationdisplay\na,b(νa−νb)/parenleftbig\n/angbracketleftˆNab−/angbracketrightˆNab+−/angbracketleftˆNab+/angbracketrightˆNab−/parenrightbig\n.\n2. Commutator with the effective magnetic\noperator\nIn this subsection, we derive the expectation value of\nthe commutator −i/angbracketleftig/bracketleftig\nˆNab±,ˆHJ/bracketrightig/angbracketrightig\n. We divide the oper-\natorˆHJinto its diagonal, Hd, and off-diagonal part, Hn,\nand derive the expectation values of the commutators\n−i/angbracketleftig/bracketleftig\nˆNab±,Hd/bracketrightig/angbracketrightig\nand−i/angbracketleftig/bracketleftig\nˆNab±,Hn/bracketrightig/angbracketrightig\nseparately.\nThecd-th matrix element of the commutator of/bracketleftig\nˆNab±,Hd/bracketrightig\nis\n/parenleftig\nˆNab±ˆHd−ˆHdˆNab±/parenrightig\ncd= (ˆNab±)cd/bracketleftig\n(ˆHd)dd−(ˆHd)cc/bracketrightig\n,\n(A6)\nwhere we designate the cd-th matrix element of an oper-\natorˆOas (ˆO)cd. Thus,/bracketleftig\nˆNab±,ˆHd/bracketrightig\n=±i(γa−γb)ˆNab∓.\nLet us now commute ˆNab±with the off-diagonal part\nˆHnand determine the expectation value ofthe commuta-\ntor−i/angbracketleft[ˆNab±,ˆHn]/angbracketright. Weconsidertheelementsofthecom-\nmutator ˆCab±= [ˆNab±,ˆHn]. We find that ( ˆCab±)cd= 0\nif neithercnordare not equal aorb. The elements in\nother cases are\n(ˆCab±)ac= (ˆNab±)ab(ˆHn)bc,\n(ˆCab±)ca=−(ˆNab±)ba(ˆHn)cb, (A7)\n(ˆCab±)bc= (ˆNab±)ba(ˆHn)ac,\n(ˆCab±)cb=−(ˆNab±)ab(ˆHn)ca\nThe expectation value /angbracketleftˆCab±/angbracketright=/angbracketleftΨg|ˆCab±|Ψg/angbracketrightis given\nby/summationtext\ncdP∗\ncPdˆCab±\ncd, which is equal to i[−2(/summationtext\ni|Pi|2νi) +\n(νa+νb)](PaP∗\nb±iP∗\naPb) as follows from Eq. (A7). Ap-\nplying that |Pa|2=/angbracketleftΨg|ˆNa|Ψg/angbracketrightand (PaP∗\nb±iP∗\naPb) =\n/angbracketleftˆNab±/angbracketright, we obtain the relation\n−i/angbracketleftig/bracketleftig\nˆNab±,ˆHJ/bracketrightig/angbracketrightig\n=/parenleftigg\n−2/summationdisplay\nkνk/angbracketleftˆNk/angbracketright+νa+νb/parenrightigg\n/angbracketleftˆNab±/angbracketright\n±(γa−γb)/angbracketleftˆNab∓/angbracketright. (A8)\nAppendix B: Spin-orbit coupling and Zeeman\ninteraction\nIn order to obtain the wave functions and the splitting\nof the 2pstate in the presence of the spin-orbit couplingand Zeeman interaction, one has to diagonilize the fol-\nlowing integral is\nˆHm=−B\n2(2ˆSx+ˆLx)−λL·S.(B1)\nThis Hamitonian has six eigenvectors and eigenenergies\nfor the 2pstate. Thus, 2 pstate is split energetically into\nsix states with energies εk±=ε2p,k±+Ek±,k= 1, 2 or\n3. The indices k±correspond to the indices jin Section\nIIIA.Ek±are the solutions of the equations\nE3\nk±±B\n2E2\nk±−/parenleftbigg3λ2\n4±Bλ\n2+B2\n2/parenrightbigg\nEk±(B2)\n+/parenleftbigg\n−λ3\n4+λB2\n4∓3λ2B\n8/parenrightbigg\n= 0.\nThe corresponding wave functions are, if λ/negationslash= 0 andB/negationslash=\n0,\nΨ2p\nk±=αk±/parenleftbig\n|Lz= 1,Sz= 1/2/angbracketright±|Lz=−1,Sz=−1/2/angbracketright/parenrightbig\n+βk±/parenleftbig\n|Lz= 1,Sz=−1/2/angbracketright±|Lz=−1,Sz= 1/2/angbracketright/parenrightbig\n(B3)\n+γk±/parenleftbig\n|Lz= 0,Sz= 1/2/angbracketright±|Lz= 1,Sz=−1/2/angbracketright/parenrightbig\n,\nwhereαk±=B/parenleftbig\n5λ/8−3Ek±/4/parenrightbig\n/[Nk±/parenleftbig\nEk±+λ/2/parenrightbig\n],\nβk±= (Ek±+B/4−λ/2)/Nk±andγk±= (Ek±−\n3λ/2−B/2)/(√\n2Nk±) andNk±is the normalisation fac-\ntor, which provides |αk±|2+|βk±|2+|γk±|2= 1.\nThe dipole matrix elements for a transition from the\nsstate with the wave function |Lz= 0,Sz= 1/2/angbracketrightto\nthe statesk±,d↑k±, are proportional to αk±for the left-\ncircularly polarized light. The dipole matrix elements for\natransitionfromthe sstatewiththewavefunction |Lz=\n0,Sz=−1/2/angbracketrightto the states k±,d↓k±, are proportional to\nβk±for the left-circularly polarized light.\nIfλ= 0 andB/negationslash= 0, then α1±=β1±= 1/(2√\n2),\nα2±=−β2±= 1/2,α3±=β3±= 1/(2√\n2). Thus, if\nλ= 0, butB/negationslash= 0, no rotation is possible, since |d↑k±|2=\n|d↓k±|2[see Eq. (20)].\nAppendix C: Antiferromagnetic system\n1. Ground and excited states\nIn Section IVA, we consider the following light field-\nfree magnetic Hamiltonians acting on the sublattices 1\nand 2, correspondingly,\nˆH(1)\nm=Jex(Jx2ˆJx1+Jy2ˆJy1+Jz2ˆJz1)\n+∆z(x)/parenleftig\n3ˆJ2\nz1(x1)−ˆJ2\n1/parenrightig\n(C1)\nˆH(2)\nm=Jex(Jx1ˆJx2+Jy1ˆJy2+Jz1ˆJz2)\n+∆z(x)/parenleftig\n3ˆJ2\nz2(x2)−ˆJ2\n2/parenrightig\n(C2)16\nThe signofthe crystalfield constants∆ zand∆xarecho-\nsen in such a way that the alignment of magnetic vectors\nof the sublattices along the xaxis is energetically prefer-\nable. Due to the exchange interaction, the state with the\nlowestenergycorrespondsto the magneticvectorshaving\nthe largestpossible amplitude and being opposite to each\nother:Jx1=−Jx2. Thus, the effective magnetic Hamil-\ntonians before the action of light can be represented as\nˆH(1)\nm(eff)= ∆z(x)/parenleftig\n3ˆJ2\nz1(x1)−ˆJ2\n1/parenrightig\n+J0ˆJx1,(C3)\nˆH(2)\nm(eff)= ∆z(x)/parenleftig\n3ˆJ2\nz2(x2)−ˆJ2\n2/parenrightig\n+J0ˆJx2\nJ0=JexJx1=−JexJx2.\nThe ground state spinors, corresponding to the fol-\nlowing expectation values of total angular momen-\ntum operator Jx1=−Jx2,Jx1>0,Jy1=\nJy2=Jz1=Jz2= 0, must fulfill the con-\nditions/angbracketleftig\nΨ(1)\n0/vextendsingle/vextendsingle/vextendsingleˆJx1/vextendsingle/vextendsingle/vextendsingleΨ(1)\n0/angbracketrightig\n=−/angbracketleftig\nΨ(2)\n0/vextendsingle/vextendsingle/vextendsingleˆJx2/vextendsingle/vextendsingle/vextendsingleΨ(2)\n0/angbracketrightig\nand/angbracketleftig\nΨ(1,2)\n0/vextendsingle/vextendsingle/vextendsingleˆJy1,y2/vextendsingle/vextendsingle/vextendsingleΨ(1,2)\n0/angbracketrightig\n=/angbracketleftig\nΨ(1,2)\n0/vextendsingle/vextendsingle/vextendsingleˆJz1,z2/vextendsingle/vextendsingle/vextendsingleΨ(1,2)\n0/angbracketrightig\n= 0,\nwhere\nˆJx1,x2=\n0√\n3\n20 0√\n3\n20 1 0\n0 1 0√\n3\n2\n0 0√\n3\n20\n, (C4)\nˆJy1,y2=\n0−i√\n3\n20 0\ni√\n3\n20−i0\n0i0−i√\n3\n2\n0 0i√\n3\n20\n,(C5)\nˆJz1,z2=\n3\n20 0 0\n01\n20 0\n0 0−1\n20\n0 0 0 −3\n2\n. (C6)\nThe spinors, which fulfill these conditions, have the fol-\nlowing form\nΨ(1)\n0=\nc\nd\nd\nc\n,Ψ(2)\n0=\nc\n−d\nd\n−c\n, (C7)\nIm(c) = Im(d) = 0, c>0, d>0.\nInthecaseofthecrystalfield ˆHcrx, theeffectiveHamil-\ntonians are diagonal in the basis with the quantization\naxis along the xaxis. Thus, the ground state state vec-\ntors are the eigenvectors of the ˆJx1,x2operators. The\nground state spinors correspond to the expectation val-\nuesJx1= 3/2 andJx2=−3/2 withc= 1/(2√\n2),\nd=√\n3/(2√\n2).\nThe situation is more complicated in the case of ˆHcrz.\nThe effective Hamiltonian is not diagonal in a basis with\nneitherxnorzquantizationaxes. The spinorsdepend on\nthe crystal field and the exchange interaction, and mustbe found numerically. If ∆ z/negationslash= 0, then the expectation\nvalues have lower values Jx1<3/2 andJx2>−3/2, and\nthe crystal field ˆHcrzleads to a partial quenching of the\nmagnetic moment.\n2. Time-dependent factors of the effective\nmagnetic operator\nIn this subsection we calculate the functions ν(1,2)\na\nandγ(1,2)\naand show that they are equal for both\nsystems. These functions are defined as ν(1,2)\na=\nRe(Y(1,2)\na),γ(1,2)\na= Im(Y(1,2)\na), where Ya=\n[ˆU(1,2)A(1,2)′]a/[ˆU(1,2)A(1,2)]a. The spinors A(1,2)are de-\nfined by Eq. (8). ˆUis the time evolution operator defined\nbyiˆU(1,2)′=ˆH(1,2)\nmˆU(1,2).\nAccording to Eqs. (5), (6) and (8) the a-th elements of\nA1,2isA(1,2)\na= 1−C(1,2)\na/P(1,2)\n0a, ifP(1,2)\n0a/negationslash= 0, otherwise,\nA(1,2)\na= 0. Here, P(1,2)\n0a=Pa(0)(1,2),C(1,2)\nais thea-\nth element of the vector obtained by the action of the\noperator in squared brackets on the initial spinor\nC(1,2)=/bracketleftigg/integraldisplayt\n−∞dt′(ˆU(1,2))−1ˆVˆU(1,2)\n×/integraldisplayt′\n−∞dt′′(ˆU(1,2))−1ˆVˆU/bracketrightigg\nΨ(1,2)\n0.(C8)\nSince the action of the exchange interaction on the ex-\ncited state is ignored, the time evolution operator, which\nacts on spinor of the excited state, can be written sim-\nply as an operator ˆUe(t) =e−iˆHcrz(crx)t. It is equal for\nthe both systems 1 and 2. The crystal field operator,\nˆHcrz, acting on the excited state characterized by the\ntermJ= 5/2, is represented by\nˆHcrz= ∆ze/parenleftbigg\n3ˆJ2\nz1−35\n4/parenrightbigg\n(C9)\n= ∆ze\n10 0 0 0 0 0\n0−2 0 0 0 0\n0 0−8 0 0\n0 0 0 −8 0 0\n0 0 0 0 −2 0\n0 0 0 0 0 10\n\nThus, the crystal field leads for both systems to the split-\nting of the excited state J= 5/2 into the following three\ndoubly degenerate states with the correspondingenergies\nεex1,2,3\n|Jz1(z2)=±5/2/angbracketright, εex1=εex+10∆ze\n|Jz1(z2)=±3/2/angbracketright, εex2=εex−2∆ze (C10)\n|Jz1(z2)=±1/2/angbracketright, εex3=εex−8∆ze,\nwhereεex= 2 eV is the energy of the excited state in the\nabsence of the crystal field.17\nThe crystal field operator, ˆHcrx, acting on the excited\nstate characterized by the term J= 5/2, is represented\nby\nˆHcrx= ∆xe/parenleftbigg\n3ˆJ2\nx1−35\n4/parenrightbigg\n(C11)\n= ∆xe\n−5 03√\n10\n20 0 0\n0 1 09√\n2\n20 0\n3√\n10\n20 4 09√\n2\n2\n09√\n2\n20 4 03√\n10\n2\n0 09√\n2\n20 1 0\n0 0 03√\n10\n20−5\n.\nThe crystal field leads for both systems to the splitting of\nthe excited state J= 5/2 into the following three doubly\ndegenerate states with the correspondingenergies εex1,2,3\n|Jx1(x2)=±1/2/angbracketright, εex1=εex−8∆xe,\n|Jx1(x2)=±3/2/angbracketright, εex2=εex−2∆xe,(C12)\n|Jx1(x2)=±5/2/angbracketright, εex3=εex+10∆xe.\nLet us examine the selection rules for the transitions\nfrom the ground state |g/angbracketrightto the excited state |ex/angbracketrightfor\nan excitation by left-circularly polarized light. A dipole\nmatrix element of a transition from a state with a total\nangular momentum Jand projection Jz=mto a state\nwithJ+1 andJz=m+1 is[62]\n/angbracketleftJ+1m+1|r+|Jm/angbracketright= (C13)\n−/radicaligg\n(J+m+1)(J+m+2)\n(J+1)(2J+1)(2J+3)/angbracketleftJ+1|r|J/angbracketright.,\nwherer+= (x+iy)/√\n2. Thus, the action of the op-\neratorˆVon the ground state and excited state can be\nrepresented as ˆV=iEd0F(t)ˆDandˆV=iEd∗\n0F(t)ˆDT,\ncorrespondingly, where\nˆD=\n−/radicalig\n2\n30 0 0\n0−/radicalig\n1\n50 0\n0 0 −/radicalig\n1\n100\n0 0 0 −/radicalig\n1\n30\n0 0 0 0\n0 0 0 0\n,(C14)\nF(t) =p(t/T)cos(ω0t), andd0is areduced dipole matrix\nelement:d0=/angbracketleftex,J= 5/2|r|g,J= 3/2/angbracketright.d0= 1 a. u. is\ntaken for simplicity. Therefore, the vectors C(1,2)can be\nwritten as\nC(1,2)=E2|d0|2/bracketleftigg/integraldisplayt\n−∞dt′F(t′)/bracketleftig\nˆU(1,2)/bracketrightig−1\n(t′)×(C15)\nˆDTˆUe(t′)/integraldisplayt′\n−∞dt′F(t′′)′ˆUe(t′′)ˆDˆU(1,2)(t′′)/bracketrightigg\nc\n±d\nd\n±c\n,As discussed in Section IVC, Jx1,y1=−Jx2,y2and\nJz1=Jz2during and after the action of light. Applying\nthese relations and investigating the relations of ν(1,2)\na\nandγ(1,2)\nato the spinors C(1,2), we obtain that ν(1)\na=ν(2)\na\nandγ(1)\na=γ(2)\na.\n3. Equations of motion\nWe solve differential equations for the expectation val-\nues\nm12±=√\n3\n2/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\n12±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n+/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\n12±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\n(C16)\nl12±=√\n3\n2/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\n12±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n−/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\n12±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\nm13±=√\n3\n2/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\n13±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n+/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\n13±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\nl13±=√\n3\n2/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\n13±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n−/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\n13±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\nm34±=√\n3\n2/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\n34±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n+/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\n34±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\nl34±=√\n3\n2/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\n34±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n−/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\n34±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\nm24±=√\n3\n2/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\n24±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n+/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\n24±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\nl24±=√\n3\n2/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\n24±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n−/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\n24±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\nm14±=3\n4/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\n14±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n+/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\n14±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\nl14±=3\n4/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\n14±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n−/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\n14±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\nm23±=/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\n23±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n+/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\n23±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\nl23±=/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\n23±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n−/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\n23±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\nma=/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\na±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n+/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\na±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\nla±=/bracketleftig/angbracketleftig\nΨ(1)\ng/vextendsingle/vextendsingle/vextendsingleˆN(1)\na±/vextendsingle/vextendsingle/vextendsingleΨ(1)\ng/angbracketrightig\n−/angbracketleftig\nΨ(2)\ng/vextendsingle/vextendsingle/vextendsingleˆN(2)\na±/vextendsingle/vextendsingle/vextendsingleΨ(2)\ng/angbracketrightig/bracketrightig\n,\nwhereais 1,2,3 or 4. The initial values of mab±(0) and\nlab±(0) are given by substituting Ψ(1,2)\ng(0) = Ψ(1)\n0. We\nobtain that m12±(0) = 0,m23±(0) = 0,m34±(0) = 0,\nm14±(0) = 0,l13±(0) = 0,l24±(0) = 0 and la(0) = 0 for\na= 1...4 and these variables remain zero at any time.\nApplying that la(0) = 0, the equations of motion can be18\nˆLxˆLyˆMz/parenleftBig\nˆM2\nz+ˆL2\nz/parenrightBig\n/2/parenleftBig\nˆM2\nx+ˆL2\nx/parenrightBig\n/2\nl′\n12+ m13−3\n2m1−3\n2m2−m13+ −l12−−2l12−l12−+3\n4l23−+l14−\nl′\n12−−3\n2m1+3\n2m2−m13+ −m13− l12+ 2l12+ l12+3\n4l23+−l14+\nl′\n23+ −m13−+m24− 2m2−2m3+m13+−m24+−l23− 0 −l12−+l34−\nl′\n23−−2m2+2m3+m13+−m24+ m13−−m24− l23+ 0 −l12++l34+\nl′\n34+ −m24−3\n2m3−3\n2m4+m24+ −l34− 2l34−−3\n4l23−−l34−−l14−\nl′\n34−−3\n2m3+3\n2m4+m24+ m24− l34+ −2l34+ −3\n4l23++l34++l14+\nl′\n14+3\n4(m13−−m24−)3\n4(m13+−m24+) −3l14− 03\n4l12−−3\n4l34−\nl′\n14−3\n4(−m13++m24+)3\n4(m13−−m24−) 3l14+ 0 −3\n4l12++3\n4l34+\nm′\n13+ l12−+l14−−3\n4l23− l12+−l14+−3\n4l23+ −2m13−−2m13− m13−\nm′\n13−−l12+−l14++3\n4l23+ l12−−l14−−3\n4l23−2m13+ 2m13+ −3\n2m1−m13++3\n2m3\nm′\n24+3\n4l23−−l14−−l34− l14++3\n4l23+−l34+ −2m24−2m24− −m24−\nm′\n24−−3\n4l23++l14++l34+ l14−+3\n4l23−−l34−2m24+ −2m24+−3\n2m2+3\n2m4+m24+\nm′\n1 l12− −l12+ 0 0 m13−\nm′\n2 l23−−l12− −l23++l12+ 0 0 m24−\nm′\n3 −l23−+l34− l23+−l34+ 0 0 −m13−\nm′\n4 −l34− l34+ 0 0 −m24−\nTABLE II: First column: k′, which is equal to −i/angbracketleftig/bracketleftig\nˆk,ˆHm+ˆHJ/bracketrightig/angbracketrightig\n, whereˆkisˆlab±or ˆmab±. Four left columns:\n−i/angbracketleftig/bracketleftig\nˆk,ˆO/bracketrightig/angbracketrightig\n, whereˆOdenotes the operators entering ˆHm:ˆLx,ˆLy,ˆMz,/parenleftig\nˆM2\nz+ˆL2\nz/parenrightig\n/2 and/parenleftig\nˆM2\nx+ˆL2\nx/parenrightig\n/2.\nwritten as\nm′\nab±=/parenleftigg\n−/summationdisplay\nkνkmk+νa+νb/parenrightigg\nmab±(C17)\n±(γa−γb)mab∓−i/angbracketleftig/bracketleftig\nˆmab±,ˆHm/bracketrightig/angbracketrightig\nl′\nab±=/parenleftigg\n−/summationdisplay\nkνkmk+νa+νb/parenrightigg\nlab± (C18)\n±(γa−γb)lab∓−i/angbracketleftig/bracketleftig\nˆlab±,ˆHm/bracketrightig/angbracketrightig\n.\nThe relations m12±(t) = 0,m23±(t) = 0 andm34±(t) = 0\nlead toMx=My= 0, andla(0) = 0 for a= 1...4 leads\ntoLz= 0. 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Lifshitz, Quantum mechanics, non-\nrelativistic theory , A-W series in advanced physics (Perg-\namon Press, Oxford, 1958)"    },    {        "title": "1009.4333v1.Direct_Measurement_of_Effective_Magnetic_Diffusivity_in_Turbulent_Flow_of_Liquid_Sodium.pdf",        "content": "arXiv:1009.4333v1  [physics.flu-dyn]  22 Sep 2010Direct Measurement of Effective Magnetic Diffusivity in Turb ulent Flow of Liquid\nSodium\nPeter Frick,1,∗Vitaliy Noskov,1Sergey Denisov,1and Rodion Stepanov1\n1Institute of Continuous Media Mechanics, Korolyov 1, Perm, 614013, RUSSIA\nThe first direct measurements of effective magnetic diffusivi ty in turbulent flow of electro-\nconductive fluids (the so-called β-effect) under magnetic Reynolds number Rm >>1 are reported.\nThe measurements are performed in a nonstationary turbulen t flow of liquid sodium, generated in\na closed toroidal channel. The peak level of the Reynolds num ber reached Re ≈3·106, which\ncorresponds to the magnetic Reynolds number Rm ≈30. The magnetic diffusivity of the liquid\nmetal was determined by measuring the phase shift between th e induced and the applied magnetic\nfields. The maximal deviation of magnetic diffusivity from it s basic (laminar) value reaches about\n50% .\nPACS numbers: 47.65.-d, 47.27.Jv, 91.25.Cw\nSmall-scale turbulence plays a crucial role in cosmic\nmagnetism, providing the small-scale (turbulent) MHD-\ndynamo and contributing a lot to the dynamics of large-\nscale magnetic fields. The mean field (large-scale) dy-\nnamoequationsarederivedbyapplyingthe Reynoldsap-\nproach to the magnetohydrodynamics (MHD) equations,\nand in the framework of the simplest case of homoge-\nneous and isotropic (but mirror asymmetric) turbulence\nthey can be reduced to [1]\n∂B\n∂t=∇×(U×B)+α∇×B+η△B,\n∇·B= 0, (1)\nwhereUandBdescribe the mean (large-scale) velocity\nand magnetic fields, η= (ρµ)−1+βis the magnetic dif-\nfusivity ( ρ- electrical resistivity, µ- magnetic permeabil-\nity), and αandβare the turbulent transport coefficients,\ndescribing the action of small-scale turbulent pulsations\non the mean field dynamics (see e.g. [2, 3]). Coefficient\nαdescribes the generation effects, and βdescribes the\ncontribution of turbulence to diffusion of the large-scale\nmagnetic field. The knowledge of the magnetic turbu-\nlent transport coefficients αandβis basic for astro- and\ngeophysical applications in dynamo theory [4].\nOver the last decade, major efforts were directed to-\nward the study of MHD-dynamo in laboratory experi-\nments (see for review [5]). The first-generation dynamo\nexperiments aredesignedon the basisofstrictly-specified\nlarge-scaleflow. The Riga dynamo is driven by the cylin-\ndrical screw flow [6], the Cadarache dynamo is based on\na von Karman flow between two counterrotating disks\n[7] and even the Karlsruhe dynamo, defined as a ”two-\nscale” dynamo, is driven by a set of strictly prescribed\nhelical jets inside 52 tubes [8]. In this sense, all labo-\nratory dynamos can be classified as quasi-laminar. In\nspite of that, the Reynolds numbers reached about 107\nand the flows were fully turbulent in all experiments.\nThus, the role of turbulence is reduced in these exper-\niments to enhancement of the diffusion of the magnetic\nfield which, with a constant magnetic permeability, canbe considered as an increase in effective resistance of liq-\nuid metal. The growth of resistivity can be crucial for\ndynamoexperimentsbecauseofthe correspondingreduc-\ntion in magnetic Reynolds number. However, no direct\nmeasurement of effective resistivity in dynamo facilities\nhas been performed up to now. An indirect indication\nof the beta-effect has been obtained in Madison sodium\nfacilities by comparison of measured magnetic field and\nmagnetic field simulated on the base of measured mean\nvelocity field [9]. An interesting scheme of eddy diffusiv-\nity estimation from hydromagnetic Taylor-Couette flow\nexperiment, recently suggested in [10].\nThe direct measurements of βare impeded by the fact\nthat the effect appears only under very large Reynolds\nnumbers, when numerous side-effects prevent the accu-\nrate isolation of the β-effect. The first attempt of such\nmeasurementswasdoneinaflowgeneratedbyapropeller\nin a vessel containing liquid sodium [11], though the au-\nthenticity of the obtained data is questionable both with\nrespect to the level of the observed conductivity varia-\ntions and the estimates of the measurement errors.\nA promising method of designing high Reynolds num-\nber flows (although nonstationary) in the limited mass of\nliquids was proposed in [12], in which the flow was gen-\nerated by the abrupt braking of a fast-rotating toroidal\nchannel. Installation of diverters in the channel made it\npossible to create a toroidal screw flow of liquid gallium,\nin which, for the first time, was observed the α-effect,\ndefined by a joint action of the gradient of turbulent pul-\nsations and large-scale vorticity [13]. The study of the\ndynamics of the nonstationary flow in a torus withoutdi-\nverters has shown that the development of the flow in\nthe channel is attended by a strong short-time burst of\nturbulent pulsations with a peak in range on the order\nof500−1000Hz[14]. This burst of small-scaleturbulence\nprovidesanopportunitytodetect the increaseineffective\nresistivity of liquid metal using the low frequency alter-\nnating magnetic field ( ∼100 Hz). The idea of such an\nexperiment has been realized in the nonstationaryflow of\nliquid gallium. The toroidal channel made from textolite2\nFIG. 1: Titanium channel and thermostatic cover.\nmade it possible to get magnetic Reynolds number less\nthan unity [15].\nIn this paper we exploit the similar experimental\nscheme using a titanium toroidal channel of larger size,\nfilled withliquid sodium, whichallowedustoincreasethe\nmagnetic Reynolds number by two orders of magnitude.\nThe apparatus is an electro-mechanical construction\nmounted on a rigid frame, which is used as a support\nfor a rotating toroidal channel (Fig. 1). The torus radius\nisR= 0.18m; the radius of the channel cross-section\nisr= 0.08m. The channel was filled with sodium in\nthe vacuum and was placed into the air thermostat. The\nchannel temperature may be stabilized in the range(50 −\n150)◦C. The temperature sensor is mounted inside the\nchannel and has good thermic contact with the sodium\nin both liquid and solid states.\nThe channel is fastened on the horizontal axis, which\nis also used for mounting a driving pulley, a system of\nsliding contacts and a disk braking system. The fre-\nquency of the channel’s rotation is up to 45 r.p.s. and\nthe flow in the channel is generated by abrupt brak-\ning – the braking time is no more than 0 .3sec. The\nmaximum velocity of the flow is reached after channel\nis stopped and achieves about 70% of the linear veloc-\nity of the channel before braking. This means that the\nReynolds number Re = Ur/ν(νis the kinematic viscos-\nity of the liquid sodium) reaches at maximum the value\nRe≈3·106, which correspondsto the magnetic Reynolds\nnumber Rm = Urρµ≈30.\nThe data-gatheringsystem is based on an NI Data Ac-\nquisition System and is a part of an electrical measuring\nsystem, whose schematic circuit is shown in Fig. 2. The\n’Generator/Amplifier’ block creates in the toroidal coil\na stabilized sinusoidal current with frequency 30 < ν <\n1000 Hz, which produces an alternating toroidal mag-\nnetic field inside the channel. Besides the toroidal coil,\ntwo diametrically located magnetic-test coils are wound\naround the channel.\nFIG. 2: Schematic circuit of the measuring system. Udp,\nUT,EandUare the driving pulse, the tachometer signal,\nthe electromotive force of magnetic-test coil and the volta ge,\ncorresponding to applied current.\n80 85 90 95 100 105/Minus1.05/Minus1.04/Minus1.03/Minus1.02/Minus1.01\nT,oC/DifferenceDeltaΘ, rad\nFIG. 3: Phase shift versus sodium temperature in the channel\nat frequency ν= 97Hz: experiment (points) and simulations\n(solid line).\nThe change in phase shift θbetween the measured\nmagnetic field and the alternating current in the toroidal\ncoil is a value, which can be treated as a measure of log-\narithmic changes of diffusivity of the sodium\n∆θ⋍C∆ρ\nρ=C∆η\nη, (2)\nwhereCis a dimensional coefficient, which depends on\nthe geometry and resistivity of the channel wall, and on\nthe frequency of the applied magnetic field. The measur-\ning system is completed with software, based on wavelet\nanalysis, which provides calculation of the time depen-\ndence of phase shift after signals recording. Wavelets are\nrequired because the variation of phase shift occurs at\ntimes comparable with the oscillation period.\nThe measurement system has been tested and cali-\nbrated by measuring the dependence of the sodium re-3\nsistivity on the temperature. The channel containing the\nsodium was cooled down from 105◦C to 80◦C. This range\nof temperature includes the sodium freezing point, which\ngives the best measure for calibration because the resis-\ntivity of the sodium decreased at that point by 31% per-\ncent, while the temperature remained constant. This ex-\ncludes the influence of resistivity variation of titanium,\ncoils, etc. Fig. 3 shows the results of phase shift mea-\nsurements performed at frequency ν= 97 Hz, together\nwith results of numerical simulations. For this frequency,\nthe skin layer thickness of titanium is about 44 mm (the\nmean thickness of titanium wall is about 10mm) and the\nskin layer thickness of sodium is about 16 mm.\nTheoretical phase shift in the skin layer of an infinite\ncylindrical solenoid, which includes a titanium cylinder\ntube with sodium, fits the experimental points well, and\nallowsustodefinethefactorofproportionalityinrelation\n(2) for each applied frequency. For the case ν= 97Hz,\nshown in Fig. 3, C= 102±3 mrad. For verification\nof the method, an alternative approach of evaluating the\nsodium resistivity was used, based on the equivalent elec-\ntrotechnical schematic of the transformer with the short-\ncircuited secondary winding, which gave close results.\nAll dynamical experiments concerning the turbulent\nflow of liquid sodium were performed under the fixed\ntemperature T= (102±1)◦C. The estimation of sodium\nheating due to energy dissipation in decaying turbulent\nflow at the highest rotational velocity f= 50 r.p.s. (con-\nsidering that its entire kinetic energy will dissipate in the\nheat) gives ∆ T≈0,8◦C, which correspondsto variations\nof resistivity less than 0 .5%.\nResults and Summary. The rotational velocity Ω\nvaried from 10 to 45 r.p.s. with a step of 5 r.p.s. Mea-\nsurements for all Ω were performed using three different\nfrequencies ν(53, 66 and 97 Hz). The evolution of the\nphase shift, measured at frequency ν= 97Hz for differ-\nent velocity of channel rotation Ω, is shown in Fig. 4.\nEach curve is the result of averaging over 10 realiza-\ntions. The end of braking is defined as the reference time\npoint (t= 0). One can see that braking generates the\nturbulent flow, the maximal intensity of which coincides\nwith the end of braking. At this moment the phase shift\nalsoreachesits maximum. Lateron, turbulent pulsations\nrapidly decay and the phase shift reduces to zero.\nThe inset of Fig. 4 shows that the measured phase\nshift decays exponentially, which contradicts the ideas\nabout the free decay of developed turbulence, which are\nrested on the power laws. The turbulent boundary layer\nin the nonstationary toroidal flow is developed in a very\nspecificway. Thiswasfound instudies ofthe dynamicsof\na similar flow of liquid gallium, which have shown that\nthe decay of the mean energy of the turbulent flow in\nthe toroidal channel follows the t−2law, while the burst\nof turbulent pulsations attends the flow formation and\ndeceases abruptly [14]. This is an additional argument\ntosupposethatthemeasuredphaseshiftismostlycaused0.0 0.5 1.0 1.5 2.0 2.5010203040\nt,s/DifferenceDeltaΘ, mrad\n0.00.51.01.52.02.51251020\nFIG. 4: Phase shift variations with flow evolution for channe l\nrotation rate Ω = 10 ,15,...,40,45r.p.s. (bottom-up) in linear\nand lin-log (inset) scales. ν= 97Hz.\n3136414753\n5966\n85\n117\n190\n253\n516\n/Minus0.2 0.0 0.2 0.4 0.6 0.8 1.001020304050\nt,s/DifferenceDeltaΘ, mrad\nFIG. 5: Phase shift variations with flow evolution for channe l\nrotation rate Ω = 40r.p.s. and different frequency ν, shown\nnear each curve.\nby small-scaleturbulence, but not by the dynamics ofthe\nmean flow.\nWe have examined the flow acrossa broad range of fre-\nquencies, 31 ≤ν≤516Hz (the skin layerthicknessvaries\nthen from 29 to 7 mm). Fig. 5 shows the phase shift evo-\nlution fordifferentfrequenciesand it confirmsthe general\nidea that the turbulent diffusivity should follow the in-\ntensity of turbulent pulsation, which grows from the wall\nof the channel to its center – at a low frequency the con-\ntribution of the central part of the flow is larger and the\nβ-effect is more pronounced. At its highest frequency the\nmeasuring system senses only the boundary layer, which\ndeveloped first: for ν= 516Hz the phase shift achieves\nthe maximum at t=−0.15sec, while for ν= 31Hz the\nmaximum appears only at t= 0.4sec.\nInFig.6weshowhowtheobserved β-effectdependson\nthe intensity of the mean flow (on the Reynolds number,\nwhich is defined by the channel rotation ratebefore brak-\ning). First, we show (in the upper panel) the maximal\ndeviation of effective magnetic diffusivity, which corre-\nsponds to the end of braking, from the basic value. Mea-\nsurements are taken using three frequencies: ν= 53, 66,4\n10 15 20 25 30 35 40 45102030405060\n/CapOmega, r.p.s./DifferenceDeltaΗΗ/Minus1,/Percent\n1/Slash12\n10 20 30 15251020\n/CapOmega, r.p.s./DifferenceDeltaΗΗ/Minus1,/Percent\nFIG. 6: Relative increase of magnetic diffusivity (percenta ge)\nversus channel rotation rate Ω at the end of braking (top) and\nat 0.7sec later (bottom): ν= 53Hz (solid, black), ν= 66Hz\n(dashed, blue), ν= 97Hz (dash-dot, red). The lower panel is\nshown in logarithmic scale; dots show the power law ”1/2”.\nand 97 Hz. Changing frequency, we vary the depth of\npenetration of magnetic field into the turbulent flow. As\nthe frequency is lowered, the thicker the skin layer be-\ncomes and the more pronounced is the observed β-effect.\nThe maximal value (for Ω = 45r.p.s. and ν= 53Hz)\nexceeds 50%. At low rotation rates the effect increases\nmonotonically, in a similar manner for all frequencies;\nhowever, with Ω >30r.p.s., the monotony is disrupted\nandthe curvesdevelopin disorder. Examiningindividual\ncurves for different realizations, it is possible to see that\nwith high rotational speeds, the structure of the curve\nnear the maximum becomes very complex – the maxi-\nmum becomes wider with a kind of plateau, against the\nbackground which appears to be separate distinct max-\nimums. All these peculiarities disappear very shortly –\nin Fig. 4 one can see that at t≈0.2−0.3, all curves\nevolve quite similar without any deviation. We show in\nthe lower panel of Fig. 6 the deviation of effective mag-\nnetic diffusivityat t= 0.7sec. Thenallthreecurvesshow\nsimilar monotonic increase of the β-effect. Shown in log-\narithmic scales, they display a tendency toward a power\nlaw ∆η∼Ω1/2at high rotational velocity.\nSo, the measurement of electric conductivity in the\nnonstationary fully developed turbulent (Re /lessorsimilar3·106)flow of liquid sodium in a closed channel shows that the\neffective magnetic diffusivity essentially increases with\nthe Reynolds number. For the maximal rotation rate\nΩ = 45r.p.s., which corresponds to Rm ≈30, the\nmaximal deviation of magnetic diffusivity reaches about\n50%. Experiments with liquid gallium at low magnetic\nReynolds number (Rm <1) revealed a quadratic like de-\npendence β∼(Rm)2[15], which corresponds to general\nconceptions of the beta-effect for low Rm. Our results\nshow that the quadratic law does not hold at moderate\nRm. Note that the turbulent viscosity in stationary pipe\nflows at high Re increases as νt∼Re1/2[16] and our re-\nsults show at the highest Reynolds numbers a tendency\nto the same power law. One should treat the obtained\ndependence to the case of stationary pipe flow, or to ho-\nmogeneous turbulence, with great caution. However, in\nview of the fact that the problem of measuring the exam-\nined characteristic in real flows is very complicated, and\nthat experimental data are completely absent, measure-\nment ofthe effective magnetic diffusivity in the turbulent\nmedium, evenin oneparticularcase, is an importantstep\ntoward the experimental substantiation of general MHD-\ndynamo conceptions.\nThis work was supported by ISTC project 3726 and\nRFBR-SNRS grant No. 07-01-92160.\n∗Electronic address: frick@icmm.ru\n[1] M. Steenbeck, F. Krause, and K. R¨ adler, Z. Naturforsch.\nA21, 369 (1966).\n[2] H. K. Moffatt, Magnetic Field Generation in Electrically\nConducting Fluids (Cambridge University Press, Cam-\nbridge, 1978).\n[3] F. Krause and K.-H. R¨ adler, Mean-field Magnetohydro-\ndynamics and Dynamo Theory (Pergamon Press, New\nYork, 1980).\n[4] Y. B. Zeldovich, A. A. Ruzmaikin, and D. D. Sokoloff,\nMagnetic Fields in Astrophysics (New York: Gordon and\nBreach, 1983).\n[5] F. Stefani, A. Gailitis, andG. Gerbeth, Zamm-Zeitschri ft\nFur Angewandte Mathematik Und Mechanik 88, 930\n(2008).\n[6] A. Gailitis, O. Lielausis, S. Dement’ev, E. Platacis,\nA. Cifersons, G. Gerbeth, T. Gundrum, F. Stefani,\nM. Christen, H. H¨ anel, et al., Phys. Rev. Lett. 84, 4365\n(2000).\n[7] R. Monchaux, M. Berhanu, M. Bourgoin, M. Moulin,\nP. Odier, J. Pinton, R. Volk, S. Fauve, N. Mordant,\nF. P´ etr´ elis, et al., Phys. Rev. Lett. 98, 044502 (2007).\n[8] R. Stieglitz and U. M¨ uller, Phys. Fluids 13, 561 (2001).\n[9] E. J. Spence, M. D. Nornberg, C. M. Jacobson, C. A.\nParada, N. Z. Taylor, R. D. Kendrick, and C. B. Forest,\nPhys. Rev. Lett. 98, 164503 (2007).\n[10] M. Gellert and G. R¨ udiger, Phys. Rev. E 80, 046314\n(2009).\n[11] A. B. Reighard and M. R. Brown, Phys. Rev. Lett. 86,\n2794 (2001).\n[12] P. Frick, V. Noskov, S. Denisov, S. Khripchenko,5\nD. Sokoloff, R. Stepanov, and A. Sukhanovsky, Magne-\ntohydrodynamics 38, 143 (2002).\n[13] R. Stepanov, R. Volk, S. Denisov, P. Frick, V. Noskov,\nand J. Pinton, Phys. Rev. E 73, 046310 (2006).\n[14] V. Noskov, R. Stepanov, S. Denisov, P. Frick, G. Verhill e,\nN.Plihon, and J. Pinton, Phys. Fluids 21, 045108 (2009).[15] S. A. Denisov, V. I. Noskov, R. A. Stepanov, and P. G.\nFrick, JETP Lett. 88, 167 (2008).\n[16] H. Schlihting, Grenzschicht-Theories (G.Braun, Karl-\nsruhe, 1964)."    },    {        "title": "2011.13562v1.Vector_Hamiltonian_Formalism_for_Nonlinear_Magnetization_Dynamics.pdf",        "content": "Vector Hamiltonian Formalism for Nonlinear Magnetization Dynamics\nVasyl Tyberkevychy, Andrei Slaviny, Petro Artemchuky, Graham Rowlandsz\u0003\nyDepartment of Physics, Oakland University, Rochester, MI 48309, USA and\nzRaytheon BBN Technologies, Cambridge, MA 02138, USA\n(Dated: November 30, 2020)\nVector Hamiltonian formalism (VHF) for the description of a weakly nonlinear magnetization\ndynamics has been developed. Transformation from the traditional Landau-Lifshitz equation, de-\nscribing dynamics of a magnetization vector m(r;t) on a sphere, to a vector Hamiltonian equation,\ndescribing dynamics of a spin excitation vector s(r;t) on a plane, is done using the azimuthal Lam-\nbert transformation that preserves both the phase-space area and vector structure of dynamical\nequations, and guarantees that the plane containing vector s(r;t) is at each value of the coordinate\nrperpendicular to the a stationary vector m0(r) describing the magnetization ground state of the\nsystem. By expanding vector s(r;t) in a complete set of linear magnetic vector eigemodes s\u0017(r)\nof the studied system, and using a weakly nonlinear approximation js(r;t)j\u001c1, it is possible to\nexpress the Hamiltonian function of the system in the form of integrals over the vector eigenmode\npro\fless\u0017(r), and calculate all the coe\u000ecients of this Hamiltonian. The developed approach al-\nlows one to describe weakly nonlinear dynamics in micro- and nano-scale magnetic systems with\ncomplicated geometries and spatially non-uniform ground states by numerically calculating linear\nspectrum and eigenmode pro\fles, and semi-analytically evaluating amplitudes of multi-mode non-\nlinear interactions. Examples of applications of the developed formalism to the magnetic systems\nhaving spatially nonuniform ground state of magnetization are presented.\nI. INTRODUCTION\nWeakly nonlinear dynamics of waves having di\u000ber-\nent physical nature is strikingly similar. Nonlinear\nresonance, parametric instabilities, self-interaction and\nself-focusing leading to the formation of one- and two-\ndimensional solitons, generation of higher harmonics {\nall these e\u000bects are common features for the dynamics of\nweakly nonlinear optical waves, waves in plasma, waves\non a liquid surface, spin waves in magnetically ordered\nmaterials, etc. Therefore, it is only natural that there\nwere many attempts to develop a generalized theoretical\ndescription of weakly nonlinear dynamics of waves, and\nthis description was based on the classical Hamiltonian\nformalism1{5. The idea of this approach is to \fnd in each\nparticular case a pair of canonically conjugated variables,\nin terms of which the natural equations of motion describ-\ning a particular wave system are transformed into a pair\nof standard Hamiltonian equations6, while the energy of\nthe system becomes a Hamiltonian function. Further-\nmore, a canonical transformation to complex canonical\nvariables (see6and section 1.1.2 in3for details) allows\none to replace two real Hamiltonian equations by one\ncomplex equation, and leaves a considerable freedom in\nthe actual choice of these complex variables, that now\nhave the same dimension. The Hamiltonian approach\nwas quite productive, and allowed to consider the weakly\nnonlinear wave process from a general point of view, in-\ndependently of the physical nature of a particular wave\nsystem.\nIn this work we are mainly interested in the nonlin-\near dynamics in the system of spin waves in magnetically\nordered materials, and will concentrate on this particu-\nlar case. The dynamics of the normalized magnetization\nvector in this case is described by the Landau-Lifshitzequation (LLE) Eq. (1) which, naturally, conserves the\nlength of the magnetization vector jmj= 1 in recognition\nof a very strong uniform internal exchange magnetic \feld\nexisting inside a ferromagnetic material. The Hamilto-\nnian approach to this important system was, \frst, in-\ntroduced by Schloemann7with the help of the Holstein-\nPrimako\u000b transformation8, and was further developed\nin Refs.2,3,9. An alternative transformation bringing the\nvectorial LLE to the complex Hamiltonian form was in-\ntroduced in Ref.10. The Hamiltonian approach developed\nfor spin waves in an unbounded ferromagnet in Refs.2,3,9\nwas quite successful. It made possible the calculation\nof explicit expressions for the spin wave spectrum and\nfor the three-wave and four-wave nonlinear coe\u000ecients of\nspin wave interactions in unbounded ferromagnetic and\nantiferromagnetic dielectrics. It also provided a quantita-\ntive theory describing the parametric excitation of spin\nwaves and nonlinear stage or a weak spin wave turbu-\nlence.\nLater, this Hamiltonian formalism was applied for\nthe case of spin waves propagating in ferromagnetic\n\flms of a \fnite thickness11{14where magnetic eigen-\nexcitations are non-uniform spin wave waves that have\na discrete spectrum and are described by plane waves\nin the \flm plane, but have well-de\fned distributions of\nmagnetization along the \flm thickness determined by\nthe boundary conditions for the magnetization at the\n\flm surfaces. Very recently this formalism has been ex-\ntended to include anti-symmetric interactions, such as\nDzyaloshinskii-Moriya interaction15.\nIt should be noted, that the technical calculations of\nthe interaction coe\u000ecients entering the spin wave Hamil-\ntonian for magnetic \flms12{15performed in the frame-\nwork of the classical Hamiltonian formalism for spin\nwaves2,3are quite cumbersome and technically ratherarXiv:2011.13562v1  [cond-mat.mtrl-sci]  27 Nov 20202\ncomplicated, because in a vectorial LLE Eq. (1) it is nec-\nessary to make a transformation to complex scalar canon-\nical variables expressed in terms of Cartesian components\nof the magnetization vector (see e.g. section 3.4.2 in3or\nEqs. (3-10){(3-13) in13). The situation here is similar\nto the situation in electrodynamics where, when switch-\ning to Cartesian projections of vectors, you get instead of\nfour vector Maxwell equations twelve scalar equations for\nthe projections of electromagnetic \feld vectors. Also, the\nstandard scalar canonical transformations bringing LLE\nto a complex Hamiltonian form are applicable only in the\ncase of spatially uniform ground state of a static mag-\nnetization. Modi\fcation of the standard approach to a\nspatially-nonuniform magnetization ground state brings\nin even more technical di\u000eculties and was used only in a\nfew special cases16.\nAt the same time, the recent progress in nano-\nmagnetism created a necessity to study nonlinear spin\nwave processes in micro- and nano-sized magnetic\nsamples that can have strongly non-uniform magnetic\nground states in the form of magnetic vortices17,18\nor skyrmions19, states containing well-de\fned domain\nwalls20, or non-uniform ground states simply related to\na \fnite lateral size of a magnetic sample21.\nThus, it is highly desirable to develop a modi\fed vari-\nant of the Hamiltonian formalism for spin waves, that\nis more compact by being able to deal directly with the\nmagnetization vector without the necessity to use com-\nplex canonical variable composed of the magnetization\nprojections, and, also, capable to deal with the cases\nwhen the magnetic ground state of a considered object is\nspatially non-uniform. The necessity of such an advanced\nHamiltonian approach is also supported by the progress\nin the research in macroscopic quantum phenomena in-\nvolving magnons22, which now has shifted in the di-\nrection of investigation of the nonlinear properties23,24\nand secondary magnetic excitations in the dense magnon\ngases25and Bose-Einstein condensates of magnons26.\nFurther development of the nonlinear theory of spin wave\ngeneration27, propagation and synchronization28in mag-\nnetic nanostructures also will strongly bene\ft from the\nintroduction of a novel vectorial Hamiltonian formalism\nfor spin waves.\nOur current work represents an attempt to develop a\nvectorial Hamiltonian formalism for spin waves. Since\nthe magnetization dynamics governed by the LLE Eq. (1)\nis dynamics on a sphere of a unit radius, our \frst goal\nwould be to map this dynamics vectorially (see Eq. (11))\non a plane tangential to this sphere, and containing two-\ndimensional vector of a magnetic excitation swhich is\neverywhere orthogonal to the coordinate-dependent vec-\ntorm0describing the spatially non-uniform magnetic\nground state of the system. This approach allows us\nto deal only with vector quantities and, eventually, ob-\ntain the expressions of all the spin wave interaction co-\ne\u000ecients in the relatively simple and compact vectorial\nform.II. VECTOR HAMILTONIAN FORMALISM\nIn this section we consider the core part of the vector\nHamiltonian formalism (VHF), namely, weakly nonlin-\near autonomous conservative dynamics of magnetic exci-\ntations. For simplicity, we shall assume that the con-\nsidered magnetic body has a \fnite volume Vs, which\nmeans that the spectrum of eigen-excitations is discrete\nand spin wave eigen-modes have \fnite support and \f-\nnite norms. This assumption is not critical for the de-\nveloped formalism and, using standard methods of solid-\nstate theory, one can easily adapt it for description of\nmagnetic excitations in in\fnite systems ( e.g., plane spin\nwaves in bulk samples or spin wave modes in thin mag-\nnetic \flms). Modi\fcations of the VHF to the case of dis-\nsipative ( e.g., Gilbert damping or spin transfer torque)\nand/or non-autonomous ( e.g., excitation of spin waves\nby a microwave magnetic \feld) interactions will be con-\nsidered in Sec. III in the framework of the general per-\nturbation theory.\nA. Landau-Lifshits Equation\nThe starting point in theoretical analysis of any mag-\nnetization dynamics problem is the Landau-Lifshits equa-\ntion (LLE), which can be written as\n@m\n@t=\r(Be\u000b\u0002m); (1)\nwherem\u0011m(t;r) is the unit vector along the magne-\ntization direction, \ris the modulus of the gyromagnetic\nratio, andBe\u000bis the e\u000bective magnetic \feld connected\nwith the energy (Hamiltonian) Hof the magnetic system\nby\nBe\u000b=\u00001\nMs\u000eH\n\u000em: (2)\nHereMsis the saturation magnetization and \u000e=\u000emde-\nnotes variational derivative with respect to the \feld m.\nNote, that, in a general case, we allow both \randMs\nto depend on position r,i.e., the developed formalism\ncan be used for description of spin dynamics in spatially\nnonuniform magnetic samples or/and magnetic systems\ncomposed of several di\u000berent magnetic materials.\nThe LLE Eq. (1) can be derived using the least action\nprinciple from the Lagrangian function\n L =Z\nLs\u0012dA\ndt\u0013\ndr\u0000H; (3)\nwhere\nLs\u0011Ms=\r (4)\nis the density of the spin angular momentum (spin den-\nsity) associated with the magnetization Msof the mag-\nnetic medium and dAis the area element encircled by3\nthe moving vector mon the unit sphere:\ndA\u0011n\u0002m\n1 +n\u0001m\u0001dm: (5)\nHerenis an arbitrary unit vector, possibly position-\ndependent (but independent of time). Di\u000berent choices\nofnlead to Lagrangian functions Eq. (3) that di\u000ber by a\ncomplete time derivative and, therefore, induce the same\nequation of motion Eq. (1).\nThe \frst term in the Lagrangian Eq. (3) is the rate of\nchange of phase-space area due to the motion of the mag-\nnetization vector mand is analogous to the term p_qin a\nstandard phase-space Lagrangian dynamics. The compli-\ncated form of the area element Eq. (5) is due to the fact\nthat the phase space of a magnetization vector is a sphere\nrather than a plane , as it is for standard Lagrangian and\nHamiltonian systems. One can signi\fcantly simplify de-\nscription of dynamics of a magnetic system by projecting\nspherical phase space of minto a plane, which is the main\nidea of the current work and previous approaches based\non classical complex Hamiltonian formalism for magne-\ntization dynamics. Our approach di\u000bers from the pre-\ndecessors by the choice of the projection function (see\nSec. II B below), which, we believe, is much better suited\nfor modern problems in magnetization dynamics.\nWe shall write the magnetic energy Hof the system in\nthe form\nH=Z\nVs\u0014\n\u0000MsBext\u0001m+1\n2m\u0001cH\u0001m\u0015\ndr: (6)\nHereBext\u0011Bext(r) is the external magnetic \feld\nandcHis a certain Hermitian operator describing self-\ninteractions in the system. Most of the common mag-\nnetic self-interactions can be written in such form. Thus,\nthe inhomogeneous exchange is described by the operator\ncHex=\u0000\u00160M2\ns\u00152\nexr2; (7a)\nwhere\u00160is the vacuum permeability and \u0015ex= p\nA=(\u00160M2s) is the exchange length ( Ais the exchange\nsti\u000bness). The operator of the dipolar interaction can be\nwritten symbolically as\ncHdip=\u00160M2\nsrr\u00002r: (7b)\nIn the case of easy-axis uniaxial anisotropy with\nanisotropy axis nanand e\u000bective \feld Ban= 2Ku=Ms\n(Kuis the energy density of uniaxial anisotropy) the in-\nteraction operator reads\ncHan=\u0000MsBannan\nnan; (7c)\nwhere\ndenotes direct vector product. An easy-plane\nanisotropy is described by the same expression Eq. (7c)\nwith negative \feld Ban<0. Finally, the Dzyaloshinskii-\nMoriya interaction (DMI), in the most general case, can\nbe described by the tensor operator\ncHDMI=b\u0000DMI\u0001r; (7d)whereb\u0000DMI is a certain third-rank tensor. Strictly\nspeaking, Eqs. (7) are valid only in the usual case of\nmagnetically-uniform medium (for example, one can eas-\nily see that the dipolar operator Eq. (7b) is not Hermitian\nifMsdepends onr), but their correction for non-uniform\ncase does not represent any di\u000eculties.\nThe only relatively common magnetic interaction that\ncannot be described by the bi-linear magnetic energy op-\neratorcHis cubic crystallographic anisotropy. Descrip-\ntion of cubic anisotropy would require modi\fcation of\nEq. (6) to include additional term proportional to m4.\nThe proposed formalism can be generalized to such cases\nwithout any principal changes, but it would lead to more\ncomplicated expressions for all linear and nonlinear coef-\n\fcients, and we will not consider such cases here.\nFor the choice of the energy functional Eq. (6), the\ne\u000bective magnetic \feld Eq. (2) takes the simple form\nBe\u000b(m) =Bext\u0000M\u00001\nscH\u0001m; (8a)\nwhich is linear in mand clearly demonstrates that non-\nlinearity of the magnetization dynamics in the system\nEq. (6) is connected solely with the curvature of the phase\nspace.\nOne can also rewrite Eq. (8a) as\ncH\u0001m=\u0000Ms[Be\u000b(m)\u0000Bext] =\u0000MsB(0)\ne\u000b(m);(8b)\nwhereB(0)\ne\u000b(m) is the self-interaction ( i.e., without ex-\nternal \feldBext) e\u000bective magnetic \feld created by the\nmagnetization distribution m. This equation shows, that\nthe operator cHcan be simply expressed through the\ne\u000bective magnetic \feld. Note, that all numerical LLE\nsolvers provides means for calculation of the e\u000bective \feld\nBe\u000b(m) and, respectively, numerical calculation of the\naction of the operator cHon any magnetization \feld m\ndoes not represent any di\u000eculty and does not require any\ncomplicated coding.\nTo proceed with the problem of weakly nonlinear mag-\nnetization dynamics, one also has to specify stationary\n(or ground) magnetization state m0(r), around which\nthe dynamics occurs. The ground state m0(r) is a sta-\ntionary solution of Eq. (1) and can be found from the\nequation\nBe\u000b(m0) =Bext\u0000M\u00001\nscH\u0001m0=B0m0;(9)\nwhereB0\u0011B0(r) is the scalar internal magnetic \feld.\nIn general, a magnetic system can have several ground\nstates for the same set of parameters. In the follow-\ning, we shall assume that the ground state of interest is\nknown (both m0andB0). We would like to emphasize,\nthat we do not assume that the ground state is spatially-\nuniform, and the developed formalism can be used for\ndescription of spin excitations of highly inhomogeneous\nmagnetic states such as e.g., a domain wall, a magnetic\nvortex, or a magnetic skyrmion.4\nNote, that the magnetic energy Eq. (6) can be written\nas\nH(m)=H(m0) (10)\n+1\n2Z\nVs(m\u0000m0)\u0001(cH+MsB0bI)\u0001(m\u0000m0)dr;\nwhereH(m0) is the ground state energy, which does\nnot in\ruence the magnetization dynamics and will be\nignored in the following, and bIis the identity operator.\nEq. (10) shows that the \frst-order (in magnetization de-\nviation (m\u0000m0)) contribution to the magnetic energy\nvanishes near the ground state m0. It also shows that\nthe statem0is stable (corresponds to an energy mini-\nmum) if the operator ( cH+MsB0bI) is positive-de\fnite\nfor allowed small deviations ( m\u0000m0). Below we shall\nassume that this condition holds unless otherwise stated.\nB. Spin Excitation Vector\nTo simplify the description of the magnetization dy-\nnamics, we shall project the spherical phase space of the\nmagnetization vector minto a plane. Namely, at every\npointrwe project the unit sphere m(t;r) into the plane\ns(t;r) orthogonal to the ground state m0(r) at this point\n(m0\u0001s\u00110) using the transformation\nm=\u0012\n1\u0000s2\n2\u0013\nm0+r\n1\u0000s2\n4s: (11)\nHeres=jsj. The vector s(t;r) will be called spin ex-\ncitation vector (SEV) below. As one can easily show,\nEq. (11) provides conservation of the length of vector m\n(m\u0001m= 1) for any choice of sorthogonal to m0.\nThe mapping Eq. (11) is known in cartography as the\nLambert azimuthal equal-area projection (see Fig. 1) and\nhas two important properties. First, it is an equal-area\ntransformation, i.e., it maps an arbitrary region on the\nsphereminto a region in the plane sof a di\u000berent shape,\nbut the same area. The equal-area property ensures that\nthe Hamiltonian nature of magnetization dynamics will\nbe preserved after the transformation to the SEV s. Sec-\nond, Eq. (11) is a simple vector transformation, which\nmeans that the vector structure of the equations of mo-\ntion is preserved by the transformation. As we shall see\nbelow, it leads to compact and coordinate-independent\nexpressions for all the coe\u000ecients and operators describ-\ning magnetization dynamics.\nVectorslies in a two-dimensional plane that is natu-\nrally embedded in the three-dimensional ( x;y;z ) space.\nThere are two methods how such vectors can be de-\nscribed in technical calculations. First, one can choose\ntwo unit vectors, e1ande2, in the SEV plane and de-\nscribesusing coordinates s1ands2. This method is\neasy to implement in the case of a uniform ground state\n(m06=f(r)), when the unit vectors e1ande2may also\nbe chosen independently of r. The choice of unit vectors\nFIG. 1. Lambert azimuthal projection Eq. (11) of the Earth\nsurface with the \\equilibrium direction\" m0at the North\npole. Red ellipses indicate projections of small-size equal-area\ndisks. Note, that noticeable distortions of the disk shapes\nstart only in the southern hemisphere.\ne1(r) ande2(r), however, is much more complicated in\nthe case of non-uniform state m0(r) and this problem\neven may not have a regular solution at all (due to the\n\\hairy ball\" theorem). In such non-uniformm cases it\nis much easier to describe sas a point in the embedding\nthree-dimensional space subject to the orthogonality con-\nstraintm0\u0001s= 0. This method is analogous to the\ndescription of the unit magnetization vector m(which\nbelongs to a two-dimensional manifold { unit sphere) us-\ning three Cartesian coordinates mx,my, andmzsubject\nto the constraint m2\nx+m2\ny+m2\nz= 1. The presented be-\nlow theory is written in a coordinate-independent vector\nform, and one can use either method in technical calcula-\ntions. All main equations of the theory were speci\fcally\nwritten in the form in which the orthogonality constraint\nis automatically satis\fed (similar to automatic satisfac-\ntion of the condition djmj=dt= 0 by the Landau-Lifshits\nequation). We would also like to note, that in the clas-\nsical complex Hamiltonian formalism of magnetization\ndynamics, which uses similar basic ideas, one always has\nto use the \frst description method (using unit vectors\nin the plane), and this is one of the reasons why this\nmethod has never been successfully used for analysis of\nspin excitations on a non-uniform background.\nThe transformation Eq. (11) maps the whole unit\nsphere into the disk jsj<2 (note that the unit sphere and\nthe disk have the same area of 4 \u0019). In the ground state\nm=m0the spin excitation vector is equal to zero, s= 0,\nthus the SEV sis a measure of deviation of magnetization\nfrom the ground state (measure of excitation of the spin5\nsystem). The mapping of the antipode point m=\u0000m0\nis not uniquely determined by Eq. (11) (it is mapped into\nthe whole circlejsj= 2), which is connected with di\u000ber-\nent topologies of a sphere and a plane. In reality, Eq. (11)\nis useful for description of weakly nonlinear magnetiza-\ntion dynamicsjsj\u001c2. Note, however, that the ground\nstatem0itself can be strongy non-uniform, i.e., the pro-\nposed approach can be used to describe weakly nonlinear\ndynamics of such magnetic objects as domain walls, vor-\ntices, or skyrmions. In the weakly nonlinear case jsj\u001c2\nEq. (11) can be expanded in the Taylor series as\nm=m0+s\u0000s2\n2m0\u0000s2\n8s+O(s5): (12)\nAs one can see, in the linear limit the SEV sis equal to\nthe magnetization deviation ( m\u0000m0), which makes it a\nvery convenient object for mixing analytical and numer-\nical approaches: practically at any point of the theoret-\nical analysis one can use interchangeably either of these\nmethods to solve a particular part of the problem, and\nswitching from one approach to another does not require\nany additional transformations.\nThe inverse to Eq. (11) transformation is given by\ns=r\n2\n1 +m0\u0001mbP0\u0001m (13)\n=bP0\u0001m+1\u0000m0\u0001m\n4bP0\u0001m+O(jm\u0000m0j5);\nwherebP0is the projection operator into the SEV plane\n(plane orthogonal to m0):\nbP0\u0011bI\u0000m0\nm0: (14)\nIn terms of the spin excitation vector s, the area ele-\nmentdAEq. (5) has a simple form\ndA=1\n2m0\u0001(s\u0002ds); (15)\nand the magnetic Lagrangian Eq. (3) can be written as\n L =1\n2Z\nVss\u0001bL0\u0001ds\ndtdr\u0000H(s); (16)\nwhere the skew-symmetric operator bL0is de\fned by\nbL0\u0001v\u0011\u0000Lsm0\u0002v: (17)\nRespectively, the equation of motion for the SEV shas\nthe form of a vector Hamiltonian equation\nbL0\u0001ds\ndt=\u000eH\n\u000es: (18)\nNote, that the operator bL0is invertible for vectors s\northogonal to m0:\nbL2\n0\u0001s=\u0000L2\nss;and Eq. (18) can also be written as\nds\ndt=\u0000L\u00002\nsbL0\u0001\u000eH\n\u000es:\nThus, Eq. (18) is a well-de\fned dynamical equation for\nthe SEVs.\nThe Hamiltonian H(s) of a magnetic system is ob-\ntained by substituting the transformation Eq. (11) into\nEq. (10). Weakly-nonlinear expansion of H(s) reads\nH=H2+H3+H4; (19)\nwhere\nH2=1\n2Z\nVss\u0001cH0\u0001sdr; (20a)\nH3=\u00001\n2Z\nVs(s2m0)\u0001cH\u0001sdr; (20b)\nH4=1\n8Z\nVsh\n(s2m0)\u0001cH\u0001(s2m0) (20c)\n\u0000(s2s)\u0001cH\u0001si\ndr;\nand the linear Hamiltonian of the system cH0is de\fned\nas\ncH0\u0011bP0\u0001(cH+MsB0bI)\u0001bP0: (21)\nWe kept in Eq. (19) only the terms up to fourth order\ninjsj, which is su\u000ecient for most weakly nonlinear prob-\nlems. We have also added bP0projectors in the de\fnition\nof the operator cH0Eq. (21). This does not change the\nquadratic part of the energy H2Eq. (20a) since bP0\u0001s=s,\nbut it is convenient for further analysis because now one\ncan \fnd the variational derivative \u000eH2=\u000es=cH0\u0001swith-\nout explicit taking into account the orthogonality con-\nditionm0\u0001s= 0. Note, also, that the operator cH0\nde\fned by Eq. (21) is a self-adjoint operator, which will\nbe important in the following.\nIt is also interesting to note, that the nonlinear en-\nergy termsH3andH4Eqs. (20b)-(20c) do not explicitly\ndepend on the internal magnetic \feld B0, and any de-\npendence of nonlinear properties of spin excitations on\nmagnetic \feld can be explained by the \feld dependence\nof the pro\fles of spin wave modes and \feld dependence\nof the ground state m0.\nIntroduction of the spin excitation vector sallowed\nus to formulate the magnetization dynamics in a \\\rat\"\nphase space, to which standard methods of weakly\nnonlinear dynamical systems can be directly applied.\nNamely, weakly-nonlinear dynamics of a magnetic sys-\ntem is most easily described in terms of amplitudes of\nlinear spin wave modes. The technical details of this ap-\nproach are derived in the rest of this Section.6\nC. Linear Eigenmodes of a Magnetic System\nIn the limit of linear excitations Eq. (18) becomes\nbL0\u0001ds\ndt=cH0\u0001s: (22)\nThe harmonic solutions ( d=dt!\u0000i!\u000b) of this equation\ns\u000bare the linear eigenmodes of magnetic excitations:\n\u0000i!\u000bbL0\u0001s\u000b=cH0\u0001s\u000b: (23)\nHere\u000bis the mode index, !\u000bis its eigenfrequency, and\ns\u000b\u0011s\u000b(r) is the complex eigenmode pro\fle.\nIn Eq. (23)bL0is a skew-symmetric operator, while cH0\nis a symmetric (Hermitian) operator. Thus, this equation\nis a generalized Hamiltonian eigenvalue problem, prop-\nerties of which are well studied. In an important case\nwhen the operator cH0is positive-de\fnite ( i.e., the mag-\nnetic ground state m0corresponds to a minimum of en-\nergy) the eigenvectors s\u000bform a complete set (basis) in\nthe space of vector functions orthogonal to the ground\nmagnetic state m0, and all the eigenfrequencies !\u000bare\nreal-valued (see Appendix A). This may also be true in a\ncase whencH0is not positive-de\fnite, although there is\nno guarantee. In the following, we shall assume that this\nimportant property holds.\nUsing Eq. (23) and symmetry properties of the oper-\natorsbL0andcH0one can derive two orthogonality re-\nlations for the mode pro\fles s\u000b(r) (see Appendix A for\nmathematical details). The \frst relation is\nZ\nVss\u0003\n\u000b\u0001bL0\u0001s\u000b0dr=i\u0016h\u000b\u0001\u000b;\u000b0; (24)\nwhere \u0001\u000b;\u000b0is the Kronecker delta and \u0016 h\u000bis the real-\nvalued norm of the \u000b-th mode:\n\u0016h\u000b\u0011\u0000iZ\nVss\u0003\n\u000b\u0001bL0\u0001s\u000bdr: (25)\nIn case of degenerate spectrum (several modes have the\nsame frequency !\u000b) the relation Eq. (24) should be un-\nderstood in the usual sense, i.e., that it is possible to\nchoose such combinations of degenerate eigenvectors that\nthe relation Eq. (24) holds.\nThe norm \u0016h\u000bhas dimensionality of action (angular mo-\nmentum) and is always real-valued and non-zero (but not\nnecessary positive). As it is clear from the de\fnition\nEq. (25) and form of the magnetic Lagrangian Eq. (16),\nthe norm \u0016h\u000bequals the reduced action corresponding to\nthe single-mode excitation and one period of oscillations\n2\u0019=!\u000b, divided by 2 \u0019. The choice \u0016 h\u000b= \u0016h(the usual re-\nduced Planck constant) corresponds to the quasi-classical\n\\magnon\" normalization, when the mode pro\fle s\u000bis a\nclassical analog of a magnon wavefunction and the mode\namplitude squared equals the number of magnons in a\ngiven quantum state. This analogy explains our choice\nof notation for the mode norm \u0016 h\u000b. Another choice of nor-\nmalization, which may be useful in certain applications,is the normalization to the total spin of the magnetic\nsystem, \u0016h\u000b=LsVs[for a system with inhomogeneous\nspin density Ls(r), \u0016h\u000b=R\nVsLs(r)dr], when the mode\namplitude is directly proportional to the magnetization\nprecession angle. We shall not specify a particular choice\nof normalization and all the expressions presented here\nare valid for any choice of \u0016 h\u000b, including the cases when\ndi\u000berent modes are normalized di\u000berently.\nThe orthogonality relations Eq. (24) allows one to\nproject an arbitrary vector function into particular eigen-\nstates and are necessary for development of a general\nperturbation theory (see Sec. III).\nAnother orthogonality-type relation which follows\nfrom Eq. (23) has the form\nZ\nVss\u0003\n\u000b\u0001cH0\u0001s\u000b0dr= \u0016h\u000b!\u000b\u0001\u000b;\u000b0: (26)\nThis relation can be used for precise (variationally-stable)\ndetermination of the eigenfrequencies !\u000bfrom approxi-\nmate spatial pro\fles s\u000b(r) of the eigenmodes.\nBoth operators bL0andcH0in Eq. (23) are real-valued.\nThen, ifs\u000b(r) is an eigenfunction with eigenvalue !\u000b\nand norm \u0016h\u000b, then the complex-conjugated vector s\u0003\n\u000b(r)\nis also an eigenfunction corresponding to the eigenvalue\n\u0000!\u000band norm\u0000\u0016h\u000b. Such \\doubling\" of eigenfunctions\nis a direct consequence of real-valuedness of the LLE\nEq. (1) and symmetry of the frequency spectrum of any\nreal process. Thus, only half of the formal eigenmodes of\nEq. (23) are independent and describe \\physical\" modes;\nthe other half are the formal \\conjugated\" modes that\nguarantee real-valuedness of the SEV s(t;r). The phys-\nical modes are the modes with positive norm \u0016 h\u000b>0; as\nit is clear from Eq. (26), such modes correspond to posi-\ntive eigenvalues !\u000b>0 if the operator cH0is positive-\nde\fnite ( i.e., the ground state m0is an energy min-\nimum). Respectively, conjugated modes have negative\nnorms \u0016h\u000b<0 and, for a positive-de\fnite cH0, negative\nfrequencies !\u000b<0.\nIn all the above equations the mode index \u000benumer-\nated all formal modes, both physical and conjugated. We\nshall keep these notations below and will use indices \u000b,\n\f,:::to enumerate or sum over all formal modes. To\nindicate only the physical modes, we shall use indices \u0017,\n\u0016,:::. The notation \u000b\u0003will be used to indicate a mode\n\\conjugated\" to the mode \u000b,i.e.,\ns\u000b\u0003\u0011s\u0003\n\u000b; !\u000b\u0003\u0011\u0000!\u000b;\u0016h\u000b\u0003\u0011\u0000\u0016h\u000b:\nFinally, we shall make a note on numerical determina-\ntion of the eigenfrequencies !\u000band eigenvectors s\u000b(r).\nThe technically simplest (although not the best) way to\n\fnd them, which is often used in modern research, is by\nusing Fourier analysis on results of direct numerical sim-\nulations of free-decaying magnetization dynamics after\nan initial low-amplitude (linear) perturbation from the\nground state. In this case the eigenfrequencies !\u000bcan\nbe identi\fed from peak positions in the magnetization7\nprecession spectrum, and the eigenvectors s\u000b(r) can be\nfound as cell-by-cell Fourier images of the magnetization\nat the Fourier frequencies !=!\u000b. The advantage of the\nVHF is that the mode pro\fles found from such numerical\nprocedure exactly coincide with the theoretical eigenvec-\ntorss\u000b, do not require any post-processing (except from\npossible normalization to provide desired norms \u0016 h\u000b), and\ncan be directly used in further analysis. This straight-\nforward method is very simple, can be performed using\nstandard micromagnetic packages and processing tech-\nniques, and often produce su\u000eciently accurate results.\nThe eigenvalue problem Eq. (23) can also be solved\ndirectly. In most cases, one is interested only in a\nsmall fraction of all formally possible magnetic modes,\nusually the modes with smallest eigenfrequencies j!\u000bj.\nThen, one can use a modi\fcation of the Ritz or gradi-\nent descent methods based on the variationally-stable ex-\npression Eq. (26). The derived orthogonality conditions\nEq. (24) and Eq. (26) also allow one to adopt Arnoldi\niteration and Lanczos algorithm for solution of magnetic\neigenvalue problems. These two methods are especially\npromising for study of magnetic excitations in large mag-\nnetic systems since they do not require explicit matrix\nrepresentation of the energy operator cH, but only cal-\nculation of action of this operator on individual SEVs\ns(r), which can be easily accomplished using standard\nmicromagnetic packages with the help of Eq. (8b).\nAs it was already mentioned above, the SEV s(r) at\nevery pointris a two-dimensional vector due to the re-\nstrictionm0\u0001s= 0. In numerical analysis, however,\nit is much more convenient to describe it as a three-\ndimensional vector. Such three-dimensional description\nwill, of course, lead to appearance of spurious unphysical\nformal modes (with s\u000bjjm0). In the VHF formulation\nEq. (23), however, these spurious modes do not repre-\nsent any problem, since all of them correspond to zero\neigenfrequency !\u000b= 0 and can be automatically \fltered\nout.\nThe described in this subsection general properties of\nthe linear eigenmodes of magnetic excitations allows one\nto formulate nonlinear magnetization dynamics using the\nstandard language of complex mode amplitudes and use\nstandard and well-developed Hamiltonian techniques for\nits analysis.\nD. Eigenmode Expansion of the Spin Excitation\nVector\nThe linear eigenmodes of a magnetic system s\u000b(r)\nform a complete set of vector functions orthogonal to\nthe ground state m0(r). Therefore, any time-dependent\nspin excitation vector s(t;r) can be expanded in a series\nover these eigenmodes:\ns(t;r) =X\n\u000bs\u000b(r)c\u000b(t) =X\n\u0017[s\u0017(r)c\u0017(t) + c:c:]:(27)Herec\u000b(t) is the complex amplitude of the \u000b-th mode\n(c\u000b\u0003=c\u0003\n\u000b) and c:c:stands for complex-conjugated part.\nThe \frst version of the expansion (sum over \u000b) uses\nsummation over all formal (\\normal\" and \\conjugated\")\nmodes, while the second version (sum over \u0017) explicitly\nsums up only the \\normal\" modes, whereas the \\con-\njugated\" modes are automatically included in the \\c :c:\"\npart. The two versions are completely mathematically\nequivalent, but di\u000ber in convenience of use: while the\n\frst (\u000b) version is more convenient in derivation of gen-\neral properties and relations, the second one ( \u0017) is prefer-\nable in practical analytical or numerical calculations of\nspin wave dynamics.\nUsing the orthogonality relations for the eigenmodes\ns\u000b, one can easily reformulate the magnetization dynam-\nics in terms of the mode amplitudes c\u000b. Thus, the La-\ngrangian Eq. (16) of the magnetic system takes the form\n L =i\n2X\n\u000b\u0016h\u000bc\u000b\u0003dc\u000b\ndt\u0000H=i\n2X\n\u0017\u0016h\u0017\u0012\nc\u0003\n\u0017dc\u0017\ndt\u0000c:c:\u0013\n\u0000H;\n(28)\nwhich induces the Hamiltonian equations of motion for\nthe amplitudes c\u000b:\ni\u0016h\u000bdc\u000b\ndt=@H\n@c\u000b\u0003: (29)\nThe dynamical equation written in this form is valid for\nboth normal and conjugated modes (due to the property\n\u0016h\u000b\u0003=\u0000\u0016h\u000b). It can also be rewritten as\ndc\u000b\ndt= [H;c\u000b]; (30)\nwhere [\u0001;\u0001] is the Poisson brackets of the system:\n[A;B]\u0011X\n\u000bi\n\u0016h\u000b@A\n@c\u000b@B\n@c\u000b\u0003: (31)\nThe Poisson-bracket form of the equations of motion can\nalso be written for any function F=F(t;fc\u000bg) of time\nand complex spin wave amplitudes c\u000b:\ndF\ndt= [H;F] +@F\n@t: (32)\nThe weakly-nonlinear expansion of the Hamiltonian H\nhas the form Eq. (19), where di\u000berent-order terms are\nexpressed through the amplitudes c\u000bas\nH2=1\n2X\n\u000b\u0016h\u000b!\u000bjc\u000bj2=X\n\u0017\u0016h\u0017!\u0017jc\u0017j2;(33a)\nH3=1\n6X\n\u000b\f\rV\u000b\f\rc\u000bc\fc\r; (33b)\nH4=1\n24X\n\u000b\f\r\u000eW\u000b\f\r\u000ec\u000bc\fc\rc\u000e: (33c)8\nHere\nV\u000b\f\r=~V\u000b\f;\r+~V\f\r;\u000b+~V\r\u000b;\f; (34a)\n~V\u000b\f;\r=\u0000Z\nVs((s\u000b\u0001s\f)m0)\u0001cH\u0001s\rdr; (34b)\nW\u000b\f\r\u000e =~W\u000b\f;\r\u000e +~W\u000b\r;\f\u000e +W\u000b\u000e;\f\r; (34c)\n~W\u000b\f;\r\u000e =Z\nVs\"\n((s\u000b\u0001s\f)m0)\u0001cH\u0001((s\r\u0001s\u000e)m0) (34d)\n\u00001\n4((s\u000b\u0001s\f)s\r)\u0001cH\u0001s\u000e\u00001\n4((s\u000b\u0001s\f)s\u000e)\u0001cH\u0001s\r\n\u00001\n4((s\r\u0001s\u000e)s\u000b)\u0001cH\u0001s\f\u00001\n4((s\r\u0001s\u000e)s\f)\u0001cH\u0001s\u000b#\ndr:\nThese expressions for the tree-magnon V\u000b\f\rand four-\nmagnonW\u000b\f\r\u000e interaction coe\u000ecients look complicated,\nhowever, their calculation requires only evaluation of var-\nious \\matrix elements\" of the energy operator cHwith\nvarious combinations of the eigenmode pro\fles s\u000b(r).\nCalculation of such \\matrix elements\" can be easily done\nnumerically once the mode pro\fles are found either an-\nalytically or from micromagnetic simulations. The in-\nteraction coe\u000ecients V\u000b\f\randW\u000b\f\r\u000e were symmetrized\nwith respect to exchange of any pair of indices ( e.g.,\nV\u000b\f\r=V\f\u000b\r=V\u000b\r\f), which is the reason for many\ncombinatorial terms in the de\fnition Eqs. (34) and com-\nplicated form of these equations.\nAccording to our convention, the nonlinear energy\nterms in Eqs. (33) are written using summation over all\nformal modes (both \\normal\" \u0017and \\conjugated\" \u0017\u0003),\nwhich leads to more compact form of these terms. How-\never, one has to remember this fact when interpreting\npossible nonlinear processes described by Hamiltonian of\ndi\u000berent orders. For instance, the three-wave Hamilto-\nnianH3contains terms proportional to c\u0003\n\u0017c\u0003\n\u00170c\u001700(\u000b=\u0017\u0003,\n\f= (\u00170)\u0003,\r=\u001700), which, in analogy with quantum\nphysics, can be interpreted as a process of parametric de-\ncay of magnon \u001700into two magnons \u0017and\u00170, and terms\nproportional to c\u0003\n\u0017c\u0003\n\u00170c\u0003\n\u001700(\u000b=\u0017\u0003,\f= (\u00170)\u0003,\r= (\u001700)\u0003),\nwhich describe creation of three magnons from vacuum\nstate. The Hamiltonian H3also contains \\conjugated\"\nprocesses (proportional to c\u0017c\u0003\n\u00170c\u0003\n\u001700andc\u0017c\u00170c\u001700), and\nthe interaction coe\u000ecients describing direct and conju-\ngated processes are complex conjugates of each other:\nV\u000b\u0003\f\u0003\r\u0003=V\u0003\n\u000b\f\r; W\u000b\u0003\f\u0003\r\u0003\u000e\u0003=W\u0003\n\u000b\f\r\u000e: (35)\nIn many cases, the nonlinear processes describing\nthree-wave interactions are non-resonant in the sense that\n!\u000b\f\r\u0011!\u000b+!\f+!\r6= 0 (36)\nfor any set of modes for which V\u000b\f\r6= 0. This is, obvi-\nously, always the case for the processes of three-magnon\ncreation from vacuum c\u0003\n\u0017c\u0003\n\u00170c\u0003\n\u001700(if the ground state corre-\nsponds to a minimum of energy), but may also be true for\nparametric decay processes c\u0003\n\u0017c\u0003\n\u00170c\u001700. Such non-resonant\nthree-magnon processes can be eliminated by a weakly-\nnonlinear canonical transformation of complex spin waveamplitudes c\u000b(see Appendix B), leading to a new Hamil-\ntonian withH3= 0 and modi\fed four-magnon interac-\ntion terms\nW0\n\u000b\f\r\u000e =W\u000b\f\r\u000e (37a)\n+\u0001W\u000b\f;\r\u000e + \u0001W\u000b\r;\f\u000e + \u0001W\u000b\u000e;\f\r;\n\u0001W\u000b\f;\r\u000e =X\n\u000fV\u000b\f\u000f\u0003V\u000f\r\u000e\n2\u0016h\u000f\u00121\n!\u000b\f\u000f\u0003\u00001\n!\u000f\r\u000e\u0013\n:(37b)\nThe four-magnon processes are always resonant, and,\ntherefore, cannot be eliminated by a similar renormaliza-\ntion procedure. For this reason, in most cases it is enough\nto take into account only three-wave H3and four-wave\nH4terms to describe dynamics of any system of weakly-\nnonlinear excitations.\nUsing the form of the Hamiltonian Eq. (33), the equa-\ntion of motion for mode amplitudes c\u000bEq. (29) can be\nwritten explicitly as\ni\u0016h\u000bdc\u000b\ndt= \u0016h\u000b!\u000bc\u000b+1\n2X\n\f\rV\u000b\u0003\f\rc\fc\r (38)\n+1\n6X\n\f\r\u000eW\u000b\u0003\f\r\u000ec\fc\rc\u000e:\nThis equation describes weakly-nonlinear magnetization\ndynamics in an arbitrary magnetic system. In most prac-\ntically interesting cases, the number of e\u000eciently (reso-\nnantly) interacting spin wave modes is limited and it is\nenough to take into account only few spin wave modes\nrelevant in a studied nonlinear process. Thus, the trans-\nformation of the original Landau-Lifshits equation to a\nsystem Eq. (38) usually allows one to substantially re-\nduce the dimensionality of the phase space of the studied\nsystem and often enables analytical analysis of rather\ncomplicated nonlinear spin wave processes.\nIII. PERTURBATION THEORY\nIn the previous section we derived Hamiltonian equa-\ntions of motion for spin wave amplitudes c\u000bin the case\nof a conservative magnetic system with time-independent\nmagnetic \feld. Here we consider modi\fcations of the\nequations of motion caused by other magnetic interac-\ntions, which may be treated perturbatively. There are\ntwo di\u000berent classes of magnetic perturbations, which we\nshall consider separately.\nThe \frst class is the conservative perturbations, which\nmay be described by an additional term \u0001 H(t;m) in the\nHamiltonian of the system. The most important exam-\nple of conservative perturbations is the interaction of a\nmagnetic system with microwave magnetic \feld b(t;r),\nwhich describes excitation of spin waves by an external\nsystem. In this case the perturbation Hamiltonian has\nthe form\n\u0001H=\u0000Z\nMsb\u0001mdr: (39)9\nThe second type of perturbations is the non-\nconservative perturbations, which may be described by\nthe additional torque \u0001 T(t;r;m) in the right-hand\nside of LLE Eq. (1). The most important example of\nnon-conservative perturbations is the dissipation of spin\nwaves, which may be described by the Gilbert damping\ntorque\n\u0001T=\u000bGm\u0002@m\n@t; (40)\nwhere\u000bGis the dimensionless Gilbert damping parame-\nter.\nBelow we shall consider these two examples.\nThe spin wave amplitudes c\u000bof the VHF are Hamilto-\nnian dynamical variables, described by the same Hamil-\ntonian as the original magnetic system. Therefore, anal-\nysis of in\ruence of any conservative perturbation within\nthe VHF is very simple: the perturbation will lead to an\nadditional Hamiltonian term in the equation of motion\ni\u0016h\u000b\u0012dc\u000b\ndt\u0013\npert=@\u0001H\n@c\u000b\u0003; (41)\nwhere the perturbation Hamiltonian \u0001 Hshould be ex-\npressed through the spin wave amplitudes c\u000b.\nWe shall consider the particular example Eq. (39) of\nexternal magnetic \feld. Substituting into Eq. (39) the\napproximate expression Eq. (12) and expanding the SEV\ns(t;r) over the spin wave modes Eq. (27), one obtains\nthe explicit expression for \u0001 H(t;c\u000b):\n\u0001H= \u0001H1+ \u0001H2+ \u0001H3; (42)\nwhere we have dropped irrelevant constant energy term\nand\n\u0001H1=X\n\u000bP\u000bc\u000b; (43a)\n\u0001H2=1\n2X\n\u000b1;\u000b2Q\u000b1\u000b2c\u000b1c\u000b2; (43b)\n\u0001H3=1\n6X\n\u000b1;\u000b2;\u000b3R\u000b1\u000b2\u000b3c\u000b1c\u000b2c\u000b3: (43c)\nHere the excitation coe\u000ecients P\u000b(t),Q\u000b1\u000b2(t), and\nR\u000b1\u000b2\u000b3(t) are given by\nP\u000b=\u0000Z\nMsb\u0001s\u000bdr; (44a)\nQ\u000b1\u000b2=Z\nMs(b\u0001m0)(s\u000b1\u0001s\u000b2)dr; (44b)\nR\u000b1\u000b2\u000b3=~R\u000b1;\u000b2\u000b3+~R\u000b2;\u000b3\u000b1+~R\u000b3;\u000b1\u000b2;(44c)\n~R\u000b1;\u000b2\u000b3=1\n4Z\nMs(b\u0001s\u000b1)(s\u000b2\u0001s\u000b3)dr:(44d)\nSimilarly to the nonlinear self-interaction coe\u000ecients\nV\u000b1\u000b2\u000b3andW\u000b1\u000b2\u000b3\u000b4, the coe\u000ecients P\u000b,Q\u000b1\u000b2, and\nR\u000b1\u000b2\u000b3, which describe interaction of the spin systemwith external \feld, are expressed as simple \\overlap in-\ntegrals\" and their calculation does not represent any dif-\n\fculty once the spin wave pro\fles s\u000b(r) are known.\nFinally, we can write explicit expression for the ex-\ncitation term in the equation of motion for spin wave\namplitudes c\u000b:\ni\u0016h\u000b\u0012dc\u000b\ndt\u0013\nexcitation=P\u000b\u0003+X\n\u000b1Q\u000b\u0003\u000b1c\u000b1+1\n2X\n\u000b1\u000b2R\u000b\u0003\u000b1\u000b2c\u000b1c\u000b2:\n(45)\nThe \frst term in the right-hand side of this equation\n(P\u000b\u0003) describes linear excitation of spin waves, the sec-\nond term (Q\u000b\u0003\u000b1) { parametric processes (second-order\nSuhl processes) and shift of spin wave frequencies due to\nmagnetic \feld modulation, and the last term ( R\u000b\u0003\u000b1\u000b2)\ndescribes nonlinear corrections to the e\u000eciency of spin\nwave excitation.\nA. Non-Conservative Perturbations { Gilbert\nDamping\nIn the case of non-conservative perturbations\n(@m=@t)nc= \u0001T(t;r;m) the additional terms in\nthe equation of motion for SEV s(t;r) can be obtained\nby di\u000berentiating approximate Eq. (13) with respect to\ntimet:\n\u0012@s\n@t\u0013\nnc=bP0\u0001\u0001T\u00001\n4\u0012\n1\u0000s2\n8\u0013\ns(m0\u0001\u0001T)+s2\n8bP0\u0001\u0001T;\n(46)\nwhere \u0001Tshould be written as a function of susing\nEq. (12).\nTo obtain equations for spin wave amplitudes c\u000b, one\nshould substitute into Eq. (46) the expansion Eq. (27)\nand apply orthogonality relation Eq. (24) by taking scalar\nproduct of this equation with s\u0003\n\u000b\u0001bL0and integrating over\nthe volume of the magnetic body.\nPreviously, this technique has been used to evaluate\ndamping rates of inhomogeneous spin wave modes caused\nby various dissipation mechanisms29. For completeness\nof the VHF presentation, we shall brie\ry repeat the\nderivation here. For simplicity, we shall consider only\nthe Gilbert damping mechanism and dissipation only\nin the linear approximation (in any case, the dissipa-\ntive torques are phenomenological, and phenomenolog-\nical nonlinear damping corrections can be added later\ndirectly into equations for c\u000b). In the limit of linear de-\nviations, Eq. (46) simpli\fes to the trivial expression\n\u0012@s\n@t\u0013\nnc= \u0001T: (47)\nIn this approximation, the Gilbert torque Eq. (40) can\nbe written as\n\u0001T=\u000bGm0\u0002@s\n@t: (48)10\nUsing the expansion Eq. (27) in Eqs. (47) and (48)\ngives\nX\n\u000bs\u000b\u0010c\u000b\ndt\u0011\nnc=\u000bGX\n\fm0\u0002s\fdc\f\ndt: (49)\nWith good accuracy, the time derivative dc\f=dtin the\nright-hand side of this equation can be replaced with its\nlinear conservative value \u0000i!\fc\f. Then, multiplying this\nequation bys\u0003\n\u000b\u0001bL0and integrating over the volume of\nthe magnetic system yields\ni\u0016h\u000b\u0010c\u000b\ndt\u0011\nnc=\u0000i\u000bGX\n\f!\f\u0014Z\nLss\u0003\n\u000b\u0001s\fdr\u0015\nc\f:(50)\nIn the case of small damping \u000bGand non-degenerate\nspectrum, one can keep only the diagonal term ( \f=\u000b)\nin the sum in the right-hand side of this equation. Then,\nthe dissipative correction to the equation of spin wave\namplitudec\u000btakes the standard form\ni\u0016h\u000b\u0010c\u000b\ndt\u0011\nnc=\u0000i\u0016h\u000b\u0000\u000bc\u000b; (51)\nwhere \u0000\u000bis the damping rate of \u000b-th spin wave mode:\n\u0000\u000b=\u000bG!\u000b\n\u0016h\u000bZ\nLsjs\u000bj2dr: (52)\nEquation (52), which was \frst derived in29, gives the\nmost general expression for Gilbert damping rate of a\nspin wave mode. This expression can be used to calcu-\nlate damping of spin wave modes in magnetic systems\nwith non-uniform ground state, in inhomogeneous sys-\ntems consisting of several di\u000berent magnetic materials,\ncan be applied for spin wave modes with non-trivial spa-\ntial structure, and so on.\nIV. EXAMPLE APPLICATION OF VHF\nHere, we shall illustrate the application of the devel-\noped vector Hamiltonian formalism using a nano-scale\nmagnetic element as a simple test system. Namely,\nthe magnetic system that we choose for the test\nprocedures is a rectangular prism with dimensions\n80 nm\u000240 nm\u00025 nm and material parameters cor-\nresponding to Permalloy. All the VHF calculations and\nnumerical simulations were performed in the absence of\nthe external bias magnetic \feld. In numerical simula-\ntions, the magnetic element was discretized with rectan-\ngular mesh with cell sizes 5 nm \u00025 nm\u00025 nm (128 cells\nin total).\nFigure IV shows the numerically calculated ground\nmagnetic state of the studied system. It is important\nto note, that, for such small magnetic prism, the edge ef-\nfects on magnetization are rather signi\fcant and, there-\nfore, the ground state is signi\fcantly non-uniform. This\nmeans, that the spin wave modes have rather complex\nFIG. 2. Ground state of the test magnetic system. From left\nto right: spatial distribution of the equilibrium magnetization\ncomponents mx,my, andmz.\npro\fles that cannot be satisfactory approximated by har-\nmonic functions and, therefore, simple analytical approx-\nimations can not be used to describe magnetization dy-\nnamics of this system.\nAt the \frst step we calculated spin wave mode pro\fles\nand frequencies of the studied system. Fig. IV shows\ndependence of several lowest spin wave eigen-frequencies\non mode index. Points show eigen-frequencies obtained\nfrom direct numerical solution of the discrete version of\nthe linear eigen-value problem Eq. (23), while solid line\ncorresponds to eigen-frequencies obtained from numerical\nspin wave pro\fles using variationally-stable calculation\nmethod Eq. (26). As one can see, spin wave frequen-\ncies calculated using these two methods coincide with\nhigh precision. This proves validity of the analytical ap-\nproach for a magnetic system with spatially-nonuniform\nground state. Also, this result demonstrates that one can\nuse variationally-stable calculations in the case when the\nspin wave mode pro\fles are known only approximately,\nwhich may be important for simulations of macro-sized\nmagnetic systems, for which direct solution of the linear\neigen-mode problem is not possible and one has to use\ncertain approximate methods.\nNext, we calculated linear damping rate Eq. (52) for all\nspin wave modes. The results of this calculation are illus-\ntrated by Fig. IV. Red points joined by the solid line show\nthe result of the VHF calculations, while green points\njoined by the dashed line correspond to the naive approx-\nimation \u0000 = \u000bG!, where\u000bG= 0:01 is the Gilbert damp-\ning constant for Py. As one can see, the two methods give\napproximately the same damping rates for higher-order\nspin wave modes, but di\u000ber by about a factor of 2 for the\nspin wave modes with lowest frequencies. This discrep-\nancy is connected with the fact, that the naive Gilbert\napproximation does not take into account non-uniform\npro\fle and ellipticity of precession of spin wave modes.\nThe in\ruence of these factors increase with the decrease\nof the spin wave frequency. It should be noted, that the\nmodes which are most important from the practical point\nof view are exactly the lowest-lying spin wave modes,\nwhich have non-zero overlap with quasi-uniform magnetic\n\feld and, therefore, can be directly excited by an external\nelectromagnetic system. Thus, Fig. IV demonstrates that\nthere is a huge di\u000berence between damping rates of prac-\ntically interesting modes calculated using the developed\nVHF approach and obtained from naive estimations.\nWe have also calculated various nonlinear interaction\ncoe\u000ecients and coe\u000ecients of interaction with external11\nFIG. 3. Lowest spin wave eigen-frequencies of the studied\nmagnetic system. Points show eigen-frequencies obtained\nfrom direct solution of the linear eigen-value problem Eq. (23),\nsolid line { eigen-frequencies obtained from numerical spin\nwave pro\fles using variationally-stable calculation method\nEq. (26).\nFIG. 4. Damping rate of the lowest spin wave modes of the\ntest magnetic system. Red points and solid line { damping\nrate with account of real spatial pro\fle of the spin wave mode\nEq. (52); green points and dashed line { naive approximation\n\u0000 =\u000bG!.\nmicrowave \feld. These parameters will be used below\nto compare results of VHF analysis with direct numer-\nical simulations for two cases { linear free magnetiza-\ntion decay and nonlinear ferromagnetic resonance. We\nwould like to stress, that VHF analysis does not have\na single \ftting parameter and that all coe\u000ecients in the\nVHF equations are calculated as various linear or nonlin-\near \\overlap integrals\" over the spin wave pro\fles s\u000b(r).Calculation of the pro\fles s\u000b(r) and eigen-frequencies !\u000b\nis the only computationally expensive part of the VHF\nprocedure.\nA. Linear Free Magnetization Decay\nTo simulate a linear free magnetization decay, we\nadded a small deviation \u000em(r) to the ground mag-\nnetic statem0(r) of the test system and used this\nnon-equilibrium magnetization distribution as the initial\ncondition in full-scale micromagnetic simulations. We\nrun micromagnetic simulations for certain time (approx-\nimately 10 ns) and calculated time dependence of yand\nzcomponents of magnetization, averaged over the vol-\nume of the magnetic prism. The same small deviation\n\u000em(r) was also used as an initial condition for simu-\nlations based on the VHF approach. In this case, we\nprojected the initial magnetization deviation into the set\nof spin wave modes s\u000b(r), which gave us the initial com-\nplex amplitudes of the spin wave modes, c\u000b(0). Then, we\nrun linear VHF solver to \fnd the time dependence of the\ncomplex amplitudes (this dependence is rather trivial in\nthe VHF representation, c\u000b(t) =c\u000b(0)e\u0000i!\u000bt\u0000\u0000\u000bt). Us-\ning the obtained time dependence of the spin wave ampli-\ntudesc\u000b(t), we restored the time dependence of the spa-\ntial pro\fle of the magnetization and found the averaged\nvalues ofyandzmagnetization components. Thus, each\nnumerical experiment provided two independent sets of\ndata formy(t) andmz(t) obtained using two di\u000berent ap-\nproaches { direct numerical simulations and simulations\nusing the VHF approach. Comparison between these sets\nof data provided information on accuracy of the VHF-\nbased simulations for linear magnetization dynamics.\nBefore demonstrating examples of the numerical ex-\nperiment, we would like to comment on the performance\nof two simulation methods. Full-scale numerical simula-\ntions took approximately the same time (about 1 minute\non a laptop we used) for each experiment. The simu-\nlations based on the VHF approach were much faster\n(about 2-3 milliseconds). The extremely small simula-\ntion time for the VHF approach is explained by the trivial\nlinear dynamics of spin wave modes in the VHF repre-\nsentation. In the case of full nonlinear VHF simulations,\nthe performance gap is smaller, but is still signi\fcant.\nIn the VHF approach, the main computational time is\nspent on the \frst VHF initialization step, at which spin\nwave pro\fles and eigen-frequencies are calculated. For\nthe chosen test system and method of eigen-problem so-\nlution, the duration of the initialization step was about\n10 seconds. This time is comparable with the full-scale\nsimulation time of one experiment, but it needs to be per-\nformed only once per experimental series. Thus, our lin-\near test demonstrated huge improvement of performance\nof VHF-based micromagnetic solver compared to tradi-\ntional approach.\nFigures IV A and IV A show simulation results for\nmy(t) andmz(t) magnetization components, respec-12\nFIG. 5. Comparison of time dependence of averaged my\nmagnetization component calculated using direct micromag-\nnetic simulations (points) and VHF-based calculations (solid\nline). Initial magnetization deviation was uniform in space\nand equal to \u000em= 0:01y.\ntively, calculated using full-scale micromagnetic solver\n(points) and the developed linear VHF solver (solid\nlines). Initial magnetization distribution was uniform in\nspace and equal to \u000em= 0:01y(magnetizatoin was ro-\ntated towards the yaxis for about 2 degrees). As one\ncan see from Fig. IV A and IV A, the VHF provides re-\nsults that are practically indistinguishable from full-scale\nmicromagnetic simulations. A minute shift of two sets of\ndata at later times t\u00191 ns is explained by the nonlinear\nfrequency shift, which is present even at such small mag-\nnetization precession angles. This e\u000bect is not described\nby the linear VHF approach (see the next subsection for\ncomparison of nonlinear magnetization dynamics).\nNote, that the time pro\fles of my(t) andmz(t) no-\nticeably deviate from a simple harmonic behavior, which\nis due to excitation of several modes in this numerical\nexperiment. The developed VHF approach correctly de-\nscribes the amplitude and phase relations between the\nexcited modes.\nWe have repeated the same numerical experiments for\nseveral di\u000berent pro\fles of the initial magnetization de-\nviation\u000em(r). As an example, we show in Fig. IV A\nresults of numerical experiment with 3 \u00023 chessboard\ninitial deviation. One can see that the agreement be-\ntween the full-scale micromagnetic simulations and VHF-\nbased calculations is as good as in the previous example\nwith uniform initial distribution. We would like to stress\none more time, that this VHF-based experiment used the\nsame information on spin wave pro\fles, frequencies, and\ndamping rates, as the previous one, so the added cal-\nculation time was of the order of few milliseconds. We\nobtained the same excellent agreement between two sim-\nulation method for all studied initial distributions of the\nFIG. 6. Comparison of time dependence of averaged mz\nmagnetization component calculated using direct micromag-\nnetic simulations (points) and VHF-based calculations (solid\nline). Initial magnetization deviation was uniform in space\nand equal to \u000em= 0:01y.\nFIG. 7. Comparison of time dependence of averaged mzmag-\nnetization component calculated using direct micromagnetic\nsimulations (points) and VHF-based calculations with n= 1\n(dashed blue line) and n= 5 (solid red line) modes. The\ninitial magnetization deviation from the ground state had the\nform of 3\u00023 chessboard pro\fle with magnitude j\u000emj= 0:01.\nnon-equilibrium magnetization, and for all cases the sat-\nisfactory results were obtained with not more than n= 5\nspin wave modes.13\nFIG. 8. Comparison of the FMR response of the test mag-\nnetic system calculated using direct micromagnetic simula-\ntions (points) and VHF-based calculations (solid line) for dif-\nferent driving \feld magnitudes hrf.\nB. Nonlinear Ferromagnetic Resonance\nWe have used the same test magnetic system to per-\nform numerical experiments on nonlinear ferromagnetic\nresonance (FMR). We used the following procedure.\nFirst, we set the magnetic state of the test system to\nthe ground state m0(r). Then, we run simulations with\nmicrowave \feld (amplitude hrf, frequency frf) switched\non for 20 ns, which was enough to reach a steady state\nprecession. After that, we run simulations for one addi-\ntional period of the microwave magnetic \feld and found\nthe magnitude of the spatially-averaged hmyicomponent\nof dynamic magnetization. For convenience, we present\nbelow the simulation results as the values of the magnetic\nsusceptibility \u001fyy=Mshmyi=hrf.\nThe described above numerical experiment was per-\nformed for several frequencies frfin the range from\n3 GHz to 5 GHz and several amplitudes of the mi-\ncrowave \feld from \u00160hrf= 12:6\u0016T (linear regime) to\n\u00160hrf= 2:51 mT (strongly nonlinear regime). The com-\nparison of the results obtained using a standard full-scale\nmicromagnetic solver and using the VHF-based analysis\nare shown in Fig. IV B.\nBlack dots and line in Fig. IV B correspond to the\nsmall magnitude of the driving microwave \feld \u00160hrf=\n12:6\u0016T. For such small-amplitude excitations the mag-\nnetization dynamics is linear with very high accuracy.\nOne can see from Fig. IV B that both full-scale and VHF-\nbased approaches correctly reproduce linear resonance\ncurve with resonance frequency corresponding to the low-\nest spin wave mode at f= 4:1 GHz.\nWith the increase of the driving \feld magnitude to\n\u00160hrf= 0:63 mT (blue dots and line in Fig. IV B),\none can clearly see nonlinear distortions of the FMR\ncurve, both in the shape of the curve and position ofthe maximum. Similarly to the previous case, VHF-\nbased and full-scale approaches practically coincide. This\nproves the validity of the VHF approach to simulations\nof weakly-nonlinear magnetization dynamics. We would\nlike to stress, that VHF results shown in Fig. IV B were\nobtained without a single \ftting parameter.\nRed and green dots and lines in Fig. IV B show the non-\nlinear FMR curves at even higher values of the driving\n\feld magnitude, \u00160hrf= 1:26 mT and \u00160hrf= 2:51 mT,\nrespectively. Such large amplitudes of the microwave\nmagnetic \feld are hardly accessible experimentally, and\nthe purpose of these simulations was to \fnd a point at\nwhich the perturbative VHF approach starts to devi-\nate from full-scale simulations. One can see, that the\nVHF-based simulations provide rather accurate quanti-\ntative description for the case \u00160hrf= 1:26 mT (red dots\nand line), but are only qualitatively correct for the case\n\u00160hrf= 2:51 mT (green). In the latter case, the average\nvalue of the y-component of dynamic magnetization was\nhmyi= 0:5, which corresponds to 30\u000eaverage preces-\nsion angle (the local precession angles were substantially\nlarger). Thus, the perturbative VHF approach is quan-\ntitatively correct up to precession angles of about 30\u000e,\nwhich is much larger than typical precession angles in\nmajority of experiments.\nV. CONCLUSIONS\nIn conclusion, we developed a new approach to in-\nvestigation of a weakly-nonlinear magnetization dynam-\nics { vector Hamiltonian formalism (VHF). The VHF\nis based on a vector transformation of a sphere to a\nplane (azimuthal Lambert projection), which preserves\nboth the Hamiltonian structure and vector character of\nthe Landau-Lifshits equation of magnetization dynamics.\nWe derived simple and compact expressions for various\nnonlinear interaction coe\u000ecients of spin wave modes in\nthe form of nonlinear \\overlap integrals\" of modes' pro-\n\fles. The developed formalism is well-suited for hybrid\nanalysis of magnetization dynamics, in which informa-\ntion about the linear dynamics of the studied magnetic\nsystem (eigen-frequencies and spin wave mode pro\fles)\nis obtained from numerical simulations, while nonlinear\ndynamics is analyzed based on quasi-Hamiltonian equa-\ntions for spin wave amplitudes. The comparison of the\nresults obtained using this method with results of full-\nscale nonlinear micromagnetic simulations demonstrates\na very good agreement for the magnetization precession\nangles of up to at least 30\u000e.\nAppendix A: Mathematical Properties of the Linear\nEigenproblem\nHere we will consider some basic mathematical proper-\nties of the linear eigenproblem Eq. (23) in the case when\nthe linear Hamiltonian of the system cH0is a positive-14\nde\fnite operator. In this case, the operator cH0can be\nrepresented as\ncH0=bA+\u0001bA; (A1)\nwherebAis a certain lower triangular matrix with real\nand positive diagonal entries and bA+denotes Hermi-\ntian conjugate of bA. Such decomposition of a Hermi-\ntian positive-de\fnite operator is known as the Cholesky\ndecomposition.\nIt should be noted, that \fnding the Cholesky decompo-\nsition Eq. (A1) is not technically much easier than solving\nthe original eigenproblem Eq. (23). Respectively, the aim\nof this section is not to provide any technical receipts for\nsolving Eq. (23), but to rigorously prove important math-\nematical properties of it. Another important note is that\nthe operator cH0is positive de\fnite (in the case when\nthe ground state m0corresponds to a minimum of en-\nergy) only for vectors s, which are orthogonal to m0. For\n\\longitudinal\" vectors f(r)m0(r) (which are parallel, in\nevery point of space, to the local direction of m0), the\naction ofcH0is zero,cH0\u0001(fm0) = 0, as it is obvious\nfrom the de\fnition of this operator Eq. (21). Therefore,\nthroughout this section we assume that the SEVs sare\ndescribed using two-dimensional coordinate approach (as\nit is explained in Sec. II B), in which operator cH0is posi-\ntive de\fnite, operator bL0is invertible, and the projection\noperatorbP0is equivalent to the identity operator.\nUsing the Cholesky decomposition Eq. (A1), the eigen-\nproblem Eq. (23) takes the form\n\u0000i!\u000bbL0\u0001s\u000b=bA+\u0001bA\u0001s\u000b:\nMultiplying both sides of this equation by ( bA\u00001)+(note,\nthat the operator bAis always invertible) and introducing\nnew eigenvectors\nu\u000b=bA\u0001s\u000b (A2)\nthis equation can be reformulated as\n\u0015\u000bu\u000b=bB\u0001u\u000b; (A3)\nwhere\u0015\u000b= 1=!\u000band\nbB=\u0000i(bA\u00001)+\u0001bL0\u0001bA\u00001: (A4)\nAs one can easily see, the operator bBis a Hermitian\noperator, so the reformulated eigenproblem Eq. (A3) is\na standard Hermitian eigen-value problem. Respectively,\nthe set of eigenvectors u\u000b(and, respectively, set of vec-\ntorss\u000b) forms a complete set of vector functions, orthog-\nonal to the ground state m0, and all eigenvalues \u0015\u000b(and\neigenfrequencies !\u000b= 1=\u0015\u000b) are real.\nMoreover, the eigenvectors u\u000bthat correspond to\ndi\u000berent eigenvalues \u0015\u000bare orthogonal to each other,\nand multiple eigenvectors corresponding to a degenerate\neigenvalue can be mutually orthogonalized. Thus,\nZ\nVsu+\n\u000b\u0001u\u000b0dr=\u000f\u000b\u0001\u000b;\u000b0; (A5)where\u000f\u000bare certain positive normalization constants and\n\u0001\u000b;\fis the Kronecker delta.\nUsing the de\fnition of the auxiliary vectors u\u000b\nEq. (A2), the orthogonality condition Eq. (A5) can be\nreformulated in terms of the SEVs s\u000b:\nZ\nVss+\n\u000b\u0001cH0\u0001s\u000b0dr=\u000f\u000b\u0001\u000b;\u000b0: (A6)\nThis condition is equivalent to Eq. (26) with \u000f\u000b= \u0016h\u000b!\u000b,\nand can also be derived in a slightly less rigorous way di-\nrectly from the eigenproblem Eq. (23). The presented\nabove derivation also proves that the product of the\nmode's norm \u0016 h\u000band its eigenfrequency !\u000bis always a\npositive quantity (for a ground state m0that corresponds\nto a minimum of energy), i.e., that the modes with pos-\nitive norms have positive eigenfrequencies.\nAnother standard property of the Hermitian eigen-\nproblem Eq. (A3) is that the operator bBis diagonal in\nthe basis of eigenvectors bu\u000b, which can be written as\nanother orthogonality condition:\nZ\nVsu+\n\u000b\u0001bB\u0001u\u000b0dr=\u000f\u000b\u0015\u000b\u0001\u000b;\u000b0: (A7)\nThis condition, also, can be rewritten in terms of the\noriginal SEVs s\u000b:\nZ\nVss+\n\u000b\u0001bL0\u0001s\u000b0dr=i\u000f\u000b\u0015\u000b\u0001\u000b;\u000b0; (A8)\nand is equivalent to Eq. (24).\nAppendix B: Elimination of Non-Resonant\nThree-Magnon Processes\nHere we shall brie\ry describe the procedure of elimina-\ntion of three-magnon processes H3in the case when such\nprocesses are non-resonant, i.e., the condition Eq. (36) is\nsatis\fed. Consider a weakly-nonlinear transformation of\nthe spin wave amplitudes c\u000b:\nc\u000b!c0\n\u000b=c\u000b+X\n\f\rA\u000b;\f\rc\fc\r+::: : (B1)\nIf this transformation is canonical, i.e., preserves the\nform of the Poisson brackets Eq. (31), the equations of\nmotion for the new amplitudes c0\n\u000bwill have the same form\nEq. (30), where the Hamiltonian function Hshould be\nwritten using the transformed amplitudes and will have\na di\u000berent functional form. Using properly chosen trans-\nformation coe\u000ecients, one may simplify the transformed\nHamiltonian, in particular, eliminate the three-magnon\ntermH3if it is non-resonant.\nThe eliminated non-resonant processes lead, in sec-\nond perturbation order, to renormalization of coe\u000ecients\nof higher-order (H4) resonant processes, i.e., to renor-\nmalization of coe\u000ecients W\u000b\f\r\u000e!W0\n\u000b\f\r\u000e =W\u000b\f\r\u000e +15\n\u0001W\u000b\f\r\u000e. In the case of magnetic systems, the correc-\ntion \u0001W\u000b\f\r\u000e can be, in general, of the same order of\nmagnitude as the original interaction coe\u000ecient W\u000b\f\r\u000e\nand, strictly speaking, cannot be ignored. To calculate\nthe correction \u0001 W\u000b\f\r\u000e, one has to use weakly-nonlinear\ncanonical transformation Eq. (B1) explicitly taking into\naccount both quadratic and cubic terms in the expansion\nc0\n\u000b(c\f), which leads to rather cumbersome and technically\ndi\u000ecult expressions.\nTherefore, instead of using explicit form of the trans-\nformation Eq. (B1), we will employ the fact that any\nHamiltonian dynamics described by equations of the form\nEq. (30) is itself a canonical transformation. Then, we\ncan consider a canonical transformation generated by cer-\ntain \\Hamiltonian function\" F:\nc0\n\u000b=c\u000b+ [F;c\u000b] +1\n2[F;[F;c\u000b]] +::: : (B2)\nChoosingFas a cubic function in spin wave amplitudes,\nF=1\n6X\n\u000b\f\rF\u000b\f\rc\u000bc\fc\r (B3)\nleads to the desired weakly-nonlinear behavior Eq. ( ??),\nwhile the \\Poisson bracket\" form of the transformation\nEq. (B2) guarantees that it is a canonical one.\nThe transformed Hamiltonian H0(c0\n\u000b) =H(c\u000b) can also\nbe written as a \\Poisson-bracket expansion\":\nH0=H\u0000[F;H] +1\n2[F;[F;H]] +::: : (B4)\nUsing weakly-nonlinear expansions of H=H2+H3+\nH4andH0=H0\n2+H0\n3+H0\n4, one can relate di\u000berent-order\nterms in the original Hand transformedH0Hamiltonian\nfunctions:\nH0\n2=H2; (B5a)\nH0\n3=H3\u0000[F;H2]; (B5b)\nH0\n4=H4\u0000[F;H3] +1\n2[F;[F;H2]]: (B5c)Thus, weakly-nonlinear transformation leaves the\nquadratic part of the Hamiltonian H2unchanged. The\nthree-magnon term H0\n3vanishes ifFsatis\fes\n[F;H2] =H3; (B6)\nin which caseH0\n4can be written in a very simple form\nH0\n4=H4\u00001\n2;[F;H3]: (B7)\nDirect evaluation of [ F;H2] gives\n[F;H2] =i\n6X\n\u000b\f\r(!\u000b+!\f+!\r)F\u000b\f\rc\u000bc\fc\r:(B8)\nThe elimination condition Eq. (B6) requires\nF\u000b\f\r=\u0000iV\u000b\f\r\n!\u000b+!\f+!\r; (B9)\nwhich can be satis\fed if all three-magnon processes are\nnon-resonant Eq. (36).\nThe transformed four-magnon Hamiltonian H0\n4\nEq. (B7) has the form\nH0\n4=H4\u0000i\n8X\n\u000b\f\r\u000e\u000fF\u000b\f\u000fV\u000f\u0003\r\u000e\n\u0016h\u000fc\u000bc\fc\rc\u000e: (B10)\nUsing the expression Eq. (B9) for coe\u000ecients F\u000b\f\r, one\ncan rewriteH0\n4in the form\nH0\n4=1\n24X\n\u000b\f\r\u000eW0\n\u000b\f\r\u000ec\u000bc\fc\rc\u000e (B11)\nwith renormalized coe\u000ecients W0\n\u000b\f\r\u000e given by the sym-\nmetrized expressions Eq. (37).\n\u0003tyberkev@oakland.edu\n1V.E. Zakharov, Radiophys Quantum Electron 17, 326{343\n(1974).\n2V.E. Zakharov, V.S. L'vov, and S.S. Starobinets, Usp. Fiz.\nNauk 114, 609 (1974) [Sov. Phys.-Usp. 17, 896 (1975)].\n3V.S. L'vov, Wave Turbulence under Parametric Excitation.\nApplications to Magnetics (Springer-Verlag, 1994).\n4V.E. Zakharov and E.A. Kuznetsov, Phys. 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Slavin,Nature 443, 430-433 (2006).\n23O. Dzyapko, I. Lisenkov, P. Nowik-Boltyk, V.E. Demi-\ndov, S.O. Demokritov, B. Koene, A. Kirilyuk, T. Rasing,\nV. Tiberkevich, and A. Slavin, Phys. Rev. B 96, 064438\n(2017).\n24I.V. Borisenko, B. Divinskiy, V.E. Demidov, G. Li, T. Nat-\ntermann, V.L. Pokrovsky, and S.O. Demokritov, Nature\nCommun. 11, 1691 (2020).\n25V. Tiberkevich, I.V. Borisenko, P. Nowik-Boltyk,\nV.E. Demidov, A.B. Rinkevich, S.O. Demokritov, and\nA.N. Slavin, Sci. Rep. 9, 9063 (2019).\n26D.A. Bozhko, A.J.E. Kreil, H.Yu. Musiienko-Shmarova,\nA.A. Serga, A. Pomyalov, V.S. L'vov, and B. Hillebrands,\nNature Commun. 10, 2460 (2019).\n27A.N. Slavin and V.S. Tiberkevich, IEEE Trans. Magn. 45,\n1875 (2009).\n28A. Awad, P. Durrenfeld, A. Houshang, et al., Nature Phys.\n13, 292{299 (2017).\n29R. Verba, V. Tiberkevich, and A. Slavin, Phys. Rev. B 98,\n104408 (2018)."    },    {        "title": "1603.09186v2.Distributed_chaos_and_solitons_at_the_edges_of_magnetically_confined_plasmas.pdf",        "content": "Distributed chaos and solitons at the edges of magnetically con\fned plasmas\nA. Bershadskii\nICAR, P.O. Box 31155, Jerusalem 91000, Israel\nIt is shown, using results of measurements of ion saturation current in the plasma edges of di\u000berent\nmagnetic fusion con\fnement devices (tokamaks and stellarators), that the plasma dynamics in the\nedges is dominated by distributed chaos with spontaneously broken translational symmetry at low\nmagnetic \feld, and with spontaneously broken re\rexional symmetry (by helical solitons) at high\nmagnetic \feld.\nINTRODUCTION\nTurbulent motion at the edges of magnetically con\fned\nplasmas in the magnetic fusion con\fnement devices de-\ngrades their performance. It is believed that space lo-\ncalized coherent structures (blobs) are the main cause\nfor the con\fnement degradation. Direct measurements\nsupport the idea that these blobs are vortex-like velocity\npatterns associated with current \flaments (with life span\n\u0018microseconds) [1].\nThe power spectra of ion saturation current is an ef-\nfective tool in order to obtain information about the dy-\nnamic processes at the edges. Fortunately these spectra\nhave a similar functional shape in most toroidal fusion\ndevices (see, for instance, Ref. [2] where a survey of the\ndata from several stellarators and JET tokamak are pre-\nsented). This observation indicates that there could be\na common underlying physical process. Presumably (see\nRef. [3] for a comprehensive analysis) this dynamics is\nchaotic rather than stochastic. The conclusion is based\non the exponential-like shape of the observed spectra.\nIndeed, the exponential power spectra are observed in\nmany dynamical systems with chaotic dynamics [4]-[6].\nThe exponential spectra can be provided [7] by pulses\nhaving a Lorentzian functional form\nL(t) =A\n2\u0014\u001c\n\u001c+i(t\u0000t0)+\u001c\n\u001c\u0000i(t\u0000t0)\u0015\n(1)\nFor a series of NLorentzian pulses the power spectrum\ncan be presented as a sum over the residues of the com-\nplex time poles\nE(!)/\f\f\f\f\fNX\nn=1exp(i!t0;n\u0000!\u001cn)\f\f\f\f\f2\n(2)\nAn exponential spectrum results from Eq. (2) if distri-\nbution of the Lorentzian pulse widths \u001cis su\u000eciently\nnarrow one\nE(!)/exp\u0000(2!\u001c) (3)\nIn the case of a broad (continuous) distribution of the\npulse widths: P(\u001c), a weighted superposition of the ex-\nponentials Eq. (3) can approximate the sum Eq. (2)\nE(!)/Z1\n0P(\u001c) exp\u0000(2!\u001c)d\u001c (4)SPECTRUM OF DISTRIBUTED CHAOS\nIt is suggested in Ref. [8] that in the distributed\nchaos the weighted superposition Eq (4) is converged to\nastretched exponential. Replacing \u001cby a dimensionless\nvariable:s=\u001c=\u001c\f(\u001c\fis a constant):\nE(!)/Z1\n0P(s) exp\u0000s(2!\u001c\f)ds (5)\nwe can write the stretched exponential as\nE(!)/exp\u0000(!=!\f)\f(6)\nwhere!\f= (2\u001c\f)\u00001.\nIn the case of \f= 1=2 (this case is of a special interest\nfor us, see below)\nP(s)/1\ns3=2exp\u0000(c=s) (7)\nwherecis a constant.\nFor other values of \fthe distribution P(s;\f) has rather\ncumbersome form [9].\nIn the experiments time signals are usually taken by\nprobes at a \fxed spatial location and re\rect character-\nistics of the spatial structures moving past the probes.\nTherefore, the frequency spectra observed in the plasma\nexperiments re\rect the wavenumber spectra (so-called\nTaylor hypothesis [10]) and the frequency spectrum Eq.\n(6) corresponds to wavenumber spectrum\nE(k)/exp\u0000(k=k\f)\f(8)\nASYMPTOTIC CONSIDERATION\nThe distribution P(s;\f) has rather simple asymptotic\nats!0 [9]\nP(s;\f)/1\ns1+\f=2(1\u0000\f)exp\u0000c\ns\f=(1\u0000\f)(9)\nLet the group velocity \u001d(\u0014) of the waves driving the dis-\ntributed chaos be scale invariant at \u0014!1\n\u001d(\u0014)/\u0014\u000b(10)arXiv:1603.09186v2  [nlin.CD]  4 Apr 20162\n5 10 15 20 25 30-22-20-18-16-14\n(λ f  [kHz]) 1/2 ln E exp-(f/f β)1/2 \nFIG. 1: Power spectrum of ion saturation current in the\nplasma edge of di\u000berent magnetic fusion con\fnement devices\n(tokamaks and stellarators). The frequency fwas rescaled\nby the authors of the Ref. [2] with the parameter \u0015: TJ-I:\n\u0015= 1, TJ-IU: \u0015= 3, JET:\u0015= 4:5,W7\u0000AS(1):\u0015= 3:5,\nW7\u0000AS(2):\u0015= 4:5. The straight line is drawn in order to\nindicate a stretched exponential decay Eq. (6) with \f= 1=2.\nand let\u001d(\u0014) to have a Gaussian distribution at \u0014!1\n:/exp\u0000(\u001d(\u0014)=\u001db)2. In this case the asymptotic of \u0014\ndistribution is\nP(\u0014)/\u0014\u000b\u00001exp\u0000\u0012\u0014\n\u0014b\u00132\u000b\n(11)\nwherekbis a constant.\nSinces/\u0014\u00001then substituting Eqs. (9) and (11) into\nequation\nP(s)ds/P(\u0014)d\u0014 (12)\none obtains\n\f=2\u000b\n1 + 2\u000b(13)\nSPONTANEOUS BREAKING OF SPACE\nTRANSLATIONAL SYMMETRY\nThere exist two main space symmetries: rota-\ntional (isotropy) and translational (homogeneity). The\nNoether's theorem relates the conservation laws of the\nangular and linear momentum to these symmetries (re-\nspectively) [11]. These conservation laws produce two\nhydrodynamic invariants: Loitsyanskii ( I4) and Birkho\u000b-\nSa\u000bman (I2) integrals [10],[12]-[15]:\nIn=Z\nrn\u00002hu(x;t)\u0001u(x+r;t)idr (14)\nat su\u000eciently rapid decay of the two-point correlation\nfunction of the \ruid velocity \feld. In the distributed\nchaos these two invariants drive two attractors. The at-\ntractors have di\u000berent basins of attraction. The basin of\n3 4 5 6 7 8 9 10-11-10-9-8-7-6-5-4\n(f [kHz] )1/2 ln E exp-(f/f β)1/2 \nlow B FIG. 2: The same as in Fig. 1 but for TJ-K stellarator (a\ntoroidal device). The data were taken from Ref. [17] and\ncorresponds to the low magnetic \feld case (the edge region of\nthe core plasma).\nattraction of the Loitsyanskii attractor is small and thin\nin comparison with that of the Birkho\u000b-Sa\u000bman attrac-\ntor. Therefore, the scaling Eq. (10) is determined by the\nBirkhof-Sa\u000bman invariant ( I2) for the statistically sta-\ntionary isotropic homogeneous turbulence [8]:\n\u001d(\u0014)/I1=2\n2\u00143=2(15)\nThen Eq. (13) provides \f= 3=4 [8].\nIt is shown in Ref. [16] that at spontaneous breaking\nof the space translational symmetry the Birkhof-Sa\u000bman\nintegralI2should be replaced in the scaling relationship\nEq. (15) by a new integral\n\r=Z\nVh!(x;t)\u0001!(x+r;t)iVdr (16)\nwhere index Vmeans integration and averaging over the\nvolume of motion and !(x;t) =r\u0002u(x;t) is vorticity.\nThen the scaling relation Eq. (15) should be replaced by\n\u001d(\u0014)/j\rj1=2\u00141=2(17)\nIn this case Eq. (13) provides \f= 1=2 [16].\nRecent generalization of the Birkho\u000b-Sa\u000bman and\nLoitsyanskii invariants for magneto-hydrodynamic\n(MHD) turbulence [14],[15] allows to generalize the\nabove consideration for MHD turbulence as well.\nFigure 1 shows frequency power spectra of ion satura-\ntion current measured in the plasma edge of di\u000berent fu-\nsion devices (tokamaks and stellarators). The frequency\nis rescaled by the authors of the Ref. [2] (by the parame-\nter\u0015, see the legend to the \fgure) and the amplitudes are\nnormalized in order to converge the data for the di\u000berent\ndevices in a single (universal) curve. These (cumulative)\ndata for the Fig. 1 were taken from the Ref. [3], where\nthe data were represented in a log-linear form. The scales\nin Fig. 1 are chosen in order to represent the Eq. (6) with3\n\f= 1=2 as a straight line. The \u0015-rescaling of frequency f\ntakes into account that the parameter f\f(or!\f= 2\u0019f\f)\ncan be di\u000berent for di\u000berent devices.\nFigure 2 shows results of more recent measurements\nperformed in the edge region of the core plasma at the\nTJ-K stellarator (a toroidal device) [17]. In this experi-\nment low magnetic \feld corresponds to 72 mT and high\nmagnetic \feld corresponds to 244 mT. Fig. 2 shows data\nobtained at the low magnetic \feld. As in Fig. 1 the\nscales in Fig. 2 are chosen in order to represent the Eq.\n(6) with\f= 1=2 as a straight line.\nSOLITON DRIVEN DISTRIBUTED CHAOS\nIn strong magnetic \feld the waves driving the dis-\ntributed chaos can be dominated by helical solitons (see,\nfor instance, Ref. [18]). It is suggested in [18] that the\nspontaneously emerging pairs of helical solitons having\nopposite helical charges (spontaneous breaking of re\rex-\nional symmetry) can serve as 'sinks' of energy. In this\ncase, unlike the Kolmogorov's-like turbulence, the energy\ndissipation rate should be taken not per volume but per\nsoliton. And, correspondingly, dimension of the parame-\nter governing this situation - G, should be m5=s3[18]. In\nthe statistically stationary isotropic homogeneous turbu-\nlence the Birkho\u000b-Sa\u000bman integral directly determines\nthe low wavenumbers power spectrum of velocity \feld\n[13]. Analogously in the soliton dominated MHD tur-\nbulence under strong magnetic \feld the low wavenum-\nbers power spectrum of the velocity \feld is directly de-\ntermined by the soliton parameter G[18]:\nE(k)/G2=3k1=3(18)\nFigure 3 (adapted from [18]) shows a streamwise power\nspectrum of the velocity in turbulent \row of mercury past\na grid (bars of the grid were inclined in direction of the\nstrong magnetic \feld). In the log-log scales the scaling\nlaw Eq. (18) corresponds to the straight line in the Fig.\n3. The data were taken from an experiment reported\nin Ref. [19]. This result was con\frmed by analogous\nexperiment reported in more recent Ref. [20].\nAs for the waves driving the distributed chaos at the\nhelical solitons domination\n\u001d(\u0014)/G1=3\u00142=3(19)\nThen Eq. (13) provides \f= 4=7.\nFigure 4 shows results of the same measurements as\nin Fig. 2 but for the high magnetic \feld. The scales in\nFig. 4 are chosen in order to represent the Eq. (6) with\n\f= 4=7 as a straight line.\n10 -8 10 -7 10 -6 \n10 2 10 31/3 \n-11/3 \nk  [m -1 ]E  [m 3/s 2]FIG. 3: Streamwise power spectrum of the velocity in turbu-\nlent \row of mercury past a grid (bars of the grid were inclined\nin direction of the strong magnetic \feld). In the log-log scales\nthe straight line with the slope equal to 1/3 corresponds to\nthe scaling Eq. (18)\n.\n2 4 6 8 10 12 14 16 18-11-10-9-8-7-6-5-4\n(f [kHz] )4/7 ln E exp-(f/f β)4/7 \nhigh B \nFIG. 4: The same as in Fig. 2 but for high magnetic \feld.\n[1] M. Spolaore et al., Phys. Rev. Lett., 102, 165001 (2009).\n[2] M. A. Pedrosa et al., Phys. Rev. Lett. 823621 (1999).\n[3] J. E. Maggs and G. J. Morales, Plasma Phys. Control.\nFusion 54124041 (2012).\n[4] U. Frisch and R. Morf, Phys. Rev., 23, 2673 (1981).\n[5] J. D. Farmer, Physica D, 4, 366 (1982).\n[6] D.E. Sigeti, Phys. Rev. E, 52, 2443 (1995).\n[7] J.E. Maggs and G.J. Morales, Phys. Rev. Lett., 107,\n185003 (2011); Phys. Rev. E 86, 015401(R) (2012).\n[8] A. Bershadskii, arXiv:1512.08837 (2015).\n[9] D. C. Johnston, Phys. Rev. B 74, 184430 (2006).\n[10] A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics,\nVol. II: Mechanics of Turbulence (Dover Pub. NY, 2007).\n[11] L.D. Landau and E.M. Lifshitz, Mechanics (Pergamon4\nPress 1969).\n[12] G. Birkho\u000b, Commun. Pure Appl. Math. 7, 19 (1954)\n[13] P. G. Sa\u000bman, J. Fluid. Mech. 27, 551 (1967).\n[14] P. A. Davidson, J. Phys.: Conference Series 318072025\n(2011).\n[15] P. A. Davidson P.A. Turbulence in rotating, strati\fed\nand electrically conducting \ruids. (Cambridge University\nPress, 2013).\n[16] A. Bershadskii, arXiv:1601.07364 (2016).[17] G. Hornung et al., Phys. Plasmas 18, 082303 (2011).\n[18] A. Bershadskii, E. Kit, and A. Tsinober, Proc. Royal\nSociety A 441, 147 (1993).\n[19] I. Platnieks and S. F. Seluto, in \"Liquid metal magneto-\nhydrodynamics\" pp. 433-439, (ed. J. Lielpeteris and R.\nMoreau; Dordrecht: Kluwer 1989).\n[20] H. Branover et al., Phys. Fluids, 16, 845 (2004)."    },    {        "title": "1904.11193v3.Electrical_response_of_S_F_TI_S_junctions_on_magnetic_texture_dynamics.pdf",        "content": "Electrical response of S-F-TI-S junctions on magnetic texture dynamics\nD. S. Rabinovich,1, 2, 3I. V. Bobkova,3, 1, 4and A. M. Bobkov3\n1Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia\n2Skolkovo Institute of Science and Technology, Skolkovo 143026, Russia\n3Institute of Solid State Physics, Chernogolovka, Moscow reg., 142432 Russia\n4National Research University Higher School of Economics, Moscow, 101000 Russia\n(Dated: March 3, 2020)\nWe consider a hybrid structure consisting of superconducting or normal leads with a combined\nferromagnet-3D topological insulator interlayer. We compare responses of a Josephson junction and\na normal junction to magnetic texture dynamics. In both cases the electromotive force resulting\nfrom the magnetization dynamics generates a voltage between the junction leads. For an open circuit\nthis voltage is the same for normal and superconducting leads and allows for electrical detection of\nmagnetization dynamics and a structure of a given magnetic texture. However, under the applied\ncurrent the electrical response of the Josephson junction is essentially di\u000berent due to the strong\ndependence of the critical Josephson current on the magnetization direction and can be used for\nexperimental probing of this dependence. We propose a setup, which is able to detect a defect\nmotion and to provide detailed information about the structure of magnetic inhomogeneity. The\ndiscussed e\u000bect could be of interest for spintronics applications.\nI. INTRODUCTION\nAt present the magnetic \feld-driven and current-\ndriven switching in spin valves and related systems, as\nwell as domain-wall (DW) and skyrmion motion is the\nfocus of research activity, which is in part motivated by\nthe promising application potential for spintronics. The\nmajority of the corresponding literature is devoted to the\ndynamics generation: structure of the driving torques\nand their ability to induce the dynamics. On the con-\ntrary, the purpose of present work is to study prospects\nof electrical detection of a given magnetization dynamics,\nwhich can be caused either by an applied magnetic \feld\nor by application of the electric current or by a magnonic\ntorque. The key element of the detection scheme is a\nhybrid structure consisting of probing leads and the fer-\nromagnet in a combination with a 3D topological insu-\nlator (3D TI), where the spin-momentum locking of 3D\ntopological insulator surface states provides a conversion\nof the magnetization dynamics in the ferromagnet into\nan electric voltage. We demonstrate that the suggested\nmethod of electrical detection is able to provide detailed\ninformation about the structure of time-dependent mag-\nnetization texture. The physical basis of the e\u000bect is the\nso-called electromotive force (emf) induced by the mag-\nnetization dynamics. The emf has been widely discussed\nin literature in the context of magnetization dynamics\nin metallic ferromagnets1{11. Due to the existence of\nthe electromotive force the DW motion leads to appear-\nance of an additional voltage drop in the region occupied\nby the moving wall. This voltage can vary in the range\nfromnVto\u0016V10and in special situations can be used\nfor electrical detection of the presence of magnetization\ndynamics12. The emf can be considered as a consequence\nof the presence of a time-dependent gauge potential in the\nlocal spin basis of a spin-textured system3,13.\nThe spin-orbit coupling can also be described in terms\nof SU(2) gauge potential14{22, which becomes time-dependent in the local spin basis in the presence of a\nmagnetization dynamics and, therefore, also results in\nthe appearance of the emf23{26. The property of spin-\nmomentum locking27{30of surface states of a 3D TI can\nbe viewed as an extremely strong spin-orbit coupling.\nProximity to a ferromagnet induces an e\u000bective exchange\n\feld in the 3D TI surface states, which follows the dy-\nnamics of the ferromagnet magnetization. The combi-\nnation of the spin-momentum locking and the induced\nexchange \feld results in the appearance of a special type\nof the emf in the 3D TI surface layer, which is deter-\nmined by time derivatives of the in-plane magnetization\ncomponents31{33. Motivated by this fact here we con-\nsider a hybrid structure in a Josephson junction geometry\nL/(F/TI)/L, where L is a lead electrode, which can be as\nnormal (N), so as superconducting (S), and F/TI is an\ninterlayer consisting of a ferromagnet (F) and a 3D topo-\nlogical insulator (TI). Here we investigate the electrical\nvoltage generated by externally induced magnetization\ndynamics at the junction and prospects of the e\u000bect for\nelectrical probing of time-dependent magnetization struc-\nture. We compare the responses of a Josephson junction\nand a normal junction to the magnetization dynamics.\nIn both cases the electromotive force resulting from the\nmagnetization dynamics generates a voltage between the\njunction leads. In the open circuit geometry the voltage\nis the same for both normal and superconducting leads\nand allows for electrical detection of magnetization dy-\nnamics. However, under the applied current the electrical\nresponse of the Josephson junction is essentially di\u000berent\ndue to the strong dependence of the critical Josephson\ncurrent on the magnetization direction and can be used\nfor experimental probing of this dependence.\nIt is timely to study F/3D TI hybrids because at\npresent there is great progress in their experimental re-\nalization. In particular, to introduce the ferromagnetic\norder into the TI, random doping of transition metal ele-\nments, e.g., Cr or V, has been employed34{37. The secondarXiv:1904.11193v3  [cond-mat.mes-hall]  1 Mar 20202\noption, which has been successfully realized experimen-\ntally, is a coupling of a nonmagnetic TI to a high Tc\nmagnetic insulator to induce strong exchange interaction\nin the surface states via the proximity e\u000bect38{41. The\nspin injection into a TI surface states via the spin pump-\ning techniques has also been realized and the resulting\nspin-electricity conversion has been measured42{45.\nThe investigated electrical response of the Josephson\njunction to the magnetization dynamics could be of in-\nterest for spintronics applications because it provides a\nway to electrically read information encoded in the mag-\nnetization. We propose a setup, which is able not only to\ndetect a defect motion but also to provide detailed infor-\nmation about the structure of magnetic inhomogeneity.\nThe paper is organized as follows. In Sec.II we describe\nthe system under consideration, formulate the necessary\nequations and calculate the general expression for the\nelectric current in the presence of magnetization dynam-\nics. In Sec.III we investigate the electrical response of\nthe system to magnetization dynamics in two particu-\nlar situations: for an open circuit and in the presence of\nthe constant applied electric current. In that section we\nalso demonstrate the possibilities of using the e\u000bect to\ndetect DW motion and DW structure. Our conclusions\nare formulated in Sec.IV.\nII. MODEL AND METHOD\nThe possible sketches of the system under considera-\ntion are presented in Fig. 1. The interlayer region of a\nS/3D TI/S Josephson (or N/3D TI/N) detecting junc-\ntion is covered by a ferromagnet. The magnetization dy-\nnamics is assumed to be induced in the ferromagnet by\nexternal means. We discuss particular examples below,\nand now we derive a formalism allowing for studying the\nelectrical response to the magnetization dynamics of a\ngeneral type and consider only the detector region, which\ncan be viewed as a S(N)/3D TI/S(N) junction (see insert\nto Fig. 1, for concreteness x-axis is chosen perpendicular\nto S/F interfaces). In principle, the ferromagnet can be\nas metallic, so as insulating, but insulating ferromagnets\nare better candidates for the measurements, as it is ex-\nplained below. We assume that the transport between\nthe leads occurs via the TI surface states. This is strictly\nthe case for insulating ferromagnets. We believe that our\nresults can be of potential interest for systems based on\nBe2Se3=YIG orBe2Se3=EuS hybrids, which were real-\nized experimentally38{41. For metallic ferromagnets the\nsituation is a bit more complicated. If the ferromagnet\nis strong, that is its exchange \feld is comparable to the\nFermi energy, then the Josephson current through the fer-\nromagnet is greatly suppressed and indeed \rows through\nthe 3D TI surface states. At the same time the normal\n(quasiparticle) current mainly \rows in the ferromagnet\nbecause its resistance is typically much smaller than the\nresistance of the conducting surface layer of the TI. This\nprovides an additional channel for the normal current,which prevents the observation of the e\u000bect under con-\nsideration, as discussed below.\nFIG. 1. (a) and (b) are possible sketches of the system under\nconsideration. Insert: detector regions in the both panels are\nmodelled by a S/F-TI/S junction.\nIt is assumed that the magnetization M(r) of the fer-\nromagnet induces an e\u000bective exchange \feld heff(r)\u0018\n\u0000M(r) in the underlying conductive TI surface layer\nandheffis small as compared to the exchange \feld in\nthe ferromagnet. The Hamiltonian that describes the TI\nsurface states in the presence of an in-plane exchange\n\feldheff(r) reads:\n^H=Z\nd2r^\ty(r)^H(r)^\t(r); (1)\n^H(r) =\u0000ivF(r\u0002ez)^\u001b+heff(r)^\u001b\u0000\u0016; (2)\nwhere ^\t = (\t\";\t#)T,vFis the Fermi velocity, ezis a\nunit vector normal to the surface of the TI, \u0016is the chem-\nical potential, and ^\u001b= (\u001bx;\u001by;\u001bz) is a vector of Pauli\nmatrices in spin space. It was shown46{48that in the\nquasiclassical approximation ( heff;\")\u001c\u0016the Green's\nfunction has the following spin structure:\n\u0014g(nF;r;\";t) = ^g(nF;r;\";t)(1 +n?\u001b)=2; (3)\nwheren?= (nF;y;\u0000nF;x;0) is the unit vector perpen-\ndicular to the direction of the quasiparticle trajectory\nnF=pF=pFand ^gis the spinless 4\u00024 matrix in the\ndirect product of particle-hole and Keldysh spaces. The\nspin structure above re\rects the fact that the spin and\nmomentum of a quasiparticle at the surface of a 3D TI\nare strictly locked and make a right angle.3\nHere we assume that the TI surface states are in the\nballistic limit because this regime is more relevant for ex-\nisting experiments. Following standard procedures49,50it\nwas demonstrated46{48,51that the spinless Green's func-\ntion ^g(nF;r;\";t) obeys the following transport equations\nin the ballistic limit:\n\u0000ivFnF^r^g=h\n\"\u001cz\u0000^\u0001;^gi\n\n; (4)\nwhere the spin-momentum locking allows for including\nheffinto the gauge covariant gradient\n^r^g=r^g+ (i=vF)[(hxey\u0000hyex);^g]\n: (5)\n[A;B]\n=A\nB\u0000B\nAandA\nB= exp[(i=2)(@\"1@t2\u0000\n@\"2@t1)]A(\"1;t1)B(\"2;t2)j\"1=\"2=\";t1=t2=t.\u001cx;y;z are Pauli\nmatrices in particle-hole space with \u001c\u0006= (\u001cx\u0006i\u001cy)=2.\n^\u0001 = \u0001(x)\u001c+\u0000\u0001\u0003(x)\u001c\u0000is the matrix structure of the\nsuperconducting order parameter \u0001( x) in the particle-\nhole space. We assume \u0001( x) = \u0001e\u0000i\u001f=2\u0002(\u0000x\u0000d=2) +\n\u0001ei\u001f=2\u0002(x\u0000d=2).\nEq. (4) should be supplemented by the normaliza-\ntion condition ^ g\n^g= 1 and boundary conditions at\nx=\u0007d=2. As a minimal model we assume no potential\nbarriers at the x=\u0007d=2 interfaces and consider these\ninterfaces as fully transparent. In this case the bound-\nary conditions are extremely simple and are reduced to\ncontinuity of ^ gfor a given quasiparticle trajectory at the\ninterfaces. However, our result for the emf remains valid\neven in a di\u000busive case and for low-transparent interfaces.\nThe density of electric current along the x-axis can be\ncalculated via the Green's function as follows:\njx=\u0000eNFvF\n41Z\n\u00001d\"\u0019Z\n\u0000\u0019d\u001e\n2\u0019cos\u001egK; (6)\nwhere\u001eis the angle the quasiparticle trajectory makes\nwith thex-axis.gKis the Keldysh part of the nor-\nmal Green's function, which can be expressed via the\nretarded, advanced parts and the distribution function '\nas follows:gK=gR\n'\u0000'\ngA. In general, the electric\ncurrent through the junction consists of two parts: the\nJosephson current jsand the normal current jn. The\nJosephson current is connected to the presence of the\nnonzero anomalous Green's functions in the interlayer\nand exists even in equilibrium. Below we calculate the\nboth contributions to the current microscopically. It is\nassumed that the e\u000bective exchange \feld in the interlayer\nof the junction is spatially homogeneous.\nHere we work near the critical temperature of the su-\nperconductors, that is \u0001 =Tc\u001c1. The Josephson current\nfor the system under consideration in this regime has al-ready been calculated52and takes the form:\njs=jcsin(\u001f\u0000\u001f0); (7)\njc=jb\u0019=2Z\n\u0000\u0019=2d\u001ecos\u001e\u0002\nexph\n\u00002\u0019Td\nvFcos\u001ei\ncosh2hxdtan\u001e\nvFi\n; (8)\n\u001f0= 2hyd=vF; (9)\nwherejb=evFNF\u00012=(\u00192T). Similar expression has al-\nready been obtained for Dirac materials51. It is seen\nfrom Eqs. (8)-(9) that the Josephson current manifest\nstrong dependence on the orientation of the ferromagnet\nmagnetization. It is sensitive to the y-component of the\nmagnetization only via the anomalous phase shift47,54,55,\ntherefore this component does not lead to superconduc-\ntivity suppression in the interlayer and does not in\ru-\nence the amplitude of the critical current. At the same\ntimex-component of the magnetization does not cou-\nple to superconductivity via the anomalous phase shift,\nbut causes the superconductivity depairing in the inter-\nlayer leading to the suppression of the critical current.\nThe suppression of the critical current as a function of\nmx\u0011Mx=Msis presented in Fig. 2. For estimates we\ntaked= 50nm,vF= 105m=s andTc= 10K, what cor-\nresponds to the parameters of Nb=Bi 2Te3=Nb Joseph-\nson junctions56. In this case \u0018N=vF=2\u0019Tc\u001912nm.\nWe have also plotted jc(mx) forTc= 1:8K, what corre-\nsponds to the Josephson junctions with Alleads.\nIt is di\u000ecult to give an accurate a-priori estimate of\nheffbecause there are no reliable experimental data on\nits value. However, basing on the experimental data\non the Curie temperature of the magnetized TI sur-\nface states40, where the Curie temperature in the range\n20\u0000150Kwas reported, we can roughly estimate heff\u0018\n0:01\u00000:1hYIG. We assume heff\u001820\u0000100Kin our\nnumerical simulations, what corresponds to the dimen-\nsionless parameter r= 2heffd=vF= 2:6\u000013:2. The\nboth parts of the strong dependence of the Josephson cur-\nrent on the magnetization orientation (the dependence\nvia anomalous phase shift and the dependence via the\namplitude of the critical current) manifest themselves in\nthe electrical response of the junction on the magnetiza-\ntion dynamics, as we demonstrate below.\nThe normal current is due to the deviation of the dis-\ntribution function from the equilibrium. It was not cal-\nculated in Ref. 52, therefore we discuss it in detail. In\norder to \fnd 'we have to solve the kinetic equation,\nwhich can be obtained from the Keldysh part of Eq. (4)\nand takes the form\n\u0000vF;x@x'= _'+_heffn?@\"': (10)\nWhen deriving this equation we have made the following\nassumptions: 1) we have neglected all the corrections to\nthe kinetic equation due to superconductivity, because\nthey lead to the corrections in the \fnal expression for4\nAl\nNb100K\nheff= 20K100K\n20Kjc/jc0\nmx\nFIG. 2.jcas a function of mxforr= 13:2,d=\u0018N= 4:1 (solid\nblue);r= 2:6,d=\u0018N= 4:1 (solid red); r= 13:2,d=\u0018N= 0:74\n(dashed blue); r= 2:6,d=\u0018N= 0:74 (dashed red). jcis\nnormalized to jc0\u0011jc(mx= 0).\nthe normal current of the order of (\u0001 =Tc)2, which can\nbe safely neglected near the critical temperature. In this\napproximation we neglect the terms of the same order\n(\u0001=Tc)2(eV=Tc) in the Josephson current, as well as in\nthe normal current; 2) the interlayer of the junction is as-\nsumed to be shorter than the inelastic energy relaxation\nlength, therefore all the inelastic relaxation processes are\nneglected in Eq. (10) and 3) we have also expanded \n-\nproducts up to the lowest order with respect to time\nderivatives: A\nB\u0019AB+ (i=2)(@\"A@tB\u0000@tA@\"B).\nThe applicability of this expansion is justi\fed by the fact\nthat the voltage, induced by the magnetization dynamics\nat the junction is small eV=(kBTc)\u001c1, what is demon-\nstrated by further numerical calculations.\nWe solve Eq. (10) neglecting the term _ 'and assuming\nthat the deviation of the distribution function from equi-\nlibrium is small '= tanh(\"=2T) +\u000e'. The last assump-\ntion is justi\fed by the condition eV=kBTc\u001c1. The term\n_'can be neglected if d=(vFtd)\u001c1, wheretdis the char-\nacteristic time of magnetization variations. For estimates\nwe taked= 50nm,vF= 105m=s andtd= 0:5\u000210\u00008s\n(this value of tdcorresponds to material parameters of\nYIG thin \flms, see below). In this case d=(vFtd)\u001810\u00004\nand the term _ 'can safely be neglected.\nThe solution should also satisfy the asymptotic val-\nues'\u0006= tanh[(\"\u0007eV=2)=2T]\u0019tanh(\"=2T)\u0007\n(eV=4T) cosh\u00002(\"=2T) atx=\u0007d=2 if we assume that\nthe leads are in equilibrium except for the voltage drop\nVbetween them. In this case the solution of Eq. (10)\ncan be easily found and takes the form:\n'\u0006= tanh\"\n2T\u0007eV\n4T1\ncosh2(\"=2T)\u0000\n(x\u0006d=2)\n2Tcosh2(\"=2T)_heffn?\nvF;x; (11)\nwhere the subscript \u0006corresponds to the trajectories\nsgnvx=\u00061.\nSubstituting Eq. (11) into Eq. (6), for the quasiparticlecontribution to the current we \fnally obtain\njn=e2NFvF\n\u0019\u0010\nV\u0000_hyd\nevF\u0011\n: (12)\nIt is seen that in the presence of magnetization dy-\nnamics there is an electromotive force E=_hyd=(evF) in\nthe TI resulting from the emergent electric \feld induced\ndue to the simultaneous presence of the time-dependent\nexchange \feld and spin-momentum locking.\nIII. ELECTRICAL RESPONSE TO\nMAGNETIZATION DYNAMICS\nA. Open circuit\nNow let us assume that the domain structure of the\nstrip is moved along y-axis. Let us consider the setup\npresented in Fig. 1(a) with an insulating ferromagnet\nand assume that an electric current is applied between\nthe S(N) leads. The total electric current jthrough the\njunction is a sum of the supercurrent contribution Eq. (7)\nand the normal current Eq. (12) \rowing via the TI sur-\nface states. It can be rewritten as follows:\nj=jcsin(\u001f\u0000\u001f0) +1\n2eRN( _\u001f\u0000_\u001f0) (13)\nFor the open circuit j= 0 and the solution of Eq. (13)\nis\u001f(t) =\u001f0(t). Therefore, the voltage generated at the\njunction due to magnetization dynamics in the ferromag-\nnet is\nVx=_\u001f\n2e=_hyd=evF; (14)\nwhere we denote the voltage at the junction presented in\nFig. 1(a) as Vx. The same physical picture is valid for\nthe junction presented in Fig. 1(b), but the voltage Vyat\nthis junction is determined as Vy=_hxd=evF. In the both\ncases the voltage is determined by the dynamics of the\nmagnetization component perpendicular to the current\ndirection. It is the same as for superconducting, so as\nfor nonsuperconducting leads at j= 0 and is determined\nonly by the emf. This is the consequence of the fact that\nthe emf and the anomalous phase shift are manifestations\nof the same gauge vector potential, which is determined\nonly by the spin-momentum locking and magnetization\nand is not in\ruenced by superconductivity.\nThe e\u000bect of this voltage generation can be used for\nelectrical detection of magnetization dynamics. Measur-\ning the voltages at the two junctions (sketched in panels\n(a) and (b) of Fig. 1), which are called by \"detectors\"\nfurther and attached to one and the same ferromagnetic\nstrip, one can obtain the full time dependence of the in-\nplane magnetization components Mx(t) andMy(t) in the\nferromagnet. The particular example of the correspond-\ning numerical simulation is demonstrated below.\nTo be concrete we consider the magnetic \feld-\ndriven DW motion along the ferromagnet strip.5\nThe \feld-induced and current induced DW motions\nhave been widely investigated both theoretically and\nexperimentally57{75. In principle, the electric current\nj6= 0 \rowing via the detector arranged as shown\nin Fig. 1(b) can generate an additional torque on the\nmagnetization76{79. For the open circuit considered here\nthis contribution is absent, and we also neglect it in the\nnext subsection, where we assume j6= 0, because we as-\nsume the current via the detector to be small and neglect\nits additional minor contribution to the external torque\nmoving the DW. We \fnd M(y;t) numerically from the\nLandau-Lifshitz-Gilbert (LLG) equation\n@M\n@t=\u0000\rM\u0002Heff+\u000b\nMsM\u0002@M\n@t; (15)\nwhereMsis the saturation magnetization, \ris the gyro-\nmagnetic ratio and Heffis the local e\u000bective \feld\nHeff=HKMy\nMsex+2A\nM2sr2\nyM\u00004\u0019Mzez+Hextex(16)\nHkis the anisotropy \feld, along the x-axis,Ais the ex-\nchange constant, Hextis the external \feld and the self-\ndemagnetization \feld 4 \u0019Mzis included. For numerical\ncalculations we use the material parameters of YIG thin\n\flms53:HK\u00180:5Oe, 4\u0019Ms= 1000Oe,\u000b\u00180:01 and\nthe domain wall width dW= 1\u0016m.\nAs it has already been de\fned above the voltage, mea-\nsured by the detector shown in Fig. 1a(b) is denoted by\nVx(y)and probes _My(x). The result of the numerical cal-\nculation ofMx;y;z(t) at the detectors is presented in pan-\nels (a) of Figs. 3-4, while the corresponding dependencies\nVx;y(t) are presented at panels (b) of these \fgures. As\nthe characteristic size of the region containing the both\ndetectors\u0018100nmis assumed to be much smaller than\nthe characteristic scale of magnetic inhomogeneity dW,\nwe can safely use our Eq. (14), which is strictly valid\nfor a homogeneous magnetization, for calculation of the\nvoltage at the detector.\nFig. 3 demonstrates the results for small enough exter-\nnal applied \feld Hext= 0:003Ms. This \feld is below the\nWalker's breakdown \feld58and in this regime the DW\nmoves keeping its initial plane structure. Having at hand\nVx;y(t) it is possible to restore the time-dependent struc-\nture of the moving wall at the detector point according\nto the relation\n\u0001Mx;y(t)=Ms= (1=V0td)tZ\ntiVy;xdt; (17)\nwhereV0=dheff=(jejvFtd) is the natural unit of the\nvoltage induced at a given detector. The characteris-\ntic time of magnetization variation can be obtained from\nthe LLG equation and takes the form td= 1=(4\u0019\u000b\rMs).\nTaking the material parameters of YIG thin \flms we ob-\ntain thattd\u00180:5\u000210\u00008s.\nFig. 4 represents the results for higher applied \feld\nHext= 0:02Ms, which exceeds the Walker's breakdown\nmxmy\nmz\n0 2 4 6 8 10 12-1.0-0.50.00.51.0\nt/tdmx,y,z(a)\nVx\nVy\n0 2 4 6 8 10 12-1.0-0.50.00.51.01.52.02.5\nt/tdVx,y/V0(b)FIG. 3. (a) Magnetization components mx;y;z =Mx;y;z=Ms\nin the region of the detectors and (b) voltages Vx;yas func-\ntions of time. Hext= 0:003Ms, for other parameters of the\nnumerical calculation see text.\n\feld. In this regime the DW initial plane shape is not\npreserved during the motion, what is re\rected in the os-\ncillations of all the magnetization components, as it is\nseen in Fig. 4(a). In this regime the magnetization at the\ndetector point as a function of time can also be found as\ndescribed above.\nObservation of a single DW traveling through the de-\ntector region gives a way to \fnd experimentally the\ne\u000bective exchange \feld heffinduced by the ferromag-\nnet in the surface layer of the TI. Indeed, in this case\nj\u0001Mx=Msj= 2 after the DW passes through the detec-\ntor region. Then heffcan be found from Eq. (17) as\nfollows:\nheff=j(evF=2d)tfZ\ntiVydtj; (18)\nwhereVyshould be integrated over the whole time region,\nwhen the voltage is nonzero.\nTypical values of the voltage induced at the detectors\nare of the order of V0. It is di\u000ecult to give an accurate a-\npriori estimate of V0because there are no reliable experi-\nmental data for heff. However, basing on the experimen-\ntal data discussed above and assuming heff\u001820\u00001000K\nandd=(vFtd)\u001810\u00004, we obtain V0\u00180:2\u000010\u0016V.\nIf the ferromagnet is metallic, there is also an addi-\ntional normal current \rowing via the ferromagnet. The6\nmxmy\nmz\n21.5 22.0 22.5 23.0 23.5 24.0 24.5-1.0-0.50.00.51.0\nt/tdmx,y,z(a)\nVx\nVy\n21.5 22.0 22.5 23.0 23.5 24.0 24.5-10-50510\nt/tdVx,y/V0(b)\nFIG. 4. (a) Magnetization components mx;y;z =Mx;y;z=Ms\nat the detectors and (b) voltages Vx;yas functions of time.\nHext= 0:02Ms, the detectors are located in the spatial region,\nwhere the DW motion is steady.\nresistance of the ferromagnet is typically much smaller\nthan the resistance of the TI surface states. For this\nreason the voltage induced at the junction due to the\npresence of the emf in the TI is suppressed by the fac-\ntorRF=(RF+RN), whereRF(N)is the resistance of\nthe ferromagnet (TI surface states). Therefore, metal-\nlic ferromagnets are not good candidates for measuring\nthe discussed e\u000bect. In addition, as it was already men-\ntioned, the time-dependent spin texture of a ferromagnet\nalso gives rise to emergent spin-dependent electric and\nmagnetic \felds and, consequently, to an additional par-\nasitic voltage in it. This voltage can interfere with the\nemf generated in the TI and the resulting e\u000bect is quite\ncomplicated.\nIn principle, a similar structure of the emf can also be\nobtained for systems where a topological insulator is re-\nplaced by a material with Rashba spin-orbit coupling or\nif the Rashba spin-orbit coupling is an internal property\nof the ferromagnetic \flm arising, for example, from struc-\ntural asymmetry in the z-direction. However, we consider\nthe TI-based systems to be a more preferable variant be-\ncause here the e\u000bect should be stronger. The emf gen-\nerated inx-direction is also predicted to be proportional\nto_hyin Rashba spin-orbit based junctions23{26, but in\naddition it should contain a reducing factor \u0001 so=\"F\u0018\n\u000bR=~vF. This factor can be estimated by taking a real-\nistic value of \u000bR\u001810\u000010eVm80,81for the interfaces ofheavy-metal systems. Then \u0001 so=\"F\u00180:1 if one assumes\nvF\u0018106ms\u00001. Therefore, it reduces the value of the\nemf, which is important for the magnetization detection.\nB. Response in the presence of the applied electric\ncurrent\nNow we turn to the case when the constant electric\ncurrentjis applied to the junction. In this case the\nvoltageV= _\u001f=2eshould be found from Eq. (13). If\njcdoes not depend on time, the solution of this equation\nrepresents the well-known voltage VJ(t) of the ac Joseph-\nson e\u000bect82shifted due to the presence of magnetization\ndynamicsV=VJ(t) + _\u001f0=2e. This leads to the appear-\nance of the nonzero resistance of the IV-characteristics\natj <jcand an additional resistance at j >jc26. How-\never, the speci\fc feature of the TI-based system is the\nstrong dependence of jcon the e\u000bective exchange \feld\ncomponent parallel to the current direction, what results\nin the strong dependence of jcon time in the presence of\nmagnetization dynamics.\nIn Figs. 5-6 we present the results of the electrical re-\nsponse of the Josephson detectors on the moving DW\nunder the constant applied current \rowing via the detec-\ntors. These \fgures are plotted for heff= 100K, when\nthe dependence of the critical current on magnetization\norientation is given by the blue solid curve in Fig. 2. We\ntake the value of the applied current j= 0:9jc0. De-\npending on the particular magnetization orientation this\nvalue of the applied current can be lower as well as higher\nthan the critical current of the junction for a given orien-\ntation, see Fig. 2. Figs. 5a-6a demonstrate the voltages\ninduced at the detectors as functions of time, while the\ncritical currents of the detectors are shown in panels (b).\nWe denote the critical current of the junction shown in\nFig. 1a(b) as jcx(cy). Fig. 5 corresponds to the param-\neters of Fig. 3a when the magnetic \feld driven the DW\nmotion is below the Walker's breakdown, and Fig. 6 rep-\nresents the case when it exceeds the Walker's breakdown\n\feld and the magnetization of the moving DW oscillates,\nas in Fig. 4a.\nAccording to Eq. (8) the critical current jcx(cy)de-\npends on time via the dependence of hx(y). This depen-\ndence is clearly seen in Figs. 5b-6b. When there is no\nDW inside the detector region, jcyis maximal, because\nthere is only heff;x6= 0, that is the component of the\ne\u000bective exchange \feld perpendicular to the Josephson\ncurrent direction, which does not suppress the value of\nthe critical current. At the same time jcxis fully sup-\npressed by this e\u000bective \feld component, because for this\ndetector it is parallel to the current. When a DW passes\nthrough the detector region the situation changes: jcyis\nsuppressed by nonzero heff;y, andjcxis restored due to\nthe decrease of heff;x in the region occupied by the DW.\nThejcy(cx)(t) dependence on time manifests itself in\nthe voltage induced at the corresponding Josephson junc-\ntion. As it is seen from Figs. 5a-6a, voltage Vxis nonzero7\nVx\nVy\n2 4 6 8 10 120.00.51.01.52.02.53.03.5\nt/tdVx,y/V0(a)\njcx\njcy\n2 4 6 8 10 120.00.20.40.60.81.0\nt/tdjcx,cy/jc0(b)\nFIG. 5. (a) Voltages Vx;yas functions of time. (b) Critical\ncurrentsjcxandjcyas functions of time. j= 0:9jc0,heff=\n100K,jc0RN= 1\u0016V,V0= 0:75\u0016V. The other parameters\nare the same as for Fig. 3.\nwhen there are no DWs in the detector region. This de-\ntector is in the resistive state because its critical current\nis less then the externally applied current. When a DW\ntravels via the detectors region we observe as Vx, so as\nVyvoltage pulses. These pulses are the results of two dif-\nferent e\u000bects: (i) the contribution due to emf, the same\nas in the open circuit discussed above and (ii) the contri-\nbution due to the dependence of the critical current on\ntime, which leads to the time dependence of the phase\ndi\u000berence\u001fbetween the superconductors. For the ex-\nample under consideration jcybecomes lower than the\napplied current when the DW passes through the detec-\ntor region. This leads to the appearance of the Joseph-\nson oscillations, which are seen in Fig. 5a. The picture of\nthe oscillating DW moving under a \feld higher than the\nWalker's breakdown, presented in Fig. 6, is more com-\nplicated. The magnetization oscillations are clearly seen\nin the dependence of the critical current on time. These\noscillations manifest themselves in the dependencies Vx;y\nand for the chosen parameters interfere with Josephson\noscillations.\nIn Figs. 7-8 we consider another regime, when the ap-\nplied current is less than the critical current of the de-\ntectors for any magnetization orientation. The param-\neters correspond to the case of lower e\u000bective exchange\n\feldheff= 20K, corresponding to the solid red curve in\nFig. 2. It is seen that in the absence of the moving DW\nVx\nVy\n21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5-10-50510\nt/tdVx,y/V0(a)\njcx\njcy\n21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.50.00.20.40.60.81.0\nt/tdjcx,cy/jc0(b)FIG. 6. (a) Voltages Vx;yas functions of time. (b) Critical\ncurrentsjcxandjcyas functions of time. j= 0:9jc0,heff=\n100K,jc0RN= 1\u0016V,V0= 0:75\u0016V. The other parameters\nare the same as for Fig. 4.\ninside the detector region the both detector junctions are\nin the dissipationless regime. The voltage pulses occur in\nVx;ywhen the DW passes through the detectors. In this\ncase there are no Josephson oscillations in Vx;y, as it is\nseen in Fig. 7a. The shape of the voltage pulses is very\nclose to the shape obtained for the open circuit, however\nthe amplitudes of the pulses are di\u000berent due to the ad-\nditional contribution from the dependence of the critical\ncurrent on time. Fig. 8 demonstrates the electrical re-\nsponse of the detectors to the oscillating motion of the\nDW above the Walker's breakdown. These oscillations\nagain manifest themselves in the dependence of the crit-\nical currents jcx(cy)on time. In the present case they di-\nrectly appear in the oscillating behavior of Vx;ywithout a\ncontamination by the Josephson oscillations because the\napplied current does not exceed the critical current of the\ndetectors.\nAs we can see, the experiment under the applied cur-\nrent is less preferable for detection of the magnetization\ndynamics because one cannot extract a pure emf sig-\nnal due to the time dependence of the critical current.\nHowever, it might be of interest for experimental deter-\nmination of jc(t). Indeed, simultaneous measurements\nof the electrical voltage at the junction in the open cir-\ncuit and under the applied current allows for the calcula-\ntion ofjc(t) and, consequently, jc(Mx;My) according to\nEq. (13).8\nVx\nVy\n2 4 6 8 10 12-1012\nt/tdVx,y/V0(a)\njcx\njcy\n2 4 6 8 10 120.60.70.80.91.0\nt/tdjcx,cy/jc0(b)\nFIG. 7. (a) Voltages Vx;yas functions of time. (b) Critical\ncurrentsjcxandjcyas functions of time. j= 0:5jc0,heff=\n20K,jc0RN= 1\u0016V,V0= 0:15\u0016V. The other parameters are\nthe same as for Fig. 3.\nIn principle, the contributions to the voltage from the\nemf and from the time-dependence of the critical current\ncan be separated. For example, if one considers the dy-\nnamics of a DW wall with perpendicular anisotropy83, lo-\ncated in the ( y;z)-plane, then for the detector presented\nin Fig. 1b there is no in-plane exchange \feld component,\nwhich is perpendicular to the current. Therefore, the emf\ncontribution does not occur at this detector. At the same\ntime, the component of heffparallel to the current is ab-\nsent at the detector in Fig. 1a. Consequently, the critical\ncurrent does not depend on time for this detector, and\nthe emf is the only contribution to the voltage.\nIV. CONCLUSIONS\nThe electrical response of the S/3D TI-F/S Joseph-\nson junction to magnetization dynamics was studied and\ncompared to the electrical response of the junction with\nnonsuperconducting leads. In 3D TI/F hybrid structures\nspin-momentum locking of the 3D TI conducting surface\nstates in combination with the induced magnetization\nleads to the appearance of a gauge vector potential. In\nthe presence of magnetization dynamics the gauge vec-tor potential becomes time-dependent and generates an\nelectromotive force. In both cases of superconducting\nVx\nVy\n21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5-10-50510\nt/tdVx,y/V0(a)\njcx\njcy\n21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.50.60.70.80.91.0\nt/tdjcx,cy/jc0(b)\nFIG. 8. (a) Voltages Vx;yas functions of time. (b) Critical\ncurrentsjcxandjcyas functions of time. j= 0:5jc0,heff=\n20K,jc0RN= 1\u0016V,V0= 0:15\u0016V. The other parameters are\nthe same as for Fig. 4.\nand nonsuperconducting leads this emf generates a volt-\nage between the leads. For an open circuit this voltage\nis the same for both normal and superconducting leads\nand allows for electrical detection of magnetization dy-\nnamics, a structure of a time-dependent magnetization\nand a measurement of the e\u000bective exchange \feld. In the\npresence of the applied current the electrical response\nof the Josephson junction contains additional contribu-\ntion from the time dependence of the critical Josephson\ncurrent, which comes from the strong dependence of the\ncritical current on the magnetization orientation. In par-\nticular geometries it complicates quantitative detection\nof the exact shape of the time-dependent magnetization\ntexture. 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We reliably calculate the dynamical e lectron mass by numerically solving the\nnonperturbative Schwinger-Dyson equations in a consisten t truncation within the lowest Landau\nlevel approximation. It is shown that the generation of dyna mical electron mass in a strong mag-\nnetic field is significantly enhanced by the perturbative ele ctron mass that explicitly breaks chiral\nsymmetry in the absence of a magnetic field.\nPACS numbers: 11.30.Rd, 11.30.Qc, 12.20.Ds\nProperties of matter in strong magnetic fields are of\nbasic interest [1, 2, 3, 4, 5] and have great potential ap-\nplications in the physics of compact stellar objects and\nthe early universe cosmology [6]. The observations of\nsoft gamma repeaters and anomalous X-ray pulsars have\nprovided compelling evidence that the magnetic fields on\nthe surface of young neutron stars are in the range of\n1014−1016G [7]. It has been suggested that at the elec-\ntroweak phase transition local magnetic fields as high as\n1022−1024G could be generated [8]. Situations of even\nstronger magnetic fields may exist in extreme astrophys-\nical and cosmological environments.\nIt has been established that the magnetic catalysis of\nchiral symmetry breaking is a nonperturbative universal\nphenomenon [9, 10, 11]. A strong magnetic field acts as a\ncatalystforchiralsymmetrybreaking,leadingtothegen-\neration of a dynamical fermion mass even at the weakest\nattractive interaction between fermions. The hallmark of\nthis effect is the dimensional reduction from (3 + 1) to\n(1+1)inthedynamicsoffermionpairinginastrongmag-\nnetic field when the lowest Landau level (LLL) plays the\ndominantrole. Therealizationofthis phenomenonin the\nchiral limit in QED (i.e., QED with massless fermions)\nhas been studied extensively in the literature over the\npast decade [10, 11, 12].\nBut until very recently, there has been no agreement\non the correct calculation of the dynamical fermion mass\ngenerated through chiral symmetry breaking in QED in\na strong magnetic field, and contradictory results have\nbeenfoundintheliterature[12, 13]. Theresolutionofthe\ncontradiction lies in the establishment of the gauge fix-\ning independence of the dynamically generated fermion\nmass calculated in the nonperturbative Schwinger-Dyson\n(SD) equations approach [14]. In particular, the study of\nRef. [14] has provided an unambiguous identification of\nthe infinite subset of diagrams that contribute to chiral\nsymmetry breaking in a strong magnetic field, and led\nto a consistent calculation of the dynamically generated\nfermion mass, reliable in the weak coupling regime and\n∗sywang@mail.tku.edu.twthe strong field limit (for a brief review, see Ref. [15]).\nIn order to highlight the most important physics re-\ngarding the mechanism of chiral symmetry breaking in a\nstrong magnetic field, the phenomenon has been stud-\nied in the literature mostly in the chiral limit. Nev-\nertheless, the universal nature of the phenomenon dic-\ntates that in realistic massive QED in a strong magnetic\nfield, the electron will acquire a dynamical mass gener-\nated through the modification of the vacuum structure\nthat is induced by the strong magnetic field. This ef-\nfect is essentially analogous to that of the approximate\nchiral symmetry breaking in QCD and the Nambu–Jona-\nLasinio model [16], where, in addition to the perturba-\ntive current quark masses, the quarks acquire nonpertur-\nbative constituent masses of dynamical origin that are\nbrought about by the breaking of chiral symmetry.\nIn this article, we extend the study of Ref. [14] to the\ncase of massive QED in a strong magnetic field. Specifi-\ncally,weconsistentlycalculatethedynamicallygenerated\nelectron mass in the weak coupling regime and the strong\nfield limit. While similar problems have been studied in\nthe past [17], to the best of our knowledge, a consistent\ncalculation of the dynamical electron mass in a strong\nmagnetic field has not appeared in the literature. Apart\nfrom its theoretical interest, this problem is of practical\ninterest and importance. In particular, sizable modifica-\ntions of the electron mass as induced by strong magnetic\nfields will find applications in neutron star astrophysics\nand early universe cosmology.\nThe Lagrangiandensity ofmassiveQEDin an external\nmagnetic field is given by\nL=−1\n4FµνFµν+ψγµ[i∂µ+e(Aext\nµ+Aµ)]ψ−mψψ,(1)\nwhereψisthequantumfermion(electron)field, Aµisthe\nAbelian quantum gauge boson (photon) field, Fµνis the\ncorresponding electromagnetic field strength, and Aext\nµ\ndescribes an external magnetic field. Here and hence-\nforth, we set ¯ h=c= 1 and use the conventions in which\ngµν= diag(−1,1,1,1) withµ,ν= 0,1,2,3. The Dirac\nmatrices satisfy {γµ,γν}=−2gµνandγ5=iγ0γ1γ2γ3.\nIntheLagrangiandensity(1), wehavenotincludedthe\ncounterterms associated with the usual ultraviolet renor-2\nmalization in QED. This is because we are solely inter-\nested in the dynamics of the electrons in the LLL, which\nis ultraviolet finite due to the effective dimensional re-\nduction as remarked above. Hence, the constants mand\nein the Lagrangian density (1) denote respectively the\nelectron mass and the absolute value of its charge that\nare defined with an appropriate (perturbative) renormal-\nization in the absence of external fields.\nWe choose the constant external magnetic field of\nstrengthBin thex3-direction. The correspondingvector\npotential is given by Aext\nµ= (0,0,Bx1,0) withB >0. A\nconvenient formalism for the study of QED in the pres-\nence of a constant external magnetic field was developed\na long time ago by Ritus [18]. The so-called Ritus Ep\nfunctions are constructed in terms of the simultaneous\neigenfunctions (eigenvectors) of the mutually commuting\noperators [γµ(i∂µ+eAext\nµ)]2, Σ3≡iγ1γ2andγ5, and\nform a complete set of Dirac matrix-valued orthonormal\nfunctions. The important advantage of the Ritus formal-\nism is that in momentum space spanned by the Epfunc-\ntions, the Dirac equation for a noninteracting fermion in\na constant external magnetic field is formally identical to\nthat in the absence of external fields. It is noted that be-\ncause the fermion mass mis proportional to the identity\noperator,whichobviouslycommuteswiththeabovethree\noperators, the Ritus formalism applies to both massless\nand massive QED.\nIt has been proved in Ref. [14] that the bare vertex\napproximation (BVA) is a consistent truncation of the\nnonperturbative SD equations within the lowest Lan-\ndau level approximation (LLLA). With a momentum\nindependent fermion self-energy that fulfills the Ward-\nTakahashi (WT) identity in the BVA within the LLLA,\nit can be shown that (i) the truncated vacuum polariza-\ntion is transverse; (ii) the truncated fermion self-energy\nis gauge independent when evaluated on the fermion\nmass shell. In particular, the would-be gauge dependent\ncontribution to the truncated fermion self-energy, which\narises from the gauge dependent term in the full pho-\nton propagator, vanishes identically on the fermion mass\nshell. As a consequence, the dynamical fermion mass,\nobtained as the solution of the truncated SD equations\nevaluated on the fermion mass shell, is manifestly gauge\nindependent. The gauge independent analysis presented\nin Ref. [14] is very general in nature and not specific to\nmassless QED. Here we indicate the crucial points in the\nanalysis extended to massive QED.\nThe motion of the LLL electrons is restricted in direc-\ntions perpendicular to the magnetic field, leading to an\neffective dimensional reduction from (3 + 1) to (1 + 1)\nin the dynamics of fermion pairing in a strong magnetic\nfield. Consistentwith theWT identityintheBVAwithin\ntheLLLA[14], thefull propagatorfortheLLLelectronin\nmomentum space (spanned by the Epfunctions) is given\nby\nG(p/bardbl) =1\nγ/bardbl·p/bardbl+m∗∆, (2)wherem∗is the (gauge independent) dynamical electron\nmass in a strong magnetic field, which should not be\nconfused with the perturbative electron mass min the\nabsence of a magnetic field. The dynamical electron\nmassm∗is yet to be determined by solving the trun-\ncated SD equations self-consistently. In the above ex-\npression,p/bardbldenotesthelongitudinalmomentum, namely,\npµ\n/bardbl= (p0,p3) and ∆ = (1+Σ3)/2 is the projection oper-\nator on the electron states with the spin polarized along\nthe external magnetic field. The projection operator ∆\nsatisfies the property ∆ γµ∆ =γµ\n/bardbl∆, which clearly re-\nflects the effective dimensional reduction from (3+1) to\n(1+1) in the dynamics of the LLL electrons.\nThe WT identity in the BVAwithin the LLLAguaran-\ntees that the vacuum polarization Π µν(q) is transverse,\ni.e.,qµΠµν(q) = 0. An explicit calculation yields\nΠµν(q) = Π(q2\n/bardbl,q2\n⊥)/parenleftbigg\ngµν\n/bardbl−qµ\n/bardblqν\n/bardbl\nq2\n/bardbl/parenrightbigg\n, (3)\nwhereq2\n/bardbl=−q2\n0+q2\n3andq2\n⊥=q2\n1+q2\n2. Eq. (3) implies\nthat the full photon propagatorin covariant gauges takes\nthe form\nDµν(q) =1\nq2+Π(q2\n/bardbl,q2\n⊥)/parenleftbigg\ngµν\n/bardbl−qµ\n/bardblqν\n/bardbl\nq2\n/bardbl/parenrightbigg\n+gµν\n⊥\nq2\n+qµ\n/bardblqν\n/bardbl\nq2q2\n/bardbl+(ξ−1)1\nq2qµqν\nq2, (4)\nwhereξis the gauge fixing parameter with ξ= 1 be-\ning the Feynman gauge. In the above expressions, the\npolarization function Π( q2\n/bardbl,q2\n⊥) is given by\nΠ(q2\n/bardbl,q2\n⊥) =2α\nπeBexp/parenleftbigg\n−q2\n⊥\n2eB/parenrightbigg\nF/parenleftbiggq2\n/bardbl\n4m2∗/parenrightbigg\n,(5)\nF(u) = 1−1\n2u/radicalbig\n1+1/ulog/radicalbig\n1+1/u+1/radicalbig\n1+1/u−1,(6)\nwhereα=e2/4πis the fine-structure constant. The di-\nmensionless function F(u) has the following asymptotic\nbehavior:F(u)≃2u/3 for|u| ≪1, andF(u)≃1 for\n|u| ≫1. Hence, photons of momenta m2\n∗≪ |q2\n/bardbl| ≪eB\nandq2\n⊥≪eBarescreenedwith a characteristicscreening\nlengthℓ= 1//radicalbig\n(2α/π)eBinduced by the strong mag-\nnetic field.\nThe self-energy of the LLL electron evaluated on the\nmass shell, p2\n/bardbl=−m2\n∗, leads to the so-called gap equa-\ntion that determines the dynamical electron mass m∗\nself-consistently. The WT identity in the BVA within\nthe LLLA guarantees that contributions to the LLL elec-\ntronself-energyfromthegaugedependent termaswellas\nfrom the terms proportional to qµ\n/bardblqν\n/bardbl/q2\n/bardblinDµν(q) vanish\nidentically on the electron mass shell (see Ref. [14] for a\ndetailed discussion). Thus, in the BVA within the LLLA3\nwe obtain the (gauge independent) on-shell electron SD\nequation\nm∗=m+ie2/integraldisplayd4q\n(2π)4m∗\n(p−q)2\n/bardbl+m2∗\n×exp(−q2\n⊥/2eB)\nq2+Π(q2\n/bardbl,q2\n⊥)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\np2\n/bardbl=−m2∗, (7)\nwhere the projection operator ∆ that multiplies both\nsides of the equation has been dropped. Using the mass\nshell condition pµ\n/bardbl= (m∗,0) that corresponds to a LLL\nelectron at rest and performing a Wick rotation to Eu-\nclidean space, we find the gap equation to be given by\nm∗=m+α\n2π2/integraldisplay\nd2q/bardblm∗\nq2\n3+(q4−m∗)2+m2∗\n×/integraldisplay∞\n0dq2\n⊥exp(−q2\n⊥/2eB)\nq2\n/bardbl+q2\n⊥+Π(q2\n/bardbl,q2\n⊥), (8)\nwhereq2\n/bardbl=q2\n3+q2\n4is the photon longitudinal momentum\nsquared in Euclidean space.\nBefore proceeding further, we discuss the solution to\nthe gapequation(8) in the chirallimit (i.e., m= 0). This\nwill be useful for our discussions below. The solution in\nthe chiral limit (denoted here and henceforth by mdyn)\nwas obtained numerically and shown to be fitted by the\nanalytic expression [14]\nmdyn=a√\n2eB αexp/bracketleftbigg\n−π\nαlog(b/α)/bracketrightbigg\n,(9)\nwhereais a constant of order one and b≃2.3. From\nEq. (9) it can be seen clearly that while on the one hand\nmdynscales as√\n2eBand increases with increase of B,\non the other hand it is exponentially suppressed at weak\ncoupling. The wide separation of scales mdyn≪√\neB,\ntogether with the gauge independence of mdyn, is at the\nheartofthefactthattheresultof mdyngivenbyEq.(9)is\nreliable in the weak coupling regime and the strong field\nlimit [14]. To get a feeling of the order of magnitudes in-\nvolved, it is instructive to note that for α= 1/137,mdyn\nis about 34 orders of magnitude smaller than the energy\nbetween adjacent Landau levels,√\n2eB, and it would re-\nquire an enormousmagneticfield ofabout 1082G to have\nmdyncomparable to m. As a result, this nonperturbative\neffect can be safely ignored in the chiral limit in QED\neven though ultrastrong magnetic fields may be under\nconsideration.\nHowever, as will be seen below, in realistic massive\nQED the generation of dynamical electron mass in a\nstrong magnetic field is significantly enhanced by the\nperturbative electron mass that explicitly breaks chiral\nsymmetry in the absence of a magnetic field. This is the\nnovel result of the present article.\nWe havenumericallysolvedthe gapequation(8) to ob-\ntainm∗as a function of B. In Fig. 1 the dynamical elec-\ntron massm∗(together with its value in the chiral limit,10010510101015102010251030\nB/B0100101102m*/mα = 1/137\nα = 0.03\nα = 0.07\nα = 0.1\nFIG. 1: Plot of the dynamical electron mass m∗(in units\nof the perturbative electron mass m) as a function of the\nmagnetic field strength B(in units of the characteristic value\nB0=m2/e≃4.4×1013G) for several values of the fine-\nstructure constant α. The thin lines represent the corre-\nsponding results in the chiral limit, i.e., mdyn. Note that\nforα= 1/137 the corresponding result in the chiral limit lies\noutside the plot range.\nmdyn) is plotted against the magnetic field strength B\nfor several values of the fine-structure constant α. While\nwe are not able to find an analytic expression that fits\nthe numerical results, an analysis of the gap equation (8)\nshows that for fixed αits solution in an asymptotically\nstrongmagneticfield is reducedto the solutionin the chi-\nral limit. In other words, for fixed αwe havem∗≈mdyn\nasB→ ∞. This asymptotic behavior is verified numeri-\ncally as can be seen clearly in Fig. 1.\nSeveral important features of the dynamical electron\nmass generated in a strong magnetic field can be gleaned\nfrom Fig. 1.\n(i) There is a wide separation of scales m∗≪√\neBas\nlong as the coupling is weak and the magnetic field\nis strong. As remarked above, together with the\ngauge independence of m∗, the wide separation of\nscales means that our results for m∗are reliable in\nthe weak coupling regime and the strong field limit.\n(ii) LetBmdenote the magnetic field for which mdynis\nequal tom(i.e., the intercept of the thin line with\nthe abscissa in Fig. 1). It is clear from the figure\nthat forB≃Bm, the corresponding m∗is about\none order of magnitude larger than m, a property\nthat is fairly independent of the values of α. This\nprovides a distinct signature that the generation of\ndynamical electron mass in a strong magnetic field\nis significantly enhanced by the perturbative elec-\ntron mass. Specifically, we note that for α= 1/137\nthere already has been a few percent increase in\nthe electron mass around 1015G, the typical mag-\nnetic fields on the surface of young neutron stars.4\nSuch an effect is within the precision of current and\nfuture astrophysical measurements. Furthermore,\nwe note that the transition of the behavior of m∗\nfromtheintermediatetotheasymptotictakesplace\naroundB≃Bm, and again is fairly independent of\nα.\n(iii) For fixed B,m∗increases with increase of α. At\nweak coupling, the dynamical contribution to the\nelectron mass is small but sizable as compared to\nits exponentially suppressed counterpart in the chi-\nral limit. Nevertheless, the dynamical contribution\nbecomes substantial and eventually dominant over\nthe perturbative electron mass as the coupling in-\ncreases. This aspect is of particular importance\nwhen the effects of the running coupling in a strong\nmagnetic field are taken into consideration.\nThe enhancement of the dynamical electron mass can\nbe understood in terms of the screening effect modified\nby the perturbative electron mass, m. First, we consider\nthe chiral limit. The corresponding polarization func-\ntion Π(q2\n/bardbl,q2\n⊥) is given by Eq. (5) with the replacement\nF(q2\n/bardbl/4m2\n∗)→F(q2\n/bardbl/4m2\ndyn). In the region of Bwhere\nmdynis exponentially small, one can make the approxi-\nmationF(q2\n/bardbl/4m2\ndyn)≃F(∞) = 1. When plugged into\nthe gap equation (8), this in turn implies that photons\nwith (Euclidean) momenta 0 < q2\n/bardbl≪eBare effectively\nscreened. The screening effect explains the smallness of\nmdyn. Away from the chiral limit, on the other hand, the\nperturbativeelectronmass mintroducesintotheprobleman additional energyscalethat is independent of the LLL\ndynamics. In the region of Bwhereδm∗≡m∗−m≪m\n(as can be seen in Fig. 1, this would be the same re-\ngion ofBconsidered above in the chiral limit), one\ncan make the replacement F(q2\n/bardbl/4m2\n∗)→F(q2\n/bardbl/4m2) in\nΠ(q2\n/bardbl,q2\n⊥). This in turn means that photons with mo-\nmentam2≪q2\n/bardbl≪eBare effectively screened. As\na result, the contribution from photons with momenta\n0<q2\n/bardbl≪m2that are not screened is responsible for the\nenhancement of the dynamical electron mass in a strong\nmagnetic field.\nThe above arguments do not depend on the specific\nvalue ofα, and remain valid up to a magnetic field\nfor whichδm∗/m∼ O(1) [or, alternatively, mdyn/m∼\nO(1)]. As we have noticed above, this takes place around\nthe magneticfield B≃Bm, for whichm∗is almostabout\none order of magnitude larger than m. This also explains\nwhy the transition of the behavior of m∗from the inter-\nmediate to the asymptotic takes place around B≃Bm.\nIn conclusion, the significant enhancement of the dy-\nnamical electron mass in QED in a strong magnetic field\nis a novel effect. We envisage a similar enhancement of\nthedynamicalquarkmassesinQCDinastrongmagnetic\nfield [19]. It would be interesting and useful to consider\napplications of these effects in astrophysics and cosmol-\nogy.\nI would like to thank C. N. Leung for a careful reading\nof the manuscript. This work was supported in part by\nthe National Science Council of Taiwan under grant 96-\n2112-M-032-005-MY3.\n[1] E. J. Ferrer, V. de la Incera, and C. Manuel, Phys. Rev.\nLett.95, 152002 (2005); E. J. Ferrer and V. de la Incera,\nibid.97, 122301 (2006).\n[2] S. Mandal, R. Saha, S. Ghosh, and S. Chakrabarty, Phys.\nRev. C74, 015801 (2006); P. Yue and H. Shen, ibid.,\n045807 (2006).\n[3] A. E. Shabad and V. V. Usov, Phys. Rev. D 73, 125021\n(2006); Phys. Rev. Lett. 96, 180401 (2006).\n[4] L. Campanelli and M. Giannotti, JCAP 0609, 017\n(2006); arXiv:astro-ph/0611207.\n[5] A. E. Shabad and V. V. Usov, Phys. Rev. Lett. 98,\n180403 (2007); Phys. Rev. D 77, 025001 (2008).\n[6] For recent reviews, see D. Grasso and H. R. Rubinstein,\nPhys. Rept. 348, 163 (2001); A. K. Harding and D. Lai,\nRept. Prog. Phys. 69, 2631 (2006).\n[7] C. Thompson and R. C. Duncan, Mon. Not. Roy. Astron.\nSoc.275, 255 (1995); Astrophys. J. 473, 322 (1996).\n[8] A. Brandenburg, K. Enqvist, and P. Olesen, Phys. Rev.\nD54, 1291 (1996); M. Joyce and M. E. Shaposhnikov,\nPhys. Rev. Lett. 79, 1193 (1997).\n[9] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy,\nPhys. Rev. Lett. 73, 3499 (1994); Phys. Lett. B 349,\n477 (1995).\n[10] V.P.Gusynin, V.A.Miransky, andI.A.Shovkovy,Phys.\nRev. D52, 4747 (1995); Nucl. Phys. B462, 249 (1996);D. K. Hong, Y. Kim, and S. J. Sin, Phys. Rev. D 54,\n7879 (1996).\n[11] D.-S. Lee, C. N. Leung, and Y. J. Ng, Phys. Rev. D 55,\n6504 (1997); E. J. Ferrer and V. de la Incera, ibid.58,\n065008 (1998).\n[12] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy,\nPhys. Rev. Lett. 83, 1291 (1999); Nucl. Phys. B563, 361\n(1999); Phys. Rev. D 67, 107703 (2003); A. V. Kuznetsov\nand N. V. Mikheev, Phys. Rev. Lett. 89, 011601 (2002).\n[13] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy,\nPhys. Rev. Lett. 90, 089101 (2003); A. V. Kuznetsov\nand N. V. Mikheev, ibid., 089102 (2003).\n[14] C. N. Leung and S.-Y. Wang, Nucl. Phys. B747, 266\n(2006); Ann. Phys. (N.Y.) 322, 701 (2007).\n[15] S.-Y. Wang, arXiv:hep-ph/0702010 (in the Proceedings\nof SCGT06, World Scientific, Singapore, 2008).\n[16] See, for instance, S. P. Klevansky, Rev. Mod. Phys. 64\n(1992) 649; T. Hatsuda and T. Kunihiro, Phys. Rept.\n247, 221 (1994).\n[17] V. P. Gusynin and A. V. Smilga, Phys. Lett. B 450, 267\n(1999); A. V. Kuznetsov, N. V. Mikheev, and M. V. Os-\nipov, Mod. Phys. Lett. A 17, 231 (2002); and references\ntherein.\n[18] V. I. Ritus, Ann. Phys. (N.Y.) 69, 555 (1972); in Is-\nsues in Intense-Field Quantum Electrodynamics , edited5\nby V. L. Ginzburg (Nova Science, Commack, New York,\n1987).\n[19] N. O. Agasian and I. A. Shushpanov, Phys. Lett. B 472,\n143 (2000); D. Kabat, K. M. Lee, and E. Weinberg,Phys. Rev. D 66, 014004 (2002); V. A. Miransky and\nI. A. Shovkovy, ibid., 045006 (2002)."    },    {        "title": "1810.02216v1.Domain_wall_dynamics_in_easy_cone_magnets.pdf",        "content": "1 \n Domain wall dynamics in easy- cone magnets \nPeong- Hwa Jang1, Se-Hyeok Oh2, and Kyung- Jin Lee1,2,3†  \n1Department of Materials Science and Engineering, Korea University, Seoul 02841 , Korea  \n2Department of Nano -Semiconductor and Engineering, Korea University, Seoul 02841, Korea  \n3KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, \nKorea  \n \n†Correspondence to:  K.-J.L. ( kj_lee@korea.ac.kr ) \n \nAbstract  \nWe theoretically and numerically investigate magnetic domain wall dynamics in a nanowire \nof easy -cone magnet. The easy -cone domain wall exhibits several distinguishing dynamic \nfeatures in comparison to the easy -axis domain wall. The features of easy -cone doma in wall \nare related to the generation of additional chiral spin textures due to the domain wall precession, \nwhich is common for various driving sources such as magnetic fields  and spin- transfer torques. \nThe unique easy- cone domain wall dynamics could enrich magnetic domain wall study and \nfind use in device applications based on easy -cone domain walls.  \n  2 \n I. Introduction \nA magnetic domain wall (DW) is a transient region between two domains, where the spin \nconfi guration continuously varies. Magnetic DWs can be used as information units in data \nstorage and logic devices [ 1,2], demanding detailed understanding of DW dynamics. In this \nrespect, the magnetic DW dynamics has been extensively studied for various classes  of \nmagnetic material such as ferromagnets [ 3-34], antiferromagnets [3 5,36], and ferrimagnets \n[37-43], and also for various driving means such as magnetic fields [ 3-8,37,38] , spin- transfer \ntorques [ 9-22], spin- orbit torques [2 3-27,35,36,39- 43], and spin waves [ 28-34]. \nTo date, most ferromagnetic DW studies have focused on ferromagnetic materials with the \neasy-axis state. In this work, we theoretically and numerically study DW dynamics in \nferromagnetic materials with the easy -cone state. When th e second -order magnetic anisotropy \nis non- negligible compared to the first -order one and satisfies a specific condition (described \nbelow), the equilibrium magnetization direction has an angle from the film normal. For the \nsystem where the cylindrical symmetry is preserved, the direction of equilibrium magnetization forms a cone with a finite angle, called the easy -cone state. Recently, the easy -cone magnet has \nattracted considerable interest for magnetic memories because of the short switching time and low switching current density [4 4-47] and for high- frequency oscillators because of its ability \nfor zero -field oscillation [48 ]. Moreover, the easy -cone magnet is able to host spin superfluids \nassociated with spontaneous breaking of the U(1) spin -rotational symmetry [ 49-54]. However, \nDW dynamics in the easy -cone magnet has not been investigated yet.  \nIn this paper, we investigate DW dynamics induced by a magnetic field or a current in a \nconically magnetized nanowire. In section II, we introduce the easy -cone state and derive its \nequilibrium DW profile. In sections III and IV, we describe easy -cone DW dynamics induced 3 \n by magnetic fi elds and spin- transfer torques, respectively . Lastly in section V, we show DW \ndynamics induced by spin injection at one side of the nanowire.  \n \nII. Equilibrium e asy-cone domain wall  \nTotal magnetic energy of the system, including the exchange, first- order and second- order \nmagnetic anisotropies, is given as  \n𝐸𝐸tot=∫𝑑𝑑𝑑𝑑 [𝐴𝐴ex(∇𝜃𝜃)2+𝐾𝐾1,effsin2𝜃𝜃+𝐾𝐾2sin4𝜃𝜃],    (1) \nwhere 𝐴𝐴ex is the exchange stiffness constant, 𝐾𝐾1,eff=𝐾𝐾1−2𝜋𝜋𝑀𝑀s2 is the first- order effective \nanisotropy energy density, 𝐾𝐾2  is the second -order anisotropy energy density, 𝑀𝑀s  is the \nsaturation magnetization, and 𝜃𝜃  is the polar angle between the magnetization and  𝐳𝐳� -axis \n(film normal). Figure 1(a) shows the phase dia gram of magnetic state as a function of the first - \nand second- order magnetic anisotropies [55 -57]. Perpendicular magnetization (perpendicular \nmagnetic anisotropy: PMA) is stabilized when 𝐾𝐾1,eff>0  and 𝐾𝐾2>−𝐾𝐾1,𝑒𝑒𝑒𝑒𝑒𝑒/2 . In-plane \nmagnetization (easy plane) is stabilized when 𝐾𝐾1,eff<0 and 𝐾𝐾2<−𝐾𝐾1,eff/2. The PMA and \neasy-plane states coexist when 𝐾𝐾1,eff>0 and 𝐾𝐾2<−𝐾𝐾1,eff/2. Finally, the easy -cone state, \nwhich is of interest in this work, is stabilized when 𝐾𝐾1,eff<0 and 𝐾𝐾2>−𝐾𝐾1,eff/2.  \nFor a single -domain easy -cone state, the energy minimization of Eq. (1) with respect to 𝜃𝜃 \ngives two equilibrium polar angles 𝜃𝜃c1=sin−1√𝜅𝜅  and 𝜃𝜃c2=𝜋𝜋−sin−1√𝜅𝜅  where κ=\n−𝐾𝐾1,eff/2𝐾𝐾2 . Figure  1(b) shows a schematic illustration of an easy -cone DW. With the \nmagnetization  𝒎𝒎=(cos𝜙𝜙sin𝜃𝜃,sin𝜙𝜙sin𝜃𝜃,cos𝜃𝜃) , the equilibrium one -dimensional DW \nprofile of easy- cone magnet is derived by solving the Euler -Lagrange equation of the total 4 \n magnetic energy [Eq. (1)] with the boundary conditions 𝜃𝜃(𝑑𝑑→∞)= 𝜃𝜃c1  and 𝜃𝜃(𝑑𝑑→\n−∞)= 𝜃𝜃c2, which is given as  \n𝜃𝜃(𝑑𝑑)=tan−1�√1−𝜅𝜅tanh��𝜅𝜅(1−𝜅𝜅)(𝑥𝑥−𝑋𝑋)𝜆𝜆⁄�\n√𝜅𝜅�+𝜋𝜋\n2,    (2) \nwhere 𝜆𝜆=�𝐴𝐴ex/𝐾𝐾2 is the DW  width and 𝑋𝑋 is the center position of the wall. In Fig. 1(c), \nwe compare Eq. (2) with the DW profile obtained from numerical calculation with the \nfollowing parameters: 𝐴𝐴ex=1.2×10−6 erg/cm , 𝐾𝐾1,eff=−3×106 erg/cm3 , 𝐾𝐾2=5×\n106 erg/cm3 , and 𝑀𝑀𝑠𝑠=1000 emu/c m3 . We find a good agreement between Eq. (2) and \nmodeling result. For a comparison, we also plot the DW profile of a PMA magnet (𝐴𝐴ex=\n1.2×10−6 erg/cm , 𝐾𝐾1=7×106 erg/cm3 , 𝐾𝐾2=0 erg/cm3 , and 𝑀𝑀s=1050 emu /cm3 ) \nin Fig. 1(c).  \nBefore ending this section, we note that there is an important difference in the equilibrium \nDW profile in between a PMA magnet and an easy- cone magnet. For a PMA DW, the in- plane \nmagnetization component is zero at 𝑑𝑑→±∞  regardless of the azimuthal angle 𝜙𝜙  of the \nmagnetization. In contrast, for an easy- cone DW, the in -plane component in the domain region \nvaries depending on 𝜙𝜙 because sin𝜃𝜃≠0. This coupling between 𝜙𝜙 and the magnetization \nprofile in the domain region r esults in unique dynamics of easy- cone DW when the DW \nprecesses, as we will explain in the next section.  \n \nIII. Domain wall dynamics induced by magnetic field  \nDynamics of easy -cone DW driven by a magnetic field applied in the  𝒛𝒛�  direction is \nstudied by solving the Landau- Lifshitz -Gilbert (LLG) equation , given as  5 \n d𝒎𝒎\n𝑑𝑑𝑑𝑑=−𝛾𝛾 𝒎𝒎×𝑯𝑯eff+𝛼𝛼 𝒎𝒎×𝑑𝑑𝒎𝒎\n𝑑𝑑𝑑𝑑,                (3)  \nwhere 𝛾𝛾  is the gyromagnetic ratio, 𝑯𝑯eff  is the effective magnetic field including the \nexchange, anisotropy, magneto- static, and external fields, and 𝛼𝛼  is the Gilbert damping \nconstant. Following Thiele ’s collective coordinate approach [5 8] for the DW position 𝑋𝑋 and \nDW angle 𝜑𝜑 and with the equilibrium DW profile [Eq. (2)], we obtain the force equation as,  \n𝛼𝛼��(1−𝜅𝜅)𝜅𝜅+(1−2𝜅𝜅)tan−1��1−𝜅𝜅\n𝜅𝜅�� 𝑋𝑋̇+2√1−𝜅𝜅 𝜆𝜆(−𝛾𝛾 𝐻𝐻𝑧𝑧+𝜑𝜑̇)=0,    (4) \nwhere 𝐻𝐻𝑧𝑧 is the magnitude of external field.  \nThe steady state solution of the DW velocity ( 𝜑𝜑̇=0) is then given as,  \n𝑣𝑣𝐷𝐷𝐷𝐷(𝐻𝐻𝑧𝑧)=2√1−𝜅𝜅\n�(1−𝜅𝜅)𝜅𝜅+(1−2𝜅𝜅)tan−1��1−𝜅𝜅\n𝜅𝜅�𝛾𝛾𝜆𝜆𝐻𝐻𝑧𝑧\n𝛼𝛼.               (5) \nAnalytic solutions of the DW velocity beyond the steady state solution are difficult to obtain \nbecause the DW profile varies both spatially and temporally, as will be discussed below.  \nIn the bottom panel of Fig. 2(a), we compare the analytic solution [Eq. (5), blue dotted line] \nof DW velocity with the velocity numerically calculated by micromagnetic simulations with \n𝛼𝛼=0.1 , nanowire length 𝐿𝐿=2 μm , width  𝑤𝑤=50 nm  , and thickness 𝑑𝑑=0.8 nm . Both \none-dimensional (1D, blue diamond symbols) and 2D (black cross  symbols, 25 discretized cells \nin the transverse direction of wire) simulations show similar results so that we focus on the 1D \nsimulation results hereafter. In Fig. 2(a), we also show the DW velocity of a PMA magnet (black open symbols ) for comparison. Numerical results for the easy -cone magnet are in \nagreement with Eq. (5) in low field regimes, whereas they largely deviate from Eq. (5) in high field regimes. This deviation is caused by the Walker breakdown [ 59], i.e., the DW precession  6 \n above a threshold field. However, field dependence of DW velocity after the Walker breakdown \nis clearly different between the easy -cone magnet and the PMA magnet. U nlike the PMA DW \nthat shows two separate regimes: steady motion below and precessional moti on above a \nthreshold field [see Fig. 2(a), black open symbols], the easy -cone DW shows three separate \nregimes: steady -state regime, intermediate regime, and precessional regime [see Fig. 2(a), solid \ndiamond symbols]. In the intermediate regime, which is absent for the PMA DW, the velocity \nof the easy -cone DW shows several up and down jumps.  \nIn the case of PMA DW, the precessional behavior after the Walker breakdown does not \ngenerate any additional magnetic textures in the uniform domain region and the azimu thal \nangle 𝜙𝜙 is spatially homogeneous even with the DW precession. In the case of easy -cone DW, \nhowever, the DW precession generates additional magnetic textures because the easy -cone \nstate stabilizes nonzero in -plane component of magnetization in the dom ain region. In Fig. 2(b), \nwe schematically describe two kinds of easy -cone DW profiles. In the absence of magnetic \nfield [the upper panel of Fig. 2(b)], the polar angle 𝜃𝜃 of magnetizations follows Eq. (2) and \nthe azimuthal angle 𝜙𝜙 is 𝜋𝜋 to minimize the exchange and shape anisotropy energies. When \napplying a magnetic field below a threshold for the precession, 𝜙𝜙 at the DW center tilts from \n𝜋𝜋 a little bit, which in turn changes 𝜙𝜙 of magnetizations near the DW  to reduce the exchange \nenergy. Even in this case, 𝜙𝜙  at 𝑑𝑑→±∞  is still 𝜋𝜋  because of the shape anisotropy. As a \nresult, 𝜙𝜙 is no longer constant and becomes inhomogeneous. When applying a magnetic field \nabove a threshold, 𝜙𝜙 at the DW center rotates by about 𝜋𝜋 and is then close to 0 [the bottom \npanel of Fig. 2(b)]. In this situation, 𝜙𝜙 changes from 𝜋𝜋 at 𝑑𝑑→−∞, through ≈ 0 at 𝑑𝑑=0, \nto 𝜋𝜋  at 𝑑𝑑→+∞ . Because of the 𝜋𝜋 -rotation of 𝜙𝜙  at the DW center, two inhomogeneous 7 \n magnetic textures are formed at the front and rear sides of DW. We call this magnetic texture \noriginating from the 𝜋𝜋-rotation of 𝜙𝜙 as a sub- DW (SD).  \nIn Fig. 2(c), we show a top view of simulated magnetization configuration in the easy -cone \nnanowire for 𝐻𝐻z=60 Oe  above the threshold for the Walker breakdown. In this case, a SD is \nformed at the front of DW and another SD is formed at the rear of DW. With increasing a \nmagnetic field, the DW center magnetization undergoes more 𝜋𝜋-rotation s of 𝜙𝜙 and as a result, \nmore SDs are generated . On the top panel in Fig. 2(a), we show the number of SDs created at \nthe front of DW as a function of 𝐻𝐻z. For magnetic fields in the intermediate regime (50 Oe \n<𝐻𝐻z< 150 Oe), the number of SDs increases discontinuously. This discontinuity is caused by \nthe fact that for an additional 𝜋𝜋 -rotation of 𝜙𝜙  of the DW center magnetization , a \ncorresponding exchange (and other) energy cost must be overcome and that the new sta te is \nstabilized by the shape anisotropy of nanowire. Whenever an additional SD is created, the DW \nvelocity shows a discontinuous drop [indicated by a black arrow in Fig. 2(a)]. We attribute this \nvelocity drop to the fact that the magnetic field must move not only the DW but also the additional SD acting as an additional energy barrier. When increasing the magnetic field, the DW velocity increases again until the field is high enough to create the next SD.  \nOn the other hand, the number of SDs at the front of DW decreases at higher fields \ncorresponding to the precessional regime [𝐻𝐻\nz> 150 Oe; the top panel of Fig. 2(a)]. Whenever \nthe magnetization at the DW center undergoes a 𝜋𝜋-rotation , it is obvious that the number of \nSDs generated at the rear of DW  incre ases. We observe that  the number of SDs at the front side, \nhowever, does not increase continuously with the field because the distance among the front \nSDs decreases  as the source of SD generation (i.e., precessing DW ) moves  faster towards the \nfront side. The d ecreased distance among SDs makes the magnetic texture energetically 8 \n unfavorable,  which limits the creation of SDs.  The newly created SD exceeding the limited \nmaximum number,  collapses and merges with the last one , resulting in the decreased SD \nnumber at the front side . We also remark that because of its unique feature of field -driven easy -\ncone DW dynamics in the presence of the shape anisotropy, multiple SDs remain in the \nnanowire even after the field  is switched off . \n \nIV. Domain wall dynamics induc ed by spin- transfer torque  \nWe investigate easy -cone DW dynamics induced by adiabatic and nonadiabatic spin-\ntransfer torques (STTs) [13-15], which is described with the modified  LLG equation as \nd𝒎𝒎\n𝑑𝑑𝑑𝑑=−𝛾𝛾 𝒎𝒎×𝑯𝑯𝑒𝑒𝑒𝑒𝑒𝑒+𝛼𝛼 𝒎𝒎×𝑑𝑑𝒎𝒎\n𝑑𝑑𝑑𝑑+𝑏𝑏j𝒎𝒎×�𝒎𝒎×d𝒎𝒎\nd𝑥𝑥�+𝛽𝛽𝑏𝑏j�𝒎𝒎×d𝒎𝒎\nd𝑥𝑥�.    (6)  \nHere 𝑏𝑏j=ℏ𝑃𝑃𝐽𝐽e/2𝑒𝑒𝑀𝑀𝑠𝑠 is the spin current velocity, ℏ is the reduced Plank constant, P  is the \nspin polarization, 𝐽𝐽e  is the current density, e is the electric charge, and 𝛽𝛽  is the non-\nadiabaticity. Following Thiele ’s approach, the steady -state solution of easy- cone DW driven \nby STT gives 𝑣𝑣DW=𝛾𝛾𝛽𝛽𝑏𝑏j/𝛼𝛼, identical to that of PMA DW [ 14,15] . We show the average \nvelocity over a time period as a function of 𝛽𝛽, in comparison to numerical results in Fig. 3(a).  \nWhen 𝛽𝛽≠𝛼𝛼 , the easy -cone DW shows processional  motion above a threshold current \ndensity. We note that once the easy -cone DW precesses, the SD generation is a common feature \nregardless of type of the driving source. Therefore, the STT is also able to create SDs. Figure 3(b) shows t emporal evolution of DW velocity  (bottom panel), the number of front SDs (middle \npanel), and magnetization configuration (top panel; top view), induced by STT above  a \nthreshold. In contrast to the field-driven generation of front SDs of which maximum number is 9 \n limited , the front SDs are continuously generated when driven by STT even right above a \nthreshold ( 𝐽𝐽e=4.8×107 A/cm2). The continuous increase in the number of front SDs in STT -\ndriven case is caused by the fact that STT induces dynamics of SDs as well as DW in the same \ndirec tion because both types of spin texture have finite spatial gradients and thus experience \nnon-zero STT. Hence the DW induced by STT does not feel the SD as an additional barrier that \nhinders its dynamics. Once a SD is created, similarly, there is a sudden drop of velocity  \n[indicated by a blue arrow in bottom panel of Fig. 3(b)]  but recovers its velocity shortly .  \n \nV.  Domain wall dynamics induced by spin injection at wire edge.   \nIn this section, we show DW dynamics driven by spin injection at one side of a nanowire \n[see Fig. 4(a)]. The coexistence of spin superfluidity and a DW in easy -cone state originates \nfrom the fact that the ground states of easy -cone magnet break U(1) and Z2 symmetries \nsimultaneously. In Ref. [53], Kim et al.  theoretically investigated easy -cone DW dynamics by \nspin injection with neglecting the non- local magnetostatic coupling, resulting in the shape \nanisotropy along the wire -length direction . They found that  the easy -cone DW moves along a \nparticular direction regardless of the magnitude of injected spin current. In this section, we \ninvestigate how the shape anisotropy, which is usually uneasy to remove from realistic \nnanowires, changes this DW dynamics. We show below that the easy -cone DW can move in \nan opposite direction, hence, change its velocity sign when we take into account the shape \nanisotropy via an additional anisotropy in the  𝒙𝒙�-direction.  10 \n To get an insight into DW dynamics by spin injection, two terms are additionally considered \nto Eq. (1), the exchange energy with respect to 𝜙𝜙 and the simplified non -local magneto -static \nenergy in a nanowire, which gives a modified total magnetic energy 𝐸𝐸𝑡𝑡𝑡𝑡𝑡𝑡′ as, \n𝐸𝐸𝑑𝑑𝑡𝑡𝑑𝑑′=∫(𝐴𝐴ex{(∇𝜃𝜃)2+sin2𝜃𝜃(∇𝜙𝜙)2}+𝐾𝐾1,𝑒𝑒𝑒𝑒𝑒𝑒sin2𝜃𝜃+𝐾𝐾2sin4𝜃𝜃+𝐾𝐾Dcos2𝜃𝜃sin2𝜙𝜙)𝑑𝑑𝑑𝑑 . (7) \nHere we include the exchange, first - and second- order anisotropy energy, and shape anisotropy \nenergy (𝐾𝐾D). DW dynamics can be interpreted by employing equations of motion of the system \nin the spherical coordinate, given as  \n𝛼𝛼 𝑠𝑠 𝜃𝜃̇−𝑠𝑠sin𝜃𝜃𝜙𝜙̇=𝜕𝜕𝐸𝐸𝑡𝑡𝑡𝑡𝑡𝑡′\n𝜕𝜕𝜕𝜕,                         (8a)  \n𝑠𝑠 sin𝜃𝜃 𝜃𝜃̇+𝛼𝛼 𝑠𝑠sin2𝜃𝜃𝜙𝜙̇=𝜕𝜕𝐸𝐸𝑡𝑡𝑡𝑡𝑡𝑡′\n𝜕𝜕𝜕𝜕,                       (8 b) \nwhere 𝑠𝑠=−𝑀𝑀s\n𝛾𝛾  is the spin moment. For the moving frame with wall velocity  𝑣𝑣SI , we \nreorganize Eq. (8) to the linear order in 𝑣𝑣SI,  \n−𝑠𝑠sin𝜃𝜃𝜔𝜔+𝛼𝛼 𝑠𝑠(−𝑣𝑣SI𝛻𝛻𝜃𝜃)\n=𝐴𝐴𝑒𝑒𝑥𝑥�−𝛻𝛻2𝜃𝜃+𝐾𝐾2\n𝐴𝐴ex𝜕𝜕𝜕𝜕(sin2𝜃𝜃−sin2𝜃𝜃c)−2𝐾𝐾D\n𝐴𝐴exsin𝜃𝜃cos𝜃𝜃sin2𝜙𝜙�,   (9a) \n𝑠𝑠 sin𝜃𝜃(−𝑣𝑣SI𝛻𝛻𝜃𝜃)+𝛼𝛼 𝑠𝑠sin2𝜃𝜃𝜔𝜔=𝐴𝐴ex�−𝛻𝛻(sin2𝜃𝜃𝛻𝛻𝜙𝜙)+2𝐾𝐾D\n𝐴𝐴excos2𝜃𝜃sin𝜙𝜙cos𝜙𝜙�,   (9b) \nwhere 𝜔𝜔≡𝜙𝜙̇, 𝜃𝜃c≡sin−1�−𝐾𝐾1,eff\n2𝐾𝐾2,𝛻𝛻𝜃𝜃≡𝜕𝜕𝑥𝑥𝜃𝜃, and 𝛻𝛻𝜙𝜙≡𝜕𝜕𝑥𝑥𝜙𝜙.  \nWe note that s pin injection i nto the conically magnetized nanowire generates SDs at the \nsource and hence 𝜙𝜙  is not a constant both temporally and spatially. This spatiotemporal \nvariation of 𝜙𝜙 makes the integration of Eq. (9) impossible. In order to make Eq. (9) integrable, 11 \n we use a crude  assumption that 𝜙𝜙 is spatially uniform  and obtain the following equations for  \n𝑣𝑣SI and 𝜔𝜔 :  \n2𝑠𝑠 cos𝜃𝜃𝑐𝑐 𝜔𝜔+𝛼𝛼 𝑠𝑠 𝐴𝐴v𝑣𝑣SI=0,                       (10a)  \n2 𝑠𝑠  cos𝜃𝜃𝑐𝑐 𝑣𝑣SI−𝛼𝛼 𝑠𝑠 𝐴𝐴ω1 𝜔𝜔=𝑗𝑗s−2𝐾𝐾D 𝐴𝐴ω2 sin𝜙𝜙cos𝜙𝜙,           (10b)  \nwher e 𝐴𝐴v=sin2𝜕𝜕𝑐𝑐+ (𝜋𝜋−2𝜕𝜕𝑐𝑐)cos2𝜕𝜕𝑐𝑐\n2𝜆𝜆 , 𝐴𝐴ω1=𝑙𝑙 sin2𝜃𝜃c+𝜆𝜆(𝜋𝜋−2𝜃𝜃𝑐𝑐) , 𝐴𝐴ω2=𝑙𝑙 cos2𝜃𝜃𝑐𝑐−𝜆𝜆(𝜋𝜋−\n2𝜃𝜃𝑐𝑐) , and 𝑗𝑗s=−𝐴𝐴ex sin2𝜃𝜃𝑐𝑐𝛻𝛻𝜙𝜙=(ℏ 𝑃𝑃  𝑗𝑗inj 𝑑𝑑inj)/(2  𝑒𝑒 𝑡𝑡EC)   is the spin current  generated \nfrom the source . Here, 𝑙𝑙 is the distance between the DW center and the injection source, 𝑗𝑗inj \nis the current density injected from a ferromagnet (FM), 𝑑𝑑inj  is the length of FM on 𝒙𝒙� -\ndirection, and 𝑡𝑡EC is the thickness of easy -cone nanowire [see Fig. 4(a)]. F rom Eq. (10), we \nderive the DW velocity 𝑣𝑣𝑆𝑆𝑆𝑆 by spin injection, given as  \n𝑣𝑣SI=𝛾𝛾�2𝐾𝐾D 𝐴𝐴ω2 sin𝜕𝜕cos𝜕𝜕−ℏ 𝑃𝑃  𝑗𝑗inj 𝑑𝑑inj \n2 𝑒𝑒 𝑡𝑡EC �\n𝑀𝑀s �2  cos𝜕𝜕𝑐𝑐+𝛼𝛼2𝐴𝐴ω1𝐴𝐴v\n2  cos𝜃𝜃𝑐𝑐�.                  (11)  \nOne finds from Eq. (11) that 𝑣𝑣SI is either positive or negative depending on 𝜙𝜙 and 𝑗𝑗inj. The \nthreshold current density 𝑗𝑗th below and above which the sign of 𝑣𝑣SI changes  can be obtained \nby maximizing sin𝜙𝜙cos𝜙𝜙 in the numerator of Eq. (11) and setting 𝑣𝑣SI=0, given as  \n𝑗𝑗th=2𝑒𝑒𝐾𝐾D𝐴𝐴ω2𝑑𝑑EC\nℏ 𝑃𝑃 𝑑𝑑inj.                             (12)  \nIn Fig. 4(a), we show a schematic view of the system containing an easy -cone DW at the \ncenter (𝑖𝑖=𝐿𝐿/2 ) of the nanowire with the spin injection source (FM1) located at 𝑖𝑖=𝐿𝐿/4 \n(source area of 20×50 nm2 ). We perform micromagnetic simulations with the following \nparameters: 𝐾𝐾1,eff=−3×106 erg/cm3 , 𝐾𝐾2=5×106 erg/cm3 , 𝑀𝑀s=1000 emu/c m3 , 12 \n polarization factor 𝑃𝑃=0.3, α=0.1, and injection length 𝑑𝑑inj=20 nm . Spins are injected \nfor a duration of 100 ns. Spin injection induces magnetization precession at the injection area \nand generates spin current proportional to 𝐴𝐴exsin2𝜃𝜃 𝛻𝛻𝜙𝜙 , which propagate and eventually \ninteract with the DW . In Fig. 4(b), we plot the time evolution of DW velocity, which shows \ntwo different DW motion with opposite sign, depending on the injected current density. \nCorresponding top views of magnetization conf iguration for low and high current densities are \nshown in Fig. 4(c) and (d), respectively.  \nFor an injected current below a threshold [ 𝑗𝑗th, Eq. (12)], the spin current is unable to precess \nthe easy -cone DW due to the shape anisotropy and thus cannot gener ate SDs at the front side \nof DW. In this case, the spin current (or, equivalently, SDs at the rear side of DW) just pushes \nthe DW, resulting the DW motion along the direction of spin current flow (i.e., a positive 𝑣𝑣SI \nin our sign convention ) [see Fig. 4(c)]. On the other hand, for an injected current above 𝑗𝑗th, \nthe spin current  induces DW precession and its angular momentum is transferred to the DW as \nexplained in Ref. [5 3]. In this case, the wall velocity shows an oscillatory behavior with a \nnegative sign in average and pulls the DW towards the spin current source [see Fig. 4(d)]. For a comparison , we estimate a theoretical 𝑗𝑗\nth  from Eq. (12) using the same parameters for \nsimulations  and obtain 𝑗𝑗th ≈ 7×107 A/cm2, which is smaller than the numerically obtained \nvalue (≈ 11×107 A/cm2), possibly due to the crude approximation adopted to derive Eq. (12). \nDespite somewhat unsatisfactory quantitative agreement, we note that Eqs. (11) and (12) reveal the underlying mecha nism of the sign change in 𝑣𝑣\nSI, depending on the magnitude of injected \ncurrent. Usually, to change the DW motion direction during the device operation, one has to \nuse a transistor that supplies bipolar currents. The above -mentioned bidirectional easy -cone   13 \n DW motion induced by a unipolar  current would be useful for device applications as one can \nuse a less expensive diode rather than a transistor.  \n \nVI. Summary  \nWe have investigated easy -cone DW dynamics induced by a magnetic field or an electric \ncurrent in conically magnetized nanowires. We find the easy- cone DW dynamics is closely \nrelated to the generation of sub- DWs caused by the combined action between the easy -cone \nground state and DW precession. For a field- driven case, the sub -DW generation results in \nunique intermediate regime where the DW velocity shows several up and down jumps. For a \nSTT-driven case, this intermediate regime is absent because STT moves not only the DW but \nalso the sub -DWs in the same direction. Lastly, for a spin  injection case, we find that the DW \nmotion direction can be controlled by varying the injected current de nsity. Our work will \nprovide a guideline for an experimental study on the easy- cone DW dynamics with various \ndriving sources.  \n \nAcknowledgement  \nWe acknowledge Se Kwon Kim for fruitful discussion. This work was supported by  the \nNational  Research Foundation of Korea (NRF) (Grants No.  2015M3D1A1070465 and No. \n2017R1A2B2006119) and the KIST Institutional Program  (Project No. 2V05750). 14 \n  \nFigure . 1. Easy- cone state and its wall profile. (a) Phase diagram of magnetic state as a function \nof effective fir st-order anisotropy K 1eff and second- order anisotropy K 2. (b) Schematic \nillustration of conically magnetized nanowire. Regions I and III are uniform domain parts \nwhere the  equilibrium polar angle s of magnetization are  𝜃𝜃=sin−1�−𝐾𝐾1,eff\n2 𝐾𝐾2  and  π−\nsin−1�−𝐾𝐾1,eff\n2 𝐾𝐾2, respectively, whereas regi on II  is domain wall part. (c) Wall profile represented \nas normalized M z component of easy -cone (blue open symbols) and PMA (black open symbols) \ndomain wall.  \n \n15 \n  \nFigure 2. Field- induced domain wall motion in a conically magnetized nanowire. (a) Domain \nwall velocity (bottom panel) and the number of SD (top panel) as a function of applied magnetic \nfield. (b) Schematic illustration of easy -cone domain wall profile in the  absence (top) and \npresence (bottom) of magnetic field. (c) Top view of magnetization configuration for |𝑯𝑯z|=\n60 Oe . Color code represents the z -component of magnetization, Mz. \n \n \n \n \n \n  \n16 \n Figure 3. Spin- transfer torque induced domain wall motion in a conically magnetized nanowire. \n(a) Domain wall velocity as a function of applied current for various non- adiabaticities 𝛽𝛽=\n0,𝛼𝛼,and 2𝛼𝛼. (b) Top view of magnetization configuration (top panel) a nd time evolution of \ndomain wall velocity and the number of SD (bottom panel) for J e=4.8 ×107 A/cm2. Color code \nrepresents the z- component of magnetization, M z. We use the same parameters of Fig. 1 and 𝑃𝑃 \n= 0.3. \n  \n \n  \n \n17 \n  \nFigure 4. Domain wall motion in a conically magnetized nanowire, induced by local spin \ninjection. (a) Schematic illustration of local spin injection in a conically magnetized nanowire. \n(b) Domain wall velocity as a function of time for various current dens ities. Easy -cone DW \nmoving (c) away from the injection source (positive velocity) for 𝑗𝑗𝑖𝑖𝑖𝑖𝑖𝑖=10.5 ×107A/\ncm2 and (d) towards the source (negative velocity in average) for 𝑗𝑗inj=11.5 ×107A/cm2. \nColor code represents the M z component of magnetization.  \n \n \n \n \n \n \n \n \n \n \n18 \n REFERENCES  \n[1] D. A. Allwood, G. Xiong, C. Faulkner, D. Atkinson, D. Petit, and R. Cowburn, Science \n309, 1688 (2005). \n[2]  S. S. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). \n[3]  G. S. Beach, C. Nistor, C. Knutson, M. Tsoi, and J. L. Erskine, Nat . Mater.  4, 741 \n(2005).  \n[4] P. Metaxas, J. Jamet, A. Mougin, M. Cormier, J. Ferré, V . Baltz, B. Rodmacq, B. Dieny, and R. Stamps, Phys . Rev. 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Phys . 45, 5406 (1974). \n "    },    {        "title": "1611.06153v1.Propagating_spin_wave_normal_modes__A_dynamic_matrix_approach_using_plane_wave_demagnetizating_tensors.pdf",        "content": "arXiv:1611.06153v1  [cond-mat.mes-hall]  18 Nov 2016Propagating spin-wave normal modes: A dynamic matrix appro ach using plane-wave\ndemagnetizating tensors\nY. Henry,∗O. Gladii, and M. Bailleul\nInstitut de Physique et Chimie des Mat´ eriaux de Strasbourg , UMR 7504,\nCNRS and Universit´ e de Strasbourg, B.P. 43, F-67037 Strasb ourg Cedex 2, France\n(Dated: November 21, 2016)\nWe present a finite-difference micromagnetic approach for de termining the normal modes of spin-\nwavespropagating in extended magnetic films and strips, which is based on the li nearized Landau-\nLifshitz equation and uses the dynamic matrix method. The mo del takes into account both short\nrange exchange interactions and long range dipole-dipole i nteractions. The latter are accounted\nfor through plane-wave dynamic demagnetization factors, w hich depend not only on the geometry\nand relative positions of the magnetic cells, as usual demag netization factors do, but also on the\nwave vector of the propagating waves. Such a numerical model is most relevant when the spin-\nwave medium is spatially inhomogeneous perpendicular to th e direction of propagation, either in\nits magnetic properties or in its equilibrium magnetic confi guration. We illustrate this point by\nstudying surface spin-waves in magnetic bilayer films and sp in-waves channelized along magnetic\ndomain walls in perpendicularly magnetized strips. In both cases, dynamic dipolar interactions\nproduce non-reciprocity effects, where counter-propagati ve spin-waves have different frequencies.\nI. INTRODUCTION\nWith the increasingly fast development of magnonics,\nwhich is the researchfield devoted to the control and ma-\nnipulation of spin-waves in magnetic materials1–3, the\nneed for theoretical tools capable of accounting for the\ncomplex physics of spin-waves is becoming more and\nmore acute. As efficient software packages4–7are now\nfreely available to the scientific community, time-domain\nnumerical micromagnetic simulations8, which are very\nversatile and allow describing systems of great complex-\nity, tend to become the tool of choice. However, because\nthey are carriedout in real space and primarily consist in\ndetermining the time evolution of the magnetization spa-\ntial distribution, these simulations are not ideally suited\nto tackle a number of problems of importance in spin-\nwave studies. One such problem is the determination of\nthe magnetic normal modes, their frequencies and their\ndispersion relations. Although micromagnetic simula-\ntions can be employed with some success for this (see,\ne.g., Refs. 9–14), through Fourier analyses of magnetiza-\ntion time series -for determining frequencies- or spatial\nprofiles -for determining wavelengths-, other approaches\nmay be more appropriate. Among the best suited ap-\nproaches are certainly those based on the dynamic ma-\ntrix method15,16. They are intrinsically in the frequency\ndomain and, unlike time-domain micromagnetic simula-\ntions which can hardly detect low frequency modes and\nfail altogether to identify degenerate ones, they have the\npotential to yield all of the normal modes.\nThe dynamic matrix method is a micromagnetic\nmethod, which involves the subdivision of the magnetic\nmedium into small cells, as most micromagnetic numer-\nical schemes, followed by the construction and diagonal-\nization of a matrix that contains all of the information\nregardingthe effective magnetic fields acting on the mag-\nnetization vectors of these cells. So far, this method has\nonly been used to study standing spin-waves in confinedmedia such as magnetic nano-elements10,15–17or arrays\nofdipolar-couplednanoparticules18. Noschemehasbeen\ndevised to describe the spin-waves travelling in magnetic\nmedia, which are very (infinitely) extended in one or\ntwo dimensions. Addressing such spin-waves by means\nofconventionalreal-spacemicromagnetic simulationsim-\npliesto simulateverylongsamples(much longerthan the\nlongest characteristic wavelength to be calculated). This\nrequires large amounts of computing power and storage\nmemory, especially when long wavelength spin-waves are\nconsidered and a good wave vector resolution is to be\nachieved. Yet, propagating spin-waves are also essential\nin magnonic applications, which rely on the capacity to\npropagate spin excitations along micron-scale circuits in\norder to process information19. The main purpose of the\npresent work is thus to deliver the full recipe for a dy-\nnamic matrix based numerical scheme adapted to plane\n(undamped) spin-waves. This is done in the first part\nof the paper (Secs. II-IV), where, after describing the\nprinciple of the method (Sec. II), we derive all the math-\nematical expressionsrequired for its implementation, i.e.,\nfor building the dynamic matrix (Sec. III), in the case of\nmagnetic media having the shape of extended films or\nthin strips. As always in micromagnetism, the main dif-\nficulty that has to be dealt with lies in the treatment\nof the long range dipole-dipole interactions. To account\nfor those, we adopt an intuitive approach using demag-\nnetizing tensors. While the concept of demagnetization\nfactors is familiar in the case of homogeneously magne-\ntized cells20, we show here (Sec. IV) that it can be ex-\ntended to cells supporting plane spin-waves, that is, cells\ninwhich themagnetizationvectoroscillatesharmonically\nin space. In this respect, our approach can be viewed\nas the finite-difference counterpart of the magnetostatic\nGreen’sfunction approachused insome(semi-)analytical\nspin-wave theories21,22.\nThe second part of the paper (Sec. V) is devoted to\napplications of our numerical scheme to questions of cur-2\nrent interest. We take this opportunity to illustrate the\nfact that its most interesting feature is certainly that it\nallows determining the propagating spin-wave modes in\nmagnetic media which are magnetically inhomogeneous\nin the plane perpendicular to the direction of propaga-\ntion, eitherbecausethematerialparametersvaryinspace\nor because the equilibrium spin configuration contains a\nnon-collinear magnetic texture. We note that all the nu-\nmerical data presented in this paper have been obtained\nwith C++ computer programsemploying the Eigen tem-\nplate library23for linear algebra operations, including\ncomplex matrix diagonalization, and the GNU Scientific\nlibrary24for numerical integration.\nII. PRINCIPLE OF THE METHOD\nIn the present work, we are not interested in stand-\ning spin-waves inside a ferromagnetic nanoobject, as in\nthe inspiring paper by Grimsditch and coworkers15, but\nrather in spin-waves that propagate as plane waves , in a\nparticular direction. This is the reason why we will as-\nsume that the magnetic media supporting the spin-waves\nare unbounded in the direction of propagation, and so\nwill necessarily be the magnetic cells used for discretiz-\ningthesemedia. Undersuchassumptions,onlysituations\nwheretheequilibrium magneticconfigurationis invariant\nupon translation along the propagation direction can be\nstudied. This is the main limitation of our approach.\nWe will consider two types of media, namely films\nof thickness T[Secs. VA] and thin strips of width W\n[Sec. VB], which are both compatible with a discretiza-\ntion by means of a one-dimensional array of Ngeo-\nmetrically identical cells. For describing our model in\nmore details, we introduce a first coordinate system,\nuvw, and the associated orthonormal direct vector basis\n{eu,ev,ew}, such that axis uis parallel to the direction\nof propagation of the spin-waves and axis vis normal to\nthe film/strip plane. With this, the magnetic cells used\nto discretize films will be slabs of thickness b=T/N,\nparallel to the ( u,w) plane [Fig. 1(a)], whereas for strips,\nthe cells will be rectangular parallelepipeds of height b\nand width c=W/N, parallel to axis u[Fig. 1(b)].\nThe starting point of our micromagnetic model is the\nLandau-Lifshitzequationdescribingthetimeevolutionof\nthe magnetization vector field M(r,t) in the absence of\nmagnetic damping, which we linearize around an equilib-\nriumMeq(r). We perform the linearization in the usual\nway25, that is, by writing both M(r,t) and the effec-\ntive magnetic field acting on it, Heff(r,t), as the sum of\na large equilibrium term and a much smaller dynamic\nterm :M(r,t) =Meq(r) +m(r,t) withMeq·m= 0\nand/bardblMeq/bardbl=MS, whereMSis the saturation magneti-\nzation, and Heff(r,t) =Heq(r) +h(r,t). Keeping only\nterms up to first order in the small parameters mandh,\nthe Landau-Lifshitz equation for a particular magneticcellαbecomes\n˙m(α)(t) =−|γ|µ0/bracketleftig\nM(α)\neq×h(α)(t)+m(α)(t)×H(α)\neq/bracketrightig\n,\n(1)\nwhereγis the gyromagnetic ratio26andµ0is the per-\nmeability of free space. Since we are concerned here with\nplanespin-wavestravelingalongaxis u, wepostulatethat\nthe variable magnetization has the form\nm(α)(u,t) =m(α)\n0ei(ωt−ku), (2)\nwherem(α)\n0is a complex amplitude vector, and kand\nωare the spin-wave wave vector and angular frequency,\nrespectively. Here, it is important to note that kcan\ntake on positive and negative values ( k=keu) in order\nto account for spin-wave propagation in both + uand−u\ndirections (assuming ω>0).\nIn addition to the uvwcoordinate system attached\nto the magnetic medium, we also introduce a coordi-\nnate system, xyz, and the corresponding vector basis\n{ex,ey,ez}, such that axis zis parallel to both H(α)\neqand\nM(α)\neq[Ref. 21], i.e., such that we have H(α)\neq=H(α)\neqez\nandM(α)\neq=MSez[Fig. 1(c)]. This, together with the\nfact that Eq. 2 implies ˙m(α)=iωm(α), allows us to re-\n(a) \nu wb v\nv \nv!T\nuv\nwz\nxy\nMeq ( )(c) (b) \nuwb\nw w!vWc\nFIG. 1: (a),(b) Geometries of the magnetic cell arrays. Ex-\ntended films are subdivided vertically into infinite horizon -\ntal slabs (a), whereas strips are subdivided transversally into\ninfinitely long rectangular parallelepipeds (b). (c) uvwand\nxyzcoordinate systems attached tothemagnetic mediumand\nequilibrium magnetization, respectively.3\nduce the 3-component vector equation for cell α[Eq. 1]\nto a set of only two equations\nω/parenleftigg\nm(α)\nx\nm(α)\ny/parenrightigg\n=−i|γ|µ0/parenleftigg\nMSh(α)\ny−H(α)\neqm(α)\ny\n−MSh(α)\nx+H(α)\neqm(α)\nx/parenrightigg\n,(3)\nin accordwith the factthat the smallamplitude magneti-\nzationdynamicsaroundthe equilibriumisconfinedinthe\n(x,y) plane. Let us call T(α)the transformation matrix\nfrom the uvwcoordinate system to the localcoordinate\nsystem,xyz, attached to M(α)\neq. It is crucial to note that\nin cases where the equilibrium magnetic configuration\nis not fully collinear throughout the entire medium (see\nSec. VB), the xyzcoordinate system, hence the matrix\nT(α), is not the same for all magnetic cells. Failure to\ntake this into account when necessary inevitably leads to\nerroneous results.\nAfter discretization, the linearized Landau-Lifshitz\nequation of the whole magnetic medium thus consists of\na set of 2 Nequations, which may finally be written in\nthe form of an eigenvalue equation\nω\nm(1)\nx\n...\nm(N)\nx\nm(1)\ny\n...\nm(N)\ny\n=\n. ... . . ... .\n...Dxx......Dxy...\n. ... . . ... .\n. ... . . ... .\n...Dyx......Dyy...\n. ... . . ... .\n\nm(1)\nx\n...\nm(N)\nx\nm(1)\ny\n...\nm(N)\ny\n\n=D\nm(1)\nx\n...\nm(N)\nx\nm(1)\ny\n...\nm(N)\ny\n, (4)\nwhereDis the so-called dynamic matrix whose dimen-\nsions are 2 N×2N. How we effectively perform this es-\nsential step will be detailed in the next section. If short\nrange exchange and long range dipole-dipole interactions\nare not included in the model, the Dmatrix is block-\ndiagonal15. If, on the contrary, those interactions are\ntaken into account, none of the off-diagonal elements is\nzero and, in general, Dhas no special properties. In\nparticular, it is neither hermitian nor (systematically)\nsparse. Numerical diagonalizationof the dynamic matrix\nis the ultimate step of the process. It yields the profiles\nof the normal modes across the magnetic medium in the\nform of ensembles of complex amplitudes m(α)\n0xandm(α)\n0y\n(α= 1..N), as well as the corresponding eigenfrequen-\ncies, which are real numbers in the absence of magnetic\ndamping.\nIt is important to note that, irrespective of the sign\nofk, the eigenvectors of Dalways come in pairs with\neigenfrequenciesof(usually) identicalabsolutevaluesbutopposite signs27, where the eigenvector with kω >0\n(resp.kω <0) corresponds to propagation in the + u\ndirection (resp. −udirection). Also, for retrieving the\ntrue spatiotemporal evolution of the dynamic magneti-\nzation in a particular mode, one has to take the real\npart ofm(α), as defined in Eq. 2. In this operation,\nall four real and imaginary parts of the m(α)\n0xandm(α)\n0y\ncomplex amplitudes a priori matter, since Re( m(α)) =\nRe(m(α)\n0)cos(ωt−ku)−Im(m(α)\n0)sin(ωt−ku). In gen-\neral, Re( m(α)\n0x), Im(m(α)\n0x), Re(m(α)\n0y), and Im( m(α)\n0y) are\nall relevant since four parameters are indeed required to\nfully characterize the precessional motion of the mag-\nnetization: three parameters -two radii and a tilt angle\nϕ(α)- are necessary to describe the elliptical time tra-\njectory of ˜m(α)= Re(m(α)) in the ( x,y) plane, while a\nfourth one, τ(α), is needed to account for the relative\nphase of the precession (see Appendix A). However, sit-\nuations are rare where the tilt angle ϕ(α)and the phase\nτ(α)vary acrossthe profileof a normal mode and thereby\ncannot be made nil for all magnetic cells28. This occurs,\nfor instance, when the spin-wave medium is magnetized\nin-plane and the orientation of the equilibrium magne-\ntization changes in space. Here, no such situation will\nbe considered. Local reference frames and phase origins\nwill be chosen so that the conditions ϕ(α)=τ(α)=0 and\nIm(m(α)\n0x)=Re(m(α)\n0y)=0 are systematically fulfilled and\nthe sole variations of Re( m(α)\n0x) and Im( m(α)\n0y) withαsuf-\nfice to characterize entirely a modal profile.\nIII. CONSTRUCTION OF THE DYNAMIC\nMATRIX\nExaminationofequation3 showsthat the construction\nof the dynamic matrix requires essentially two things.\nThe first one is to evaluate the magnitude of the static\npart of the effective magnetic field in each cell, H(α)\neq,\nknowing what the whole equilibrium magnetic configura-\ntionM(β)\neq(withβ= 1..N) is. This is rather trivial. As\nin usual micromagnetic simulations, one needs to take\ninto account contributions from the external magnetic\nfield (H0), crystal anisotropy ( HK), exchange interac-\ntions between nearest neighbor cells ( Hex), and dipolar\ninteractions ( Hd). All of these are more easily evaluated\nin theuvwcoordinate system.\nFor crystal anisotropy, we use the general expression\nH(α)\nK=2Ku\nµ0M2\nS/parenleftig\nM(α)\neq·a/parenrightig\na\n−2Kc\nµ0M4\nS3/summationdisplay\ni=1\n/summationdisplay\nj/negationslash=i(M(α)\neq·cj)2\n/parenleftig\nM(α)\neq·ci/parenrightig\nci.\n(5)\nThe first term accounts for a uniaxial anisotropy of con-\nstantKuand axis a, while the second stands for a cubic\nanisotropy of constant Kcand axes c1,c2, andc3such4\nthatci·cj=δij, whereδijis Kronecker’s delta. Both\nterms are first-order.\nIn the continuous-medium approximation, the usual\nexpression for isotropic exchange is\nHex(r) = Λ2∆Meq(r), (6)\nwhere ∆ is the Laplacian operator and Λ =/radicalig\n2A\nµ0M2\nSis\nthe exchange length, with Athe exchange stiffness con-\nstant. Using a discrete expression of the second central\nderivative based on second-order Taylor expansion and\nimplementing free boundary conditions in the simplest\npossible way (see. Sec. VI), it becomes\nH(α)\nex=Λ2\nξ2/bracketleftig/parenleftig\nM(α−1)\neq−M(α)\neq/parenrightig\n(1−δ1α)\n−/parenleftig\nM(α)\neq−M(α+1)\neq/parenrightig\n(1−δNα)/bracketrightig\n(7)\nwhereξ=b(films) or c(strips), depending on the geom-\netry of the magnetic cell array [Fig. 1(a,b)].\nFinally, for dipole-dipole interactions, we follow a con-\nventional micromagnetic approach, where the dipolar\nfield experienced by the magnetization of cell αis related\nto the magnetizationvectorsof cells β= 1..N, which cre-\nate the field, through demagnetizing tensors20\nH(α)\nd=−N/summationdisplay\nβ=1N(αβ)·M(β)\neq. (8)\nThe dimensionless tensor N(αβ)is necessarily symmet-\nric and it depends on the shape and relative position of\nthe source ( β) and target ( α) cells. In the case of in-\nfinitely extended slabs [Fig. 1(a)], mutual ( α/ne}ationslash=β) demag-\nnetizing effects are strictly nil and only the v-component\nof the magnetization creates a self ( α=β) demagnetiz-\ning field, which is along v. Thus, in the uvwcoordinatesystem,N(αβ)takes the trivial form\nN(αβ)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle0 0 0\n0δαβ0\n0 0 0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (9)\nInthecaseofinfinitelylongcellswith rectangular( b×c)\ncross section, general analytical expressions can be de-\nrived for the non-zero components of N(αβ), which are\nquite cumbersome (see Appendix B). Under the assump-\ntion that the cells are arranged in a one-dimensional ar-\nray [Fig. 1(b)], they become somewhat simpler and the\ndemagnetizing tensor reduces to\nN(αβ)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle0 0 0\n0N(αβ)\nvv0\n0 0 N(αβ)\nww/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (10)\nMoreover, as a consequence of tr( N(αβ)) =δαβ[Ref. 20],\nthe two non-zero diagonal components are conveniently\nrelated to each other by\nN(αβ)\nvv+N(αβ)\nww=δαβ. (11)\nThus, calculating N(αβ)simply amounts to evaluating\na single quantity, for instance N(αβ)\nww. When the source\nand target cells coincide (self demagnetizing tensor), this\ntensorelementdependssolelyontheaspectratio p=c/b.\nIt is given by the equation\nNww(p) =1\nπ/bracketleftbigg1−p2\n2pln/parenleftbig\n1+p2/parenrightbig\n+plnp+2arctan/parenleftbigg1\np/parenrightbigg/bracketrightbigg\n(12)\nderived by Brown29and Aharoni30. When, on the con-\ntrary, the source cell is located at a finite distance ncc\n(withnc∈N∗) from the target cell (mutual demagnetiz-\ning tensor), it becomes\nN(αβ)\nww(p,nc) =nc+1/summationdisplay\nn=nc−1(−2)1−|n−nc|\n2π/bracketleftbigg1−n2p2\n2pln/parenleftbig\n1+n2p2/parenrightbig\n+n2pln|np|+2narctan/parenleftbigg1\nnp/parenrightbigg/bracketrightbigg\n, (13)\nwhich follows from Eq. B2 in the particular case δv= 0,\nδw=ncc.\nThe second task we need to perform for building D\nis to express the xandycomponents of the dynamic\nfield in each cell {h(α)\nx,h(α)\ny}in terms of the xand\nycomponents of the variable magnetization in every\ncells{m(1)\nx,..,m(N)\nx,m(1)\ny,..,m(N)\ny}, in an explicit man-\nner. This is a slightly more complex task, especially be-\ncause dipole-dipole interactions couple all the individual\nLandau-Lifshitz equations [Eq. 3] together. Special at-tention must also be paid since we now have to express\nall vector quantities in the more relevant xyzcoordinate\nsystem.\nAs for the equilibrium magnetic field, we must add up\nseveralcontributionsto obtain the full dynamic magnetic\nfieldh(α)produced when the magnetization departs from\nequilibrium ( m(β)/ne}ationslash=0). With the exception of the exter-\nnalfieldH0, whichisassumedtobetime-independent, all\ncontributions to H(α)\neqhave a dynamic counterpart. The\ncontribution of crystal anisotropy to h(α), which corre-\nsponds to the equilibrium anisotropy field in Eq. 5, is315\nh(α)\nK=2Ku\nµ0M2\nS/parenleftig\nm(α)·a/parenrightig\na\n−2Kc\nµ0M4\nS3/summationdisplay\ni=1\n\n\n/summationdisplay\nj/negationslash=i/parenleftig\nM(α)\neq·cj/parenrightig2\n/parenleftig\nm(α)·ci/parenrightig\nci+2\n/summationdisplay\nj/negationslash=i/parenleftig\nM(α)\neq·cj/parenrightig/parenleftig\nm(α)·cj/parenrightig\n/parenleftig\nM(α)\neq·ci/parenrightig\nci\n\n,(14)\nin a compact vector form. Getting the xandycomponents of h(α)\nKas a function of those of m(α)from Eq. 14 is a\nmatter of simple arithmetics. We obtain\n(l=x,y)el·h(α)\nK=2Ku\nµ0M2\nS/bracketleftig\nalaxm(α)\nx+alaym(α)\ny/bracketrightig\n−2Kc\nµ0M2\nS/braceleftig/bracketleftbig\nc1lc1x/parenleftbig\nc2\n2z+c2\n3z/parenrightbig\n+2c2zc3z(c2xc3l+c2lc3x)\n+c2lc2x/parenleftbig\nc2\n3z+c2\n1z/parenrightbig\n+2c3zc1z(c3xc1l+c3lc1x)\n+c3lc3x/parenleftbig\nc2\n1z+c2\n2z/parenrightbig\n+2c1zc2z(c1xc2l+c1lc2x)/bracketrightig\nm(α)\nx\n+/bracketleftbig\nc1lc1y/parenleftbig\nc2\n2z+c2\n3z/parenrightbig\n+2c2zc3z(c2yc3l+c2lc3y)\n+c2lc2y/parenleftbig\nc2\n3z+c2\n1z/parenrightbig\n+2c3zc1z(c3yc1l+c3lc1y)\n+c3lc3y/parenleftbig\nc2\n1z+c2\n2z/parenrightbig\n+2c1zc2z(c1yc2l+c1lc2y)/bracketrightbig\nm(α)\ny/bracerightig\n, (15)\nwhere (ax,ay,az) and (cix,ciy,ciz) are the coordinates of\nthe unit vectors aandciin the{ex,ey,ez}local basis.\nBy definition of the transformation matrix T(α), those\nare related to the coordinates in the {eu,ev,ew}global\nbasis by vl=/summationtext3\nj=1T(α)\nljvj(v=a,ci), where the indices\nl,j= 1,2,3 stand for x,y,zandu,v,w, respectively.The contribution of isotropic exchange to the dynamic\nmagneticfield, hex(r), in thecontinuous-mediumapprox-\nimation, maybe obtainedby replacing Meq(r) withm(r)\nin Eq. 6. Applied to plane spin-waves of the form given\nin Eq. 2 and introducing free boundary conditions, the\nexpression one obtains becomes\nh(α)\nex=Λ2\nξ2/bracketleftig/parenleftig\nm(α−1)−m(α)/parenrightig\n(1−δ1α)−/parenleftig\nm(α)−m(α+1)/parenrightig\n(1−δNα)/bracketrightig\n−Λ2k2m(α)(16)\nafter discretization. Deriving correct expressions for the\nxandycomponents of h(α)\nexas a function of those of the\nvariable magnetization vectors m(α−1),m(α), andm(α+1)\nfrom Eq. 16 requires to take into account the possible\nnon-collinear character of the equilibrium, i.e., the fact\nthat the local xyzreference frames in the nearest neigh-\nbor cells α±1 are not necessarily the same as in cell α.This can be achieved by introducing the relevant trans-\nformation matrices, T(β), withβ=α−1,α,α+1, and\ntheir inverse matrices ¯T(β)=/parenleftbig\nT(β)/parenrightbig−1. WithT(β)\nij(resp.\n¯T(β)\nij) denoting the elements of matrix T(β)(resp.¯T(β)),\nwe have6\n(l=x,y)el·h(α)\nex=Λ2\nξ2/bracketleftigg/parenleftigg3/summationdisplay\ni=1T(α)\nli¯T(α−1)\ni1/parenrightigg\n(1−δ1α)m(α−1)\nx+/parenleftigg3/summationdisplay\ni=1T(α)\nli¯T(α−1)\ni2/parenrightigg\n(1−δ1α)m(α−1)\ny\n−/parenleftigg3/summationdisplay\ni=1T(α)\nli¯T(α)\ni1/parenrightigg\n/parenleftbig\n2+ξ2k2/parenrightbig\nm(α)\nx−/parenleftigg3/summationdisplay\ni=1T(α)\nli¯T(α)\ni2/parenrightigg\n/parenleftbig\n2+ξ2k2/parenrightbig\nm(α)\ny\n+/parenleftigg3/summationdisplay\ni=1T(α)\nli¯T(α+1)\ni1/parenrightigg\n(1−δNα)m(α+1)\nx+/parenleftigg3/summationdisplay\ni=1T(α)\nli¯T(α+1)\ni2/parenrightigg\n(1−δNα)m(α+1)\ny/bracketrightigg\n,(17)\nwhere the indices l=xandl=yare to be understood\nasl= 1 and l= 2, respectively, when it comes to the\nelements of matrix T(α).\nFinally, for describing dynamic dipole-dipole interac-\ntions between magnetic cells, we use also demagnetizing\ntensors, as in the static case. By analogy with Eq. 8, we\nwrite the dipolar contribution to h(α)as\nh(α)\nd=−N/summationdisplay\nβ=1n(αβ)·m(β). (18)\nWewillseeinthenextsectionthatthecomponentsofthe\nnewly introduced plane-wave demagnetizing tensor n(αβ)\ndepend not only on the geometry and relative positions\nof the magnetic cells, as usual (static) demagnetizationfactors do, but also on the wave vector kof the propa-\ngating spin-waves. We will also see that rather simple\nanalytical expressions can be derived for n(αβ)when the\ncells are extended slabs [Fig. 1(a)] but that complex in-\ntegral expressions must be dealt with in case the cells\nare rectangular parallelepipeds [Fig. 1(b)]. When deriv-\ning expressions for the xandycomponents of h(α)\ndas\na function of those of the variable magnetization vectors\nm(β)(β= 1..N)fromEq.18, wemustonceagainaccount\nforthe possible non-collinearcharacterofthe equilibrium\nby introducing the appropriate transformation matrices.\nWith tensors n(αβ)expressed in the uvwcoordinate sys-\ntem, we have h(α)\nd=−/summationtextN\nβ=1T(α)n(αβ)¯T(β)·m(β)in the\nxyzreference frame. It readily follows\n(l=x,y)el·h(α)\nd=−N/summationdisplay\nβ=1\n\n\n3/summationdisplay\ni=1\n3/summationdisplay\nj=1T(α)\nljn(αβ)\nji\n¯T(β)\ni1\nm(β)\nx+\n3/summationdisplay\ni=1\n3/summationdisplay\nj=1T(α)\nljn(αβ)\nji\n¯T(β)\ni2\nm(β)\ny\n\n,(19)\nwhere indices i,j= 1,2,3 stand for u,v,wwhen it comes\nto the elements of tensor n(αβ)and, as before, the indices\nl=xandl=yare to be understood as l=1 and l=2\nwhen it comes to the elements of matrix T(α).\nFor the sake of simplicity, we have implicitly assumed\nso far that the magnetic parameters of the media sup-\nporting the spin-waves were homogeneous. While ac-\ncounting for space variations of magnetic anisotropy is\nstraightforward (anisotropy constants and axes just need\nto be made α-dependent in Eqs. 5, 14, and 15), introduc-\ning space variations of saturation magnetization and/or\nexchange stiffness is not. Indeed, to allow for an α-\ndependence of these two parameters, the expressions of\nthe static [Eq. 7] and dynamic [Eq. 16] exchange fields\nmust be transformed in a way which is not totally triv-\nial, as we will discuss in Sec. VA.IV. DYNAMIC DEMAGNETIZING TENSORS\nOur goal in this key section is to derive mathematical\nexpressions for the non-zero components of the dynamic\ndemagnetizing tensors n(αβ)introduced earlier [Sec. III,\nEq. 18]. Since these expressions are among the most\nimportant results of the present work, we shall give a\nrather detailed description of how they can be obtained.\nIn short, what needs to be done is the following. First,\nthe natureofthe magneticchargesthat the variablemag-\nnetization of the source cell creates must be determined.\nNext, the magnetic field that these chargesproduce must\nbe evaluated and averaged throughout the target cell.\nLast, the tensor element n(αβ)\nijmust be identified with\nthe negative of the proportionality factor between the i-\ncomponent of the averaged field and the j-component of\nthe variable magnetization.\nAt this point, it isimportant to note that n(αβ)has the\nsame intrinsic properties as all demagnetizing tensors20:7\nIt is symmetric and obeys tr( n(αβ)) =δαβ. Furthermore,\nas its static counterpart, n(αβ)is necessarily diagonal\nwhen the source and target cells coincide ( α=β). All\nthese requirements reduce very strongly the number of\nindependent tensor elements that need to be calculated\nto fully determine n(αβ), in general.\nA. Infinite slabs\nIn the case where the magnetic cells are slabs with infi-\nnite dimensions along both uandw, the in-plane compo-\nnent of the variable magnetization perpendicular to the\ndirection of propagation, i.e., the w-component, never\nproduces any magnetic charge. Therefore, the corre-\nsponding tensor elements n(αβ)\niw=n(αβ)\nwi(withi=u,v,w)\nare always nil and n(αβ)contains at most four non-zero\ncomponents. Volume charges −∇·m(β)=ikm(β)\nuare\ncreated by the u-component of m(β), as a result of its\nspace oscillatory nature [Eq. 2], and surface charges ex-\nist at the top (+ m(β)\nv) and bottom ( −m(β)\nv) surfaces of\nthe source cell as soon as m(β)has a finite vertical com-\nponent. With m(β)having the form of a plane wave,\nall those magnetic charges vary harmonically along the\ndirection of propagation.\nLet us then consider a magnetic surface charge distri-\nbution parallel to the ( u,w) plane, located at the vertical\nposition v0, and harmonic in the u-direction, as defined\nby\nσ2D(v0,r,t) =σ0δ(v−v0)ei(ωt−ku),(20)\nwhereσ0is a complex amplitude and δis Dirac’s func-\ntion. The magnetostatic potential created by such a\ncharge distribution writes32\nφ2D(σ0,v0,r,t) =σ0\n2|k|e−|k(v−v0)|ei(ωt−ku),(21)\nand the magnetic field that derives from it is\nh2D(σ0,v0,r,t) =−∇φ2D(σ0,v0,r,t)\n=σ0\n2e−|k(v−v0)|ei(ωt−ku)\n×[isgn(k)eu+sgn(v−v0)ev].(22)\nWith Eq. 22, we can evaluate any component of n(αβ).\nWe shall illustrate this by calculating two particular dy-\nnamic demagnetization factors, n(αβ)\nuuandn(αβ)\nuv, which\nis a priori enough to fully determine the tensor in case\nthe magnetic cells have the shape of infinitely extended\nslabs.\nLetvβandvαbe the vertical coordinates of the source\nand target cells, respectively. To derive the expression\nofn(αβ)\nuu, we need to calculate the u-component of the\nmagnetic field produced by the volume charges ikm(β)\nu\nin the source cell and then average it over the thick-\nness of the target cell. The first step amounts to inte-grating the contributions of all surface charge distribu-\ntionsikm(β)\nuδ(v−v0) =ikm(β)\n0uδ(v−v0)ei(ωt−ku)such that\nvβ−b/2/lessorequalslantv0/lessorequalslantvβ+b/2. Therefore, we can write\n−n(αβ)\nuum(β)\nu=1\nbvα+b\n2ˆ\nvα−b\n2vβ+b\n2ˆ\nvβ−b\n2eu·h2D/parenleftig\nikm(β)\n0u,v0,r,t/parenrightig\ndv0dv.\n(23)\nSubstituting Eq. 22 in Eq. 23, it comes\nn(αβ)\nuu=|k|\n2bˆvα+b\n2\nvα−b\n2ˆvβ+b\n2\nvβ−b\n2e−|k(v−v0)|dv0dv.(24)\nThe result of this double integration, which can be car-\nried out analytically, is different depending on whether\nα=β, or not. If the source and target cells coincide, we\nobtain\n(α=β)nuu= 1−1−e−|k|b\n|k|b,(25)\nwhereas, if they are disjoint, we have\n(α/ne}ationslash=β)n(αβ)\nuu=2 sinh2/parenleftbigkb\n2/parenrightbig\n|k|be−|k(vα−vβ)|.(26)\nAs we could have anticipated, when α=β,nuucorre-\nsponds to the well-known P00coefficient, which appears\nin the expression of the matrix elements of the dipole-\ndipole interaction in the popular spin-waves theory of\nKalinikos and Slavin21,33, as the result of the integration\nover the film(cell) thickness of the magnetostatic Green’s\nfunction for a plane spin-wave having a uniform profile.\nTo calculate n(αβ)\nuv, we have to consider this time\ntheu-components of the magnetic fields produced by\nthe two surface charge distributions ±m(β)\n0vδ(v−vβ∓\nb/2)ei(ωt−ku), add them up, and average the sum over\nthe target cell. Therefore, we can write\n−n(αβ)\nuvm(β)\nv=1\nbvα+b\n2ˆ\nvα−b\n2eu·/bracketleftbigg\nh2D/parenleftbigg\n+m(β)\n0v,vβ+b\n2,r,t/parenrightbigg\n+h2D/parenleftbigg\n−m(β)\n0v,vβ−b\n2,r,t/parenrightbigg/bracketrightbigg\ndv.\n(27)\nSubstituting Eq. 22 in Eq. 27, it comes\nn(αβ)\nuv=−i\n2bsgn(k)\n׈vα+b\n2\nvα−b\n2/bracketleftig\ne−|k(v−vβ−b\n2)|−e−|k(v−vβ+b\n2)|/bracketrightig\ndv.\n(28)\nAs forn(αβ)\nuu, the integration can be performed analyti-\ncally and we find that the result is different depending8\non whether the source and target cells coincide or not.\nWhen they do coincide, n(αβ)\nuvis zero, which means that\nthe dynamic self demagnetizing tensor is diagonal, as ex-\npected. It can be written as\n(α=β)n(self)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglenuu0 0\n0 1−nuu0\n0 0 0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,(29)\nwithnuugiven by Eq. 25. When the source and target\ncells are disjoint, we obtain\nn(αβ)\nuv=−isgn(k)sgn(vα−vβ)2sinh2/parenleftbigkb\n2/parenrightbig\n|k|be−|k(vα−vβ)|.\n(30)Comparing Eqs. 26 and 30, it appears that n(αβ)\nuuand\nn(αβ)\nuvare related to each other by\nn(αβ)\nuv=−isgn(k)sgn(vα−vβ)n(αβ)\nuu.(31)\nThe dynamic mutual demagnetizing tensor can then be\nwritten as\n(α/ne}ationslash=β)n(αβ)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen(αβ)\nuu −isgn(k)sgn(vα−vβ)n(αβ)\nuu0\n−isgn(k)sgn(vα−vβ)n(αβ)\nuu −n(αβ)\nuu 0\n0 0 0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (32)\nwithn(αβ)\nuugiven by Eq. 26. Thus, for magnetic cells\nin the shape of extended slabs, a single demagnetization\nfactor is always sufficient to fully determine the whole\ntensorn(αβ).\nB. Rectangular parallelepipeds\nIn the case where the magnetic cells are rectangular\nparallelepipeds with infinite length along u, the situation\nis more complex, both physically and mathematically. In\naddition to the volumes charges −∇·m(β)=ikm(β)\nuand\nsurface charges on the top (+ m(β)\nv) and bottom ( −m(β)\nv)\nfaces of the source cell created, as before, by the uandv\ncomponents of m(β), surface charges are also generated\nnow on the left (+ m(β)\nw) and right ( −m(β)\nw) faces of the\ncell by the w-component of m(β). Therefore, none of theelements of n(αβ)is a priorizero, atleast when the source\nand target cells are disjoint.\nFor calculating the dynamic demagnetization factors\nin the same manner as above [Sec. IVA], we now need\nto know the expression of the magnetic field, h1D, ema-\nnating from a one-dimensional harmonic distribution of\nmagnetic charges parallel to axis uand located at the\ntransverse position ( v0,w0), as defined by\nσ1D(v0,w0,r,t) =σ0δ(v−v0)δ(w−w0)ei(ωt−ku),(33)\nwhereσ0is again a complex amplitude. To our knowl-\nedge, no such expression is available in the literature. It\nis derived from basic magnetostatics in Appendix C. In\ntheuvwreference frame, the three components of h1D\nare\neu·h1D(σ0,v0,w0,r,t) =iσ0\n2πk K0(|k|/radicalbig\n(v−v0)2+(w−w0)2)ei(ωt−ku)(34a)\nev·h1D(σ0,v0,w0,r,t) =σ0\n2π|k|(v−v0)K1(|k|/radicalbig\n(v−v0)2+(w−w0)2)/radicalbig\n(v−v0)2+(w−w0)2ei(ωt−ku)(34b)\new·h1D(σ0,v0,w0,r,t) =σ0\n2π|k|(w−w0)K1(|k|/radicalbig\n(v−v0)2+(w−w0)2)/radicalbig\n(v−v0)2+(w−w0)2ei(ωt−ku), (34c)\nwhereKndenotesthe n-thordermodifiedBesselfunction of the second kind.9\nWith these expressions, we are equipped to calcu-\nlate all elements of the dynamic demagnetizing tensor in\nthe case where the magnetic cells are rectangular paral-\nlelepipeds. However,onlyaveryfewisactuallyneeded to\nfully determine n(αβ)under the assumption that the cells\nare arranged in a one-dimensional array, with the same v\ncoordinate. For such a cell array, indeed, even if all com-\nponentsofthedynamicmagnetizationeffectivelyproduce\nmagnetic charges, four of the six off-diagonal elements\nsystematically vanish. These are the uv,vu,vw, andwv\ncomponents. Taking also into account the requirements\non the symmetry and trace of n(αβ)mentioned in the in-\ntroduction of Sec. IV, one sees that no more than three\ndynamic demagnetization factorsneed to be known (only\ntwo ifα=β). Below, we derive mathematical expressions\nfor a particularset ofthree factors, the diagonalelements\nn(αβ)\nuuandn(αβ)\nwwand the off-diagonal element n(αβ)\nwu, withwhich we can write n(αβ)as\nn(αβ)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen(αβ)\nuu 0 n(αβ)\nwu\n0δαβ−/parenleftig\nn(αβ)\nuu+n(αβ)\nww/parenrightig\n0\nn(αβ)\nwu 0 n(αβ)\nww/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\n(35)\nLetwβandwαbe the horizontal coordinates of the\nsource and target cells, respectively. To calculate n(αβ)\nuu,\nwe have to consider once again the u-component of\nthe magnetic field produced by the volume charges in\nthe source cell, that is here, all charge distributions\nikm(β)\n0uei(ωt−ku)δ(v−v0)δ(w−w0) such that −b/2/lessorequalslantv0/lessorequalslant\n+b/2 andwβ−c/2/lessorequalslantw0/lessorequalslantwβ+c/2, and average it over\nthe cross section of the target cell. This translates into\n−n(αβ)\nuum(β)\nu=1\nbcˆwα+c\n2\nwα−c\n2ˆ+b\n2\n−b\n2ˆwβ+c\n2\nwβ−c\n2ˆ+b\n2\n−b\n2eu·h1D/parenleftig\nikm(β)\n0u,v0,w0,r,t/parenrightig\ndv0dw0dvdw, (36)\nand, after substituting Eq. 34a in Eq. 36, we get\nn(αβ)\nuu=k2\n2πbcˆwα+c\n2\nwα−c\n2ˆ+b\n2\n−b\n2ˆwβ+c\n2\nwβ−c\n2ˆ+b\n2\n−b\n2K0(|k|/radicalbig\n(v−v0)2+(w−w0)2)dv0dw0dvdw. (37)\nTo calculate n(αβ)\nww, we must consider this time the w-\ncomponent of the magnetic field produced by the surface\ncharges on the lateral faces of the source cell, i.e., all\ncharge distributions + m(β)\n0wei(ωt−ku)δ(v−v0)δ(w−c/2)(left face) and −m(β)\n0wei(ωt−ku)δ(v−v0)δ(w+c/2) (right\nface) such that vβ−b/2/lessorequalslantv0/lessorequalslantvβ+b/2, and averageit over\nthe target cell. This writes\n−n(αβ)\nwwm(β)\nw=1\nbcˆwα+c\n2\nwα−c\n2ˆ+b\n2\n−b\n2ˆ+b\n2\n−b\n2ew·/bracketleftig\nh1D/parenleftig\n+m(β)\n0w,v0,+c\n2,r,t/parenrightig\n+h1D/parenleftig\n−m(β)\n0w,v0,−c\n2,r,t/parenrightig/bracketrightig\ndv0dvdw, (38)\nand, after substitution of Eq. 34c in Eq. 38, we obtain\nn(αβ)\nww=−|k|\n2πbcˆwα+c\n2\nwα−c\n2ˆ+b\n2\n−b\n2ˆ+b\n2\n−b\n2/bracketleftigg\n(w−c\n2)K1(|k|/radicalbig\n(v−v0)2+(w−c\n2)2)/radicalbig\n(v−v0)2+(w−c\n2)2\n−(w+c\n2)K1(|k|/radicalbig\n(v−v0)2+(w+c\n2)2)/radicalbig\n(v−v0)2+(w+c\n2)2/bracketrightigg\ndv0dvdw. (39)\nFinally, calculating n(αβ)\nwurequires to consider the w-component of the magnetic field produced by the volume charges\nikm(β)\nuin the source cell. Then, we have\n−n(αβ)\nwum(β)\nu=1\nbcˆwα+c\n2\nwα−c\n2ˆ+b\n2\n−b\n2ˆwβ+c\n2\nwβ−c\n2ˆ+b\n2\n−b\n2ew·h1D/parenleftig\nikm(β)\n0u,v0,w0,r,t/parenrightig\ndv0dw0dvdw, (40)\nand, after substituting Eq. 34c in Eq. 40, we find\nn(αβ)\nwu=−ik|k|\n2πbcˆwα+c\n2\nwα−c\n2ˆ+b\n2\n−b\n2ˆwβ+c\n2\nwβ−c\n2ˆ+b\n2\n−b\n2(w−w0)K1(|k|/radicalbig\n(v−v0)2+(w−w0)2)/radicalbig\n(v−v0)2+(w−w0)2dv0dw0dvdw. (41)10\nWe note that, using the same kind of reasoning, it\nwould be rather straightforward to derive mathematical\nexpressions for the redundant elements n(αβ)\nuwandn(αβ)\nww,\nand for all elements in the more general case of par-\nallelepipedic cells arranged in a two-dimensional array\n(not considered here). We note also that the integrands\nand the integration domains are such that none of the\nmultiple integrals that appear in the expressions of the\ndynamic demagnetization factors [Eqs. 37, 39, and 41]\ncan be calculated analytically. Then, numerical meth-\nods must be employed to compute these factors. In cases\nwhere the source and target cells are totally disjoint, any\ninterpolatory cubature rule (a multidimensional Simp-\nson’s rule in our case) may be used effectively. When\nthey either coincide or when they share a face or an edge,\nhowever, one has to cope with the difficulty that the in-\ntegrands have singularities at some points of the inte-\ngration domains. In situations like these, Monte Carlo\nbased algorithms such as the Vegas algorithm34, which\nis implemented in the GNU Scientific Library, are better\nsuited. As a concluding remark to both Secs. IVA and\nIVB, we note finally that the diagonal elements of the\ndynamic demagnetizating tensors are always real num-\nbers whereas, as long as the magnetic cells are arranged\nin a one-dimensional array, the non-zero off-diagonal ele-\nments are systematically imaginary. A partial graphical\nexplanation of why this is so will be given in Sec. IVC.\nC. Discussion\n1. Wave vector dependence of the dynamic demagnetization\nfactors\nAs illustrated in Fig. 2(a,b), the uuelements of the\nself and mutual dynamic demagnetizing tensors of ex-\ntended magnetic slabsshowverydifferentvariationswith\nk. The self demagnetization factor varies monotonously\n[Fig. 2(a)]. It goes from zero at low wave vector, where\nthe magnetic charges −∇·m(β)=ikm(β)\nucreated by\nm(β)\nu=m(β)\nueuare extremely dilute, to unity at high\nwave vectors, where successive spin-wave wavefronts are\nso close to one another that these volumes charges be-\ncome spatially distributed very much like surface charges\nin an ensemble of closely packed, perpendicularly magne-\ntized thin magnetic films, sitting vertically. The transi-\ntion between the two regimes occurs for k≃b−1. In con-\ntrast, the mutual demagnetization factor vanishes both\nin the low wave vector limit (for the same reason as the\nself demagnetization factor) and in the high wave vector\nlimit [Fig. 2(b)]. In the latter case, this occurs because\nthe alternated positive and negative charges associated\nwith the spatial variation of m(β)\nubecome indistinguish-\nable, as viewed from the target cell, and average out to\nzero. In between these two limits [Fig. 2(c)], the mag-\nnetic charges are arranged so that the dynamic dipolar\ncoupling between the source and target cells is sizeable.\n /2 !\"\nwvk(f) \nhd\nmu\nuv+\nhd!\n\"k (c) \n /2 01\n01\n10 -1 10 010 110 210 310 4-0.02 0.02 \n0\n10 -1 10 010 110 210 310 4-0.001 0.001 \n0(a) (d) ##\n    Re (nuu )\n    Re (nvv ) \n    Im (nvu ) Re( nuu ), Re( nvv ), Im( nvu )\nk (rad/µm) (b)    Re (nuu )\n   Re (nvv )\n   Re (nww ) \n   Im (nwu )\n(e) Re( nuu ), Re( nvv ), Re( nww ), Im( nwu )\nk (rad/µm) ##\nFIG. 2: (a),(b) uu,vv, andvucomponents of the self (a) and\nmutual (b) dynamic demagnetizing tensors as a function of\nthe wave vector kfor extended magnetic slabs of thickness\nb= 2 nm separated by 40 nm ( vα> vβ). (c),(f) Schematic\nrepresentationsofthedipole field, hd, createdoutsideasource\ncell (β) by the u-component of the variable magnetization\nin a plane spin-wave propagating along u, for slabs (c) and\nparallepipeds (f). (d),(e) uu,wwandwucomponents of the\nself (d) and mutual (e) dynamic demagnetizing tensors as a\nfunction of the wave vector kfor magnetic parallepipeds of\nheightb= 5 nm and width c= 2 nm, separated by 40 nm\n(wα>wβ).\nn(αβ)\nuureaches a maximum value for k=|vα−vβ|−1, which\nis inversely proportional to the cell separation and ap-\nproximately amounts to1\n2b|vα−vβ|−1. Figure 2(c) il-\nlustrates the reason why the vuelement of the mutual\ndemagnetizing tensor is imaginary while the uuone is\nreal [Eqs. 26, 31]: this reflects the fact that the uand\nvcomponents of the dipole field created by m(β)\nuat the\nlocation of the target cell oscillate in quadrature and are\nmaximum in places separated by a quarter of spin-wave\nwavelength.\nMovingtoparallepipeds[Fig.2(f)], thatis, limitingthe\nsize of the magnetic cells in both dimensions perpendicu-\nlar to the direction of spin-wavepropagation, induces ad-\nditional finite size effects as compared to extended slabs.\nFor the description of the changes produced, which are\nqualitative as well as quantitative, one should keep in\nmind that moving from our film geometry [Fig. 1(a)]\nto our strip geometry [Fig. 1(b)] also requires to rotate\nthe magnetic medium by 90oabout axis u, i.e., to in-\nterchange vandw. First of all, the factors related to\nthe newly confined dimension (the self and mutual nvv)\nare no longer systematically nil [Fig. 2(d,e)]. This obvi-11\nously followsfrom the creationofsome dipole field by the\nso far ”silent” component of the variable magnetization\n(mwin the film geometry). Second, the k-dependence\nof some of the factors is altered, especially in the low- k\nlimit. For instance, the self demagnetization factor nww\n(the pendant of nvvin the film geometry) tends towards\na value, which is reduced below unity [Fig. 2(d)] all the\nmore strongly, the smaller the aspect ratio b/c. Con-\ncomitantly, its mutual counterpart, while still exhibiting\na local optimum, takes on a non-zero value when k→0\n[Fig. 2(e)]. Third, as a comparison of the vertical scales\nin panels (b) and (e) of Fig. 2 reveals, the reduction of\nthe second lateral cell dimension, at fixed cell separation,\nis accompanied by a global decrease of the magnitude of\nall the mutual dynamic demagnetization factors. This is\nthe natural consequence of an increased spatial dilution\nof the dynamic stray field produced by the source cell as\nthe target cell subtends a smaller solid angle [Fig. 2(f)].\n2. Validation of the method\nAs reported in Appendix D, a number of tests have\nbeen performed in order to check the validity of our the-\noretical results regarding the plane-wave demagnetiza-\ntion factors [Secs. IVA and IVB]. None of these tests\nconsisting in comparisons with results from analytical\nmodels and examination of various limiting cases has\nproved our results wrong. As an introduction to the\nuse of our method for exploring propagating spin-wave\nphysics, we describe hereafter another demanding test,\nwhich demonstratesthe validity ofournumericalscheme.\nInthistest, dispersionrelationscomputedforanisotropy-\nfree homogeneous extended films in the dipole-exchange\nregime are compared to predictions of the perturbation\ntheory of Kalinikos and Slavin21. The latter uses the\nmagnetostatic Green’s function method to account for\ndipolar interactions and allows for the derivation of ex-\nplicit, thoughapproximate,expressionsforthedispersion\nrelations33. At order zero in perturbation and in the ab-\nsence of surface pinning, the dispersion relations of the\nn-th mode in the backwardvolumewave(BVW, Meq/bardblu)\nand surface wave (SW, Meq/bardblw) configurations are\n(BVW) ω2\nn=/parenleftbig\nωH+ωMΛ2k2\nn/parenrightbig\n×/parenleftbig\nωH+ωMΛ2k2\nn+ωM(1−Pnn)/parenrightbig\n(42a)\n(SW)ω2\nn=/parenleftbig\nωH+ωMΛ2k2\nn+ωMPnn/parenrightbig\n×/parenleftbig\nωH+ωMΛ2k2\nn+ωM(1−Pnn)/parenrightbig\n(42b)\nwherenis the quantization number along the film thick-\nness, Λ still denotes the exchange length, k2\nn=k2+\n(nπ/T)2, andPnnis given by\nPnn=k2\nk2n/bracketleftbigg\n1−/parenleftbigg2\n1+δ0n/parenrightbiggk2\nk2n/parenleftbigg1−(−1)ne−|k|T\n|k|T/parenrightbigg/bracketrightbigg\n.\n(43)\nFigure 3(a) illustrates the fact that, as long as the film\nthickness remains moderate, the dispersion relations of0 50 100 150 200 020 40 60 (b)  !\"#   (GHz) \nk (rad/µm) n = 1 \nn = 0 \n0 50 100 150 200 020 40 100 120 \nSW \nBVW \nn = 0 n = 1 (a)  !\"#   (GHz) \nk (rad/µm) \n0 2 4 6 8 10 -0.9531 0.3024 0.3028 Re( m0x), Im( m0y)  (arb. units) \n(e) \nBVW \nv (nm) -0.87 0.48 0.54 \n n = 0 (c) SW \n0 2 4 6 8 10 -1 01\n (f) Re( m0x), Im( m0y)  (arb. units) \nv (nm) -1 01\n (d) \n n = 1 \nFIG. 3: (a),(b) Dispersion relations of the first two spin-\nwave branches in 10 nm (a) and 20 nm (b) thick films with\nA=11 pJ/m and MS=800 kA/m. Symbols and lines corre-\nspond to results of our numerical approach ( b= 0.2 nm) and\npredictions of the zero order perturbation theory of Kalini kos\nand Slavin [Eq. 42], respectively. The films are magne-\ntized in-plane, along u(red circles) or w(black squares).\nµ0H0=20 mT. (c)-(f) v-profiles of the n=0 (c,e) and n=1\n(d,f) backward volume wave modes (c,d) and surface wave\nmodes (e,f) with k=50 rad/ µm in the 10 nm thick film [in-\ndicated with solid symbols in (a)]. Open and solid symbols\nrepresent the out-of-plane ( y=v) and in-plane components of\nthe dynamic magnetization, respectively. In (c,d) x=−w.\nIn (e,f)x=u.\nthe first two modes ( n=0,1) computed with our finite-\ndifference approach match those calculated with Eqs. 42\nquite closely, for both magnetic configurations. This is a\nproofthatournumericalschemeiscorrect,notonlyasfar\nas dipolar interactions are concerned, but also regarding\nhow the exchange interactions are treated.\nSome clear discrepancy however appears at large film\nthickness [Fig. 3(b)]. This does not come as a surprise\nsince, on increasing T, the frequency distance between\nthe spin-wave branches with n=0 andn=1 decreases so\nthat their dipole-dipole hybridization may become sig-\nnificant, which is not accounted for by the zero-order\napproximation21. Noticeably, deviations from the com-\nputed data are observed earlier for surface waves35than\nfor volume waves. This reduction of the thickness range\nof applicability of the analytical model for surface waves12\nis likely related to their specific modal profile. Unlike\nvolume waves, which have well defined profile symmetry\n[Fig. 3(e,f)], surface waves are neither fully symmetric\nnor fully antisymmetric [Fig. 3(c,d)]. As a consequence,\nhybridization between branches of odd and even indices,\nwhich is not permitted for volume waves, is allowed for\nsurfacewavesandhybridizationisthusgenerallystronger\nin the SW configuration. We have checked that on in-\ncluding explicitly the hybridization between the n= 0\nandn=1 surface wave branches, as is done for example\nin Refs. 36 and 37, the data produced by the analytical\nmodel are lying significantly closer to those computed\nwith our numerical approach (not shown).\nV. APPLICATIONS\nThe physical situations where the finite-difference ap-\nproach that we propose should prove most useful are ei-\nther those where the material parameters vary through-\nout the magnetic medium or those where the medium\nis not homogeneously magnetized, two possibilities that\nare difficult to include in an analytical spin-wave theory\nsuch as the one developed for films21,33. In order to il-\nlustrate this point and, simultaneously, give examples of\napplication of our numerical model in the two geometries\nconsidered here, we will address below two questions of\ncurrent interest: i) the frequency non-reciprocity of sur-\nface waves in films with heterogenous magnetic proper-\nties (Sec. VA) and ii) the channeling of spin-wavesinside\nmagnetic domain walls (Sec. VB). It should be noticed\nthat all the non-collinear equilibrium spin configurations\ndiscussed hereafter have been determined by solving nu-\nmerically overdamped Landau-Lifshitz-Gilbert equations\nwith a fourth order Runge-Kutta method.\nA. Inhomogeneous magnetic films\nOwing to their largest group velocity, surface waves\n(Meq/bardblw) are often considered as the most relevant spin-\nwavesfor magnonic applications38. They are also special\nin that they are the only standard spin-waves for which\ni) two components of the dynamic dipole field hd, one\nin-plane ( u) and one out-of-plane ( v), contribute to the\ntorque acting on the dynamic magnetization and ii) the\noff-diagonal elements of the mutual demagnetizing ten-\nsor [Eq. 32], which change sign on reversing the direction\nof propagation, play an important role. As illustrated\nin Fig. 4(a,b), these peculiarities lead to the formation\nof asymmetric distributions of dynamic magnetization\nacross the film thickness, such that waves propagating\nin opposite directions have larger amplitudes near op-\nposite surfaces. Because of this specific character, also,\ncounter-propagating spin-waves of a given wave vector\n|k|have different frequencies as soon as the film exhibits\nvertically asymmetric properties like, for instance, in-\nequivalent magnetization pinning (anisotropy) at the top0 10 20 30 01\n0 10 20 30 01Im (m0v)\nv (nm) |k| = 50 rad/µm |k| = 10 rad/µm \n(f) (b) (a) \nv (nm)   (d) \n(e) -1 01Re (m0u) (arb. units) (c) \n-1 01 \n 01\nk < 0 \nk > 0 |m0|\n 01 \n \nFIG. 4: Profiles of the fundamental surface wave modes with\nk >0 (solid symbols) and k <0 (open symbols) in a 30 nm\nthick film of permalloy ( A=11 pJ/m, MS=800 kA/m), for\n|k|= 10 rad/ µm (left column) and |k|= 50 rad/ µm (right\ncolumn): (a),(b) total amplitude, (c),(d) in-plane compon ent\nand (e),(f) out-of-plane component of the dynamic magneti-\nzation. The external magnetic field is µ0H0= 50 mT. All\nplotted quantities are in arbitrary units.\nand bottom surfaces25,37. Our numerical approach is\nparticularly well suited to compute the frequency non-\nreciprocities produced by all sorts of magnetic symmetry\nbreaking. Here, we will consider the case of a bilayer\nfilm made of two ferromagnetic materials with different\nexchange stiffness and saturation magnetization.\nBefore proceeding with the description of our results,\na technical remark must be made. When AandMS\nvary in space, the exchange interaction must be treated\ncarefully. Its contributions to the equilibrium and dy-\nnamic magnetic fields [Eqs. 7 and 16] can no longer be\nexpressedin terms ofexchange length, which is a concept\nonly valid inside a homogeneous magnetic material. New\nexpressions must be used, where AandMSappear ex-\nplicitly. Starting from the Heisenberg formulation of the\nexchange energy and assuming that the angle between\nadjacent spins remain small, one may easily show that\nEq. 7 becomes\nH(α)\nex=2A(α−)\nµ0M(α)\nSξ2/parenleftigg\nM(α−1)\neq\nM(α−1)\nS−M(α)\neq\nM(α)\nS/parenrightigg\n(1−δ1α)\n+2A(α+)\nµ0M(α)\nSξ2/parenleftigg\nM(α+1)\neq\nM(α+1)\nS−M(α)\neq\nM(α)\nS/parenrightigg\n(1−δNα),\n(44)\nwhereA(α±)denotes the value ofthe exchangecoefficient13\n(a) \nw uvMeq \ntFe tPy \nH0\n0 100 200 10 20 30 40 50 \n k > 0 \n k < 0  /2 ! (GHz) \n|k| (rad/µm) (b) \nFIG. 5: (a) Schematic representation of a transversally mag -\nnetized (001)Fe/Py bilayer film. (b) Dispersion relation of\nthe fundamental surface wave mode in a film with tFe=tPy=\n7.5nmsubmittedtoanexternalmagnetic field µ0H0=50mT,\nfor positive (solid line) and negative (dashed line) wave ve c-\ntors. The symbols indicate the modes whose v-profiles are\nshown in Fig. 6.\nbetween cell αand cell α±1, which we choose here to\nexpress as the harmonic mean of the exchange stiffness\nconstants in the volume of the cells, A(α±)=2A(α)A(α±1)\nA(α)+A(α±1).\nSimilarly, Eq. 16 becomes\nh(α)\nex=2A(α−)\nµ0M(α)\nSξ2/parenleftigg\nm(α−1)\nM(α−1)\nS−m(α)\nM(α)\nS/parenrightigg\n(1−δ1α)\n+2A(α+)\nµ0M(α)\nSξ2/parenleftigg\nm(α+1)\nM(α+1)\nS−m(α)\nM(α)\nS/parenrightigg\n(1−δNα)\n−2A(α)\nµ0M(α)\nS2k2m(α), (45)\nand equations 17 should be modified accordingly.\nThe system we consider now consists of a permalloy\n(Py) layer ( A= 11 pJ/m, MS= 800 kA/m) of thick-\nnesstPylying on top of and exchange coupled to a single\ncrystalbccFe layer ( A= 20 pJ/m, MS= 1700 kA/m,\nKc=50 kJ/m3) of thickness tFe[Fig. 5(a)]. The Fe crys-\ntal is oriented so that {c1,c2,c3}={eu,ev,ew}and the\nexternal magnetic field is applied parallel to ew=c3,\nwhich is an easy direction of magnetization for the Fe\ncomponent, in order to magnetize the film at right an-\ngle to the propagation direction, {x,y,z}={u,v,w}. As\nmay be seen in Fig. 6(a,b), the bi-component character0 5 10 15 -1 01\n0 5 10 15 -1 01(A/MS)d m0/d v\nv (nm) k < 0 k > 0 \n(f) (b) \n(c) (a) \nv (nm)   (d) \n(e) -1 01m0/MS (arb. units) \n-1 01 \n -1 01\n Re( m0u)\n Im( m0v)m0\n -1 01 \n \nFIG. 6: (a),(b) v-profiles of the fundamental surface wave\nmodes with |k|= 50 rad/ µm through a Fe/Py bilayer film\nwithtFe=tPy= 7.5 nm submitted to an external field\nµ0H0=50 mT. (c)-(f) Variations of the quantitiesm0\nMS(c,d)\nandA\nMS∂m0\n∂v(e,f) with the v-coordinate, as deduced from\nthe mode profiles shown in (a) and (b). In each panel, the\nin-plane ( u) and out-of-plane ( v) components are shown as\nblack squares and red circles, respectively. The left and ri ght\ncolumns correspond to modes with k<0 (f=20.0 GHz) and\nk>0 (f=21.1 GHz), respectively. All plotted quantities are\nin arbitrary units.\nof the film strongly manifests itself in the profile of the\nnormal modes, in the form of discontinuities at the lo-\ncation of the Fe/Py interface. As expected, the ratios\nRe(m0u)/MSand Im(m0v)/MS, which are measures of\nthe precession angles of the magnetization, remain con-\ntinuous there but they exhibit clear changes of slope\n[Fig. 6(c,d)]. The latter are necessary to fulfil the mi-\ncromagnetic boundary condition39, which requires that\nA\nMS∂m0\n∂vbe continuous across the interface [Fig. 6(e,f)].\nAs illustrated in Fig. 5(b), a difference in the fre-\nquencies of counter-propagating spin-waves is observed\nas soon as the wave vector is not zero and the film\nis indeed magnetically asymmetric ( tPytFe/ne}ationslash= 0). No-\nticeably, the frequency non-reciprocity effect appears as\nmaximum for tPy/tFeof order unity, irrespective of the\ntotal film thickness T=tPy+tFe[Fig. 7]. Neverthe-\nless, a rich behavior is observed when varying T. While\nin thin films ( T/lessorequalslant15 nm), the frequency difference\n∆f=f(−|k|)−f(|k|) for the fundamental SW mode\nis always negative [Fig. 7(a,b)], in thick films, it goes\nfrom negative to positive with increasing k[Fig. 7(c,d)].\nThis sign reversal occurs for a wave vector k∗which de-\ncreases fast with increasing T[Fig. 8] but is only weakly14\n/s49\n/s50/s51\n/s45/s48/s46/s53/s48/s48/s46/s53\n/s45/s48/s46/s53/s49\n/s45/s49/s45/s48/s46/s53/s45/s49\n/s45/s48/s46/s53/s45/s49\n/s48/s46/s53 /s49/s46/s48 /s48/s46/s49/s40/s100/s41\n/s116\n/s70/s101/s47/s40 /s116\n/s70/s101/s43 /s116\n/s80/s121 /s41/s32/s51\n/s50\n/s49\n/s48\n/s45/s49\n/s45/s50\n/s45/s51/s102/s32/s40/s71/s72/s122 /s41\n/s48/s46/s49 /s48/s46/s53 /s48/s46/s57/s40/s99/s41\n/s116\n/s70/s101/s47/s40 /s116\n/s70/s101/s43 /s116\n/s80/s121 /s41/s32\n/s48/s46/s49 /s48/s46/s53 /s48/s46/s57/s40/s98/s41\n/s116\n/s70/s101/s47/s40 /s116\n/s70/s101/s43 /s116\n/s80/s121 /s41/s32\n/s48/s46/s48 /s48/s46/s53 /s48/s46/s57/s48/s49/s48/s48/s50/s48/s48\n/s116\n/s70/s101/s47/s40 /s116\n/s70/s101/s43 /s116\n/s80/s121 /s41/s107 /s32/s40/s114/s97/s100/s47/s181/s109/s41/s40/s97/s41\nFIG. 7: Frequency non-reciprocity ∆ f=f(−|k|)−f(|k|) of the fundamental surface wave mode as a function of the wav e vector\nkand composition, for Fe/Py bilayer films of varying total thi cknessT: (a)T=10 nm, (b) T=15 nm, (c) T=20 nm, and (d)\nT=25 nm ( µ0H0=50 mT).\ndependent on the film composition [Figs. 7(d) and 8], at\nleast in the range 0 .2< tFe/T <0.8. This is an indi-\ncation that the change of sign of ∆ fis not due to the\nmagnetic asymmetry itself. It is rather related to an\nintrinsic phenomenon, which occurs also in symmetric\nfilms and just gets highlighted when the magnetic sym-\nmetry is broken. We note that, in thick films with modal\nprofiles not perturbed by any kind of magnetic asym-\nmetry, the overall localization of the fundamental SW\nmode (looking at |m0|) does not reversewhen kincreases\n[Fig. 4(a,b)] but the side of the film where the out-of-\nplane component of the dynamic magnetization ( m0v) is\nthelargestdoes[Fig.4(e,f)]. Furthermore, theparameter\nSv=Im/parenleftig\nm0v(0)−m0v(T)\nm0v(0)+m0v(T)/parenrightig\n, which measures the intrinsic\ndegreeofasymmetryoftheprofileof m0vinhomogeneous\nfilms, varies with kin the same qualitative manner as ∆ f\ndoes for composite films [Fig. 9]. Then it seems that the\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s32/s32/s32/s32/s32/s32/s116\n/s70/s101/s47/s84\n/s32/s48/s46/s51\n/s32/s48/s46/s53\n/s32/s48/s46/s55/s107 /s42/s32/s40/s114/s97/s100/s47/s181/s109/s41\n/s84 /s32/s40/s110/s109/s41\nFIG. 8: Wave vector k∗at which the frequency non-\nreciprocity ∆ fof the fundamental surface wave mode in\nFe/Py bilayer films changes sign as a function of the film\nthickness T, for three values of the relative fraction of Fe,\ntFe/T=0.3 (squares), tFe/T=0.5 (circles), and tFe/T=0.7\n(diamonds). The shaded zone indicates the thickness range\nwhere no change of sign occurs. The line is a guide to the eye.\nµ0H0= 50 mT.behavior of the frequency non-reciprocity in bilayer films\nis somehow related to that of m0v. It is however beyond\nthe scope of the present paper to elucidate why this is so.\nFor that, a dedicated analytical theory would certainly\nbe necessary, such as the one developed in Ref. 37 to ac-\ncount for the effect of a difference in anisotropy at the\ntwo films surfaces.\nIn thick films, the overall magnitude of the frequency\nnon-reciprocity effect increases monotonously with in-\ncreasing T[Figs. 7 and 9(a)]. This originates essentially\nfromthe combinationoftwofactors: i) the largerthe film\nthickness the larger the intrinsic modal profile asymme-\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s45/s50/s48/s50/s52/s54\n/s48 /s53/s48 /s49/s48/s48/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s51/s48\n/s50/s56\n/s50/s54\n/s50/s52\n/s50/s50\n/s50/s48\n/s49/s56\n/s49/s54/s102/s32/s32/s40/s71/s72/s122/s41\n/s107 /s32/s40/s114/s97/s100/s47 /s109 /s41/s40/s97/s41\n/s50/s50/s50/s52/s50/s54/s50/s56/s51/s48\n/s50/s48/s40/s98/s41/s83\n/s118\n/s107 /s32/s40/s114/s97/s100/s47 /s109 /s41\nFIG. 9: (a) Variation of the frequency non-reciprocity ∆ f=\nf(−|k|)−f(|k|) of the fundamental surface wave mode as\na function of the wave vector kin Fe/Py bilayer films with\nthickness Tvaryingfrom 16 nm (bottommost curve) to 30 nm\n(topmost curve). All bilayers are such that tFe/T=0.5. (b)\nVariation of the asymmetry parameter Svwithkin homoge-\nnous films of thickness Tvarying from 20 nm (bottommost\ncurve) to 30 nm (topmost curve). The films are supposed to\nbe made of a hypothetical material with mean magnetic pa-\nrameters A= 15.5 pJ/m and MS= 1250 kA/m. The data\nshown in panels (a) and (b) do not correspond one-to-one.\nThey have been chosen so as to best reveal the similitude\nbetween the behaviors of the two quantities, ∆ fandSv.15\n-40 -20 0 20 40 12 16 20 24 28 \n-40 -20 0 20 40 812 16 20 24 0.5  \n 0.6 0.7 0.8 \n(c) (d) (b) (a) \nk (rad/ !m)   5 mT 50 mT 100 mT -100 0 100 10 20 30 40 \"#$%  (GHz) \n \n-100 0 100 10 20 30 40 50 \"/2 % (GHz) \nk (rad/ !m) n = 0 n = 1 \nFIG. 10: Dispersion relations of surface wave modes in 34 nm\nthick films. (a) Fundamental mode in pure Fe (open squares)\nand Py (open diamonds) films, and in a Fe/Py bilayer with\ntFe/T= 0.6 (solid circles), in a transverse magnetic field\nµ0H0=50mT.(b)FundamentalmodesinFe/Pybilayerswith\nFe contents tFe/Tvarying from 0.5 to 0.8 ( µ0H0=50 mT). (c)\nFundamental mode in a Fe/Py bilayer with tFe/T=0.6 sub-\nmitted to different magnetic fields. (d) First two SW modes\nin a Fe/Py bilayer with tFe/T= 0.6 (µ0H0=50 mT).\ntry and ii) the larger the modal profile asymmetry the\nlarger ∆ f, for a given magnetic asymmetry. Thus, for\nT/greaterorequalslant30 nm, extremely large effects with ∆ fs of several\nGHz can be obtained. They are associated with disper-\nsion relations, which fall in between those of pure Fe and\nPy films of the same thickness [Fig. 10(a)], but end up\nbeing extremely asymmetric because each of their pos-\nitive and negative- kbranches tends to follow the ω(k)\ncurve of the material where the mode is more strongly\nlocalized: the negative- kbranch is pulled upwards, to-\nwards the ω(k) curve of Fe, while the positive- kbranch\nis pulled downwards, towardsthat of Py. As a byproduct\nofthisskewing, thedispersionrelationsofthickFe/Pybi-\nlayers quite systematically show a well defined frequency\nplateau, that is, a range of positive kvalues where the\ngroup velocity vg=∂ω\n∂kis close to zero. For spin-waves\nof the corresponding frequencies, effective propagation is\nonly possible with a negative wave vector, i.e., in the\n−udirection ( ω>0). The narrow frequency window in\nquestion can be widely tuned by changing the compo-\nsition of the film [Fig. 10(b)] or the magnitude of theexternal magnetic field [Fig. 10(c)]. Such an usual be-\nhavior might be useful in applications, for instance, to\nbuild narrow band microwave isolators. Of course, all\nthe non-reciprocity phenomena discussed above switch\nsign or invert when H0is reversed and the bilayer film is\nmagnetized along −w. As may be seen in Fig. 10(d), fre-\nquency non-reciprocities also qualitatively invert when\nmoving from the first SW mode ( n= 0) to the second\n(n=1).\nB. Inhomogeneously magnetized strips\nAs demonstrated recently using time-domain micro-\nmagnetic simulations14and even more recently through\nBrillouin light scattering experiments40, a magnetic do-\nmain wall can act as a magnonic waveguide. The rea-\nsons for this are essentially twofold. First, a domain wall\nquite systematically hosts a spin-wave mode, which is\nstrongly localized sidewise by the confining potential of\nthe magnetic texture but free to propagate in the direc-\ntion parallel to the wall. Second, this bound spin-wave\nmode lies in the energy gap of the usual extended (bulk)\nspin-wave modes and is therefore spectrally isolated, at\nleast at low k. This remarkable ability of domain walls\nto channel spin-waves provides an efficient solution to\nthe difficult problem of guiding spin-waves along curved\nand/or reprogramable paths14,40,41. It is believed that\nit could play a crucial role in the future development of\nmagnonic circuits.\nBelow, we use our dynamic matrix approach to study\ndomain-wall channelized spin-wave (DWCSW) normal\nmodes. We consider the case of Bloch walls formed\nin an hypothetical material with strong perpendicular-\nto-plane uniaxial magnetic anisotropy ( A= 15 pJ/m,\nMS=1 MA/m, Ku=1 MJ/m3,a=ev)14. We examine\nthe situation where a unique and straight wall runs along\nthe entire length of a magnetic strip, dividing it later-\nally in two oppositely magnetized domains [Fig. 11(a,b)].\nWe assume zero external magnetic field so that the wall\nsits at the center of the strip. For W≫b, the equilib-\nrium magnetic configuration follows Walker’s profile42,\ni.e.,zw=ez·ew= 0,zv=ez·ev=pvtanh/parenleftig\nw−w0\n∆w/parenrightig\n,\nzu=ez·eu=pusech/parenleftig\nw−w0\n∆w/parenrightig\n, wherew0=W/2 and ∆ w\nare the position and width of the domain wall, respec-\ntively,pvis the circulationnumber, which takes the value\n±1 depending on whether the wall is ”down-up” or ”up-\ndown”, and puis the polarity number, which amounts to\n±1 depending on whether Meqpoints along + uor−u\nat the domain wall center. As expected, we find that the\nspin-wavenormalmodeoflowestfrequencyinthisconfig-\nuration is a mode bound to the domain wall [Fig. 11(d)],\nwhereas the next one, lying at much higher frequency\n(ω/2π>20 GHz, see Fig. 11(c)), is a bulk-like mode with\nmaximum amplitude near the center of the magnetic do-\nmains and zero amplitude at the domain wall location\n[Fig. 11(e)].16\n0 128 256 -1 01-4 04\n(e) m0 (arb. units) \nw (nm)    |m0|              Re (m0v)\n   Re (m0u)               Im (m0w)(d) m0 (arb. units) -1 01ez(b) \n  zw  zv  zu\n-200 -100 0 100 200 020 40 60 \n(c)  !\"#   (GHz) \nk (rad/µm )$\n$ \nwv u\nW/2 \n(a) \nFIG. 11: (a) Schematic representation of a magnetic strip\nwith perpendicular-to-plane anisotropy containing a sing le\nBloch wall in its centre. (b) Variation of the equilibrium ma g-\nnetization direction ez(zi=ez·ei,i=u,v,w) across such a\nstrip, 1 nm thick and 256 nm wide. Results of numerical sim-\nulations (symbols) are compared to predictions of Walker’s\nanalytical model with pu=pv=+1 and ∆ w=6.1 nm (lines),\nsee text for details. (c) Dispersion relations of the propag at-\ning spin-wave normal modes of lowest and second lowest fre-\nquencies in the magnetic configuration shown in (b). (d),(e)\nw-profiles of these two modes at the points marked with sym-\nbols in (c), i.e., for k= +50 rad/ µm (x=−w).\nIn an infinite defect free magnetic medium, the disper-\nsion relation of the DWCSW mode bound to a unique\ndomainwallisgaplesssincetheenergycostofmovingthe\nwall as a block, which is what the DWCSW mode with\nk=0isallabout, iszero43. Here, agapisobservedwhose\nsize, ∆ DWCSW, increases with decreasing strip width W\n[Fig. 12(a)] and increasing strip thickness b[Fig. 12(b)].\nThis gap is a measure of the restoring force that brings\nthe domainwall backto its equilibrium position in caseit\nis shifted sidewards, which increases dipolar energy. As\ndemonstrated by the scalingof∆ DWCSWwith the inverse\nofW[Fig. 12(c)], the opening of the gap is a finite size\neffect, whichmay be viewedasoriginatingfrom the inter-\naction between the domain wall and the lateral edges of\nthe magnetic medium. We note in passing that the data\nin Fig. 12(c) provethe ability of our numericalmethod to\ndetermine accurately normal mode frequencies as small\nas 20 MHz.\nFrom Fig. 12(a,b), it is clear that the dispersion re-\nlation of the DWCSW mode bound to a Bloch wall is\nnot symmetric about k=0, even in a magnetic medium\nof thickness as small as 1 nm. The degree of asymme-\ntry at large kincreases with increasing thickness but\nis independent of the strip width. This suggests that0 4 8 12 16 0300 600 900 \n-40 -20 0 20 40 0.0 0.5 1.0 1.5 2.0 \n 1 nm \n 3 nm \n 5 nm \n(c)  DWCSW  (MH z) !\nW -1  (µm -1 ) p v = +1 \n p v = -1 \n(d)  \nk (rad/µm) \"#$%   (GHz) !-40 -20 0 20 40 0.0 0.5 1.0 1.5 2.0 \n(a)  128 nm \n 256 nm \n 1024 nm \"#$%   (GHz) \nk (rad/µm) -40 -20 0 20 40 0123\"#$%   (GHz) (b)  1 nm \n 3 nm \n 5 nm  \nk (rad/µm) \nFIG. 12: (a),(b) Dispersion relations of the DWCSW modes\nbound to single Bloch walls with pu=pv=+1 in magnetic\nstrips with differentwidth W(a) and thickness b(b). The val-\nues of the varying parameter are indicated in the legends. In\n(a), thethicknessis b=1nm. In(b), thewidth is W=512nm.\n(c) Variation of the frequency gap ∆ DWCSWwith the width\nof the strip Wfor different thickness b. The lines are linear\nfits. (d) Dispersion relations of the DWCSW modes bound to\nsingle Bloch walls with the same polarity but opposite circu -\nlations, in a 1 nm thick 512 nm wide strip: solid (resp. open)\nsymbols correspond to pv=+1 (resp. pv=−1). The line is\nthe prediction of Garcia-Sanchez et al. [Ref. 14], see text for\ndetails.\nthe asymmetry is intrinsic in the sense that its source\nis localized within the domain wall region. Interestingly,\nalso, the dispersion curveis transformedinto its symmet-\nric about the frequency axis when the circulation pvis\nchanged [Fig. 12(d)], but it is unaffected when the po-\nlaritypuis reversed. This shows that the asymmetry of\nthe dispersion curve is not directly linked to the domain\nwall chirality, since the latter obeys the same symmetry\nrules as pu×pv. Finally, we note that, as bapproches\nzero, only one of the two branches of the computed ω(k)\ncurve, either the positive k-branch or the negative k-\nbranch depending on pv, follows quite closely the rela-\ntionω(k) =/radicalbig\nωk(ωk+ω⊥), withωk=2|γ|Ak2/MSand\nω⊥=|γ|µ0NwMS, derived in Ref. 14 by treating the do-\nmain wall as a magnetic object with effective demagneti-\nzation factor Nw=b/(b+π∆w) along the perpendicular-\nto-wall axis w. Whether this is a coincidence or whether\nthere are good physical reasons for that is a question left\nto future investigations.17\nHere, unlike in other works14, no interaction\nthat produces a chiral symmetry breaking, like the\nDzyaloshinskii-Moriya interaction, is considered. There-\nfore, the non-reciprocal character of the spin-wave prop-\nagation must originate from dipole-dipole interactions,\nas in the case of surface waves [Sec. VA]. As a mat-\nter of fact, there exists a rather strong similitude be-\ntween a perpendicularly magnetized strip (with or with-\nout a Bloch wall) and a transversally magnetized film.\nIn both cases, indeed, the medium is magnetized in such\na way that Meqhas a (large) component in the plane\nperpendicular to the direction of spin-wave propagation,\nu, and, conversely, mpossesses a non-zero u-component.\nThis is the first necessaryingredient for observingdipole-\ninduced non-reciprocity since, as an examination of the\nmutual demagnetizing tensors [Eqs. 32 and 35] reveals,\nno dipolar coupling depending on the sign of kcan ever\nexist ifmu=0. For frequency non-reciprocity to occur, a\nsecondingredientisnecessary: themagneticsystemmust\nnot be mirror symmetric about its midplane normal to\neu×ez[Ref. 44]. If it is mirror symmetric, non-reciprocal\ndipolar couplings may play a significant role (they pro-\nduce asymmetric modal profile in the SW configuration)\nbut they cannot yield any difference in the frequency of\ncounter-propagating spin-waves as their average effect is\nquantitatively the same for both positive and negative k.\nHere, it is the very presence of the Bloch wall in the strip\nwhichbreakstheleft/rightsymmetryaboutthemidplane\nnormal to eu×ez=ew. With the wall sitting at the cen-\ntre of the strip, there exist no symmetry operation which\nchangeskinto−kwhile leavingthe equilibrium magnetic\nconfiguration unchanged. To some extent, the presence\nof the wall is equivalent to having MS>0 in one half of\nthe strip and MS<0 in the other.\nLet us examine in detail how non-reciprocal dipolar\ncouplings are affected when either the circulation or the\npolarity of the wall is changed. For this, we refer to\nFig. 13wherethe essentialfeatures ofthe DWCSWmode\n[Fig. 11(d)], as deduced from numerical simulations, are\nsketched : muandmwoscillate in quadrature; m0wis\nmaximum at the center of the wall whereas m0ushows\ntwo maxima of opposite signs located symmetrically on\neither side of the wall center; mvplays no decisive role.\nIn this figure, one sees that the dipolar field hw\ndu(blue\narrows) created by mw(solid red arrows) and acting\nonmu(open red arrows) reverses when the direction of\npropagation is reversed [Fig. 13(b)]. This is the essence\nof the non-reciprocity phenomenon, which is reflected in\nthe change of sign of nuwon reversing k[Eq. 41]. One\nalso sees that, as far as dynamic dipolar interactions are\nconcerned, changing the polarity of the wall [Fig. 13(c)]\nhas no effect since the relative orientation of muand\nhw\nduremains the same, whereas changing the circula-\ntion [Fig. 13(d)] is equivalent to reversing the direction\nof propagation (see grey boxes). This explains why the\ndispersion curves for pu=+1 and pu=−1 are identical\nand those for pv=+1 and pv=−1 are symmetric to each\nother [Fig. 12(d)]. u\nvw0+ /4 + /2 \n- /4 \n- /2 \n(a)\n(d) pv=-1 (b)\n(c) pu=-1 k <0      \n          \n     \nFIG. 13: Schematic representation of the DWCSW mode\nhosted by a Bloch wall along a full spin-wave wavelength.\nThe black arrows pointing in and out of the figure represent\nthe out-of-plane component of the equilibrium magnetizati on\nMeq. The open and solid red arrows show the uandw-\ncomponents of the dynamic magnetization, muandmw, re-\nspectively, as deduced from the numerically determined nor -\nmal modes. The solid blue arrows represent the dipolar field\ncreated by mwand acting on mu. (a) Reference case with\npu=pv=+1 and k>0. (b) Reversed direction of propagation\n(k <0). (c) Reversed polarity ( pu=−1). (d) Reversed cir-\nculation ( pv=−1). In all four cases, time tis such that mv\n(not shown) points out of the figure at the center of the wall\n(marked with a black dot), in u=0.\nVI. POSSIBLE EXTENSIONS OF THE MODEL\nA first possible extension of the model described in\nthis paper would consist in implementing more general\nand accurate boundary conditions39. Here, for the sake\nof simplicity, we have assumed so called free boundary\nconditions, which arise from the sole symmetry break-\ning of the exchange interactions at the surfaces of the\nmagnetic medium. Moreover, as written in Eqs. 7 and\n16, these conditions ( ∂Meq/∂n= 0 and ∂m/∂n= 0,\nwherenis the normal to the surface) are implemented\nin the crudest possible way: instead of using accuracy\npreserving expansions of the spatial derivatives for mag-\nnetic cells sitting at or close to the surfaces39, we sim-\nply forget altogether, in the expressions of the static and\ndynamic exchange fields based on second-order Taylor18\nexpansions, those pair-interaction-like terms of the form\nΛ2\nξ2(M(α−1)\neq−M(α)\neq) orΛ2\nξ2(m(α−1)−m(α)) which involve\nmissing magnetic cells ( α/lessorequalslant1 orα>N). We wish to em-\nphasize however that, with small enough cells, this crude\napproximation has very little influence on the computed\nmode profiles and usually none on the frequencies.\nEven with free boundary conditions, a number of sur-\nface phenomena not discussed above may be included\nin the model, especially in the film geometry. Sur-\nface anisotropies may be introduced as bulk anisotropies\npresent only in the magnetic cells sitting next to the\ntop and/or bottom surfaces. For small enough cells,\nthis is quite equivalent to introducing them through\nproper boundary conditions. Similarly, an interfa-\ncial Dzyaloshinskii-Moriya (DM) interaction, as result-\ning from a perpendicular-to-plane symmetry breaking45,\nmay also be included. To do so only requires to intro-\nduce a new contribution to the dynamic magnetic field\nexperienced by the cell(s) sitting next to the surface(s)\nsincethe DM interactiondoesnotcontributetothe static\neffective field Hequnder the assumption that the orien-\ntation of the equilibrium magnetization depends only on\nthev-coordinate. Startingfromthe expressionofthe DM\nenergy density given in Eq. 2 of Ref. 46 and taking intoaccount the plane wave nature of the spin-waves [Eq. 2],\none easily shows that this new contribution has the form\nh(α)\nDM=−i2D(α)\nµ0M2\nSk/parenleftbig\new×m(α)/parenrightbig\n, whereD(α)isthe contin-\nuous effective DM constant, in J/m2, possibly different\nat the top ( α=N) and bottom ( α=1) surfaces. In the\nxyzcoordinate system, this yields\n(α= 1,N)ex·h(α)\nDM=i2D(α)\nµ0M2\nSkw(α)\nzm(α)\ny(46a)\ney·h(α)\nDM=−i2D(α)\nµ0M2\nSkw(α)\nzm(α)\nx,(46b)\nwithw(α)\nz=T(α)\n33thez-coordinate of unit vector ew. We\nnote that introducing the same DM interaction in the\nstrip geometry can only be achieved by simultaneously\nadding a new contribution to Heqand implementing\nspecific exchange-DM boundary conditions at the strip\nedges47–49.\nTaking into account magnetic damping is another pos-\nsible extension of the method. With a damping torque of\nthe formproposedbyGilbert, i.e.,α∗\nMS(M×˙M) (damping\nconstant α∗), equation 3 becomes\nω/parenleftigg\nm(α)\nx\nm(α)\ny/parenrightigg\n=−i|γ|µ0\n1+α2∗/parenleftigg\nMS(h(α)\ny+α∗h(α)\nx)−H(α)\neq(m(α)\ny+α∗m(α)\ny)\n−MS(h(α)\nx−α∗h(α)\ny) +H(α)\neq(m(α)\nx−α∗m(α)\ny)/parenrightigg\n. (47)\nThis shows that the construction of the dynamic matrix\ndoes not require to evaluate new quantities, just to ar-\nrange those considered in the present work in a slightly\ndifferent manner. With damping included, the eigenfre-\nquencies become complex numbers and their imaginary\nparts are the inverses of the relaxation times ( T2) of the\nnormal modes. Together with the group velocity vgde-\nrived from the dispersion relation, T2yields the attenu-\nation length Latt=vgT2of a spin-wave mode, which is a\nparameter of great interest in magnonics.\nFinally, moving from a one-dimensional to a two-\ndimensional array of parallelepipedic cells would be the\nultimate extension. It would allow one to describe more\naccurately what happens in thick strips where the mag-\nnetic configuration and/or properties are also inhomo-\ngeneous through the thickness of the medium, not just\nacross its width. In practice, this would essentially re-\nquire to take into account not just two but four nearest\nneighboring cells in the expressions of the static and dy-\nnamic exchange fields.VII. CONCLUSION\nThe full recipe has been given fora finite-difference nu-\nmericalschemededicatedtothedeterminationofthenor-\nmal modes of spin-waves propagating as plane-waves in\nextended magnetic films and strips, in the linear regime.\nThe approach, based on the dynamic matrix method,\nheavily relies on the use of plane-wave (dynamic) de-\nmagnetization factors, for which mathematical expres-\nsions have been derived. As illustrated through two\nexamples in the paper, it is well suited to study mag-\nnetic media whose material parameters vary in space,\nlike multilayered films, or contain non-collinear micro-\nmagnetic textures such as magnetic domain walls. It\nwould allowone exploringspin-wavephysics in verycom-\nplex systems, which are doubly inhomogeneous (both in\ntheir magnetic parameters and in their equilibrium mag-\nnetic configuration) like, for instance, thin-film hard-soft\nexchange-spring magnets where planar domain walls can\nbe formed52.\nThe main limitation of the presented micromagnetic\nmodel resides in the assumption that the equilibrium\nmagnetic configuration is invariant along the direction of\nspin-wavepropagation. This makes the model unsuitable19\nfor studying how spin-waves propagate in the presence of\ncomplex magnetic microstructures which never fulfil this\ncondition, like crossties, vortices, or skyrmions. In such\nsituations, onewouldhavetoresorttousualtime-domain\nmicromagneticsimulationsortootherrecentlydeveloped\nspecificmethods53. Webelievethatthislimitationisam-\nply counterbalanced by the wealth of accurate informa-\ntion that can easily be obtained in situations where the\nmodel is applicable, which includes the spatial profiles,\nfrequencies, and dispersion relations of virtually all the\npropagating spin-wave modes. Besides, a way has been\noutlined to obtain yet even more micromagnetic infor-\nmation about these modes, by accounting for the effect\nof magnetic damping and thereby getting access to their\nrelaxation time and attenuation length.\nAppendix A: Parameters describing the precessional\nmotion of magnetization\nHereafter, we give mathematical expressions for the\nfour practical parameters that best describe the preces-\nsional motion of the magnetization in a given magnetic\ncellαas a function of the not-so-convenient complex am-\nplitudesm(α)\n0xandm(α)\n0y. Assuming u=0, the time trajec-\ntory of the true variable magnetization ˜m(α)= Re(m(α))\nof cellα, in the ( x,y) plane, is an ellipse [Fig. 14], whose\nparametric equations are\n/braceleftigg\n˜m(α)\nx(t) = Re(m(α)\n0x)cosωt−Im(m(α)\n0x)sinωt\n˜m(α)\ny(t) = Re(m(α)\n0y)cosωt−Im(m(α)\n0y)sinωt.(A1)\nComparing them to the general form for an ellipse cen-\ntered at the origin\n\n\n˜m(α)\nx(t) =a(α)cosϕ(α)cos(ωt+τ(α))\n−b(α)sinϕ(α)sin(ωt+τ(α))\n˜m(α)\ny(t) =a(α)sinϕ(α)cos(ωt+τ(α))\n+b(α)cosϕ(α)sin(ωt+τ(α))(A2)\nand introducing the intermediate variables\nη(α)\n±= Re/parenleftig\nm(α)\n0x/parenrightig\n±Im/parenleftig\nm(α)\n0y/parenrightig\n(A3a)\nζ(α)\n±= Re/parenleftig\nm(α)\n0y/parenrightig\n±Im/parenleftig\nm(α)\n0x/parenrightig\n,(A3b)\nwe find\na(α)=/parenleftigg/vextendsingle/vextendsingle/vextendsinglem(α)\n0/vextendsingle/vextendsingle/vextendsingle2\n+/parenleftig\nη(α)\n+2\n+ζ(α)\n−2/parenrightig1\n2/parenleftig\nη(α)\n−2\n+ζ(α)\n+2/parenrightig1\n2/parenrightigg1\n2\n√\n2\nand (A4a)\nb(α)=/vextendsingle/vextendsingle/vextendsinglem(α)\n0/vextendsingle/vextendsingle/vextendsingle2\n−η(α)\n+2−ζ(α)\n−2\n2a(α), (A4b)/s45/s49 /s48 /s49/s45/s49/s48/s49\n/s124/s98/s40 /s41/s40 /s41/s126\n/s109\n/s121\n/s109\n/s120/s126/s40 /s41\n/s97/s40 /s41\nFIG. 14: Time trajectory of the magnetization for m(α)\n0x=\n−0.51+0.72i,m(α)\n0y= 0.38−0.07i(u= 0,ω >0), which\ncorrespond to a(α)= 0.93,b(α)= 0.26,ϕ(α)=−0.33 rad,\nτ(α)= 2.28 rad. The red solid circle marks the position of\nthe magnetization vector at t=0 and the arrow indicates the\ndirection of precession, which is determined by the sign of\nωa(α)/b(α).\nfor the ellipse semi-axes a(α)>0 andb(α)(|b(α)|/lessorequalslanta(α)),\nϕ(α)=1\n2/bracketleftig\nArg/parenleftig\nη(α)\n−+iζ(α)\n+/parenrightig\n+Arg/parenleftig\nη(α)\n++iζ(α)\n−/parenrightig/bracketrightig\n,\n(A5)\nfor the tilt angle of the ellipse major axis with respect to\nthex-axis, and\nτ(α)=/braceleftbigg1\n2/bracketleftig\nArg/parenleftig\nη(α)\n−+iζ(α)\n+/parenrightig\n−Arg/parenleftig\nη(α)\n++iζ(α)\n−/parenrightig/bracketrightig\n+2πn|n∈Z/bracerightig\n. (A6)\nfor the phase of the precessional motion. We note that\nthis set of equations [Eqs. A4-A6] is not unique and that\nRe(m(α)\n0y)=Im(m(α)\n0x)=0 implies ϕ(α)=τ(α)=0 and vice\nversa. Also, while a tilt angle outside the range ( −π,π]\nwould bear no physical meaning, τ(α)can take on values\noutside this rangein orderto account forrelative changes\nof phase exceeding 2 πacross a mode profile. Such a sit-\nuation may indeed arise in specific circumstances, for in-\nstance, in the case of the DWCSW mode associated with\na N´ eel wall in an in-plane magnetized strip.\nAppendix B: Static mutual demagnetization factors\nof rectangular parallelepipeds with infinite length\nIn this appendix, we derive analytical expressions for\nthe static mutual demagnetization factors between par-\nallelepipedic magnetic cells with infinite length in the u-\ndirection and rectangular( b×c) crosssection in the ( v,w)\nplane. It is assumed that the source cell, which creates\nthe dipolar field, is centered in ( v,w) = (0,0), whereas\nthe target cell, which experiences it, is centered at the\nrelative coordinates ( δv,δw). The starting point of the20\ncalculation is the well-known expression of the magnetic\nfield created by a one-dimensional distribution of mag-\nnetic charges σ0δ(v−v0)δ(w−w0) with linear density\nσ0, parallel to axis u, that is,\nH1D(σ0,ρ) =σ0\n2πρeρ, (B1)\nwhereρ=/radicalbig\n(v−v0)2+(w−w0)2is the radial distance to\nthe line of charges and eρ=v−v0\nρev+w−w0\nρewis a unit\nvector in the radial direction.If the source cell is saturated along w, surface charges\n±MSarecreatedonits verticalfaces. The strayfieldpro-\nduced is obtained by integrating H1Doverv0∈[−b\n2,+b\n2],\nwithσ0=±MSdv0inw0=±c\n2. Then two more integra-\ntions over v∈[δv−b\n2,δv+b\n2] andw∈[δw−c\n2,δw+c\n2] are\nnecessary to calculate its average value over the volume\nof the target cell, Hd. Finally, the demagnetization fac-\ntorNiw(withi=u,v,w) may be identified as the factor\nthat makes the i-component of Hdequal to −NiwMS.\nThe full calculation is long but straightforward. It yields\nNww(δv,δw) =1\n2πbc1/summationdisplay\nn=−11/summationdisplay\nm=−1(2−3|n|)(2−3|m|)/braceleftbigg\n(δv+nb)(δw+mc) arctan/parenleftbiggδv+nb\nδw+mc/parenrightbigg\n+(δv+nb)2−(δw+mc)2\n4ln/bracketleftbig\n(δv+nb)2+(δw+mc)2/bracketrightbig/bracerightbigg\n(B2)\nand\nNvw(δv,δw) =1\n4πbc1/summationdisplay\nn=−11/summationdisplay\nm=−1(2−3|n|)(2−3|m|)/braceleftbigg\n(δv+nb)2arctan/parenleftbiggδw+mc\nδv+nb/parenrightbigg\n+(δv+nb)(δw+mc) ln/bracketleftbig\n(δv+nb)2+(δw+mc)2/bracketrightbig\n+ (δw+mc)2arctan/parenleftbiggδv+nb\nδw+mc/parenrightbigg/bracerightbigg\n. (B3)\nNuwand, more generally, all Nuielements ( i=u,v,w)\nare nil since H1Dhas no component along u. From\nEq. B3, one may see that Nvwis also nil as soon as either\nδvorδwis zero.\nTheNivelements can be calculated in a similar man-\nner, by assuming that the source cell is saturated along\nvand that surface charges ±MSare therefore created on\nits horizontal faces v0=±b\n2. Alternatively, they can also\nbe deduced by using the intrinsic properties of the de-\nmagnetizing tensor, namely the fact that it is symmetric\nand that its trace equals the fraction of the volume of\nthe source cell which overlaps that of the target cell20.\nFor totally disjoint rectangular parallepipeds with infi-\nnite length, this means Niu=Nui= 0 (i=u,v,w),\nNwv=Nvw, andNvv=−Nww.\nAppendix C: Magnetic field from a one-dimensional\nharmonic distribution of magnetic charges.\nOur goal here is to derive an analytical expression for\nthe magnetic field h1Dcreated by a one-dimensional har-\nmonic distribution of magnetic charges parallel to axis\nuand located at the transverse position ( v0,w0), as de-\nfined by Eq. 33. We start by looking for the correspond-\ning magnetostatic potential φ1D, which obeys Laplace’sequation ∆ φ1D= 0 everywhere in space but at the posi-\ntion of the line of charges. To solve this problem, cylin-\ndrical coordinates ( ρ,θ,u) are more appropriate than the\ncartesian coordinates ( u,v,w). Moreover, the charge dis-\ntribution is such that the solution is expected to be of\nthe form\nφ1D(ρ,u,t) =˜φ1D(ρ)ei(ωt−ku), (C1)\nwhereρ=/radicalbig\n(v−v0)2+(w−w0)2is once again the radial\ndistance to the line of charges. Introducing this trial so-\nlution into Laplace’sequation and performingthe change\nof variable ǫ=kρ, we find that ˜φ1Dmust obey\nǫ2∂2˜φ1D\n∂ǫ2+ǫ∂˜φ1D\n∂ǫ−ǫ2˜φ1D= 0.(C2)\nGeneralsolutionstoEq.C2arelinearcombinationsofthe\nzero-th order modified Bessel functions of the first ( I0)\nand second ( K0) kinds. However, I0cannot be part of a\nphysicalsolutionsinceit divergeswhen itsargumentgoes\nto both positive infinity and negative infinity. As for K0,\nit takes on complex values for negative real arguments\nanddivergesatnegativeinfinity. Then themagnetostatic\npotential φ1Dmust be of the form\nφ1D(ρ,u,t) =AK0(|k|ρ)ei(ωt−ku)(C3)21\nand the magnetic field deriving from it must write\nh1D(ρ,u,t) =−∇φ1D(ρ,u,t)\n=Akei(ωt−ku)\n×[iK0(|k|ρ)eu+sgn(k)K1(|k|ρ)eρ],\n(C4)\nwhereK1(ǫ) =−∂K0(ǫ)\n∂ǫis the first-order modified Bessel\nfunction of the second kind and eρ=v−v0\nρev+w−w0\nρew,\nas before.\nTo determine the unknown prefactor A, we may use\nGauss’s theorem. To this end, we construct a Gauss\nvolume consisting of a cylinder of radius R, length L,\nand axis merged with the line of magnetic charges. This\ncylinder is bounded by three surfaces: The two circular\nend surfaces denoted Σ 1and Σ 3, and the lateral sur-\nface called Σ 2. If we make the cylinder radius tend to\nzero, the flux of h1Dthrough Σ 1and Σ 3vanishes be-\ncause lim\nR→0[|k|RK0(|k|R)] = 0, whereas the flux of h1D\nthrough Σ 2is\nΦΣ2= lim\nR→0‹\nΣ2(eρ·h1D)dΣ2\n= lim\nR→0ˆu+L\nuˆ2π\n0A|k|RK1(|k|R)ei(ωt−ku)dθdu\n= 2πAˆu+L\nuei(ωt−ku)du, (C5)\nusing lim\nR→0[|k|RK1(|k|R)] = 1. Equating Φ Σ2with the\ntotal magnetic charge contained in the cylinder\nQM=σ0ˆu+L\nuei(ωt−ku)du, (C6)\nwe readily find\nA=σ0\n2π. (C7)\nAppendix D: Test of the plane-wave demagnetizing\ntensor approach\nIn order to demonstrate the correctness of our theo-\nretical results concerning the plane-wave demagnetizing\ntensor of magnetic cells having the shape of extended\nslabs [Sec. IVA], normal mode profiles computed for ho-\nmogeneous extended films in the purely magnetostatic\n(exchange-free) case have been compared to predictions\nof the exact analytical model developed by Damon and\nEshbach50,51. For the lowest-order even backward vol-\nume wave mode ( ez=eu), this model adapted to our\ngeometry [Fig. 1(a)] and conventions predicts\nm0v=−iφ0νχkvcos(kv(v−T/2)) (D1a)\nm0w=−φ0κkvcos(kv(v−T/2)),(D1b)/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s45/s49/s48/s49\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s45/s49/s48/s49\n/s107 /s32/s60/s32/s48/s107 /s32/s62/s32/s48 /s40/s97/s41 /s40/s99/s41/s73/s109/s40 /s109\n/s48 /s118/s41/s44/s32/s82/s101/s40 /s109\n/s48 /s119/s41/s32/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s118 /s32/s40/s110/s109/s41/s40/s98/s41/s40/s100/s41/s82/s101/s40 /s109\n/s48 /s117/s41/s44/s32/s73/s109/s40 /s109\n/s48 /s118/s41/s32/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s118 /s32/s40/s110/s109/s41\nFIG. 15: Profiles of the lowest-order even backward vol-\nume wave mode (a,b) and surface wave mode (c,d) with\n|k|= 60 rad/ µm, in a 40 nm thick film with A= 0 and\nMS= 800 kA/m ( µ0H0= 0.1 T). Symbols and lines corre-\nspond to results of our numerical approach ( b= 0.5 nm) and\npredictions of the Damon-Eshbach analytical model [Eqs. D1\nand D3], respectively. The complex amplitudes m0u(blue\ndiamonds), m0v(red circles), and m0w(black squares) are\nshown for both k >0 (a,c) and k <0 (b,d).\nwhereφ0is a constant which depends on the normaliza-\ntion conditions, ν=sgn(k), and\nχ=ωMωH\nω2\nM−ω2, κ=ωMω\nω2\nM−ω2, kv=−k2\n1+χ,(D2)\nwithωM=|γ|µ0MSandωH=|γ|µ0H0. For the surface\nwave mode ( ez=ew), the model yields\nm0u=φ0|k|(νχ−κ)e|k|v+p(ν)(νχ+κ)e−|k|v\nA(ν)\n(D3a)\nm0v=iφ0|k|(νχ−κ)e|k|v−p(ν)(νχ+κ)e−|k|v\nA(ν),\n(D3b)\nwhere\np(ν) =χ+2−νκ\nχ+νκ(D4)\nand\nA(ν) =/braceleftigg\n2e|k|T/2ν= 1\ne−|k|T/2/bracketleftbig\n(χ+2+κ)e2|k|T−(χ+κ)/bracketrightbig\nν=−1.\n(D5)\nFigure 15 shows the mode profiles calculated using the\nabove two sets of analytical expressions (lines), Eqs. D1\nand D3, together with results of our numerical ap-\nproach (symbols), for a particular wave vector value\n|k|=60 rad/ µm. The two types of data match each other\nperfectly, for both surface and volume waves. Since such\na test is rather demanding, we may conclude that dipole-\ndipole interactions are correctly accounted for by using\nthe dynamic demagnetizing tensors derived in Sec. IVA.22\n/s48/s49\n/s48/s49\n/s49/s48/s45/s50\n/s49/s48/s45/s49\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50/s45/s48/s46/s48/s56/s48/s46/s48/s48/s48/s46/s48/s56\n/s49/s48/s45/s49\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51/s45/s48/s46/s48/s54/s48/s46/s48/s48/s48/s46/s48/s54/s82/s101/s40 /s110\n/s117/s117/s41\n/s82/s101/s40 /s110\n/s118/s118/s41/s40/s97/s41 /s40/s99/s41/s82/s101/s40 /s110\n/s117/s117/s41/s44/s32/s82/s101/s40 /s110\n/s118/s118/s41/s44/s32/s82/s101/s40 /s110\n/s119/s119/s41/s44/s32/s73/s109/s40 /s110\n/s119/s117/s41\n/s98 /s47/s99/s40/s98/s41/s82/s101/s40 /s110\n/s119/s119/s41\n/s73/s109/s40 /s110\n/s119/s117/s41\n/s40/s100/s41/s82/s101/s40 /s110\n/s117/s117/s41/s44/s32/s82/s101/s40 /s110\n/s118/s118/s41/s44/s32/s82/s101/s40 /s110\n/s119/s119/s41/s44/s32/s73/s109/s40 /s110\n/s119/s117/s41\n/s107 /s32/s40/s114/s97/s100/s47/s181/s109/s41\nFIG. 16: (a) Self and (b) mutual ( wα−wβ= +2c) dynamic\ndemagnetization factors ofparallelepipedic magnetic cel ls ver-\nsus cell aspect ratio ( b= 10 nm), as calculated numerically for\nk= 10−12rad/µm. (c) Self and (d) mutual ( wα−wβ=−3c)\ndynamic demagnetization factors of parallelepipedic cell s ver-\nsus wave vector, as calculated numerically for b= 200µm\nandc=20 nm. Data computed using the integral expressions\nderived in Sec. IVB (symbols) are compared to analytical re-\nsults (lines) for (a,b) k= 0 [Eqs. 11-13] and (c,d) b→+∞\n[Eqs. 25, 26, and 30, with bandvreplaced with candw].\nIn the case of parallelepipedic magnetic cells [Sec. VB]\nand of spin-wave medium having the shape of a strip,\nsuch demanding tests as reported above for films could\nnot be performed since fully analytical theories are not\navailable, which could be used for comparison. The onlytests we could devise consist in examining limiting cases.\nA first natural test is to check that the dynamic de-\nmagnetization factors defined by the integral expressions\nEqs. 37, 39, and 41 behave properly when the wave vec-\ntorktends to zero. Figure 16(a,b) shows that this is\nindeed the case: all factors become equal to their static\ncounterparts given by Eqs. 11-13. Another possibility is\ntoinvestigatewhat happens whenthe height bofthe cells\nbecomes much larger than both the cell width cand the\nspin-wave wavelength λ=2π/|k|. Figure 16(c,d) shows\nthat, as expected, the dynamic demagnetization factors\nare then very close to those of extended slabs and obey\nthe analytical expressions derived in Sec. IVA (Eqs. 25,\n26, and 30 with bandvreplaced with candw, respec-\ntively). Although these two tests are not as stringent\nas those performed for the film geometry, they support\nour claim that dipolar interactions can also be well de-\nscribed by dynamic demagnetization factors in the strip\ngeometry.\nAcknowledgments\nThe authors thank Joo-Von Kim, Felipe Garcia-\nSanchez,RiccardoHertel, andAndr´ eThiavilleforfruitful\ndiscussions, and acknowledge financial support from the\nFrenchNationalResearchAgency(ANR) underContract\nNo. ANR-11-BS10-0003 (NanoSWITI). O. G. thanks\nIdeX Unistra for doctoral funding.\n∗Electronic address: yves.henry@ipcms.unistra.fr\n1S. Neusser and D. Grundler, Adv. Mater. 21, 2927 (2009).\n2V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D: Appl. Phys. 43, 264001 (2010) and references\ntherein.\n3B. Lenk, H. Ulrichs, F. Garbs, M. M¨ unzenberg, Phys. 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V.\n28When the magnetization precesses in phase all along a\nmode profile, it is always possible to make the condition\nτ(α)=0 true for all α. Should it not be automatically ful-\nfilled by the numerical routine used for diagonalizing the\ndynamic matrix, it can always be realized afterwards, by\nmultiplying all complex amplitudes m(α)\n0xandm(α)\n0yby an\nappropriate eiθfactor. This does not change the physi-\ncal nature of the eigenmodes since these are defined up\nto a constant scalar multiplier. Making ϕ(α)= 0, on the\nother hand, is a matter of choosing adequate local refer-\nence frames (which are free to rotate about axis z) such\nthat axis xalways corresponds to the major axis of the\nelliptical trajectory followed by M(α). For that, a good\nchoice is usually to have axis xlying in the plane of the\nmagnetic medium.\n29W. F. Brown, Jr., Magnetostatic Principles in ferromag-\nnetism(North Holland, Amsterdam, 1962), Appendix.\n30A. Aharoni, J. Appl. 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R. Eshbach, J. Phys. Chem. Solids\n19, 308, (1961).\n51D. D. Stancil and A. Prabhakar, Spin waves, theory and\napplications (Springer, 2009), Chapter 5.\n52J.McCord, Y.Henry,T.Hauet, F.Montaigne, E.E.Fuller-\nton, S. Mangin, Phys. Rev. B 78, 094417 (2008)\n53F. J. Buijnsters, A. Fasolino, M. I. Katsnelson, Phys. Rev.\nB89, 174433 (2014)."    },    {        "title": "1611.02847v1.Simultaneous_imaging_of_strain_waves_and_induced_magnetization_dynamics_at_the_nanometer_scale.pdf",        "content": "Simultaneous imaging of strain waves and induced magnetization\ndynamics at the nanometer scale\nMichael Foerster*,1Ferran Maci\u0012 a*,2, 3,\u0003Nahuel Statuto,3Simone Finizio,4, 5\nAlberto Hern\u0013 andez-M\u0013 \u0010nguez,6Sergi Lend\u0013 \u0010nez,3Paulo V. Santos,6Josep\nFontcuberta,2Joan Manel Hern\u0012 andez,3Mathias Kl aui,4and Lucia Aballe1\n1ALBA Synchrotron Light Source, 08290 Cerdanyola del Valles, Spain\n2Institut de Ci\u0012 encia de Materials de Barcelona (ICMAB-CSIC),\nCampus UAB, 08193 Bellaterra, Spain\n3Dept. of Condensed Matter Physics,\nUniversity of Barcelona, 08028 Barcelona, Spain\n4Institut f ur Physik, Johannes Gutenberg Universit at Mainz, 55099 Mainz, Germany\n5Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland\n6Paul-Drude-Institut fur Festk orperelektronik,\nHausvogteiplatz 5-7, 10117 Berlin, Germany\n(Dated: October 8, 2018)\n1arXiv:1611.02847v1  [cond-mat.mes-hall]  9 Nov 2016Abstract\nChanges in strain can be used to modify electronic and magnetic properties in crystal\nstructures1,2, to manipulate nanoparticles and cells3, or to control chemical reactions4. The\nmagneto-elastic (ME) e\u000bect|the change of magnetic properties caused by the elastic deforma-\ntion (strain) of a magnetic material|has been proposed as an alternative approach to magnetic\n\felds for the low power control of magnetization states of nanoelements since it avoids charge cur-\nrents, which entail ohmic losses. Multiferroic heterostructures5and nanocomposites have exploited\nthis e\u000bect in search of electric control of magnetic states, mostly in the static regime. Quantitative\nstudies combining strain and magnetization dynamics are needed for practical applications and\nso far, a high resolution technique for this has been lacking. Here, we have studied the e\u000bect of\nthe dynamic strain accompanying a surface acoustic wave on magnetic nanostructures. We have\nsimultaneously imaged the temporal evolution of both strain waves and magnetization dynamics of\nnanostructures at the picosecond timescale. The newly developed experimental technique, based\non X-ray microscopy, is versatile and provides a pathway to the study of strain-induced e\u000bects\nat the nanoscale. Our results provide fundamental insight in the coupling between strain and\nmagnetization in nanostructures at the picosecond timescale, having implications in the design of\nstrain-controlled magnetostrictive nano-devices.\n2Magnetization states in magnetic materials are fundamental building blocks for con-\nstructing memory, computing and further communication devices at the nanoscale. Static\nstates such as magnetic domains are being used in non-volatile memories6, whereas dynamic\nexcitations|spin-waves|might serve to transmit signals and encode information in future\nelectronic devices7. Collective magnetization states, which result from electron exchange\ncoupling, are traditionally modi\fed through magnetic \felds created with electrical currents,\ngiving rise to heat dissipation and stray \felds. The spin-transfer-torque e\u000bect8{10, which can\nbe originated from pure spin currents o\u000ber promising pathways towards the control of mag-\nnetic states at the nanoscale without using magnetic \felds. Another promising strategy for\nhandling high-speed magnetic moment variation at the nanoscale together with low-power\ndissipation is the use of electric \felds. Although direct e\u000bects of electric \felds on magnetic\nstates are weak, electric \felds can be used to induce strain and elastic deformations in a\nnanoscale magnetic material that might result in changes of magnetic properties as shown\nmostly by static experiments11{15.\nSurface acoustic waves (SAWs) are propagating strain waves that can be generated\nthrough oscillating electric \felds at the surface of piezoelectric materials. SAWs have been\nused to induce magnetization oscillations in magnetic materials and to achieve assisted rever-\nsal of the magnetic moment16{21. However, SAW induced magnetization dynamics is mostly\ntreated as an e\u000bective variation in the magnetic energy, providing thus little information\nregarding the physical coupling between phononic and magnetization modes.\nIn this letter we report an experimental study that probes the dynamic coupling of SAWs\nwith the magnetization dynamics of nano-elements. The study provides a simultaneous di-\nrect observation of both strain waves and magnetization modes with high spatial and tempo-\nral resolution. Our technique combines time and spatially resolved X-ray magnetic circular\ndichroism (XMCD)22and Photoemission Electron Microscopy (PEEM)23. While XMCD\nprobes the magnetization, the low-energy electrons detected by PEEM yield information on\nthe piezo-electric potential caused by the strain wave in the piezoelectric substrate, thus\nproviding a local measurement of strain strength. Stroboscopic XMCD and PEEM images\nsynchronized with the SAWs allow us to correlate the local changes in magnetization with\nthe spatial variation of the strain \feld.\nA schematic plot of the measurement is shown in Fig. 1. Micrometric Nickel (Ni)\nsquares were deposited onto piezoelectric LiNbO 3substrates containing interdigital trans-\n3ducers (IDTs) for the excitation of SAWs. The IDTs were designed to launch SAWs of a\nfrequencyfSAW= 499:654 MHz at room temperature, which is exactly the repetition rate\nof X-ray bunches at the ALBA Synchrotron in multibunch mode. By using an electronic\nphase locked loop (PLL) between the synchrotron master clock and the rf-excitation signal\napplied to the IDT, we achieved phase synchronization between the SAW and the X-ray\nlight pulses illuminating the sample24(see, Methods and Supplementary Materials). For\neach phase delay between the SAW and the X-ray pulses, we recorded PEEM images that\nprovided magnetic contrast of the sample surface through the XMCD e\u000bect. These strobo-\nscopic measurements allowed us to reconstruct the strain wave propagation and its e\u000bect on\nthe magnetic structures with a time resolution of \u001980 ps.\nIn Fig. 2A we show a PEEM image with a \feld of view of 50 \u0016m containing Ni squares of\n2\u00022\u0016m2in presence of SAWs. We observe bright and dark stripe lines with the periodicity\nof the SAW excitation (wavelength, \u0015SAW\u00198\u0016m). The SAW produces a contrast in\nthe PEEM images because the piezoelectric voltage associated with the wave shifts the\nenergy of the secondary electrons that leave the sample surface. Imaging with a \fxed phase\ndelay and a slightly detuned (sub Hz) SAW frequency con\frmed the SAW propagation\ndirection by direct observation of the displacement of the stripes in consecutive PEEM\nimages (see, videos in Supplementary Material). Figure 2B shows the number of secondary\nelectrons (photoemission intensity) as a function of the electron kinetic energy, recorded\nby our detector at two surface areas corresponding to opposite phases of the wave, cf. the\narrows in Fig. 2A. The energy shift between the two spectra corresponds to the peak-to-peak\namplitude of the SAW-induced piezoelectric potential added to the 10 keV applied at the\nsample surface for PEEM detection (2.6 V for the rf-power used in Fig. 2A). A schematic\nplot of the piezoelectric SAW is presented in Fig. 2C showing the intensity of the strain\nmodulation in the x-zplane of a SAW propagating along x. Strain arises from the spatial\nvariation of the displacements; the SAW decays exponentially with depth with a decay length\nof the order of the SAW wavelength25,26. Figure 2C shows, as well, the electric \feld (blue\narrows) corresponding to the modulation along xof the piezoelectric voltage at the sample\nsurface (dashed blue line).\nThe measurement of the amplitude of the surface electric potential associated with the\nSAW allows for a quanti\fcation of the strain applied to the Ni nanostructures (see, Methods).\nWe plot in Fig. 2D the oscillation along xof the piezoelectric potential with a peak-to-peak\n4amplitude of 2.6 V (blue dashed curve) measured in Fig. 2A, together with its correspond-\ning calculated longitudinal in-plane strain component, Sxx(solid curve), which is the strain\ncomponent responsible for the variations of the in-plane magnetic anisotropy in our struc-\ntures. We notice that the piezoelectric potential, and therefore the out-of-plane electric \feld,\nEz=\u0000@z\u001eSAW, is in phase with Sxx. Once we have shown that PEEM images provide a\ndirect visualization of the surface potential associated to the SAW and thus a quanti\fcation\nof the dynamic strain, we now focus on the response originated by the SAW on the magnetic\nstructures.\nThe intensity of the XMCD images is proportional to the component of the Ni magneti-\nzation along the X-ray incidence direction, represented in intensity gray scale. In order to\nquantify the ME-induced anisotropies, we chose polycrystalline Ni squares of 2 \u00022\u0016m2size\nand 20 nm thickness, having a four-domain Landau \rux-closure state15(see, images in Fig.\n3). We \frst studied samples with the Ni squares' sides aligned with the SAW propagation\ndirection. Figure 3A shows a temporal reconstruction of the e\u000bect of a SAW on a single Ni\nsquare; we plotted the direct PEEM images (top row panel) to see the SAW propagation and\nthe PEEM/XMCD images (lower row panel) to show the magnetic domain con\fguration.\nThe dynamical process of the magnetization is precisely observed in this \fgure where images\ncorrespond to intervals of 333 ps (1/6 of the SAW period): gray domains are \frst favored\n(magnetization perpendicular to SAW propagation) whereas black and white domains grow\nat larger values of the phase (magnetization aligned with the SAW propagation).\nWe have analyzed the response to SAW of domain con\fgurations from multiple squares\nwithin the same piezoelectric substrate by acquiring 20 \u0016m2size XMCD images at di\u000berent\nphase delays between SAW and X-ray pulses. At each phase delay, we calculated the area\noccupied by black and white domains in each square. This corresponds to the total area\nwith magnetization oriented along the xdirection, which is the one modulated by the SAWs.\nFigure 3B shows a summary of the obtained values as a function of the SAW phase acting\non each square (measured from the PEEM image). The ensemble of points obtained from all\nanalyzed squares is well \ftted by a sinusoidal function (red curve) with the same periodicity\nas the one of the SAW (green shadowed curve). We can translate the variations in the\nmagnetic-domain con\fguration observed in the XMCD images into variations of magnetic\nanisotropy by means of micromagnetic simulations (see, Supplementary Material). In the\ncase shown in Fig. 3B, the oscillation of the domain areas is well reproduced by a strain-\n5induced modulation of the magnetic anisotropy of amplitude kME,ac\u00191 kJ/m3superimposed\nto a preexisting uniaxial anisotropy of kU\u00191:2 kJ/m3caused by the deposition process. The\nstatic value of the preexisting uniaxial anisotropy has been con\frmed by direct ferromagnetic\nresonance spectroscopy (FMR) measurements on the same \flms. We also estimated the in-\nplane strain at the surface of the substrate from the SAW piezoelectric potential as described\nin Fig. 2. For the experimental values reported in Fig. 3B, the corresponding value of the\nstrain modulation is Sxx= 4:5\u000210\u00004.\nFrom the correlation between in-space variations of strain and variation of magnetic\nanisotropy, we obtained a value for the parameter \f=kME,ac=Sxx= 2:2\u0002106J/m3at 500\nMHz. The value of this ME coupling coe\u000ecient is similar to the reported values measured\nwith static strain15. We expected the value \fto be similar to the static case because\nstrain-induced changes of magnetic anisotropy are related to the modi\fcations of electron\norbitals and thus these electronic properties must respond much faster than the 500 MHz\nstrain oscillation used in our experiment. However, we observe in Fig. 3B (and also in\n3A) a considerable delay between the magnetization oscillation and the strain wave that\namounts to\u0019270 ps (phase delay of \u001950 deg), which cannot be attributed to a delay\nin the sound propagation from the LiNbO 3substrate to the Ni structures. Phase delays\ncan be expected if induced excitations (SAW) are coupled to internal resonances of the\nsystem27. We have identi\fed three di\u000berent magnetic resonance processes in the studied\nsample con\fguration with micromagnetic modeling (see, Supplementary Material), which\ncorrespond to i)precessional motion within the magnetic domain, ii)domain-wall precession,\nand iii)vortex motion. We found that domain-wall resonances have excitation frequencies\nsimilar to the SAW excitation frequency and thus there might be a coupling between them\nthat causes the delay.\nFinally we explore a judiciously selected di\u000berent geometry consisting of Ni squares ro-\ntated 45 degrees with respect to the SAW propagation direction in order to avoid the e\u000bect\nof domain wall and vortex resonances. Such con\fguration has four magnetic domains en-\nergetically equivalent with respect to the uniaxial varying anisotropy and thus no domain\ngrowth (shrinking) or domain-wall displacement occur; instead there is a coherent rotation\nof the magnetization within each of the domains. A study of this second con\fguration using\nNi squares of the same size and thickness is plotted in Fig. 3C. Indeed, in this con\fguration\nthe magnetic response is much faster, showing a sizable decrease in the delay between the\n6SAW and the magnetization oscillation, from 270 ps down to about 90 ps (phase delay of \u0019\n15 deg).\nIn summary, we have resolved simultaneously at the nanometer scale the strain caused\nby a SAW of\u0019500 MHz and the response of magnetic domains by using XMCD-PEEM\nmicroscopy, unveiling the dynamic response of the magneto-elastic e\u000bect. We found that\nmanipulation of magnetization states in ferromagnetic structures with SAW is possible at the\npicosecond scale with e\u000eciencies as high as for the static case. The magnetization dynamics\nis governed by the intrinsic con\fguration of the magnetic domains and by their orientation\nwith respect to the SAW-induced strain, which has to be considered in the design of the\nmagnetic devices. 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Sune, Proceedings of ICALEPCS , 1423 (2012).\n29\\Roditi international, http://www.roditi.com/singlecrystal/linbo3,\" (2016).\n30A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. VanWaeyen-\nberge, AIP Adv. , 107133.\n9FIGURES\n10 kV (PEEM)\n             +\n500 MHz (ac signal) X-rays\n500MHz\ne-\nIDT1 \nLiNbO3SAWNi Squares\nx\nzyPEEM Image XMCD Image \nIDT2 \nFrequency (MHz)S12(dB)\n460 480 500 520-90-50-3010\n540-70\nEz\n10 kV\nFIG. 1. Schematic plot of the experimental set-up. Circularly polarized X-rays illuminate the\nsample in the form of 20 ps pulses with a repetition rate of f0\u0019500 MHz. The IDT 1 of the\nhybrid device receives an AC electric signal of the same frequency, which is phase locked to the\nsynchrotron repetition rate, generating a piezoelectric surface acoustic wave (SAW) that propagates\nthrough the LiNbO 3substrate and interacts with the magnetic nanostructures. The phase-resolved\nvariation of the piezoelectric voltage at the surface sample is probed with the PEEM, as well as the\nmagnetization contrast along the X-ray propagation direction arising from the XMCD e\u000bect. Top-\nleft: PEEM image with a \feld of view of 20 \u000220\u0016m2shows four 2\u00022\u0016m2Ni squares in presence\nof a piezoelectric wave|black and white stripes indicate the sign of the piezoelectric voltage. Top-\nright: XMCD image of the same structures showing magnetic domain structures within the Ni\nsquares. Inset: Experimental rf-power spectrum of the transmission coe\u000ecient between IDT1 and\nIDT2,S21, tuned to have a maximum at f0.\n10Photoemission int. (Arb.u.)\nRela�ve Photoelectron Energy (V)2 0 8 4 6Photoelectron Energy = 4.8 V\nA\nB\nC\nDλsaw=8 micron\nλsawWave\npropaga�on\nλsaw\nLiNbO3Piezoelectric field\nSxx ϕ\nPiezoelectric voltage  \n ϕ (V)\nStrain(%) (x10-2)\n2 0 8 4 6\nx(μm)-0.50.5\n-1.51.0\n-1.01.5\n0.0 01\n-2-12\nx\nz\nEzPEEMFIG. 2. (A) PEEM image of multiple Ni squares of 2 \u00022\u0016m2with a \feld of view of 50 \u0016m recorded\nat a photoelectron energy of 4.8 V. The piezoelectric voltage produces periodic dark and bright\nzones in the PEEM image. (B) Two photoelectron energy scans of the photoemission intensity\ncorresponding to a bright and a dark zone (indicated in A). The vertical line indicates the electron\nenergy at which A was acquired. (C) Schematic of the in-plane strain induced modulation in the\nLiNbO 3caused by the acoustic wave. Blue arrows indicate the oscillating piezoelectric voltage\nassociated with the strain modulation. (D) Calculation of the in-plane strain component (right-\nhand-side axis) and the piezoelectric voltage (left-hand-side axis) at the sample surface for a SAW\nwith a 8\u0016m wavelength ( \u0015SAW). Calculations of the strain are done to match the measured\npiezoelectric voltage of Fig. 2A. The magnetization changes are driven basically by the in-plane\nstrain component along the SAW propagation direction. The piezoelectric voltage is in phase with\nthe in-plane strain component, albeit with an opposite sign.\n11B C Config 1: Domain-wall mo�on\nSAW\nX-raysConfig 2: Domain magne�za�on\n                 Rota�on \nX-raysSAW\nτ=48 deg\nPhase (deg)-90 0 90 180 360 270\n0.5\n0.6\n0.7\n0.8B & W frac�on of total area\nPhase (deg)-90 0 90 180 360 270\n-0.2\n0.0\n0.2\n0.40.6Normalized G2-G1\nτ=15 degB\nG1 G2W\nSAW propaga�on direc�onA\n30o90o150o210o270o330o\nphase\nFIG. 3. (A) PEEM (top row) and XMCD-PEEM (lower row) images of a 2 \u00022\u0016m2Ni square at\ndi\u000berent phases of the SAW. Images correspond to phase lapses of 60 deg (that correspond to 333\nps). PEEM images are 8 \u00028\u0016m2; XMCD-PEEM images are 4 \u00024\u0016m2. (B) Analysis of the domain\ncon\fguration from multiple (4) Ni squares of 2 \u00022\u0016m2as a function of the individual phase with\nrespect to the SAW for a con\fguration with square sides aligned with SAW propagation direction.\nSchematic plots of the e\u000bect of SAW on Ni squares shown in top panels. We computed the area\nof black and white domains (amount of magnetization along the X-ray incidence direction). (C)\nAnalysis of the domain con\fguration from multiple (9) Ni squares of 2 \u00022\u0016m2as a function of the\nindividual phase with respect to the SAW for a con\fguration with square sides rotated 45 degrees\nwith SAW propagation direction. Schematic plots of the e\u000bect of SAW on Ni squares shown in top\npanels. We computed the intensity of the relative normalized grey domains, (I(G1)-I(G2))/(I(W)-\nI(B)). A best \ft to the data with a sinusoidal function is plotted in red in both (B) and (C). A\nschematic strain wave is plotted in green (with no scale) to indicate the phase values corresponding\nto maximum and minimum strain corresponding to a response without delay.\n12METHODS\nSample fabrication and characterization. Unidirectional interdigital transducers (IDT)\nwere patterned with photolithography and deposited with electron beam evaporation (10 nm Ti j\n40 nm Alj10 nm Ti) on the 128 YX cut of LiNbO 3piezoelectric substrates. The transducers\ngenerate multiple harmonics of a fundamental frequency, f0, which was tuned such that the 4th\nharmonic matches the synchrotron frequency of 499.654 MHz. 20 nm thick Ni nanostructures\nwere de\fned with electron beam lithography and deposited by means of e-beam evaporation onto\nthe piezoelectric substrate and in the acoustic path between the two IDTs (see Fig. 1). The\nacoustic wave transmission from IDT1 to IDT2 was characterized both at atmospheric pressure\nwith pico-probes and in the UHV chamber in the beamline with a network analyzer. The Ni\n\flms were characterized with SQUID magnetometry to determine the saturation magnetization,\nMs= 490\u0002103A m\u00001and with FMR spectroscopy to determine the damping parameter and the\nuniaxial anisotropy, \u000b= 0:03 andkU= 1200 Jm\u00003.\nExperimental realization The experiments were performed at the CIRCE beamline of the\nALBA Synchrotron Light Source23. The beamline employs an Elmitec spectroscopic low-energy\nelectron and photoemission electron microscope (SpeLEEM/PEEM) operating in ultra-high vac-\nuum. In order to generate XMCD-PEEM images, two PEEM images at the energy of the Ni\nL3absorption edge are acquired with opposite photon helicity (circular polarization), and then\nsubtracted pixel-by-pixel to provide images with XMCD magnetic contrast as intensity. Samples\nwere mounted on in-house designed sample holders24and the IDT contacted with wire-bonds.\nIDT were several mm away from the sample center and the Ni nanostructures, thus allowing to\nscreen them from the high electric \feld of the objective lens by the raised sample holder cap. After\nintroduction in vacuum each sample was degassed at low temperature ( <100 C) for at least 1\nh in order to reduce the risk of arcs between sample (at high voltage) and microscope objective\n(at ground). A reduced acceleration voltage of 10 kV (standard is 20 kV) was used to further\nreduce the risk of discharges. The beamline intensity was adjusted in order to avoid excessive sur-\nface charging of the LiNbO 3substrate. For the synchronized excitation, the digital timing signal\nprovided by the ALBA timing system28was converted into a phase locked 499.654 MHz (referred\nto as 500 MHz throughout the text) analog signal with a Keysight EXG Vector signal generator\n13(model N5172B with option 1ER). The phase with respect to the master clock and the amplitude\nof the signal can be adjusted at this level as the experiment requires. The analog signal is then\ntransmitted by a custom optical \fber system into the PEEM high voltage rack and ampli\fed24.\nThe phase or temporal resolution depends on the size of the zone analyzed as the phase changes\nwith\u0015approx 8\u0016m. However, an upper limit of the total temporal smearing (electronic jitter\nplus photon distribution from bunch length and bunch dephasing) for a small enough zone can be\nderived from the sharpest possible step feature of the SAW that can be resolved, which corresponds\nto ca. 80 ps. This is a good value for time resolved PEEM, which is helped by the continuous\nelectrical AC excitation of a resonator structure (IDT), for which jitter is more easily controlled,\nbut also re\rects the quality of the ALBA beam in multi-bunch mode. All data presented was taken\nin thermal equilibrium. When the SAW were switched on, a slow (time scale of minutes up to\ntens of minutes) drift in the PEEM imaging showing small changes in the SAW wavelength from\nLNO surface was observed and indicated a change in temperature. Data was taken after a long\nperiod of thermalization and we compared snapshots at di\u000berent instants within the 2 ns SAW cycle.\nPiezoelectric voltage to strain conversion . We calculated the amplitude of the piezoelectric\npotential and strain tensor components by numerically solving the coupled di\u000berential equations\nof the mechanical and electric displacement for an acoustic wave propagating along the xdirection\nof a semi-in\fnite 128 Y-cut LiNbO 3substrate. To obtain surface modes, we looked for solutions\nthat decay towards z >0 and satisfy the stress and electric displacement boundary conditions at\nthe surface, z= 0. We have used the same SAW wavelength as the experiment, and have selected\npower density so that the amplitude of the simulated piezoelectric potential at z= 0 coincides\nwith the measured one29.\nMicromagnetic simulations . Numerical simulations were performed using a MuMax3\ncode30on a graphics card with 2048 processing cores. A full code is appended in the Supple-\nmentary Material. We considered a two-dimensional layer and integrated the Landau-Lifshitz-\nGilbert-Slonczewski equation to describe the magnetization dynamics. We computed di\u000berent\nsample geometries with a resolution of 4 nm in the grid size. Thermal e\u000bects are neglected. The\nparameters of the magnetic layer, a Nickel \flm, were taken from the sample characterization: sat-\nuration magnetization Ms= 490\u0002103A m\u00001, Gilbert damping constant \u000b= 0:03, and exchange\nconstant 10\u000011Jm\u00001. (The exchange constant does not a\u000bect signi\fcantly the simulations. The\n14damping constant plays a role for values larger than 0.1 inducing a delay of more than 5 deg. under\na SAW excitation of 500 MHz. Lower damping values do not a\u000bect neither the e\u000eciency nor the\ndelay times but sharpen the internal resonances). A \fxed uniaxial anisotropy is introduced in the\nsimulations with a value kU= 1200 J m\u00003with an additional oscillating term of kME;ac having\nthe wavelength set by the SAW ( \u0015SAW\u00198\u0016m); simulations covered frequencies from tens of MHz\nto tens of GHz.\nACKNOWLEDGEMENTS\nThe authors thank Jordi Prat for technical help on the beam line and with the data analy-\nsis, Hermann Stoll and Rolf Heidemann for advice on electronics, Abel Fontsere, Bernat Molas\nand Oscar Matilla from Alba electronics for the development of the 500 MHz synchronous exci-\ntation setup, and Werner Seidel from PDI for assistance in the preparation of the acoustic delay\nlines on LiNbO3. The project was supported by the ALBA in-house research program through\nIH2015PEEM and the allocation of in-house beamtime as well as with proposal 2016021647. FM\nacknowledges \fnancial support from the Ram\u0013 on y Cajal program through RYC-2014-16515. FM\nand JF acknowledge support from MINECO through the Severo Ochoa Program for Centers of\nExcellence in R&D (SEV-2015-0496). Funding from MINECO through MAT2015-69144 (JMH, NS\nand FM) and MAT2015-64110 (LA and MF) is acknowledged. SF and MK aknowledge Graduate\nSchool of Excellence Materials Science in Mainz (Grant No. GSC 266), the Swiss National Science\nFoundation (SNF), The German Research Foundation DFG (TRR 173 Spin+X), the ERC (ERC-\n2014-PoC 665672 MULTIREV), The EC (NMP3-LA-2010 246102 IFOX, FP-PEOPLE-2013-ITN\n608003 WALL) and the Center for innovative and Emerging Materials at the Johannes Guten-\nberg Universit at Mainz. SF acknowledges the \fnancial support from the EU Horizon 2020 Project\nMAGicSky (Grant No. 665095)\nAUTHOR CONTRIBUTIONS\nMF conceived the PEEM experiment with input from SF, JF and MK. MF, FM and LA planned\nand directed the project. LA, MF, FM, NS, SL, JMH, and SF performed XPEEM measurements.\nLA, MF, FM, and SF analyzed the data. NS, JMH and FM performed micro-magnetic simulations.\n15AH and PS designed, prepared and characterized frequency tuned IDT on LiNbO 3, SF prepared the\nNi microstructures. All authors discussed the results and contributed to drafting the manuscript.\n16Supplementary Materials\nVIDEOS DESCRIPTION\n\u000fSTVscan.avi : The SAW produce a contrast in the PEEM images because the piezo-\nelectric voltage associated with the strain wave shifts the energy of the secondary\nelectrons. Thus, bright and dark stripe lines with the periodicity of the SAW excita-\ntion (wavelength, \u0015SAW\u00198\u0016m) appear in the PEEM images. We compiled a video\nwith PEEM images corresponding to di\u000berent electron kinetic energy (that were con-\ntrolled in our detector). Note the contrast inversion during the scan corresponding\nto the ranges highlighted in Figure 2b. A single image is presented in Fig. 3A in the\nmain manuscript.\n\u000fDetuned.avi We recorded PEEM images with a SAW frequency having a small (sub-\nHz) detuning with respect to the synchrotron bunch frequency to con\frm the SAW\npropagation direction by direct observation of the displacement of the stripes in the\nPEEM images. This video presents a 0.01 Hz detuning both positive and negative\nthat con\frm the wave propagation.\n\u000fXMCD.avi PEEM and PEEM/XMCD images are combined in a video that shows the\nsimultaneous evolution of the piezoelectric voltage and the magnetic domain con\fgu-\nration in the hybrid sample of LiNbO 3with Ni squares.\nMICROMAGNETICS RESULTS\nWe modeled the dynamic anisotropy variations in the Ni nanostructures with micro-\nmagnetic simulations using the open-source MuMax3 code30on a graphics card with 2048\nprocessing cores. Simulation parameters are reported in the Methods section and a simpli\fed\ncode is listed in the following section.\nWe can estimate the induced anisotropy of a given magnetic-domain con\fguration and\nthus we can translate the variations observed in the experiments into variations of the\nmagnetic anisotropy. Magnetic domain con\fgurations corresponding to di\u000berent uniaxial\nanisotropies are shown in Fig. 4. The top panels show magnetic-domain con\fguration on a\n17Ni square 2\u00022\u0016m2considering that the anisotropy axis is along the square sides whereas\nthe lower panels present the case where anisotropy axis is along the diagonal of the square.\n0 kJ/m31 kJ/m32 kJ/m33 kJ/m3\n0 kJ/m31 kJ/m32 kJ/m33 kJ/m3kU\nkU\nFIG. 4. Magnetic domain con\fgurations corresponding to di\u000berent uniaxial anisotropies. Top pan-\nels show magnetic-domain con\fguration on a Ni square 2 \u00022\u0016m2with 20 nanometer in thickness,\nconsidering that the anisotropy axis is along the square sides. Lower panels present the case where\nanisotropy axis is along square diagonal on the same micrometric structures\nThe studied magnetic nanostructures have internal resonances in the magnetic domains,\nin the domain walls, and even in the vortex formed in the center of the Landau \rux closure\nstates. Raabe et al.27experimentally identi\fed three di\u000berent dynamic processes|with dif-\nferent timescales|in a Ni micrometric squares under a short magnetic pulse; i) precessional\nmotion within the magnetic domain, ii) domain-wall precession and iii) vortex motion. In\nour experiment we studied two con\fgurations that precisely can serve to separate the dy-\nnamics involving domain wall motion (square sides aligned with the SAW) from dynamics\nthat only involves magnetization rotation within a domain (square diagonal aligned with\nthe SAW).\nWe have introduced in the simulations a time varying anisotropy with a \fxed wavelength\n\u0015SAW= 8\u0016m and measured the magnetic response of the Ni squares under di\u000berent frequen-\ncies of the oscillating anisotropy. In order to quantify the dynamic state of the Ni squares we\ncalculated the averaged magnetic energy at each frequency. Fig. 5 plots the system energy\nin the Ni square as a function of the SAW frequency (an oscillating uniaxial anisotropy) for\nthe two studied con\fgurations: SAW aligned with square sides (in black) and SAW aligned\nwith square diagonal (in red).\nResonance frequencies produce variation in the magnetic-domain con\fguration having a\n18higher energy and can be observed as peaks in the plot of Fig. 5. The con\fguration with\nthe anisotropy aligned along the square sides presents a richer response to SAW frequencies;\ntwo resonance peaks at \u001945 MHz and\u001990 MHz corresponding to vortex motion, one\npeak at\u00191 GHz corresponding to the domain-wall resonance and one peak at \u00192 GHz\ncorresponding to domain resonance. The con\fguration with the anisotropy aligned with\nthe square diagonal is much simpler because the domain walls remain static and only the\nmagnetization within the domains varies; there is a resonance at \u001945 MHz corresponding to\nthe vortex oscillation and a resonance at \u00193:5 GHz corresponding to the domain resonance.\nDomain wallDomain\nVortex\n0,01 0,1 1 10-50050100150200250300350400450\nfrequency (GHz)System Energy (eV)SAW along square sides\nSAW along square diagonal\nFIG. 5. Energy of the magnetic domain con\fguration of a Ni square 2 \u00022\u0016m2with 20 nanometer\nin thickness as a function of the SAW frequency perturbation. The anisotropy is modulated with\na \fxed wavelength \u0015SAW= 8\u0016m to emulate the e\u000bect of the SAW.\n19MICROMAGNETICS CODE\n// mumax3 is a GPU-accelerated micromagnetic simulation open-source software\n// developed at the DyNaMat group of Prof. Van Waeyenberge at Ghent University.\n// The mumax3 code is written and maintained by Arne Vansteenkiste.\n//GRID\nCellSize:=4.e-9\nNumCells:=512\nSetGridSize(NumCells, NumCells, 1)\nSetCellSize(CellSize, CellSize, 20.e-9)\nSetgeom(universe())\n//MATERIAL PARAMETERS FOR STANDARD Ni\nMsat=490e3\nAex=1e-11\nAlpha = 0.03\n//INITIAL MAGNETIZATION STATE\nm = vortex(1,1)\n//REGIONS\nMaxRegion:=200\nCellsPerRegion:=NumCells/MaxRegion\nRegionWidth:=CellsPerRegion*CellSize\nSampleCenter:= CellSize*NumCells/2.\nfor i:=0; i<=MaxRegion; i++ f\ndefregion(i, xrange(i*RegionWidth-SampleCenter,1))\ng\n//DEFINING ANISOTROPY VECTOR\nfor i:=0; i<=MaxRegion; i++ f\nAnisU.SetRegion(i,vector(1.,0.,0.))\ng\n//DEFINING ANISOTROPY CONSTANT\nfreq:=500000. //SAW freq\nKuav:=1.2e3 //Nominal Anisotropy\nKumod:=1.e3\nLambd:=4. //in sample width units\nfor i:=0; i<=MaxRegion; i++ f\nku1.SetRegion(i, Kuav + Kumod * cos( 2*pi*((i* 1/MaxRegion -0.5)/Lambd - freq*t)))\ng\nrelax()\nrun(20e-9)\n20"    },    {        "title": "2203.16008v1.Atomistic_modeling_of_spin_and_electron_dynamics_in_two_dimensional_magnets_switched_by_two_dimensional_topological_insulators.pdf",        "content": "Atomistic modeling of spin and electron dynamics in two-dimensional magnets\nswitched by two-dimensional topological insulators\nSabyasachi Tiwari1;2;3, Maarten L. Van de Put1;3, Kristiaan Temst4, William G. Vandenberghe1, and Bart Sor\u0013 ee3;5;6\n1Department of Materials Science and Engineering, The University of Texas at Dallas,\n800 W Campbell Rd., Richardson, Texas 75080, USA\n2Department of Materials Engineering, KU Leuven, Kasteelpark Arenberg 44, 3001 Leuven, Belgium\n3Imec, Kapeldreef 75, 3001 Heverlee, Belgium\n4Quantum Solid State Physics, Department of Physics and Astronomy,\nKU Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium\n5Department of Electrical Engineering, KU Leuven,\nKasteelpark Arenberg 10, 3001 Leuven, Belgium and\n6Department of Physics, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium.\nTo design fast memory devices, we need material combinations which can facilitate fast read\nand write operation. We present a heterostructure comprising a two-dimensional (2D) magnet and\na 2D topological insulator (TI) as a viable option for designing fast memory devices. We theo-\nretically model spin-charge dynamics between the 2D magnets and 2D TIs. Using the adiabatic\napproximation, we combine the non-equilibrium Green's function method for spin-dependent elec-\ntron transport, and time-quanti\fed Monte-Carlo for simulating magnetization dynamics. We show\nthat it is possible to switch the magnetic domain of a ferromagnet using spin-torque from spin-\npolarized edge states of 2D TI. We further show that the switching between TIs and 2D magnets is\nstrongly dependent on the interface exchange ( Jint), and an optimal interface exchange depending\non the exchange interaction within the magnet is required for e\u000ecient switching. Finally, we com-\npare the experimentally grown Cr-compounds and show that Cr-compounds with higher anisotropy\n(such as CrI 3) results in lower switching speed but more stable magnetic order.\nI. INTRODUCTION\nThanks to the recent discovery of two-dimensional\n(2D) magnets e.g., CrI3[1], CrBr 3[2], and CrGeTe 3[3],\nresearch into 2D magnetics has garnered unprecedented\nattention. 2D magnetic materials open a plethora of\nopportunities in their use in future application in de-\nvices including spintronics [4, 5], valleytronics [6], and\nskyrmion [7]-based magnetic memories [8]. However,\nmany of the devices require low-dimensional magnets in-\nterfaced with semiconductors to function as memory de-\nvices [9{11].\nAn interesting avenue of designing electronic devices\nusing low-dimensional magnets lies in interfacing low-\ndimensional ferromagnets (FM) with topological insu-\nlators (TI) [9, 10]. The surface states of topologi-\ncal insulators can act as spin-channels with high spin-\npolarizability. Moreover, depending on the direction of\napplied bias, the spin of the edge states can be switched\nthanks to spin-momentum locking [12].\nAlthough there have been some experimental works on\nrealizing magnetic devices using TI-FM interfaces [9{12],\na solid theoretical understanding of the interfacial physics\nis missing. Most of theoretical works have either inves-\ntigated equilibrium TI-FM interfaces with a FM having\na \fxed magnetic orientation [13, 14] or investigated the\nimpact of magnetic materials on the topological order of\nthe TIs [15].\nFor the technological application of TI-FM material\nsystems, it is necessary to understand how the motion\nof spin-polarized charge carriers in TIs impact the spin\ndynamics of the FM and vice-versa. To understand such\na coupled spin-charge e\u000bect, it is necessary to model thecoupled spin-charge dynamics of TIs and FMs. There\nhave been recent theoretical works on modelling FM-\nsemiconductor interfaces [16{18], however a major issue\nlies with the description of the 2D magnets.\nMost of the current methods use\nLandau{Lifshitz{Gilbert (LLG) [19] equation to\ndescribe magnetization dynamics, coupled with quan-\ntum transport methodologies, e.g., non-equilibrium\nGreen's function (NEGF) [20]. The largest limitation\nlies in the magnetization dynamics of the magnetic\nmaterial because the LLG equation assumes a continu-\nous description of the magnetic structure [16] and the\natomistic description is lost. Moreover, interactions\nsuch as exchange anisotropy, which play a major role\nin determining the magnetic order of low-dimensional\nmagnets [21{25] are hard to include in the LLG equa-\ntion, and most frequently a scalar approximation to the\nexchange interactions is assumed [26].\nOn the other hand, Monte-Carlo (MC) simulations are\nshown to be very accurate in predicting temperature de-\npendent observables for 2D magnets taking into account\nfull anisotropy [21, 22, 27]. Unfortunately, MC simula-\ntions do not have a standard method of quantifying time\nto obtain time dependent observables such as magnetic\nswitching. There have been previous works on quanti-\nfying time within MC simulations for systems with in-\nplane rotational invariance [28, 29], called time-quanti\fed\nMonte-Carlo (TQMC) simulations. Thankfully, most of\nthe 2D magnetic materials yet discovered experimentally\nhave an in-plane rotational invariance, and TQMC can\nbe applied for such materials for magnetization dynam-\nics.\nWe present a theoretical study of spin-dynamicsarXiv:2203.16008v1  [cond-mat.mes-hall]  30 Mar 20222\n𝑒𝑒−𝑒𝑒−\n𝑒𝑒−𝑒𝑒−\n𝑒𝑒−𝑒𝑒−\n𝑒𝑒−𝑒𝑒−\n𝑒𝑒−𝑒𝑒−\n𝑒𝑒−𝑒𝑒−\n𝑒𝑒−𝑒𝑒−\n𝑒𝑒−𝑒𝑒−𝑒𝑒−\n𝑒𝑒−𝑒𝑒−\n𝑒𝑒−𝑒𝑒−\n𝑒𝑒−𝑒𝑒−𝑀𝑀(𝑡𝑡+Δ𝑡𝑡)\n𝑚𝑚(𝑡𝑡+Δ𝑡𝑡)2D Magnet\n2D TI\nInjection\nFIG. 1: The interface between a 2D TI and a layered magnet. The electrons are injected from the contacts\n(illustrated by the blue arrow below right contact) at time t+ \u0001t. The \rowing electrons carry a small magnetic\nmoment (illustrated by the small blue arrows) exerting a torque due to electronic magnetic con\fguration: m(t+ \u0001t)\non the interfacing magnet with magnetic con\fguration: M(t+ \u0001t) (illustrated by the red arrows).\nand spin-induced switching in a TI-FM heterostruc-\nture. We combine the non-equilibrium Green's function\nmethod for spin-dependent electron transport with time-\nquanti\fed Monte-Carlo for simulating magnetic systems.\nWe use our method to study spin-induced switching in\na heterostructure of a two-dimensional topological insu-\nlator and a two-dimensional ferromagnet. We show that\nit is possible to change the magnetic-domain structure\nin the ferromagnet using spin-injection from TIs, which\ncan be used to design high-speed memory devices. We\nthen show that the switching can only be achieved ef-\n\fciently for an optimal interfacial exchange interaction\nbetween TI and FM. Finally, we compare the switch-\ning time for four experimentally grown Cr-compounds:\nCrI3, CrBr 3, CrCl 3, and CrGeTe 3. We show that the\nhigher anisotropy of CrI 3results in a much larger switch-\ning time compared to CrBr 3and CrCl 3, which have lower\nanisotropy.\nII. METHODOLOGY\nII.1. The Heisenberg Hamiltonian\nWe model the 2D magnetic structure using the Heisen-\nberg Hamiltonian [21]\nHm=\u00001\n2X\ni6=jSi\u0001Jij\u0001Sj+DX\ni(Sz\ni)2(1)\nwhere, S=Sxx+Syy+Szzis the spin-vector with mag-\nnitudejSj=p\n(Sx)2+ (Sy)2+ (Sz)2. The exchange in-\nteraction strength Jijbetween spins at site iandjis a\n3\u00023 tensor [22] whose parameters are obtained by \ftting\nto DFT calculations [21, 22]. The second term in Eq. (1)\nis the single-ion anisotropy with strength D. We assume\nin-plane rotational invariance for the 2D magnetic mate-rials which leads to the Hamiltonian reducing to,\nHm=X\ni;jJij\n2\u0002\nSi\u0001Sj+ \u0001ij(Sz\niSz\nj\u0000Sx\niSx\nj\u0000Sy\niSy\nj)\u0003\n+DX\ni(Sz\ni)2;(2)\nwhere the exchange anisotropy (\u0001 ij) accounts for the dis-\ntinct values for the in-plane and out-of-plane anisotropic\nexchange strength, Jxx\nij=Jyy\nij=Jij(1\u0000\u0001ij) and\nJzz\nij=Jij(1 + \u0001ij) [21], respectively.\nMoreover, we de\fne the total magnetization per atom\nin the x,y, and zdirection as,\nSz=y=x=1\nNatomX\niSz=y=x\ni: (3)\nHere,Natom is the total number of atoms.\nII.2. Electronic Hamiltonian\nWe model the electronic structure of the topological\ninsulators using a tight-binding Hamiltonian,\nHelec=\u0000tX\nhr;r0i;\u000bcy\nr;\u000bcr0;\u000b\n+ i\u0003 soX\nhhr;r0ii;\u000b;\fvr;r0cy\nr;\u000b\u001bz\n\u000b;\fcr0;\f:(4)\nHere the \frst term, with coupling strength t, accounts for\nthe nearest neighbor hopping, hr;r0i, between adjacent\nlattice sites randr0, with respective electron creation\nand annihilation operators cy\nr;\u000b; cr0;\u000b.\u000band\frepresent\nspin degrees of freedom, i.e.,\u000b2f\";#gand\f2f\";#g.\nThe second term is the next nearest neighbor spin-orbit\ncoupling term with strength \u0003 so.\u001bzis the zthcompo-\nnent of the Pauli matrices. The parameter vr;r0is +1\nwhen the shortest next nearest neighbor path from rto\nr0with respect to atom ris clockwise, and \u00001 if it is\ncounterclockwise.3\nII.3. Combined Hamiltonian\nWe combine the model for the Heisenberg Hamilto-\nnian and the electronic Hamiltonian using the interaction\nHamiltonians\nH0\nm=JintX\nr;imr\u0001Si; (5a)\nH0\nelec=Je\nintX\ni;r;\u000b;\fcy\nr;\u000bSi\u0001\u001b\u000b;\fcr;\f: (5b)\nHere, theJintandJe\nintare the interface interactions at\nthe semiconductor-FM interface. The electron magneti-\nzation, m=mxx+myy+mzzis the expectation value\nof the Pauli spin matrix: \u001b=\u001bxx+\u001byy+\u001bzz. Explic-\nitly, we calculate the electron magnetization mby taking\nthe trace of the density matrix over the spin degrees of\nfreedom,\nm(r) =~\r\n4\u0019Tr[\u001b\u0001\u001a(r;r0)]spin; (6)\nwith the electronic density matrix,\n\u001a(r;r0) =Z\nG<(r;r0;E)dE; (7)\nwhere,G<(E) is the lesser Green's function. As is\nstandard in ballistic NEGF, the lesser Green's func-\ntion is obtained from, G<(r;r0;E) =AL(r;r0;E)f(E\u0000\n\u0016L) +AR(r;r0;E)f(E\u0000\u0016R). Here,AL=R(r;r0;E) are\nthe spectral functions for the left (L) and the right (R)\ncontacts, and \u0016L=Rare the respective chemical poten-\ntials withf(E) being the Fermi-Dirac distribution func-\ntion. The spectral functions are obtained using, AL=R=\nGr\u0000L=RGa, with \u0000 L=R=i(\u0006L=R\u0000\u0006y\nL=R).Gr(E) =\n(E\u0000H0\nelec\u0000\u0006L\u0000\u0006R)\u00001is the retarded Green's function.\nThe advanced Green's function is Ga= (Gr)y. The con-\ntact self-energies \u0006 L=Rare obtained using the quantum\ntransmitting boundary method (QTBM) [30, 31]. Simi-\nlar to magnet, we de\fne the magnetization per atom in\nTIs as,\nmz=y=x=1\nNTI\natomX\nimz=y=x\ni: (8)\nHere,NTI\natom is the total number of atoms in TI.\nII.4. Magnetization dynamics and\ntime-quanti\fcation\nTo simulate the magnetization dynamics as a function\nof time, we use the time-quanti\fed Monte-Carlo (MC)\napproach [28, 29]. Within TQMC, one step of a MC\nsimulation is assigned a time step \u0001 tof,\nR2=20kBT\u000b\r\n(1 +\u000b2)M\u0001t: (9)\n𝑅𝑅\n𝐗𝐗𝐙𝐙 𝑆𝑆1FIG. 2: The choice of spin in a MC step using TQMC.\nThe SpinS1shows the initial spin state which is\nallowed to take a value within the radius Rwith spin\nstate ranging between S1toS2(illustrated in black\ncircle). The spin state S2shows an intermediate spin\nstate which can take a value within the the radius R\n(illustrated in blue) with spin state switching from S2to\nS3. The radius Ris determined by the Gilbert damping.\nHere,Ris a cone radius up to which a trial spin-\nrotation is allowed in each MC step. kBis the Boltz-\nmann constant, \u000bis the Gilbert damping parameter,\nM=q\nS2x+S2y+S2zis the magnetic moment, and \r\nis the gyromagnetic ratio.\nFigure 2 shows a single magnetic atom with spins cho-\nsen for three successive time steps using TQMC ( S1,S2,\nS3). Within TQMC for each time step, we make a cone\nof radius of Raround the spin-vector of the present spin\ncon\fguration ( S1, green). The next spin-con\fguration\n(S2, black) is chosen within the cone using the Metropolis\nalgorithm [22, 32]. Once, the spin-con\fgration is updated\ntoS2, we choose the next spin con\fguration ( S3, blue)\nby using the same method of making a cone of radius R\naroundS2.\nThe expression in Eq. (9) for time-discretization has\nbeen obtained by assuming only single-spin interactions\nand Langevin dynamics [28, 29]. Therefore, this time dis-\ncretization is valid only under the condition that the mag-\nnetization dynamics can be approximated using Langevin\ndynamics, and is more accurate for materials with in-\nplane rotational invariance and Gilbert damping: \u000b\u0014\n1 [28, 29]. The approximation of magnetization dynam-\nics to Langevin dynamics is only valid under high out-\nof-plane anisotropy or very low temperatures ( T < 10\nK) where the spins start behaving collectively as a sin-\ngle unit. Due to the above reason, we perform all our\ncalculations below T < 10 K.\nII.5. Algorithm for spin-charge dynamics\nThe energy of the magnetic interaction is of the or-\nder of meV, while the electronic interaction energy is of\nthe order of eV. Therefore, the time scales of magnetiza-\ntion dynamics are in ps, whereas that of electrons is in4\nSISC 2021\nDrain Source𝑉𝑉ds\nXZ\nY\nX(a.1) Side View\n(a.2) Top View\n(a)\n−1000100Vds[mV](b.1)\n−2.50.02.5M[µB](b.2)Sx\nSy\nSz\n0.0 0.2 0.4 0.6\nTime [ns]−2.50.02.5m[µB] (b.3) mx\nmy\nmz (b)\nFIG. 3: (a) The device con\fguration studied for TI-FM switching. The 2D TI is contacted with a drain and a\nsource contact and a voltage ( Vds) is applied between the contacts. (b) The magnetization of the 2D FM as a\nfunction of time. (b.1) The bias is applied at t= 0 ns and switched to 100 meV from -100 meV at 0.25 ns. (b.2) The\nmagnetization of the 2D FM in the z-direction ( Sz) switches its direction (from 3 \u0016Bto\u00003\u0016B). (b.3) The induced\nmagnetization in the 2D TI.\n𝑡𝑡=0.23ns 𝑡𝑡=0.27ns 𝑡𝑡=0.41ns\nXY\n𝑚𝑚z/|𝑚𝑚| 𝑆𝑆z/|𝑀𝑀|\nFIG. 4: The magnetic con\fguration ( M(t)) of the FM\n(upper three panels), and the electronic spin\ncon\fgurations ( m(t)) of the TI (lower three panels) at\nt= 0:23ns;0:27 ns, and 0 :41 ns while the voltage pulse\nis switched at t= 0:25 ns. The atomic structure of the\nFM is shown as an imprint within the magnetic\ncon\fguration of the TI.\nfs. Within the adiabatic approximation, we assume that\ndue to the large di\u000berences in the time scale, the elec-\ntrons are considered to only see an adiabatic change in\nmagnetic moment and are thus always found to occupy\ninstantaneous eigenstates.\nHence, as shown in Fig. 1, we start with a magnetic\norientation of the magnetic material, and at each time-\nstep (t+ \u0001t), we rotate the spin of the top magnetic\nlayer using MC sampling. After the MC step, we cal-\nculate the magnetic moment of the magnet M(t+ \u0001t).\nFor the same time-step, we obtain the lesser Green's func-\ntion using the NEGF method and calculate the electronicmagnetization m(t+ \u0001t). For the next time step, the\nmagnetization of the electronic system is an input to the\nmagnetic system, and the loop continues.\nII.6. Computational Details\nFor all calculations unless mentioned speci\fcally, we\nhave used values \u000b= 0:05,kBT= 0:5 meV,Jint=\nJe\nint= 25 meV. We have used the parameters of a\nCrI3monolayer to parameterize the interface 2D ferro-\nmagnet, which we obtained from DFT calculations using\nthe procedure in Ref. 21. The saturation magnetization\nused for the Cr-atoms is 3 \u0016B. To model the TIs we have\nused the parameters of the tight-binding Hamiltonian for\nstanene [31].\nIII. RESULTS AND DISCUSSION\nIII.1. TI-FM interface operation\nFigure 3 (a) shows the TI-FM con\fguration we are in-\nvestigating. For the TI we use a 5 nm wide stanene rib-\nbon, for the 2D FM we use CrI 3with size: 3 nm\u00023:5 nm.\nThe 2D FM is positioned near one of the edges of the 2D\nTI as shown in the top view in Fig. 3 panel (a.2). A\npotential (Vds) is applied across the TI ribbon to inject\ncharge carriers in the TI ribbon. For Vds>0, the edge\nstates on the top edge of the TI have up-spin, whereas\nthe bottom edge exhibit down-spin. For Vds<0, spin po-\nlarization of the edge states is \ripped. Therefore, the in-\nterfaced FM experiences positive or negative spin-torque\nfrom the underlying charge carriers of the TI, depending\non the applied Vds.\nFigure 3 (b) panel (b.1) shows the applied Vdsas a\nfunction of time. We apply a step Vdspulse of 100 mV at5\n050100Vds[mV]\n0.00 0.25 0.50 0.75 1.00 1.25\nTime [ns]−101Sz/|M|\nα= 0.02\nα= 0.03\nα= 0.04\nα= 0.05(a.1)\n(a.2)\n(a)\n050100Vds[mV]\n0.0 0.2 0.4 0.6 0.8 1.0\nTime [ns]−101Sz/|M|Jint= 5.0 meV\nJint= 25.0 meV\nJint= 50.0 meV\nJint= 100.0 meV\nJint= 150.0 meV(b.2)(b.1) (b)\n20 40 60 80 100 120 140\nJint[meV]0.050.100.150.200.250.300.35Transition time [ns]\n(c)\n𝑆𝑆z/|𝑀𝑀|𝐽𝐽int=25meV 𝐽𝐽int=150meV\n𝑚𝑚z[𝜇𝜇B] (d)\nFIG. 5: (a) The applied Vds(panel a.1) and the magnetization of the top 2D FM for \u000b= 0:02;0:03;0:04;0:05\n(panel a.2). (b) The applied Vdsand the magnetization of the top 2D FM for interface interaction strengths\nJint= 5;25;50;100;150 meV (panel b.2). (c) The transition time for the applied pulse in panel (b.1) as a function\nofJint. (d) The magnetization at t= 0:45 ns forJint= 25 meV and Jint= 150 meV. The upper two panels show the\nmagnetization of the 2D FM in the z-direction ( Sz=jMj), and the lower two panels show the induced magnetization\nin the 2D TI ( mz).\nt= 1:75 ns. Figure 3 (b) panel (b.2) shows the average\nmagnetization in the x;yandz-direction of the top FM\nlayer as a function of applied Vds. We observe that with\na change in the applied Vdsfrom -100 mV to 100 mV\natt= 0:5 ns, the magnetization in the z-direction ( Sz)\nswitches from 3 \u0016Bto -3\u0016B. When the applied voltage\nswitches from 100 mV to -100 mV, we observe that the\nmagnetization switches again from -3 \u0016Bto 3\u0016B.\nFigure 3 (b) panel (b.3) shows the induced magnetiza-\ntion of the TI ribbon in the x,y, and zdirection as a\nfunction of time. We observe that due to the magnetiza-\ntion of the top FM, there is a \fnite induced magnetiza-\ntion in the TI ribbons when the applied bias is non-zero.\nInterestingly, we \fnd that whenever the bias switches\nand the magnetization of the top FM transitions, the to-\ntal induced magnetization of the TI ribbon also shows a\npeak.To further analyze the magnetization dynamics of the\nTI-FM interface, we plot the out-of-plane projection of\nthe magnetization of the entire TI and FM sample at\nvarious time steps in Fig. 4. Figure 4 shows the magneti-\nzation in the z-direction for the 2D magnet (top panels,\nSz=jMj) and the TI (bottom panels, mz=jMj), respec-\ntively. We observe that at t= 0:23 ns, the TI has a\npositive bias Vds>0 and both the TI and the FM have\npositive magnetization at the interface. Note that we\nhave taken the interface parameter to be positive, mean-\ning the interaction is ferromagnetic.\nAtt= 0:27 ns, the bias of the TI has opposite polar-\nizationVds<0. We observe that the magnetization of the\nTI reverses. The reversal of the magnetization in TI in-\nduces a spin-torque on the interfacing FM, which results\nin the formation of a small region of magnetization, with\nopposite orientation as that of the entire FM ( ^Sz<0).6\n0.00 0.05 0.10 0.15 0.20\n∆NN0.060.080.100.12Transition time [ns]\n(a)\n050100Vds[mV]\n0 1 2 3 4\nTime [ns]−101Sz/|M| CrI3\nCrBr 3\nCrCl 3\nCrGeTe 3(b.1)\n(b.2) (b)\nFIG. 6: (a) Transition time as a function of nearest-neighbor anisotropy \u0001 NN. We perform the same sweep of \u0001 NN\nfor 10 di\u000berent starting con\fgurations. The solid line shows the median and the shaded region shows the 25th-75th\npercentile. (b) Comparison of switching in CrI 3, CrBr 3, CrCl 3, and CrGeTe 3. Panel (b.1) shows the applied\nVdsand panel (b.2) shows the magnetization ( Sz=jMj) for CrI 3, CrBr 3, CrCl 3, and CrGeTe 3. We use an interface\nexchangeJint= 25 meV, and \u000b= 0:005.\nAtt= 0:41 ns, we observe that the domain with ^Sz<0\ngrows and changes completely to a FM con\fguration with\na magnetization of the 2D magnet equal to \u00003\u0016B. Inter-\nestingly, we \fnd that the top magnetization of the FM\ndoes not have a signi\fcant impact on the magnetization\nof the TI. The reason for such behavior is because we have\nassumed in our calculations that the interface interaction\nstrengthJint=Je\nint= 25 meV. The interaction strength\nfelt by the TI is of the order of meV, whereas the topo-\nlogical energy gap of TIs is of the order of eV. Hence, the\nweak interaction strength will always lead to a negligible\nresponse of the TI magnetization due to magnetization\nat the interface with the FM.\nIII.2. Impact of Jintand\u000b\nFigure 5 (a) panel (a.1) shows the applied bias across\nthe 2D TI. We apply a Vdspulse with a pulse width\nof 0.4 ns and an amplitude of 100 mV at t= 0:8 ns.\nFigure 5 (a) panel (a.2) shows the normalized magneti-\nzation in the z-direction ( Sz=jMj) as a function of time\nof the interface 2D FM for Gilbert damping parameter\n\u000b= 0:02;0:03;0:04;0:05 for a pulse bias. With the\ntransition in applied Vds, the normalized magnetization\nof the 2D magnet transitions from 1 to \u00001. We also ob-\nserve that with increasing \u000b, the transition occurs faster,\nsuggesting that the switching speed increases.\nFigure 5 (b) panels (b.1) and (b.2) show a pulse bias\napplied accross the 2D TI with a pulse width of 0.2 ns\napplied at 0.4 ns as a function of time, the normalized\nmagnetization in the z-direction ( Sz=jMj) as a function\nof time of the interface 2D FM for interface exchange\nparameterJint= 20;25;50;75;100, and 150 meV. We\nobserve that with increasing Jint, the transition occursfaster up to Jint= 50 meV. For Jint>50 meV, we see\nthat the magnetization does not switch smoothly, and\nthe magnetization stabilizes for the entire duration of\nthe pulse. After the applied pulse returns to 0 mV, the\nmagnetization still switches for all Jint>50 meV, albeit\nwith some delay.\nFigure 5 (c) shows the transition time for the applied\npulse in Fig. 5 (b) panel (b.1) as a function of Jint. We\nde\fne the transition time as t[Sz<\u00000:7M]\u0000t[Sz>\n0:7M]. We observe that the transition time goes down as\na function of Jintup toJint= 50 meV and then increases\nforJint>50 meV. Therefore, we \fnd that an optimal\nvalue of interface exchange ( Jint) is required for obtaining\na small transition time.\nTo further understand the reason for the low transi-\ntion rate at higher Jint, we compare the magnetization\natt= 0:45 ns in Fig. 5 (d) for Jint= 25 meV and\nJint= 150 meV. The upper two panels show the mag-\nnetization of the 2D FM in the z-direction ( Sz=jMj), and\nthe lower two panels show the induced magnetization in\nthe 2D TI ( mz). We observe that for Jint= 150 meV, mz\nis lower than the mzforJint= 25 meV. Therefore, the\nlower magnetization of the TI results in a lower torque on\nthe 2D FM, causing a delay in transition. The reason for\nthe lower magnetization due to higher Jintis because the\nmagnet opens a bandgap in the 2D TI, resulting in a su-\nperposition of up and down spins [14]. The superposition\nof up and down spins in TI leads to a reduced magne-\ntization (For further details, see Supplementary Fig. 1\nand Fig. 2).7\nTABLE I:J-parameters and anisotropies of\nexperimental Cr-compounds\nParameters JNNJNNNJNNNN \u0001NN\u0001NNN \u0001NNNN\n(meV) (meV) (meV)\nCrI3 2.21 0.75 - 0.029 0.045 -\nCrBr 3 1.38 0.44 - 0.010 0.012 -\nCrCl 3 1.31 0.24 - 0.001 0.008 -\nCrGeTe 35.87 -0.28 0.345 0.02 0.0 0.028\nIII.3. Impact of anisotropy and Cr-compounds\nWe compare the transition time for the device in Fig. 3\n(a) as a function of nearest-neighbor anisotropy (\u0001 NN) of\nthe FM as shown in Fig. 6. We perform the same sweep\nof \u0001 NNfor 10 di\u000berent starting con\fgurations. The solid\nline shows the median and the shaded region shows the\n25th-75thpercentile. We use Jint= 0:025 eV and \u000b=\n0:05 and nearest neighbor exchange for the 2D magnet\nto beJNN= 2:5 meV.\nWe \fnd that with increasing \u0001 NNthe transition time\nincreases. Therefore, for faster switching of the TI-FM\ndevice, it will be ideal to use materials with lower \u0001 NN.\nHowever, lower \u0001 NNresults in a lower Curie tempera-\nture [21]. Hence, the choice of the FM material should\ntake into account the temperature of operation and the\nrequired switching time for the device.\nFor the device shown in Fig. 3 (a), we use CrI 3, CrBr 3,\nCrCl 3, and CrGeTe 3as the 2D magnetic material. Al-\nthough, many studies in previous works have used a para-\nmetric value of \u000b[33] for Cr-compounds, recent reports\non the measurement of intrinsic Gilbert damping of a sis-\nter compound CrCl 3have found a value of \u000b= 0:002 [34].\nUnfortunately, there is no similar report for other Cr-\ncompounds , therefore we have used an \u000bof similar or-\nder,\u000b= 0:005, for all the three Cr-compounds for a\nfair comparison. The parameters used for modeling their\nmagnetic structure of the Cr-compounds is shown in Ta-\nble I.\nFigure 6 (b) panel (b.1) shows the applied Vdsand\npanel (b.2) shows the out-of-plane magnetization Szfor\nCrI3, CrBr 3, CrCl 3, and CrGeTe 3. We \fnd that the\nmagnetization of CrI 3is the most stable among the Cr-\ncompounds followed by CrGeTe 3, CrBr 3, and CrCl 3, re-\nspectively. However, CrI 3is also the slowest to respond\nto a change in the applied bias and has the highest tran-\nsition time.\nComparing the parameters for the Cr-compounds in\nTable I, we \fnd that the transition time is inversely pro-\nportional to the nearest-neighbor anisotropy (\u0001 NN). The\nhigher the anisotropy, the higher the transition time. On\nthe other hand, the stability of magnetization is directly\nproportional to the anisotropy.IV. CONCLUSION\nWe have presented a method to model spin-charge dy-\nnamics at a magnetic-topological insulator interface. Our\nmodel is general and not limited to only the TI-FM inter-\nface. Our method combines NEGF and TQMC to model\nthe spin-charge dynamics. The bene\ft of TQMC+NEGF\nover conventional methods such as LLG+QTBM or\nLLG+NEGF lies in the atomistic description of the spins,\nwhich allows one to implement the full exchange tensor\nfor modelling the magnetic exchange interactions.\nUsing our method, we have theoretically investigated\nthe spin-charge dynamics in a 2D TI-FM heterostructure\nwhere the size of the 2D FM was smaller than the 2D TI\nand placed on one of the edges of the 2D TI. We have\nshown that by electrically biasing the 2D TI, it is pos-\nsible to switch the magnetization of the 2D FM without\ndestroying the edge state of the 2D TI. The most im-\nportant result of our work is that the switching of the\n2D FM using 2D TI spin-torque can only be achieved\ne\u000eciently in TI-FM material combinations that have an\noptimal interface exchange interaction Jint. We have also\ncompared experimentally grown 2D Cr-compounds. We\nhave shown that the transition rate of magnetization sig-\nni\fcantly depends on the anisotropy, and low anisotropic\nCr-compounds (CrBr 3and CrCl 3) show faster switching\nin comparison to the higher anisotropic Cr-compounds\n(CrI 3and CrGeTe 3). Finally, there are recent reports\nof experimentally growing 2D FM-semiconductor het-\nerostructures for spintronic devices [35], and given the\ndevice design presented in our paper is feasible to make\nexperimentally, and we believe it should be explored ex-\nperimentally.\nV. ACKNOWLEDGEMENTS\nThe project or e\u000bort depicted was or is sponsored by\nthe Department of Defense, Defense Threat Reduction\nAgency. 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China\n3Key Laboratory of Solar Activity, National Astronomical Ob servatories of Chinese Academy of Science, Beijing\n100012, China\n4University of Chinese Academy of Sciences, 19A Yuquan Road, Shijingshan District, Beijing 100049, China\n(Received; Revised; Accepted)\nSubmitted to ApJ\nABSTRACT\nUsing high spatial and temporal data from the Solar Dynamics Observatory (SDO)\nand theInterface Region Imaging Spectrograph (IRIS), several observational signatures\nof magnetic reconnection in the course of magnetic flux cancellation are presented,\nincluding two loop-loop interaction processes, multiple plasma blob eje ctions, and a\nsheet-like structure that appeared above the flux cancellation sit es with a Y-shaped and\nan inverted Y-shaped ends. The IRIS1400˚A observations show that the plasma blobs\nwere ejected from the tip of the Y-shaped ends of the sheet-like s tructure. Obvious\nphotospheric magnetic flux cancellation occurred after the first lo op-loop interaction\nand continued until the end of the observation. Complemented by t he nonlinear force-\nfree field extrapolation, we found that two sets of magnetic field line s, which reveal\nan X-shaped configuration, align well with the interacted coronal lo ops. Moreover, a\nmagnetic null point is found to be situated at about 0 .9 Mm height right above the flux\ncancellation sites and located between the two sets of magnetic field lines. These results\nsuggest that the flux cancellation might be a result of submergence of magnetic field\nlines following magnetic reconnection that occurs in the lower atmosp here of the Sun,\nand the ejected plasma blobs should be plasmoids created in the shee t-like structure\ndue to the tearing-mode instability. This observation reveals detaile d magnetic field\nstructure and dynamic process above the flux cancellation sites an d will help us to\nunderstand magnetic reconnection in the lower atmosphere of the Sun.\nCorresponding author: Bo Yang\nboyang@ynao.ac.cn2\nKeywords: Sun: activity – Sun: atmosphere – Sun: magnetic fields\n1.INTRODUCTION\nMagnetic reconnection is a process by which magnetic field lines with an tiparallel components\nare brought together in a current sheet or at a magnetic null point , where they break up and\nreconnect to form new magnetic field lines( Priest & Forbes 2000 ;Yamada et al. 2010 ). Dur-\ning this process magnetic energy is thereby converted into plasma k inetic and thermal energy.\nIt is widely accepted that magnetic reconnection is the cause of var ious types of solar activi-\nties, such as solar flares( Shibata 1996a ), coronal mass ejections(CMEs; Lin & Forbes 2000 ), fila-\nment eruptions( Chen & Shibata 2000 ;Shen et al. 2012 ;Zhou et al. 2017 ), jets(Shibata et al. 1996b ;\nJiang et al. 2013 ), explosive events( Innes et al. 1997 ), and coronal bright points( Priest et al. 1994 ).\nTo date, many signatures that are probably related to magnetic re connection have been reported, in-\ncluding hot cusp-shaped structures( Tsuneta et al. 1992 ), loop-top hard X-ray sources( Masuda et al.\n1994;Sui & Holman 2003 ), reconnection inflows( Yokoyama et al. 2001 ;Li & Zhang 2009 ;Su et al.\n2013;Sun et al. 2015 ;Yang et al. 2015 ) and outflows( Asai et al. 2004 ;Savage et al. 2010 ;Liu et al.\n2013;Chen et al. 2016 ), current sheets( Webb et al. 2003 ;Lin et al. 2005 ;Liu et al. 2010 ;Xue et al.\n2016;Yan et al. 2018 ), plasmoid ejections( Shibata et al. 1995 ;Nishizuka et al. 2010 ;Takasao et al.\n2012), loop-loopinteractions( Sakai & de Jager 1996 ;Li et al. 2014 ), and drifting pulsating structures\nobserved in radio waves( Kliem et al. 2000 ;Ning et al. 2007 ). Through decades of observations, a lot\nof evidence for the reconnection scenario has been obtained. How ever, most evidence was indirect\nand was detected in the solar corona, and direct observational ev idence that characterizes the recon-\nnection in the lower atmosphere has been poorly reported.\nMagnetic flux cancellation, which observationally describes the mutu al disappearance of converg-\ning magnetic patches of opposite polarities in the photospheric longit udinal magnetograms( Livi et al.\n1985;Martin et al.1985 ), isconsidered tobeevidence ofmagneticreconnectionoccurring inthelower\natmosphere of the Sun( Priest et al. 1994 ). A “U-loop emergence” scenario and an “Ω-loop submer-\ngence” scenario were proposed by Zwaan(1987) to account for magnetic flux cancellation. Two\nunconnected magnetic patches of opposite polarities could build up c onnection by magnetic recon-\nnection during flux cancellation( Wang & Shi 1993 ), and whether a “U-loop emergence” scenario or\nan “Ω-loop submergence” scenario could contribute to flux cancella tion depends on the height that\nreconnection is initiated. The “U-loop emergence” will be dominant du ring the cancellation when\nmagneticreconnectiontakesplacebelowthephotosphere. Onthe contrary,the“Ω-loopsubmergence”\nwill be dominant when magnetic reconnection occurs above the phot osphere. By investigating the\nevolution of the photospheric and chromospheric magnetograms s imultaneously, Harvey et al. (1999)\nproposed strong evidence that suggests an “Ω-loop submergenc e” scenario at flux cancellation sites.\nTransverse magnetic field and Doppler velocity field around flux canc ellation sites are usually utilized\nto study flux cancellation events. During and after flux cancellation , it is usually found that hori-\nzontal field at the flux cancellation sites enhanced significantly( Wang & Shi 1993 ;Yang et al. 2016 ).\nHowever, Wang & Shi (1993) implied that the change of the horizontal field at flux cancellation sit es\ncould not fit the quite popular view of interpreting flux cancellation th at mentioned above. They put\nforward that the association of flares to flux cancellation seems to represent the coupling of a slow\nreconnection in the lower atmosphere to a fast reconnection in the upper atmosphere. Chae et al.3\n(2004) andIida et al. (2010) verified that both red shifts and strong horizontal field at flux ca n-\ncellation sites support the “Ω-loop submergence” scenario. Zhang et al. (2009) reported extremely\nlarge Doppler blue-shifts at flux cancellation sites and interpreted t he cancellation as a “U-loop\nemergence”. Kubo & Shimizu (2007) found that there are both blue and red shifts at flux cancella-\ntion sites, indicating that magnetic reconnection between the conv erging magnetic patches occurs at\nmultiple locations with different heights. Nevertheless, Yang et al. (2009) investigated the emerged\ndipoles in a coronal hole and found that the submergence of the eme rged original loops can also lead\nto flux cancellation. Therefore, to understand the physical natu re of flux cancellation, the detailed\nmagnetic structures above the flux cancellation sites in the upper a tmosphere need to be investigated\nin detail.\nInthispaper, withhighresolutionobservationsacquiredbythe Solar Dynamics Observatory (SDO;\nPesnell et al. 2012 ) and the Interface Region Imaging Spectrograph (IRIS;De Pontieu et al. 2014 ),\nwe present clear and direct observational evidence showing magne tic reconnection associated with\nphotospheric magnetic flux cancellation. This is an exemplary event w ith which to show in detail the\nrelationship between magnetic reconnection and photospheric mag netic flux cancellation. In Section\n2, we describe the detailed observations and methods that we used . The results are shown in Section\n3. The conclusion and the discussion are given in Section 4.\n2.OBSERVATIONS AND METHODS\nThe detailed reconnection process associated with magnetic flux ca ncellation on 2015 January 9\nwas captured by the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012 ) and the Helioseismic\nand Magnetic Imager (HMI; Schou et al. 2012 ) on board the SDO. The AIA instrument observes\nfull-disk images of the Sun in 10 ultraviolet (UV) and extreme ultraviole t (EUV) wavelengths with a\nspatial resolution of1.′′5 (0.′′6 pixel−1) anda high cadence ofup to 12 s. In this study, we mainly used\nthe Level 1.5 images observed in 304 ˚A (HeII, 0.05 MK), 171 ˚A (FeIX, 0.6 MK), 94 ˚A (FeXVIII, 7\nMK), and 1600 ˚A (CIV+ cont., 0.01 MK). HMI measures the full-disk continuum intensity imag es\nand line of sight (LOS) magnetic field for the Fe Iabsorption line at 6173 ˚A with a spatial sampling\nof 0.′′5 pixel−1and a cadence of 45 s. The AIA data used in this study were taken be tween 2015\nJanuary 9 19:30 UT and 21:30 UT, and the HMI data were taken betwe en 2015 January 9 18:00\nUT and 22:00 UT. This event was also observed by IRISslit jaw imager (SJI) in 1400 ˚A during two\nperiods (19:03-20:00 UT; 20:40-21:32 UT). The time cadence and the spatial resolution of the SJIs\nare 9 s and 0.′′332 pixel−1, respectively. Using full-disk soft X-ray (SXR) images from the X-R ay\nTelescope (XRT) aboard the Hinodesatellite ( Kosugi et al. 2007 ), the associated coronal structures\nwere also examined. All images were then aligned by differentially rotat ing to a reference time (20:40\nUT on 2015 January 9).\nIn addition, continuous photospheric vector field ( Turmon et al. 2010 ), which has a pixel scale\nof about 0.′′5 pixel−1and a cadence of 12 minutes, in the so called HMI Active Region Patche s\n(HARPs) region is also provided by HMI. The Very Fast Inversion of t he Stokes Vector algorithm\n(Borrero et al. 2011 ) is utilized to compute the vector field data, and the Minimum Energy m ethod\n(Metcalf 1994 ;Metcalf et al. 2006 ;Leka et al. 2009 ) is used to resolve the remaining 180◦azimuth\nambiguity. In order to remove the projection effect, the HARP vec tor field data are remapped to a\nLambert Cylindrical Equal-area (CEA) projection and then transf ormed into standard heliographic\nspherical coordinates. To obtain the magnetic field topology of the flux cancellation event, we\ncarried out a nonlinear force-free magnetic field (NLFFF) extrapo lation to reconstruct the coronal4\nfields. To perform the NLFFF extrapolation, the ”weighted optimiza tion” method ( Wheatland et al.\n2000;Wiegelmann 2004 ) is used. Before the extrapolation, a preprocessing procedure, which drives\nthe observed non-force-free data towards suitable boundary c onditions for a force-free extrapolation\n(Wiegelmann et al. 2006 ), is applied to the bottom boundary vector data.\n3.RESULTS\n3.1.Cancellation of Photospheric Magnetic Field\nOn 2015 January 9, AR NOAA 12257 was located at about N5◦W29◦with aβmagnetic config-\nuration. As shown in Figure 1( a), the magnetic flux cancellation region of interest is enclosed by a\nred rectangle, and the detailed magnetic flux cancellation process is shown in the zoomed view in\npanels(bf). The cancelling magnetic flux patches “p” and “n1” existed from th e beginning of the\nobservations, and a transverse field, which was emanated from p a nd connected to n1, indicates that\nthere was a connectivity between p and n1(panel ( b)). Note that flux emergence happened before\n20:00 UT (panels( bc)). The positive flux patches of the emerged flux were mixed with p, w hile its\nnegative flux patches were composed of “n2” and “n3”. In particu lar, during its emerging process, n3\nmoved toward and merged with n1. As a result, the flux density and t he area of p and n1 increased,\nalthough the flux cancellation occurring between p and n1. A remark able decrease of the flux density\nand the area of n1 was observed from 20:00 UT to 21:40 UT (panels( dg)), and the unsigned neg-\native flux was dropped by 8 .0×1019Mx, corresponding to an approximate flux cancellation rate of\n4.8×1019Mxh−1. At the end of the observations, n1 almost disappeared (panel( f)). Different from\nmany flux cancellation events observed before( Wang & Shi 1993 ;Yang et al. 2016 ), the change of the\ntransverse field was not obvious, and the flux cancellation was acco mpanied by the flux emergence\nat the same region. Therefore, it is difficult to confirm which mechanis m could account for the flux\ncancellation. Investigating the coronal structures and activities above the flux cancellation sites may\nshed light on the understanding of the physical nature of the flux c ancellation.\n3.2.The First Loop-Loop interaction Process\nScrutinizing the observations from SDO/AIA and IRIS, it is found that two loop-loop interaction\nprocesses and two plasma blob ejection processes were closely rela ted to the flux cancellation. The\nfirst loop-loop interaction process is displayed in Figure 2 (see also th e animation, loop-loop1.mpeg).\nJust prior to the interaction, at about 19:40 UT, an IRIS1400˚A image shows the general appearance\nofthe two sets ofinteracted loops, “L1”and“L2” (panel( a)). Remarkably, asshown by the contoured\nHMI magnetogram, L1 connected the positive flux patch “p1” to a n egative flux patch n1, whereas\nL2 connected the positive flux patch p to a negative flux patch n2. T hus, the adjacent endpoints\nof L1 and L2 were co-spatial with the cancelling flux patches p and n1 . By about 19:43 UT, the\nloop-loop interaction started, and a set of rising loops, “L3”, which connected p1 to n2, was formed\n(panels(bc)). At the same time, four footpoint brightenings, which were exac tly coincident with\nthe footpoints of L1 and L2, appeared (panel( candf)). Note that L1 could not be detected by\nthe AIA observations before the interaction. However, after th e interaction, L1 and L3 were clearly\npresented by the AIA 94 ˚A and SXR images (panels( de)), implying that L1 and L3 might be\nheated during the interaction and the connectivity of L1 were part ially changed. Furthermore, a set\nof loops, “L4” , which connected p to n1, was also observed by the A IA 94˚A and SXR observations.\nThese observations indicate that magnetic reconnection may take place between L1 and L2. The\nreconnection changed the connectivity of L1 and L2, resulting in th e formation of L3 and L4.5\n3.3.Successive Ejection of Plasma Blobs from the Flux Cancellat ion Sites\nIt is widely accepted that plasma blob ejections and magnetic flux can cellation are evidence of\nmagnetic reconnection. Currently, plasma blob ejections were fre quently observed in different types\nof reconnection events; however, successive plasma blob ejectio ns associated with flux cancellation\nhas rarely been observed. In our observations, from 20:25 UT to 2 0:40 UT, about half an hour after\nthe first loop-loop interaction, we found that a chain of plasma blobs were ejected from the flux\ncancellation sites successively. The detailed ejection process is disp layed by the selected AIA 304\n˚A images in Figure 3 (see also the animation, blobs1.mpeg). Before the initiation of the plasma\nblob ejection at about 20:21 UT, it is found that two adjacent bright streaks were rooted in the flux\ncancellation patches p and n1, respectively (panel( a)). Those bright streaks may represent two sets of\nloopswithopposite directions. Inparticular, thebright streak roo tedinn1 hadthe sameconnectivity\nas L1. Hereafter, we call the loops, which connect p1 to n1, as L1. As soon as those bright streaks\napproached to each other, the plasma blobs, as indicated by the ar rows, were ejected from the flux\ncancellation sites, propagated along L1 and finally stopped at the fa r ends of L1 (panels ( bd)).\nThe plasma blob ejections observed here are quite similar to that rep orted by Zhang et al. (2016).\nGenerally, it is believed that those ejected blobs are formed by a tea ring-mode instability occurring\nin a current sheet structure( Furth et al. 1963 ;Shibata & Tanuma 2001 ). Thus, our observations may\nalso imply that magnetic reconnection and a tearing process may occ ur in a current sheet between\nthe two bright streaks.\n3.4.The Second Loop-Loop Interaction Process\nImmediately after the plasma blob ejection, an intense activity was f ollowed by (see also the ani-\nmation, loop-loop2.mpeg). This activity is quite obvious in the AIA 304 ˚A images in Figure 4( ac).\nAt about 20:40 UT, when the plasma blob ejections stopped, a compa ct brightening appeared at the\nflux cancellation sites (panel( a)). Simultaneously, relatively weak remote brightening appeared at\nthe location corresponding to the negative footpoint of L1. Subse quently, from 20:42 UT to 20:50\nUT, mass flows, which originated from the compact brightening regio n, spread along two arched\ntrajectories in opposite directions (panels( bc)). Careful inspecting the AIA 171 ˚A difference im-\nage (panel( d)) found that mass flows moved along the two arched trajectories and traced out the\nappearance of two sets of loops, L1 and “L5”. Supplemented by th e contoured HMI magnetogram,\none can see that the adjacent ends of L1 and L5 are rooted in the c ancelling flux patches p and n1\n(panel (d)), respectively. Moreover, the negative footpoint of L5 rooted in a plage region (labeled\nas “n”). These observational signatures may suggest that the c ompact brightening and the plasma\nflows are the results of the interaction occurring between L1 and L 5.\n3.5.Successive ejection of Plasma Blobs from a Sheet-Like Struc ture Observed by IRIS\nAt 21:03 UT, about 13 minutes after the second loop-loop interactio n, it is particularly remarkable\nthat a sheet-like structure with a Y-shaped and an inverted Y-sha ped ends appeared above the flux\ncancellation sites (Figure 5( a)). Afterwards, multiple plasma blobs stemmed likely from the tip of\nthe Y-shaped end and ejected successively along L1. This is evidenc ed by the sequential IRIS1400\n˚A images in Figure 5( ad) (see also the animation, blobs2.mpeg). The zoomed view (panel( e))\ndisplays the morphology of the sheet-like structure more clearly. T his sheet-like structure, which is\nsimilar to the sheet-like structure reported by Singh et al. (2012) andLi et al.(2016), lasted about 8\nminutes and finally disappeared at about 21:11 UT. Singh et al. (2012) andLi et al.(2016) suggested6\nthat this structure should be a current sheet. Inour observatio ns, however, there is no direct evidence\nto confirm that this sheet-like structure is a current sheet apart fromits morphology. Fortunately, the\nvector field data obtained by HMI is conducive to extrapolate and re construct the coronal magnetic\nfield over the flux cancellation region, and is helpful for us to unders tand the event.\n3.6.Magnetic Topology of the Flux Cancellation Region\nWith the aid of the HMI vector magnetograms, we carried out an NLF FF extrapolation to recon-\nstructthecoronalmagneticfieldofthefluxcancellationregion. Fig ure5(gf))showtheconsequence\nof the NLFFF extrapolation. The red and blue lines, which traced fro mthe photospheric flux patches\np and n1, delineate the extrapolated coronal field lines. It is evident that the red and blue field lines\nreveal an X-shaped configuration. Previous theoretical and obs ervational studies( Priest et al. 1994 ;\nJiang et al. 2017 ) suggested that such a configuration should contain a magnetic nu ll point, which\nis in favour of the reconnection. Employing a trilinear null finding meth od(Haynes & Parnell 2007 )\nto scan the NLFFF-modeled field, we indeed find that a magnetic null p oint is located between the\nred and blue filed lines (as indicated by the green arrows in Figure 5( fg)). The magnetic null\npoint is situated at ∼0.9 Mm height right above the flux cancellation sites. It separates the red and\nblue field lines into two distinct connections, one connects p1-n1, an d the other connects p-n. From\nFigure 4( d) and Figure 5( g), it is found that the red and blue field lines match strikingly well with L1\nand L5. Magnetic field near an X-type null point would collapse and evo lve to a field with a current\nsheet(Priest & Forbes 2000 ). In our event, the X-shaped magnetic filed configuration may imply that\nthe magnetic null point is an X-type null point. Moreover, the spatia l location of the magnetic null\npoint and the observed sheet-like structure is almost overlapping. Thus, our observations strongly\nsuggest that the reconnection occurring between L1 and L5 was t riggered at the magnetic null point,\nand the magnetic field near the null point collapsed during the reconn ection, resulting in the forma-\ntion of a current sheet. Accordingly, we speculate that the sheet -like structure observed by the IRIS\nmay represent a current sheet. A tearing-mode instability ( Furth et al. 1963 ;Priest & Forbes 2000 )\nmay further develop in the sheet-like structure, creating the mult iple plasma blobs.\n4.CONCLUSION AND DISCUSSION\nIn this paper, we present two unambiguous loop-loop interaction pr ocesses and two plasma blob\nejection processes, which are closely related to magnetic flux canc elation in the same location. The\nfirst loop-loop interaction took place between a set of pre-existing loops(L1) and a set of emerging\nsmall loops (L2). Half an hour after the first loop-loop interaction, a chain of plasma blobs were\nejected from the flux cancellation sites and spread along L1. Immed iately after the plasma blobs\nejection, the second loop-loop interaction initiated. Compact brigh tening resided at the flux cancel-\nlation region and mass flows spread in opposite directions were obser ved. The mass flows traced out\nthe interacted loops L1 and another set of loops (L5). Following the second loop-loop interaction,\nIRIS1400˚A images show that a sheet-like structure with a Y-shaped and an inv erted Y-shaped ends\nappeared above the flux cancellation sites and a chain of plasma blobs were ejected successively from\nthe tip of the Y-shaped ends and moved along L1. It is evident from H MI vertical magnetograms\nthat obvious flux cancellation occurred after the first loop-loop int eraction and continued till the end\nof the observation. Supplemented by an NLFFF extrapolation, two sets of coronal field lines, which\nalign with L1 and L5 very well, are extrapolated. Moreover, it is found that a magnetic null point is\nlocated between the two sets of coronal field lines. Based on the ob servations, we suggest that the7\nfirst loop-loop interaction may due to the magnetic reconnection be tween L1 and L2, while the sec-\nond loop-loop interaction may due to the magnetic reconnection bet ween L1 and L5. Furthermore, a\ntearing-mode instability might be further developed in the course of the interaction between L1 and\nL5 in a current sheet, creating the ejected plasma blobs. Our obse rvations not only provide evidence\nof a submergence of “Ω-loop ” following magnetic reconnection at th e flux cancellation sites, but also\nshed new light on magnetic reconnection in the lower atmosphere of t he Sun.\nPrevious theoretical models have suggested that there should be magnetic null point and\ncurrent sheet around the flux cancellation sites in the upper atmos phere(Priest et al. 1994 ;\nvon Rekowski et al. 2006 ). Numerous observations mainly focused on the change of the velo city\nfield and the transverse field around flux cancellation sites, while dire ct observation of the detailed\nstructure andthe dynamic process above the flux cancellation site s were extremely rare. In our event,\nthe loop-loop interactions should be the evidence of magnetic recon nection, the sheet-like structure\nrevealed by the IRIS1400˚A images should be a current sheet resided above the flux cancellatio n\nsites. Moreover, the extrapolated coronal field lines and the dete cted magnetic null point may further\nevidence that magnetic reconnection would occur above the flux ca ncellation sites. These results are\ncomparable with the theoretical models of Zwaan(1987) andPriest et al. (1994), and reveal detailed\nmagnetic field structure above the flux cancellation sites. Accordin g to the theoretical models of\nZwaan(1987) andPriest et al. (1994), we can naturally explain our observations as follow: as L1\ncontacts L2 or L1 contacts L5 in a magnetic null point, magnetic field near the null point would\ncollapse and evolve to a field with a current sheet (as shown by the sh eet-like structure in Figure\n5(e)). Magnetic reconnection between L1 and L2 or between L1 and L5 happened at the null point\nor inside the current sheet, leading to the formation of a set of long loops that connects their far\nends of the interacted loops and a set of short loops (L4) that con nects their adjacent ends. Caused\nby magnetic tension, L4 further submerged and resulted in the flux cancellation. The theoretical\nmodels of Zwaan(1987) andPriest et al. (1994) are suitable for interpreting the two loop-loop in-\nteractions and the associated flux cancellation. However, the det ailed dynamic processes above the\nflux cancellation sites, for instance, the plasma blob ejections, nee d further investigation.\nAn important observational signature in our event is the ejection o f plasma blobs. As men-\ntioned above, these plasma blobs were ejected from the tip of the Y -shaped ends of the sheet-like\nstructure, and the extrapolated coronal field lines reveal an X-s haped configuration containing a\nmagnetic null point. Accordingly, we inferred that the ejected plas ma blobs should be plasmoids,\nwhich are created by the tearing-mode instability occurring in the cu rrent sheet( Furth et al. 1963 ;\nBhattacharjee et al. 2009 ). Previously, plasmoids are frequently observed in the coaxial brig ht rays\nthatappearsinwhitelightimagesinthewakeoftheCMEs( Lin et al.2005 ), inthecurrent sheet ofso-\nlar flares( Takasao et al. 2012 ;Kumar & Cho 2013 ), and in some jets ( Singh et al. 2012 ;Zhang et al.\n2014,2016;Zhang & Zhang 2017 ). More recently, a detailed formation and evolution process of plas -\nmoids was reported by Li et al.(2016). They found that the plasmoids appeared within the current\nsheets at the interfaces between an erupting filament and nearby coronal loops and propagate bidi-\nrectionally along them, and then further along the filament or the loo ps. In the lower atmosphere,\ncontinuous ejections of plasmoids from the flux cancellation sites ha ve seldom been observed directly.\nThrough the numerical simulation method, Ni et al. (2015) simulated magnetic reconnection process\nin partially ionized solar chromosphere and confirmed that fast magn etic reconnection mediated by\ntearing-mode instability could be indeed triggered. In particular, by analysing the Si IVline profiles8\nobtained from the flux cancellation sites of some small-scale reconne ction events, Innes et al. (2015)\nsuggested that a fast reconnection proceeding via tearing-mode instability may play a central role in\nthose small-scale reconnection events. In the present case, the continuous ejection of plasma blobs\nabove the flux cancellation sites from the tip of the Y-shaped ends o f the sheet-like structure are\nthe direct observational evidence that support the idea of Innes et al. (2015). This observation dis-\nplays the detailed dynamic process above the flux cancellation sites, and has a significant physical\nimplication for the magnetic reconnection in the lower atmosphere of the Sun.\nBefore the first loop-loop interaction, we notice that there is conn ectivity between the cancelling\nflux patches p and n1, and there is lack of velocity field information ar ound the flux cancellation\nsites. Moreover, new magnetic flux emerged beside the cancelling flu x patches, and the emerged\npositive flux patches mixed with p, while parts of its negative flux patc hes moved and merged with\nn1. Thus, it is difficult to absolutely rule out the possibility that the sub mergence of original loops\nconnecting p to n1 may also contribute to the magnetic flux cancellat ion. However, it is clear from\nthe time profile of flux changes (Figure 1( g)) that obvious flux cancellation was observed after the\nfirst loop-loop interaction and continued till the end of the observa tion. This time interval covers the\nsecond loop-loop interaction process and the two plasma blob eject ion processes (as indicated by the\npink shadow in Figure 1( g)). Therefore, our observations support a causal relationship a mong the\nloop-loop interactions, the plasma blob ejections, and the flux canc ellation.\nWe thank the anonymous referee for useful comments and sugge stions that have improved the\nquality of the manuscript. We are very grateful to the AIA, HMI, IRIS, andHinodeteams for\nfree access to data. 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M. 2009, ApJ,\n703, 1012\nYang, S., Zhang, J., & Xiang, Y. 2015, ApJL, 798,\nL11\nYokoyama, T., Akita, K., Morimoto, T., Inoue,\nK., & Newmark, J. 2001, ApJL, 546, L69\nZwaan, C. 1987, ARA&A, 25, 83\nZhang, Q. M., Chen, P. F., Ding, M. D., & Ji,\nH. S. 2014, A&A, 568, A30\nZhang, Q. M., Ji, H. S., & Su, Y. N. 2016, SoPh,\n291, 85910\nZhang, J., Yang, S.-H., & Jin, C.-L. 2009,\nResearch in Astronomy and Astrophysics, 9, 921\nZhang, Y., & Zhang, J. 2017, ApJ, 834, 79Zhou, G. P., Zhang, J., Wang, J. X., &\nWheatland, M. S. 2017, ApJL, 851, L111\nFigure 1. SDO/HMI vertical images displaying the general appearance of t he NOAA AR 12257 at 18:00\nUT on 2015 January 9 (panel ( a)) and the cancellation of opposite polarities (panels ( bf)). Negative and\npositive magnetic flux patches are denoted as “n1”, “n2”, “n3 ”, and “p”, respectively. Panel ( g) showing\nthe changes in negative magnetic flux in the blue box in panel ( e). In panel ( b), the transverse fields\nare overplotted as red and yellow arrows, which originate fr om a positive and negative longitudinal field,\nrespectively. The field of view (FOV) of panels ( b)(f) is outlined by the red rectangle in panel ( a). The\nvertical pink shadow in panel ( g) denotes the time when the flux cancellation occurs obviousl y.12\nFigure 2. IRIS1400˚A SJI images ( a)(c) showing the first loop-loop interaction process that took p lace\nbetween two sets of loops,“L1” and “L2”. AIA 94 ˚A image ( d) andHinode/XRT SXR image ( e) showing\nthe coronal connectivity after the interaction. “L3” and “L 4” denote the newly formed loops. Brightenings\nlocated at the footpoints of L1 and L2 are distinctly showed i n the AIA 1600 ˚A difference image ( f). The\norange andred contours overplotted on panels ( a), (d), and(f) represent theintensity contours of the positive\nand negative magnetic fields, with contour levels of 200 G and -100 G, respectively. Likewise, “n1”, “n2”,\n“n3”, “p”, and “p1” denote negative and positive magnetic flu x patches. An animation of panels ( ac) and\n(d) is available. The animation is 2s in duration, covering 19: 39:25 UT to 19:49:49 UT. (An animation of\nthis figure is available.)13\nFigure 3. AIA 304 ˚A images ( a)(d) present the successive ejection of plasma blobs (as indica ted by the\narrows in panels ( bd)) from the flux cancellation sites. Two bright streaks, whic h may imply two sets of\ninteracted loops, are clearly seen in panel ( a). Iso-Gauss contours of ±100Gare superposed by green and\nblue lines on panel ( a). An animation of this figure is available. The animation is 1 s in duration, covering\n20:19:07 UT to 20:36:55 UT. (An animation of this figure is ava ilable.)14\nFigure 4. AIA 304 ˚A images ( a)(c) and a 171 ˚A difference image ( d) showing the second loop-loop\ninteraction process that occurs between two sets of loops, L 1 and “L5” (as indicated by the red arrows). “n”\ndenotes a plage region where the negative footpoints of L5 ar e rooted. Iso-Gauss contours of ±100Gare also\nsuperposed by green and blue lines in panel ( a) and panel ( d). An animation of this figure is available. The\nanimation is 1s in duration, covering 20:37:19 UT to 20:54:5 9 UT. (An animation of this figure is available.)15\n\tG\n \tH\n \nOVMM\u0001QPJOUOVMM\u0001QPJOU \n\u0013\u0011\u001b\u0015\u0019\u0001\u000165 \u0013\u0011\u001b\u0015\u0019\u0001\u000165Q\u0012 O\u0012 Q\nO\nFigure 5. Sequence of IRIS1400˚A SJI images ( a)(d) exhibit the successive ejection of plasma blobs\nfrom a sheet-like structure above the flux cancellation site s. (e) A zoomed view corresponding to the green\nbox in (b). Iso-Gauss contours of ±30Gare also superposed by green and blue lines in panel ( a) and panel\n(e). NLFFF magnetic field extrapolation showing a top view (pan el (f)) and a side view (panel ( g)) of\nthe coronal magnetic field over the flux cancellation region a t 20:48 UT. The red and blue lines represent\nfield lines rooted in the cancelling flux patches n1 and p, resp ectively. A magnetic null point (indicated by\ngreen arrows) lies above the flux cancellation sites. It is lo cated between the red and blue field lines. The\nbackground images are HMI vertical field images. An animatio n of panels ( ae) is available. The animation\nis 3s in duration, covering 20:59:51 UT to 21:15:08 UT. (An an imation of this figure is available.)"    },    {        "title": "1202.4048v1.Measuring_spectrum_of_spin_wave_using_vortex_dynamics.pdf",        "content": "Measuring spectrum of spin wave using vortex dynamics\nShi-Zeng Lin and Lev N. Bulaevskii\nTheoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n(Dated: July 13, 2021)\nWe propose to measure the spectrum of magnetic excitation in magnetic materials using motion of vortex\nlattice driven by both ac and dc current in superconductors. When the motion of vortex lattice is resonant with\noscillation of magnetic moments, the voltage decreases at a given current. From transport measurement, one\ncan obtain frequency of the magnetic excitation with the wave number determined by vortex lattice constant.\nBy changing the lattice constant through applied magnetic fields, one can obtains the spectrum of the magnetic\nexcitation up to a wave vector of order 10 nm\u00001.\nPACS numbers: 74.25.Uv, 74.25.F-, 74.25.Ha\nI. INTRODUCTION\nMeasurement of the spectrum of magnetic excitation is cru-\ncial for understanding magnetic properties of magnetic mate-\nrials. Neutron scattering has been widely used to measure the\nspectrum. However, the source of neutron is not easy to ac-\ncess. It also requires large samples. For example, the neutron\nscattering is not applicable to ultra-thin films because the in-\nteraction between the spin wave and neutrons is week1. On the\nother hand, conventional techniques such as the ferromagnetic\nresonance and Brillouin light scattering can only measure the\nenergy gap of spin wave.\nAbrikosov vortices being magnetic excitation in supercon-\nductors are expect to interact strongly with the magnetic mo-\nments, which points a possible way to measure the magnetic\nexcitation using vortex dynamics. It was proposed that one\ncan use vortex dynamics driven by a dc current to measure the\nspin-wave excitation through transport measurement of the IV\ncharacteristics2. One may be able to obtain the spectrum of\nspin wave up to the wave number 1 =\u0018with\u0018being the coher-\nence length. The proposed technique can be applied to super-\nconductors with coexistence of magnetic and superconduct-\ning order3,4. One can also measure the spectrum of magnetic\nexcitation in conventional magnetic materials by fabricating\nartificial bilayer systems, consisting of the magnetic material\nto be measured and a superconducting layer5.\nIn this work, we propose to measure the spectrum of mag-\nnetic excitation in magnetic materials using motion of vortex\nlattice driven by both ac and dc current through transport mea-\nsurement. The advantages with ac current are: 1) it does not\nrequire large current to reach the resonances between the vor-\ntex motion and spin-wave excitations, thus the nonequilibrium\ne\u000bect that leads to the instability of vortex lattice and heating\ne\u000bect can be minimized; 2) the e \u000bect of random pinning cen-\nters can be minimized with an ac current; 3) additional infor-\nmation such as wave vector of the vortex lattice can also be\nobtained.\nII. PROPOSAL\nWe consider a magnetic superconductor where magnetic or-\ndering coexists with superconductivity as shown in Fig. 1(a)\nor a bilayer system with magnetic film and superconductingfilm as shown in Fig. 1(b). We assume that the magnetic sys-\ntem has an in-plane easy axis ( xaxis). The spectrum of spin\nwave \n(k) is gapped due to the anisotropy, where kis the\nwave vector. A magnetic field to create vortex lattice is ap-\nplied perpendicular to the easy axis of the magnetic moments\n(z-axis). A transverse current both with dc, Idc, and ac com-\nponent, Iacsin(!It), then is applied to drive the vortex lattice.\nDriven by the Lorentz force, vortex lattice moves with a dc\nvelocity vdc, and also oscillates due to the ac current and the\ninteraction with the magnetic moments, v=vdc+vac. The mo-\ntion of vortex lattice perturbs the magnetic moments and ex-\ncites magnons. The resonance between the motion of vortex\nlattice and precession of magnetic moments is achieved un-\nder appropriate condition [see Eq. (14)], which can be probed\nfrom IVcurve.\nUsing the vortex-as-particles approximation, the motion of\nvortex is described by an over-damped Langevin equation\n\u0011@tri=Fp+FL+Fvv\u0000@riHint(r\u0000ri); (1)\nwhere ri=(xi;yi) is the vortex coordinate and \u0011=\nBzHc2\u001bn=c2is the Bardeen-Stephen damping constant with\nHc2the upper critical field and \u001bnthe normal state conduc-\ntivity just above the critical temperature. Fvvis the repul-\nsive force between vortices. FL=Fdc+Facsin(!It) is the\nLorentz force due to the ac and dc current. Fpis the pinning\nforce, andHintis the Zeeman interaction between magnetic\nmoments and vortices\nHint(r\u0000ri)=\u0000Z\nBv;z(r\u0000ri)Mz(r)dr2; (2)\nFIG. 1. (color online) (a) and (b): Schematic view of vortex dynam-\nics in (a) magnetic superconductors (MSC) where superconducting\nand magnetic ordering coexist, (b) artificial bilayer systems consist-\ning of superconducting (SC) and magnetic (M) layers.arXiv:1202.4048v1  [cond-mat.supr-con]  18 Feb 20122\nwhere Bv;zis the magnetic field associated with a vortex and\nMzis the magnetic moment along the z-axis. We have used\na continuum description of the magnetic subsystem because\nthe vortex size is much larger than the lattice constant of spin\nsubsystem. The random pinning centers distort lattice order\nin static6. However in the flux flow region, the vortex lattice\norder is recovered because the motion of lattice quickly aver-\nages out the e \u000bect of random pinning centers7,8. In this region,\nit is safe to approximate straight vortex line along the z-axis\nand the problem becomes two dimensional.\nIn the following calculations, we will focus on the reso-\nnance between motion of vortex lattice and magnetic subsys-\ntem, and neglect the resonance due to pinning centers. We use\nan approximation that the motion of vortex lattice is not af-\nfected by the magnetic moments. The motion of vortex lattice\nin the presence of the dc and ac current thus is described by\nri(t)=Ri\u0000vdct\u0000racsin(!It+\u001e) (3)\nwhere Riforms a regular lattice with a wave vector Gand\u001e\nis an arbitrary phase. The vortex lattice oscillates with fre-\nquency!Iand amplitude rac=Jac\b0=(c\u0011!I) due to the ac\ninput current Jac.\nWe use the quasistatic approximation that the structure of\nvortex driven by the Lorentz force remains the same as the\nstatic one. The magnetic induction Bof the moving vortex\nlattice is described by the London equation taking the mag-\nnetic moments Minto account3,9–11\n\u00152\nLr\u0002r\u0002 (B\u00004\u0019M)+B= \b 0X\ni\u000e[r\u0000ri(t)]ˆz;(4)\nwith\u0015Lthe London penetration depth without magnetic sub-\nsystem, \b0=hc=(2e) the quantum flux and ˆzthe unit vector\nalong the z-axis.\nThe magnetic field ”seen” by the magnetic subsystem is\nHz=Bz\u00004\u0019Mz. Using the linear response approximation,\nMz(k;!)=\u001fzz(k;!)Hz(k;!) with\u001fzz(k;!) the magnetic sus-\nceptibility, we obtain the induced magnetization Mz(k;!)=\n\u001fzz(k;!)Bz(k;!)=[1+4\u0019\u001fzz(k;!)]. We then obtain the mag-\nnetic fields associated with the vortex lattice from Eq. (4)\nBz(G;!)=B0K(G;!)0BBBB@G2\u00152\nL\n1+4\u0019\u001fzz(G;!)+11CCCCA\u00001\n;(5)\nwith B0the average magnetic induction. The London pen-\netration depth is renormalized in the presence of magnetic\nsubsystem3,9–11. The functionK(G;!) is\nK(G;!)=Z1\n0dtexp\u0002iG\u0001(vdct+racsin(!It+\u001e))\u0000i!t\u0003\n(6)\nUsing the Fourier expansion of the Bessel function, we have\nK(G;!)=n=+1X\nn=\u00001Jn(G\u0001rac)exp(in\u001e)\u000e(G\u0001vdc\u0000!+n!I)\n(7)\nwhere\u000e(x) is the Dirac delta function and Jnis the Bessel\nfunction of the first kind with an integer n. Due to the pinningcenters, the Bragg peaks of the vortex lattice are smeared out.\nThe velocity of vortex is also fluctuating. These two e \u000bects\ncan be taken into account by replacing the \u000efunction in Eq.\n(7) by\nW(G;!)=1\nG\u0001vdc\u0000!+n!I+i\r; (8)\nwhere\raccounts for the broadening of the resonance peaks\nby the pinning centers.\nTheIVcharacteristics of the system can be derived from the\npower balance equation. The energy input per unit volume is\nJEwith Jthe external current and Ethe electric field. The\npower dissipated per unit volume by quasiparticles due to the\nmotion of vortex lattice is \u0011v2. The velocity of vortex lattice\ncan be measured from voltage Eaccording to v=Ec=B0.\nThe energy per unit volume P(E) transferred from the vortex\nlattice into the magnetic subsystem is2,12\nP(E)=\u0000hMz(r;t)@tHz(r;t)i\n=\u0000R\nIm[\u001fzz(k;!)]!jHz(\u0000k;\u0000!)j2d2kd!(9)\nwhereh\u0001\u0001\u0001irepresents time and space average. The trans-\nferred energy P&0 finally dissipated through magnetic\ndamping. Using Hz=Bz=(1+4\u0019\u001fzz), we have\nP=\u0000X\nG;nZ\nd!Im\u0002\u001fzz(G;!)\u0003!\f\f\f\f\f\fB0Jn(G\u0001rac)W(G;!)\n\u00152\nLG2+1+4\u0019\u001fzz(G;!)\f\f\f\f\f\f2\n(10)\nFrom the power balance condition JE=\u0011v2+P, we derive\ntheIVcharacteristics\nJ=\u0011c2\nB2\n0E+1\nEP(E): (11)\nIn the presence of magnetic subsystem, the e \u000bective viscosity\nof vortex is enhanced\n\u0011e\u000b=\u0011+P(E)\nv2: (12)\n02468n\n=3n=2n=1dI/dV0\n3 6 9 1 21 5036(b)(\na)n=1n=2Current IV\noltage Vn=3\nFIG. 2. (color online) (a): Schematic view of the IVcurve and (b)\ndI=dVcurve.3\nFor a given current, voltage drops because of the energy ex-\nchange between the vortex lattice and magnetic subsystem.\nThe susceptibility of the magnetic subsystem is13\n\u001fzz(!;k)=!M\n(k)\n\n2(k)\u0000!2+i!\f; (13)\nwhere!M=\u00162nM=(2~) with\u0016the magnetic moment and nM\nthe density of magnetic moment. \n(k) is the dispersion of\nspin-wave excitation and \fthe relaxation rate of spin wave.\nTherefore the resonance takes place when\nn!I+G(B)\u0001vdc= \n(G); (14)\nis satisfied. The principal axis of the moving vortex lattice is\nalong the driving direction, and G(B)\u0001vdc=2\u0019vdcpB0=\b0\nfor a square lattice. The resonance amplitude depends on\nthe amplitude of the ac current according to [Jn(G\u0001rac)]2.\nBoth the random pinning centers and magnetic damping con-\ntribute to the broadening of resonance, yielding a linewidth\nof resonance of order \f0=\f+\r. Away from the resonance\nj\n2(G)\u0000(!0\nn)2j\u001d!0\nn\f0with!0\nn\u0011G\u0001vdc+n!I, we have\nIm[\u001fzz]\u00190. The current is then given by J=\u0011c2E=B2\n0, be-\ncause there is no spin wave excitation with frequency !0\nnat the\nwave vector of vortex lattice G. At resonance, the current is\nenhanced for a given E,J=\u0011c2E=B2\n0+ \u0001J. Here we estimate\nthe current enhancement \u0001J.\nWe consider frequency slightly away from the resonant fre-\nquency \u0001!= \n\u0000(n!I+G\u0001vdc)with\f0<<\u0001! << \n. We\nthen have Re\u0002\u001fzz\u0003=!M/(2\u0001!) and Im\u0002\u001fzz\u0003=!M\f=(2\u0001!)2.\nFor a square vortex lattice, we have G=2\u0019pB0=\b0\u0010\nlx;ly\u0011\nwith integers lxandly. In the interval of frequency deviation\n\u0001!where\u00152\nLG2>>4\u0019\u001fzz(G;!0\nn), we estimate the enhance-\nment of current over the linear background when the current\nis injected along the xdirection\n\u0001J\nJ=16\u00192~nM\f0B0\n!0n\u0011\b0l2\nxJ2\nn(G\u0001rac)\n\u0010\nl2x+l2y\u00112: (15)\nFor the HoNi 2B2C magnetic superconductor2,14,15, we have\nHc2=10 T,\u001bn\u0018107(\n\u0001m)\u00001,nM=1022cm\u00003and\n\f\u0018106s\u00001. As\rdepends on the concentration of pin-\nning centers and is not known for magnetic superconductors,\nhere we neglect \rand only keep \ffor the estimation of \u0001J.\nWe then estimate \u0001J=J\u00190:8l2\nxJ2\nn(G\u0001rac)\u0010\nl2\nx+l2\ny\u0011\u00002at a fre-\nquency!0\nn\u001910 GHz.\nTo achieve measurable enhancement of current \u0001Jat reso-\nnances, one requires G\u0001rac\u00181. Using rac=Jac\b0=(c\u0011!I),\nwe estimate the amplitude of the ac current that yields mea-\nsurable resonances\nJac\u0018\u0011!Ic\nG\b0=!I\nGBc2\u001bn\nc: (16)\nFor HoNi 2B2C we estimate Jac\u0018109A=m2when!I\u0019\n10 GHz, G\u00190:3nm\u00001, which is much smaller than the de-\npairing current Jdp=cBc2\u0018.\u0010\n6p\n3\u0019\u00152\u0011\n\u00191013A=m2with\n\u0018\u001950 nm and\u0015L\u0019100 nm.15Thus the measurable current\nenhancement \u0001Jcan be realized experimentally.Here we present a procedure to extract \n(G) from the IV\ncurve. One measures IVcurve at a given vortex density B0\nand ac current. When the resonance condition Eq. (14) is\nsatisfied, the current is enhanced according to Eq. (15) for a\ngiven voltage, see Fig. 2(a). This enhancement can be seen\nclearly from the curve of dI=dVas a function of V, see Fig.\n2(b). The linewidth of the resonance is due to the magnetic\ndamping and random pinning centers. The voltage where cur-\nrent is enhanced is the resonant voltage. One then changes\nthe frequency of the ac current !I, and measures the resonant\nvoltage as a function of !I. From the voltage, one knows the\nvelocity of vortices lattice at resonance. One then plots !I\nas a function of the resonant velocity. From the interception\nwith the vertical axis, one obtains \n(G) according to Eq. (14).\nFrom the slope, one obtains Galong the driving direction. By\nchanging Gthrough external magnetic fields, one then obtains\nthe spectrum of spin wave \n(G). The spectrum can also be\ndetermined from the sub-resonance with n=2.\nIII. DISCUSSION\nHere we discuss the region that Gcan be tuned. The mag-\nnetic field associated with vortices cants the magnetic mo-\nments and induces magnetization along the vortex axis ¯Mz\nwith ¯Mz=\u001fzzB0=(4\u0019\u001fzz+1), thus the spectrum measured\nis\n(G;¯Mz). To get the spectrum at ¯Mz=0, we need\n¯Mz=Ms\u001c1 with Msthe saturation magnetization, which\ngives an upper bound for B0. Since B0\u0019\b0=a2with the\nvortex lattice constant a, this limits the maximal wave vector\nthat can be achieved G\u00192\u0019=a. For the HoNi 2B2C magnetic\nsuperconductor2,14,15,Ms\u00191000 G and \u001fzz\u00190:03, which\ngives maximal Gmax\u00190:3 nm\u00001. For a large susceptibility\n\u001fzz\u00181, it was shown that vortices may form clusters due to\nthe attraction between vortices12. To create vortex lattice, the\nminimal vortex density is nv\u00191=\u00152\nL. Thus the lower bound of\nGin this case is G=2\u0019=\u0015 L. For\u001fzz\u001c1,Gcan be tuned to 0\ncorresponding to a dilute vortex lattice.\nThe magnetic correlation length \u0018mis much smaller than the\nmaximal vortex lattice constant, \u0018mGmax\u001c1. Therefore we\ncan expand \nas\n(G)\u0019!g+~2G2=(2ms) for a ferromagnet,\nwith!gthe energy gap and msthe mass of spin wave. For an\nantiferromagnet, \n(G)\u0019!g+vs\u0001Gwith vsthe spin wave ve-\nlocity. Although the present method can only measure small\nportion of the Brillouin zone compared with the neutron scat-\ntering, addition information such as the mass msor velocity vs\nof the spin wave can be extracted, which is advantageous over\nthe ferromagnetic resonance measurement.\nFor magnetic materials, typically the energy gap is !g\u0018\n10 GHz to 100 GHz. Experimentally, one can achieve !I\u0018\n100 GHz by microwave. Thus one can measure the spin-wave\nspectrum at a low velocity of the vortex lattice. The nonequi-\nlibrium e \u000bect that destabilizes the vortex lattice, caused by the\nLarkin-Ovchinnikov mechanism16, can be avoided.\nThe ac current itself generates ac magnetic field Hac=\n2\u0019JacL=cwith Lbeing the linear size of the system. The in-\nduced magnetization is Mac=\u001fzzHac. We estimate Mac=Ms\u0019\n0:1 for a system with linear size 100 \u0016m for HoNi 2B2C. Thus4\nthe e\u000bect of ac magnetic field due to the ac current does not af-\nfect the orientations of the magnetic moments. The ferromag-\nnetic resonance by Hacis avoided because !I<! g. The para-\nmetric resonance17–19below!gcan be avoided with a small\nHac<<Ms.\nIn the presence of pinning centers, it requires a critical cur-\nrent density to depin the vortex for a dc current. The ac force\ndue to the ac current helps vortices to depin from the pinning\ncenters, similar to thermal noise.20–22. For Pb-In and Nb-Ta\nalloys20, it was found experimentally that the pinning e \u000bect\nbecomes very weak for an ac current with frequency above\n!c=10 MHz. Thus the pinning e \u000bect can be greatly sup-\npressed with the ac current with frequencies !I\u001d!cused in\nour proposal.\nWhen the motion of vortex lattice experiences a quenched\npinning center, it can be e \u000bectively modelled as a particle\nmoving in a periodic potential. When an ac current is ap-\nplied in additional to the dc current, there will be current steps\nknown as the Shapiro steps in the IVcurve23,24. The Shapiro\nsteps occur when the frequency of the ac current matches\nthe washboard potential for vortex due to the pinning center,\nn!I=G\u0001vdc. The Shapiro steps thus can be easily distin-\nguished from the resonance with the spin-wave excitations,\nsee Eq. (14).\nThe current in Eq. (11) does not depend on the phase di \u000ber-\nence between the ac current and spin wave \u001e, thus precludesthe existence of the Shapiro steps due to the magnetic mo-\nments. The reasons for the absence of the Shapiro steps are as\nfollows. First, the lattice constant of spin subsystem is much\nsmaller than \u0015L, thus the vortex lattice does not feel the wash-\nboard potential due to the magnetic moments. Secondly, the\nresponse of magnetic moments to the magnetic field associ-\nated with the vortex lattice is linear, thus no dc current is in-\nduced in this linear region.\nIn summary, we have proposed to measure the spectrum\nof magnetic excitation in magnetic materials using motion of\nvortex lattice driven by both ac and dc current in superconduc-\ntors. The resonance between the motion of vortex lattice and\nspin wave manifests as an enhancement of current at a given\nvoltage. Thus the frequency of the magnetic excitation with\nthe wave number determined by vortex lattice constant can be\nextracted from transport measurement. By changing the lat-\ntice constant through applied magnetic fields, the spectrum of\nthe magnetic excitation up to a wave vector of order 10 nm\u00001\ncan be obtained.\nIV . ACKNOWLEDGEMENT\nWe are indebted to C. D. Batista, M. Ross and A. V . Os-\ncar for helpful discussion. Research are supported by the Los\nAlamos Laboratory directed research and development pro-\ngram with project number 20110138ER.\n1R. V ollmer, M. Etzkorn, P. S. A. Kumar, H. Ibach, and\nJ. Kirschner, Phys. Rev. Lett. 91, 147201 (2003).\n2L. N. Bulaevskii, M. Hruska, and M. P. Maley, Phys. Rev. Lett.\n95, 207002 (2005).\n3L. N. Bulaevskii, A. I. Buzdin, M. L. Kulic, and S. V . Panjukov,\nAdv. Phys. 34, 175 (1985).\n4A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n5I. F. Lyuksyutov and V . L. Pokrovsky, Adv. Phys. 54, 67 (2005).\n6G. Blatter, M. V . Feigelman, V . B. Geshkenbein, A. I. Larkin, and\nV . M. Vinokur, Rev. Mod. Phys. 66, 1125 (1994).\n7A. E. Koshelev and V . M. Vinokur, Phys. Rev. Lett. 73, 3580\n(1994).\n8R. Besseling, N. Kokubo, and P. H. Kes, Phys. Rev. Lett. 91,\n177002 (2003).\n9M. Tachiki, H. Matsumoto, and H. Umezawa, Phys. Rev. B 20,\n1915 (1979).\n10K. E. Gray, Phys. Rev. B 27, 4157 (1983).\n11A. I. Buzdin and L. N. Bulaevskii, Sov. Phys. Usp. 29, 412 (1985).\n12S. Z. Lin, L. N. Bulaevskii, and C. D. Batista, arXiv:1201.4195\n(2012).13L. P. L ´evy, Magnetism and Superconductivity (Springer, New\nYork, 2000).\n14H. Eisaki, H. Takagi, R. J. Cava, B. Batlogg, J. J. Krajewski,\nJ. Peck, W. F., K. Mizuhashi, J. O. Lee, and S. Uchida, Phys.\nRev. B 50, 647 (1994).\n15K. H. M ¨uller and V . N. Narozhnyi, Rep. Prog. Phys. 64, 943\n(2001).\n16A. I. Larkin and Y . N. Ovchinnikov, Sov. Phys. JETP 41, 960\n(1976).\n17H. Suhl, J. Phys. and Chem. Solids 1, 209 (1957).\n18E. Schl ¨omann, J. Appl. Phys. 32, 1006 (1961).\n19M. Chen and C. E. Patton, Nonlinear Phenomena and Chaos in\nMagnetic Materials, Editor: P . E. Wigen (World Scientific Pub.\nCo. Inc., Singapore, 1994).\n20J. I. Gittlema and B. Rosenblu, Phys. Rev. Lett. 16, 734 (1966).\n21A. Glatz, T. Nattermann, and V . Pokrovsky, Phys. Rev. Lett. 90,\n047201 (2003).\n22W. P. Cao, M. B. Luo, and X. Hu, New J. Phys. 14, 013006\n(2012).\n23A. T. Fiory, Phys. Rev. Lett. 27, 501 (1971).\n24A. Schmid and W. Hauger, J. Low. Temp. Phys. 11, 667 (1973)."    },    {        "title": "2202.01394v2.Magnetic_domain_wall_dynamics_under_external_electric_field_in_bilayer_CrI__3_.pdf",        "content": "Magnetic domain wall dynamics under external electric field in bilayer CrI 3\nSahar Izadi Vishkayi1and Reza Asgari1, 2,\u0003\n1School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran\n2School of Physics, University of New South Wales, Kensington, NSW 2052, Australia\nMotivated by manipulating the magnetic order of bilayer CrI 3, we carry out microscopic calculations to find\nthe magnetic order and various magnetic domains of the system in the presence of an electric field. Making\nuse of density functional simulations, a spin model Hamiltonian is introduced consisting of isotropic exchange\ncouplings, Dzyaloshinskii-Moriya (DM) interaction, and on-site magnetic anisotropy. The spin dynamics of\ntwo well-known states of bilayer CrI 3, low temperature (LT) and high temperature (HT) phases, are obtained\nby solving the Landau-Lifshitz-Gilbert equation. We show that the magnetic texture is stacking-dependent in\nbilayer CrI 3and stable magnetic domains can appear in the HT stack which are tunable by external electric and\nmagnetic fields. Therefore, we suggest that the HT phase represents a promising candidate for data storage in\nthe modern generation of spintronic devices working on magnetic domain engineering.\nI. INTRODUCTION\nTwo-dimensional (2D) magnetic materials have been in-\nvestigated by experimental [1–6] and theoretical [7–11] re-\nsearchers to analyze their attractive properties from the ini-\ntial report of the successful synthesis of thin layer CrI 3[1].\nThe presence of a substantial magnetocrystalline anisotropy\nwas discovered, thus breaking the invariance under spin ro-\ntations, and the crystal magnetic ordering was observed to\nbe a layer-dependent phenomenon. Because of the existence\nof metastable magnetic domains with ultra-thin domain walls\n(like permanent nanoparticle magnets [12]) which possess\nspin dynamics [13–16], CrI 3, in monolayer and multilayer\nforms, has recently been described as a viable contender for\nthe modern generation of memory devices. The microscopic\nnature of these magnetic domains in the presence of external\nfields is still poorly understood, offering a significant chal-\nlenge for theoretical materials science.\nTwo geometrically stable stacks of bilayer CrI 3structures\nare known as low-temperature (LT) and high-temperature\n(HT) phases (in more than 210K) [8, 17, 18]. The LT phase\nis a ferromagnetic (FM) ground state in which the magnetic\nmoments of the Cr atoms are aligned in both layers, whereas\nthe magnetic moments of the Cr atoms in distinct layers of\nthe HT phase are in an antiferromagnetic (AFM) configura-\ntion [8, 19–22]. The competition between orbitally depen-\ndent interlayer AFM super-super-exchange and interlayer FM\nsuper-super-exchange causes the magnetic ordering. In the\nHT bilayer CrI 3, a vertical external electric field generates an\nAFM to an FM magnetic transition [8, 23, 24]. Because of\ntheir sensitivity to the electric field, the HT phase of the bi-\nlayer CrI 3can be more efficient in ultrafast memory devices,\nand it could be the first device for the engineering of magnetic\ndomains in the world of 2D magnets. For a monolayer up\nto bulk structures, the Curie temperature of a thin layer CrI 3\nranges from 45K to61K [1, 25]. The magnetic ordering tem-\nperature of a thin layer CrI 3is raised by the number of layers,\naccording to experiments [1]. The N ´eel temperature of 45K\n\u0003Electronic address: asgari@ipm.iris reported for the HT bilayer [26] and it is expected that the\nLT bilayer maintains a larger Curie temperature due to strong\nFM ordering between the layers [27].\nAlthough the Ising Hamiltonian model is a classic model\nfor explaining the magnetic behavior of 2D CrI 3, it cannot ex-\nplain the experimentally observed spin wave [28]. To explain\nthe spin canting physics, an extended Heisenberg Hamilto-\nnian containing Kitaev [29], biquadratic anisotropy exchange\nterm [13], or Dzyaloshinskii-Moria (DM) [30, 31] interaction\nterms was proposed. The spin wave in a CrI 3structure [32]\nis described by a Heisenberg Hamiltonian with Kitaev or DM\ninteractions, and a small portion of the DM interactions in 2D\nCrI3is responsible for a topological gap [31].\nThe ability to alter and adjust magnetic domains of 2D bi-\nlayer magnets to make them useful in technology and spin-\ntronics is a key question here. We focus on the bilayer CrI 3in\nthe presence of an external electric field [8, 19, 23, 24, 26] to\nanswer this specific topic.\nIn this study, we use a spin model Hamiltonian in the pres-\nence of external fields, with isotropic exchange and DM inter-\nactions. We use density functional theory (DFT) computations\nto derive a suitable spin model Hamiltonian for the bilayer\nCrI3that accurately reproduces the magnetic characteristics\nof the system. The isotropic exchange coupling parameters\nof the bilayer CrI 3have been investigated in the presence of\nan external electric field [19], but the Dzyaloshinskii-Moria\n(DM) interaction has yet to be investigated. In contrast to what\nwe found for a monolayer CrI 3[30], our calculations suggest\nthat the DM interaction in the HT bilayer CrI 3is substan-\ntially connected to the applied electric field. The spin dynam-\nics of the system are calculated using the Landau-Lifshitz-\nGilbert formalism [33] with DFT-derived exchange parame-\nters as starting inputs. To obtain a temperature dependent\nmagnetization, we also employ the Monte Carlo Metropolis\ntechnique with 106time steps for a 50\u000250nm2layer.\nThe weak strength of the isotropic intralayer exchange cou-\npling signifies the magnetic ordering of the layers, which can\nbe modified by an external electrical field, according to our\nfindings. The findings also reveal that an electric field, like\na magnetic field, alters the magnetic texture of the system.\nFurthermore, by applying an electric field resulting from an\nincrease in the intralayer isotropic exchange coefficient, thearXiv:2202.01394v2  [cond-mat.mtrl-sci]  16 Jun 20222\n(a) (b)\n(c) (d)1234\n2 13 4βα\nθ\nFIG. 1: (Color online) The top view ((a) and (c)) and side view ((b)\nand (d)) of the HT and LT bilayer CrI 3structures, respectively. The\nblack lozenges show the unit cell of bilayers. The blue (pink) cir-\ncles represent chromium (iodine) atoms. In (b) and (d), chromium\natoms are labeled by 1-4 in the unit cell. In (b) the bonding angles\nbetween the atoms of each layer are shown by \u000b,\fand\u0012. The crystal\nparameters are given in Table I.\ntemperature of the magnetic order of the HT bilayer is raised.\nThe electric field also affects the magnetic texturing on the\ndomain walls. The results reveal that the magnetic textures\nin the studied systems can be manipulated using an external\nelectric field. To be more precise, in the LT bilayer, we show\nthat the DM interactions lead to the creation of a quasi-circular\ndomain shape in the layers, but its value is extremely smaller\nthan the intralayer exchange coupling to stabilize it in the sys-\ntem, therefore, the obtained magnetic domain is unstable and\ndisappearing quickly.\nThis paper is organized as follows. We commence with a\ndescription of our theoretical formalism in Sec. II, followed\nby the details of the DFT simulations, the spin model Hamil-\ntonian and spin dynamics. Numerical results for the spin-spin\ninteraction parameters in the presence of an electric field, the\nexchange coefficients in both the HT and LT phases and spin\ndynamics in two cases are reported in Sec. III. We summarize\nour main findings in Sec. IV .\nII. THEORY AND MODELS\nA. DFT calculations\nWe use Quantum Espresso package [34] to simulate\ndensity-functional based calculations. The generalized gra-\ndient exchange-correlation within Perdew-Burke-Ernzerhof\n[35] functional is considered to expose the electronic and\nmagnetic ground states of two bilayer CrI 3structures. An en-\nergy cut-off of 50Ry for wave vector and an 8\u00028\u00021grid\nmesh ofk-points within the first Brillouin zone are definedas the converged input parameters. To avoid repeats effects,\nwe consider a vacuum of 20˚A along the z-direction. The\nvan der Waals Grimme-D2 [36] correction is used to consider\nthe interaction between layers. DFT +Ucalculations [37] are\nalso needed to find correct magnetic ground states of CrI 3bi-\nlayers. At this point, we consider U= 3 eV as the on-site\nHubbard parameter similar to Refs. [15, 17, 19, 38]. To ob-\ntain a reliable total ground-state energy, we maintain a high\ndegree of accuracy of 10\u000010eV . The bilayers are relaxed until\nthe maximum force on each atom is 0:01eV/˚A even under\nthe electric field conditions. It should be noted that the con-\nsideration of the spin-orbit coupling (SOC) and noncollinear\nspin-polarization are essential to obtain the DM interaction\nand magnetic anisotropic energy (MAE).\nB. Spin-model Hamiltonian\nA spin model Hamiltonian for a 2D magnetic hexagonal\nlattice in each layer is utilized to provide spin-spin interactions\nin a magnetic 2D bilayer CrI 3system:\nH=X\ni;j[1\n2JijSi\u0001Sj+\rijSizj2+\n1\n2Dij\u0001(Si\u0002Sj) +X\ni\u0016BBext\u0001Si]; (1)\nwhereJijis the isotropic exchange coupling parameter be-\ntweeniandjatoms,\ridenotes a magnetic anisotropy co-\nefficient, Dijis the DM interaction vector and the last term\nshows the effect of an external magnetic field, Bext, and\u0016B\nis the magnetic Bohr. The magnetic moment of ithCr atom\nis indicated by Siin Eq. (1). It is intriguing to compare the\nstrength of the interlayer and intralayer exchange coefficient\nparameters which are respectively referred by vandtindices\nin this paper.\nC. Spin Dynamics\nThe Landau-Lifshitz-Gilbert (LLG) equation [33] is used to\natomistic model the spin dynamics of the system after the co-\nefficients of the spin Hamiltonian are established. We follow\nthe method introduced by Evans et al. [39] implemented in\nV AMPIRE code. The time evolution of the spin direction is\ngiven by LLG equation according to\n@Si\n@t=\u0000\r\n1 +\u00152[Si\u0002Hi\neff+\u0015Si\u0002(Si\u0002Hi\neff)];(2)\nwhere\ris the gyromagnetic ration and \u0015is microscopic\ndamping equal to unity for finding the equilibration states\nquickly. The effective field applied on each spin is defined\nbyHi\neff=\u0000@H=@Si+Hi\nthand the effective thermal field,\nHi\nth, is counted by Langevin dynamic approach [40] to con-\nsider thermal spin fluctuations.\nIn order to obtain the spin dynamics simulations, a 100 nm\n\u0002100 nm square bilayer CrI 3was kept in thermal equilib-\nrium above the critical (N ´eel or Curie) temperature and then3\nlinearly cooled to 0 K in 2 ns. The magnetic moments of\natoms are randomized at a temperature above the critical tem-\nperature, so we chose random initial spin directions for the\nspin dynamics simulation. The time step of 1 fs is considered\nin the integration scheme of Heun’s method [41] for solving\nLLG equations, suitable for materials with low critical tem-\nperature and large dimensions to obtain stable results [39].\nThe long-range demagnetization field is also considered in\nthe spin dynamics calculations. The magnetic dipole-dipole\ninteraction induces local spin-flip in the magnetic materials\nand leading to the creation of complicated spin texture even in\nthe materials with a FM ground state [42, 43]. We discretize\nthe layer to macrocells with 1nm3volume to avoid expenses\nof the demagnetization field computing at the atomistic level.\nThe magnetic moments of macrocells, mmc, are defined by\nthe sum of atomic magnetic moments within the macrocell\nand the demagnetization field of ithmacrocell is given by\nHmci\nd=\u00160\n4\u0019(X\ni6=jMij\u0001mj\nmc)\u0000\u00160\n3mi\nmc\nVimc; (3)\nwhereVi\nmcis the volume of ithmacrocell,\u00160= 4\u0019\u000210\u00007is\nthe vacuum permeability and Mijis the dipole-dipole inter-\naction matrix between iandjmacrocells defined as [39],\nMij=2\n6643rxrx\u00001\nr3\nij\u00001\n33rxry 3rxrz\n3rxry3ryry\u00001\nr3\nij\u00001\n33ryrz\n3rxrz 3ryrz3rzrz\u00001\nr3\nij\u00001\n33\n775;(4)\nwhere the positions of macrocells are obtained by the mag-\nnetic center of the mass relation and the vector between the\nposition ofiandjmacrocells is shown by rij=rij(rx^i+\nry^j+rz^k). We consider that the sheet is linearly cooled in 2\nns from an equilibrium temperature, upper the Curie temper-\nature, to zero Kelvin. To obtain temperature dependent mag-\nnetization, we use the Monte Carlo Metropolis algorithm with\n106time steps for a 50\u000250nm2sheet.\nIII. NUMERICAL RESULTS AND DISCUSSIONS\nThe magnetic ground state, symmetric and asymmetric ex-\nchange coefficients, and temperature-dependent magnetiza-\ntion of bilayer CrI 3in both the HT and LT phases in the\nground state are presented in this section. The effects of an\nexternal electric field on the coefficients are then studied. Fi-\nnally, we investigate the spin dynamics of both phases in the\npresence of a perpendicular electric field.\nThe HT phase of bilayer CrI 3is in the form of the mono-\nclinic structure, while the LT bilayer possesses a rhombohe-\ndral crystal structure as displayed in Figs. 1(a)-(d). The differ-\nent crystal structures leads to the unique magnetic behaviors.\nBoth the HT and LT phases are relaxed by spin-dependent\nDFT calculations and their optimized lattice constant, bond-\ning lengths and angles are reported in Table I. The results re-\nveal that the bonding angles and lengths of both layers are the\nsame, and that the geometry of the bilayer is unaffected by theelectric field, which is in good agreement with those described\nin Ref. [8]. To obtain a formation of the energy, we use the\nstraight formula of EF=Et(bilayer )\u0000nCrE(Cr)\u0000nIE(I)\nwhereEt(bilayer )is the total energy of the bilayer CrI 3,\nnCr(nI)is the number of the chromium (iodine) atoms in the\nunit cell, and E(Cr)andE(I)are the energy of free Cr and\nI atoms, respectively. The formation energies of the HT and\nLT bilayers are given in Table I indicating that the both phases\nare energetically favored to occur.\nA. Exchange coefficients in the HT phase\nIn the HT phase of bilayer CrI 3, there are four Cr atoms\nin the unit cell; two in each layer. At zero external magnetic\nfield, if the nearest neighbor interlayer and intralayer interac-\ntions are considered, the extended first two terms of the model\nHamiltonian (Eq. (1)) in the unit cell can be written as\nH=1\n2(\u0011tJt[(S1\u0001S2) + (S2\u0001S1) + (S3\u0001S4) + (S4\u0001S3)]+\n\u00111vJ1v[(S1\u0001S3) + (S3\u0001S1) + (S2\u0001S4) + (S4\u0001S2)]+\n\u00112vJ2v[(S1\u0001S4) + (S4\u0001S1) + (S2\u0001S3) + (S3\u0001S2)])+\n\u0011\rjSzj2(5)\nwhere\u0011t= 3 is the number of the nearest neighbors in each\nlayer and\u00111v=\u00112v= 1 are the number of type- 1(be-\ntween atoms numbered by 1(2) and 3(4) in Fig. 1 (b))\nand type- 2(between atoms numbered by 1(2) and 4(3) in\nFig. 1 (b)) neighbors in the interlayer. \u0011= 4 is the number\nof Cr atoms in the unit cell. The exchange coupling coeffi-\ncients are obtained by mapping the model Hamiltonian onto\nthe DFT-based calculations which are computed for various\nmagnetic configurations. For calculation of the isotropic ex-\nchange coefficient, 4-distinct spin configurations are consid-\nered as; FM ( S1=S^k;S2=S^k;S3=S^k;S4=S^k),\nAFM1 ( S1=S^k;S2=S^k;S3=\u0000S^k;S4=\u0000S^k), AFM2\n(S1=S^k;S2=\u0000S^k;S3=\u0000S^k;S4=S^k) and AFM3\n(S1=S^k;S2=\u0000S^k;S3=S^k;S4=\u0000S^k) in which all the\nmagnetic moments of Cr atoms are aligned in the z-direction\nandS= 3=2. In Fig. 2, the considered magnetic orders are\nrepresented in bilayer CrI 3. In these configurations, the DM\nterms of Eq. (1) are vanished owing to the parallel magnetic\nmoments of Cr atoms, and their total energies are obtained by\nEFM= 2S2(\u0011tJt+\u00111vJ1v+\u00112vJ2v) +\u0011\rS2; (6)\nEAFM 1= 2S2(\u0011tJt\u0000\u00111vJ1v\u0000\u00112vJ2v) +\u0011\rS2;\nEAFM 2= 2S2(\u0000\u0011tJt\u0000\u00111vJ1v+\u00112vJ2v) +\u0011\rS2;\nEFM3= 2S2(\u0000\u0011tJt+\u00111vJ1v\u0000\u00112vJ2v) +\u0011\rS2:\nWe carry out spin-dependent DFT calculations to find the to-\ntal energies of the considered magnetic configurations and as\na consequence the exchange coupling parameters of the HT4\nTABLE I: Lattice constant, a, Cr-Cr (I-I) distance between the layers, hCr\u0000Cr(hI\u0000I), Cr-I (Cr-Cr) bonding length, dCr\u0000I(dCr\u0000Cr), two\ndifferent I-Cr-I bonding angles, \fI\u0000Cr\u0000Iand\u000bI\u0000Cr\u0000I, and Cr-I-Cr bonding angle, \u0012Cr\u0000I\u0000Cr, and formation energy, Efof the HT and LT\nbilayer CrI 3structures. The bonding angles are shown in Fig. 1 (b).\nPhasea(˚A)hCr\u0000Cr(˚A)hI\u0000I(˚A)dCr\u0000I(˚A)dCr\u0000Cr(˚A)\fI\u0000Cr\u0000I(o)\u000bI\u0000Cr\u0000I(o)\u0012Cr\u0000I\u0000Cr(o)EF(eV/per Cr)\nHT 6.977 7.24 3.99 2.81 4.03 177.4 89.9 91.6 -8.81\nLT 6.979 6.59 3.35 2.81 4.03 177.6 89.9 91.7 -8.86\n(a) (b)\n(c) (d)\nJt\nJv\nFIG. 2: (Color online) The magnetic ordering of Cr atoms in (a)\nFM, (b) AFM1, (c) AFM2 and (d) AFM3 spin configurations of HT\nbilayer CrI 3. The dark (light) balls represent the Cr atoms in the\ntop (bottom) layer and arrows show the magnetic moment vectors\nof Cr atoms in the z-direction. The dashed lines connect the atoms\nwith intralayer and interlayer exchange coupling which are indicated,\nrespectively, by t and v indices.\nbilayer are given by\nJt=(EFM+EAFM 1)\u0000(EAFM 2+EAFM 3)\n8S2\u0011t; (7)\nJv=EFM\u0000EAFM 1\n4S2\u0011v; (8)\nwhereJv=J1v+J2vis the interlayer exchange coupling\nand\u0011v=\u00111v=\u00112v. The intralayer exchange coupling is\nJt=\u00007:00meV which presents the strong ferromagnetic\ncoupling of Cr atoms in the layers. Jv= +0:08meV indi-\ncates the layers are coupled in a form of antiferromagnetic.\nFurthermore, the weak strength of the isotropic interlayer ex-\nchange coupling denotes the magnetic order of the layers can\nbe tuned by the external electrical field [8].\nWe observe the variation of the isotropic interlayer and in-\ntralayer exchange coupling by DFT-based calculations, shown\nin Fig. 3. It is observed that the sign of the interlayer exchange\ncoupling is changed between 2-2:5V/nm, hence a magnetic\ntransition from the AFM to the FM occurs between layers\nwhile the strength of the intralayer exchange coefficient is no\nlonger altered by the electric field. This means that the ferro-\n0 1 2 3 4\nEl (V/nm)-7.5-7.25-7Jt (meV)(a)\n-0.3-0.2-0.100.10.2\nJv (meV)\n0 1 2 3\nEl (V/nm)-60-40-200204060DM (eV)(b)\nDtz\nDvz\nDvy\nDvxFIG. 3: (Color online) (a) The intralayer (squares) and interlayer\n(lozenges) isotropic exchange coefficients as a function of the ex-\nternal electric field, El. The change of the sign of the interlayer\nexchange shows a magnetic transition from the AFM to a FM be-\ntween the layers. The ferromagnetic coupling of Cr atoms in the\nlayer is extremely strong. (b) The non-zero component of interlayer\nand intralayer DM vectors of the bilayer CrI 3structures as a function\nofEl. The intralayer exchange coupling is changed significantly by\nthe electric field. Although the zandxcomponents of the interlayer\nDM interactions are independent of the electric field, the sign and the\nvalue ofDy\nvvary.\nmagnetic coupling of the Cr atoms in the layer is extremely\nstrong.\nThe extended Hamiltonian for the DM term is given by\nHDM=1\n2[Dt12\u0001(S1\u0002S2) +Dt120\u0001(S1\u0002S20) +Dt1200\u0001\n(S1\u0002S200) +Dt21\u0001(S2\u0002S1) +Dt210\u0001(S2\u0002S10)\n+Dt2100\u0001(S2\u0002S100) +Dt34\u0001(S3\u0002S4) +Dt340\u0001\n(S3\u0002S40) +Dt3400\u0001(S3\u0002S400) +Dt43\u0001(S4\u0002S3)\n+Dt430\u0001(S4\u0002S30) +Dt4300\u0001(S4\u0002S300)+\nDv13\u0001(S1\u0002S3) +Dv31\u0001(S3\u0002S1)+\nDv14\u0001(S1\u0002S4) +Dv41\u0001(S4\u0002S1)+\nDv23\u0001(S2\u0002S3) +Dv32\u0001(S3\u0002S2)+\nDv24\u0001(S2\u0002S4) +Dv42\u0001(S4\u0002S2)]; (9)\nwhere Si0andSi00are equal to the magnetic moment of ithCr\natom, and Si0=Si00=Si. On the other hand, the DM vector\nbetween two atoms can be obtained by Dij=Dij^uij, where\nDij=Djiis a scalar and ^uijis a unit-vector in the direction\nof the line connecting iandjatoms [44]. Therefore, Dij=\n\u0000Dji,Dt120=Rz(120)Dt12,Dt1200=Rz(240)Dt12and\nRz(\u0012)is a\u0012degree rotation around the z-axis. Three first5\n01234\nEl (V/nm)4045505560 T*(K)(b)\n-8-7-6-5-4-3-2-1\nMAE (meV)\n0 20 40 60\nTemperature (K)00.20.40.60.81M/MS(a)\nEl=0\nEl=2 V/nmEl=0.5 V/nm\nEl=1 V/nm\nEl=3.5 V/nm\nEl=4 V/nmEl=3 V/nmEl=2.5 V/nmEl=1.5 V/nm\nFIG. 4: (Color online) (a) The normalized magnetization of the HT\nbilayers as a function of temperature under various external electric\nfield. (b) The magnetic order temperature, T\u0003(T\u0003=TNforEl62\nandT\u0003=TCforEl>2:5), (lozenges) and magnetic anisotropy\nenergy (squares) of the HT bilayer CrI 3versus the external electric\nfield. The magnetic order temperature of the HT bilayer is increased\nby applying the electric field that arises from the increasing of the\nintralayer isotropic exchange coefficient. Furthermore, the magnetic\nmoments of the Cr atoms are in out-of-plane direction and it is un-\nchangeable by the external electric field.\nterms of Eq. (9) can be written as\nHDM12= (10)\n(Dt12+Rz(120)Dt12+Rz(240)Dt12)\u0001(S1\u0002S2):\nIfDt12= (Dx\nt12^i+Dy\nt12^j+Dz\nt12^k), then (Dt12+\nRz(120)Dt12+Rz(240)Dt12) = 3Dz\nt12^k, hence only the\nz-component of the DM vector between two Cr atoms in\nthe layers of the bilayer CrI 3can be entered in the model\nHamiltonian of Eq. (1). In the similar way, we can write\n(Dt34+Rz(120)Dt34+Rz(240)Dt34) = 3Dz\nt34^kfor the\nDM interaction between Cr atoms in the top layer. Due to\nthe symmetries between the layers, it is possible to consider\nDz\nt12=Dz\nt34=Dz\nt. On the other hand, Dt12=\u0000Dt21and\nDt13=\u0000Dt31, hence for the first five lines of Eq. (9), we\nhave the intralayer terms of the DM interaction as,\nHDMt =1\n2[6Dz\nt^k\u0001(S1\u0002S2) + 6Dz\nt^k\u0001(S3\u0002S4)]:(11)\nTherefore, the symmetries of the HT phase dictate that just the\nz-component of the intralayer DM interaction vector, Dz\nt, has\na non-zero value.\nMoreover, the geometry of the bilayer crystal indicates that\n^u13=^u24and^u32=Ry(180) ^u14, then we can assume\nDv13=Dv24andDv32=Ry(180)Dv14. To obtain the\ninterlayer contribution of the DM vector, we have to focus on\nthe last four lines of Eq. (9) which can be summarized in the\nfollowing form,\nHDMv =Dv13\u0001(S1\u0002S3) +Dv13\u0001(S2\u0002S4) +Dv14\u0001\n(S1\u0002S4) +Ry(180)Dv14\u0001(S3\u0002S2) (12)\nwhere Dv13= (Dx\nv1;Dy\nv1;Dz\nv1),Dv14= (Dx\nv2;Dy\nv2;Dz\nv2)\nandRy(180)Dv14= (\u0000Dx\nv2;Dy\nv2;\u0000Dz\nv2). Eight different\nspin configurations are needed to obtain enough informationabout the interlayer and intralayer DM vectors given in Table\nII.\nThe DM interactions are calculated by mapping the consid-\nered spin configurations on the model Hamiltonian and com-\nparing the spin-dependent DFT-obtained total energies. At\nthis place, we want to focus on the DM terms described in\nEqs. (11) and (12).\nHDMt (1st)= 6S[(Dz\nt^k)\u0001(sin(\u0019=6)^k)];\nHDMt (2nd)=\u00006S[(Dz\nt^k)\u0001(sin(\u0019=6)^k)];\nHDMv (1st)=HDMv (2nd)= 0: (13)\nBy imposing the above relations into Eq. (1), the total energy\nequations are defined and the z-component of the intralayer\nDM vector is given by\nDz\nt=E1st\u0000E2nd\n12S2sin(\u0019=6): (14)\nIn a similar way, we can find the DM terms of model Hamilto-\nnian for the other configurations and the interlayer DM vector\nof the HT bilayer is achieved by DFT-obtained total energies\naccording to\nDz\nv=Dv1z+Dv2z=E3rd\u0000E4th\n4S2sin(\u0019=6);\nDy\nv=Dv1y=E5th\u0000E6th\n4S2sin(\u0019=6);\nDx\nv=Dv1x+Dv2x=E8th\u0000E7th\n4S2sin(\u0019=6): (15)\nWe obtainDz\nt= 1:1\u0016eV ,Dz\nv= 3:5\u0016eV ,Dy\nv=\u000038:8\n\u0016eV andDx\nv= 3:1\u0016eV for the HT bilayer. Accordingly,\nthe intralayer DM interaction is ignorable in comparison with\nthe interlayer one. The results show the strongest component\nof the interlayer DM vector is in the y-direction; this means\nthe magnetic moments of Cr atoms in the layers tend to rotate\nalong thex-zplane, as long as they align in the z-direction\nbecause of the high MAE of the bilayer.\nThe interlayer and intralayer DM interactions can be tuned\nby an external electric field, very similar to the isotropic ex-\nchange coupling. In Fig. 3 (b), it is observed that the in-\ntralayer exchange coupling is changed significantly by the\nelectric field. Although the zandxcomponents of the inter-\nlayer DM interactions are independent of the electric field, the\nsign and the value of Dy\nvvary. The sign of the y-component of\nthe DM changes from negative to positive values between 2-\n2:5V/nm where a transition from the AFM to FM is achieved\nfor the HT bilayer. In fact, a phase transition needs a spin-\ncanting between layers in the x-zplane satisfied by the non-\nzeroDy\nvwhich is in the same order of the Jvin the HT bilayer.\nThe application of the electric field shifts the atomic positions\nslightly, resulting in a change in the orbital couplings between\natomic neighbors. Indeed, if the eg\u0000t2gcoupling, the re-\nsulting FM exchange will larger than that causing the AFM\nexchange (eg\u0000egorbital coupling), the bilayer is in the FM\nground states and vice versa [15, 22]. Upon increasing the\nelectric field, an AFM to FM transition is observed in the HT6\nTABLE II: Eight spin configurations for calculation of the interlayer and intralayer DM vectors of the HT bilayer.\nConfig. 1st2nd3rd4th5th6th7th8th\nS1S^iS^iS^iS^iS^k S^k S^k S^k\nS2S(cos(\u0019\n6)^i+S(cos(\u0019\n6)^i\u0000S^iS^iS^k S^k S^k S^k\ncos(\u0019\n6)^j) sin(\u0019\n6)^j)\nS3S^iS^iS(cos(\u0019\n6)^i+S(cos(\u0019\n6)^i\u0000S(cos(\u0019\n6)^k+S(cos(\u0019\n6)^k\u0000S(cos(\u0019\n6)^k+S(cos(\u0019\n6)^k\u0000\nsin(\u0019\n6)^j) sin(\u0019\n6)^j) sin(\u0019\n6)^i) sin(\u0019\n6)^i) sin(\u0019\n6)^j) sin(\u0019\n6)^j)\nS4S(cos(\u0019\n6)^i+S(cos(\u0019\n6)^i\u0000S(cos(\u0019\n6)^i+S(cos(\u0019\n6)^i\u0000S(cos(\u0019\n6)^k+S(cos(\u0019\n6)^k\u0000S(cos(\u0019\n6)^k+S(cos(\u0019\n6)^k\u0000\nsin(\u0019\n6)^j) sin(\u0019\n6)^j) sin(\u0019\n6)^j) sin(\u0019\n6)^j) sin(\u0019\n6)^i) sin(\u0019\n6)^i) sin(\u0019\n6)^j) sin(\u0019\n6)^j)\nbilayer CrI 3, showing the dominant effect of eg\u0000t2gcoupling.\nWe expect these changes to be accompanied by spin tilt, and\nconsequently, the DM interactions representing the spin tilting\nability are tuned by the electric field.\nTo calculate the magnetic order temperature, Curie TC,\nor N ´eelTNtemperature of the FM or AFM bilayer, a sam-\nple of 50\u000250 nm2is considered and the metropolis Monte\nCarlo algorithms are used for the calculation of temperature-\ndependent magnetization (see Fig. 4 (a)). According to Fig.\n4 (b), the magnetic order temperature of the HT bilayer is in-\ncreased by applying the electric field that arises from the in-\ncreasing of the intralayer isotropic exchange coefficient.\nThe variation of MAE = (E?\u0000Ek)=\u0011S2as a function of\nthe electric field is shown in Fig. 4 (b). It shows that the mag-\nnetic moments of the Cr atoms are in out-of-plane direction,\nand it is unchangeable by the external electric field.\nB. Exchange coefficients in the LT phase\nNow we turn our attention to another phase of the bilayer\nsystem. Four Cr atoms in a unit cell of the LT-phase bilayer\nCrI3prefer to determine a parallel spin orientation in which\nthe minimized total energy belongs to the FM magnetic con-\nfiguration [8]. Here, similar to the HT phase, we are interested\nin obtaining the symmetric and antisymmetric exchange coef-\nficients of the LT bilayer. We should note that the intralayer\nnearest neighbors of the LT is equivalent to that in the HT\nphase, while there is only an interlayer nearest neighbor bond\nbetween the Cr atoms numbered by 2and3in Fig. 1 (d). In\nthe case of the LT bilayer CrI 3, the first two terms of Eq. (1)\nare extended to be as\nH=1\n2(\u0011tJt[(S1\u0001S2) + (S2\u0001S1) + (S3\u0001S4) + (S4\u0001S3)]+\n\u0011vJv[(S2\u0001S3) + (S3\u0001S2)]) +\u0011\rjSzj2; (16)\nwhere\u0011t= 3 and\u0011v= 1 are the number of the intralayer\nand interlayer nearest neighbors. We consider four different\nmagnetic configurations of Cr atoms, as a FM ( S1=S^k,\nS2=S^k,S3=S^k,S4=S^k), AFM1 ( S1=S^k,S2=S^k,\nS3=\u0000S^k,S4=\u0000S^k), AFM2 ( S1=S^k,S2=\u0000S^k,\nS3=\u0000S^k,S4=S^k) and AFM3 ( S1=S^k,S2=\u0000S^k,\nS3=S^k,S4=\u0000S^k), and thus able to find interlayer and\nintralayer symmetric exchange coefficients. By applying thespin vectors on Eq. (16), the relations for the total energies of\nthe magnetic configurations are given by,\nEFM=E0+ 2S2\u0011tJt+S2\u0011vJv+\u0011\rS2;\nEAFM 1=E0+ 2S2\u0011tJt\u0000S2\u0011vJv+\u0011\rS2;\nEAFM 2=E0\u00002S2\u0011tJt+S2\u0011vJv+\u0011\rS2;\nEAFM 3=E0\u00002S2\u0011tJt\u0000S2\u0011vJv+\u0011\rS2: (17)\nThe DFT-obtained total energies give us the symmetric ex-\nchange coefficients as\nJt=(EFM+EAFM 1)\u0000(EAFM 2+EAFM 3)\n8S2\u0011t;\nJv=(EFM\u0000EAFM 1)\n2S2\u0011v: (18)\nThe results displayed in Fig. 5 (a) show the interlayer and in-\ntralayer symmetric exchange couplings possess negative and\nconsistent values with the FM ground state of the LT bilay-\ners. The sign of JtandJvremain negative by applying the\nelectric field, while their amounts are slightly increased. In\nfact, the electric field develops the ferromagnetic coupling of\nlayers increase in the LT bilayer.\nThe third term of the Hamiltonian in Eq. (1) can be written\nfor the LT bilayer as,\nHDM=1\n2[Dt12\u0001(S1\u0002S2) +Dt120\u0001(S1\u0002S0\n2) +Dt1200\n\u0001(S1\u0002S00\n2) +Dt21\u0001(S2\u0002S1) +Dt210\u0001(S2\u0002S0\n1)\n+Dt2100\u0001(S2\u0002S00\n1) +Dt34\u0001(S3\u0002S4) +Dt340\u0001(S3\n\u0002S0\n4) +Dt3400\u0001(S3\u0002S00\n4) +Dt43\u0001(S4\u0002S3) +Dt430\n\u0001(S4\u0002S0\n3) +Dt4300\u0001(S4\u0002S00\n3) +Dv23\u0001(S2\u0002S3)\n+Dv32\u0001(S3\u0002S2)]: (19)\nSimilar to the HT phase, the symmetric of atomic positions\ndictates to have only the z-component of the intralayer DM\ninteraction in the Hamiltonian which is given by Eq. (11). For\nthe interlayer DM interaction of the LT phase\nHDMv =Dv23\u0001(S2\u0002S3): (20)\nIt should be noticed that there is a C3rotation axis on the\nbonding length between Cr atoms numbered by 2and3in the7\n01234\nEl (V/nm)-7.4-7.2-7-6.8J (meV)(a)\nJt\nJv\n0 1 2 3\nEl (V/nm)-60-40-20020DM ( 7eV)(b)\nDtz\nDvz\nFIG. 5: (Color online) (a) The intralayer (squares) and interlayer\n(lozenges) isotropic exchange coefficients and (b) the z-component\nof interlayer and intralayer DM vectors of the LT bilayer CrI 3as a\nfunction of the external electric field, El. The sign of the exchange\ncoefficients remain negative by applying the electric field, while their\namounts are slightly increased showing that the electric field makes\nthe ferromagnetic coupling of layers stronger. Furthermore, the z\u0000\ncomponent of the DM interaction of the LT bilayer increases by the\nelectric field meaning that the spin-canting are in the plane of the Cr\natoms.\nTABLE III: Four spin configurations for calculation of the interlayer\nand intralayer DM vectors of the LT bilayer CrI 3.\nConfig. 1st2nd3rd4th\nS1S^iS^iS^iS^i\nS2S(cos(\u0019\n6)^i+S(cos(\u0019\n6)^i\u0000S^iS^i\nsin(\u0019\n6)^j) sin(\u0019\n6)^j)\nS3S^iS^iS(cos(\u0019\n6)^i+S(cos(\u0019\n6)^i\u0000\nsin(\u0019\n6)^j) sin(\u0019\n6)^j)\nS4S(cos(\u0019\n6)^i+S(cos(\u0019\n6)^i\u0000S(cos(\u0019\n6)^i+S(cos(\u0019\n6)^i\u0000\nsin(\u0019\n6)^j) sin(\u0019\n6)^j) sin(\u0019\n6)^j) sin(\u0019\n6)^j)\nunit cell of the LT bilayers (see Fig. 1 (d)). According to\nthe Moriya symmetry rules [45], the DM interaction between\natoms 2and3is parallel to the C3axis which is along the\nz-direction. Therefore, we need to find the z-component of\nthe interlayer DM interaction which can be obtained by two\nspin configurations (see Table III). Finally, the z-component\nof intralayer and interlayer DM interactions are respectively\ngiven by\nDz\nt=(E1st\u0000E2nd) + (E3rd\u0000E4th)\n12S2sin(\u0019=6);\nDz\nv=E3rd\u0000E4th\n2S2sin(\u0019=6): (21)\nThe negligible values of Dz\nt=\u00000:4\u0016eV andDz\nv=\u00000:3\n\u0016eV are obtained using our spin-dependent DFT-based calcu-\nlations. It should be noted that there is an inversion symmetry\nin the center of the 2and3atomic bond in the LT bilayer,\nsoDz\nvshould be zero, and therefore, there is no any DM in-\nteraction between the layers. We investigate the effect of the\nexternal electric field on the interlayer and intralayer DM in-\nteractions of the LT phase, and the results are reported in Fig.5 (b). It is observed that the interlayer and intralayer DM in-\nteractions of the LT phase are increased by developing the\nelectric field. It is interesting that the order of DM interac-\ntion variation is similar for both LT and HT phases, but their\nbehavior is different. In the presence of an electric field, just\nthez-component of the DM interaction of the LT bilayer in-\ncreases; this means that the spin-canting are in the plane of\nthe Cr atoms. On the other hand, the interlayer symmetric ex-\nchange coupling, Jv, is much stronger than Dz\nv, subsequently,\nthe FM coupling survives in a competition between FM cou-\npling of layers and in-plane canting of spins.\nIn the HT bilayer, the amount of the interlayer DM vector\nand interlayer symmetric exchange coupling are in the same\norder. In fact, they are in a close competition with each other\nand it remains in the external electric field. It is intriguing that\nthe both of them leads to a transition from the AFM to FM\nin the HT phase; the interlayer DM interaction is dominant\nin they-direction which leads to the spin canting in the x-\nz-plane and, as a consequence, helps to the rotation of the\nspin-direction from the AFM to FM and also the sign of Jvis\nchanged by applying the electric field.\nIt should be noticed that the interlayer and intralayer\nisotropic exchange coupling in the Heisenberg Hamiltonian\nof the bilayer CrI 3have been calculated in Ref. [19] which\nagrees well with our work. They obtained Jt=\u00006:49meV\n(by DFT) and Jv= +0:10meV (by experimental results from\nRef. [1]) which are close to our findings ( Jt=\u00007:00meV\nandJv= +0:8meV) in order of magnitude and sign for\nHT bilayer CrI 3. They have also shown that the isotropic ex-\nchange couplings are tuned by the electric field. There are\nDFT-based calculations [8, 17, 19] and experimental results\n[23, 24, 26] showing that the interlayer exchange coupling\nsign of HT phase of bilayer CrI 3is changed by the electric\nfield. One can find the DFT-obtained exchange couplings of\ndifferent stacks of bilayer CrI 3in the absence of an electric\nfield in Refs. [15, 17, 22, 38]. In contrast, the interlayer and\nintralayer DM interactions of the bilayer CrI 3have not been\ncalculated in previous works, and we report here their electric\nfield tunable behavior.\nThe temperature dependence of the magnetic moments of\nthe Cr atoms, M(T)in the LT bilayer is illustrated in Fig.\n6 (a). It is shown that M(T)is nearly independent of the\nelectric field. The variation of the Currie temperature, TC, by\nthe electric field arises directly from the variation of Jtand it\nis gradually increased in a more intense electric field (see Fig.\n6 (b)). The MAE of the LT phase as a function of the electric\nfield is shown in Fig. 6 (b). The MAE remains negative by\nvarying the electric field so the easy axis of the LT bilayer lies\nin thez-direction and its value does not change significantly.\nWe obtained a N ´eel temperature of 44 K for the AFM-HT\nbilayer and a Curie temperature of 59 K for the FM-LT bilayer\nCrI3, which are tuned by the electric field due to the variation\nin the parameters of the spin model Hamiltonian. Increas-\ning the number of layers from monolayer to bulk increases\nthe critical temperature between 45 K and 61 K [1, 25]. A\nN´eel temperature of 45 K was reported experimentally for the\nHT bilayer [26]. The larger Curie temperature is due to the\nstronger FM interlayer exchange coupling in LT bilayer CrI 38\n01234\nEl (V/nm)565860626466 TC (K)(b)\n-4-3-2-10\nMAE (meV)\n0 20 40 60\nTemperature (K)00.20.40.60.81M/MS(a)\nEl=0\nEl=0.5 V/nm\nEl=1 V/nm\nEl=3.5 V/nm\nEl=4 V/nmEl=3 V/nmEl=2.5 V/nmEl=2 V/nmEl=1.5 V/nm\nFIG. 6: (Color online) (a) The magnetization of the LT bilayers as\na function of temperature under different external electric field. (b)\nThe Curie temperature (lozenges) and magnetic anisotropy energy\n(squares) of LT bilayer CrI 3versus external electric field. Notice that\nthe MAE remains negative by changing the electric field, therefore,\nthe easy axis of the LT bilayer is in the z-direction and its value does\nnot change significantly.\n[27].\nAlthough the magnetic coupling coefficients of both the LT\nand HT bilayers are tuned, the magnetic ground state of the\nHT bilayer can be manipulated by the electric field. To per-\nceive the possibility of manipulating magnetic textures in the\nbilayer CrI 3, we simulate their spin dynamics in the presence\nof the external electric and magnetic fields and the emergence\nof the magnetic domains and skyrmion patterns are explored.\nTo obtain the spin dynamics of the bilayer, the atomic posi-\ntion of the system is constant while different atomic magnetic\nmoments are appeared by time evolutions [39]. Here, we ob-\ntained that the formation energy is 1000 times more than the\nexchange energy in the bilayer CrI 3which guarantees the sta-\nbility of the atomic structure under spin evolutions. In fact,\nwe want to mention that the spin evolution happens in a con-\nstant atomic structure in any electric field. Under the electric\nfield, the atomic positions are relaxed by DFT-based calcu-\nlations and leading variation of the interband and intraband\nisotropic exchange and DM coefficients, so the spin dynamics\nare calculated in the new structure.\nC. Spin dynamics of the HT phase\nIn the HT bilayer, the AFM coupling between layers is dic-\ntated in the spin dynamics of the system. The spin dynamics\nof the top and bottom layers are plotted in Fig. 7. In T= 0:3\nK, it is observed that the bilayer is divided into two areas in\nwhich layers are coupled in form of an anti-ferromagnetic.\nWhile the bottom layer has a magnetic moment in the +z-\ndirection, the top layer has a magnetic moment in the \u0000z-\ndirection and vice versa; this means that the AFM order is\nconstant between the layers. This regularity is also established\non the domain walls, where the perpendicular components of\nthe magnetic moments, M?, of two layers are in opposite di-\nrections. The results of the simulations show that the created\nmagnetic texture is stable in size, but M?is slowly changed to\nbe perpendicular to the domain wall by the time evolution as\nFIG. 7: (Color online) The magnetic texture of the top and bottom\nlayers of the HT bilayer in El= 0 V/nm,B= 0, (a)T= 0:3K,\ntime= 2 ns, (b)T= 0, time = 2:2ns and (c)T= 0, time = 3 ns.\nThe blue (red) areas show where the magnetic moments of Cr atoms\nare in the -z(+z)-direction. M?is represented by black arrows on the\ndomain walls. The created magnetic texture is stable in size, but M?\nis slowly changed to be perpendicular to the domain wall by the time\nevolution.\nshown in Figs. 7 (a)-(c). Thus, an ultrathin N ´eel-type domain\nwall appears on both the top and bottom layers. It is clear\nthat the zero magnetic moments are obtained for the HT bi-\nlayer, but there is an interesting microscopic magnetic pattern\non each layer.\nWe obtain the symmetric and asymmetric exchange cou-\npling coefficients of the spin Hamiltonian depending on the\nexternal electric field, therefore, we explore the variation of\nthe spin dynamics of the bilayer by the electric field shown in\nFigs. 8 (a)-(c). In El= 1 V/nm, the magnetic order of the\nHT bilayer is the AFM and JtandJvare near to their val-\nues in the absence of the electric field case, while the value\nofDz\ntis increased to 25:5\u0016eV , therefore the new spin texture\nis created (see Fig. 8 (b)). If we focus on one of the layers,\nwe see that the wider magnetic domains in El= 0case (Fig.\n8 (a)) changes to two smaller quasi-circle magnetic domains\nwith finite diameters (nearly 25nm and 10nm). Skyrmion\npatterns occur around the magnetic moments, while their chi-\nrality is different for the top and bottom layers. The places and\nsizes of magnetic domains are the same in the top and bottom9\nFIG. 8: (Color online) The magnetic texture of the top and bottom\nlayers of the HT bilayer in (a) El= 0, (b)El= 1 V/nm and (c)\nEl= 2 V/nm,T= 0:3K,B= 0 and time = 2 ns. The skyrmion\npatterns occur around the magnetic moments, while their chirality\nis different for the top and bottom layers. The places and sizes of\nmagnetic domains are the same in the top and bottom layers, how-\never, the direction of the magnetic moments are exactly opposite for\ninside and outside of the domain walls.\nlayers, however, the direction of the magnetic moments are\nexactly opposite for inside and outside of the domain walls.\nWe obtain these stable quasi-circle magnetic domains under\nthe time evolution.\nFor the HT bilayer, the amount of the intralayer DM inter-\naction reaches to 41\u0016eV by increasing the electric field to\nEl= 2V/nm, and the interlayer isotropic exchange coupling\nis equal to\u000020\u0016eV . Although the value of Jvis reduced,\nthe negative sign shows the bilayer is still in the AFM phase.\nThese changes lead to a new spin texture in the bilayer. In\nFig. 8 (c), 30 nm-diameter quasi-circle magnetic domains\nare created in the both top and bottom layers with opposite\nspin orientation, so the AFM configuration of the bilayer is\nconserved. Most importantly, the diameter, position, number\nof the magnetic domain and the domain wall chirality are re-\nlated to the external electric field. In fact, the magnetic texture\nof the HT bilayer can be manipulated by an external electric\nfield owing to the variation of the DM and isotropic exchange\ncoupling interactions. Therefore, stable magnetic domains are\nconstituted in the HT bilayer that possess fascinating micro-\nscopic information, while the total magnetism is zero in the\n100 \n50 100 0 \n0 \n0 50 100 0 X (run) 50 100 \n5 100 x (nm) FIG. 9: (Color online) The magnetic texture of the top and bottom\nlayers of HT bilayer in El= 3 V/nm,T= 0:3K,B= 0 and\ntime= 2 ns. Notice that the the magnetic domain in the FM bilayer\nis stable under the time evolution.\nmacroscopic level.\nInEl= 3 V/nm, the layers are coupled in a form of a\nferromagnetic. This phase transition is obviously observed in\nthe spin dynamics simulation which both layers have the same\nspin orientation (see Fig. 9) and originates from the positive\nsign ofJv. On the other hand, the perpendicular-component\nof the magnetic moments of the top and bottom layers are\nsimilar on the domain walls. The spin dynamics simulations\nshow that the obtained magnetic domain in the FM bilayer is\nstill stable under the time evolution.\nAccording to Figs. 10 (a)-(c), our simulation results show\nthat the external magnetic field changes the spin dynamic of\nthe bilayer. In the absence of an external electric field, Fig.10\n(b) shows that there is an area with FM configurations in the\nAFM background of the bilayer. Indeed, the external mag-\nnetic field is in competition with the interlayer exchange cou-\npling and desire to accompany the spin of the Cr atoms paral-\nlel to the external magnetic field. It is not completely winner\nin the competition when Bextis less than 0:5T. By increasing\nthe external magnetic field to Bext= 2T, the layers gain par-\nallel magnetic moments in the direction of the applied mag-\nnetic field.\nAnother example is the system when exposed to an exter-\nnal electric field, El= 2 V/nm. InBext= 0:5T, the AFM\npattern is completely disappeared and the magnetic moments\nof the Cr atoms are parallel. In the competition between ex-\nchange coupling coefficients and the Zeeman energy, the Zee-\nman energy of the strong magnetic field has conquered. In\nthe presence of 0:01T external magnetic field, the simulation\nresults (Figs. 11(a)-(c)) show that the quasi-circular magnetic\ndomain form converts to a quasi-elliptical shape, but the AFM\nconfiguration between the top and bottom layers is conserved\nbecause the applied magnetic field is not large enough to cre-\nate a phase transition in the magnetic ordering of the bilayer.\nInterestingly, the obtained magnetic domain and almost do-\nmain wall do not change by the time evolution after cooling\ntime (2 ns). Accordingly, a stable magnetic texture can be ob-\ntained and tuned by a weak external magnetic field, while the\nstrong magnetic field leads to the magnetic phase transition in\nthe bilayer.10\nFIG. 10: (Color online) The spin dynamic results of the HT bilayer\ninEl= 0 V/nm, (a)B= 0,T= 0:3K andtime = 2 ns, (b)\nB= 0:5T,T= 0:3K and time = 2ns and (c)B= 0:5T,T= 0K\nand time = 3 ns. the external magnetic field is in competition with\nthe interlayer exchange coupling and desire to bring the spin of the\nCr atoms parallel to the external magnetic field.\nD. Spin dynamic of the LT phase\nIn the LT bilayer, a strong negative interlayer exchange cou-\npling leads to the creation of the FM magnetic configuration.\nThe spin dynamic simulation, in Fig. 12 (a) shows two wide\nareas with opposite spin directions in each layer while the\nmagnetic moments of the layers are actually analogous. In the\nboundary of the two areas, the perpendicular magnetic mo-\nments of the Cr atoms are dominated and their chirality is the\nsame in both layers. In fact, a 1D spin-wave is created at the\nboundary and the results show that the chirality of the spin-\nwave is changed smoothly in time (compare Fig. 12 (a) and\n(b)). On the other hand, the created areas sizes are stable by\nthe time evolution, so we can consider that the 1D in-plane\nspin-wave is metastable (due to the change of chirality). In\nthe absence of an electric field, the ignorable DM interaction\ncannot create a magnetic domain, and the observed spin-wave\nis due to the demagnetization field. For clarity, we report the\nresults of the spin dynamic simulation in the absence of a de-\nmagnetization field in Fig. 12 (c). It is obviously observed\nthat the magnetic domain is disappeared and all the Cr atoms\nof both layers have parallel magnetic moments due to strong\nFIG. 11: (Color online) The spin dynamic results of the HT bilayer\ninEl= 2 V/nm,B= 0:01T (a)T= 0:3K,time = 2 ns, (b)\nT= 0 K, time = 2:2ns and (c)T= 0 K, time = 3 ns. Notice,\nhere, the applied magnetic field can not create a phase transition in\nthe magnetic order of the HT bilayer.\ninterlayer and intralayer isotropic exchange couplings of the\nLT bilayer CrI 3.\nThe interlayer and intralayer DM interactions are increased\nby applying the external electric field, while the sign of the\nexchange coupling and their coefficients values remain practi-\ncally constant in the LT bilayer CrI 3. The spin dynamic simu-\nlations show the existence of a similar quasi-circular magnetic\ndomain shape in the top and bottom layers in the presence of\nthe electric field. Hence, the meta-stable magnetic domain is\ndisturbed by increasing the DM interaction in El= 1 V/nm\nandEl= 2V/nm, see Figs. 13 (a)-(c). The results show that\nthis quasi-circular magnetic domains are disappeared by cool-\ning the system and the magnetic moments of the Cr atoms\ngained parallelized, see Fig. 14 for the LT bilayer under\nEl= 1V/nm. We can assume that the DM interactions leads\nto the creation of a quasi-circular domain shape in the layers,\nbut its value is extremely smaller than the exchange coupling\nto stabilize it in the system, therefore, the obtained magnetic\ndomain is unstable and disappears rapidly in the LT bilayer.\nThe spin dynamics in the twisted bilayer CrI 3was recently\ncalculated using an arbitrary DM interaction and a local moir ´e\nfield instead of an isotropic exchange coupling between the\nlayers [15] while ignoring the demagnetization field. They11\n100 \n100 \n0 \n0 \n100 \n0 \n0 50 \n50 \n50 \nX (run) (a) \n100 \n0 \n(b) \n100 0 \n(c) \n100 0 50 100 \n50 100 \n50 100 \nx (nm) \nFIG. 12: (Color online) The spin dynamic results of the LT bilayer\ninEl= 0 V/nm,Bext= 0 T (a)T= 0:3K andtime = 2 ns, (b)\nT= 0K and time = 3ns and (c)T= 0:3K and time = 2ns in the\nabsence of a demagnetization field. A 1D metastable spin-wave is\ncreated at the boundary and the chirality of the spin wave is changed\nsmoothly in time. In this case, the Cr atoms have parallel magnetic\nmoments due to strong interlayer and intralayer isotropic exchange\ncouplings.\nshowed that the twisted texture depends on the twist angle.\nIt is worth noting that applying an electric field is easier than\ntwisting the double layer CrI 3in experiments. It is experimen-\ntally shown that the moir ´e magnetic domain in AFM twisted\nbilayer magnetic crystals can be controlled by gate voltages\n[16]. Despite a recent interest in studying spin dynamics in 2D\nmagnets [13, 15, 16, 46–49] and their promising application\nin new generation magnetic domain-based devices [50–52],\nthere is no report of spin texture formation in the bilayer CrI 3\nby considering all DFT-obtained isotropic and anisotropic ex-\nchange couplings in the presence of an electric field. Here\nwe calculate the spin dynamics of a large bilayer CrI 3as an\nexperimental domain by solving the LLG equations where\nthe demagnetizing field is important to find the best magnetic\nground state of the system. The results show that the magnetic\ndomain in the HT bilayer CrI 3can be tuned by the electric\nfield, which is consistent with the experimental results [16].\nFIG. 13: (Color online) The magnetic texture of the top and bottom\nlayers of the LT bilayer in (a) El= 0, (b)El= 1 V/nm and (c)\nEl= 2 V/nm,T= 0:3K,B= 0 and time = 2 ns. The existence\nof a similar quasi-circle magnetic domain in the top and bottom lay-\ners in the presence of the electric field are seen and the meta-stable\nmagnetic domain is disturbed by increasing the DM interaction.\nIV . CONCLUSION\nThe interlayer and intralayer isotropic exchange coupling\ncoefficients and DM interactions are obtained by DFT-based\ncalculations for bilayer CrI 3. We have shown that in the pres-\nence of an external electric field, interlayer DM interactions\nhave a critical impact on the magnetic phase transition of the\nHT bilayer. The effects of the external electric field on the\nparameters of the spin-model Hamiltonian and temperature-\ndependent magnetic moments of bilayers are explored. Most\nimportantly, a stable magnetic domain in the HT bilayer CrI 3\nwhich can be manipulated by the electric and magnetic fields\ncan be constituted owing to the tunable interlayer and in-\ntralayer exchange coupling and DM interactions. 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Andruh 6, C.Maxim 6 \n \n1Dipartimento di Fisica” A.Volta” e Unita’ CNISM-CNR , Universita’ di Pavia, I-27100 \nPavia, Italy \n2Department of Physics, Indian Institute of Technolo gy Bombay, Powai, Mumbai-\n400076, India \n3Laboratory for Molecular Magnetism and INSTM Resear ch Unit, University of \nFlorence, I-50019 Sesto Fiorentino, Italy \n4Dipartimento di Scienze Molecolari Applicate ai Bio sistemi DISMAB, Università di \nMilano, I-20134 Milano, Italy \n5 S3-CNR-INFM, I-41100 Modena, Italy   \n6Inorganic Chemistry Laboratory, Faculty of Chemistr y, University of Bucharest, Str. \nDumbrava Rosie 23, 020464 Bucharest, Romania  \n \n \nAbstract \n \nThe magnetic properties and the spin dynamics of tw o molecular magnets have been \ninvestigated by magnetization and d.c. susceptibili ty measurements, Electron \nParamagnetic  Resonance (EPR) and proton Nuclear Ma gnetic Resonance (NMR) over a \nwide range of temperature (1.6-300K) at applied mag netic fields, H=0.5 and 1.5 Tesla. \nThe two molecular magnets consist of Cu II (saldmen)(H 2O)} 6{Fe III (CN) 6}](ClO 4)3·8H 2O \nin short Cu 6Fe and the analog compound with cobalt, Cu 6Co. It is found that in Cu 6Fe \nwhose magnetic core is constituted by six Cu 2+ ions and one Fe 3+ ion all with s=1/2, a \nweak ferromagnetic interaction between Cu 2+  moments through the central Fe 3+  ion with \nJ = 0.14 K is present, while in Cu 6Co the Co 3+  ion is diamagnetic and the weak \ninteraction is antiferromagnetic with J = -1.12 K. The NMR spectra show the presence of \nnon equivalent groups of protons with a measurable contact hyperfine interaction  2consistent with a small admixture of s-wave functio n with the d-function of the magnetic \nion. The NMR relaxation results are explained in te rms of a single ion (Cu 2+ , Fe 3+ , Co 3+ ) \nuncorrelated spin dynamics with an almost temperatu re independent correlation time due \nto the weak magnetic exchange interaction. We concl ude that the two molecular magnets \nstudied here behave as single molecule paramagnets with a very weak intramolecular \ninteraction, almost of the order of the dipolar int ermolecular interaction. Thus they \nrepresent a new class of molecular magnets which di ffer from the single molecule \nmagnets investigated up to now, where the intramole cular interaction is much larger than \nthe intermolecular one. \n \nI) Introduction \n              The development of molecular chemistr y in synthesizing transition metal-ion \nbased molecular clusters whose properties are midwa y between atoms and bulk systems \nprovides a unique opportunity to the scientific com munity for the study of nanoscale \nmagnetism [1]. Because of the presence of non-magne tic organic ligands that prevent \nmagnetic interactions, the intermolecular interacti ons are weak in comparison to \nintramolecular super-exchange interactions. Hence, the molecules are isolated \nmagnetically from each other and it is of great int erest to investigate spin dynamics of \nthese nanomagnets, often called single molecule mag nets (SMM). In the SMM reported \nup to now, a strong exchange magnetic interaction e xists among the magnetic moments \nwithin each individual molecule which leads to a lo w temperature ground state \ncharacterized by either a high total moment S or a singlet antiferromagnetic (AFM) state \nS=0 depending on the topology of the magnetic ions and on their mutual magnetic \ncoupling. In the case of high spin ground state and  high magnetic anisotropy, quantum \ntunneling of magnetization and quantum coherence at  low temperature have been \nobserved, making these nanomagnets promising candid ates for magnetic storage and \nquantum computing among other applications [2,3]. \n          In this paper we present the magnetic pro perties of heptanuclear molecular magnets \nCu 6Fe and Cu 6Co with very small intramolecular magnetic coupling . These molecules are \nthus a prototype of single molecule paramagnets. As  it will be shown by the experimental  3results, the Cu 6Fe compound consists of six Cu 2+ magnetic ions with a weak \nferromagnetic interaction via the bond to a central  Fe 3+  ion. The isostructural Cu 6Co \ncompound, instead, appears to be formed by six Cu 2+  magnetic ions and a central \ndiamagnetic Co 3+  ion with a small antiferromagnetic coupling betwee n Cu 2+  ions via the \nbond to the central Co 3+  ion. \n         The paper is organized as follows. In sect ion II we summarize briefly the synthesis \nof the compounds and their crystal structure. The d etailed description of the results of this \nsection will be presented in a separate publication  [4]. In Section III we present the \nexperimental results and data analysis. The results  are presented in separate subsections \nfor the magnetization, the EPR, the static and the dynamic NMR. Section IV contains a \ncomparison between Cu 6Fe and Cu 6Co and a discussion of the results obtained with th e \ndifferent techniques and the relevant conclusions. \n         \nII) The Samples \n(A) Cu 6Fe  \n         Polycrystalline sample[{Cu II (saldmen)(H 2O)} 6{Fe III (CN) 6}](ClO 4)3·8H 2O (i.e, \nC72 H118 C13 Cu 6FeN 18 O32 ) was synthesized from the reaction of binuclear co pper(II) \ncomplex, [Cu 2(saldmen) 2(µ-H2O)(H 2O) 2](ClO 4)2·2H 2O, with K 4[Fe(CN) 6] (H saldmen is \nthe Schiff base resulted by reacting salicylaldehyd e with N,N-dimethylethylenediamine \nas will be described elsewere [4]). In this molecul e, 16 out of 118 protons belong to 8 \ncrystallization water molecules and the remaining 1 02 protons arise from the organic \nligands and from six water molecules co-ordinated t o six Cu 2+  ions. \nThe lattice is of hexagonal symmetry ( R-3c) with cell constants a=27.8777(16) Å, \nb=27.8777(16) Å, c=21.369(13) Å, α=β=90 ° and γ=120 °. The six Cu 2+  ions are located at \nthe corners of an octahedron and are connected by t he cyano groups and one Fe 3+  at the \ncenter of the octahedron. The Cu-Fe-Cu angles are 1 80˚ and the Fe-Cu angles across the \nCN bridges are Cu-N-C=171.76(54)˚ and Fe-C-N=176.54 (57)˚. \nThe nearest neighbor bond distances are Fe-H=4.0482 Å and Cu-H=2.9649Å. \n  4\n \n \n \n \n \n \n \n \n \n \n \nFig. 1 . Crystal structure of Cu 6Fe . Cu 6Co is isostructural, with Fe replaced by Co \n \n(B)  Cu 6Co  \n     The Cu 6Co crystals were obtained by adding  an acetonitril e-water (1:1) solution (20 \nmL) containing 0.3 mmol of [Cu 2(saldmen) 2(µ-H2O)(H 2O) 2](ClO 4)2·2H 2O, 10 mL \nacetonitrile-water (1:1) solution containing 0.1 mm ol of K 3[Co(CN) 6] under stirring. \nGreen crystals suitable for X-ray diffraction were obtained directly from the reaction \nmixture, by slow evaporation of the filtrate at roo m temperature [4]. \nThe lattice is also of hexagonal symmetry ( R-3c) with cell constants a=27.9545(19)Å, \nb=27.9545(19)Å, c=21.3938(16)Å, α=β=90 ° and γ=120 °. The six Cu 2+  ions are located at \nthe corners of an octahedron and are connected by t he cyano groups and one Co 3+  at the \ncenter of the octahedron. The Cu-Co-Cu angles are 1 80˚ and the Co-Cu angles across the \nCN bridges are Cu-N-C=174.21˚ and Co-C-N=173.712˚. \nThe nearest neighbor bond distances are Co-H=3.9836 Å and Cu-H=2.9628Å. \n \nIII) Experimental results and analysis  5A. Magnetic susceptibility  \n             The temperature dependence of the magnetic suscepti bility ( χ=M/H) in the \ntemperature range 2-210 K at two applied magnetic f ields, 0.1 Tesla and 1Tesla for \nCu 6Fe, and in the temperature range 2-160 K at 1Tesla for Cu 6Co, was measured with a \nSuperconducting Quantum Interference Device (SQUID)  magnetometer. The raw data \nwere corrected by the sample holder and the single ion diamagnetic contributions before \nanalysis . \n     The results of the susceptibility measurements  are shown in fig. 2 for both Cu 6Fe and \nCu 6Co samples. Over most of the temperature range the χT vs T data show a simple \nparamagnetic behavior. At very low temperature it i s evident that a departure from the \nsimple Curie law due to a small ferromagnetic (FM) coupling for Cu 6Fe and a small \nantiferromagnetic (AFM) coupling for Cu 6Co. \nThe data for Cu 6Fe can be fitted with a Curie-Weiss law with C = 2. 72 ±0.2 (emu.K /mol) \nand T F =+0.07 K. This corresponds to an average g factor for the seven spins s=1/2 per \nmolecule of g = 2.035. This is surprisingly low, gi ven the supposedly unquenched orbital \nmomentum characterizing the 2T2g  state of low spin Fe(III) in octahedral symmetry w hich \nshould lead to a much larger average g value [5]. T he same behavior has been recently \nreported for a linear, cyanide bridged, CuFeCu comp lex, and attributed to the peculiar \ngeometrical distortion of Fe(CN) 6 unit, leading to an almost complete quench of the \nangular momentum [6]. The obtained value of the Wei ss constant correspond, in the \nframework of simple Molecular Field Approximation ( MFA),\nBF\nkJszs \n3) 1(2+=θ , to a \nweak ferromagnetic interaction J F = 0.14 K.  \n     On the other hand the data for Cu 6Co were fitted with a Curie-Weiss law with C = \n2.44 ±0.06 (emu K /mol) and T N = -0.56 K. This correspond to six spins s=1/2 with  an \naverage g factor g = 2.075. Again, by using the MFA  expression for the Weiss constant \none finds an antiferromagnetic interaction J AF  = -1.12 K.  6     \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig.2 . Magnetic susceptibility times the temperature vs temperature for (a) Cu 6Fe and (b)  \nCu 6Co. The solid lines are theoretical fits in terms o f a Curie-Weiss law as discussed in \nthe text.  \nAs a whole, these results point to the existence of  only a very weak exchange coupling \ninteraction between the magnetic centers. This conc lusion is reinforced by the isothermal \nmagnetization curves which can be fitted reasonably  well in terms of non-interacting \nparamagnetic ions [4].  0 50 100 150 200 250 2.5 2.6 2.7 2.8 2.9 3.0 \n(a) \n  χχχχT ( emu K /mol) \nT(K) Cu 6Fe \n Exp. \n Fit \n0 20 40 60 80 100 120 140 160 180 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 \n(b) \n  χT ( emu K /mole) \nT (K) Cu 6Co \n Exp. \n Fit  7While this is not much surprising for the Cu 6Co derivative, for which the interacting \ncenters are located far apart from each other, and mutually counterbalancing interactions \nmay occur, the situation is more puzzling for the C u 6Fe derivative. For this system the \nmagnetic orbitals of Cu(II) and those of Fe(III), r espectively e g and t 2g  in octahedral \nsymmetry, should be orthogonal, leading to a substa ntial ferromagnetic interaction. While \nthe observed interaction is indeed of the expected sign, its magnitude is much lower than \nexpected. It is however to be noted that a negligib ly small value of the exchange coupling \nof Cu(II) with Fe(CN) 63- units has been recently reported [7]. \n \nB. EPR spectra                                       \nElectron Paramagnetic Resonance (EPR) measurements were carried out at 9.45 \nGHz (X band) at room temperature with a Bruker spec trometer, equipped with a standard \nmicrowave cavity. A modulation field of 0.05 mT and  a microwave power of about 1.86 \nmW were used. \nThe room temperature EPR spectrum of Cu 6Fe and of Cu 6Co are shown in Fig. 3. \nThe shape of the signal for both systems is typical  of octahedral Cu 2+  ions with axial \ndistortion environment, leading to a g // >g ⊥>2.00 pattern. This is in agreement with the \nfindings of crystal structure solution, which indic ated a square pyramidal coordination \nenvironment for Cu(II) [8]. The experimental spectr um could be satisfactorily simulated \n(as a powder spectrum resulting from the superposit ion of spectra of axial sites with \nangular orientations randomly distributed) by assum ing an anisotropic g-factor with a \nLorentzian line shape . The values obtained from the simulation of the spec trum are \ng// =2.172 and g ⊥= 2.085. This confirms that the unpaired electron i s located, as expected, \nin a d x2\n-y2\n orbital, so that the absence (or weakness) of the e xchange coupling between \nFe(III) and Cu(II) should be regarded as accidental . Finally, we note that the absence of \nthe EPR signal arising from Fe 3+  ion in the corresponding derivative at room temper ature \nis most likely due to the fast relaxation time of l ow spin Fe(III) at this temperature, \nleading to an exceedingly broad line. Further studi es at lower temperatures are currently \nin progress to clarify this issue.  8 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 3  Experimental (black line) and computed from numeri cal analysis (red line) \nderivative EPR signals in Cu 6Fe (a) and in Cu 6Co (b). \n \nC. Proton NMR spectra                                       \nNuclear Magnetic Resonance (NMR) measurements on po lycrystalline Cu 6Fe and Cu 6Co \nsamples were performed with a standard TecMag Fouri er transform pulse NMR \nspectrometer using short π/2-π/2 radio frequency (r.f) pulses (1.9-2.2 µs) in the \ntemperature range 1.6 K to 300 K at two applied mag netic fields,  H=0.5 T and 1.5 T. We \nemployed a continuous flow cryostat in the temperat ure range 4.2 to 300 K and a bath \ncryostat in the temperature range 1.6 to 4.2 K. Fou rier transform of the half echo spin \nsignal of the NMR spectrum was taken in the case wh ere the whole line could be \nirradiated with one r.f pulse. The low temperature broad spectra were obtained by the 2800 3000 3200 3400 3600 -20 -10 010  \n  (b) Derivative EPR Signals (arb. units) \nB (G) -100 -50 050 \n   \n (a)  9convolution of lines obtained from several Fourier transforms each one collected at \ndifferent values of the irradiation frequency keepi ng the external field constant. \n       Proton NMR spectra for Cu 6Fe and Cu 6Co were collected as a function of frequency \nat constant applied magnetic field H=1.5 T at diffe rent temperatures. The spectra thus \nobtained are shown in Fig.4. They are found to broa den progressively with decreasing \ntemperature and to develop a structure due to the p resence of a shifted small component. \nThe spectra at low temperatures could be fitted wel l with two Gaussian functions having \ndifferent width and shift. In the analysis of the d ata which follows we use as experimental \nresults for the full width at half maximum (FWHM) a nd for the paramagnetic shift \nLps KννΔ= (νL is the Larmor frequency ) the values used for the fitted Gaussian lines. \n      \n \n.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n-2.0x10 60.0 2.0x10 64.0x10 60.0 2.0x10 54.0x10 56.0x10 58.0x10 5\nT=1.6 K \n T= 5 K \n T= 10 K \n T= 20 K \n Fits to Gaussian \n(b) 1H NMR, Cu 6Co, H=1.5 T  \n Intensity (arb. units) \nFrequency(Hz) -2.0x10 60.0 2.0x10 64.0x10 60.0 2.0x10 64.0x10 66.0x10 6\n T= 1.6 K \n T=3.8 K \n T=6.9 K \n T=20 K \n Fits to Gaussian \n(a) 1H NMR, Cu 6Fe, H=1.5 T \n  Intensity (arb. units) \nFrequency(Hz)  10  \nFig.4    Representative spectra of proton NMR at different t emperatures with fitting \ncurves made up of the superposition of two Gaussian  lines at different resonance \nfrequency for Cu 6Fe (a) and for Cu 6Co (b).  \n  \nThe shape and width of the proton NMR spectrum is d etermined by two main \ninteractions: (i) the nuclear-nuclear dipolar inter action, (ii) the hyperfine interaction of \nthe proton with the neighboring magnetic ions. The first interaction generates a \ntemperature and field independent broadening [9] wh ich depends on the hydrogen \ndistribution in the molecule and is thus similar in  all molecular magnets independently of \ntheir magnetic properties [10].   \nThe hyperfine field resulting from the interaction of protons with local magnetic \nmoments of Cu 2+ may contain contributions from both the classical d ipolar interaction \nand from a direct contact term due to the hybridiza tion of proton s-wave function with the \nd-wave function of magnetic ions. The dipolar contr ibution has tensorial character and is \nthus responsible for the inhomogeneous width of the  line due to the random distribution \nof orientations in a powder sample and to the many non-equivalent proton sites. The \ncontact interaction, on the other hand, has scalar form and it can generate a shift of the \nline for certain groups of equivalent protons in th e molecule [11]. \n     In the usual simple Gaussian approximation for  the NMR line shape, the line width is \nproportional to the square root of the second momen t, which in turn is given by the sum \nof the second moments due to the two interactions d escribed above [9]: \n \nm d FWHM 〉Δ〈+〉Δ〈∝2 2ν ν      (1)                \n                              \n where <Δν2>d  is   the intrinsic second moment due to nuclear d ipolar interactions, and \n<Δν2>m is the second moment of the local frequency-shift distribution (due to nearby \nelectronic moments) at the different proton sites o f all molecules. The relation between \n<Δν2>m and local Cu 2+  electronic moments for a simple dipolar interactio n is given by [9] \n  11 2\n, 3\n,,22\n0,2)( 1∑ ∑∑ ∑ ∑\n\n\n\n〉〈 =\n\n〉−〈 =〉Δ〈\n∈ ∈Δ\n∈Δ\nR Ri Rjtjz\njiji\nR Rit iR m m\nrA\nN Nϑ γνν ν          (2) \n \nwhere R labels different molecules, i and j span di fferent protons and Cu 2+  ions within \neach molecule, N is the total number of probed prot ons. In Eq.2 , νR,i  is the NMR \nresonance frequency of nucleus i and  νL = γ H is the bare Larmor resonance frequency. \nThe difference between the two resonance frequencie s represents the shift for nucleus i \ndue to the local field generated by the nearby mome nts j. A ( ϑi,j ) is the angular dependent \ndipolar coupling constant between nucleus i and mom ent j and r i,j  the corresponding \ndistance. < mz,j > is the component of the Cu 2+  moment j in the direction of the applied \nfield, averaged over the NMR data acquisition times cale. In a simple paramagnet one \nexpects \nAjzNmχ=〉〈, where χ is the SQUID susceptibility in emu/mole and N A is \nAvogadro’s number.  \nWe can thus write approximately: \n \nχ νz mA FWHM =〉Δ〈=2    (3) \n \nwhere A z is the dipolar coupling constant averaged over all  protons and all orientations . \nThe experimental results for the magnetic contribut ion to the line width are plotted as a \nfunction of the magnetic susceptibility in Fig.5 fo r both Cu 6Fe and Cu 6Co. The linear \nrelation predicted by Eq.3 is well verified and the  values obtained from the fit for the \naverage dipolar coupling are A z =  2.53 ×10 22  cm -3 ( for Cu 6Fe ) and A z = 3.44 ×10 22  cm -3 \n(for Cu 6Co) which are consistent with the dipolar interacti on of protons not directly \ncoupled to the Cu 2+  magnetic ions at a mean distance of 3 Å from Cu 2+ .  12                    0.0 0.2 0.4 0.6 0.8 1.0 1.2 0100 200 300 400 500 600 700 800 900 \n Cu 6Fe \n Cu 6Co  \n (FWHM) m(kHz) \nχχ χχ(emu/mole) \n \nFig.5.  Magnetic inhomogeneous broadening of the proton NM R line plotted vs. magnetic \nsusceptibility in Cu 6Co and Cu 6Fe. The straight lines are curve fits according to Eq.3 . \n \n    We turn now to the analysis of the small shifted li ne observed in the NMR spectra of \nboth Cu 6Fe and Cu 6Co (see Fig.4). The paramagnetic shift is defined a s \nLLR\nps Kννν−= , \nwhere νR is the resonance frequency and νL is the proton Larmor frequency . It can be \nexpressed as [11]: \n \n  )(TNHK\nBAeff \nps χµ=                                                                      (4)  \n \nwhere µB= Bohr magneton and χ(T)=paramagnetic susceptibility per mole, \nNA=Avogadro’s number, H eff = local hyperfine field. The hyperfine field, which  \ngenerates the line shift, is due to a contact scala r interaction arising from the electron \ndensity associated with the s- part of the wave fun ction at the proton site. Thus H eff  can \nbe expressed in terms of the atomic hyperfine coupl ing constant, a(s), multiplied by a \ncorrection factor, ξ, which gives the fraction of s-character of the wa ve function of the \nmagnetic electron at the proton site [11]:  13  \n)(sa HB eff ξµ=          (5) \n \nFor an atom the hyperfine constant can be expressed  as ABnP sa µγπh316 )(= , with  \n2) 0 (A APψ=  the electron probability density at the nucleus fo r the free atom . \n     The experimental results for the shift of the satellite line in Fig.4 are shown in Fig.6 \nfor both Cu 6Fe and Cu 6Co plotted also as a function of the magnetic susce ptibility. As \nseen in the figure the prediction of Eq.4 is well v erified. From the slope of the plot of the \nshift vs. the susceptibility one derives a value of  506.5 G for the hyperfine magnetic field \nat the proton site for hydrogen bonded to the Cu 2+ for Cu 6Fe and a value of 462.4 G in the \ncase of Cu 6Co. \n      The theoretical hyperfine constant for H atom  is a(s)= 0.0473cm -1 [11] close to the \nvalue reported for the molecular hydrogen ion H 2+ [12] and corresponding to an hyperfine \nfield at the proton site of about 28 Tesla. Thus th e contact term for the bridging \nhydrogen’s in Cu 6Fe and Cu 6Co is only about 0.17% of the atomic hyperfine fiel d for 1s \nelectron in hydrogen atom consistent with a very sm all overlap of d and s wave functions \nof the magnetic ion and the hydrogen respectively.  14 0 5 10 15 20 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 \n0,0 0,2 0,4 0,6 0,8 1,0 1,2 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6  Cu 6Fe \n Cu 6Co \n  Kps (%) \nT(K) \n  Kps (%) \nχχ χχ(emu/mole) \n \nFig. 6  Temperature dependence of paramagnetic shift of th e satellite line (see Fig.4) in \nCu 6Fe and Cu 6Co. The inset shows the linear behavior of K ps  vs. χ with temperature as an \nimplicit parameter. \n \n D. Proton NMR signal intensity, T 2 and wipeout effects \n      The normalised signal intensities for protons  studied as a function of temperature in \nCu 6Fe and Cu 6Co are shown below. The signal intensity was measur ed by the area under \nthe echoes collected at different delay times, obta ined from then usual Hahn-echo \nsequence [13]. The M xy (t) vs. t curve giving the spin-spin relaxation rec overy law was \nextrapolated at t=0 and normalised by multiplying b y T to compensate for the Boltzmann \nfactor. At low temperature the spectrum broadens an d so it was acquired point by point \nby sweeping the resonance frequency at fixed magnet ic field. As shown in Fig.7 the \ndecrease of the normalized intensity in the interme diate temperature regime indicates a \nloss of signal. The loss of signal is a phenomenon,  which has been observed in many \nmolecular nanomagnets [14]. The explanation of this  “wipe-out” effect rests in the very  15 short T 2 attained by the nuclei closer to the magnetic ions . T2 was measured in our \nsystems from the exponential decay of the echo ampl itude as a function of time delay \nbetween two rf pulses and the results are shown in Fig. 8. The very short value of  T 2 and \nthe  broad maximum observed in the T dependence of 1/T 2 in Fig.8 are in qualitative \nagreement with the loss of signal intensity observe d in the same temperature range.  \n                       \n0 50 100 150 200 250 300 0.0 0.2 0.4 0.6 0.8 1.0  Cu 6Co, H=1.5T \n Cu 6Co, H=0.5T \n Cu 6Fe, H=1.5T \n  I*T/I max *T max \nT(K) \n \nFig. 7 . Temperature dependence of normalised NMR signal i ntensity multiplied by \ntemperature  \n \n \n  16  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 8 . Temperature dependence of spin-spin relaxation ra te in Cu 6Fe at H=1.5T and at \nH=1.5T and at H= 0.5T in Cu 6Co. \n \nThus the results of the temperature and field  depe ndence of the relaxation rate, which \nwill be discussed in the following paragraph, refer  only to the average proton relaxation \nrate of the nuclei which can be detected. Since a l arge number of nuclei escape detection \n( i.e, the above cited “wipeout” effect) the absolu te values of 1/T 1 are clearly not very \nsignificant. However, the relative temperature and field dependence should not be \naffected by the wipe-out effect. \n \nE. Temperature and field dependence of NSLR \n               The proton nuclear spin lattice rela xation rate (NSLR), T 1-1 , was obtained by \nmonitoring the recovery of the nuclear magnetizatio n following a long comb of π/2 radio \nfrequency (r.f ) pulses in order to obtain the best  initial saturation conditions. The \nrecovery was found to be strongly non exponential a t all temperatures and magnetic \nfields. This is a common situation in molecular nan omagnets [10] since the protons are 0 50 100 150 200 250 300 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 \n  \n T 2-1 _Cu6Co at 0.5 T \n T 2-1 _Cu6Co at 1.5 T \n T 2-1 _Cu6Fe at 1.5 T T2-1 (µµµµs-1 )\nT(K)  17 located at different distances and angles from the relaxing magnetic ions. Since the spin \ndiffusion is not sufficiently fast to ensure a comm on spin temperature during the recovery \nprocess, the recovery curve is a superposition of m any exponential curves each one \nrepresenting the relaxation of a given proton. By m easuring the initial recovery or tangent \nat the origin one measures the average relaxation r ate, which is dominated by the fast \nrelaxing protons (the nearest to the magnetic ions) . The shape of the recovery curve may \nchange as a function of temperature and magnetic fi eld as the result of the spin diffusion \neffect [15]. Thus for better consistency we measure d the NSLR from the time at which \nthe recovery curve has reduced to 1/e of the initia l value. The measured parameter is in \nany case proportional to the average relaxation rat e of the protons detected in the NMR \nsignal [15]. \n    \n  The results for the field dependence of the proton relaxation rate at three different \ntemperatures in both compounds are shown in Fig.9 a nd 10. \n \n \n \n \n \n \n           \n \n \n \n \n \n \n \n \nFig. 9. Field dependence of spin- lattice relaxation rate in Cu 6Fe at \nvarious temperatures with fit according to Eq(7) ( see text). 0 1 2 3 4 50.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1H NMR, Cu 6Fe \n  \n T=300 K \n T=77 K \n T=4.2 K 1/T 1(ms  -1 )\nH(Tesla)  18  \n \n                          \n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 \n1H NMR, Cu 6Co \n T= 4.2 K \n T= 77 K \n T= 300 K \n  T1-1 (ms -1 )\nH(T) \n \nFig. 10  Field dependence of spin-lattice relaxation rate i n Cu 6Co at various temperatures \nwith fit according to Eq(7). ( see text). \n \n                  The results for the temperature dependence of proton NSLR at two external \nmagnetic fields, H=1.5 T and H= 0.5 T are shown in Fig.11. \n  19         \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 11  Temperature dependence of spin-lattice relaxation rate at H=1.5T in Cu 6Fe and \nat H=1.5T, 0.5T in Cu6Co. The inset shows the low t emperature behavior.  \n \nThe weak temperature dependence is at variance with  the pronounced peak observed in \nstrongly exchange coupled molecular nanomagnets [10 ]. This is consistent with the \nsimple paramagnetic behaviour observed in the magne tization measurements. One can \nconclude that the spin dynamics reflects the fluctu ations of the single  magnetic moments \nof the ions in the molecule without effects associa ted to the collective spin dynamics \nexcept for the very low T region where an upturn of  1/T 1 is observed for the FM coupled \nCu 6Fe and a downturn of 1/T 1 is observed in AFM coupled Cu 6Co (see inset of Fig.11). \nThe weak decrease of the relaxation rate in the int ermediate temperature range is most \nlikely due to the decrease of spin diffusion time d ue to the inhomogeneous broadening of \nthe proton NMR line [15]. 0 50 100 150 200 250 300 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 \n0 5 10 15 20 25 30 35 40 0.0 0.6 1.2 1.8 \n  \n T 1-1 _Cu 6Co at 0.5 T \n T 1-1 _Cu 6Fe at 1.5 T \n T 1-1 _Cu 6Co at 1.5 T T1-1 (ms -1 )\nT(K) \n  T1-1 (ms -1 )\nT(K)  20 A more quantitative analysis of the data can be don e on the basis of Moriya’s theory for \nNSLR in Heisenberg isotropic three dimensional para magnets [16,17] \n In three dimensional paramagnets the spectral dens ity J ( ω) of the electronic spin \nfluctuation is Lorentzian with a correlation freque ncy given by [16,17]: \n \n) 1(2+ = szs Jk B\nexc hω                                                                                                (6) \nwhere z is the number of nearest neighbors and for both the systems Cu 6Fe and Cu 6Co, \nz=1 \nThe NSLR is proportional to the spectral density at  both the electronic Larmor frequency \nωe and at the nuclear Larmor frequency ωN [10,16,17]  \n \n()\n\n+ =±±)()(21) 0 (\n41\n222\n1nzz\ne B\nBneJA JATk\ngTω ω χ\nµπγγh                                             (7) \n \n where A ±  and A z are the Fourier transforms of the spherical compon ent of the product of \ntwo dipolar interaction tensors describing the hype rfine coupling of a given proton to the \nparamagnetic ion along transverse and longitudinal directions, respectively, with respect \nto the external magnetic field averaged over all pr otons and all directions [10,16]. \n     In three dimensional paramagnets with strong e xchange interaction J the exchange \nfrequency ωexc is much larger than both ωe and ωn and thus one finds that the relaxation \nrate is field independent since \nexc e exc exc \neJωωωωω1)(2 2≅\n+=±    and  \nexc n exc exc \nnzJωωωωω1)(2 2≅\n+= . On the other hand in the present case, since the exchange \ncoupling is very small, a field dependence is possi ble from the first term  in Eq.7. \nWe have fitted the results in Fig.9 and 10 with Eq. 7 which can be rewritten for a \nLorentzian spectral density and ωexc >>  ωn as:                                  \n  21 \n\n\n\n+\n+=±\nexc z\ne exc exc A AKT ω ωωω 1\n211\n2 2\n1                                                                 (8)              \n \n                         \nThe constant K can be estimated from the known valu e of the susceptibility χ(0). The \nvalues are K =1.7 ms -1 for Cu 6Fe and K =1.4 ms -1 for Cu 6Co. The exchange frequency \nshould be of the order of the value obtained using Moriya’s formula [16,17] by using the \nmeasured exchange interactions J i.e. ωexc = 1.3×10 10  Hz for Cu 6Fe (J= 0.14 K, \nferromagnetic) and ωexc = 1.036×10 11  Hz for Cu 6Co ( J= -1.12 K, antiferromagnetic).  \n     The experimental data in Fig.10 and 11 can be fitted by Eq.8 with values for the \nhyperfine constants: A ± ≅ 1.4×10 46  cm -6 and A z ≅ 0.31×10 46  cm -6 for Cu 6Fe and A ± ≅ \n2.07×10 46  cm -6 and A z ≅ 0.4×10 46  cm -6 for Cu 6Co . The fitting parameters are of the \ncorrect order of magnitude as obtained in the case of many other molecular nanomagnets \n[10,18]. In particular, the values of A z, which depend only on the tensorial dipolar \ninteraction [9], are consistent with a dipolar inte raction of protons with nearest neighbor \nand next nearest neighbor magnetic ions. The coupli ng constant A ± , on the other hand, \ncan contain contributions from both the dipolar int eraction and the scalar contact \nhyperfine interaction. In both cases we found that A ± > Az, which is an indication of the \npresence of an additional contribution  due to a co ntact interaction arising from the \nhybridization of hydrogen s wave function with the d wave function of Cu ions as found \nin the analysis of the spectra ( see section C ).   The exchange frequencies which best fits \nthe data are : a) for Cu 6Fe , 1.0 x 10 11  Hz , 1.3 x 10 11  Hz and 1.4 x 10 11  Hz at 300K, 77K \nand 4.2K respectively b) for Cu 6Co , 1.2 x 10 11  Hz , 1.7 x 10 11  Hz, 1.9 x 10 11  Hz for \n300K,77K and 4.2 K respectively. In both cases the estimated error is ± 10% .  The weak \ntemperature dependence is probably irrelevant since  the recovery of the nuclear \nmagnetization and thus the value of the measured  N SLR can be affected in a different \nway at different temperatures by spin diffusion eff ects which are too difficult to account \nfor. The order of magnitude of the exchange frequen cy extracted from the data is in \nexcellent agreement with the theoretical value from  Moriya’s Eq.6 ( ωexc = 1.036×10 11  Hz \n) only for Cu 6Co . For Cu 6Fe the experimental value is one order of magnitude  larger.  22 This could be due to a much faster fluctuation rate  for the Fe +3  magnetic moment as also \nsuggested by the impossibility of detecting the EPR  signal.  \n                  \nIV) Summary and Conclusions  \n          We have shown that Cu 6Fe and Cu 6Co are novel magnetic molecular clusters in \nthe sense that, contrary to most of the molecular n anomagnets [1], the magnetic centres \nare very weakly coupled within the cluster. Thus a crystal of Cu 6Fe (Co) is made up of \nidentical single molecule paramagnets. In Cu 6Fe the Cu 2+ ions (s=1/2) are found to be \ncoupled in pairs via the magnetic Fe 3+  (s=1/2) ion by a super-exchange ferromagnetic \ninteraction with J F = 0.14 K. In view of the weakness of the coupling constant it could \nalso be a simple dipolar coupling between the Cu 2+ ion and the Fe 3+ ion. On the other \nhand in Cu 6Co, the Cu 2+  ions appear to be coupled via the diamagnetic Co 3+ ion by a \nsuper-exchange antiferromagnetic interaction with J AF  =-1.12 K. \nIn both compounds the EPR spectra are indicative of  an octahedral Cu 2+ site with axial \ndistortion (i.e. g // =2.172 and g ⊥= 2.085). The proton spin–lattice relaxation time i s \nconsistent with an almost temperature independent s ingle correlation frequency ωexc  = \n10 11 Hz related to the fluctuations of the electron spin  due to the T 2 –type flip-flop \ntransitions associated to the weak exchange couplin g as predicted by Moriya [16,17]. \nHowever, a quantitative disagreement with the simpl e Moriya’s prediction  is found for \nCu 6Fe for which we cannot find the EPR signal of the F e 3+  ion a circumstance which \nsuggest a fast fluctuation of this ionic magnetic m oment not accounted for by Moriya’s \ntheory which applies to isotropic Heisenberg intera ctions only. Below about 2 K the \nproton relaxation rate shows an increase in Cu 6Fe due to the ferromagnetic correlations \nand a decrease in Cu 6Co due to antiferromagnetic correlations (see inset  Fig.11). \nMeasurements at much lower temperature are necessar y to investigate the possible \npresence of long range magnetic order.  \n           In conclusion the magnetic molecular clu sters studied here appear to be very \nsuitable candidates to investigate magnetic orderin g at very low temperature ( millikelvin \nrange) where the competition between the weak intra molecular exchange interaction and \nthe even weaker intermolecular dipolar interaction may lead to some novel kind of \nmagnetic order.  23 Acknowledgements \n \nWe acknowledge support from the EU Network of Excel lence MAGMANet and from \nGrant PRIN N.2006029518 of the Italian Ministry of Research. \n \n \n \nReferences: \n[1] D. Gatteschi, R. Sessoli,and J.Villain , Molecular Nanomagnets , (Oxford Univeristy \nPress, 2006). \n[2] M. N. Leuenberger and D. Loss, Nature (London)  410 , 789 (2001) \n[3] F. Troiani, M. Affronte, S. Carretta, P.Santini  and G. Amoretti, Phys. Rev. Lett. 94 , \n190501 (2005). \n[4] C. Maxim, M. Andruh, L. Sorace, A. Caneschi P. Khuntia, A.Lascialfari et al, to be \npublished. \n[5] M. Atanasov, P. Comba, S. Forster, G. Linti, T.  Malcherek, R. Miletich, A. I. \nPrikhod'ko, H. Wadepohl,  Inorg. Chem. , 45 , 7722, (2006). \n[6]  M. Atanasov, C. Busche, P. Comba, F. El Hallak , B. Martin, G. Rajaraman, J. van \nSlageren and H. Wadepohl,  Inorg. Chem. , 47 , 8112, (2008). \n[7]  M. Atanasov, P. Comba, Y.D. Lampecka, G. Linti , T.Malcherek, R. Miletich, A. I. \nPrikhod'ko, H. Pritzkow,  Chem. Eur. J.  12, 737 (2006). \n[8] S. Golhen, L. Ouahab, D. Grandjean and P. Molin ie, Inorg. Chem. 37 , 1499 (1998) \n[9] A. Abragam, The Principles of Nuclear Magnetism  (Clarendon,Oxford, 1961). \n[10] F. Borsa, A. Lascialfari and Y. Furukawa, in “ Novel NMR and EPR       Techniques” \n, eds. J. Dolinsek, M.Vilfan and S. Zumer   , Springer (2006)   \n[11] G. C. Carter, L. H. Bennett and D. J. Kahan,  Metallic Shifts in NMR part 1 , \n(Pergamon, Oxford, London, 1971). \n[12]  J. F. Babb and A. Dalgarno, Phys Rev. A 46 , R5317 (1992). \n[13] C.P.Slichter, Principles of Magnetic Resonance , Springer-Verlag, Berlin (1990), , \np.p 131.  24 [14] M. Belesi, A. Lascialfari, D. Procissi, Z. H. Jang, F. Borsa, Phys. Rev. B 72 , 014440 \n(2005). \n[15] I. J. Lowe and D. Tse, Phys. Rev. 166 , 279 (1968). \n [16]  T. Moriya, Prog. Theor. Phys. 28 , 371 (1962) \n[17] T. Moriya, Prog. Theor. Phys. 16 ,  23 (1956) \n[18] A. Lascialfari, D. Gatteschi, F. Borsa, A. Sha stri, Z. H. Jang and P. Carretta, Phys. \nRev. B,  57 , 514 (1998). \n            \n \n \n \n \n \n "    },    {        "title": "1901.09610v1.Study_of_Dynamo_Action_in_Three_Dimensional_Magnetohydrodynamic_Plasma_with_Arnold_Beltrami_Childress_Flow.pdf",        "content": "Study of Dynamo Action in Three Dimensional Magnetohydrodynamic Plasma with\nArnold-Beltrami-Childress Flow\nRupak Mukherjee\u0003and Rajaraman Ganeshy\nInstitute for Plasma Research, HBNI, Bhat, Gandhinagar - 382428, India\nFor a three dimensional magnetohydrodynamic (MHD) plasma the dynamo action with ABC \row\nas initial condition has been studied. The study delineates crucial parameter that gives a transition\nfrom coherent nonlinear oscillation to dynamo. Further, for both kinematic and dynamic models\nat magnetic Prandtl number equal to unity the dynamo action is studied for driven ABC \rows.\nThe magnetic resistivity has been chosen at a value where the fast dynamo occurs and the growth\nrate shows no further variation with the change of magnetic Reynold's number. The exponent\nof growth of magnetic energy increases, indicating a faster dynamo, if a higher wave number is\nexcited compared to the one with a lower wave number. The result has been found to hold good\nfor both kinematic and externally forced dynamic dynamos where the backreaction of magnetic\n\feld on the velocity \feld is no more negligible. In case of an externally forced dynamic dynamo,\nthe super Alfvenic \rows have been found to excite strong dynamos giving rise to the growth of\nmagnetic energy of seven orders of magnitude. The back-reaction of magnetic \feld on the velocity\n\feld through Lorentz force term has been found to a\u000bect the dynamics of the velocity \feld and\nin turn the dynamics of magnetic \feld, leading to a saturation, when the dynamo action is very\nprominent.\nI. INTRODUCTION\nOne of the most interesting open questions of astro-\nphysics is the birth of magnetic \feld in the cosmos.\nThere are several theoretical models [1, 2] and some\nof them are tested in laboratory [3{6] also, mimicing\nsome aspects of the astrophysical plasma. Amongst\nthe zoo of theoretical models, E N Parker's [7] theory\nofdynamo action is one widely celebrated model. The\nlarge-scale magnetic \feld generation in the `Sun' or in\ngalaxies are mostly attributed to mean-\feld-dynamo.\nOn the other hand there are astrophysical evidences of\nsmall scale magnetic \feld generation through turbulent\n\ructuation dynamo [8{14]. The origin of dynamo in\nthree dimensional plasma is still poorly understood\nand still a matter of debate [15]. In general, the\nmost of the theoretical models employed in this study\nuse the basic equations of MagnetoHydroDynamics\n(MHD). The model governs the dynamics of each ` \ruid\nelement ' - a collisional enough fundamental block of\nthe medium. However, MHD equations describing\nthe plasma in the continuum limit o\u000ber fundamental\nchallenges to the analytical solution of the basic equa-\ntions [16]. Hence it is interesting to ask whether there\nis any \fnite dimensional description of the subject exists?\nThe authors have shown that in two spatial di-\nmensions for incompressible \rows a \fnite dimensional\napproach exists and the analytical results were found to\n\ft well with the numerical results obtained earlier [17].\nHowever, the authors also delineate the regimes where\nthe analytical description does not hold good [17]. In\n\u0003rupak@ipr.res.in; rupakmukherjee01@gmail.com\nyganesh@ipr.res.inthree spatial dimensions, the problem becomes more\ncritical to analyse analytically. The phase space of the\nsystem being in\fnite dimensional, in three dimensions,\nlong time prediction of the chaotic trajectories are\nextremely challenging. But, it was previously shown\nby the authors that for some typical chaotic \rows in\nthree spatial dimensions, the \row and the magnetic \feld\nvariables are found to reconstruct back to their initial\ncondition - thereby getting trapped in the phase space of\nthe system [18]. The cause of such recurrence is believed\nto be the low dimensional behaviour of the single \ruid\nplasma medium for some typical parameters. Most of\nthe short scales were not excited in the system and thus\nthe continuum was acting like a low degrees of freedom\nmedium.\nTherefore in is natural to ask, is there any way to\nexcite the short scales in a regulated manner as we\ncontinuously move in the parameter scale? Finally the\nquestion becomes, what happens when all the scales are\nexcited?\nIn the \frst part of this paper, we address the above\nquestions. We propose a model distinctly showing a\ncontinuous transition to self-consistent dynamo from a\nnon-linear coherent oscillation [17]. Though, an analyti-\ncal description identifying the exact process is still under\ndevelopment, the direct numerical simulation studies of\nthree dimensional chaotic \rows support the conjecture.\nIt is known that, in case of a short-scale dynamo, the\nmagnetic \feld lines frozen to the plasma \rows get \frst\nstretched along the chaotic velocity \rows - then gets\ntwisted and folded back [19]. Such processes introduce\ngeneration of short scales into the system giving birth\nof dynamos classi\fed as STF dynamo [20, 21]. Though\nit is well known that a small but non-zero resistivity\na\u000bects the plasma relaxation because of reconnectionarXiv:1901.09610v1  [physics.plasm-ph]  28 Jan 20192\nprocess [22{27], we choose \rows with \fnite viscosity and\nresistivity showing the robustness of our results. We see\na continuous growth of magnetic energy at the cost of\nkinetic energy and thereafter decrease of the magnetic\nenergy through reconnection process and converting\nback to kinetic energy. We move in parameters and\n\fnd the reconnection to occur with less probability\nand growth of magnetic energy. Thus we move in\nparameter space and in one limit observe coherent\nnonlinear oscillation of kinetic and magnetic energy\nwithin the premise of single \ruid magnetohydrodynam-\nics and in the other limit observe dynamo action to occur.\nFrom our previous study [18], we choose \rows that\ndo not recur and thus even though there is an energy\nexchange between the kinetic and the magnetic modes,\nthe trapping in phase space does not occur even in the\nopposite limit of the dynamo.\nAs a test case we choose Arnold-Beltrami-Childress\n(ABC) \row since it is already known to produce fastest\ndynamos in the kinematic regimes [28] and is non-\nrecurrent [18]. We reproduce the works of Galloway and\nFrisch [29] in the kinematic regime and further check\nour results in the self-consistent regime with the work\nof Sadek et al [30]. We use the well-benchmarked three\ndimensional MHD code G-MHD3D [31, 32] capable\nof direct numerical simulation of weakly compressible\nsingle \ruid MHD equations.\nIn the second part of the paper, the growth of magnetic\nenergy (dynamo action) under the action of externally\ndriven ABC \row has been studied. The analysis can be\ndivided into two parts, i)the linear kinematic regime\nwhere the plasma \row stretches the magnetic \feld lines\ngiving rise to an exponential growth of magnetic \feld\nand ii)the nonlinear regime where the magnetic \feld\ngenerates Lorentz force strong enough to modify the\ntopology of the background plasma \row, resulting in\na saturation of the growth rate of the magnetic \feld\n[33, 34].\nIn the linear kinematic regime, the growth of magnetic\n\feld is found to exponentially rise without bound after\ncrossing a critical threshold Reynold's number ( Rmc).\nWhen the backreaction is not negligible, (we dub this\ncase as dynamic dynamo ) the dynamo saturates when\nit enters the nonlinear regime. At very low magnetic\nPrandtl number ( Pm\u001c1) the dynamic dynamo has\nbeen studied in detail [33, 35{43]. A detailed review\nof the ABC \row leading to dynamo action is due to\nGalloway [44]. However, in the solar convection zone,\nunprecedentedly high resolution simultion with Prandtl\nnumber unity ( Pm= 1) has been shown to produce\nglobal-scale magnetic \feld even in the regime of large\nReynolds numbers [45]. For Pm= 1 and Reynold's\nnumberRm > Rm c, we extend our search for faster\ndynamos (now dynamic one) when the initial velocityand forcing scales contain higher wave-number. We also\naddress the cases where the Alfven speed and sonic speed\ndi\u000ber signi\fcantly. We see that the backreaction of the\nmagnetic \feld alters the ABC \row pro\fle and thereby\nthe growth of magnetic energy itself gets a\u000bected. We\nnotice three distinct growth rates in the magnetic energy\nsolely because of the inclusion of the backreaction of the\nmagnetic \feld on the velocity \feld and vice-versa.\nIn particular, the following aspects of dynamo action\nunder ABC \row have been studied in detail:\n\u000fWe identify crucial parameter that controls the gen-\neration of short scales and thereby lead to dynamo\naction.\n\u000fWe \fnd that, above the critical magnetic Raynold's\nnumber (Rmc), the growth rate of kinematic dy-\nnamo process increases with velocity scale contain-\ning higher wave numbers.\n\u000fWe also \fnd that, above Rmc, the growth rate of\nself-consistent or dynamic dynamo also increases,\nwith velocity and forcing scales containing higher\nwave numbers.\n\u000fFor super Alfvenic \rows, when strong dynamo ac-\ntion occur, the e\u000bect of interplay of energy between\nmagnetic and kinetic modes leads to saturation of\nthe growth of magnetic energy at late times.\n\u000fFor ABC \row to start with, both for kinematic\nas well as dynamic dynamo action, the magnetic\nenergy is primarily contained in the intermediate\nscales.\nII. GOVERNING EQUATIONS\nThe single \ruid magnetohydrodynamic (MHD)\ndescription of a plasma is quite incomplete but has\nbeen found to serve aptly to explain many phenomena\nobserved in laboratory and astrophysical systems. Thus\nunder certain criteria, the plasma dynamics is believed\nto be well modelled through single \ruid MHD equations.\nThe two di\u000berent charge species (electrons and ions) are\nassumed to form a single \ruid because of the negligible\nmass of the electrons. A \ruid element is assumed to be\nmuch larger than the length scale of separation between\nthe two di\u000berent charge species. Also the timescale at\nwhich the phenomena are observed are quite longer than\nthe gyrofrequency of each of the charge species. Thus\nno large scale electric \feld is produced or sustained in\nthe timescale of interest.\nThe basic equations governing the dynamics of the3\nmagnetohydrodynamic \ruid are as follows:\n@\u001a\n@t+~r\u0001(\u001a~ u) = 0 (1)\n@(\u001a~ u)\n@t+~r\u0001\u0014\n\u001a~ u\n~ u+\u0012\nP+B2\n2\u0013\nI\u0000~B\n~B\u0015\n=\u0016r2~ u+\u001a~f (2)\n@~B\n@t+~r\u0001\u0010\n~ u\n~B\u0000~B\n~ u\u0011\n=\u0011r2~B (3)\nwhereP=C2\ns\u001aand~f=0\n@Asin(kfz) +Ccos(kfy)\nBsin(kfx) +Acos(kfz)\nCsin(kfy) +Bcos(kfx)1\nA:\nIn the above system of equations, \u001a,~ u,Pand~Bare\nthe density, velocity, kinetic pressure and the magnetic\n\feld of a \ruid element respectively. \u0016and\u0011denote\nthe coe\u000ecients of kinematic viscosity and magnetic\nresistivity. We assume \u0016and\u0011are constants throughout\nspace and time. The symbol \\ \n\" represents the dyadic\nbetween the two vector quantities.\nThe kinetic Reynold's number ( Re) and magnetic\nReynold's number ( Rm) are de\fned by Re=U0L\n\u0016\nandRm =U0L\n\u0011whereU0is the maximum velocity of\nthe \ruid medium to start with and Lis the system length.\nWe also de\fne the sound speed of the \ruid medium\nasCs=U0\nMs, where,Msis the sonic Mach number of\nthe \ruid. We assume it to be uniform throughout the\nspace and time. The Alfven speed is determined from the\nrelationVA=U0\nMAwhereMAis the Alfven Mach number\nof the plasma medium. The initial magnetic \feld present\nin the plasma is determined from relation B0=VAp\u001a0,\nwhere,\u001a0is the initial density pro\fle of the \ruid.\nIII. PARAMETER DETAILS\nThe \frst results of Galloway and Frisch [46] showed\ncritical dependency on the magnitude of magnetic resis-\ntivity (Rm). Later this result was further tested and\nreproduced with much greater accuracy and resolution\nby in several other independent studies [47, 48]. The ob-\nservation that, within a 2 \u0019periodic box, for algorithms\ndepending on spectral solvers, the smallest features in\nmagnetic \feld are on scales of order Rm\u00001=2indicates\nthat the grid size required to resolve a given Rmscales\nlikeRm1=2[29]. We choose, N= 64 which resolves Rm\nupto 4096 and keep our parameters \fxed at Rm= 450\n(where the growth rate of dynamo was found to get sat-\nurated [29]) well within the resolution threshold. Also\nthe result of Sadek et al [30] con\frms that most of the\nkinetic and magnetic energy content remains within the\nlarge scales, even when the driving wave-number is kept\nat intermediate scales (at least upto kf= 16). This sets\nlimit to our choice of maximum driving wave number\n(kf= 16) at the grid resolution N= 64.Throughout our simulation, we set N= 64,L= 2\u0019,\n\u000et= 10\u00004,\u001a0= 1. For some test runs the grid resolution\nis increased to N= 128 for both kinematic and dynamic\ncases but we found no signi\fcant variation of the physics\nresults. The initial magnitude of density ( \u001a0) is known to\na\u000bect the dynamics and growth rate of an instability in\na compressible neutral \ruid [49, 50]. However, in present\ncase we keep the initial density \fxed ( \u001a0= 1) for all the\nruns. We check our code with smaller time stepping ( \u000et)\nkeeping the grid resolution N= 64. No deviation from\nthe results were observed with such test runs.\nThe kinematic viscosity is controlled through the pa-\nrametersReand to guarantee similar decay of kinetic\nand magnetic energy, we set Re=Rmeverywhere. Next\nwe vary the Alfven speed through MAand observe the\ne\u000bect of these parameters on the dynamo action. We also\nchange the magnitude of forcing by controlling the values\nofA;B;C and the length-scale of forcing through kf.\nThe OpenMP parallel MHD3D code is run on 20 cores\nfor 9600 CPU hours for a single run with parameters\nmentioned above and got the following results.\nIV. SIMULATION RESULTS\nA. Initial Pro\fle of Density, Velocity and Magnetic\nField\nWe start with the initial condition \u001a=\u001a0as a\nuniform density \ruid, the initial velocity pro\fle as\nux=U0[Asin(kfz) +Ccos(kfy)],uy=U0[Bsin(kfx) +\nAcos(kfz)],uz=U0[Csin(kfy) +Bcos(kfx)] and the\ninitial magnetic \feld as Bx=By=Bz=B0. We keep\nthe initial pro\fles of all the \felds identical throughout\nour paper unless otherwise stated.\nB. Transition to Dynamo\nForMA\u00181, a coherent nonlinear oscillation is repro-\nduced as reported earlier [17]. As MAis moved from\nunity, the oscillation persists alongwith the generation of\nother modes into the system. Thus as can be found from\nFig. 1, the magnetic energy does not come back to its\ninitial value after one period of oscillation. Upon further\nincrement of Alfven Mach number, the linear dependency\nof the frequency of oscillation breaks down and persistent\nmagnetic \feld starts to generate. Finally, the growth\nof magnetic energy reaches a maximum. From Fig. 1,\nit can be seen that, the normalised magnetic energy at\nMA= 102& 103does not di\u000ber signi\fcantly, indicating a\nsaturation of the growth. However, such saturation does\nnot occur in the driven cases where, the plasma is driven\ncontinuously using an external drive which pumps in ki-\nnetic energy to the system. Such phenomena is further\nexplored in the next section of the paper.4\n 1 10 100\n 0  20  40  60  80  100Normalised Magnetic Energy [B2(t)/B2(0)]\nTimeMA = 0.5\nMA = 1\nMA = 1.5\nMA = 3\nMA = 5\nMA = 7\nMA = 10\nMA = 30\nMA = 50\nMA = 70\nMA = 100\nMA = 1000\nFIG. 1. (Color online) Transition to dynamo with the increase\nofMAfrom coherent nonlinear oscillation.\nC. Kinematic Dynamo\nThe phenomenon of magnetic energy growing exponen-\ntially with time for a statistically steady \row, where the\nvelocity \feld is held \fxed in time, is called, kinematic dy-\nnamo action. Arnold-Beltrami-Childress (ABC) \row be-\ning a steady solution of Euler equation, sets the premise\nto study the kinematic dynamo problem. For ABC \row\nkinematic dynamo was \frst obtained by Arnold et al [51]\nat magnetic Reynold's number ( Rm) between 9 and 17 :5.\nGalloway et al [46] found a more e\u000ecient dynamo e\u000bect\nwith much higher growth rate after Rm= 27 breaking\ncertain symmetries of the \row. Later on, the study had\nbeen extended for the parameters where A,B,Care not\nequal [29]. The threshold Rmfor a kinematic dynamo\nhas been well explored [38, 52{54]. The real part of the\ngrowth rate of the magnetic energy for increasing Rm\nis found to increase while the imaginary part decreases\ncontinuously [29]. ABC \rows with di\u000bernet forcing scales\n(kf6= 1) providing kinematic dynamo has been explored\nby Galanti et al [40] and more recently by Archontis et\nal[42].\nFirst we reproduce the previous results and then\nchoose an optimal set of working parameters. The moti-\nvation behind choosing the parameters are explained in\nthe previous section. We give the following runs (Table I)\nto explore the parameter regime of a kinematic dynamo\nproblem.\n1. E\u000bect of Magnetic Resistivity\nE\u000bect of magnetic resistivity ( \u0011) through the magnetic\nReynold's number ( Rm) has been widely studied in past\n[29, 40] and in recent years [47]. First we reproduce\nthe previous results by Galloway et al [29] using our\ncode [Runs: KR1, KR2, KR3]. Similar to the previous\nstudy [29] we choose U0= 1,A=B=C= 1,\nkf= 1. We time evolve only Eq. 3 for the initial timeName kfMARm\nKF1 11000 450\nKF2 21000 450\nKF3 41000 450\nKF4 81000 450\nKF5 161000 450\nKM1 1100 450\nKR1 110450\nKR2 110200\nKR3 110120\nKM2 11450\nKM3 10:1450\nTABLE I. Parameter details with which the simulation has\nbeen run for kinematic dynamo problem.\n 1e-05 1 100000 1e+10 1e+15 1e+20 1e+25\n 0  50  100  150  200  250  300Magnetic Energy\nTimeRm = 120\nRm = 200\nRm = 450\nFIG. 2. Kinematic Dynamo e\u000bect reproduced using the iden-\ntical parameter regime ( A=B=C= 1,kf= 1) by Galloway\net al [29]. The grid resolution is 643which is close to the value\n603that was taken by Galloway et al [29]. The growth rates\nof magnetic energy\u0012P\nVB2(x;y;z )\n2\u0013\nof kinematic dynamo are\nfound to increase as Rm is increased. The oscillation fre-\nquency of the magnetic energy is also found to be similar as\nGalloway et al [29]\ndata mentioned above for magnetic Reynold's number\nRm = 120;200;450 and obtain the identical growth\nof magnetic \feld as Galloway et al [29]. This result\nis shown in Fig 2. We also reproduce the real and\nimaginary part of the eigenvalue obtained previously\n[29]. The critical value of onset of kinetic dynamo action\nis found to be Rm= 27.\nWe derive the energy spectra of the kinematic dynamo\nfrom ABC \row [Fig.(3)] and observe energy is not only\ncontained in large scales rather, the energy contained in\nthe intermediate scles are quite large. We observe a k0:7\nscaling of magnetic energy.5\n 1e-20 1e-15 1e-10 1e-05 1 100000 1e+10 1e+15 1e+20\n 1  10B(k)\nktime = 0\ntime = 10\ntime = 20\ntime = 30\ntime = 40\ntime = 50\ntime = 100\ntime = 150\ntime = 200\nx0.7\nFIG. 3. Magnetic energy spectra at di\u000berent time for ABC\n\row with the identical parameter described in Fig. (2) with\nRm= 450. The energy contained in the large scales at late\ntimes shows that the short scales are equally important for a\nkinematic dynamo obtained from ABC \row.\n2. E\u000bect of Forcing Scale\nThe e\u000bect of forcing scale on the growth rate of mag-\nnetic energy has been earlier studied by Galanti et al [40]\nforkf= 1 to 10 for Rm values upto Rm = 45. For\nthe kinematic dynamo case, Galloway and Frisch ([29])\nhave shown that, even though the critical value of Rm\nfor kinematic dynamo action for ABC \row is Rmc= 27,\nthe growth rate monotonically increases until Rm= 350.\nIn the past work of Galanti et al [40] it was found that\nkf= 2 has higher growth rate than kf= 1 forRm= 45.\nAlso changing Rm from 12 to 20 did not a\u000bect the\ngrowth rate for di\u000berent kfmuch. In our case, we keep\nthe forcing length scale at kf= 1;2;4;8;16 holding the\nRm= 450 much above the critical value ( Rmcrit= 27)\nof onset of kinetic dynamo for kf= 1 (where imaginary\npart of the eigenvalue (2 \r) is undetectably small) and in\nthe regime where the growth rate does not vary much\nwith the further increment of Rm. [Runs: KF1, KF2,\nKF3, KF4, KF5] The growth rate of normalised mag-\nnetic energy,B2\nB2\n0, is found to increase as kfis increased\n[Fig.4, 5]. However, the growth rate (2 \r) saturates as kf\nis increased for U0= 0:1[5. A similar saturation was also\nobserved earlier though at Rm= 12 and 20 [40].\nThe late time dynamics is found to be widely di\u000ber-\nent for di\u000berent driving frequencies ( kf) [Fig.4,5]. For\na kinetic dynamo problem similar transient behaviour\n(kf= 16 and 8 in Fig. 5) starting from a typical initial\ncondition has been addresses previously in detail [47].\nIt was found that when the fastest growing eigenmode\nis not excited, it takes some time for the fastest eigen-\nmode to overcome the initially excited mode and hence\nthe crossover happens at a later time. Even for a dy-\nnamic dynamo under external forcing, similar result was\nearlier obtained by Galanti et al [40] forA=B=C= 1,\n 1 1e+20 1e+40 1e+60 1e+80 1e+100\n 0  10  20  30  40  50Magnetic Energy\nTimekf = 1\nkf = 2\nkf = 4\nkf = 8\nkf = 16FIG. 4. Kinematic Dynamo e\u000bect for di\u000berent driving fre-\nquency ( kf). The magnetic energy is normalised with the\ninitial magnitude at time t= 0. The parameters chosen are\nA=B=C= 1 and Re=Rm= 450 with U0= 1. The\ninitial growth rates are found to grow as kfis increased.\n 1 10 100\n 0  10  20  30  40  50Normalised Magnetic Energy\nTimekf = 1\nkf = 2\nkf = 4\nkf = 8\nkf = 16\nFIG. 5. Kinematic Dynamo e\u000bect for di\u000berent driving fre-\nquency ( kf). The normalised magnetic energy is de\fned as\u0012P\nVB2(x;y;z;t )\n2\u0000P\nVB2(x;y;z; 0)\n2\u0013\n. The parameters chosen are\nA=B=C= 1 and Re=Rm= 450 with U0= 0:1. The\ninitial growth rates are found to grow and saturate as the kf\nis increased.\nkf= 1 andRe=Rm= 12.\n3. E\u000bect of Alfven Speed\nThe Alfven speed is de\fned as VA=U0\nMA. IfMA<1;\nVA<U0and the plasma is called Sub-Alfvenic. Similarly\nifMA>1, the plasma is Super-Alfvenic. For kinetic dy-\nnamo problem, the growth rate of magnetic energy is\nfound to be independent of the magnitude of MA. [Runs:\nKF1, KM1, KR1, KM2, KM3] We check the growth\nrate forMA= 0:1;1;10;100;1000 and for every case the\ngrowth rate of dynamo is found to be identical as shown6\n 1e-06 0.0001 0.01 1 100 10000 1e+06 1e+08 1e+10\n 0  10  20  30  40  50Magnetic Energy\nTimeMA = 1000\nMA = 100\nMA = 1\nMA = 0.1\nMA = 0.01\nFIG. 6. Kinematic Dynamo e\u000bect for di\u000berent driving fre-\nquency ( MA). The parameters chosen are A=B=C= 1\nandRe=Rm = 450 with U0= 1. The growth rates are\nfound to be identical for di\u000berent MAvalues.\n 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100\n 0  10  20  30  40  50Magnetic Energy\nTimeMA = 1000\nMA = 100\nMA = 10\nMA = 1\nMA = 0.1\nFIG. 7. Kinematic Dynamo e\u000bect for di\u000berent driving fre-\nquency ( MA). The parameters chosen are A=B=C= 1\nandRe=Rm= 450 with U0= 0:1. The growth rates are\nfound to be identical for di\u000berent MAvalues.\nin Fig. 6 and 7 unlike the dynamic case discussed in the\nnext subsection.\nD. Dynamo with Back-reaction\nA dynamic dynamo represents a situation where\nthe magnetic energy grows exponentially for a plasma\nwhere the plasma itself evolves in time. Hence the\nvelocity \feld is not externally imposed like a kinematic\ndynamo, rather it has a dynamical nature. The time\nevolution of the velocity \feld is generally governed\nby the Navier-Stokes equation including the magnetic\nfeedback on the velocity \feld. In order to simulate such\na scenario, we time evolve all the three equations, viz.\nEq. 1, 2, 3. A result for parameters MA= 1000 and\nkf= 1 for initial \row pro\fle ABC is given in Fig. 8.\n 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100\n 0  50  100  150  200  250  300Energy\nTimeKinetic Energy\nMagnetic EnergyFIG. 8. Dynamic dynamo growth of kinetic\u0012P\nVU2(x;y;z )\n2\u0013\nand magnetic\u0012P\nVB2(x;y;z )\n2\u0013\nenergy for ABC \row with MA=\n1000 and kf= 1 for a long time with initial \row pro\fle as\nABC \row.\nWe change the forcing scale, Alfven velocity and\ncompressibility to observe the e\u000bect on the dynamics of\nthe \felds.\nWe turn on external forcing to the velocity\n\feld. We keep the nature of forcing as fx=\nAsin(kfz) +Ccos(kfy),fy=Bsin(kfx) +Acos(kfz),\nfz=Csin(kfy)+Bcos(kfx). We keep A=B=C= 0:1\nandU0= 1 throughout all the calculations and \fx\nRe=Rm = 450. In case of an external forcing the\ninitial memory is lost and hence the sensitivity to the\ninitial condition is expected to be lost. We redo our\nnumerical calculations for an initial random velocity\n\feld pro\fle and \fnd that the basic nature of dynamo\ne\u000bect does not get a\u000bected as shown in Fig. 9. The\nsaturation regime for both the kinetic (sum over all\nvelocity modes) and magnetic (sum over all magnetic\nmodes) energies remain the same though the two systems\nare evolved from di\u000berent initial conditions. We perform\nthe following runs (Table. II) using our code to un-\nderstand the externally forced ABC \row dynamo process.\nNow we vary U0and the magnitude of A;B;C keeping\nA=B=Cfor all the cases. We run our simulation\nforU0= 0:1;0:2;0:3;0:4;0:5 keepingA=B=C= 0:1\nand see the trend of dynamo action is identical for all\nvalues ofU0[Fig. 10]. Next we vary the values of A=\n0:1;0:1;0:3 keepingA=B=CandU0= 0:1. We see\nfaster growth of dynamo with higher values of forcing\nthrough the magnitudes of A,BandC[Fig.11].7\n 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100\n 0  10  20  30  40  50Energy\nTimeIC: ABC; Kinetic\nIC: ABC; Magnetic\nIC: Random; Kinetic\nIC: Random; Magnetic\nFIG. 9. Dynamic dynamo growth of kinetic and magnetic\nenergy for two di\u000berent initial conditions with MA= 1000\nandkf= 1. The initial growth rate of magnetic energy for\nABC initial \row pro\fle is found to be identical with that of\nrandom \feld pro\fle.\nName kfMAMs\nFDF1 11000 0:1\nFDF2 21000 0:1\nFDF3 41000 0:1\nFDF4 81000 0:1\nFDF5 161000 0:1\nFDMA1 1100 0:1\nFDMA2 1100:1\nFDMA3 110:1\nFDMA4 10:10:1\nFDMS1 1100 0:2\nFDMS2 1100 0:3\nFDMS3 1100 0:4\nFDMS4 1100 0:5\nTABLE II. Parameter details with which the simulation has\nbeen run for the externally forced dynamic dynamo problem.\n1. E\u000bect of Forcing scale\nA dynamic dynamo with external forcing has been\nstudied earlier by Galanti et al [40] for incompressible\nplasma with U0= 1A=B=C= 0:1,Re&Rmupto\n20 (belowRmc= 27), and kf= 1;2;4. We change the\nlength scale of forcing ( kf) on the velocity \feld keeping\nU0= 0:1,A=B=C= 0:1,Re=Rm= 450,Ms= 0:1\nandMA= 1000 as \fxed parameters. [Runs: FDF1,\nFDF2, FDF3, FDF4, FDF5] From Fig. 12 we \fnd, the\ngrowth rate of magnetic energy increases while that of\nkinetic energy decreases as kfis increased. The case\nkf= 16 in Fig. 12 shows a delayed dynamo action. A\npossible explanation of this late time dynamo action is\nthe excitation of a slow eigenmode to start with, which\ngets overpowered by the fastest eigenmode excited later.\nThe identical phenomena we have seen in the kinematic\ndynamo section [Fig. 5]. From Fig. 12 we also note\nthat though externally forced, the saturation regime of\n 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100\n 0  10  20  30  40  50Energy\nTimeU0 = 0.1; Kinetic Energy\nU0 = 0.1; Magnetic Energy\nU0 = 0.2; Kinetic Energy\nU0 = 0.2; Magnetic Energy\nU0 = 0.3; Kinetic Energy\nU0 = 0.3; Magnetic Energy\nU0 = 0.4; Kinetic Energy\nU0 = 0.4; Magnetic Energy\nU0 = 0.5; Kinetic Energy\nU0 = 0.5; Magnetic EnergyFIG. 10. Dynamic dynamo growth of kinetic and mag-\nnetic energy for \fve di\u000berent initial velocities viz. U0=\n0:1;0:2;0:3;0:4;0:5, with MA= 1000 and kf= 1. The initial\ngrowth rate of magnetic energy for ABC initial \row pro\fle is\nfound to be identical for all initial magnitude of velocities.\n 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100\n 0  20  40  60  80  100Energy\nTimeA = 0.1; Kinetic Energy\nA = 0.1; Magnetic Energy\nA = 0.2; Kinetic Energy\nA = 0.2; Magnetic Energy\nA = 0.3; Kinetic Energy\nA = 0.3; Magnetic Energy\nFIG. 11. Dynamic dynamo growth of kinetic and magnetic\nenergy for three di\u000berent forcing magnitudes A= 0:1;0:2;0:3\nhaving A=B=C, with MA= 1000 and kf= 1. The initial\ngrowth rate of magnetic energy for ABC initial \row pro\fle\nis found to be increase with the increament of magnitude of\nforcing.\nboth kinetic and magnetic eneries goes downwards as\nkfis increased. This is so, because the forcing scale\nalso has a wave number term within it which helps to\ndrain out energy through viscous dissipation, if a higher\nwavenumber ( kf) is excited.\n2. E\u000bect of Alfven Speed\nWe change the Alfven Mach number ( MA) of the\nplasma, keeping U0= 0:1A=B=C= 0:1,Re=\nRm= 450,Ms= 0:1 andkf= 1.\nWe analyse the runs: FDF1, FDMA1, FDMA2,8\n 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100\n 0  10  20  30  40  50Energy\nTimekf = 1; Kinetic\nkf = 1; Magnetic\nkf = 2; Kinetic\nkf = 2; Magnetic\nkf = 4; Kinetic\nkf = 4; Magnetic\nkf = 8; Kinetic\nkf = 8; Magnetic\nkf = 16; Kinetic\nkf = 16; Magnetic\nFIG. 12. Dynamic dynamo growth of kinetic and magnetic en-\nergy for kf= 1;2;4;8;16 with MA= 1000. The growth rate\nof magnetic energy is found to be steeper with the increase of\nkf. The growth rate of kinetic energy due to external forcing\ndecreases as kfis increased.\nFDMA3. By choosing MA= 1;10;100 and 1000 we\nset the Alfven Velocity VA=U0\nMA= 10\u00001;10\u00002;10\u00003\nand 10\u00004respectively. For \u001a0= 1,VA=B0, the initial\nmagnitude of the seed magnetic \feld pro\fle. As we start\nfrom a lower value of B0, the growth rate of the magnetic\nenergy increases rapidly. This is quite similar to the\nkinematic dynamo action with a distinct di\u000berence. In\nkinematic dynamo there was no saturation of magnetic\nenergy. On the other hand, in forced dynamic dynamo,\nthere is a saturation value of the magnetic \feld. This\nsaturation is believed to be due to the backreaction of the\nmagnetic \feld on the velocity \feld through the Lorentz\nforce term. The strong magnetic \feld generated through\nthe dynamo process, starts a\u000becting the topology of the\nvelocity \feld in turn a\u000becting its dynamics. Thus the\nmodi\fed velocity \feld no longer remains a ABC \row\nand \fnally the dynamo saturates. The e\u000bect of such\nmagnetic feedback on the velocity \feld is shown in Fig\n13 forMA= 1;100;1000.\nWe do the following observations from Fig 13.\n\u000fWe notice that, for both the case MA= 100 and\n1000 there exists three distinct slopes. At the begin-\nning, the magnetic energy starts exponentialy increasing\nwith time. Once it gets ampli\fed by around four orders\nof magnitude, the exponent of increament suddenly falls\ndown for both the cases MA= 100 and 1000. After that,\nthe magnetic energy again starts increasing with higher\nexponent.\n\u000fIt is also note-worthy that, the initial growth rate of\nthe magnetic energy for MA= 100 and 1000 are identi-\ncal though they di\u000ber later on. Thus we understand it as\nsimilar to Kinematic dynamo (7) where the backreaction\nis negligible. However, at later time because of the dif-\nference in the strength of the backreaction, the slopes of\nincreament of the magnetic energy in logarithmic scale\n 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100\n 0  10  20  30  40  50Energy\nTimeMA = 1000; Kinetic\nMA = 1000; Magnetic\nMA = 100; Kinetic\nMA = 100; Magnetic\nMA = 1; Kinetic\nMA = 1; MagneticFIG. 13. Dynamic dynamo growth of kinetic and magnetic\nenergy for MA= 1;100;1000. The back-reaction of magnetic\n\feld on velocity \feld is found to a\u000bect the growth rate and\ndynamics of velocity \feld. This e\u000bect is captured in the time\nevolution of kinetic energy for di\u000berent MA.\ndi\u000bers.\n\u000fWe also \fnd that when the growth of the dynamo is\nseveral orders of magnitude (for higher values of MA) the\nkinetic energy also grows faster though ultimately both\nkinetic and magnetic energies saturate at the same value.\n\u000fWe see that, independent of the strength of the seed\nmagnetic \feld, the saturation regime of the kinetic and\nmagnetic energies are the same.\nThus we conclude from the above observations that,\nif the velocity \feld is ABC forced, whatever be the seed\nmagnetic \feld, the dynamo e\u000bect becomes possible in\nsuper Alfvenic systems and the dynamo action is quite\nstrong leading the \fnal magnetic energy comparable to\nthe kinetic energy.\n3. Energy Spectra\nNow we analyse the kinetic and magnetic energy\nspectra of the dynamic dynamo action at di\u000berent times\nforU0= 0:1,A=B=C= 0:1,kf= 1,MA= 1000,\nRe=Rm= 450. Initially the energy content was limited\nto the fundamental mode only. But in course of time the\nkinetic energy shows a k\u00005=3spectra while the magnetic\nenergy shows a k0:7spectra identical to the kinematic\ndynamo phenomena. However it is worth notable that\nthe growth of magnetic energy in intermediate scales\nis much slower than the kinematic dynamo as can be\nfound in Fig.39\n 1e-18 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1 100\n 1  10E(k)\nktime = 0\ntime = 10\ntime = 20\ntime = 30\ntime = 40\ntime = 50\ntime = 100\ntime = 150\ntime = 200\nx-5/3\nFIG. 14. Kinetic energy spectra for dynamic dynamo with\nU0= 0:1,A=B=C= 0:1,kf= 1, MA= 1000, Re=\nRm= 450.\n 1e-25 1e-20 1e-15 1e-10 1e-05 1\n 1  10B(k)\nktime = 0\ntime = 10\ntime = 20\ntime = 30\ntime = 40\ntime = 50\ntime = 100\ntime = 150\ntime = 200\nx0.7\nFIG. 15. Kinetic energy spectra for dynamic dynamo with\nU0= 0:1,A=B=C= 0:1,kf= 1, MA= 1000, Re=\nRm= 450.V. SUMMARY AND FUTURE WORKS\nIn this work we have analysed several phenomena of a\nmagnetohydrodynamic plasma under ABC \row.\n\u000fFirst we study the kinematic dynamo e\u000bects where\nthe velocity \feld is the ABC \row - a known solution\nof Euler equation. At di\u000berent wave-numbers of the\n\row, we see that the growth rate of the magnetic energy\nin the kinematic dynamo case increases as kfis increased.\n\u000fIn case of an ABC forced velocity \feld for the dy-\nnamic dynamo problem seems to show similar variation\nwithkf, though now the dynamo action becomes very\nprominant. The magnetic energy grows upto the order\nof kinetic energy when we remain in the super Alfvenic\nregime.\n\u000fThe magnetic energy is found to be contained\nprimarily in the intermediate scales in wave-number.\nThe compressibility however has not been found to\na\u000bect the results for the weakly compressible cases. The\ne\u000bect of variation of initial density ( \u001a0) on the dynamo\ne\u000bect can be an interesting piece of study and will be\nexplored elsewhere.\nVI. ACKNOWLEDGEMENT\nR.M. acknowledges several insightful discussions with\nAkanksha Gupta, Vikrant Saxena and Abhijit Sen at In-\nstitute for Plasma Research, India. The development as\nwell as benchmarking of MHD3D has been done at Udb-\nhav and Uday clusters at IPR.\n[1] I. B. Zeldovich, A. A. Ruzmaikin, and D. D. Sokolov, in\nNew York, Gordon and Breach Science Publishers (The\nFluid Mechanics of Astrophysics and Geophysics. Volume\n3), 1983, 381 p. Translation. (1983), vol. 3.\n[2] H. K. Mo\u000batt, Cambridge University Press, Cambridge,\nLondon, New York, Melbourne (1978).\n[3] A. Gailitis, O. Lielausis, E. Platacis, S. Dement'ev,\nA. Cifersons, G. Gerbeth, T. Gundrum, F. Stefani,\nM. Christen, and G. Will, Physical Review Letters 86,\n3024 (2001).\n[4] R. Stieglitz and U. M uller, Physics of Fluids 13, 561\n(2001).\n[5] R. 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Proctor, Physical review letters 98, 208501\n(2007)."    },    {        "title": "1105.2200v1.Dynamic_fields_in_the_partial_magnetization_plateau_of_Ca3Co2O6.pdf",        "content": "arXiv:1105.2200v1  [cond-mat.str-el]  11 May 2011Dynamic fields in the partial magnetization plateau\nof Ca 3Co2O6\nP. J. Baker1, J. S. Lord1, and D. Prabhakaran2\n1ISIS Facility, STFC Rutherford Appleton Laboratory, Didco t, OX11 0QX,\nUnited Kingdom\n2Oxford University Department of Physics, Clarendon Labora tory, Parks Road,\nOxford OX1 3PU, United Kingdom\nE-mail:peter.baker@stfc.ac.uk\nAbstract. Fluctuation dynamics in magnetization plateaux are a relat ively\npoorly explored area in frustrated magnetism. Here we use mu on spin relaxation\nto determine the fluctuation timescale and associated field d istribution width\nin the partial magnetization plateau of Ca 3Co2O6. The muon spin relaxation\nrate has a simple and characteristic field dependence which w e model and by\nfitting to the data at 15 K extract a fluctuation timescale τ= 880(30) ps and\na field distribution width ∆ = 40 .6(3) mT. Comparison with previous results on\nCa3Co2O6suggests that this fluctuation timescale can be associated w ith short-\nrange, slowly fluctuating magnetic order.\nPACS numbers: 76.75.+i, 75.40.Gb, 75.50.Ee\nSubmitted to: JPCMDynamic fields in the partial magnetization plateau of Ca 3Co2O6 2\n1. Introduction\nWhen magnetic ions are arranged in certain structures their intera ctions can act to\nimpede magnetic ordering. This is referred to as magnetic frustrat ion and is widely\nstudied [1, 2]. A well known example of a frustrated structure is ant iferromagnetically\ncoupled spins on a triangular lattice [3]. An alternative version of this is the\nantiferromagnetic coupling of ferromagnetic chains on a triangular lattice when the\nintrachain coupling is far stronger than the interchain coupling. This is realized in\ncompounds such as CoNb 2O6[4] and Ca 3Co2O6[5], where the moments within the\nchains have Ising (easy-axis) anisotropy.\nIn applied magnetic fields frustrated magnets can undergo transit ions between\ndifferent magnetic structures. In between such transitions the m agnetization of the\nsample stays fixed over a significant range of fields: a magnetization plateau. These\nshow a simple fraction of the saturation magnetization such as 1 /3 because the\nindividualatomicspinsareeitheralignedoranti-alignedwiththeexter nalfield. Sucha\npartial magnetization plateau is observed in the intermediate, part ially ordered phase\nof Ca3Co2O6between 10 and 25 K [6, 7, 8, 9, 10]. A field of ∼0.5 T is sufficient to\nenter the 1 /3 magnetization plateau and then a further transition into the satu rated\nmagnetic state occurs at 3 .6 T. Below ∼10 K a series of magnetization steps are seen\nin single crystals for fields applied along the c-axis, with pronounced hysteresis [7, 10].\nThat these magnetization steps occur in multiples of 1 .2 T has led to the suggestion\nthat they represent quantum tunnelling of the magnetization [6, 7]\nMuon spin relaxation ( µSR) can be used to investigate both the static and\ndynamic properties of frustrated magnets [11, 12, 13]. This is most commonly carried\nout in zero applied field, since the technique is rare in allowing this and is e xquisitely\nsensitive to small magnetic fields that could emerge due to weak magn etic order. It\nis sensitive to fluctuating magnetic fields on time scales ranging from a round 10−11\nto 10−5s, intermediate between neutron scattering and ac magnetic susc eptibility.\nMeasurements on magnetic systems in applied fields along the initial dir ection of the\nmuon spin allow the implanted muon spins to be decoupled from the field d istribution\nthey experience at their stopping site, providing a further window o n the static and\ndynamic properties [12].\nIn this paper we describe the results of a µSR investigation of the magnetization\nplateau in Ca 3Co2O6. The broad and experimentally accessible plateau in the\nintermediate temperature region offers an ideal ground for quant itative comparison\nbetween models for the relaxation dynamics and experimental data . We propose a\nsimple model to describe the field dependence of the muon spin relaxa tion rate, find\nthat it describes the data successfully in the plateau region, and us e it to estimate the\ntime scale and field distribution associated with the magnetic fluctuat ions that the\nmuon probes in this system.\n2. Experimental\nStoichiometricCa 3Co2O6hasbeen synthesisedbythe conventionalsolid statereaction\ntechnique. High purity ( >99.99 %) starting materials CaCO 3and Co 3O4were dried\nat 150◦C for 6 h before weighing. Mixed chemicals were calcined and sintered a t\n810−860◦C for 48 h with intermediate grinding. Finally the sintered powder was\npressed into cylindrical rods and sintered at 925◦C for 48 h in air.\nOurµSR experiment [11, 12, 13] was carried out on the newly constructe d HiFiDynamic fields in the partial magnetization plateau of Ca 3Co2O6 3\n0.00.20.40.60.81.0Pz(t)\n0 1 2 3 4 5\nt(s)3.8 T\n1 T\n0 T\n15 K\nFigure 1. Examples of the Pz(t) data recorded at different fields for Ca 3Co2O6\nat 15 K. The lines plotted are fits of the data to Equation 3, add ing a slowly\nrelaxing background at 0 T.\nspectrometer [14] at the ISIS Pulsed Muon Facility. This has a 5 T sup erconducting\nmagnet, almost ideally suited to the field range needed for this exper iment. The\npowder sample was mounted in a 25 µm silver foil packet on a silver backing plate\nand cooled to 15 K using a helium flow cryostat. Silver is used because t he muon spin\nrelaxation rate for muons stopping outside the sample will be close to zero over the\nfield range of interest here.\nThe fully spin polarized muons are implanted into the sample and genera lly stop\nat interstitial positions within the crystal structure without signifi cant loss of spin\npolarization. The spin polarization Pz(t) of the muon depends on the magnetic\nenvironment of the of the muon stopping site and can be measured u sing the\nasymmetric decay of the muon into a positron, which is detected, an d two neutrinos.\nAround 10 million decay positrons were detected in each data set. Th e emittedDynamic fields in the partial magnetization plateau of Ca 3Co2O6 4\npositronsaredetected inscintillationcountersaroundthesamplep osition[14], divided\nintotwobanksforward(F)andbackward(B)relativeoftheinitialm uonspindirection.\nThe time-dependent asymmetry A(t) betweenthe countratesin the banksofdetectors\nis:\nA(t) =NF(t)−αNB(t)\nNF(t)+αNB(t), (1)\nwhere the parameter αdescribes the relative counting efficiency of the two banks of\ndetectors, depending on the sample and detector geometry. Fro mA(t) it is possible\nto infer the spin polarization Pz(t) of the implanted muon ensemble:\nPz(t) =A(t)−Abg\nA(0)−Abg. (2)\nThe measured asymmetries A(0) and Abgare the initial and background values\nrespectively and depend on the detector geometry and the fract ion of the muon beam\nincident on the sample. The field variation of α,A(0), and Abgrequired careful\nconsideration in fields /greaterorsimilar1 T because the decay positrons spiral significantly in the\nfield. This was corrected for by comparison with data from referen ce samples [14].\nFollowing on from earlier muon work on Ca 3Co2O6in the intermediate\ntemperature region [8, 9], we expect that in close to zero applied field the muon spin\nrelaxationwillnotshowthe full initialasymmetryobservedinthe par amagneticregion\nbecause the internal fields are too large to be observed at ISIS. ( The pulsed nature\nof the muon beam gives a range of arrival times for muons entering t he sample which\nlimits the maximum observed relaxation rate.) Takeshita and co-work ers investigated\nthe form of the spin relaxation up to around 1 T in this temperature r egion, finding\nthat the initial asymmetry had been recovered by ∼0.5 T and that the relaxation was\nexponential over the whole field range. This agrees with the form of our data from 0\nup to 3.8 T, as shown in Fig. 1 and we therefore fitted our data to the funct ion:\nPz(t) = exp(−λt), (3)\nwhereλis the relaxation rate. For λ/greaterorsimilar0.1 MHz,λcan be extracted from the data\nindependent from any field variation of α,A(0), and Abg. This is the case throughout\nour measured field range. In zero field Abgdecreases slowly with time due to muons\nstopping in the silver sample holder and any weak fields perpendicular t o the initial\nmuon spin direction, but in applied field Abgis effectively constant.\nTo model the field dependence of λwe make the assumption that the distribution\nof fluctuating magnetic fields approximates to a Lorentzian [12, 13] with a width ∆\nand that one correlation time τis an adequate description of the fluctuation rate. We\nfurther assume that within the partial magnetization plateau thes e two parameters\ncanbe takentobeconstant. Makingtheseassumptionsallowsus to simplify Redfield’s\nequation [15] to the form:\nλ=2γ2\nµ∆2τ\n1+γ2µB2\nLFτ2. (4)\nThe parameter γµ= 2π×135.5 MHzT−1is the muon gyromagnetic ratio.\n3. Results\nIn Fig. 2 (a) we show the relaxation rate λextracted from the data at 15 K as a\nfunction of the magnetic field BLF=µ0H. Three behaviours are apparent in theDynamic fields in the partial magnetization plateau of Ca 3Co2O6 5\n0.00.51.01.52.02.53.03.54.0-1(s)\n0 2 4 6 8 10 12 14\n2\n0H2(T2)(b)Plateau0246810(MHz)\n0 1 2 3 4\n0H(T)(a) Plateau\nFigure 2. Parameters extracted from the µSR data using Equation 3: (a)\nRelaxation rate λplotted against the applied field µ0H. (b) Inverse of the\nrelaxation rate λ−1plotted against the square of the applied field µ2\n0H2. The\ntrend obtained by fitting the data to Equation 4 is shown in the shaded\nmagnetization plateau region.\ndata: At fields below 0 .5 T there is a rapid decrease in λ, between 0 .5 and 3.6 T\nthe decrease in λwith increasing field is slower, and above 3 .6 T any change in λis\nsmaller than the experimental error. These regions correspond t o the initial increase\nin bulk magnetization, the magnetization plateau, and the saturate d magnetic phase\nrespectively.\nTo make a more obvious comparison with the Redfield equation (4) disc ussed\nabove we plot λ−1againstµ2\n0H2in Fig. 2 (b). This rearrangement means that\nthe trend for constant ∆ and τwill appear as a straight line, as is observed in the\ndata in the magnetization plateau region highlighted. Fitting equation 4 to the data\nshown in Fig. 2 between 0 .5 and 3.6 T gives the parameters: ∆ = 40 .6(3) mT and\nτ= 880(30) ps. This gives the line plotted in Fig. 2 (b). These paramete rs areDynamic fields in the partial magnetization plateau of Ca 3Co2O6 6\ninsensitive to small reductions in the fitting range at either end of th e plateau region.\n4. Conclusion\nUsing the partial magnetization plateau in Ca 3Co2O6as a model system in which\nto test a simple model for the field-dependence of the muon spin rela xation rate in\na magnetization plateau we find very good agreement between the f orm predicted\nby the model and the experimental data. We are able to obtain a field distribution\nwidth and fluctuation timescale for the dynamic magnetism within the p lateau. This\nsuggests that this model could also be applied to other magnetic sys tems that exhibit\na magnetization plateau and provide new information about their dyn amic behaviour.\nThe width of the field distribution that we obtain is comparable with the\n∼100 mT static internal field observed in zero-field measurements [8]. However,\nwere the muon probing such static fields they would have been decou pled from the\nmuon spin at fields lower than those in the magnetization plateau. Bot h the previous\nµSR results [8] and neutron diffraction work [10] show strong evidenc e for shorter\nranged, dynamic correlations coexisting with the long-range magne tic order at this\ntemperature. Such dynamic correlations would not be decoupled fr om the muon spin\nso easily and it is therefore this behaviour that we are probing in this e xperiment.\nSince the field distribution width will be determined by the same magnet ic moments\nthe same distance from the muon stopping site(s), and this is entire ly consistent with\nour results.\nThe timescale we find for the fluctuations /lessorsimilar1 ns sits comfortably within the\nsensitivity range of µSR and suggests that these fluctuations would be static on the\ntimescale of neutron diffraction. Again, this is consistent with the pr evious neutron\ndiffraction measurements where the short-rangemagnetic order is quasistatic [10], and\nthe previous µSR results where a significant strongly-relaxingcomponent sugges tiveof\nslowingfluctuating short-rangedorderisobservedat thistemper ature[8, 9]. Earlierac\nsusceptibility measurements found a significantly longer timescale, ∼0.4 s [7], which\nis far outside the window we can probe using muons, and is likely to be as sociated\nwith a separate process within the system.\nComparing our model to the data in the region B <0.5 T we find far poorer\nagreement. Of course, the assumption that ∆ is independent of fie ld is unlikely to\nbe appropriate here and we would only get some approximation to the average value.\nThe parameters coming from our data for B <0.5 T are: ∆ ≃30 mT and τ≃5 ns,\nwhich suggest that the decoupling from longer ranged order remain s relevant in this\nfield range. Takeshita et al.carried out a similar analysis of their data at 20 K and\nobtained larger, although comparable, values for both parameter s [9].\n5. Acknowledgements\nWe thank the Science and Technology Facilities Council for a Facility De velopment\ngrant (FDPG/082) and the European Commission under the 7thFramework\nProgramme through the Key Action: Strengthening the European Research Area,\nResearch Infrastructures: Contract no: CP-CSA INFRA-2008 -1.1.1 Number 226507-\nNMI3. We acknowledge EPSRC support for sample growth. Data wer e collected with\nassistance from: Roxana Dudric, Hassan Saadaoui, Robert Taras enkko, Lee Iverson,\nCalin Rusu, Adam Berlie, Edwin Kermarrec, and Neda Nikseresht.Dynamic fields in the partial magnetization plateau of Ca 3Co2O6 7\nReferences\n[1] Ramirez A P 2003 Geometric frustration: Magic moments Nature421483\n[2] Frustrated Spin Systems, ed. Diep HT, World Scientific, ( 2004).\n[3] Collins MF and Petrenko OA 1997 Triangular Antiferromag netsCan. J. Phys. 75605\n[4] ColdeaR et al.2010 Quantum CriticalityinanIsingChain: ExperimentalEv idence forEmergent\nE8Symmetry Science 327177\n[5] Aasland S, Fjellv˚ ag H, and Hauback B 1997 Magnetic prope rties of the one-dimensional\nCa3Co2O6Solid State Communications 101187\n[6] Maignan A et al.2004 Quantum tunneling of the magnetization in the Ising cha in compound\nCa3Co2O6J. Mater. Chem. 141231\n[7] Hardy V F et al.2004 Temperature and time dependence of the field-driven mag netization steps\nin Ca3Co2O6single crystals Phys. Rev. B 70064424\n[8] Sugiyama J et al.2005 Appearance of a two-dimensional antiferromagnetic or der in quasi-one-\ndimensional cobalt oxides Phys. Rev. B 72064418\n[9] Takeshita S et al.2006 Muon Spin Relaxation Study of Partially Disordered Sta te in Triangular-\nLattice Antiferromagnet: Ca 3Co2O6J. Phys. Soc. Jpn 75034712\n[10] Fleck C L et al.2010 Field-driven magnetisation steps in Ca 3Co2O6: A single-crystal neutron-\ndiffraction study EPL9067006\n[11] Blundell S J 1999 Spin polarized muons in condensed matt er systems Contemp. Phys. 40175\n[12] Dalmas de R´ eotier P and Yaouanc A 1997 Muon spin rotatio n and relaxation in magnetic\nmaterials J. Phys.: Condens. Matter 99113\n[13]Muon Spin Rotation, Relaxation, and Resonance , A.Yaouanc andP.DalmasdeR´ eotier, (Oxford\nUniversity Press, Oxford, 2011)\n[14] Lord J S et al.in preparation\n[15] C. P. Slichter, Principles of Magnetic Resonance (3rdedition, Springer-Verlag, New York, 1996)."    },    {        "title": "2005.02713v1.Inverse_Magnetic_Billiards_on_a_Square.pdf",        "content": "INVERSE-MAGNETIC BILLIARDS ON A SQUARE\nANDRES PERICO\n1.Introduction\nWe work in a unit square \n \u001aR2where we have a constant magnetic \feld\noutside of \n with magnitude B, inside there is no magnetic \feld. A particle at\nan initial position in the boundary of \n starts moving towards the interior of the\nsquare. Without loss of generality we can assume that the square is in canonical\nposition with vertices (0 ;0);(1;0);(1;1);(0;1) and that the particle starts at some\npoint on the bottom side (on the x\u0000axis).\nFigure 1. Initial conditions\n1.1.Description of dynamics. Say that the initial conditions are ( s;\u0012) wheres2\n(0;1),\u00122(0;\u0019) (we are going to omit the corners for now). Following the billiards\nnotation, these coordinates for initial conditions are called Birkho\u000b coordinates.\nThe particle moves inside \n in straight line at an angle \u0012with respect to the\nboundary until it hits the boundary again at ( s1;\u00121) on the exiting side. It has\nthree options for exiting side, the other three sides of the square. We are considering\nan electron with charge \u00001, this is why the circular motion outside of \n corresponds\nto a counterclockwise movement. After hitting the boundary it moves on a circle\nof radiusr= 1=Btangent at the exit point to the line of its previous trajectory.\nThe particle will hit \n again at ( s2;\u00122) on the entering side, depending on Bthe\nradius of rotation will be smaller or bigger giving di\u000berent return points on di\u000berent\nsides. The map ( s1;\u00121)!(s2;\u00122) will be referred to as the magnetic bounce . As\nsoon as it hits \n again the particle moves in straight line, this line is tangent to the\ncircle of its previous trajectory at the point of entry (see \fgure 2).\n1arXiv:2005.02713v1  [math.DS]  6 May 20202 ANDRES PERICO\nFigure 2. One bounce in the magnetic billiard\nThis dynamics can be described by a map\nF: \u00062!\u00062\nF(s;\u0012) = (s2;\u00122)\nwhere \u00062=f(s;\u0012)j0\u0014s\u00144;0< \u0012 < \u0019gis the space of directed unit vectors\ntoward the interior of \n with initial points on the boundary.\nWe want to study the dynamics of this billiard type map. The most challenging\naspect of this investigation concerns trajectories which go around a corner. See \fg-\nure 2. The map is apparently chaotic and all the chaos seems to come from turning\naround corners.\nHere are some of the questions we want to answer:\n\u000fAre there any periodic orbits?\n\u000fFor anyB, is there a periodic orbit?\n\u000fCan we classify the periodic orbits?\n\u000fCan we classify non periodic orbits?\n\u000fIs any orbit dense in \n?\n\u000fIs any orbit dense in \u00062?\n\u000fAs a mapF: \u00062!\u00062symplectic integrable.\n\u000fHow do orbits look like?INVERSE-MAGNETIC BILLIARDS ON A SQUARE 3\n1.2.Extreme cases: B!0;1.\nClassic billiards is the limit case r= lim\nB!1rB, so the particle is trapped in the\nsquare and bounces in the classic way.\nThe case when r= lim\nB!0rBrefers when the particle continues an straight trajectory\nall the time, so the only part of the trajectory inside the square is the initial segment.\nThis orbit has only one exit and one entry point: the entry point is exactly the\nsame starting point, the particle get there again after in\fnite time.\n2.No turning corners\nWe are assuming our initial condition is on the bottom side of the square\n(0< s < 1), we look for the returning map Fdepending on the exiting side:\nright, top or left side. Assume that the entry point is on the same side as the exit\npoint, this means that the trajectory outside \n doesn't go around a corner.\nFigure 3 shows an example of this case, \u0012=\u0019\n4or initial slope equal to 1 and\nradius small enough (depending on our initial conditions) so the magnetic bounce\nhappens on the same side of the square.\nFigure 3. Example of no turning corners4 ANDRES PERICO\nBouncing on the right side of the square\nFigure 4. Bouncing on the right side\nsn+1= (1\u0000sn) tan\u0012; sn+2=sn+1+2\nBcos\u0012; \u0012n+2=\u0019\n2\u0000\u0012n:\nBouncing on the top side of the square\nFigure 5. Bouncing on the top side\nsn+1= 1\u0000sn\u0000cot\u0012; sn+2=sn+1+2\nBsin\u0012; \u0012n+2=\u0019\u0000\u0012n:INVERSE-MAGNETIC BILLIARDS ON A SQUARE 5\nBouncing on the left side of the square\nFigure 6. Bouncing on the left side\nsn+1= 1\u0000sntan\u0012; sn+2=sn+1+2\nBcos\u0012; \u0012n+2=3\u0019\n2\u0000\u0012n:\n2.1.Rational slope (tan \u00122Q).When the slope is a rational numberp\nqwithp\nandqrelative primes, for classical billiards the orbit is periodic. For our inverse\nmagnetic billiard we'd like the nearby periodic orbits, to understand the problem\nwe want conditions where the particle never turns a corner.\nNot turning a corner is the same as saying that the unfolded trajectory will cross\nthe same sides that the classical one, a straight line in Z\u0002Z, the will never be at\nopposite sides of a vertex.\nSince the slope isp\nqwe will have 2( p+q) bounces, for us these bounces are\nactually exits and returns from and to the unit square. The \fgure 7 shows the case\nof slope 2=3, where we have 2(2 + 3) = 10 bounces in the classical billiard.6 ANDRES PERICO\nFigure 7. Case of slopep\nq=2\n3\nThe unfolded lattice for the magnetic billiard with these conditions is as shown\nin the \fgure 8\nFigure 8. Case of slopep\nq=2\n3\nLemma: With initial conditions (s0;\u00120)with tan\u00120=p\nq2Q,s06=Zpand\n2=B < minfjs0\u0000k=qj;js0p=q\u0000k=pj:k2Zg, then the orbit is periodic.\nThe condition s06=Zpdeletes the cases where the orbit hits corners. We can\ntakeB > 2=minfjs0p\nq\u00001\nqj;js0\u00001\npjgso we actually insure that the radius of the\ncircle is small enough to stay in the same side of the square. Since2\nBis the diame-\nter of the circle, with this condition we are securing that the exiting point ( s1;\u00121)\nis at least at one diameter distance from all the corners. This choice of Bmakes\nthe diameter of the circle smaller than all the distances between the orbit and theINVERSE-MAGNETIC BILLIARDS ON A SQUARE 7\npoints in the lattice Z\u0002Z.\nThe closest that a straight line that starts at the origin with slopep\nqgets to a\npoint in the lattice is min fjk=qj;jk=pj:k2Zg. Then a line that starts at ( s0;0)\nwill get shifted at the intersection points (with the lines in the lattice) by s0to\nthe right on the horizontal lines, and by s0p\nqdownwards in the vertical lines. This\nmeans that we can get as close as we want to the points in the lattice depending\nons0. Once you \fx s0, the closest you get is min fjs0\u0000k=qj;js0p=q\u0000k=pj:k2Zg.\nIn our setting we will have 2( p+q) shiftings of the classical trajectory, these\nshifting can be \u000f1=2\nBcos\u0012nor\u000f2=2\nBsin\u0012ndepending of the side of bounce:\n\u000f1for a vertical side (red or orange in the grid) and \u000f2for horizontal sides (blue\nand green on the grid). These numbers correspond to the formulas in the previous\nsection.\nIn the unfolded lattice Z\u0002Zthe trajectory will cross 2 qvertical lines and 2 p\nhorizontal lines. pof those crossing correspond to the lower side of the square, pto\nupper side, qto the left side and qto the right side. In our notation this means 2 p\nshifts of\u000f2units horizontally and 2 qshifts of\u000f1units vertically.\nSince the rotation is counter clockwise, the shifting on the lower side is always to\nthe right, on the right side is upwards, on the top side is to the left and on the left\nside is downwards. Since the shifting is towards opposite directions on parallel sides\nthey will cancel each other. With this we have that ( s0;\u00120) = (s2(p+q);\u00122(p+q)),\nthis proves the lemma.\nThe \fgure shows the \frst bounces for the example p=q= 2=3. Every shift has\nanother one that cancels it out, after 2( p+q) bounces we are at the initial condi-\ntions again.\nFigure 9. Case of rational slopep\nq8 ANDRES PERICO\n2.1.1. Irrational slope. This case is impossible to have with all bounces on the same\nexiting side. The particle must turn a corner: the same argument as before works\nhere, the shifting of a classical dense orbit in our tessellation will give us a turn of\na corner.\nExamples of this case will be shown in the next section.\n2.2.Lemma,B > 1.Lemma : For any radius smaller than 1 (i.e. for any magnetic\n\feld with magnitude bigger than 1), there exist a periodic orbit.\nProof : Pick the initial condition ( r;\u0019=2).\n3.Numerics: Turning corners, turning chaotic\nIn this section we give numerical evidence that our billiard map tends to be\nchaotic.\nAfter a turn of a corner the set of \u0012values is dense, the slope can be rational or\nirrational. If rational, we are not in the conditions of our lemma, we have violated\nthe condition of being away from a corner by the minimum established distance.\n3.1.B > 1.In this case we are demanding the radius of the circle to be small\nenough to go around at most one corner, so the entering side is the same or adja-\ncent to the exiting side.\nWe calculate the map F: \u00062!\u00062with di\u000berent initial conditions (on the\nBottom side) and radius. The pictures show Fn(s0;\u00120) = (s2n;\u00122n) for di\u000berent\nnumbers of iterations. Here we use the notation u= cos\u0012.\n3.1.1. Radius 0:02.Let's look to a periodic orbit \frst:\nFigure 10. r= 0:02;s= 0:9;\u0012=\u0019=4INVERSE-MAGNETIC BILLIARDS ON A SQUARE 9\nThis one is similar to \fgure 3 with slope 1.\nNow four di\u000berent orbits (4 di\u000berent initial conditions):\nFigure 11. r= 002; (s;u) = (0:1;0);(0:3;0:8);(0:5;\u00000:5);(0:09;p\n2=2)\nNow with more iterations (1000) was done with several initial conditions with\nthe same result:\nFigure 12. r= 0:02;s= 0:001;u=\u00000:99 REP = 1300\nGoing for 10000 bounces10 ANDRES PERICO\nFigure 13. r= 0:02;s= 0:001;u=\u00000:99 REP = 10000\n3.1.2. Radius 0:49.Now we try with a di\u000berent radius. Here we encounter a piece-\nwise linear behavior. If you increase the radius, the pattern becomes clearer.\nFigure 14. r=0.49, s=0.1 u=0, s=0.2 u=0.8, s=0.5 u=-0.5,\ns=0.09 u=0.45INVERSE-MAGNETIC BILLIARDS ON A SQUARE 11\nFigure 15. r=0.49, s=0.3, u=0.8\n3.1.3. Radius 0:85.More clear that there is a piecewise linear behavior.\nFigure 16. r=0.85, s=0.33, u=0.8, 3000 repetitions\n4.Conclusion\nInverse magnetic billiard dynamics limits to regular billiards on the square, but\nturns out to be much more intricate and complicated than regular billiards. Nu-\nmerical experiments suggest that the dynamics is ergodic for strong magnetic \felds,\nwhile it has unexplained sawtooth type patterns in the intermediate range of \felds.\nAs far as our original questions, listed at the the end of section 1, we have\nanswered the \frst three, those regarding periodic orbits, a\u000ermatively. We are\nunable to classify periodic orbits that turn corners, although by symmetry these\nexist. We also found dense orbits in both spaces \n and \u00062, we didn't classify these.\nThe remaining questions remain open.12 ANDRES PERICO\n5.\nReferences\n[1] V. Arnold, Mathematical methods of classical mechanics , Springer, 1989.\n[2] G.D. Birkho\u000b, Dynamical Systems , American Mathematical Society / Providence, RI, Amer-\nican Mathematical Society, 1927.\n[3] N. Berglund and H. Kunz, Integrability and ergodicity of classical billiards in a magnetic\n\feld, Journal of Statistical Physics 83, no. 1-2, 81{126, 1996.\n[4] N. Chernov and Ro. Markarian, Chaotic Billiards , American Mathematical Soc., 2006.\n[5] G. Datseris, L. Hupe,and R. Fleischmann, Estimating Lyapunov exponents in billiards , Chaos:\nAn Interdisciplinary Journal of Nonlinear Science, 29 no.9, p.093115, 2019.\n[6] S. Gasiorek, On the Dynamics of Inverse Magnetic Billiards , Ph.D. thesis, University of\nCalifornia Santa Cruz, 2019\n[7] M. Robnik and M. V. Berry, Classical billiards in magnetic \felds , J. Phys. A: Math. Gen.\n18, no. 9, 1361{1378, 1985.\n[8] M. Robnik, Regular and chaotic billiard dynamics in magnetic \felds , Nonlinear Phenomena\nand Chaos 1, 303{330, 1986.\n[9] S. Tabachnikov, Billiards , Soc. Math. France, 1995.\n[10] S. Tabachnikov. Outer billiards , Russ. Math. Surv., 48, 75-102, 1993.\n[11] S. Tabachnikov, Remarks on magnetic \rows and magnetic billiards, Finsler metrics and a\nmagnetic analog of Hilbert's fourth problem , Modern dynamical systems and applications,\npp.233-250, 2004.\n[12] T. Tasn\u0013 adi, Hard Chaos in Magnetic Billiards (On the Euclidean Plane) , Communications\nin Mathematical Physics 187, no. 3, 597{621, 1997.\n[13] T. Tasnadi, The behavior of nearby trajectories in magnetic billiards , J. Math. Phys. 37,\n5577-5598, 1996.\n[14] Z. V or os, T. Tasn\u0013 adi, J. Cserti, and P. Pollner, Tunable Lyapunov exponent in inverse mag-\nnetic billiards , Physical Review E 67, no. 6, 065202, 2003.\nE-mail address :aperico@ucsc.edu\nDepartment of Mathematics, University of California, Santa Cruz, CA 95060"    },    {        "title": "2209.02266v1.Temperature__and_field_angular_dependent_helical_spin_period_characterized_by_magnetic_dynamics_in_a_chiral_helimagnet__MnNb_3S_6_.pdf",        "content": "Temperature - and field angular -dependent helical spin period \ncharacterized by magnetic dynamics in a chiral helimagnet  MnNb 3S6 \nLiyuan Li1, Haotian Li1, Kaiyuan Zhou1, Yaoyu Gu1, Qingwei Fu1, Lina Chen1,2,*, \nLei Zhang3,*, and R. H. Liu1,* \n 1 National Laboratory of Solid State Microstructures, School of Physics and Collaborative Innovation \nCenter of Advanced Microstructures, Nanjing University, Nanjing 210093, China  \n2 New Energy Technology Engineering Laboratory of Jiangsu Provence & School of Science, Nanjing \nUniversity of Posts and Telecommunications, Nanjing 210023, China  \n3 Anhui Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field \nLaboratory, Hefei Institutes of Physical Science, Chinese Academy of Sciences, Hefei 230031, China  \n \n \n* Corresponding authors:  \nRonghua Liu, rhliu@nju.edu.cn ,  \nLina Chen, chenlina@njupt,edu.cn  \nLei Zhang , zhanglei@hmfl.ac.cn    Abstract:  The chiral magnets with topological spin textures provide a rare platform to \nexplore topology and magnetism for potential application implementation. Here, we \nstudy the magnetic dynamics of several spin configurations on the monoaxial chiral \nmagnetic crystal MnNb 3S6 via broadband ferromagnetic re sonance (FMR) technique at \ncryogenic temperature. In the high-field forced ferromagnetic state (FFM) regime, the \nobtained frequency f vs. resonance field Hres dispersion curve follows the well -known \nKittel formula for a single FFM, while in the low-field c hiral magnetic soliton lattice \n(CSL) regime, the dependence of Hres on magnetic field angle can be well -described by \nour modified Kittel formula including the mixture of a helical spin segment and the \nFFM phase. Furthermore, compared to the sophisticated L orentz micrograph technique, \nthe observed magnetic dynamics corresponding to different spin configurations allow \nus to obtain temperature - and field -dependent proportion of helical spin texture and \nhelical spin period ratio L(H)/L(0) via our modified Kitte l formula. Our results \ndemonstrated that field - and temperature -dependent nontrivial magnetic structures and \ncorresponding distinct spin dynamics in chiral magnets can be an alternative and \nefficient approach to uncover ing and control ling nontrivial topolo gical magnetic \ndynamics.   \n \nKeywords : chiral helimagnets, magnetic solitons , ferromagnetic resonance , phase \ndiagram  \n \nPACS number(s):  81.05.Xj, 75.30. -m, 76.50.+g , 75.30.Kz  \n  1. Introduction  \nChiral helimagnets  (CHM)  possess nontrivial spin -textures with spiral or rotary \nalignment of spin moments , such as  topological spin textures of magnetic skyrmions , \nwhich provide a platform to study the interesting topological physics  and potential \napplications for spintronics [1-5]. MnNb 3S6 and CrNb 3S6 are typical chiral helimagnet s \nwith the same lattice structure [6,7] , analogous electronic [8,9] , and magnetic structu res \n[10,11] . In the monoaxial chiral helimagnet s[12], all spins are in the ab-plane and rotat e \nat a definite angle along  the c-axis due to the  competi tion among magnetocrystalline \nanisotropy , the interlayer  Heisenberg  and Dzyaloshinskii -Moriya (D M) interaction s. \nThe DM interaction aris es from losing  the inversion center in the magnetic atoms \nsublattice . The Heisenberg  interaction  (coefficient  J) prefers all  spins  forming collinear \narrangements (ferromagnetic or antiferromagnetic alignments) . In contrast,  the chiral \nDM interaction  (coefficient D) favors the non -collinear alignment of spins and  \nfacilitate s chiral magnetic order s[12]. Thus, their compet ition generates a chiral \nhelimagnet with  a fixed spin helix period L(0) determined by the ratio of two \ninteraction s L(0) = tan-1 (D/J)[13,14] .  \nHowever, under an  external field H, the field-dependent  Zeeman interaction  will \nalso compete  with the above two magnetic interactions and c an be used to achieve field -\ncontrol lable  spin texture s[6,15,16] . Therefore , by tuning magnetic field or/and \ntemperature, the chiral  helimagnets can evolve from a CH M into a chiral magnetic  \nsoliton lattice (CSL)  or forced fe rromagnetic state (FFM) to achieve the minimal total \nenergy in terms of the competition of several magneti c interactions [6,17,18] . \nAdditionally, there exist  several specific chiral spin textures d eviating from the ideal \nhelical state  [Figure  1(a)], e.g., chiral conical phase (CCP) [19], tilted chiral magnetic  \nsoliton lattice  (TCSL) [17], and CSL[13] depending on not only the amplitude but also \nthe angle of the  external  field to the ab -plane for the  strong  easy-plane  anisotropy  in \nthese monoaxial hexagonal crystals [17,20] . As the s chematics are shown i n Figure  1(a), \nthe external field H tilts the spin direction of the spin soliton lattice , modulate s the spin \nhelix period L(0) to L(H) at H < Hc, and finally turn s it into the FFM regime  at H > Hc. The previous theoretical investigations of CHM [21,22]  reported  that the period of CSL \ncan be described by  the 1D chiral sine -Gordon model, which  generally follows the \nformula L(Hin)/L(0) = 4 K(k)E(k)/π2 [23-26]. where K(k) and E(k) are the elliptic \nintegrals of the  first and second kinds with modulus  k (0 ≤ k ≤1), respectively , and Hin \nis the in -plane component of an external magnetic field . The elliptic modul us k is given \nby k/E(k) = (Hin/Hc)1/2 to minimiz e the CSL formation energy . The static and dynamic \nmagnetic properties experiments confirm that the nontrivial spin configurations of these \nchiral helimagnets highly depend on the external magnetic field, dimensi onality, and \ntemperature [1,7,17,27 -29]. Moreover, the Lorentz transmission electron microscopy \nalso directl y observed the temperature -dependent CSL state and its period  in \nCrNb 3S6[13,14,30,31] . However, for MnNb 3S6 helimagnet with the same lattice \nstructure as CrNb 3S6, the Lorentz transmission electron microscopy measurement  failed \nto identify the spatial period of CSL  because MnNb 3S6 has a much lower magnetic \norder temperature Tc ~ 45 K and the weak field modulation  of the helix period [32]. \nTherefore, a high sensitivity  technique that can catch  the spiral period information of \nMnNb3S6 and its evolution with the external magnetic field and temperature is urgently \nneeded . \nHere, we perform the systematic ferromagnetic resonance experiment to \ninvestigate thoroughly the detailed dependence of magnetic dynamics corresponding to \nthe nontrivial CSL in MnNb 3S6 on the field magnitude, angle, and temperature.  We find \nthat chiral helimagnet MnNb 3S6 exhibits  a distinct field angular  dependen ce of  spin \nresonance  in low -field nontrivial CSL from the uniform FMR  in high -field FFM. Then, \nwe propose  a modified Kittel mo del considering  partial helix spin texture s, which can \nsuccessfully describ e the experimentally observed spin dynamics of the low -field \nnontrivial CSL at different temperatures. More over, the modified Kittel model also \nenables us to extract temperature - and field -dependent proportion of the helical spin \ntexture and helical spin period ratio L(H)/L(0), like the sophisticated Lorentz \nmicrograph technique  in most c hiral helimagnets . The demonstrated  method can \ngenerally be used as an al ternative and easy-access approach to explore interesting magnetic dynamics not just in MnNb 3S6 and other  topologically  nontrivial chiral \nmagnets.  \n \nFigure  1 Several nontrivial spin configurations and their phase diagrams of MnNb 3S6. \n(a) Schematic  of spin configuration of several magnetic orders in monoaxial chiral \nhelimagnet s under the external magnetic field: CHM state at H = 0, CCP, TCSL, CSL \nstates at 0 < H < Hc and FFM state at H > Hc, respectively. The orange  and green  arrows \nrepresent the c-axis of the MnNb 3S6 crystal  and the direction of the  magnetic  field H, \nrespectively.  (b) - (c) Phase diagram of the specific magnetic orders in monoaxial chiral \nhelimagnet  MnNb 3S6 crystal with H // ab-plane  (b) and H // c axis (c) . The boundaries  \namong CSL (blue region) and FFM states  (green region)  were determined by critical \nfield (squares)  obtained from  the quasi -static magnetization  hysteresis loops.  The \ncritical field data (solid circle) reported by others is als o shown in the phase diagram [7]. \nTc represents  the Curie temperature  45 K of MnNb 3S6, determined from the M - T curve s. \nPM (orange region) represents paramagnetism.   \nFigure 1 (b) and (c) show t he phase diagram of  spin textures in MnNb 3S6 \ndetermined from the stat ic field- and temperature -dependent magnetic susceptibility  \nresults  with H in the ab-plane and H parallel to the c -axis of the single -crystal sample,  \nrespectively [see the Supporting  Information  (SI)[33]]. Note that  the critical fields \nobtained by the static magnetization loop have some deviations from previous \nreports [11,34]  due to different definition criteria a nd broad transition region s in M(H) \ncurves . More specifically,  the critical field of the phase diagram in Figure 1(b) is slightly \nhigher than in our previous reports [11]. One can find t he detailed M(H), M(T) curves, \nand the definition criteria of the critical field in the S I [33]. The phase diagram of the \nstudied chiral magnet MnNb 3S6 shows  two dominated  spin configuration regions: a \nlow-field CSL and a high -field FFM  below its critical magnetic order temperature Tc = \n45 K, consistent with the previous reports  [solid circle  in phase diagram] [7]. \n2. Experimental Section  \nThe differential ferromagnetic resonance ( FMR ) spectroscopy  is based on  a coplanar \nwaveguide (CPW) , illustrated  in Figure 2(a). A 1×1 mm square -shaped single -crystal \nMnNb 3S6 with ~ 10 um thickness was fixed to the S -pole of the CPW by using the \napiezon N -grease  with high thermal conductivity  and its c axis align s along the z-axis \n[Figure 2(a)] . All cryogenic -temperature FMR spectra data were collected using a \nhomemade differential FMR measurement system combining the lock -in tec hnique and \na closed -cycle G -M refrigerator -based cryostat. Static magnetic field  H can rotate in the \ny-z plane  [Figure 2(a)] and be modulated  with an amplitude of 1  - 2 Oe  by a pair of \nsecondary Helmholtz coils powered by an alternating current source with  a low audio \nfrequency of 129.99  Hz.  \n3. Results and Discussion  \n3.1 Spin resonance of the high -field FFM regime  \nFigure 2 (b) shows the representative pseudocolor plot of  normalized  magnetic field -\ndependent FMR spectra obtained at excit ation  frequency f varying from 5 to 20 GHz \nwith 1 GHz steps, oblique field angle θH = 45° and cryogenic temperature T = 4.5 K. \nThe inset of Fig. 2(b) exhibits a  representative differential FMR spectrum with f = 19 \nGHz, which can be well fitted using  a differential Lorentzian function . The \ncharacteristic dynam ic properties , e.g., the resonan ce field Hr and linewidth, can be \nextracted accurately from the fitting parameters of the experimental FMR spectra. As \nmentioned above, the external magnetic field can change the spin texture of MnNb 3S6. \nFor instance,  the low-field CSL with a nontrivial topolog ical property  will be driven \ninto the trivial  FM state  by an in -plane magnetic field Hin ≥ Hc ~ 0.5 1 kOe at T = 4.5 K . \nTherefore, it is expected that the different dynamic properties corresponding to two \ndistinct spin textures could be observed in our broadband FMR spe ctra.   \nFigure  2 Broadband differential FMR of flake crystal sample.  (a) Schematic diagram \nof the high -sensitivity differential FMR  experimental  setup  combining a coplanar \nwaveguide technique . The green arrow represents the injection of microwave current.  \n(b) Pseu docolor plot of the  representative  normalized magnetic field -dependent FMR \nspectra obtained at  frequency  f between 5 and 20 GHz increased in 1 GHz steps , oblique \nfield angle  θH = 45° and temperature T = 4.5 K . The i nset is a representative differential  \nFMR spectrum (squares)  obtained  at f = 19 GHz  and fitted by the differential Lorentz ian \nfunction  (solid black  line). \nTo systematically explore the specific dynamics of  chiral helimagnet MnNb 3S6, \nwe measure d the broadband FMR spectra carefully  at several different temperatures  T \n= 4.5  K, 10  K, 30  K, and  45 K. Figure 3(a) - (d) show  the frequency -dependent \nresonance field Hres extracted by  fitting experimental  FMR spectra  with a differential \nLorentz ian function [35,36] . For high-field range  H > H c ~ 0.5 1 kOe, we found that the \ndispersion curves of  f vs 𝐻res obtained at all four different temperatures can be  well-\nfitted with the well -known  Kittel formula as follow s [see SI for specific derivation \nprocess] :  \n𝑓=𝛾√(𝐻𝑟𝑒𝑠𝑐𝑜𝑠(𝜃𝑀−𝜃𝐻)+4𝜋𝑀𝑒𝑓𝑓cos(2𝜃𝑀))\n∗(𝐻𝑟𝑒𝑠𝑐𝑜𝑠(𝜃𝑀−𝜃𝐻)−4𝜋𝑀𝑒𝑓𝑓𝑠𝑖𝑛2𝜃𝑀)                          (1) \nwhere γ = 2.9 kOe/MHz  is the gyromagnetic ratio,  4πMeff = 4πMs + Hk is the effective  \nmagnetization,  Ms is the saturation magnetization determined from static magnetization \nmeasurements, Hk is the effective anisotropy field, the out -of-plane angle of the external \nfield θH = 30o and magnetization  θM. For monoaxial chiral helimagnet MnNb 3S6 with \neasy-plane  anisotropy , the magnetocrystalline anisotropy constant s Ku1, Ku2 are defined \nas Hk = - [2(Ku1+2Ku2)]/(μ0Ms), where  Ku2 can be  neglected  here because it is a fourth -\norder small item . Thus, Ku1 can be calculated by Ku1 = - (μ0MsHk)/2. The Kittel formula \n(eq. (1)) can well fit the high-field dispersion relation, indicating  that all spins have a \nuniform precession under the high-field range, consistent with discussed magnetic \nfield-forced FM state at H > H c in the H - T phase diagram above  [Figure  1(b)]. Note \nthat, for T = 45 K, only f vs. 𝐻res data in the high field range was used to be fitted \nbecause it exhibits a significant deviation at the low field range due to the strong spin \nfluctuation near its critical magnetic order temperature Tc = 45 K.  \nFuthermore , we can  obtain temperature -dependen t effective magnetization Meff, \nout-of-plane angle of magnetization θM, and magnetocrystalline anisotropy constant Ku1, \nwhich were together with the saturation magnetization Ms measured by SQUID \nmagnetometer shown  in Figure  3(e) and (f). Analogous to  Ms, Ku1 exhibits a monotonic \ndecreas e with increasing temperature and rapidly reduces to near zero wh ile \ntemperature approaches Tc = 45 K. Moreover , the temperature -dependent equilibrium \nposition of magnetization θM shows that the magnetic moment is more accessible to \nfollow external magnetic field  H due to  the decrease of Hk and demagnet ized field with \nincreasing temperature .  \nFigure  3 Temperature dependence of the uniform FMR of the high -field FFM regime.  \n(a) – (d) Symbols: f vs Hres experimental data obtained  at oblique field  angle θH = 30° \nand temperature  T = 4.5 K (a), 10 K (b), 30 K (c), and 45 K (d). The solid lines are the  \nfitting result s of the FMR data at the high -field range  with the Kittel formula eq. (1). \nThe regions with green and blue backgrounds represent the high -field FFM regime and \nlow-field CSL regime, respectively. (e) – (f) Temperature -dependent  effective  \nmagnetization Meff and saturation magnetization  Ms (e), the out-of-plane  angle of \nmagnetization θM and magnetocrystalline anisotropy constant Ku1 (f) were determined \nby fitting the FMR data using eq. (1) with best -fit parameters.  \nIn addition to  the discussed f vs. Hres dispersion relation above , the linewidth ∆H, \ncharacterized by using the full width at h alf maximum ( FWHM ), can be used to analyze \nthe Gilbert damping constant. FWHM  is determined by fitting experimental FMR \nspectra with a differential Lorentzi an function [35,36] . Figure 4(a)  - (f) show the \ndependence of FWHM  on the excitation frequency f at several different temperatures T \n= 4.5 K, 10 K, 20 K, 30 K, 40 K and 45 K  with H in the  ab-plane ( θH = 0°). For the \nhigh-field FFM regime  (H > H c), the relation of linewidth  ∆H vs. f can be well -fitted \nwith the following  formula  ∆H = ∆H0+αf/γ, where ∆H0 is the inhomogeneous linewidth \nbroadening  constant , α is the Gilbert damping factor . One can easily see that the \nlinewidth obviously deviates from the linear fitting in the low-field range , which is \ncaused by the emerging CSL phase in the low field, well consistent with the discussed \nf vs. Hres dispersion relation in Fig ure 3 abov e. The temperature dependence of the \nGilbert  damping constants α corresponding to  high-field FFM regime  [Inset of Fig ure \n4(f)] shows a gradual enhancement with increasing temperature at the low -temperature \nrange far below Tc = 45 K, and then suddenly reaches 0.11 at 40 K from 0.05 at 30 K \nwhen the temperature approaches to Tc. The significant broadening of the linewidth \nnear Curie temperature Tc is related to the thermal effect -induced strong spin fluctuation . \n \nFigure 4  Temperature - and field dependence of the FMR linewi dth. (a) – (f) The FMR \nlinewidth FWHM  vs. f experimental data (symbols) obtained  at in-plane  field H (θH = \n0°), T = 4.5 K (a), 10 K (b), 20 K (c) , 30 K ( d), 40 K ( e) and 45 K ( f). The linear  lines \nare the fitting result s of the FWHM  data with ∆H = ∆H0+αf/γ. The dash ed lines are  the \nextension of the linear fitting as guides to the  eye. The regions with green and blue \nbackgrounds represent the high -field FFM regime and low -field CSL regime, \nrespectively. Inset in ( f) Dependence of the Gilbert damping constant α on temperature \ndetermined by the linear fittings of the data in (a) – (f). \n3.2 Spin resonance  of the low-field CSL regime  \nUnlike the high-field FFM regime , the low-field CSL regime  includes  two magnetic \nstructures , helical spin texture and FM phase . Compar ed to a single FM state, t he \nmixture of  helical spin segment  and FM part in the CSL regime is expected to exhibit \ndistinct magnetic dynamics due to the  change  of various magnetic  interaction  energ ies \nof the whole system.  As discussed  in Figure  3(a) and (d), the experimentally obtained \ndispersion results devia te significantly  from the Kittel  formula eq. (1) at the low-field \nrange. Analogous  to the unsaturated magnetic domain system [37], we derived a  \nmodified  Kittel formula  (eq. (2)) for this mixture of spin textures  by reconsidering  the \ntotal magnetic interaction energy  of the system  via setting the proportion s of the helical \nspin segment and FM phase as q and p = 1- q, respectively [see SI for specific derivation \nprocess]  [33]. The modified FMR Kittel formula is given as follow s: \n𝑓=𝛾√(𝑞∗𝐻𝑠𝑖𝑛𝜃𝑀𝑠𝑖𝑛𝜃𝐻+𝑝∗𝐻𝑐𝑜𝑠(𝜃𝑀−𝜃𝐻)+4𝜋𝑀𝑒𝑓𝑓cos(2𝜃𝑀))\n∗(𝑞∗𝐻𝑠𝑖𝑛𝜃𝑀𝑠𝑖𝑛𝜃𝐻+𝑝∗𝐻𝑐𝑜𝑠(𝜃𝑀−𝜃𝐻)−4𝜋𝑀𝑒𝑓𝑓𝑠𝑖𝑛2𝜃𝑀)        (2) \nwhere 1/ q = L(Hin)/L(0) can be proved strictly . Setting q = 0 in the modified Kittel \nmodel can be returned to the standard Kittel formula  (eq. (1))  for the pure ferromagnetic \nstate.  As mentioned above,  the proportion of helical spin texture q depends significantly \non in-plane field component Hin [13,14] . Therefore, it is difficult to  get a reliable fitting \nresult about  the experimentally obtained  fFMR vs. Hres dispersion relation  in the low-\nfield CSL regime . Because  the spin helix period L(H) (proportional to 1/q) shows a \nsigni ficant in -plane magnetic field dependent divergency around  critical  field Hc \n[13,14] . \nTo further investigate spin dynamics in the low -field nontrivial CSL regime, we \nadopted out -of-plane angular -dependent FMR spectra. Because the critical field Hc \nfrom CSL transferring to FFM is expected to be higher at large θH due to the \ndemagnetization field and strong  easy-plane magnetic an isotropy.  More specifically , we \nquantitatively calculate the in -plane component of the resonance field  obtained in the \nout-of-plane angular -dependent FMR spectra and find it only  changes by 2.6% at f = 6 \nGHz, T = 5 K (see detail s in SI  [33]), avoiding the in-plane field-induced  significant \nmodulation of q. Figure 5 shows the dependence of resonance field Hres on out -of-plane angle θH from 0o to 90o with T = 5 K at different resonant frequencies . Figure 5(a) - (h) \nshow that the experimental angular -dependent Hres results can be well fitted by the \nmodified Kittel formula eq. (2) [see the detailed  fitting process in the SI][33] . The non -\nzero helical spin proportion q under low excitation frequency (less 8 GHz) indicates the \nexistence of the CSL state in the studied oblique field range with a low in -plane \ncomponent field Hin < Hc, consistent with the discussion of f vs. Hres curves at the in -\nplane field above. Figure 5(i) and ( j) show the field dependence of  the obtained fitting \nparameter q and the helical spin  period ratio L(Hin)/L(0). Helical spin proportion  q \ngradually decreases  with increasing field and reaches zero corresponding to the \ndisappearance of helical spin texture when the resonant field is above its critical field \nHc ~ 5.1 kOe at T = 5 K, similar to the previously reported field -dependent  spin helix \nperiod L(H) of CSL state in chiral helimagnet CrNb 3S6 by using the Lorentz micrograph \ntechnique [13,14]  \n \nFigure 5  Out-of-plane angular dependence of FMR spectra at 5 K. (a) – (f) The angular \ndependence of resonance field Hres at T = 5 K , fext = 4 GHz  (a), 6 GHz  (b), 7 GHz  (c), \n8 GHz  (d), 10 GHz  (e), 12 GHz  (f), 15 GHz  (g), 17 GHz  (h), respectively. The red solid  \nlines  are the results of fitting with the  modified Kittel formula eq. (2) described in the \nmain text. ( i) – (j) Temperature dependence of  the fitting parameter q (i) and the helical \nspin period ratio L(Hin)/L(0) (j), respectively.  The error bar of q is defined  in the SI [33] . \nTo further investigate the temperature effect on the helical spin  period of  the low-\nfield CSL regime , we also adopted out -of-plane angular -dependent  FMR spectra  at \ndifferent temperatures below Tc. Figure 6 shows the dependence of Hres on out -of-plane \nangle θH from 0o to 90o with f = 6 GHz.  Similarly, the obtained angular -dependent Hres \ndata can also be well fitted with the modified Kittel formula eq. (2) shown as  red solid \nfitting curves in Figure  6(a) - (f). Figure 6(g) and (h) show the temperature dependence \nof the obtained fitting para meter  q and L(Hin)/L(0). L(Hin)/L(0) gradually increases  with \nincreasing temperature  and reaches  infinity  when the resonant field  is above its critical \nfield Hc at T ≥ 40 K , also consistent with  the previous report [13,14] . \n \nFigure  6 Temperature -dependence of spin dynamics in the low -field CSL state.  (a) – (f) \nThe angular dependence of resonance fie ld Hres at fext = 6 GHz , T = 5 K  (a), 10 K (b), \n20 K  (c), 30K  (d), 40 K  (e), 45 K (f), respec tively. The red solid  lines are the results of \nfitting with the modified Kittel formula eq. (2) described in the text . (g) – (h) \nTemperature dependence of  the obtained fitting paramete r q (g) and the helical spin  \nperiod ra tio L(Hin)/L(0) (h), respectively . \n \n3.3 Phase diagram determined by spin dynamics  \nIn addition to the  phase diagram consist ing of the FFM and CSL  state, as shown \nin Figure 1(b), determined by the static magnetization characteristics,  the dynamic \nanalysis can also provide us with a detailed  phase diagram of the CSL state. We  \nquantitatively estimate the proportion of the helical spin texture q (or helical period \nL(H)) from the angular -dependent dispersion relation of spin dynamics . We measure  a \nseries of out-of-plane angular -dependent FMR spectra  with f = 4, 7, 8, 10, 12, 15, 17 \nGHz at different temperatures T = 5, 10, 20, 30, 40, 45 K (see the detail in SI [33]) . \nAfter the analysis of dispersion relations  as discussed above, we obtain the contour plot \nin terms of the component of the helical spin texture q (equal to L(0)/L(H)) in the plane \nof temperature and in -plane fi eld [Figure  7(a)], being overall consistent with the two -\ndimension phase diagram  [Figure 1(b)] determined by static magnetic susceptibility  \nmeasurement s. In the low-field CSL regime, the  spin helix period L(H) gradually \nincreases with increasing the applied external in -plane magnetic field because an in-\nplane field can help to enhance the FM segment in CSL due to the Zeeman effec t. Figure \n7(b) shows that the helical spin proportion q vs. normalized in -plane field H/Hc curves \nobtained at different temperatures collapse into a single field dependence curve. Our \nresult s are consistent with the field dependence of L(H) obtained by  the Lorentz \nmicrograph technique [13,14,31] , confirming that analysis of out-of-plane field angular \ndependence of spin resonance using  the modified Kittel formula can be regarded as  \nanother valid approach to probe  the topological spin texture period in chiral magnets.  \n \nFigure  7 Phase diagram determined by spin dynamics. (a) The contour plot in term s \nof the  proportion of helical spin texture q in the plane of  temperature T and in -plane \nmagnetic field H. (b) Univ ersal  field scale  dependence of q obtained at different \ntemperatures.  \n \n4. Conclusion  \nIn summary , several specific spin textures and their distinct dynamics of the chiral \nhelimagnet  MnNb 3S6 have been characterized detailly by field- and temperature -\ndependent static magnetization and broadband different ial FMR spectr oscopy . The \nhigh-field FFM  follows the standard Kittel dispersion relation  of a single domain FM \nstate. In contrast,  the low -field nontrivial CSL prefers the modifi ed Kittel formula \nincluding  the partial  helix spin texture . Furthermore , like the sophisticated Lorentz \nmicrograph technique, the modified  Kittel  model  proposed in this work as an \nalternative  and easy access  approach  enable s us to  extract the temperature - and field -\ndependent helical spin  period ratio L(H)/L(0) quantitatively from the angular -dependent \nFMR dispersion relation  obtained at different temperatures . Our results find that the  \nspecific angular -dependent magnetic dynamics  of nontrivial magnetic states proved in \nour work provide a vital clue  to explor ing interesting  magneti c dynamics in other \ntopologically  nontrivial chiral magnets.  \nAcknowledg ments  \nProject supported by the National Natural Science Foundation  of China (Grant Nos. \n11774150, 12074178, 12004171 , 12074386, and 11874358 ), the Applied Basic \nResearch Programs of Science and Technology Commission Foundation of  Jiangsu \nProvince, C hina (Grant No. BK20170627),  the Open Research Fund of Jiangsu \nProvincial Key Labor atory for Nanotechnology , and the Scientific Foundation of \nNanjing University of Posts and Telecommunication s (NUPTSF) (Grant No. \nNY220164) . \n  Reference  \n1 S. Muhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, \nP. Boni, Science 323, 915 (2009).  \n2 X. Z. Yu, Y . Onose, N. Kanazawa, J. H. Park, J. H. Han, Y . Matsui, N. 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Mater. 528, 167772 (2021).  \n Supporting Information  \n 1 / 12 \n Temperature - and field angular -dependent helical spin period \ncharacterized by magnetic dynamics in a chiral helimagnet MnNb 3S6 \nLiyuan Li1, Haotian Li1, Kaiyuan Zhou1, Yaoyu Gu1, Qingwei Fu1, Lina Chen1,2,*, \nLei Zhang3,*, and R. H. Liu1,* \n 1 National Laboratory of Solid State Microstructures, School of Physics and Collaborative Innovation \nCenter of Advanced Microstructures, Nanjing University, Nanjing 210093, China  \n2 New Energy Technology Engineering Laboratory of Jiangsu Provence & Sch ool of Science, Nanjing \nUniversity of Posts and Telecommunications, Nanjing 210023, China  \n3 Anhui Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field \nLaboratory, Hefei Institutes of Physical Science, Chinese Academy of Sci ences, Hefei 230031, China  \n \n \n* Corresponding authors:  \nRonghua Liu, rhliu@nju.edu.cn ,  \nLina Chen, chenlina@njupt,edu.cn  \nLei Zhang, zhanglei@hmfl.ac.cn  \n Supporting Information  \n 2 / 12 \n 1. The FMR Kittel formula for a single ferromagnet  \nTo systematically characterize the magnetization dynamics of the chiral helimagnet \nsingle -crystal MnNb 3S6, we firstly studied the uniformed ferromagnetic resonance \n(FMR) of the field -forced ferromagnet (FFM) state under the high field range. For a \nferromag netic crystal with a single magnetic domain, the total free energy E generally \nincludes the magnetocrystalline anisotropic energy Ek, the external field Zeeman \nenergy EH, the demagnetized field energy ED and the exchange energy Eex . In addition, \nsince the uniform precession of magnetic moment in a single magnetic domain crystal, \nthe exchange energy Eex only exists near the edge of the crystal and can be neglected. \nFor the monoaxial chiral helimagnet MnNb 3S6 with an easy -planar magnet ocrystalline \nanisotropy, the whole free energy of the FFM state could be written as:  \n𝐸=𝐸𝑘+𝐸𝐻+𝐸𝐷 \n=𝐾0+𝐾𝑢1𝑐𝑜𝑠2𝜃−𝜇0𝑀𝐻∙[𝑠𝑖𝑛𝜉𝑠𝑖𝑛𝜃 +𝑐𝑜𝑠𝜉𝑐𝑜𝑠𝜃 cos(𝜂−𝜑)]            (A1)  \n+𝑀2\n8𝑁𝑦𝑐𝑜𝑠2𝜃+𝑀2\n8𝑁𝑧𝑠𝑖𝑛2𝜃 \nwhere θ = θM, ξ = θH with the same definition with the main text, φ and η are the \nazimuth angle of the magnetization M and the external magnetic field H, while Ny and \nNz are the demagnetization factor in the y -axis and z -axis directions, respectively.  The \nfirst two terms at t he right side of the above formula eq. (A1) are the magnetocrystalline \nanisotropic energy Ek; the third term is the external field Zeeman energy EH; the last \ntwo terms are the demagnetized field energy ED. The e quilibrium position of the M can \nbe determine d by ∂E/∂ θ = ∂ E/∂φ = 0 , which leads to  \n{−[𝐾𝑢1+𝑀2\n8(𝑁𝑦−𝑁𝑧)]𝑠𝑖𝑛2𝜃−𝜇0𝑀𝐻𝑠𝑖𝑛(𝜉−𝜃)=0\n𝜂=𝜑                   (A2)  \nTherefore, the dispersion relation of the resonate frequency fFMR on the magnetic field \nH  is  \n𝑓=𝛾\n𝜇0𝑀𝑐𝑜𝑠𝜃√[𝜕2𝐸\n𝜕𝜃2∙𝜕2𝐸\n𝜕𝜑2−(𝜕2𝐸\n𝜕𝜃𝜕𝜑)2\n] \n=𝛾√(𝐻𝑐𝑜𝑠(𝜃𝑀−𝜃𝐻)+4𝜋𝑀𝑒𝑓𝑓cos(2𝜃𝑀))\n∗(𝐻𝑐𝑜𝑠(𝜃𝑀−𝜃𝐻)−4𝜋𝑀𝑒𝑓𝑓𝑠𝑖𝑛2𝜃𝑀)                            (A3)  Supporting Information  \n 3 / 12 \n where the effective magnetization 4 πMeff = -2[Ku1 + M2(Ny - Nz)/8]/μ0M = (Nz – \nNy)M/4μ0 - 2Ku1/μ0M. The effective magnetocrystalline anisotropy field Hk also can be \ndefined as Hk = - (2Ku1)/(μ0M). Additionally, the effective magnetization is equal to  \n4πMs + Hk because the studied magnetic crystal is a thin flake sample.  \n \n2. The modified Kittel formula for a chiral magnetic soliton lattice  \nTo further study the magnetization dynamics of the chiral magnetic soliton lattice CSL \n(consisting of the mixture of a helical spin segment and an FM block), we performed \nthe additional FMR experiments of the out-of-plane ma gnetic angular -dependent \nresonance field Hres with a certain excitation frequency. To better analyze the dynamics \nof the nontrivial CSL, we rewrite the whole free energy  by adding a term  contributed \nfrom the partial helical spin segment  in a similar way used in dealing with  the \nunsaturated magnetic domain system [1]. Setting the proportions of the helical spin \nsegment and the FM block in the CSL phase as q and p = 1 - q, respectively , we can get  \n𝐸𝑘=𝐾𝑢0+𝐾𝑢1𝑐𝑜𝑠2𝜃                                              (A4a)  \n𝐸𝐻=−𝜇0𝑀𝐻∙[𝑞𝑠𝑖𝑛𝜉𝑠𝑖𝑛𝜃 +𝑝𝑐𝑜𝑠(𝜃−𝜉)]                            (A4b)  \n𝐸𝐷=𝑝𝑀2\n8𝑁𝑦𝑐𝑜𝑠2𝜃+𝑁𝑧\n2(𝑝𝑀sin𝜃\n2+𝑞𝑀sin𝜃\n2)2=𝑝𝑀2\n8𝑁𝑦𝑐𝑜𝑠2𝜃+𝑀2\n8𝑁𝑧𝑠𝑖𝑛2𝜃    (A4c)  \n𝐸=𝐸𝑘+𝐸𝐻+𝐸𝐷                                                 (A5)  \nSince only FMR acoustic mode (magnetization processions of different \nferromagnet blocks is in -phase) was observed in the FMR experiments, the internal \nmagnetic field between the ferromagnet segments (or the effective bias field caused \nby the helical spin segment) arise from the nonlinear spin configuration ( the interlayer \nHeisenberg and Dzyaloshinskii -Moriya interactions ) should  keep a constant. \nTherefore, the interlayer exchange interaction will do not contribute to the dispersion \nrelation of the experimentally observed FMR acoustic mode. Following the same \nprocedure mentioned in the above part, the dispersion relation of fFMR vs. H is given \nby: \n𝑓=𝛾√(𝑞∗𝐻𝑠𝑖𝑛𝜃𝑀𝑠𝑖𝑛𝜃𝐻+𝑝∗𝐻𝑐𝑜𝑠(𝜃𝑀−𝜃𝐻)+4𝜋𝑀𝑒𝑓𝑓cos(2𝜃𝑀))\n∗(𝑞∗𝐻𝑠𝑖𝑛𝜃𝑀𝑠𝑖𝑛𝜃𝐻+𝑝∗𝐻𝑐𝑜𝑠(𝜃𝑀−𝜃𝐻)−4𝜋𝑀𝑒𝑓𝑓𝑠𝑖𝑛2𝜃𝑀)       (A6)  Supporting Information  \n 4 / 12 \n Where 4 πMeff = -2[Ku1 + M2(pNy - Nz)/8]/μ0M = (Nz – pNy)M/4μ0 - 2Ku1/μ0M is the \nsame as the effective magnetization 4 πMeff in the high -field forced FM regime above \nbecause Ny is 0 for a thin flake sample. Therefore, in fitting the spin -dynamic data \nobtained in the low -field CSL regime, we used the effective magnetization 4 πMeff \ndetermined from the high -field forced FM regime. Based on the above modulated \nFMR Kittle formula, we can obtain the proportions of the helical spin segment q and \nthe FM block p=1-q in the  studied CSL phase from the experimental \nangular -dependent resonance frequency. Furthermore, we can get the helix period \ninformation of the nontrivial chiral magnetic soliton lattice  through 1/q = L(Hin)/L(0) \nand its temperature dependence.  \n3. Static magne tization measurement  \nWe charactered  the temperature and field -dependent  static magnet ization  of the single \ncrystal  MnNb 3S6 by superconducting  quantum  interference  device  (SQUID)  \nmagnetometer  with H in the ab -plane and H parallel to the c -axis of the single -crystal \nsample, respectively . The phase diagram with H in the ab -plane (and H parallel to the \nc-axis) [Figure  1(b)] (and [Figure 1(c)])  of spin textures in MnNb 3S6 was determined  by \nthe field- [Figure  S1(a) and (b)] (and [Figure S1(e)])  and temperature -dependent \nmagnetic susceptibility [ Figure  S1(c)] (and [Figure S1(f)]) curves  with the applied field  \nH in the ab -plane  (and H parallel to the c -axis)  of the sample . The critical field Hc of \nthe transition from the CSL state to  the FFM state was defined  where the \nmagnetization M approaches its saturation in the M-H curve s [top inset in Figure  \nS1(a)] and the first -order differentiation d M/dH begins to deviate from the level line \nat the dc field Hdc [bottom inset in Figure  S1(b)] [2]. The field cooling (FC) and \nzero-field cooling (ZFC) curves with H = 40 Oe applying in the ab-plane and  parallel \nto the c -axis are shown in Figure S1(c) and Figure S1(f), respectively. M (T) curves \nshow the Curie temperature Tc ~ 45 K of MnNb 3S6. An anomalous sharp peak is \nobserved in the ZFC curve, consistent with the characteristic peak of chiral \nhelimagnets and antiferromagnets. All behaviors are well consistent with the \npreviously reported results of the chiral  helimagnets [3]. Figure S1(d) shows the Supporting Information  \n 5 / 12 \n out-of-plane  angular -dependent  magnetization with the external field Hext rotated from \nthe ab -plane to the c -axis at 2K, 40 K, and 60 K .  \n \n \nFigure  S1. (a) Magnetization curve s measured at dc field parallel to ab -plane of \nMnNb 3S6 at temperature T = 2 K. The bl ack arrow in the top inset marks the critical \nfield Hc of the transition from the CSL state to the FFM state.  The bottom inset is the \nfirst-order differentiation d M/dH, which also can determine the critical field Hc, \nmarked by the black arrow . (b) M-H curve s obtain ed with H in the ab-plane at T \nbetween 2 and 40 K increased in 2 K steps . (c) FC and ZFC curves with H = 40 Oe in \nthe ab-plane . (d) Magnetization as a function of the out-of-plane angle measured at H \n= 200 Oe, T = 2 K, 40 K, 60 K. (e) M-H curves with H parallel to the c -axis at T = 2 K, \n20 K, 40 K, 60 K . (f) FC and ZFC curves with H = 40 Oe parallel to the c -axis. \n4. Analysis of out -of-plane angular -dependent FMR spectra  \nWe systematically performed and analyzed the  out-of-plane  field angular -dependent \nFMR spectra  with the excitation frequency  f = 4, 7, 6, 8, 10, 12, 15, 17 GHz at different \ntemperatures T = 5, 10, 20, 30, 40, 45 K.  Based on  the 1D chiral sine -Gordon model  \nL(Hin)/L(0) = 4 K(k)E(k)/π2 in the main text, we can quantitatively estimate  the spiral \norder proportional q(θH) with the out -of-plane angle at a fixed excitation frequency \n(the in -plane component of the resonance field Hres). Taking out-of-plane \nangular -dependent FMR spectra  at f = 6 GHz, T = 5 K as an example, we found that \nSupporting Information  \n 6 / 12 \n the in -plane component of the resonance field only changes by 2.6% [Figure S2 (a)]. \nTherefore, w e adopted the out -of-plane angular -dependent FMR spectra  at a fixed \nfrequency can avoid a field -induced significant change of q [Fig. S2 (b)]. \nFor the low-field CSL re gime , we fit the angular -dependent Hres with a fixed  \nexcitation frequency by using the modified Kittel formula with three free parameters : \nthe effective magnetization Meff, the helical portion q, and the magnetization angle  θM. \nHowever, w e can independently determine  the effective magnetization Meff by the  \nhigh-field FFM state using the standa rd Kittel formula  because Meff should be  equal \nfor these two states under different fields at the same temperature . Second, to further \nminimize the deviation of q( θH) caused by the slight change of the in -plane component \nHin of the resonance field, we included the empirical relation of q( θH) using the 1D \nchiral sine -Gordon model L(Hin)/L(0) = 4 K(k)E(k)/π2 in fitting the angular dependence \nof FMR results, The following  Fig. S2(b) shows that the spiral order proportional q \nonly has less 9 % change from θH = 0 to 90o. Although the Hres changes dramatically for \nthe out -of-plane angular -dependent FMR spectra, the helical proportion q or L(H) still \nkeeps almost no change (< 9 %) under the series out -of-plane resonance field Hres with a \nconstant excitation frequency (6 GHz) because the in -plane component of Hres did not \nhave significant change . Therefore, we use the average value q as the spir al phase \nproportional under a fixed excitation frequency corresponding to a certain in -plane \ncomponent of the resonance field.  The error bar of q was determined by the fitting \ndeviation and slight change of  Hin. \nThe equilibrium angle  θM of the magnetic moment M was determined together by \nexternal magnetic field H, a sharp -induced demagnetized field, and an effective \nanisotropy field .   We can obtain the θM vs. θH curve  from the experimentally obtained \nf vs H  dispersion curve at a certain θH with the FMR Kittel formula. , as shown in \nFig.S2(c).   \n Supporting Information  \n 7 / 12 \n  \nFigure S2. (a) -(c) the in -plane component of an external magnetic field Hin (a), helical \nspin portion q (b), and out -of-plane angle of magnetization θM (c) with the  out-of-plane \nangle θH of the external field.  \n \nFurthermore, to intuitively prove  the reliab ility and accuracy of the dynamic \nbehavior analysis of the low -field CLS regime, we provided a more detailed analysis of \nout-of-plane angular -dependent FMR spectra  at frequency f = 4 GHz and T = 5 K  by \nusing  different helical proportions q = 0, 0.1, 0.2, 0 .3, 0.35 as the fitting parameters , as \nshown in Fig. 3S.  The fitting curve with q less 0.3 significantly deviates from the \nexperimental data, indicating that the dynamic behaviors analysis of the CLS state \nusing the modified Kittel model has good accuracy and sensitivity to estimate the \nhelical portion q. \n \n \nFigure 3S  The angular dependence of resonance field Hres at fext = 4 GHz, T = 5 K. The \ncolored solid lines are the results of fitting with the modified Kittel formula with a \nfixing q = 0.35, 0.3, 0.2, 0.1, 0, respectively.  \n \nBased on the discussion above, we fit the  angular -dependent Hres data obtained at  \nf < 8 GHz , H < Hc using the modified Kittel formula eq. (2) in the main text . The solid \nred curve represent s the best fitting curve. The non -zero q indicates the existence of the \nSupporting Information  \n 8 / 12 \n CSL state in the studied oblique field range with a low er in-plane  field component  than \nthe critical field Hc. The temperature dependence of  the obtained fitting parameter q \nand the helical spin  period ratio L(Hin)/L(0) are summarized in Figs. S4(g) – S6(g) and \nFigs. S4(h) – S6(h), respectively . For f > 10 GHz,  the solid red curves in Figs. S7 – \nS10 represent the fitting curves with the Kittel formula eq. (1) (equal to q = 0 case in \nmodified Kittel formula  eq. ( 2)) instead of the modified Kittel formula eq. (2), \nindicating that  the CSL state was destroyed or degraded by enhancement of magnetic \nfield exceeding the critical field Hc. \n \nFigure  S4. (a) – (f) The angular dependence of resonance field Hres at fext = 4 GHz , T = 5 \nK (a), 10 K (b), 20 K (c), 30K (d), 40 K (e), 45 K (f), respectively. The solid re d lines are \nthe results of fitting with the  modified Kittel formula eq. (2) described in the text.  (g) – \n(h) The temperature dependence of  the obtained fitting parameter q (g) and the helical \nspin period ratio L(Hin)/L(0) (h), respectively.  \n \nFigure  S5. (a) – (f) The angular dependence of resonance field Hres at fext = 7 GHz , T = 5 \nSupporting Information  \n 9 / 12 \n K (a), 10 K (b), 20 K (c), 30K (d), 40 K (e), 45 K (f), respectively. The solid re d lines are \nthe results of fitting with the  modified Kittel formula eq. (2) described in the  main  text. \n(g) – (h) The temperature dependence of  the obtained fitting parameter q (g) and the \nhelical spin  period ratio L(Hin)/L(0) (h), respectively.  \n \nFigure  S6. (a) – (f) The angular dependence of resonance field Hres at fext = 8 GHz , T = 5 \nK (a), 10 K (b), 20 K (c), 30K (d), 40 K (e), 45 K (f), respectively. The solid re d lines are \nthe results of fitting with the  modified Kittel formula eq. (2) described in the  main  text. \n(g) – (h) The temperature dependence of  the obtained fitting parameter q (g) and the \nhelical spin  period ratio L(Hin)/L(0) (h), respectively.  \n \nFigure  S7. (a) – (e) The angular dependence of resonance field Hres at fext = 10 GHz , T = \n5 K (a), 10 K (b), 20 K (c), 30K (d), 40 K (e), respectively. The solid re d lines are the \nresults of fitting with the  standard  Kittel formula eq. (1) described in the main text. \nSupporting Information  \n 10 / 12 \n  \nFigure  S8. (a) – (e) The angular dependence of resonance field Hres at fext = 12 GHz , T = \n5 K (a), 10 K (b), 20 K (c), 30K (d), 40 K (e), respectively. The solid re d lines are the \nresults of fitting with the  Kittel formula eq. (1). \n \nFigure  S9. (a) – (e) The angular dependence of resonance field Hres at fext = 15 GHz , T = \n5 K (a), 10 K (b), 20 K (c), 30K (d), 40 K (e), respectively. The solid re d lines are the \nresults of fitting with the Kittel formula eq. (1). \nSupporting Information  \n 11 / 12 \n  \nFigure  S10. (a) – (e) The angular dependence of resonance field Hres at fext = 17 GHz , T \n= 5 K  (a), 10 K (b), 20 K (c), 30K (d), 40 K (e), respectively. The solid re d lines are the \nresults of fitting with the  Kittel formula eq. (1).  \nSupporting Information  \n 12 / 12 \n References:  \n1 X. Shen, H. R. Chen, Y . Li, H. Xia, F. L. Zeng, J. Xu, H. Y . Kwon, Y . Ji, C. Won, W. \nZhang, Y . Z. Wu, J. Magn. Magn. Mater. 528, 167772 (2021).  \n2 N. J. Ghimire, M. A. McGuire, D. S. Parker, B. Sipos, S. Tang, J. Q. Yan, B. C. \nSales, D. Mandrus, Phys. Rev. B 87, 104403 (2013).  \n3 Y . H. Dai, W. Liu, Y . M. Wang, J. Y . Fan, L. Pi, L. Zhang, Y . H. Zhang, J. Phys. \nCondens. Matter. 31, 195803 (2019).  \n "    },    {        "title": "2103.13099v1.Current_induced_magnetization_dynamics_in_single_and_double_layer_magnetic_nanopillars_grown_by_molecular_beam_epitaxy.pdf",        "content": " 1 Current -induced magnetization dynamics in single and double \nlayer magnetic nanopil lars grown by molecular beam epitaxy  \n \nN Müsgens1, E Maynicke1, M Weidenbach1, C J P Smits1, M Bückins2, J Mayer2, B Beschoten1, and  \nG Güntherodt1 \n1Physikalisches Institut II A, RWTH Aachen University, 52056 Aachen, and Virtual Institute for \nSpinelectronics (VISel)  \n2Gemeinschaftslabor für Elektronenmikroskopie, RWTH Aachen University, 52056 Aachen  \n E-mail: bernd.beschoten@physik.rwth- aachen.de  \n \n \n \nAbstract  \n \nMolecular beam epitax y is used to fabricate  magnetic single and double layer  junctions  which are \ndeposited in prefabricated nanostencil masks.  For all Co | Cu | Co double layer junctions we observe a \nstable intermediate resistance state which can be reached  by current starting  from the parallel \nconfiguration of the respective ferromagnetic layers. The generation of spin waves is investigated at \nroom temperature in the frequency domain by spectrum analysis, demonstrating both in- plane and out -\nof-plane precessions of the magnetiz ation of the free magnetic layer. Current -induced magnetization \ndynamics in magnetic single layer junctions of Cu  | Co | Cu has been investigated in magnetic fields \nwhich are applied perpendicular to the magnetic layer. We find a hysteretic switching in th e current \nsweeps with resistance changes significantly larger than the anisotropic magnetoresi stance effect.  \n \nPACS: \n72.25.Ba , 75.47.De, 72.25.Mk, 75.30.Ds  \n \n \n \n \n \n \n \n \n \n \n \n1. Introduction  \n  2 Spin-transfer -induced magnetization dynamics has been studied experimentally  [1-7] since the \ntheoretical predictions by  Slonczewski  [8] and Berger  [9]. A standard spin transfer device consists of \ntwo ferromagnetic (FM) layers of unequal thicknesses, sepa rated by a nonmagnetic layer (NM) \nforming a nanopillar with a lateral dimension s on the 100 nm  scale. Electrons flowing perpendicular  to \nthe sample plane get spin -polarized by passing one of the ferromagnet s and then repolarized at the \nother NM/FM interfac e. Thereby the spin component transverse to the second FM layer’s \nmagnetization is absorbed and acts as a torque on its  magnetization . Because of the difference in layer \nthicknesses,  only the magnetization of the thin ferromagnetic layer gets destabilized and can either be \nswitched or excited in a precessional mode depending on the applied field and the passing current  [10-\n12]. \nA sufficient ly large electric current can affect the magnetization state of a ferromagnet not only in \ndouble ferromagnetic layer sy stems, but also in single ferromagnetic nanopillars [ 13-19]. A current \nflow generates a spin accumulation at both interfaces of the N M/FM/N M trilayer. To obtain an \nunequal torque on the magnetization the spin accumulations have to differ. This can be achie ved by \nasymmetric NM leads. Again,  the interface plays a n important role  for both switching behaviour  and \nmagnetization dynamics. One approach to better control the interfaces is the use of molecular beam \nepitaxy (MBE) [ 20]. We therefore fabricated magneti c nanopillars consisting of both magnetic single \nand double layer structures by MBE and analyze d their current -induced switching behaviour  by \nmagneto -transport and by high -frequency probes.  \n \n2. Samples and experimental setup  \n \nFocused -ion-beam  (FIB)  milling is  used to fabricate nanostencil templates which are suited for the \npreparation of multilayered spin- valve structures in the current perpendicular to plane (cpp) geometry  \n[4, 21- 22]. This approach allows to quickly modify and optimize both material combinati ons and \ngrowth conditions. The device dimensions are defined prior to the thin film deposition. In f igure 1 we \nsummarize the relevant process steps. First , a bottom electrode is fabricated by optical lithography and \nsubsequent Ar+ etching of  an extended Pt  layer, which was sputtered onto a Si substrate. Subsequently, \nan insulator SiO 2 and a second Pt layer are sputtered. Next  FIB (the FEI Strata 205 with Ga+ liquid \nmetal ion source has been operated at 30 keV with a beam current of 1 pA) is used to open up an \naperture in the top Pt layer. The size of the aperture directly defines the diameter of the magnetic \nnanopillar device. The aperture in the hard mask gives access to the underlying insulator for the \nsubsequent HF dip, which yields an isotropic selective  wet etching of the SiO 2. The resulting undercut \n(for schematic picture see f igure 1(a)) can easily be imaged by scanning electron microscopy  in top \nview of the apertures (f igure 1(b)). The nanostencil is then transferred into an MBE chamber for the \ngrowth  of the desired thin film stack  (figure 1(c)) . Details about the crystallinity of the pillars will be  3 discussed elsewhere [23].  As a final step the undercut is filled up with a thick Cu contact layer. O ptical \nlithography and Ar+ milling  is then used to def ine a top electrode in cross -point geometry which \nguarantees electrical access to both top and bottom electrodes.  \nTransport measurements were performed at room temperature in an external magnetic field. The \ndifferential resistance d V/dI was measur ed in four -point geometry (see f igure 2) by a lock -in technique \nwith a 100 µA modulation current at f = 1132 Hz which is superimposed to a DC current. The sample \nis connected to a high frequency (HF) sample holder. It consists of a flexible HF cable and a co planar \nwaveguide (not shown in f igure 2) with a total bandwidth of ~18 GHz. In order to detect the \nmicrowave emission of the junction we used a bias tee which separates DC from HF signals. The lat ter \nis amplified by 40 dB and analyzed by a 44 GHz bandwidth spectr um analyzer.  \nPositive current is defined by electron flow from the thin to the thick ferromagnetic layer in magnetic \ndouble layer samples while in magnetic single layer samples positive current is given by electron flow \nfrom the thin to the thick Cu metal  layer.   \n \n 3. Spin transfer studies in magnetic double layer systems  \n \nWe first focus on magneto -transport and microwave emission data on magnetic double layer samples. \nThe stack sequence of the pillar junction is |  3 nm  Co | 25 nm  Cu | 15 nm  Co |. As an ex ample, we \ndiscuss results on a 50  × 150 nm\n2 junction in more detail. In f igure 3 we compare the magnetic \nswitching as obtained in the differential resistance from magnetic field sweeps ( figure  3 (a)) and from \ncurrent sweeps ( figure  3(b)). Data are taken at  room temperature with the magnetic field oriented in \nthe sample plane close to the easy axis direction. The magneto- resistance curve was taken at a small \nDC current of 0.1 mA ( figure  3(a)). The device shows a clear hysteretic switching between a low \nresistive parallel ( P) and high resistive antiparallel ( AP) state  with a magneto- resistance value of  \n∆R/R ≈ 3.1 %  (see black curve in f igure 3(a)). Sweeping the magnetic field from large positive \n(H > Hc) to negative values the junction first remains in the low -resistance state for H  > 0 and switches \ncompletely into the high- resistance state at small negative fields. Note, that in contrast to the switching \nat the outer coercive fields the low field switching is not abrupt. It rather appe ars in two steps (see \narrow in f igure 3(a)).  \nWe now discuss the current -induced switching behaviour  for positive magnetic fields. The junction \nwas first set into the P state by a large positive magnetic field. Current sweeps were then \nsystematically recorded upon lowering the mag netic field. As an example, we sh ow a current sweep at \n300 Oe  in figure 3(b). Hysteretic switching can clearly be observed. Note, that the current -induced \nchange in resistance is significantly smaller ( ∆R/R ≈ 1.9 %) than the values obtained in the magnetic  \nfield sweep (see green dotted line s in figure 3 as guide to the eye). For comparison, w e added the  4 current -induced high resistanc e values obtained at 0.1  mA in f igure 3(a) as red open circles. It is \nobvious that we cannot reach the AP state of the junctio n at any magnetic field. We rather switch into \na different magnetic state of the device. This new magnetic state is stable even for current values up to \n20 mA. Note that the critical field value below which we observe current -induced switching does not \nmatch the coercive field of the free layer ( Hc = 1200 Oe) , which again indicates that we are not \nswitching into the AP state. We want to emphasize that we observe such a switching into an \nintermediate state in all MBE grown samples independent of the junction  area (ranging from \n30 × 60 nm2 to 50 ×  150 nm2) and the interlayer Cu thickness (ranging from 10 nm to 25 nm). In \ncontrast, we have never observed the I state in junction, which were fabricated by sputter deposition \n[22]. A detailed analysis of the magnet ic configuration of the intermediate state is beyond the scope of \nthe paper and will be published elsewhere [ 23].  \nIn figure 4 we summarize the threshold currents for current -induced magnetization reversal in a false-\ncolour  plot of the differential resista nce as a function of both the current and the magnetic field . In \norder to easier visualize the switching behaviour  we subtract the parabolic background in the dV /dI vs. \nI curves (see f igure 3(b)). We determine  a parabolic fit of the low resistance P state,  which  we \nextrapolate to the full current range and subtract from the measured data. We apply this method for \nboth sweep directions  (+16  mA to -16 mA and -16 mA to +  16 mA) and thereafter average the \ndifferential resistance for each magnetic field . Using t his method, we can easily distinguish between \nthe low resistance P state (dark blue regime, f igure 4) and the high resistance intermediate state I \n(green regime) as well as the hysteretic switching regime (light blue regime).  Four different regimes can be identified in the phase diagram . For large negative currents the \nmagnetization s of both Co layers are aligned in a parallel configuration . For magnetic fields smaller \nthan 600 Oe the thin Co layer can hysteretically be switched by the current from the par allel into the \nintermediate state . The critical switching currents for both the P to I and the I  to P transitions merge \nnear 600 Oe (see dotted lines in f igure 4 as guide to the eye). No switching can be observed above \n600 Oe. Instead, non- hysteretic peaks are observed in the differential resistance (see inset of f igure 6), \nwhich indicates an unstable regime. Although we never reach the AP  state by current sweeps in our \ndevices, our phase diagram has striking similarities to p revious results on sputtered Co  | Cu | Co \nsamples [11-12,22,24] . The main difference for our MBE grown samples is that we switch into a \nstable intermediate state and not into the AP state. Note, that in all other studies , current -induced \nswitching is observed right below the coercive fi eld of the free layer and the junctions can be switched \ninto the AP state at all positive fields.  \nAlthough we do not reach the AP state by current, our phase diagram also shows an unstable regime. \nIn comparison to previous microwave studies on the sputter ed samples, it is therefore interesting to \nexplore the magnetization dynamics of our samples in this regime . The microwave emission of the \njunction has been detected in the frequency domain by spectrum analysis. F igure 5(a) depicts selected  5 HF spectra measured at 740  Oe on the iden tical junction as discussed in f igures 3 and 4. The spectra \nare corrected by a background reference spectra of the transmission line which gives signals on the \norder of – 60 dBm. For better illustration, a false colour  plot of all  spectra is plotted in f igure 5(b). No \npeaks are observed in the spectra for low currents.  As the current is increased into the unstable regime \nup to 10 mA, a peak appears at f = 4.5 GHz. When further increasing the current, the peak shifts \nlinearly to lar ger frequency . The power output also increases and does not even saturate at large \ncurrents. The observed blue shift indicates an out -of-plane precession of the free layer magnetization \nvector  [25-26]. This behaviour  we only observe in a narrow field regim e. A detailed analysis of the \noverall spectra in the phase diagram will be given elsewhere [ 23]. The existence of these narrow \nspectra furthermore indicates that the intermediate state is given by a well -defined magnetization and \ndoes not originate from a simple multi- domain configuration.  \nTypically, we observe the frequency blue shift in samples with a thick Cu spacer layer thickness. In \ncontrast, samples with thinner Cu layer thickness show a frequency red shift, i.e. a decrease of \nspinwave frequency wit h increasing curr ent. As an example, we show in f igure 6 microwave spectra \nof a Co/Cu/Co bilayer sample with a Cu layer thickness of 15  nm and a cross section of 50  × 100 nm2. \nAs for the spectra in f igure 5, we chose the same magnetic field strength of 740 Oe. The decrease of \nthe spin wave frequency with increasing current can clearly be seen in the spectra. The corresponding \nnon-hysteretic ∆dV/dI vs. I curve is plotted in the inset of f igure 6(a). There is a direct correlation \nbetween the emitted microwave power and the differential resistance measurement shown by the \ncolour  code in figure 6 (a) with the corresponding resistance data in the inset. Consistent with previous \nstudies the onset of the magnetization dynamics typically occurs only near the peak in the differential \nresistance, while the relative position of this onset varies with the magnetic field (not shown) . Such a \nred shift in frequency is most commonly observed for in- plane magnetic fields [ 25]. Note, that the \nemitted microwave power for the jun ction with in -plane precession (f igure 6) is an order of magnitude \nless than for the junction showing the out -of-plane precession (f igure 5) , which indicates a larger \nprecessional angle for latter case.   \n \n4. Spin transfer studies in magnetic single layer systems \n \nMost spin transfer devices consist of at least two ferromagnetic layers. One of these layers provides  \nthe spin- polarized current and at the same time may act as a fixed reference layer for the detection of \nthe magnetization reversal of a free secon d ferromagnetic layer. Recently , it has been demonstrated \nthat the distinction between a fixed and a free ferromagnetic layer is not necessary  [16]. Current -\ninduced reversible changes in the resistance have  been observed in junctions with only one \nferromag netic layer for magnetic fields perpendicular to the plane of the layer . These resistance \nchanges have been attributed to the onset of non uniform spin wave modes. Even hysteretic switching  6 has been observed at smaller perpendicular magnetic fields. The im portance of an asymmetric spin \naccumulation on both  sides of the ferromagnetic layer - which can be realized by asymmetric leads -  \nhas been elaborated both theoretically and experimentally  [13-14,16] . However, t he actual  magnetic \nmicrostructure in these ju nctions is not yet known. As crystallinity and interface roughness may affect \nthe generation of nonuniform spin waves, we investigate the current -induced switching behaviour in \nMBE grown single -layer junctions at room temperature.  \nIn figure 7 we show magneto -transport measurements on a single layer junction with a stack sequence \nof 5 nm  Cu | 8 nm  Co | 100 nm  Cu and a junction area  of 30 ×  60 nm2. The data were taken at room \ntemperature with magnetic fields applied perpendicular to the thin film plane.  The current sweeps \nshow pronounced reversible dips in th e differential resistance (see f igure 7(a)), which may occur for \nboth current polarities. The dips move to larger current values with decreasing magnetic field (see \nfigure 7(b)). We observe hysteretic swi tching in magnetic field ranges between - 4.4 and -3 kOe and \n2.8 and 4.8 k Oe. The resistance changes by ~ 0.25 %. This effect cannot be explained by the \nanisotropic magnetoresistance (AMR) which is significantly smaller (~ 0.1%, data not shown). Similar \nresults have been published for junctions, which had been sputtered [ 16-17] or had been deposited by \ne-beam evaporation [ 18].  \n \n5. Concl usion  \n \nIn summary, we have used nanostencil mask templates to prepare magnetic nanopillars by molecular beam epitaxy. We f abricated b oth single layer and double layer junctions  with a stacking sequence of \nCu | Co | Cu and Co | Cu | Co, respectively. D ouble layer junctions show a magnetoresistance of up to \n3 % at room temperature . By current sweeps t hese junctions cannot be sw itched from a parallel low -\nresistance state into the antiparallel high -resistance state. Instead, the free layer switches into a new \nmagnetization state which results into an intermediate stable resistance of the device.  This \nintermediate state has previou sly not been observed in sputtered samples. Furthermore we observed \ntwo distinct spin wave modes indicating both in-plane and out -of-plane precession s of the thin layer \nmagnetization. For asymmetric single layer junctions  our results give indirect evidence for a \nnonuniform magnetization distribution resulting in current -induced hysteretic switching with a \nmagnetoresistance effect larger than expected from the AMR  effect .  \n We acknowledge helpful discussions with B. Özyilmaz. Work was supported by the DFG th rough SPP \n1133 and by the HGF. \n \n References:  \n[1] Tsoi M, Jansen A G M, Bass  J, Chiang W- C, Seck M, Tsoi V and Wyder P   7   1998 Phys. Rev. Lett.  80 4281 \n \n[2] Katine J A, Albert F J, Buhrman R A, Myers E B and Ralph D C  \n  2000 Phys. Rev. Lett.  84 3149 \n \n[3] Grollier J, Cros V, Hamzi c A, George J M, Jaffrès H, Fert A, Faini G, Youssef J B and  \n  Le Gall H 2001 Appl. Phys. Lett.  78 3663  \n \n[4] Sun J Z, Monsma D J, Abraham D W, Rooks M J and Koch R H   2002 Appl. Phys. Lett. 81  2002 \n [5] Kiselev S I, Sankey J C, K rivorotov I N, Emley N C, Schoelkopf R J and Buhrman R A  \n  2003 Nature  425 380 \n [6] Rippard W H, Pufall M R, Kaka S, Russek S E and Silva T J  \n  2004 Phys. Rev. Lett.  92 027201  \n \n[7] Kiselev S I, Sankey J C, Krivorotov I N, Emley N C, Garcia A G F, Buhrman R A and  \n  Ralph D C 2005 Phys. Rev. B  72 064430  \n \n[8] Slonczewski J C 1996 J. Magn. Magn. Mater. 159  L1 \n[9] Berger L 1996 Phys. Rev. B 54 9353  \n[10] Grollier J, Cross V, Jaffres H, Hamzic A, George J M, Faini G, Ben Youssef J, Le Gall H and \nFert A 2003 Phy s. Rev. B  67 174402  \n [11] Xiao J, Zangwill A and Stiles M D 2005 Phys. Rev. B  72 014446  \n \n[12] Urazhdin S 2004 Phys. Rev. B  69 134430  \n \n[13] Stiles M D, Xiao J and Zangwill A 2004 Phys. Rev. B  69 054408  \n \n[14] Polianski M L and Brouwer P W 2004 Phys. Rev. Let t. 92 026602  \n [15] Adam S, Polianski M L, Brouwer P W 2006 Phys. Rev. B  73 024425  \n \n[16] Özyilmaz B, Kent A D, Sun J Z, Rooks M J and Koch R H  \n   2004 Phys. Rev. Lett.  93 176604  \n \n[17] Özyilmaz B and Kent A D 2006 Appl. Phys. Lett. 88  152506  \n \n[18] Parge A, Niermann T, Seibt M and Münzenberg M 2007 J. Appl. Phys. 101 104302 \n \n[19] Stiles M D and Zangwill A 2002 J. Appl. Physt. 91 6812  \n \n[20] Dassow H, Lehndorff R, Bürgler D E, Buchmeier M, Grünberg P A, Schneider C M and  \nvan der Hart A 2006 Appl. Phys. Lett.  89 222511  \n \n[21] Sun J Z, Monsma D J, Kuan T S, Rooks M J, Abraham D W, Oezyilmaz B, Kent A D  \n and Koch R H 2003 J. Appl. Phys. 93  6859  \n \n[22] Özyilmaz B, Richter G, Müsgens N, Fraune M, Hawraneck M, Beschoten B, Güntherodt  \n  G, Bückins M and Mayer J  2007 J.  Appl. Phys.  101 063920   8  \n[23] Müsgens N, Maynicke E, Weidenbach M, Smits C J P, Bückins M, Mayer J, Beschoten B and \nGüntherodt  G to be published  \n \n[24] Özyilmaz B, Kent A D, Monsma D, Sun J Z, Rooks M J and Koch R H  \n  2003 Phys. Rev. Lett.  91 067203  \n \n[25] Stiles M D and Miltat J 2006  in Spin Dynamics in Confined Magnetic Structures III , Springer \nSeries Topics in Applied Physics Vol. 101 (Springer -Verlag, Berlin, Heidelberg ) \n \n[26] Slavin A N and Tiberkevich V S 2005 Phys. Rev. B  72 094428  \n \n \n \n \n \n \n \n \n \n \n  9 \n \n \n \nFigure 1: N anostencil mask fabrication. (a) A prefabricated template, consisting of a patterned bottom \nelectrodes (Pt) covered by an insulator (SiO 2) and a Pt hard mask is opened up using focused ion beam \n(FIB) milling. A selective wet- etching generates an und ercut and gives access to the bottom electrode. \n(b) SEM top view of the FI B-generated  holes after the selective wet -etching. (c) The desired thin film \nstack ( here Co/Cu/Co) and Cu  top electrode is grown using molecular beam epitaxy.   10 \n \n \n \nFigure 2:  Schemati c of the sample and the circuit used for differential resistance and high -frequency \nmeasurements.   11 \n \n \nFigure 3:  Differential resistance measurements of a magnetic double layer junction \n(3 nm Co | 25 nm  Cu | 15 nm  Co, cross- sectional area 50 × 150 nm2) at T = 300 K. (a) \nMagnetoresistance loop with the external magnetic field applied in the sample plane along the easy \naxis direction (black) . Red dots represent the high resistance resulting from current sweeps. (b) \nCurrent sweep at H = 300 Oe . The dotted green  lines are guides to the eye and represent the respective \nlow and high resistance state at 0.1  mA.   12 \n \n \nFigure 4 : False colour  plot of the differential resistance dV/dI, which is subtracted from  the parabolic \nbackground (see f igure 3(b) ) and averaged over both current sweep directions for a magnetic double - \nlayer junction (3 nm  Co | 25 nm  Cu | 15 nm  Co, cross -sectional area 50 × 150 nm2). Dashed lines \nindicate the boundaries between different magnetization configurations of the junction: low resistance \n(parallel alignment P), high resistance (intermediate state I), hysteretic regime (P/I) and unstable \nregime (U) for large fields and large positive currents.  \n  13 \n \n \nFigure 5:  Microwave emission spectra from  a magnetic double layer junction \n(3 nm Co | 25 nm  Cu | 15 nm  Co, cross- sectional area 50 × 150 nm2) at H = 740 Oe and T = 300 K for \nlarge  positive currents. ( a) Selected spectra at different current values. (b) two-dimensionl false colour  \nplot of the emitted microwave power as a function of current.  \n \n  14 \n \n \nFigure 6:  Microwave emission spectra from a magnetic double layer system \n(3 nm  Co | 15 nm  Cu | 15 nm  Co; 50 ×  100 nm2) at H  = 740 Oe and T  = 300 K. (a) Selected spectra at \ndifferent current values which are marked in the differential resistance measurement (inset). (b) two-\ndimensionl false colour  plot of the emitted microwave power as a function of current.  \n  15 \n \n \nFigure 7:  (a) Current sweeps at various magnetic fields for a single magnetic layer junction \n(3 nm Co | 25 nm  Cu | 15 nm  Co, 50 × 150 nm2) with the magnetic field aligned in the out- of-plane \ndirection. Data were taken at room temperature. (b) False colour  plot of differential resistance vs. \ncurrent.  \n \n \n "    },    {        "title": "1906.05133v1.Dynamics_of_magnetic_flux_tubes_in_accretion_discs_of_T_Tauri_stars.pdf",        "content": "arXiv:1906.05133v1  [astro-ph.SR]  12 Jun 2019MNRAS 000,1–18(2018) Preprint 13 June 2019 Compiled using MNRAS L ATEX style file v3.0\nDynamics of magnetic flux tubes in accretion discs of\nT Tauri stars\nA. E. Dudorov1⋆, S. A. Khaibrakhmanov1,2†, A. M. Sobolev2‡,\n1Chelyabinsk state university, 129 Br. Kashirinykh str., Ch elyabinsk 454001, Russia\n2Ural Federal University, 51 Lenin str., Ekaterinburg 62000 0, Russia\nABSTRACT\nDynamics of slender magnetic flux tubes (MFT) in the accretion discs of T Tauri\nstars is investigated. We perform simulations taking into account bu oyant, aerody-\nnamic and turbulent drag forces, radiative heat exchange betwee n MFT and ambient\ngas,magneticfieldofthedisc.TheequationsofMFTdynamicsareso lvedusingRunge-\nKutta method of the fourth order. The simulations show that ther e are two regimes of\nMFT motion in absence of external magnetic field. In the region r<0.2au, the MFT\nof radii 0.05/lessorequalslanta0/lessorequalslant0.16H(His the scale height of the disc) with initial plasma beta\nof 1 experience thermal oscillations above the disc. The oscillations d ecay over some\ntime, and MFT continue upward motion afterwards. Thinner or thick er MFT do not\noscillate. MFT velocity increases with initial radius and magnetic field st rength. MFT\nrise periodically with velocities up to 5-15km s−1and periods of 0.5−10yr determined\nby the toroidal magnetic field generation time. Approximately 20% of disc mass and\nmagnetic flux can escape to disc atmosphere via the magnetic buoya ncy over charac-\nteristic time of disc evolution. MFT dispersal forms expanding magne tized corona of\nthe disc. External magnetic field causes MFT oscillations near the dis c surface. These\nmagnetic oscillations have periods from several days to 1-3 months atr<0.6au. The\nmagnetic oscillations decay over few periods. We simulate MFT dynamic s in accretion\ndiscs in the Chameleon I cluster. The simulations demonstrate that M FT oscillations\ncan produce observed IR-variability of T Tauri stars.\nKey words: accretion, accretion discs; diffusion; MHD; stars: circumstellar ma tter;\nISM: evolution, magnetic fields.\n1 INTRODUCTION\nA number of observations show that young stellar objects\n(YSO) have large-scale magnetic field. Investigations of Ze e-\nman splitting and broadening of spectral lines of classical T\nTauri stars have shown that the stars have magnetic field\nwith strength of 1−3kG at their surface ( Guenther et al.\n1999;Johns-Krull 2007 ).Donati et al. (2005) reported the\nregistration of amagnetic field with strength of ∼103G near\nthe inner edge of the accretion disc in FU Orionis using Zee-\nman splitting of spectral lines.\nIt is now possible to make polarization maps of the ac-\ncretion discs at millimeter and (sub)millimeter wavelengt hs\nwithALMA. Interpretation of the polarization maps in\nframe of Davis-Greenstein mechanism allows to determine\ngeometry of the magnetic field in the disc ( Stephens et al.\n⋆E-mail: dudorov@csu.ru (AED)\n†E-mail: khaibrakhmanov@csu.ru (SAKh)\n‡E-mail: andrej.sobolev@urfu.ru (AMS)2014;Li et al. 2016 ,2018). Although there are other\npossible interpretations of the polarization maps (see\nLazarian & Hoang 2007 ;Tazaki et al. 2017 ;Kataoka et al.\n2017;Stephens et al. 2017 ).\nAccording to modern theory of star formation, young\nstars with the accretion discs form as a result of gravita-\ntional collapse of rotating molecular cloud cores with mag-\nnetic field (see reviews by Inutsuka 2012 ;Li et al. 2014 ).\nNumerical simulations show that the magnetic flux of the\nmolecular cloud cores is partially conserved during the col -\nlapse, so itis naturaltoassume that themagnetic fieldof the\naccretion discs of young stars is the fossil one (see reviews\nof the theory of the fossil magnetic field by Dudorov 1995 ;\nDudorov & Khaibrakhmanov 2015 ). The magnetic field of\naccretion discs can also be a result of dynamo (see, for\nexample, Brandenburg et al. 1995 ;Gressel & Pessah 2015 ;\nMoss et al. 2016 , and references therein).\nEvolution of magnetic field in the accre-\ntion discs was usually investigated in kinematic\napproximation for prescribed uniform diffusiv-\n©2018 The Authors2A. E. Dudorov, S. A. Khaibrakhmanov, A. M. Sobolev\nity (Bisnovatyi-Kogan & Ruzmaikin 1976 ;Lubow et al.\n1994;Agapitou & Papaloizou 1996 ;Guilet & Ogilvie 2014 ;\nOkuzumi et al. 2014 ).Dudorov & Khaibrakhmanov (2014)\nandKhaibrakhmanov et al. (2017) developed a magneto-\nhydrodynamic (MHD) model of the accretion discs taking\ninto account Ohmic, ambipolar diffusion and the Hall\neffect. They have shown that the magnetic field geometry\nvaries through the disc. Inside the region of low ionization\nfraction (‘dead’ zone ( Gammie 1996 )), Ohmic diffusion\nhinders amplification of the magnetic field, so that the\nmagnetic field has poloidal geometry. Ambipolar diffusion\noperates in outer regions of the accretion discs, where\nthe magnetic field asquires quasi-radial or quasi-azimutha l\ngeometry depending on the intensity of ionization and\ngrain parameters. The Hall effect operates near the borders\nof the ‘dead’ zones and leads to redistribution of the\npoloidal and toroidal components of the magnetic field. The\nmagnetic field is frozen in gas near the inner edge of the\naccretion discs, where thermal ionization operates. Effect ive\ngeneration of the toroidal magnetic field is possible in this\nregion.Dudorov & Khaibrakhmanov (2014) have suggested\nthat magnetic buoyancy instability can solve the problem\nof runaway growth of toroidal magnetic field.\nThe magnetic buoyancy instability leads to formation\nof separate magnetic flux tubes (MFT) from the regular\nmagnetic field (see Parker 1979 ). The MFT can float from\nthe region of their formation to the surface under the ac-\ntion of buoyant force. The instability has been found in nu-\nmerical simulations of unstable gas layers with planar mag-\nnetic field ( Cattaneo & Hughes 1988 ;Matthews et al. 1995 ;\nWissink et al. 2000 ;Fan 2001 ).Vasil & Brummell (2008)\nhave shown that the instability also arises in the gas layer\nwith the magnetic field that is generated out of perpen-\ndicular magnetic field by a shear flow in the plane of the\nlayer.Takasao et al. (2018) detected formation of MFT in\n3D MHD simulations of the inner regions of the magnetized\naccretion discs of young stars.\nMagnetic buoyancy instability develops and MFT form\nif the magnetic pressure is of the order of the gas pressure.\nNonlinear evolution of the instability can lead to formatio n\nof the MFT with strong magnetic field (plasma β<1). For\nexample, Machida et al. (2000) performed MHD simulations\nof magnetic buoyancy instability in differentially rotatin g\nmagnetized disks. They reported about the formation of a\nfilamentary-shaped intermittent magnetic structures ( β<1)\ninside the disc. Inside the disc, magnetic buoyancy instabi l-\nity will lead to formation of magnetic rings of the toroidal\nmagnetic field. Major radius of the rings will be equal to the\ndistance to the star, while minor radius will be limited by\nthe pressure scale height of the disc.\nThe MFT rise upwards in the disc under the action of\nbuoyant force, because their density is less than the densit y\nof ambient gas. The MFT dynamics in the discs has been\ninvestigated numerically in frame of slender magnetic flux\ntube approximation ( Sakimoto & Coroniti 1989 ;Torkelsson\n1993;Chakrabarti & D’Silva 1994 ;Schramkowski 1996 ;\nAchterberg 1996 ).Sakimoto & Coroniti (1989) investigated\nthe dynamics of MFT inside the radiation pressure domi-\nnated regions of accretion discs of quasi-stellar objects t ak-\ning into account aerodynamic drag, MFT shear, and heat\nexchange between the MFT and external gas in radiative\ndiffusion approximation. Torkelsson (1993) considered simi-lar problem for the case of Stokes’ drag law. Schramkowski\n(1996) studied MFT dynamics in optically thick radiation\npressure dominated discs paying special attention to the\nrole of shear and magnetic tension. He have found that\nstrong shear can lead to formation of coronal loops from the\ninitially horizontal toroidal MFT. Chakrabarti & D’Silva\n(1994) investigated dynamics of toroidal MFT inside the ge-\nometrically thick radiation pressure dominated discs arou nd\nblack holes. They have shown that magnetic tension leads\nto MFT collapse. Deb et al. (2017) extended model of\nChakrabarti & D’Silva (1994) to take into account time-\ndependent evolution of the flow around the MFT.\nDynamics of slender MFT in accretion discs of young\nstars, and effect of external magnetic field on the dynam-\nics of the MFT have not been investigated yet. Some of\nthe results of simulations in slender flux tube approximatio n\nwere confirmed in 2D and 3D MHD simulations of the mag-\nnetic flux escape from the accretion discs ( Matsumoto et al.\n1988;Shibata et al. 1990 ;Ziegler 2001 ) and stratified gas\nlayer (Mart´ ınez-Sykora et al. 2015 ).\nAfter rising from the disc, the MFT can lead to\nvarious effects, such as outflows, variability and bursts.\nDudorov (1991) proposed that MFT can be a part of\nthe molecular outflows in the star formation regions.\nChakrabarti & D’Silva (1994) andDeb et al. (2017) argued\nthat the MFT can play a role in acceleration and colli-\nmation of jets from accretion discs of black holes. Forma-\ntion of looplike magnetic structures above the discs due\nto magnetic buoyancy instability and magnetic reconnec-\ntion can lead to burst acitivity and heating of the region\nabove the disc ( Galeev et al. 1979 ;Stella & Rosner 1984 ;\nSchramkowski 1996 ;Miller & Stone 2000 ;Hirose & Turner\n2011;Uzdensky 2013 ).Mazets & Bykov (1993) proposed\nthat the MFT can carry away angular momentum from the\naccretion discs.\nPresent paper concerns mass and magnetic flux es-\ncape from the accretion discs of young stars due to mag-\nnetic buoyancy, formation of expanding magnetized ‘corona ’\nabove the disc, and connection between rising MFT and\nIR-variability of the accretion discs. We investigate MFT\ndynamics in the accretion discs of young stars taking into\naccount the radiative heat exchange between MFT and\nambient gas, effects of the aerodynamic and turbulent\ndrag, and external magnetic field. Some particular aspects\nof MFT dynamics in the accretion discs of young stars\nwere considered by Dudorov & Khaibrakhmanov (2016)\nandKhaibrakhmanov et al. (2018). Structure of the ac-\ncretion disc is calculated with the help our MHD model\nof the accretion discs ( Dudorov & Khaibrakhmanov 2014 ;\nKhaibrakhmanov et al. 2017 ).\nThe paper is organized as follows. In section 2.1, we dis-\ncuss approximations of the model. In section 2.2, the gov-\nerning equations are derived. The governing equations are\nwritten in terms of non-dimensional variables in section 2.3.\nSection 2.4is devoted to solution methods of the model\nequations. The model of the disc is described in section 2.5.\nFiducial results are presented and discussed in section 3.1.\nIn section 3.2, we investigate influence of the model param-\neters on the MFT dynamics. We estimate mass loss rates\ndue to rising MFT in section 3.3. The MFT dynamics tak-\ning into account magnetic pressure of the disc is investigat ed\nin section 3.4. We apply our model for interpretation of the\nMNRAS 000,1–18(2018)Flux tube dynamics in accretion discs 3\nobservational data on IR-variability in section 3.5. Section 4\nsummarize results and conclusions.\n2 MODEL\n2.1 Problem statement\nWeconsider geometrically thin,optically thickaccretion disc\nof a young star (see Figure 1). The disc has pressure Pe,\ndensityρe, temperature Te, and magnetic field with strength\nBe. Disc mass is small compared to the mass of the star\nM⋆, and therefore self-gravity of the disc can be neglected.\nWe use the cylindrical system of coordinates (r,ϕ,z). The\nvertical axis zis directed along the angular velocity vector\nof the disc Ω=(0,0,Ω). The accretion disc is considered\nto be in hydrostatic equilibrium in the z-direction. Vertical\ncoordinate of the surface of the disc is zs.\nWe assume that magnetic buoyancy instability leads to\nformation of a MFT in the form of a torus out of toroidal\nmagnetic field in the disc. In the case of axial symmetry,\nwe investigate the dynamics of the unit length cylindrical\nelement of this magnetic torus. The MFT is located at the\ndistance rfrom the rotation axis at a coordinate z0in the\nbeginning. It has velocity v, cross-section radius a, pressure\nP, densityρ, temperature T, and magnetic field strength B.\nSchematic problem statement is shown in Figure 1.\nThe MFT is in pressure equilibrium with the external\ngas,\nP+B2\n8π=Pe.\nMagnetic pressure outside the MFT is not considered in this\nequation. The effect of external magnetic field is considered\nin Section 3.4. We assume that temperatures inside and out-\nside the MFT are equal to each other initially. For the ideal\ngas with equation of state\nP=Rg\nµρT (1)\nthe density difference equals\n∆ρ=ρe−ρ=B2\n8πv2s, (2)\nwhere\nvs=/radicalBigg\nRgTe\nµ(3)\nis the sound speed, Rgis the universal gas constant, µ=2.3\nis the mean molecular weight of the gas. The gas inside the\nMFT has smaller density comparing to the surrounding gas,\nasP<Pe. Due to positive density difference ∆ρ>0, the\nbuoyant force\nfb=−∆ρgz, (4)\ncauses rise of the MFT in the z-direction, where gz<0is\nthe vertical component of stellar gravity. Drag force coun-\nteracts the motion of the MFT. We consider drag forces of\ntwo types, turbulent and aerodynamic drag.2.2 Basic equations\nOur model of MFT dynamics is based on the slen-\nder flux tube approximation. Similar models have been\nused by Sakimoto & Coroniti (1989),Torkelsson (1993),\nChakrabarti & D’Silva (1994),Schramkowski (1996). We\nwrite the equations of the MFT dynamics follow-\ningDudorov & Kirillov (1986),\ndv\ndt=/parenleftbigg\n1−ρe\nρ/parenrightbigg\ng+fd(v,ρ,T,a,ρe),(5)\ndr\ndt=v, (6)\nMl=ρπa2=const, (7)\nΦ=πa2B=const, (8)\ndQ=dU+PedV, (9)\nP+B2\n8π=Pe, (10)\ndPe\ndz=−ρegz, (11)\nU=Pe\nρ(γ−1)+B2\n8πρ. (12)\nIn equation of motion ( 5)fdis the drag force per unit\nmass of the flux tube, ( 6,7,8) are the equations defining\nthe velocity v, mass Mlper unit length and magnetic flux\nΦof the MFT, ( 9) is the first law of thermodynamics ( Q\nis the quantity of heat per unit mass, Uis the energy of\nMFT per unit mass, V=1/ρis the specific volume), ( 10)\nis the pressure balance equation, ( 11) is the equation of the\nhydrostatic equilibrium of disc in the z-direction, ( 1) is the\nequation of state, γis the adiabatic index.\nWe consider MFT motion in the z-direction, v=\n(0,0,v),fd=(0,0,fd). In this case, Equations ( 5,6) are\nreduced to\ndv\ndt=/parenleftbigg\n1−ρe\nρ/parenrightbigg\ngz+fd, (13)\ndz\ndt=v. (14)\nThe vertical component of stellar gravity acceleration\ngz=−zGM⋆\nr3/parenleftbigg\n1+z2\nr2/parenrightbigg−3/2\n. (15)\nIn the case of aerodynamic drag (see Parker(1979))\nfd=−ρev2\n2Cd\nρπa, (16)\nwhere Cdis the dragcoefficient ∼1. The turbulentdragforce\ncan be calculated as ( Pneuman & Raadu 1972 )\nfd=−πρe/parenleftBig\nνtav3/parenrightBig1/2\nρπa2, (17)\nTurbulent viscosity νtcan be estimated as ( Shakura 1972 ;\nShakura & Sunyaev 1973 )\nνt=αvsH, (18)\nwhereαis non-dimensional parameter characterizing the\nturbulence efficiency, His the scale height of the disc. Drag\nforce is evaluated using formula ( 17) inside the disc, z/lessorequalslantzs,\nand using formula ( 16) above the disc, z>zs.\nMNRAS 000,1–18(2018)4A. E. Dudorov, S. A. Khaibrakhmanov, A. M. Sobolev\nFigure 1. Panel (a): general picture of the accretion disc (orange col or) with magnetic field Beand toroidal magnetic flux tube with B\n(green color). Panel (b): cross-section of the disc and magn etic flux tube in the r−zplane. Dynamics of slender cylindrical MFT in the\nz-direction under the action of buoyant force, fb, and drag force, fd, is investigated. (color figure online)\nEquations ( 7) and (8) give\na=a0/parenleftbiggρ\nρ0/parenrightbigg−1/2\n, (19)\nB=B0ρ\nρ0, (20)\nwhere a0,ρ0and B0are the initial radius, density and mag-\nnetic field strength of the MFT, respectively.\nTaking time derivative of Equations ( 9) and (10) we\nobtain\ndU\ndt−Pe\nρ2dρ\ndt=hc, (21)\nd\ndt/parenleftbigg\nP+B2\n8π/parenrightbigg\n=vdPe\ndz, (22)\nwhere\nhc=dQ\ndt(23)\nis the rate of heat exchange. Energy and pressure are the\nthermodynamic functions of density and temperature, U=\nU(ρ,T)and P=P(ρ,T). Then equations ( 21) and (22) can\nbe written as\nUTdT\ndt+/parenleftbigg\nUρ−Pe\nρ2/parenrightbiggdρ\ndt=hc, (24)\n/parenleftbigPρ+Cmρ/parenrightbigdρ\ndt+PTdT\ndt=vdPe\ndz, (25)\nwhere the superscript (...)Tmeans derivative with respect to\nT(with constantρ) and the superscript (...)ρmeans deriva-\ntive with respect to ρ(with constant T). Term Cmρis the\nderivative of magnetic pressure B2/8πwith respect toρ, and\nCm=B2\n0\n4πρ2\n0.\nSolvingequations( 24,25)for timederivativesof ρandT\nand using ( 11), we derive equations describing the evolutionof density and temperature of the MFT\ndρ\ndt=hcPT+UTρegzv\nPT/parenleftbigg\nUρ−Pe\nρ2/parenrightbigg\n−UT/parenleftbigPρ+Cmρ/parenrightbig,(26)\ndT\ndt=ρegzv/parenleftbigg\nUρ−Pe\nρ2/parenrightbigg\n+hc/parenleftbigPρ+Cmρ/parenrightbig\nUT/parenleftbigPρ+Cmρ/parenrightbig−PT/parenleftbigg\nUρ−Pe\nρ2/parenrightbigg.(27)\nNow consider equations of the accretion disc structure.\nWe solve equation of the hydrostatic equilibrium ( 11) using\npolytropic dependence of pressure on density together with\nequation of state ( 1), and get the vertical profiles of density\nand temperature inside the disc\nρe(z)=ρm/parenleftbigg\n1−k−1\n2k/parenleftBigz\nH/parenrightBig2/parenrightbigg1\nk−1\n, (28)\nTe(z)=Tm/parenleftbigg\n1−k−1\n2k/parenleftBigz\nH/parenrightBig2/parenrightbigg\n, (29)\nwhereρm=ρe(z=0),Tm=Te(z=0)are the density and\ntemperature in the midplane of the disc, k=1+1/n,nis the\npolytropic index, scale height H=vs/Ωk,\nΩk=/radicalbigg\nGM⋆\nr3(30)\nis the Keplerian angular velocity.\nWe set temperature Taof the gas above the disc, z>zs,\nequal to the effective temperature of the disc\nTa=Teff=280/parenleftbiggL\nL⊙/parenrightbigg1/4/parenleftBigr\n1 au/parenrightBig−1/2\nK, (31)\nwhere Lis the luminosity of the star. Formula( 31) is derived\nunder the assumption that the gas in the photosphere of the\ndisc is heated by stellar radiation ( Hayashi 1981 ). We deter-\nmine the coordinate of the disc surface from the equality of\n(29) and effective temperature of the disc ( 31)\nzs=H/radicalBigg\n2k\nk−1/parenleftbigg\n1−Ta\nTm/parenrightbigg\n. (32)\nMNRAS 000,1–18(2018)Flux tube dynamics in accretion discs 5\nWe assume that temperature is constant Taabove the\ndisc surface, z>zs, and density falls down with zaccording\nto Equation ( 11) to the point where ρebecomes equal to the\ndensity of molecular cloud core ρism=3.8×10−20g cm−3.\nHeating rate of the MFT can be evaluated from heat\ntransfer equation (see Zel’dovich & Raizer 1967 )\nhc=1\nρdivq, (33)\nwhereqis the vector of the heat flux density. We con-\nsider the heat flux driven by the radiative heat conductiv-\nity (Mihalas 1978 )\nq=−κ∇T. (34)\nκ=4σRT3\n3κRρ, (35)\nwhereκRis the Rosseland mean opacity, σRis the Stefan-\nBoltzmann constant. The heat exchange occurs through the\nsurface of the MFT. Let us introduce cylindrical coordinate s\n(r′,ϕ′,z′), where r′is the distancefrom theaxis of theMFT,\nϕ′is the azimuthal angle, z′is the coordinate along the axis\nof the MFT. Then\ndivq=1\nr′∂\n∂r′/parenleftbigr′q/parenrightbig≈qext−qin\na, (36)\nwhere qextand qinare external and internal heat flux densi-\nties, correspondingly. Then\nhc≃ −4\n3κRρ2σRT4−σRT4e\na2. (37)\nWe determineκRas the power-law function of gas density\nand temperature following Dudorov & Khaibrakhmanov\n(2014).\n2.3 Non-dimensional variables\nLet us introduce following non-dimensional variables:\nu=v/va, ˜z=z/H, ˜T=T/Tm,\n˜ρ=ρ/ρm, ˜t=t/tA, ˜hc=hc/hm,\n˜B=B/Be, ˜g=gz/fa, ˜fd=fd/fa,\n˜P=P/(ρmv2a), (38)\nwherevais theAlfv´ enspeed, tA=H/vais theAlfv´ encrossing\ntime, hm=εm/tA,εmis the energy density of magnetic field,\nBeis the magnetic field strength, fa=va/tA. All scales are\ndefined at the midplane of the disc.\nEquations ( 13,14,26,27,19,20)inthenon-dimensionalvariables (tilde signs are omitted):\ndu\ndt=/parenleftbigg\n1−ρe\nρ/parenrightbigg\ng+fd, (39)\ndz\ndt=u, (40)\ndT\ndt=2(γ−1)\nβ×\nhc/parenleftbiggβ\n2T+Cmρ/parenrightbigg\n+ρegu/parenleftbiggCm\n2−Pe\nρ/parenrightbigg\n3−γ\n2Cmρ+β\n2T+(γ−1)Pe\nρ,(41)\ndρ\ndt=−ρegu+(γ−1)hcρ\n3−γ\n2Cmρ+β\n2T+(γ−1)Pe\nρ, (42)\na=Caρ−1/2, (43)\nB=CBρ, (44)\nwhereβis the plasma beta calculated for the parameters of\nthe disc in the midplane,\nCB=˜B0\n˜ρ0, (45)\nCa=˜a0˜ρ1/2\n0. (46)\n2.4 Method of solution and initial conditions\nThe system of dynamic equations ( 39-42) is solved with\nthe help of the explicit Runge-Kutta method of the fourth\norder of accuracy. Automatic step selection is used, and\nadopted relative accuracy equals 10−4. At each time step,\nthe radius and magnetic field strength of the MFT are cal-\nculated from Equations ( 43-44).\nAt the initial moment of time, the MFT with coordi-\nnates rand z0has zero velocity u0=0. The MFT is in\nthermal equilibrium with surrounding gas, T(z0)=Te. We\nspecify the initial magnetic field strength of the MFT B0\nwith the help of plasma beta\nB0=/radicalBigg\n8πP0\nβ0, (47)\nwhereβ0is the initial plasma beta inside the MFT. The ini-\ntial density is determined from the condition of the pressur e\nequilibrium ( 10) using plasma beta,\nρ0=Pe(z0)\nRgT(z0)\nµ/parenleftbigg\n1+1\nβ0/parenrightbigg. (48)\n2.5 Model of the disc\nWe use our MHD model of the accretion discs to\ncalculate the structure and magnetic field of the ac-\ncretion disc. Let us describe briefly the features of\nthe model (see Dudorov & Khaibrakhmanov (2014) and\nKhaibrakhmanov et al. (2017) for details).\nThe model is MHD-generalization\nofShakura & Sunyaev (1973) model. We solve MHD\nequations in the approximation of a geometrically thin\nMNRAS 000,1–18(2018)6A. E. Dudorov, S. A. Khaibrakhmanov, A. M. Sobolev\nand optically thick stationary disc. It is assumed that\nthe turbulence is the main mechanism of the angular\nmomentum transport in the disc. Turbulent viscosity is\nestimated according to expression ( 18). The tempera-\nture of the disc is calculated from the balance between\nturbulent ‘viscous’ heating and radiative cooling follow-\ningShakura & Sunyaev (1973). We use low-temperature\nopacities from Semenov et al. (2003). The heating of the\nouter parts of the disc by stellar radiation and cosmic rays\nis also taken into account following D’Alessio et al. (1998).\nThese two mechanisms determine the temperature of the\ndisc in the regions, where the turbulence can be weak. The\nmodel has two main parameters: αand accretion rate ∝dotaccM.\nIn addition to equations of Shakura & Sunyaev (1973),\nwe solve the induction equation taking into account Ohmic\ndiffusion, magnetic ambipolar diffusion, magnetic buoyancy\nand the Hall effect. Ionization fraction is determined from\nthe equation of collisional ionization (see Spitzer 1978 ) con-\nsidering the ionization by cosmic rays, X-rays and radioac-\ntive decay, radiative recombinations and the recombinatio ns\non dust grains. The evaporation of dust grains and thermal\nionization of hydrogen and metals are included in the model\nfollowing Dudorov & Sazonov (1987).\nInner radius of the disc is assumed to be equal to the\nradius of stellar magnetosphere. Outer radius of the disc,\nrout, is determined as the contact boundary, where the disc\npressure equals the pressure of the external medium.\n3 RESULTS\nWe consider accretion disc of classical T Tauri star with\nmass M⋆=1M⊙, radius R⋆=2R⊙, surface magnetic field\nstrength B⋆=2kG, liminocity L⋆=1L⊙. The disc is char-\nacterized by turbulence parameter α=0.01and mass accre-\ntion rate ∝dotaccM=10−7M⊙yr−1. In this case, the inner radius of\nthe disc lies at rin=0.027au from the star, and the outer\nradius rout=320au. Ionization fraction xin the disc is cal-\nculated for dust grain radius 0.1µm, cosmic rays ionization\nrateξ=10−17s−1and attenuation length ΣCR=100 g cm−2.\nIn Figure 2a, we plot radial profiles of gas surface den-\nsity and ionization fraction of the disc calculated using ou r\nmodel for adopted parameters. Surface density decreases\nwith rfrom≈2.5×104g cm−2at the inner boundary of the\ndisc to≈5 g cm−2near its outer boundary. Typical slope of\nΣ(r)dependence is −0.7in the region r=[1,100]au. The\nradial profile of ionization fraction is non-monotonic. In t he\ninnermost part of the disk, r<0.6au, the ionization frac-\ntion is high, x>10−10, due to thermal ionization. A plateau\ninx(r)profile at r=[0.03,0.2]au reflects total ionization of\nPotassium. The ionization fraction is minimumat r≈0.8au.\nAt further distances, ionization fraction increases with rdue\nthe decrease of surface density and corresponding more effi-\ncient ionization by cosmic rays.\nIn Figure 2b, we plot radial profiles of Bzand corre-\nsponding plasma beta calculated as βz=8πρmv2s/B2z. Mag-\nnetic field strength decreases with distance r. Near the in-\nner edge of the disk, Bz≈170G, which is nearly equal to\nthe stellar magnetic field at this distance. In the region of\nthermal ionization, r=[0.027,0.6]au, the magnetic field is\nfrozen into gas, and the strength of Bzis proportional to\ngas surface density. Plasma beta decreases from 300 to 30/uni00000014/uni00000013/uni00000013/uni00000014/uni00000013/uni00000014/uni00000014/uni00000013/uni00000015/uni00000014/uni00000013/uni00000016/uni00000014/uni00000013/uni00000017/uni00000014/uni00000013/uni00000018Σ/uni00000003/uni0000003e/uni0000004a/uni00000003/uni00000046/uni00000050−2/uni00000040/uni0000000b/uni00000044/uni0000000c\n/uni00000014/uni00000013/uni00000010/uni00000015/uni00000014/uni00000013/uni00000010/uni00000014/uni00000014/uni00000013/uni00000013/uni00000014/uni00000013/uni00000014/uni00000014/uni00000013/uni00000015\nr/uni00000003/uni0000003e/uni00000044/uni00000058/uni00000040/uni00000014/uni00000013/uni00000010/uni00000016/uni00000014/uni00000013/uni00000010/uni00000015/uni00000014/uni00000013/uni00000010/uni00000014/uni00000014/uni00000013/uni00000013/uni00000014/uni00000013/uni00000014/uni00000014/uni00000013/uni00000015/uni00000014/uni00000013/uni00000016Bz/uni00000003/uni0000003e/uni0000002a/uni00000040/uni0000000b/uni00000045/uni0000000c/uni00000014/uni00000013/uni00000010/uni00000014/uni00000018/uni00000014/uni00000013/uni00000010/uni00000014/uni00000017/uni00000014/uni00000013/uni00000010/uni00000014/uni00000016/uni00000014/uni00000013/uni00000010/uni00000014/uni00000015/uni00000014/uni00000013/uni00000010/uni00000014/uni00000014/uni00000014/uni00000013/uni00000010/uni00000014/uni00000013/uni00000014/uni00000013/uni00000010/uni0000001c/uni00000014/uni00000013/uni00000010/uni0000001b/uni00000014/uni00000013/uni00000010/uni0000001a/uni00000014/uni00000013/uni00000010/uni00000019/uni00000014/uni00000013/uni00000010/uni00000018\nx\n/uni00000014/uni00000013/uni00000013/uni00000014/uni00000013/uni00000014/uni00000014/uni00000013/uni00000015/uni00000014/uni00000013/uni00000016/uni00000014/uni00000013/uni00000017/uni00000014/uni00000013/uni00000018/uni00000014/uni00000013/uni00000019\nβz\nFigure 2. Panel (a): radial profiles ofthe surface density, Σ(black\nline, left y-axis), and ionization fraction, x(grey line, right y-\naxis), in the MHD model of the accretion disk for the adopted\nparameters. Panel (b): radial profiles of Bz(black line, left y-\naxis) and corresponding plasma beta (grey line, right y-axis).\nin this region. Magnetic field strength abruptly decreases b y\ntwo orders of magnitude at r≃0.6au. This is the transition\nto the ‘dead’ zone, region of low ionization fraction where\nOhmic and ambipolar diffusion prevent magnetic field am-\nplification. The ‘dead’ zone occupies region from 0.6 to 30\nau for considered parameters. Plasma beta is 103−105inside\nthe ‘dead’ zone. Further Bzandβzdecrease with rto the\nvalues 5×10−3G and 6, respectively, near the outer edge of\nthe disc.\nIn the following we consider dynamics of the MFT that\ncan form in the inner region of the disc, r<0.6au. We carry\nout simulations of the MFT dynamics for the various initial\nradii a0=[0.01,0.1,0.2,0.4]H, plasma betasβ0=[0.01,0.1,\n1,10], coordinates rin range 0.027÷0.6au and coordinates\nz0=[0.5,1]H. Adiabatic index of the gas γ=7/5.\nDensity, temperature and magnetic field strength of the\naccretion disc are listed in Table 1(column 1: r-distance,\ncolumn 2: midplane density, column 3: midplane tempera-\nture, column 4: effective temperature of the disc, column 5:\nscale height, column 6: magnetic field strength Be=Bz\nin the disc, column 7: z-coordinate of the surface of the\ndisc). Table 1shows that density, temperature and mag-\nMNRAS 000,1–18(2018)Flux tube dynamics in accretion discs 7\nTable 1. The characteristics of the accretion disc\nr[au]ρm[g cm−3]Tm[K]Ta[K]H[au] Be[G]\n(1) (2) (3) (4) (5) (6)\n0.027 2.0×10−64830 1700 6 .2×10−4170\n0.15 4.1×10−82015 715 5 .3×10−329.5\n0.2 2.1×10−81840 625 7 .7×10−322.4\n0.4 3.7×10−91430 445 2 .0×10−29.9\n0.6 1.4×10−91240 360 3 .3×10−26.3\nneticfieldstrengthdecrease with r-distance inthedisc. Scale\nheight of the disc increases with r. We choose polytropic\nindex n=3. In this case, the coordinate of the disc sur-\nface varies from zs=2.28H≈0.0014au at r=0.027au to\nzs=2.38H≈0.08au at r=0.6au.\n3.1 Fiducial run\nIn this section, we present and discuss the simulations of\nMFT dynamics for the following representative parameters:\nr=0.15au,β0=1,a0=0.1H, and z0=0.5H. We perform\ntwo sets of simulations to study the role of radiative heat\nexchange. First, we simulate the dynamics of MFT evolving\nin thermal equilibrium with ambient gas (Section 3.1.1). In\nthis case, Equations ( 41) and (42) are excluded, equality\nT=Teis adopted, and density is determined from pressure\nbalance Equation ( 10). Second, we carry out the simulations\ntaking intoaccount the radiative heat exchange according t o\nEquations ( 41) and (42) (Section 3.1.2).\n3.1.1 Thermal equilibrium\nIn this section, we discuss general features of the MFT dy-\nnamics in the case of thermal equilibrium. In Figures 3(a-c),\nwe plot the dependences of velocity, temperature and radius\nof the MFT on its z-coordinate. Corresponding dependences\nofz-coordinate, drag and buoyant forces, internal end exter-\nnal densitieson time are depictedin Figures 3(d-f).Absolute\nvalue of the drag force is plotted for convenience.\nFigure3(a) shows the MFT begins to rise with high ac-\nceleration. The velocity of the MFT grows very fast from 0\nto0.5 km s−1, because the buoyant force fbis much stronger\nthan the drag force fdin the beginning of motion (see Fig-\nure3(e)). After that, the buoyancy and drag forces become\nnearly equal to each other, |fd|/lessorsimilarfb, and the velocity mono-\ntonically increases up to ≈1.8 km s−1at the surface of the\ndisc zs≈2.27H. The MFT rises to the surface during time\n≈0.8tA≈18d, as Figure 3(d) shows. Absolute values of\nthe buoyant and drag forces become small, acceleration of\nthe MFT vanishes and it acquires nearly steady velocity\nv≈1.8 km s−1further.\nThe MFTisinthermal equilibriumwithambientgas, so\nthat T=Teduring its motion, as Figure 3(b) demonstrates.\nFigure3(c)shows thattheMFT expandsin thecourse of the\nrise, i.e. its radius increases with z. Correspondingly, density\nof the MFT decreases, as Figure 3(f) shows. Internal density\nalways stays less than the external one in thermal equilib-\nrium. Ultimately radius of the MFT exceeds the thicknessof the disc, a>zs, at z≈2.8H. We do not simulate fur-\nther motion of the MFT, as the slender tube approximation\nviolates under such circumstances. The process of substan-\ntial expansion of MFT above the disc can be interpreted\nas a transformation of rising slender flux tubes into non-\nuniform expanding magnetized corona of the disc. Further\nwe will call this process as a dispersal of the MFT. Effect of\nmagnetic flux escape from the disc with subsequent forma-\ntion of magnetized corona has been found by Miller & Stone\n(2000);Machida et al. (2000);Johansen & Levin (2008);\nTurner et al. (2010);Romanova et al. (2011);Takasao et al.\n(2018) in the MHD simulations of the accretion discs.\nTakasao et al. (2018) found formation of the flux tubes from\nthe regular magnetic field of the disc, that confirms our as-\nsumptions.\n3.1.2 Role of heat exchange\nConsider dynamics of the MFT in the case, when radia-\ntive heat exchange is taken into account self-consistently\n(Figure 4). Dynamics of the MFT inside the disc, z<zs\n(t<1tA), is similar to the one discussed in Section 3.1.1\n(Figure3). The MFT rises with increasing speed, expands\nand its density decreases. It accelerates to v≈1.8 km s−1\nnear the surface of the disc. Temperature Tis nearly equal\ntoTe, and its vertical profile matches the polytropic profile\n(29). Approximate equality of the temperatures inside and\noutside the MFT reflects fast heat exchange at z<zs.\nFigure4(a) shows that the MFT starts to decelerate\nafter rising from the disc. The velocity goes to zero, when\nthe MFT reaches point z≈2.7Hatt≈1tA. After that,\nthe velocity becomes negative, and the MFT starts to move\ndownwards, i.e. it ‘sinks’. When the MFT reaches point z≈\n2.2H, its velocity changes sign again, and the MFT starts\nto move upwards in the second time. Such an up and down\nmotion is observedwithinthetime period from t≈1tAtot≈\n2tA. Velocity periodically changes its sign, while its absolut e\nvalue decreases at this part of the trajectory. Further, at t>\n2tA, the MFT rises from the disc and continues monotonic\nupward motion with small nearly steady velocity of about\n0.04 km s−1.\nThe oscillatory motion of the MFT near the surface of\nthe disc in the time interval from t≈1tAtot≈2tAis ex-\nplained by the following. The internal temperature of the\nMFT decreases with respect to constant temperature out-\nside, when the MFT rises from the disc (see Figure 4(b)).\nThe temperature takes minimum value Tmin≈380K, when\nthe MFT rise to the point z≈2.7H. The cooling of the MFT\nis caused by its practically adiabatic expansion. The radia -\ntive heat exchange appears to be inefficient to compensate\nadiabatic cooling at this part of the trajectory. Decrease o f\ninternal temperature with respect to the external one cause s\nthe MFT to expand more slowly than in the case of ther-\nmal equilibrium. The density of the MFT decreases with z\nmore slowly than the external density, and ρexceedsρe, i.e.\nthe MFT loses buoyancy at the point z≈2.4H. The MFT\nmoves by inertia upwards for some time, until it stops at\nz≈2.7H. Then the MFT starts to ‘sink’ due to negative\nbuoyancy. The MFT contracts a little, and its temperature\nand density grow in a process of downward motion (see Fig-\nures4(b, c, f)). The buoyancy restores ( ρ<ρ e), when the\ninternal and external temperatures become nearly equal to\nMNRAS 000,1–18(2018)8A. E. Dudorov, S. A. Khaibrakhmanov, A. M. Sobolevv [km s-1]\n00,511,52\nz [H]0,5 1 1,5 2 2,5 3(a)\nT[K]\n6008001 0001 2001 4001 6001 8002 000\nz [H]0,5 1 1,5 2 2,5 3(b)\nT\nTe\na [H]\n0123\nz [H]0,5 1 1,5 2 2,5 3(c)z [H]\n0,511,522,53\nt[tA]0 0,2 0,4 0,6 0,8 1(d)\nf[dyn g -1]\n00,20,40,60,81\nt [tA]0 0,2 0,4 0,6 0,8 1|fd|\nfb(e)\nρ [g cm -3]\n1e-111e-101e-091e-081e-07\nt [tA]0 0,2 0,4 0,6 0,8 1ρ\nρe(f)\nFigure 3. Dynamics of the MFT in thermal equilibrium with ambient gas f orr=0.15au,β0=1,a0=0.1H,z0=0.5H. Panel (a):\nvertical profile of velocity. Panel (b): vertical profile of i nternal and external temperatures (black solid and blue das hed lines, respectively).\nPanel (c): vertical profile of MFT radius. Panel (d): depende nce of MFT z-coordinate on time. Panel (e): dependence of drag and buoya nt\nforces on time (black solid and blue dashed lines, respectiv ely). Absolute value of the drag force is depicted. Panel (f) : dependence of\ninternal and external density on time (black solid and blue d ashed lines, respectively). Vertical lines in panels (a-c) and horizontal line\nin panel (d) show the surface of the disc, zs=2.27H. The Alfv´ en crossing time tA=1.05Pk, where Pkis the Keplerian period equal to\n0.06yr for the adopted parameters. (color figure online)\neach other. This leads to deceleration of the MFT and to\nchange of the velocity sign near the point z=2.2H. The ra-\ndiative heat exchange leads to equalization of internal and\nexternal temperatures in a process of further periods of up\nand down motion with respect to the point of zero buoyancy\n(ρ=ρe). Ultimately, temperatures Tand Tebecome nearly\nequal to each other, and the MFT oscillations change onto\nmonotonic upward motion with steady velocity above the\ndisc surface at t>2tA. The oscillations decay due to equal-\nization of the temperatures, and corresponding decrease of\ndensity difference and buoyant force. The MFT is optically\nthick during its motion. We call the oscillations discussed in\nthis section as the thermal ones.\n3.2 Dependence on parameters\nWe investigate the dynamics of MFT for various initial radii\na0, distance rand plasma beta β0in this section. As it\nwas mentioned in the introduction, MFT are likely form\nwithβ0∼1. In this section we consider MFT dynamics for\nplasma beta in range [0.01,10]to study the dependence of\nMFT characteristics on the initial magnetic field strength.\nKhaibrakhmanov et al. (2018) have shown that the charac-\nteristics of MFT near disc surface practically do not depend\non the initial position z0. In this section, we present calcu-\nlations for z0=0.5H. Radiative heat exchange is taken into\naccount in all considered runs. Internal and external tem-\nperatures are equal to each other initially.\nIn Figure 5, we plot the dependences of MFT z-\ncoordinate and temperature on time, and velocity versusz-coordinate , for r=0.15au,β0=1,z0=0.5Hand various\ninitial radii a0. Black lines correspond to the fiducial case.\nFigure5(a) shows that rise time increases with decreas-\ning radius of MFT. For example, the MFT with a0=0.01H\n(blue line) rises to the surface of the disc over time of 7tA,\nwhile the MFT with a0=0.4H(magenta line) reaches disc\nsurface within time of 0.4tA(tAis the Alfv´ en crossing time).\nMFT with larger radius moves faster, because the buoyant\nforce is proportional to MFT volume ( ∝a3), while the drag\nforce is proportional to MFT surface area ( ∝a2) (see Fig-\nure5(c)).\nMFT with initial radius a0=0.1Hin Figure 5exhibits\nthermal oscillations. Our simulations show that the MFT\nwith initial radius a0>0.17H(see green and magenta lines\nin Figure 5) disperse fast above the disc before reaching\nthe point of zero buoyancy. The MFT with initial radius\na0<0.06H(see orange and blue lines in Figure 5) move\nslowly, so that the radiative heat exchange effectively equa l-\nizes internal and external temperatures (see Figure 5(b)),\npreventing the loss of buoyancy.\nIn Figure 6, we plot dependence of MFT z-coordinate\non time for a0=0.1H,β0=1,z0=0.5Hand various r-\ncoordinates. Black lines correspond to the fiducial run dis-\ncussed in Section 3.1.2.\nFigure6shows that time of MFT rise to the surface\nof the disc increases with r-distance. The MFT float up to\nthe surface over time of 2d at r=0.027au, and 140d at\nr=0.6au, andvelocityoftheMFTdecreases from 2.8 km s−1\nto1.5 km s−1.\nMFT experience the thermal oscillations only in the re-\ngion r/lessorequalslant0.2au (blue and black lines in Figure 6). Oscillation\nMNRAS 000,1–18(2018)Flux tube dynamics in accretion discs 9v [km s -1]\n-2-1012\nz [H]0,5 1 1,5 2 2,5 3(a)\nT[K]\n5001 0001 5002 000\nz [H]0,5 1 1,5 2 2,5 3T\nTe(b)\na [H]\n00,511,522,5\nz [H]0,5 1 1,5 2 2,5 3(c)z [H]\n0123\nt [tA]0 1 2 3 4 5(d)\nf [dyn g -1]\n-1-0,500,51\nt [tA]0 1 2 3 4 5fd\nfb\n(e)\nρ [g cm-3]\n1e-111e-101e-091e-081e-07\nt [tA]0 1 2 3 4 5ρ\nρe\n(f)00,511,522,5\nt [tA]0 1 2 3 4 5\n-1,5-1-0,500,511,52\nt [tA]0 1 2 3 4 55001 0001 5002 000\nt[tA]0 1 2 3 4 5\nFigure 4. Same as in Figure 3, but for the case when radiative heat exchange is taken into a ccount. Insets in panels (a), (b) and (c)\nshow dependence of MFT velocity, temperature and radius on t ime, respectively. (color figure online)z [H]\n01234\nt [tA]0 2 4 6 8(a)\na0=0.01 H\na0=0.05 H\na0=0.1H\na0=0.2H\na0=0.4H\nT [K]\n05001 0001 5002 000\nt [tA]0 2 4 6 8(b)v[km s -1]\n-2-101234\nz [H]0,5 1 1,5 2 2,5 3 3,5(c)\nFigure 5. Dynamics of the MFT in run with r=0.15au,β0=1,z0=0.5Hand various initial radii a0(lines of different colors). Panel\n(a): dependence of MFT z-coordinate on time. Panel (b): dependence of MFT temperatu re on time. Panel (c): vertical profiles of MFT\nvelocity. Horizontal line in panel (a) and vertical line in p anel (c) depict the surface of the disc. Horizontal dashed li ne in panel (b) shows\ntemperature of the gas above the disc, Te=715K. The Alfv´ en crossing time tA=1.05Pkand the Keplerian period Pk=0.06yr for the\nadopted parameters. (colour figure online)\nperiod increases with r-distance. It is approximately equal\nto1d at the distance r=0.027au (blue line), and 10d\nat the distance r=0.15au (black line). The simulations\nof MFT dynamics for various initial plasma beta show that\nonly the MFT with β0=1experience thermal oscillations in\nthe region r/lessorequalslant0.2au.\nIn the region r>0.2au, the MFT rise so slowly that\nthe radiative heat exchange is able to equalize internal and\nexternal temperatures, preventing the loss of buoyancy.\nIn Table 2, we present initial parameters and some re-\nsults of the simulations of the MFT dynamics at the dis-\ntances r=0.027and r=0.6au. Initial coordinate z0is0.5H\nin all presented runs. Column 1 shows number of run. Col-\numn 2 contains the values of r-distance. Initial plasma betaβ0and radius of the MFT a0are listed in columns 3 and 4,\nrespectively. The MFT mass, M, and magnetic flux, Φ, are\ngiven in columns 5 and 6, respectively. We list velocity, vsurf,\nradius, asurf, and magnetic energy, Em, of the MFT in the\nmoment, when it crosses the disc surface, in columns 7, 8\nand 9. Time of rise to the surface of the disc, tsurf, is given\nin column 10. Column 11 gives time of the toroidal magnetic\nfield generation (see discussion in Sections 3.3below). Col-\numn 12 contains the rate of mass loss due to buoyancy (see\ndiscussion in Section 3.3below). In column 13, we list ratios\nof the minimal MFT temperature during its rise with re-\nspect to corresponding external temperature. Typically, t he\nminimal temperature is achieved at z≈2.3−3H.\nTable2shows that masses of MFT range from 6.3×\nMNRAS 000,1–18(2018)10A. E. Dudorov, S. A. Khaibrakhmanov, A. M. Sobolev\nTable 2. Parameters of runs\nrunr[au]β0a0[H]M[M⊙]Φ[Mx] vsurf[km s−1]asurf[H]Em, [erg] tsurf[yr]tgen[yr]∝dotaccMb,[M⊙yr−1]Tmin/Te\n(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)\n1 0.6 0.01 0.01 2.8×10−112.9×10210.6 0.034 2.2×10342.0 10.6 2.6×10−121.0\n2 0.6 0.1 0.01 2.5×10−102.7×10210.5 0.035 1.7×10342.2 10.2 2.5×10−111.0\n3 0.6 1 0.01 1.4×10−92.0×10210.2 0.046 5.5×10333.5 7.6 1.8×10−101.0\n4 0.6 10 0.01 2.5×10−98.6×10200.05 0.059 4.8×103412.0 3.2 1.8×10−101.0\n5 0.6 0.01 0.1 2.8×10−92.9×10235.2 0.336 2.1×10360.2 10.6 2.6×10−101.0\n6 0.6 0.1 0.1 2.5×10−82.7×10233.6 0.355 1.7×10360.2 10.2 2.5×10−90.95\n7 0.6 1 0.1 1.4×10−72.0×10231.5 0.463 5.6×10350.4 7.6 1.8×10−80.78\n8 0.6 10 0.1 2.4×10−78.6×10220.3 0.578 6.6×10341.4 3.2 7.9×10−80.91\n9 0.6 0.01 0.2 1.1×10−81.1×10249.0 0.673 8.4×10360.1 10.6 1.0×10−91.0\n10 0.6 0.1 0.2 1.0×10−71.1×10245.5 0.709 7.0×10360.1 10.2 9.8×10−90.83\n11 0.6 1 0.2 5.6×10−78.1×10232.2 0.925 2.2×10360.3 7.6 7.4×10−80.58\n12 0.027 0.01 0.01 6.3×10−137.0×10190.9 0.028 2.8×10330.02 0.69 9.1×10−131.0\n13 0.027 0.1 0.01 5.8×10−126.8×10190.8 0.029 2.4×10340.02 0.66 8.8×10−121.0\n14 0.027 1 0.01 3.2×10−115.0×10190.4 0.036 8.7×10320.03 0.49 6.5×10−111.0\n15 0.027 10 0.01 3.2×10−112.3×10190.07 0.047 1.0×10320.10 0.20 2.9×10−100.98\n16 0.027 0.01 0.1 6.3×10−117.0×10218.6 0.276 2.8×10350.002 0.69 9.1×10−110.98\n17 0.027 0.1 0.1 5.8×10−106.8×10216.5 0.290 2.3×10350.002 0.66 8.8×10−100.34\n18 0.027 1 0.1 3.2×10−95.0×10212.8 0.356 8.4×10340.004 0.49 6.5×10−90.48\n19 0.027 10 0.1 5.8×10−92.3×10210.5 0.431 1.0×10340.13 0.2 2.9×10−80.89\n20 0.027 0.01 0.2 2.5×10−102.8×102215.6 0.554 1.1×10360.001 0.69 3.6×10−100.58\n21 0.027 0.1 0.2 2.3×10−92.7×102210.3 0.576 9.4×10350.0011 0.66 3.5×10−90.35\n22 0.027 1 0.2 1.3×10−82.0×10224.3 0.708 3.4×10350.0024 0.49 2.7×10−80.39z [H]\n0,511,522,53\nt [d]0 50 100 150\nr \u0000 0 \u0001 \u0002 \u0003 \u0004 au\u0005 \u00060.15 au\u0007 =0.6 au2.8 km s-11.8 km s -11.5 km s -1\n0,511,522,53\n0 5 10 15\nFigure 6. Dependence of the MFT z-positionon time in run with\na0=0.1H,β0=1,z0=0.5Hat various r-coordinates (lines of\ndifferent colors). The inset shows zoomed-in region of the tr ajec-\ntory of the MFT for r=0.027au. Numbers near the curves show\nmaximum velocity of the MFT (blue for r=0.027au, black for\nr=0.15au, green for r=0.6au). The Keplerian period equals 1.6,\n21and 170d atr=0.027,0.15and 0.6au, respectively. (colour\nfigure online)\n10−13M⊙(run 12) to 1.0×10−7M⊙(run 10). The MFT mass\nincreases with its radius. The density and, therefore, MFT\nmass increase with β0(it follows from Equation ( 2)). The\nMFT transfer magnetic fluxes in range from 2.3×1019Mx\n(run 15) to 1.1×1024Mx (run 9). Velocity of the MFT\nwithβ0=1has the values in interval 0.2−4.3 km s−1that\nis comparable with local sound speed of 1−2.5 km s−1. The\nMFT with smaller β0accelerate to higher velocities and canhave supersonic speed. Maximum velocity ∼15.6 km s−1is\nachieved in run 20. In this case, bow shocks will probably\nform during motion of the MFT, which can change drag\nlaw and lead to additional heating. In the following section s\nwe will present and discuss the simulations, in which the\nvelocity of the MFT does not exceed significantly the sound\nspeed.\nWhentheMFTcrosses thediscsurface, its radiusis sev-\neral times smaller than zs. For example, a(z=zs)=0.035H\nfor run 1. The MFT continue to expand, when they move\nabove the disc (see Figure 3(c)). The MFT radius becomes\nlarger than the disc height at z∼2.7−3.5Hin performed\nruns. For instance, the MFT radius exceeds the disc height\natz=2.7Hin run 8, and at z=3.5Hin run 9. As it was\nstated in Section 3.1, we interpret the dispersal of the rising\nMFT as a formation of expanding magnetized corona above\nthe surface of the disc.\nThinnest MFT with a0/lessorequalslant0.01Hstay in thermal equi-\nlibrium with external gas, T≈Te, as column 13 of Table 2\nshows. Thicker MFT cool down in comparison to the exter-\nnal gas during their motion. For example, the MFT with\nβ0=1,a0=0.2and r=0.6au cools down to minimal tem-\nperature ≈0.58Te(run 11). Among the runs presented in\nTable2, prominent thermal oscillations are found only in\nruns 18 and 19, in agreement with the discussion of the Fig-\nure6. Oscillation behaviour is also observed in run 8 with\nβ0=10,a0=0.2,r=0.6, but the oscillations rapidly decay\nwithin three periods in this case. In the other runs, the MFT\ndisperse fast above the disc.\nMNRAS 000,1–18(2018)Flux tube dynamics in accretion discs 11\n3.3 Mass and magnetic flux loss due to buoyancy\nof flux tubes\nAs it has been shown in sections 3.1and3.2, the MFT rise\nfromthedisctoitsatmospherecarryingawaymassandmag-\nnetic flux. The rising MFT can be the seed for the formation\nof jets and outflows from accretion discs. Similar idea was\nconsidered by Chakrabarti & D’Silva (1994) andDeb et al.\n(2017) in application to the accretion discs around black\nholes. The rate of vertical mass transport via buoyancy can\nbe estimated as a mass of the flux tubes rising from the disc\nper unit of time. We consider times of the MFT formation\nandrise as characteristic time scales. The characteristic time\nof MFT formation is comparable to the time of the toroidal\nmagnetic field generation. To estimate efficiency of the mass\ntransport form disk interior to its atmosphere via buoyancy ,\nwe compare the characteristic times in this section.\nThe time of the azimuthal magnetic field generation can\nbe estimated from the ϕ-component of induction equation in\nthe approximations of the accretion disc model,\n∂Bϕ\n∂t=Bz∂vϕ\n∂z. (49)\nThis equation shows that Bϕis generated from Bzby the dif-\nferential rotation of the disc. The velocity vϕis determined\nfrom the balance between the gravity and centrifugal force\nin the r-direction. In our case,\nvϕ=/radicalbigg\nGM⋆\nr/parenleftbigg\n1+z2\nr2/parenrightbigg−3/4\n=rΩk/parenleftbigg\n1+z2\nr2/parenrightbigg−3/4\n, (50)\nso Equation ( 49) in the case z2/r2≪1turns to\n∂Bϕ\n∂t≃ −3\n2z\nrBzΩk. (51)\nTherefore, the time of the azimuthal magnetic field genera-\ntion up to a given value Bϕ\ntgen=2\n3Bϕ\nBzΩ−1\nk/parenleftBigz\nr/parenrightBig−1\n≃2.12Pk/parenleftbiggz/r\n0.05/parenrightbigg−1Bϕ\nBz, (52)\nwhere Pkis the Keplerian period. Typical time scale of Bϕ\namplification up to the value Bϕ=Bzis nearly two Keple-\nrian periods for z=0.05r. Keplerian period increases with\ndistanceas Pk∝r3/2,therefore fastest generation of Bϕtakes\nplace in the innermost region of the disc.\nMFT rise and generation times for considered parame-\nters are listed in columns 10 and 11 of Table 2, respectively.\nWe plot dependence of Pk,tgenand tsurfon the r-coordinate\nin Figure 7(a). Table 2and Figure 7(a) show that charac-\nteristic times increase with distance. The rise time tsurfin-\ncreases from 0.5 d at 0.027au to 200d at r=0.6au. It is\nnearly equal to Keplerian period and order of magnitudeless\nthan generation time tgen(52). This means that the dynam-\nics of the MFT will occur in two stages. At the first stage\nhavingduration tgen, toroidal magnetic fieldis generated and\nMFT form due to magnetic buoyancy instability. At the sec-\nond stage, the MFT rise from the disc over the time tsurfand\ncarry away some mass and magnetic fluxtodisc atmosphere.\nAfter that, this two-stage process repeats. Since tsurf≪tgen,\nthe process of mass and magnetic flux transport from disc to\nits atmosphere is periodic with typical period tgen. Table2\nand Figure 7(a) show that tgen≈ (0.5−0.7)yr at r=0.027au\nand tgen≈ (8−10)yr at r=0.6au.The MFT can form inside the region r=0.027÷0.6au.\nResultspresentedinFigure 7(a)indicatethatMFTwillform\ninside the disc and rise from it first at small rand then at\nfarther distances from the star.\nThe rate of vertical mass transport via the buoyancy,\n∝dotaccMb, can be estimated by division of the MFT mass by the\ncharacteristic time of the magnetic field amplification tgen.\nValuesof ∝dotaccMblieinrange ∼10−12−10−7M⊙yr−1(column12of\nTable2).This valueincreases withplasmabetaandradiusof\nMFT. For example, maximum rate ∝dotaccMb=7.9×10−8M⊙yr−1\nis found in run 8 with a0=0.1Handβ0=10. Rate of\nmagnetic flux transport via buoyancy can be estimated in\nsimilar way, ∝dotaccΦb=Φ/tsurf.\nIn Figure 7(b), we plot the dependences of ∝dotaccMband∝dotaccΦb\non the r-coordinate for the MFT with initial parameters\na0=0.1H,β0=1,z0=0.5H. Figure 7(b) shows that ∝dotaccMbin-\ncreases with distance, from ≈6×10−9M⊙yr−1atr=0.027au\nto≈2×10−8M⊙yr−1atr=0.6au. Therefore, the periodic\nprocess of mass and magnetic flux transport from the disc to\nits atmosphere caused by magnetic buoyancy is the most ef-\nficient near the outer edge of thermal ionization zone, where\nthe MFT with large radius form. The rate of vertical mass\ntransport via the buoyancy at r=0.6au is five times smaller\nthan the mass accretion rate in the disc ∝dotaccM, i.e. 20%of the\naccreted mass can be transported from the disc to its atmo-\nsphere via buoyancy.\nIt should be noted that ‘dead’ zone is situated in the\nregion between r=0.6au and r=33au for the con-\nsidered parameters of the disc. The rate ∝dotaccMbwill decrease\nrapidly with rbeyond r=0.6au, since amplification of the\ntoroidal magnetic field is hindered by Ohmic diffusion and\nMFT cannot form inside the ‘dead’ zone. MFT can form\nonly in the surface layer of the disc above the ‘dead’ zone.\nTypical surface density of this layer is 5−10 g cm−2(e.g.,\nDudorov & Khaibrakhmanov (2014)), and the MFT form-\ning inside this layer have small radius of ∼0.01Hand carry\naway tiny mass.\nFigure7shows that rate of magnetic flux transport via\nbuoyancy also increases with r, from∝dotaccΦb≈1×1022Mx yr−1\natr=0.027au to∝dotaccΦb≈2.7×1022Mx yr−1atr=0.6au.\nMinimum and maximum magnetic fluxes are 2.3×1019and\n1.1×1024Mx(runs15 and9), andmagnetic energies are 3.0×\n1033and3×1037erg(see columns 6 and 9 of Table 2). Total\nmagnetic flux of the disc equals 5×1029Mx for considered\nparameters. Therefore, nearly 20%of the disc magnetic flux\ncan be lost via the magnetic buoyancy over ∼1Myr.\n3.4 Effect of external magnetic field\nInsections 3.1-3.3,we investigatedthedynamics oftheMFT\nin the disc without external magnetic field. The magnetic\nfield of the disc can influence the MFT dynamics mainly\nthrough the magnetic pressure. In this section, we discuss\nthe MFT dynamics in the case, when the magnetic pressure\noutside the MFT is taken into account in pressure equilib-\nrium Equation ( 10), i.e.\nP+B2\n8π=Pe+B2e\n8π. (53)\nFirst, we discuss the dynamics of the MFT for the fidu-\ncial parameters r=0.15au,β0=1,a0=0.1H,z0=0.5H\n(see Section 3.1.1). In Figure 8, we plot the vertical profiles\nMNRAS 000,1–18(2018)12A. E. Dudorov, S. A. Khaibrakhmanov, A. M. Sobolev\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000014 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000016 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000019\nr/uni00000003/uni0000003e/uni00000044/uni00000058/uni00000040/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001b/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000015/uni00000014/uni00000011/uni00000017/uni00000014/uni00000011/uni00000019/uni00000014/uni00000011/uni0000001b/uni00000015/uni00000011/uni00000013˙Mb/uni00000003/uni0000003e10−8M⊙yr−1/uni00000040/uni0000000b/uni00000045/uni0000000c\n/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013/uni00000015/uni00000011/uni00000018\n˙Φb/uni00000003/uni0000003e1022Mx yr−1/uni00000040/uni00000014/uni00000013/uni00000013/uni00000014/uni00000013/uni00000014/uni00000014/uni00000013/uni00000015/uni00000014/uni00000013/uni00000016/uni00000014/uni00000013/uni00000017t/uni00000003/uni0000003e/uni00000047/uni00000040/uni0000000b/uni00000044/uni0000000c\nPk\ntgen\ntsurf\nFigure 7. Panel (a): dependences of Keplerian period Pk(grey\nline), MFT generaton time tgen(orange line) and MFT rise time\ntsurf(green line) on the r-coordinate. Panel (b): dependences of\nthe rates of vertical mass transport (black line, left y-axis) and\nmagnetic flux transport (grey line, right y-axis) due to buoyancy\non ther-coordinate. Initial parameters of the MFT: a0=0.1H,\nβ0=1. (colour figure online)\nof velocity, temperature and radius of the MFT (panels (a),\n(b) and (c), respectively), as well as dependence of its z-\ncoordinate, drag and buoyant forces, internal and external\ndensities on time (panels (d), (e) and (f), respectively). I n-\nsets in panels (a-c) show dependences of v,Tand aon time.\nSimulation is performed for the case of thermal equilibrium\nto eliminate effects of thermal oscillations.\nFigure8(a) shows that the MFT floats with increasing\nvelocity at the initial part of rise, z<2H(t<0.75tA).\nVelocity of the MFT reaches the value of 1.35 km s−1atz=\n2H. After that the MFT decelerates, and its velocity goes to\nzero at z=2.35H. Then the velocity becomes negative, and\nthe MFT starts to move downwards. Its velocity changes\nsign again at z=2H, and the MFT starts to move upwards.\nIn the following, the MFT experiences oscillatory motion up\nand down with respect to the point z≈2.15H. Maximum\nvalue of the velocity decreases during these oscillations, i.e.\nthe oscillations decay. Figures 8(b, c) show that the MFT\npulsatesintheprocess oftheoscillations, i.e. its temper ature\nand radius periodically change with respect to the values of\n860K and 0.25H, respectively. Period of the oscillations,\nPosc, equals 0.4tA.\nThe oscillations of the MFT are explained as follows.Figures8(e, f) demonstrate that at the initial part of the\ntrajectory, t<0.75tA, the MFT density is less than the ex-\nternal density, and the buoyant force is positive. Figures 8(c,\nd) show that the MFT expands and its magnetic field weak-\nens. Strength of the external magnetic field, Be, is assumed\nto be constant in our simulations. Internal magnetic field\nstrength Bbecomes less than Beat the point z=2.18H\n(t=0.8tA). Increasing magnetic pressure outside the MFT\ncauses ittoexpandslower thaninthecase ofrise withoutex-\nternal magnetic field (see Figure 3(f)). Density of the MFT\nis greater than the external density, and the buoyant force i s\nnegative, since Bis less than Bein this point. The MFT rise\nto the point z≈2.35Hby inertia, and then start to move\ndownwards. The MFT contracts during the downward mo-\ntion, i.e. its radius decreases, while the temperature, den sity\nand magnetic field strength increase. The buoyancy become\npositive again, and the MFT starts to move upwards after\nBexceeds Be. In the following, the MFT oscillates near the\npointofzerobuoyancy,thatisdeterminedbyequality B=Be\nin this case. We call these oscillations as magnetic ones.\nDrag force reduces the kinetic energy of the MFT, and\nleads to the decay of oscillations. Magnitude of the velocit y\noscillations reduces by factor of ten over five periods of os-\ncillations. The MFT radius approaches the value of 0.25H\nin this case.\nThe magnetic oscillation has one significant difference\nform the thermal ones discussed in Section 3.1.2. The ther-\nmal oscillations are caused by departure from thermal equi-\nlibrium. Therefore, they cease, and the MFT start to move\nupwards monotonically after the temperatures inside and\noutside the MFT become equal to each other. The magnetic\noscillations are caused by the action of the external mag-\nnetic pressure. In the considered case, constant Beprevents\nfurther rise of the MFT. Dynamics of the MFT is character-\nizedbydecayingoscillations near thepointof zerobuoyanc y.\nIn the case, when external magnetic field Bedecreases with\nheight, the MFT would float farther.\nWe performed the simulation for the same parameters\nas discussed above, but taking into account the radiative\nheat exchange. The simulations show that the picture of the\nMFT dynamics is practically the same, as in Figure 8, i.e.\ndeparture from thermal equilibrium does not influence the\nmagnetic oscillations.\nFigure9demonstrates dynamicsoftheMFTtakinginto\naccount external magnetic field for various initial radii an d\nplasma betas at r=0.2au.\nFigure9shows that the magnetic oscillations are ob-\nservedforallconsideredparameters.TheMFTwithstronger\nmagnetic field (small β0) experience oscillations at higher al-\ntitude. For example, the MFT with β0=0.1and a0=0.2H\n(orange lines) oscillate above the surface of the disc, near the\npoint z=2.5H. Oscillation period in this case is less than\nin the caseβ0=1(black lines), while magnitude of velocity\noscillations is higher, and magnitude of radius variations is\nsmaller.\nThe MFT with a smaller initial radius move slower, but\noscillate at about the same height as the tubes of a larger\nradius at the same initial field strength (compare black and\ngreen lines in Figure 9).\nWe carried out the simulations of MFT dynamics tak-\ning into account external magnetic pressure for a full set of\nparameters a0andβ0considered in Table 2and for various\nMNRAS 000,1–18(2018)Flux tube dynamics in accretion discs 13\b [km s -1]\n-101\nz [H]0,5 1 1,5 2 2,5(a)\nT[K]\n1 0001 5002 000\nz [H]0,5 1 1,5 2 2,5T\nTe(b)\na [H]\n0,10,20,30,4\nz [H]0,5 1 1,5 2 2,5(c)z [H]\n0,511,522,5B\tG\n050100150200\nt [tA]0 1 2 3 4 5\n( \u000b \f\nf [\nd\r n g-1]\n-1-0,500,51\nt [tA]0 1 2 3 4 5f\nfd\n(e)\n\u000e [g cm -3]\n1e-091e-081e-07\nt [tA]0 1 2 3 4 5\n\u000f\u0010e\n(f)0,10,150,20,250,30,35\nt [tA]0 1 2 3 4 5\n-1,5-1-0,500,511,5\nt [tA]0 1 2 3 4 55001 0001 5002 000\nt[tA]0 1 2 3 4 5\nFigure 8. Same as in Figure 3, but for the case, when effect of the external magnetic field is taken into account. Magnetic field strength\nof the MFT is depicted in panel (d) with the solid blue line (ri ghty-axis). The horizontal dashed blue line in panel (d) shows th e strength\nof external magnetic field. (colour figure online)\n/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013/uni00000015/uni00000011/uni00000018/uni00000016/uni00000011/uni00000013/uni00000016/uni00000011/uni00000018z/uni00000003/uni0000003eH/uni00000040/uni0000000b/uni00000044/uni0000000c\nβ0= 1/uni0000000f/uni00000003a0= 0.2H\nβ0= 0.1/uni0000000f/uni00000003a0= 0.2H\nβ0= 1/uni0000000f/uni00000003a0= 0.1H\n/uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018\nt/uni00000003/uni0000003e/uni00000050/uni00000052/uni00000051/uni00000057/uni0000004b/uni00000056/uni00000040/uni00000013/uni00000011/uni00000014/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000016/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001aa/uni00000003/uni0000003eH/uni00000040/uni0000000b/uni00000048/uni0000000c\n/uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018\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\n/uni00000013 /uni00000014 /uni00000015 /uni00000016 /uni00000017 /uni00000018\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−1/uni00000040/uni0000000b/uni00000045/uni0000000c\n/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001b∆ρ/uni00000003/uni0000003eρ0/uni00000040/uni0000000b/uni00000046/uni0000000c\nFigure 9. The dependences of z-coordinate (panel (a), horizontal dashed line is the surfa ce of the disc), velocity (panel (b)), density\ndifference (panel (c)), magnetic field strength (panel (d), h orizontal dashed line is the external magnetic field Be), radii (panel (e)) and\ntemperature T(panel (f), horizontal dashed line is the external temperat ure above the disc Ta) on time for the case when external\nmagnetic pressure is taken into account. Black line: β0=1anda0=0.2H, grey line: β0=0.1anda0=0.2H, green line: β0=1and\na0=0.2H. Distance to the star is r=0.2au. The Alfv´ en crossing time tA=0.94Pkand the Keplerian period Pk=2.7months for the\nadopted parameters. (colour figure online)\nr-coordinates in range [0.027,0.6]au. The simulations show\nthat the altitude, at which MFT oscillate, increases with in i-\ntial magnetic field strength of the MFT. The oscillations of\nthe MFT withβ0=1take place under the surface of the\ndisc, while the MFT with β0<1oscillate above the surface.\nTypical radii of the MFT are of order of 0.5H, so that the\noscillations of the MFT will lead to periodical changes of th edisc structure near its surface, in the region from z≈1.5H\ntoz≈3H.\nIn Figure 10, we plot the periods of MFT oscillations\nPofor various r-coordinates. The results are obtained for the\nMFT with initial radius a0=0.1H, coordinate z0=0.5H,\nplasma betaβ0=1(black line) and β0=0.1(grey line).\nFigure10shows that oscillation period increases with the\nMNRAS 000,1–18(2018)14A. E. Dudorov, S. A. Khaibrakhmanov, A. M. Sobolev\n/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000014 /uni00000013/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000016 /uni00000013/uni00000011/uni00000017 /uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000019\nr/uni00000003/uni0000003e/uni00000044/uni00000058/uni00000040/uni00000013/uni00000015/uni00000013/uni00000017/uni00000013/uni00000019/uni00000013/uni0000001b/uni00000013/uni00000014/uni00000013/uni00000013Posc/uni00000003/uni0000003e/uni00000047/uni00000040β0= 1\nβ0= 0.1\nFigure 10. Dependences of the period of the MFT’ oscillations\non ther-coordinate. Unit of time is day. Initial parameters of the\nMFT:a0=0.1H,z0=0.5H. Black line: β0=1, grey line: β0=0.1.\nradial distance from 1d≈0.6Pkatr=0.027au to 100d\n≈0.6Pkatr=0.6au in the caseβ0=1. The oscillation\nperiod of the MFT with β0=0.1are smaller and vary form\n0.5d at r=0.027au to 45d at r=0.6au.\nComparison of toroidal magnetic field generation and\nMFT rise times depicted in Figure 7(a) and oscillation pe-\nriods in Figure 10shows that Posc≪tgen. We conclude that\nthe dynamics of the toroidal magnetic field in the consid-\nered region proceeds in two stages: slow generation of the\ntoroidal magnetic field ( tgen=0.5−10yr) with subsequent\nfast rise and oscillations of the MFT ( Posc=1−100d).\n3.5 Comparison with observations\nAs it has been shown in Sections 3.1.2and3.4, thin MFT os-\ncillate near the surface of the disc after rising from interi or.\nDensity, radius and temperature periodically change durin g\nthe oscillations. The oscillations can lead tovariability of the\naccretion disc radiation. MFT contain both gas and refrac-\ntory dust particles in the considered region. Therefore, th e\noscillations can cause the IR-variabilityof the disc radia tion.\nBased on the resistive MHD simulations of the inner region\ndynamics of the minimum mass solar nebula, Turner et al.\n(2010) proposed that magnetic activity can lift dust grains\nintothediscatmosphere andcause theIR-variabilityofYSO\n(see also discussion in Flaherty et al. (2011)). In this paper,\nwe study similar effect in the frame of slender magnetic flux\ntube approximation.\nIR-variability has been observed in many YSO.\nFlaherty et al. (2016) have found that the stars in the\nChameleon I cluster exhibit variability on time-scales of\nmonths (20-200 days). Magnitude of the fluctuations ranges\nfrom 0.05 to 0.5 mag. In order to test the hypothe-\nsis that rising MFT cause the IR-variability of YSO, we\nperform the simulations of MFT dynamics in two discs\nof classical T Tauri stars from the sample presented in\nFlaherty et al. (2016). We have chosen stars J11092266-\n7634320 and J11100369-7633291 (indexed as 439 and 530,\nrespectively, in Table 1 from Flaherty et al. (2016)). Masses,\naccretion rates, luminosities and effective temperatures o fthese stars are given in columns 2-5 of Table 3. Stellar radii\nare estimated from the relation L⋆=σRT4\neff4πR2\n⋆(column\n6), where Teffis the stellar effective temperature, R⋆is the\nstellar radius. The periods of IR-variations, ∆t, measured by\nFlaherty et al. (2016) are listed in column 7. Column 8 gives\nvalues of considered r-distances from the star. For each star,\nwe simulate the MFT dynamics at two distances: ri(values\nwith symbol ‘i’ in brackets in column 8), where tempera-\nture is nearly equal to temperature of silicate dust grains\nevaporation ∼1500K (seePollack et al. 1994 ), and ro(val-\nues with symbol ‘o’ in brackets in column 8), determin-\ning the outer boundary of the region of thermal ionization.\nRadii riand robound the region of the efficient toroidal\nmagnetic field amplification, where formation of the MFT\nwith both gas and dust is possible. We take magnetic field\nstrength at the stellar surface to be 2 kG for both stars (see\nYang & Johns-Krull 2011 ). Using these stellar parameters,\nwe calculate the structure of the accretion discs using ac-\ncretion disc model of Dudorov & Khaibrakhmanov (2014).\nMidplane density, temperature, scale height and magnetic\nfield strength of the discs of stars 439 and 530 are presented\nin columns 9-12 of Table 3.\nThe simulations are carried out for β0=[0.1,1]and\na0=0.01H÷0.4H. The simulations show that oscillation\nperiods Pweakly depend on the MFT radius. As an ex-\nample, in Figure 11we plot dependences of z-coordinate ,\nradius aand temperature Ton time for the MFT in the disc\nof star 439, at riand ro. Initial MFT parameters: z0=0.5H,\na0=0.1H,β0=1. Figure 11shows that dynamics of the\nMFT is similar to that considered in Section 3.4. The MFT\nrise and then oscillate near the surface of the disc. Oscilla -\ntion period at ri(20 d) is less than the oscillation period\natro(70 d). The MFT experience temperature pulsations\naround value T∼700K at riand around T∼500K at\nro. With these temperatures, maximum of emission peaks\nat wavelengthsλ≈ (4−6)µm, according to Wien’s displace-\nment law. Maximal temperature variations are 100K at ri\nand200K at ro.\nThe dependences of oscillation periods Poscon the r-\ndistance for each star are shown in Figure 12. The results\nare obtained for a0=0.1H,z0=0.5H, plasma betaβ0=1\n(star 439) andβ0=5(star 530). Star 530 has small accretion\nrate, and the region of thermal ionization is situated close to\nthe star, r<0.1au. The disc of star 439 has more extended\nregion of thermal ionization, r<0.42au. In both cases, the\noscillation periods increase with distance from the star.\nFor the adopted MFT parameters, the range of oscilla-\ntion periods is in agreement with the observational values o f\nvariability times. Observed period for star 439, P439=32d\ncorresponds to oscillation of the MFT at r≈0.25au for\nβ0=1. Period of star 530, P530=35d, corresponds to the\nMFT oscillating at r=0.1au forβ0=5. Thus, our simula-\ntions confirm the hypothesis that rising magnetic fields can\nbe a source of the IR-variability of YSO.\nTo determine whether the MFT can contribute to IR\nradiation of considered T Tauri stars or not, we estimate\noptical depthof the MFT with respect toIR radiation as τ=\n2aρκ, where opacityκis determined as a function of density\nand temperature according to Dudorov & Khaibrakhmanov\n(2014). We get thatτhas the values between 20 and 70\nduring oscillations at r=rofor star 439. Optical thickness\nMNRAS 000,1–18(2018)Flux tube dynamics in accretion discs 15\nTable 3. Parameters of YSO in Chameleon.\nNoM⋆[M⊙]∝dotaccM[M⊙yr−1]L⋆[L⊙]Teff[K]R⋆[R⊙]P[d]r[au]ρm[g cm−3]Tm[K]Tirr[K]H[au]Be[G]\n(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)\n439 0.6 4.8×10−80.8 3669 2.2 32 0.2 (i) 1.3×10−81590 590 0.007 18.0\n0.42 (o) 2.0×10−91200 410 0.019 5.0\n530 0.63 2×10−90.66 3955 1.7 35 0.07 (i) 1.6×10−81460 950 0.0013 20.4\n0.1 (o) 5.6×10−91220 800 0.0024 4.0\n/uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013\nt/uni00000003/uni0000003e/uni00000050/uni00000052/uni00000051/uni00000057/uni0000004b/uni00000056/uni00000040/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018/uni00000015/uni00000011/uni00000013/uni00000015/uni00000011/uni00000018z/uni00000003/uni0000003eH/uni00000040/uni0000000b/uni00000044/uni0000000c\nri= 0.2/uni00000003/uni00000044/uni00000058\nro= 0.42/uni00000003/uni00000044/uni00000058\n/uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013\nt/uni00000003/uni0000003e/uni00000050/uni00000052/uni00000051/uni00000057/uni0000004b/uni00000056/uni00000040/uni00000013/uni00000011/uni00000014/uni00000013/uni00000013/uni00000011/uni00000014/uni00000018/uni00000013/uni00000011/uni00000015/uni00000013/uni00000013/uni00000011/uni00000015/uni00000018/uni00000013/uni00000011/uni00000016/uni00000013/uni00000013/uni00000011/uni00000016/uni00000018a/uni00000003/uni0000003eH/uni00000040/uni0000000b/uni00000045/uni0000000c\n/uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013\nt/uni00000003/uni0000003e/uni00000050/uni00000052/uni00000051/uni00000057/uni0000004b/uni00000056/uni00000040/uni00000017/uni00000013/uni00000013/uni00000019/uni00000013/uni00000013/uni0000001b/uni00000013/uni00000013/uni00000014/uni00000013/uni00000013/uni00000013/uni00000014/uni00000015/uni00000013/uni00000013/uni00000014/uni00000017/uni00000013/uni00000013/uni00000014/uni00000019/uni00000013/uni00000013T/uni00000003/uni0000003eK/uni00000040/uni0000000b/uni00000046/uni0000000c\nFigure 11. The dependences of the coordinate (panel (a)), radius (pane l (b)) and temperature (panel (c)) on time for the MFT with\nβ0=1,a0=0.1H,z0=0.5Hin the disc of star 439 in Cha-1. Black line: r=ri, grey line: r=ro(see Table 3).\n/uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000014/uni00000013 /uni00000013/uni00000011/uni00000014/uni00000018 /uni00000013/uni00000011/uni00000015/uni00000013 /uni00000013/uni00000011/uni00000015/uni00000018 /uni00000013/uni00000011/uni00000016/uni00000013 /uni00000013/uni00000011/uni00000016/uni00000018 /uni00000013/uni00000011/uni00000017/uni00000013 /uni00000013/uni00000011/uni00000017/uni00000018\nr/uni00000003/uni0000003e/uni00000044/uni00000058/uni00000040/uni00000013/uni00000014/uni00000013/uni00000015/uni00000013/uni00000016/uni00000013/uni00000017/uni00000013/uni00000018/uni00000013/uni00000019/uni00000013/uni0000001a/uni00000013/uni0000001b/uni00000013Posc/uni00000003/uni0000003e/uni00000047/uni00000040/uni00000017/uni00000016/uni0000001c/uni0000000f/uni00000003a0= 0.1H/uni0000000f/uni00000003β0= 1\n/uni00000017/uni00000016/uni0000001c/uni0000000f/uni00000003/uni00000029/uni00000014/uni00000019\n/uni00000018/uni00000016/uni00000013/uni0000000f/uni00000003a0= 0.1H/uni0000000f/uni00000003β0= 5\n/uni00000018/uni00000016/uni00000013/uni0000000f/uni00000003/uni00000029/uni00000014/uni00000019\nFigure 12. The dependence of MFT oscillation periods on the\nr-coordinate in the discs of star 439 (black line), and star 53 0\n(grey line, see Table 3). Black and grey dashed horizontal lines\nshow corresponding observed periods from Flaherty et al. (2016).\nfor star 530 is of the same order. We conclude that the MFT\nare very optically thick.\nThere may be two possible processes causing the vari-\nability. First, MFT temperature fluctuations (like in Fig-\nure11) can cause the radiation variability. Second, the oscil-\nlating MFT can intercept stellar radiation and periodicall y\ncast a shadow on the outer disc regions. Stellar radiation\nreprocessed by the dust in outer disc regions will have an\noscillating IR-component.4 CONCLUSION AND DISCUSSION\nWe investigated numerically the dynamics of slender MFT\nin the accretion discs of T Tauri stars. We considered the\nMFT forming in the region of intense generation of the\ntoroidal magnetic field due to magnetic buoyancy instabil-\nity. We formulate the equations of slender MFT dynamics\ntaking into account the aerodynamic and turbulent drags.\nKhaibrakhmanov et al. (2018) have investigated the MFT\ndynamics taking into account the heat exchange with the\nexternal medium of constant temperature. The radiative\nflux has been estimated in the diffusion approximation. In\npresent work, we investigated additionally influence of the\nmagnetic field of the disc on the MFT dynamics. The verti-\ncal structure of the disc was calculated using the polytropi c\nequation of state. In particular, the thermal and magnetic\noscillations of the MFT was investigated in this paper.\nSimulations were performed for initial radii of the MFT\na0=0.01H÷0.4Hand radial distances r=0.027÷0.6au in\nthe accretion disc of solar mass T Tauri star. Slender flux\ntube approximation allows us to investigate the dynamics\nof ‘soft’ MFT with weak magnetic field ( β/greaterorsimilar1) and ‘stiff’\nMFT with strong magnetic field ( β<1). Latter case cannot\nbe considered using multidimensional numerical simulatio ns\nin frame of classical MHD because of strict limitations on\nthe time step. For comparison, we investigated dynamics of\nthe MFT with plasma betas β0=0.01÷10. The structure of\nthe accretion disc was calculated with the help of the model\nofDudorov & Khaibrakhmanov (2014).\nFirst of all, we considered the dynamics of the MFT in\nabsence of external magnetic field. In this case, we found two\nregimes of the MFT dynamics. Thin MFT with initial radius\na0/lessorequalslant0.05Hand thick MFT with a0/greaterorequalslant0.16Hrapidly accel-\nerate, then rise with slowly increasing velocity, decelera te a\nMNRAS 000,1–18(2018)16A. E. Dudorov, S. A. Khaibrakhmanov, A. M. Sobolev\nlittle and acquire nearly steady velocity above the surface of\nthe disc. The MFT of intermediate radii a0∼0.1Hexperi-\nence thermal oscillations during some time after rising fro m\nthe disc. The oscillations are due to adiabaticity and slow\nheat exchange with the external gas. These oscillations are\nfound only for the MFT with β0=1formed at r-distances\nless than 0.2au. After radiative heat exchange equalizes in-\nternal and external temperatures, the oscillations decay a nd\nthe MFT continue to move upwards above the disc.\nWe studied dependence of the MFT velocity, mass and\nmagnetic flux on its initial parameters. Velocity of the MFT\nincreases with initial radius and magnetic field strength.\nTypical rise velocities are of several km s−1. The MFT\nwith weak magnetic field ( β0=10) have velocities of 0.05−\n0.5 km s−1. The MFT with plasma β=1accelerate to veloc-\nity≈0.2−4km s−1comparable with sound speed, in agree-\nment with analytical estimates of Khaibrakhmanov et al.\n(2018). The MFT with strong initial magnetic field (plasma\nbeta<1) can reach supersonic velocity up to ∼10−15km\ns−1. In this case, accurate investigation of MFT supersonic\nmotion with bow shocks can be done by including depen-\ndence of the aerodynamic drag on the Mach number. We\nplan to do this in future works.\nThe dynamics of the MFT consists in two stages. At\nthe first stage, toroidal magnetic field is generated and MFT\nform due to magnetic buoyancy instability over time tgen. At\nthe second stage, the MFT rise from the disc to its suface\nover the time tsurf. In a process of further upward motion\nthe MFT carry away some mass and magnetic flux to disc\natmosphere. Our calculations show that tsurf≪tgen. There-\nfore, the process of mass and magnetic flux transport from\nthe disk to its atmosphere due to buoyancy is periodic with\ntypical period tgen∼0.5−10yr. The vertical mass transport\nrate due to buoyancy ∝dotaccMb∼10−12−10−7M⊙yr−1. Our calcu-\nlations show that approximately 20% of disc mass flux can\ncome outfrom thedisc viabuoyancy.The rising MFTcan be\ntheseedfor theformation ofjets andoutflows from accretion\ndiscs. The MFT carry magnetic fluxes Φb∼1019−1024Mx,\nso that magnetic flux of Φb≈1030Mx, that comprises 20%\nof total disc magnetic flux, can be carried away from the\ndisc by rising MFT during period of 1Myr. We assume that\nmagnetic buoyancyis the mechanism responsible for the effi-\ncient magnetic flux escape from the accretion discs of young\nstars.Khaibrakhmanov & Dudorov (2017) have shown that\nformation and rise of the MFT with initial radius a0=0.1H\nstabilize the strength of the toroidal magnetic field at the\nlevel of the poloidal magnetic field strength.\nThe buoyancy decreases and MFT acquires steady\nspeed above the disc. This is explained by the fact that the\nexternal density ρeand the density of MFT decrease with\nz-coordinate. Correspondingly, the buoyant and drag forces\nalso reduce, acceleration of MFT approaches to zero, MFT\nmoves by inertia and significantly expands above the disc.\nWe interpret this process as a formation of an expanding\nmagnetized ‘corona’ above the disc. Expansion of the MFT\ncan be reduced by the internal magnetic tension. We will\nconsider this possibility in future works.\nSecond important case investigated by us concerns the\neffect of the magnetic field of the disc on the dynamics of\nMFT. In this case, MFT rise from the disc and start to oscil-\nlate at height zosc≈2−2.5Habove the midplane of the disc.\nThe magnetic oscillations take place near the point wherestrengths of internal, B, and external, Be, magnetic fields\nare nearly equal. Above this point, the buoyant force is neg-\native since B<Beand the MFT has greater density than the\nexternal gas. The oscillation periods Poscincrease with the\ndistancefromthestar, Posc=(1−10)daysat r=0.027auand\nPosc=(1−3)months at r=0.6au. The MFT with stronger\ninitial magnetic field (smaller plasma beta) rise to higher a l-\ntitudes and experience oscillations with smaller periods. The\nmagnetic oscillations decay with time, and occur in the disc\nover time of the toroidal magnetic field generation. The os-\ncillations of the MFT found in our simulations indicate that\nthe mass carried by the MFT rsing from the disc can come\nback to the disc. Accumulation of the magnetic flux near\nthe surface of the discs can further lead to the burst release\nof the magnetic energy as it was suggested by Shibata et al.\n(1990). Magnetic energy of the MFT of ∼1033−1037erg\ncan be released due to such bursts. Magnetic field Becan\ndecrease with height, and the MFT probably can rise to\nhigher altitudes before reaching the point of zero buoyancy\nin this case.\nPeriodic formation, rise and thermal or magnetic oscil-\nlations of the MFT can be interpreted as periodical changes\nin the accretion disc structure. This process is similar to\nperiodic ejection of buoyant magnetic fields from the disc\nthat has been found in frame of numerical MHD simula-\ntions by (e.g. Turner et al. 2010 ;Takasao et al. 2018 ). At\nthe distance r=0.6au, temperature is ∼1000K, and the\nrefractory dust grains are present in this region. Therefor e,\nbuoyantMFT contain dust, and the oscillations can produce\nIR-variability of the disc.\nIR-variability is the common feature of YSO (see\nFlaherty et al. (2013);Flaherty et al. (2016) and references\ntherein). Flaherty et al. (2016)havefoundthatthelow-mass\nYSO in the Chameleon I star forming region exhibit vari-\nability over time scales of months (20-200 days). Classical\nT Tauri stars also exhibit variability in optical wavelengt hs.\nFor example, Rigon et al. (2017) reported the detection of\nthe optical variability with periods of 20−60d of classical\nT Tauri stars in Taurus-Auriga region.\nWe simulated MFT dynamics in the discs of two\nclassical T Tauri stars J11092266-7634320 and J11100369-\n7633291 (439 and 530) from the sample presented in\nFlaherty et al. (2016). The magnetic oscillation periods\nfound in our simulations are in good agreement with the\nobserved periods for considered stars. Our simulations hav e\nshown that variability period of star 439 (32 days) is con-\nsistent with the oscillations of the MFT with initial β0=1\nand a0=0.1H. Observed period for star 530 (35 days) can\nbe explained by the oscillations of the MFT with a0=0.1H\nandβ0=5.\nOur estimations have shown that the MFT are opti-\ncally thick with respect to IR-radiation. The variability c an\nbe caused by MFT temperature fluctuations, as well as by\nperiodical screening the outer disc regions from stellar ra -\ndiation by the MFT. It should be noted that Takasao et al.\n(2018) found formation of the MFT in 3D MHD simulations\nof the inner regions of the accretion discs and also argued\nthat periodically rising MFT may contribute to the variabil -\nity of discs radiation.\nThe MFT can form in the region ri<r<roau\n(ri=r(T=1500 K),rois the outer boundary of the zone\nof the thermal ionization), where magnetic field is efficientl y\nMNRAS 000,1–18(2018)Flux tube dynamics in accretion discs 17\namplified and temperature is lower than 1500 K, so that\nthe MFT contain the refractory dust particles. For stars 439\nand 530 radii ri=0.2au,0.07au and ro=0.42au,0.1au,\nrespectively. Therefore, the oscillations occur in a range of\nradii, which is consistent with conclusions of Flaherty et al.\n(2016). Magnetic oscillations decay with time. For example,\nmagnitude of temperature oscillations reduces by 100K over\n1.3months at ri=0.2and over 4.7months at ro=0.42in\nthe disc of star J11092266-7634320. Thus, significant tem-\nperature variations of oscillating MFT, ∆T∼100K, can be\ndetected within time of 200d corresponding to observation\ntime inFlaherty et al. (2016).\nTambovtseva & Grinin (2008) suggested that inhomo-\ngeneities in the disc centrifugal winds containing both gas\nand dust can cause variations in circumstellar extinction o b-\nserved in T Tauri stars. Our simulations show that magnetic\nbuoyancy can transport mass from the accretion disc to its\natmosphere. We propose that such this process can be re-\nsponsible for the variations in circumstellar extinction o b-\nserved in T Tauri stars. Similar hypothesis was proposed\nbyMiyake et al. (2016), who simulated the time evolution\nof the dust grain distribution in the vertical direction ins ide\nthe minimum mass solar nebula.\nAsSchramkowski (1996) have shown, longitudinal per-\nturbations of the magnetic flux tubes can lead to the forma-\ntion of thearcs above the disc. We assume that such orbiting\narcs can cast shadows on the outer regions of the accretion\ndiscs. This problem needs further investigation.\nToroidal magnetic field in the disc is generated from the\npoloidal one over the time scale of rotation period. Tension\nof the poloidal magnetic field lines can slow down the disc\nrotation, and, as a consequence, hinder the generation of th e\ntoroidal magnetic field in the considered region. This effect\ntakes place if the energy of the poloidal magnetic field is\ncomparable with the rotational energy. This condition for\nthe keplerian disc can be written as\nβ∼2×10−4/parenleftbiggH/r\n0.01/parenrightbigg2\n, (54)\ni.e. the magnetic field with β∼10−4will influence signifi-\ncantly the disc rotation. Kinematic approximation β>1is\nused in our model of the accretion disc, and the poloidal\nmagnetic field cannot be strong enough to stop generation\nof the toroidal magnetic field and formation of the MFT.\nWe did not consider the effect of the azimuthal veloc-\nity shear on the MFT dynamics. Shibata et al. (1990) have\nfound that rise velocity of the MFT in the presence of ve-\nlocity shear is smaller than in the case of no shear. MHD\nsimulations of MFT rising from the upper convection zone\nof the Sun to the solar atmosphere have shown that the\nMFT dynamics is sensitive to the twist of the MFT (see\nFan et al. 1998 ;Magara 2001 ;Mart´ ınez-Sykora et al. 2015 ).\nStrongly twisted MFT retain their coherent structures dur-\ningthe rise, while the MFT withouttwist splits intoavortex\npair and lose significant amount of their magnetic flux. Up\nto date, there are no detailed simulations of the MFT for-\nmation in the accretion discs. It is hard to make conclusions\nabout degree of their twist. In our simulations we implicitl y\nassumed that the MFT retain their coherence and do not\nlose the magnetic flux. Probably, loss of the magnetic flux\nwould lead to decrease of the MFT velocity.\nThe magnetic pressure acts only in the direction per-pendicular to the magnetic lines. Uniform magnetic field of\nthe disc Beconsidered in this work will lead to flattening\nof the MFT in the direction of motion. Generally speaking,\nthe toroidal magnetic field is generated not only in the re-\ngion of MFT formation, but in the entire volume of the disc.\nTherefore, toroidal magnetic field of the disc will influence\nMFT dynamics together with the poloidal magnetic field.\nAssuming that intensities of the toroidal and poloidal mag-\nnetic fields are comparable, we adopted in this work that\nthe magnetic pressure is isotropic, and the cross-section o f\ntheMFT remains nearly round.Effects ofnon-uniformMFT\nexpansion can be investigated in two-dimensional model of\nthe MFT dynamics, that we aim to elaborate in future. We\nplan to develop more detailed model of the vertical structur e\nof the disc in the future. Interesting task is investigation of\nthe dynamics of the magnetic rings in the accretion discs of\nyoung stars, i.e. study the evolution of major radius of the\ntoroidal MFT.\nACKNOWLEDGEMENTS\nThe work of AED is supported by the Russian Founda-\ntion for Basic Research (project 18-02-01067). The work of\nSAKh and AMS is supported by the Ministry of Science\nand High Education (the basic part of the State assign-\nment, RK No. AAAA-A17-117030310283-7) and by the Act\n211 Government of the Russian Federation, contract No.\n02.A03.21.0006. The authors thank prof. Dmitry Bisikalo,\nDr. Sergey Parfenov, Dr. Anna Evgrafova and Dr. Vitaly\nAkimkin for useful comments. We are also grateful to Lyud-\nmila Lapina for checking the English language in the paper.\nWethankanonymousreferee for adetailed review anduseful\ncomments.\nREFERENCES\nAchterberg A., 1996, A&A, 313, 1008\nAgapitou V.,Papaloizou J. C. B.,1996, Astrophysical Lette rs and\nCommunications, 34, 363\nBisnovatyi-Kogan G. 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In particular, their dynamics under\nexternal \felds is an issue of vital importance both for fundamental science and\nfor technical application. Whereas observations of magnetic skyrmions had been\nlimited to metallic magnets so far, their realization was discovered also in a\nchiral-lattice insulating magnet Cu 2OSeO 3in 2012. Skyrmions in the insulator\nturned out to exhibit multiferroic nature with spin-induced ferroelectricity.\nStrong magnetoelectric coupling between noncollinear skyrmion spins and electric\npolarizations mediated by relativistic spin-orbit interaction enables us to drive\nmotion and oscillation of magnetic skyrmions by application of electric \felds\ninstead of injection of electric currents. Insulating materials also provide an\nenvironment suitable for detection of pure spin dynamics through spectroscopic\nmeasurements owing to absence of appreciable charge excitations. In this article,\nwe review recent theoretical and experimental studies on multiferroic properties\nand dynamical magnetoelectric phenomena of magnetic skyrmions in insulators.\nWe argue that multiferroic skyrmions show unique coupled oscillation modes of\nmagnetizations and polarizations, so-called electromagnon excitations, which are\nboth magnetically and electrically active, and interference between the electric\nand magnetic activation processes leads to peculiar magnetoelectric e\u000bects in a\nmicrowave frequency regime.\nKeywords : skyrmion, multiferroics, magnetoelectirc coupling, electromagnonarXiv:1511.07761v1  [cond-mat.str-el]  24 Nov 2015Dynamical magnetoelectric phenomena of multiferroic skyrmions 2\nFigure 1. (color online). (a) Schematic of the original\nhedgehog-type skyrmion. Its magnetizations point in all\ndirections wrapping a sphere. (b) Schematic of the vortex-\ntype skyrmion discovered in chiral-lattice ferromagnets, which\ncorresponds to a projection of the hedgehog-type skyrmion\nonto a two-dimensional plane. Its magnetizations also point\neverywhere wrapping a sphere. (c) Schematic of a skyrmion\ncrystal realized in chiral-lattice ferromagnets under an external\nmagnetic \feld, in which skyrmions are hexagonally packed to\nform a triangular lattice. (d) Helical spin structure (or proper\nscrew spin structure) with magnetizations rotating within a\nplane normal to propagation vector q.\n1. Introduction\n1.1. Magnetic Skyrmions\nNoncollinear spin textures in magnets often host\nintriguing physical phenomena and useful device\nfunctions through coupling to charge degree of\nfreedom, and thereby have been subject to intensive\nstudies. For instance, it is known that magnetic\ndomain walls and magnetic vortices can be driven by\nelectric currents via a spin transfer torque mechanism,\nwhich points to their application for spin-electronics\ndevices such as race track memory. Multiferroics\nwith magnetically-induced ferroelectricity is another\ntypical example [1, 2, 3, 4, 5, 6]. Magnetic spirals in\ninsulating magnets with frustrated interactions often\ninduce asymmetry of charge distribution and resultant\nferroelectric polarization Pvia relativistic spin-orbit\ninteractions [7, 8] in compounds such as TbMnO 3.\nThe magnetoelectric coupling between spins and\nelectric polarizations gives rise to rich cross-correlation\nphenomena such as magnetic-\feld switchings of\nferroelectric polarization [9], and electric-\feld controls\nof spin chiralities [10, 11], magnetic modulation\nvectors [12], and magnetic domain distributions [13].\nIn addition to these magnetic structures, magneticskyrmions as nanometric spin whirls are recently\nattracting intensive research interest [14, 15, 16].\nSkyrmion was originally proposed by Tony Skyrme in\n1960s to account for stability of baryons in particle\nphysics as a topological solution of a nonlinear sigma\nmodel in three dimensions [17, 18]. In the original\nsense, a magnetic skyrmion comprises spins pointing in\nall directions wrapping a sphere similar to a hedgehog\nas shown in Fig. 1(a).\nLater Bogdanov and his collaborators theoreti-\ncally predicted that skyrmions should appear in ferro-\nmagnets without spatial inversion symmetry as vortex-\nlike spin structures shown in Fig. 1(b), which cor-\nresponds to a projection of the original hedgehog\nskyrmion onto a two-dimensional plane [19, 20, 21].\nThey also predicted that skyrmions are often crystal-\nlized to form a triangular lattice as shown in Fig. 1(c),\nwhich is called skyrmion crystal.\nMagnetic interactions between two neighboring\nmagnetizations in magnets are mostly categorized into\ntwo types, i.e., mi\u0001mjtype symmetric exchange\ninteractions and mi\u0002mjtype Dzyaloshinskii-Moriya\ninteractions. In magnets with broken inversion\nsymmetry, there exists a \fnite net component of the\nDzyaloshinskii-Moriya interaction [22, 23], which is\ngiven in a continuum from as,\nHDM/Z\ndrM\u0001(r\u0002M); (1)\nwhereMis a classical magnetization vector. This in-\nteraction favors noncollinear alignment of magnetiza-\ntions with 90\u000erotation with \fxed handedness (spin\nchirality) and, thus, competes with the ferromagnetic\nexchange interaction which favors parallel (collinear)\nalignment of magnetizations. Consequently, a ground\nstate of these chiral-lattice ferromagnets in the ab-\nsence of external magnetic \feld is a helical state, so-\ncalled proper screw state, in which magnetizations ro-\ntate within a plane normal to the propagation vector\nqas shown in Fig. 1(d). An increase of the mag-\nnetic \feld at certain temperatures changes the helical\nphase to a skyrmion-crystal phase and eventually to\na \feld-polarized ferromagnetic phase. A size of such\nskyrmions is typically 5-100 nm, which is determined\nby the ratio between strengths of the ferromagnetic\ninteraction and the Dzyaloshinskii-Moriya interaction.\nThe size becomes smaller for stronger Dzyaloshinskii-\nMoriya interaction.\nA skyrmion is characterized by a topological\ninvariantQ, so-called skyrmion number, which\nrepresents how many times the magnetizations wrap\na sphere. The skyrmion number Qis de\fned as,Z\nd2r\u0012@^n\n@x\u0002@^n\n@y\u0013\n\u0001^n=\u00064\u0019Q; (2)\nwhere ^nis a unit vector pointing in the local\nmagnetization direction. The left-hand side of thisDynamical magnetoelectric phenomena of multiferroic skyrmions 3\nFigure 2. (color online). (a) Crystal structure of MnSi with\nchiral cubic P2 13 symmetry. (b) Chiral crystal structure of MnSi\nviewed along the [111] direction.\nequation represents a sum of solid angles spanned\nby three neighboring magnetizations. Because\nthe magnetizations in a skyrmion point everywhere\nwrapping a sphere once, its value becomes +4 \u0019or\n\u00004\u0019. The sign is determined by the magnetization\norientation at the core, that is, Q= +1 (Q=\u00001) for\ncore magnetizations pointing upward (downward).\nThis \fnite topological number indicates that\nskyrmions belong to a topological class distinct from\nnon-topological classes, to which usual magnetic\nstructures such as ferromagnetic states, helices and\ndomain walls belong. This means that we cannot\ncreate a skyrmion starting from uniformly magnetized\nferromagnetic state or annihilate a skyrmion through\ncontinuous modulation of the spatial magnetization\nalignment. Instead, a local \rop of magnetization is\nrequired to create or annihilate a skyrmion, which\ncosts rather large energy whose order is determined\nby the exchange interaction. Owing to this topological\nprotection, skyrmion spin textures are very robust.\n1.2. Skyrmions in metallic B20 compounds\nIn 2009, the realization of skyrmion crystals was\ndiscovered in a metallic B20 compound MnSi under\nan external magnetic \feld by the small-angle neutron\nscattering (SANS) measurements [24]. Subsequently,\nreal-space images of skyrmion spin structures and\nhexagonal skyrmion crystals were obtained by a\nLorentz transmission electron microscopy (LTEM)\nfor thin-\flm samples of Fe 1\u0000xCoxSi [25]. The\nskyrmion-crystal phase was also observed in other\nB20 compounds such as FeGe and Mn 1\u0000xFexGe by\nSANS [26, 27, 28, 29, 30] and LTEM [31, 32, 33, 34].\nThese compounds have a chiral crystal structure [see\nFigs. 2(a) and (b)], which belongs to the cubic P2 13\nspace group. In these materials, triangular skyrmion\ncrystals appear on a plane normal to the applied\nmagnetic \feld. Magnetizations in each skyrmion\npoint antiparallel to the magnetic \feld at the center\nFigure 3. (color online). Three-dimensional structure of\nmagnetic skyrmion crystal where the skyrmion spin structures\nare stacked to form vortex tubes.\nand gradually rotate upon propagating along the\nradial directions towards the periphery at which the\nmagnetizations are parallel to the magnetic \feld. Such\ntwo-dimensional vortex-like textures are stacked to\nform tube-like structures as shown in Fig. 3.\nShown in Fig. 4(a) is the experimental phase\ndiagram for bulk samples of MnSi in plane of\ntemperature Tand magnetic \feld B[24]. The\nskyrmion-crystal phase is positioned in a tiny window\nofTandBon the verge of the phase boundary between\nthe paramagnetic and the helical (longitudinal conical)\nphases. This indicates that the skyrmion-crystal phase\nis rather unstable in the bulk samples. However\nthe stability of the skyrmion-crystal state essentially\ndepends on the dimension of the system, and it attains\ngreater stability when the sample becomes thinner [31,\n35, 36, 37]. Figure 4(b) displays the experimental phase\ndiagram for thin-\flm samples of MnSi [32]. The area\nof the skyrmion-crystal phase noticeably spreads over\na wideT-Brange, and is realized even at the lowest\ntemperatures.\nThe enhanced stability of skyrmion crystal in thin-\n\flm samples can be understood as follows. When\na magnetic \feld Bis applied to a bulk sample, the\nconical spin structure propagating parallel to Bwith\nuniform magnetization component due to the spin\ncanting towards the Bdirection is stabilized owing\nto energy gains from both the Dzyaloshinskii-Moriya\ninteraction and the Zeeman coupling. The skyrmion-\ncrystal state is usually higher in energy than this\nconical state. However, when the sample thickness\nbecomes comparable to or thinner than the conical\nperiodicity, the conical state can no longer bene\ft\nfrom the energy gain of the Dzyaloshinskii-Moriya\ninteraction, and thus is destabilized. Instead theDynamical magnetoelectric phenomena of multiferroic skyrmions 4\nFigure 4. (color online). Experimental phase diagrams of MnSi\nin plane of temperature Tand magnetic \feld Bfor (a) bulk\nsamples [24] and (b) thin-\flm samples [32]. For the bulk sample,\nthe skyrmion-crystal phase appears only in a tiny region inside\nthe conical phase at \fnite TandBon the verge of the boundary\nto the paramagnetic phase. In contrast, the skyrmion-crystal\nphase spreads over a wide T-Brange for thin-\flm samples.\nThe enhanced stability of skyrmion-crystal phase is attributed\nto destabilization of the longitudinal conical phase in thin-\flm\nsamples (see text). Inset in (a) shows the magnetic structure of\nthe longitudinal conical state.\nskyrmion-crystal state attains relative stability against\nthe conical state. It has been argued that the uniaxial\nanisotropy, inhomogeneous chiral modulations, and the\ndipolar interaction can also stabilize skyrmions in thin-\n\flm samples [38, 39, 40, 41, 42, 43].\nSkyrmions in chiral-lattice ferromagnets appear\nnot only in the crystallized form but also as isolated\ndefects in the ferromagnetic state [25]. Such\nskyrmion defects are also stable because of the\ntopological protection, and behave as particles since\ntheir spin textures are closed within a nanometre-\nscale region with peripheral magnetizations parallel\nto those of the outside ferromagnetic background.\nThe isolated skyrmion defects are attracting a great\ndeal of interest because they have turned out\nto possess numerous advantageous properties for\ntechnical application to information carriers for high-\ndensity and low-energy-consuming magnetic memories,\nFigure 5. (color online). (a) Crystal structure of the chiral-\nlattice magnetic insulator Cu 2OSeO 3, which belongs to the\nchiral cubic P2 13 space group. (b) Magnetic structure of\nCu2OSeO 3composed of tetrahedra of four Cu2+ions (S=1/2)\nwith three-up and one-down spins. (Reproduced from Ref. [60].)\nthat is, (1) topologically protected robustness, (2)\nsmall nanometric size, (3) rather high transition\ntemperatures, and (4) ultra-low energy costs for\ndriving their motion. Concerning the last property,\nit was experimentally revealed that one can drive their\nmotion by injection of electric currents via the spin-\ntransfer torques, and its threshold current density is\n\fve or six orders of magnitudes smaller than that for\nother noncollinear magnetic textures such as domain\nwalls and helical structures [44, 45, 46, 47]. Subsequent\ntheoretical study attributed this high mobility or\nnearly pinning-free motion of skyrmions to their\nparticle-like nature and \fnite topological number [48,\n49, 50]. Skyrmions in metallic system have attracted\nresearch interest also for intriguing electron-transport\nphenomena due to the emergent electromagnetic \feld\ninduced by their topological nature [51, 52, 53, 54,\n55, 56, 57, 58, 59]. The skyrmion number directly\ncorresponds to the gauge \rux through the quantum\nBerry phase which gives rise to the topological Hall\ne\u000bect of conduction electrons.\n1.3. Skyrmions in magnetic insulator Cu 2OSeO 3\nWhile the observations of skyrmions in the early\nstage had been limited to speci\fc metallic magnets\nwith chiral B20 structure such as MnSi, FeGe, and\nFe1\u0000xCoxSi, formation of the skyrmion crystal was\ndiscovered also in an insulating magnet Cu 2OSeO 3via\nthe Lorentz transmission electron microscopy for thin-\n\flm samples [60] and the small angle neutron scattering\nexperiments for bulk samples [61, 62].\nThe crystal structure of Cu 2OSeO 3belongs to\nthe chiral cubic P2 13 space group, which is equivalent\nto the B20 compounds [see Fig. 5(a)]. However\nthe coordination of atoms in Cu 2OSeO 3is quite\ndi\u000berent from that in the B20 compounds. There\nare two inequivalent Cu2+sites with di\u000berent oxygen\ncoordination. One is surrounded by a square pyramid\nof oxygen ligands, while the other is surrounded by\na trigonal bipyramid. The nominal ratio betweenDynamical magnetoelectric phenomena of multiferroic skyrmions 5\nFigure 6. (color online). Experimental T-Bphase diagrams of\nthe copper oxoselenite Cu 2OSeO 3for (a) bulk samples and (b)\nthin-\flm samples with sample thickness of \u0018100 nm. In spite of\nthe di\u000berent origin of magnetism between metallic and insulating\nmagnets, the phase diagrams of Cu 2OSeO 3are similar to those\nof MnSi shown in Fig. 4. Whereas the skyrmion-crystal phase\nis restricted to a narrow T-Bwindow just below the magnetic-\nordering temperature for the bulk samples, it spreads over the\nwide area and even to the lowest temperatures for the thin-\flm\nsamples. Real-space images of the skyrmion crystal and the\nhelical structure obtained by the Lorentz transmission electron\nmicroscopy are also displayed. (Reproduced from Ref. [60].)\nthe former and the latter Cu2+ions is 3:1. The\nmagnetic structure of Cu 2OSeO 3consists of a network\nof tetrahedra composed of four Cu2+(S=1/2) ions at\ntheir apexes as shown in Fig. 5(b). Powder neutron\ndi\u000braction [63] and nuclear magnetic resonance [64, 65]\nexperiments suggested that three-up and one-down\ntype ferrimagnetic spin arrangement is realized on each\ntetrahedron below Tc\u001858 K.\nFigures 6(a) and (b) show experimental T-Bphase\ndiagrams for bulk samples and thin-\flm samples of\nCu2OSeO 3, respectively [60]. In spite of the di\u000berent\norigin of magnetism between metallic and insulating\nmagnets, the phase diagrams of the insulating copper\noxoselenite Cu 2OSeO 3are quite similar to those of\nthe metallic B20 compounds. For bulk samples, the\nskyrmion-crystal phase takes place only as a smallpocket (so-called A phase) in the phase diagram\nat \fniteTandB. On the other hand, its area\nspreads over a wide T-Brange and even to the\nlowest temperatures for thin-\flm samples. Magnetic\nmodulation periods in the helical and skyrmion-crystal\nstates are\u0018630\u0017A, which is much longer than the\ncrystallographic lattice constant \u00188:9\u0017A.\n2. Magnetoelectric properties\n2.1. Multiferroic nature\nIn usual materials, their magnetic properties are\na\u000bected or manipulated by magnetic \felds, while\ntheir dielectric properties are by electric \felds. In\ncontrast, the control of magnetism by electric \felds\nand inversely the control of dielectricity by magnetic\n\felds are called magnetoelectric e\u000bect, which was\n\frst predicted by P. Curie more than a century\nago [66]. The electric control of magnetism is one\nof the key issues in the \feld of spintronics aiming for\nlow-energy-consuming magnetic devices because Joule-\nheating energy losses due to electric currents injected\nfor generating a magnetic \feld or driving magnetic\ndomains can be overcome by employing electric \felds\nin insulators. The \frst experimental observation of the\nmagnetoelectric e\u000bect was reported for Cr 2O3where\nthe linear magnetoelectric e\u000bect ( Mi=\u000bjiEjandPi=\n\u000bijHj) shows up due to the simultaneous breaking\nof time-reversal and space-inversion symmetries but\nwith a rather small coe\u000ecient \u000bij[1, 6]. In order to\nenhance the magnitude of the magnetoelectric e\u000bect,\nthe employment of multiferroics, that is, materials\nendowed with both ferroelectric and magnetic orders,\nis particularly promising when these two orders are\nstrongly coupled to each other. Recently several\ninsulating magnets with noncollinear spin texture\n(such as TbMnO 3, MnWO 4, CuO, and a series\nof hexaferrites etc) have been reported to host\nferroelectricity of magnetic origin [67]. In these\nmaterials, the low-symmetry of spin texture breaks\nspatial inversion symmetry inherent to the original\ncrystal lattice, and causes a polar distribution of\nelectric charges. The strong coupling between the spin\ntexture and the ferroelectricity in such systems enables\nlarge and versatile magnetoelectric responses.\nWhereas observations of magnetic skyrmion\ncrystal had been limited to metallic B20 alloys,\nits realization was discovered also in a magnetic\ninsulator Cu 2OSeO 3(copper oxoselenite) with a chiral\ncrystal symmetry in 2012. Noncollinear skyrmion\nspin textures in the insulator can induce ferroelectric\npolarization via relativistic spin-orbit interactions\ndepending on the direction of external magnetic \feld.\nA presence or absence of the magnetism-induced\nferroelectric polarization in the skyrmion-crystal phaseDynamical magnetoelectric phenomena of multiferroic skyrmions 6\nFigure 7. (color online). (a) Symmetry axes of the Cu 2OSeO 3\ncrystal, which belongs to the chiral cubic P2 13 space group:\nfour three-fold rotation axes, 3, along h111i, and three two-fold\nscrew axes, 2 1, alongh100i. (b) Magnetization con\fguration\nof the skyrmion crystal formed within a plane normal to the\napplied magnetic \feld H, which possesses a six-fold rotation\naxis, 6, and six two-fold rotation axes followed by time reversal,\n20. The in-plane magnetization components are represented by\narrows, while the out-of-plane components are by colors. (c)-\n(e) When the skyrmion crystal is formed, most of the symmetry\nelements become broken, and the system can be polar to induce\nthe ferroelectric polarization Pdepending on the direction of H\n(see text).\nand, if any, its direction can be known from the\nsymmetry argument [60, 68]. Although the spatial\ninversion symmetry is broken, the original crystal\nlattice of Cu 2OSeO 3belongs to the non-polar space\ngroup P2 13, and thus does not host ferroelectricity. In\naddition, the skyrmion-crystal spin structure itself as\nwell as the conical and ferromagnetic (ferrimagnetic)\nspin orders are not polar, either. However combination\nof the crystal and the magnetic symmetries renders the\nsystem polar, and allows the emergence of ferroelectric\npolarization.\nAs shown in Fig. 7(a), the crystal structure of\nCu2OSeO 3possesses four three-fold rotation axes, 3,\nalongh111i, and three 2 1-screw axes along h100i. On\nthe other hand, the magnetic structure of the skyrmion\ncrystal has a six-fold rotation axis, 6, along the external\nmagnetic \feld H, and two-fold rotation axes followed\nby time reversal, 20, normal toHas shown in Fig. 6(b).\nNote that skyrmion crystals appear always on the\nplane normal to external magnetic \feld. When such a\nskyrmion spin texture is formed on the crystal lattice of\nCu2OSeO 3, most of the symmetry elements are broken,\nand eventually the system can become polar depending\non the direction of H.\nWe discuss three cases with di\u000berent Hdirectionsshown in Figs. 7(c)-(e). When a skyrmion crystal\nsets in under Hk[110] [see Fig. 7(c)], only the 20\n1-\naxis (k[001]) normal to Hsurvives. Consequently the\nsystem becomes polar along [001], and emergence of\nferroelectric polarization Pk[001] is allowed. Likewise,\nin the case of Hk[111] [see Fig. 7(d)], only the three-\nfold rotation axis parallel to Hremains unbroken.\nSubsequently, emergence of PkHk[111] is allowed.\nIn contrast, in the case of Hk[001] as shown in\nFig. 7(e), orthogonal arrangement of screw axes along\nh001iremains, and thus no ferroelectric polarization\ncan be expected. This argument holds also for the\nhelimagnetic and ferrimagnetic states, and one can\nexpect emergence of ferroelectric polarizations Pk[001]\nandPk[111] underHk[110] andHk[111], respectively,\nwhereasPis zero under Hk[001] for all these magnetic\nstates.\nFigures 8(a)-(i) show H-dependence of [111]\ncomponent of net magnetization M[111], ac magnetic\nsusceptibility \u001f0, and [111] component of ferroelectric\npolarization P[111] forHk[111] measured at di\u000berent\ntemperatures, that is, 5 K, 55 K and 57 K [60]. The\nsystem goes through several magnetic phases as the\nmagnetic \feld increases. At T=5 K and 55 K, the\nhelimagnetic state (multiple q-domain), helimagnetic\nstate (single q-domain), and collinear ferrimagnetic\nstate successively emerge with increasing magnetic\n\feld. On the other hand, the phase evolution at\nT=57 K with increasing magnetic \feld is as follows:\nhelimagnetic state (multiple q-domain)!helimagnetic\nstate (single q-domain)!skyrmion-crystal state !\nhelimagnetic state (single q-domain)!ferrimagnetic\nstate. Namely the skyrmion-crystal phase takes\nplace within the helimagnetic phase at T=57 K. The\npro\fle of\u001f0shows clear anomalies at the magnetic\ntransition points. Note that these magnetic-phase\nevolutions are not a\u000bected by the H-direction. We\n\fnd that all the magnetically ordered states can induce\n\fnite ferroelectric polarization Palong [111] direction\nunderHk[111] in agreement with the above symmetry\nanalysis, but they have di\u000berent signs and magnitudes.\nFigures 9(d)-(l) show H-dependence of net\nmagnetization M, ac magnetic susceptibility \u001f0, and\nferroelectric polarization Pmeasured at 57 K (just\nbelowTc\u001858 K) for di\u000berent H-directions, that\nis,Hk[001],Hk[110], andHk[111] [68]. A presence\nor absence of ferroelectric polarization Pand its\norientation for each Hdirection are totally consistent\nwith the above symmetry argument [see Figs. 9(a)-\n(c)]. The phase evolutions and magnetoelectric\nnature of Cu 2OSeO 3discussed here have been\ncon\frmed by several experimental techniques such\nas electron spin resonance (ESR) measurements [69],\nmagnetoelectric susceptibility measurements [70, 71],\nresonant soft x-ray scatterings [72], and muon-spinDynamical magnetoelectric phenomena of multiferroic skyrmions 7\nFigure 8. (color online). (a)-(c) Magnetic-\feld dependence of (a) [111] component of net magnetization M[111], (b) ac magnetic\nsusceptibility \u001f0, and (c) [111] component of ferroelectric polarization P[111] for bulk samples of Cu 2OSeO 3measured under Hk[111]\natT=5 K. (d)-(f) Corresponding pro\fles at T=55 K. (g)-(i) Corresponding pro\fles at T=57 K. Letter symbols f, s, h, and h' stand\nfor ferrimagnetic, skyrmion-crystal, helimagnetic (single q-domain), and helimagnetic (multiple q-domains) states, respectively. At\nzero magnetic \feld, the measured P[111] is zero because of cancellation of electric polarizations from multiple q-domains. If one\ncould obtain a single domain by \feld-cooling procedure, \fnite values of P[111] should be observed as indicated by dashed red lines.\n(Reproduced from Ref. [60].)\nFigure 10. (color online). Schematics of the spin-\ndependent metal-ligand hybridization mechanism as an origin\nof magnetism-induced electric polarizations. Local electric\npolarizations pijemerge along the bond vector eijconnecting\ntheith magnetic metal ion Cu2+and thejth ligand ion\nO2\u0000whose magnitude depends on the relative direction of\nmagnetization miagainst the bond.\nrotation measurements [73].2.2. Spin-dependent metal-ligand hybridization\nmechanism\nA theoretical study based on the \frst-principles\ncalculation suggested a crucial role of relativistic\nspin-orbit interactions for magnetoelectric coupling\nin Cu 2OSeO 3[74]. The electric polarizations\nin Cu 2OSeO 3are microscopically induced via the\nso-called spin-dependent metal-ligand hybridization\nmechanism. Local electric polarizations pijinduced\nby this mechanism are given by [75, 76, 77],\npij/(eij\u0001mi)2eij: (3)\nHereiandjare indices of the transition-metal ions\nCu2+and the oxygen O2\u0000ions, respectively. This\nmechanism assumes a single pair of adjacent magnetic\n(Cu2+) and ligand (O2\u0000) ions witheijandmibeing\nthe unit vector along the bond connecting them and the\nmagnetization direction at the transition-metal site,\nrespectively. The covalency between these two sites\nis governed by the relative direction of miagainst the\nbond through the spin-orbit interaction, and the local\npolarizationpijis induced along the bond direction\neij.\nBy taking a summation of contributions pijDynamical magnetoelectric phenomena of multiferroic skyrmions 8\nFigure 9. (color online). (a)-(c) Magnetically-induced ferroelectric polarization Pin Cu 2OSeO 3for various directions of Hpredicted\nby the symmetry argument (see text). (d)-(f) Magnetic-\feld dependence of (a) net magnetization M, (b) ac magnetic susceptibility\n\u001f0, and (c) ferroelectric polarization Pfor bulk samples of Cu 2OSeO 3measured at 57 K for Hk[001]. (g)-(i) Corresponding pro\fles\nforHk[110]. (j)-(l) Corresponding pro\fles for Hk[111]. Letter symbols f, s, h, and h' stand for ferrimagnetic, skyrmion-crystal,\nhelimagnetic (single q-domain), and helimagnetic (multiple q-domains) states, respectively. The measured Pis always zero at H=0\nbecause of cancellation of contributions from multiple q-domains. Finite values of P[001] andP[111] indicated by dashed lines should\nbe observed in a single-domain sample after \feld-cooling procedure. (Reproduced from Ref. [68].)\ngiven by Eq. (3) for all Cu-O bonds within a\ncrystallographic unit cell, the local polarization p(r)\nfrom the crystallographic unit cell can be evaluated.\nSince the modulation period of the magnetic skyrmion\nlattice is much longer than the crystallographic lattice\nconstant in Cu 2OSeO 3, the local spin structure within\na crystallographic unit cell can be regarded as nearly\ncollinear. Consequently one-by-one correspondence\nbetween the local magnetization m(r) and the local\nelectric polarization p(r) can be obtained as shown in\nFigs. 11(a) and (b) [68].\nFor the collinear ferrimagnetic state with spatially\nuniformm(r) andp(r), one can know the dependenceof ferroelectric polarization Pon theH-direction.\nShown in Figs. 11(c) and (d) are the H-direction\ndependence of the [110] and [001] components of P\n(P[110] andP[001]) measured at 2 K with H= 0:5 T,\nnamely, in the ferrimagnetic phase. Here Hrotates\naround the [ \u0016110]-axis, and \u0012is de\fned as an angle\nbetweenH-direction and the [001] axis [see Fig. 11(f)].\nThe development of P[001] measured for Hrotating\naround the [001] axis is also shown in Fig. 11(e) where\nan angle between Hand the [100] axis is de\fned as \u001e\n[see Fig. 11(g)]. Both of the P-pro\fles show sinusoidal\nmodulation with a period of 180\u000eas a function of\nrespectiveH-rotation angle. On the other hand,Dynamical magnetoelectric phenomena of multiferroic skyrmions 9\nFigure 11. (color online). (a), (b) Three-dimensional\nrepresentation of general correspondence between (a) M- and\n(b)P-directions in the collinear spin state where arrows at\nthe same position in (a) and (b) represent the M-vector and\ncorresponding induced P-vector, respectively. (c) [110] and (d)\n[001] components of ferroelectric polarization Psimultaneously\nmeasured under a magnetic \feld Hrotating around the [ \u0016110]\naxis. (e) [001] component of Punder Hrotating around the\n[001] axis. Both measurements are performed for the collinear\nferrimagnetic state at 2 K with H=0.5 T. Dashed lines indicate\nthe theoretically expected behaviors from Eq. (3), and arrows\ndenote the direction of H-rotation. (f),(g) Experimentally\nobtained relationships between the directions of PandMin the\nferrimagnetic state and de\fnitions of \u0012and\u001e(the angle between\ntheH-direction and the speci\fc crystal axis) are summarized for\n(f)Hrotating around [ \u0016110] axis and (g) Hrotating around [001]\naxis. Here the directions of MandPare indicated by thick\narrows, while the cross symbol \u0002denotesP= 0. (Reproduced\nfrom Ref. [68].)\nthe calculation using Eq. (3) predicts P[110]/sin 2\u0012\nandP[001]/1\u0000cos 2\u0012for the former case, while\nP[001]/sin 2\u001efor the latter case. These behaviors\nare indicated by dashed lines in Figs. 11(c)-(e), which\nperfectly reproduce the experimentally observed P\npro\fles and hence strongly suggests a validity of the\nspin-dependent metal-ligand hybridization mechanism\nas an origin of the electric polarizations in Cu 2OSeO 3.\nBy identifying the microscopic magnetoelectric\ncoupling given by Eq. (3), spatial distributions of\nelectric polarizations p(r) and electric charges \u001a(r) =\n\u0000r\u0001p(r) can be obtained for a single skyrmion. The\nspatial distribution of m(r) in the skyrmion crystal is\napproximately given by,\nm(r)/ezM0+3X\nn=1[ezcos(qn\u0001r+\u0019)+ensin(qn\u0001r+\u0019)];(4)\nwhereqndenotes three magnetic modulation vectors\nFigure 12. (color online). Calculated spatial distributions of\n(a)-(c) local electric polarization vectors p, and (d)-(f) local\nelectric charges \u001afor the skyrmion-crystal state (see text).\nMagnetic \feld His applied along the out-of-plane direction.\nThey are for Hk[001] [(a),(d)], Hk[110] [(b),(e)], and Hk[111]\n[(c),(f)]. The background color represents relative values of pz\nfor (a)-(c) and \u001afor (d)-(f), respectively. Here mzandpzstand\nfor the out-of-plane components of mandp, respectively. The\ndashed hexagon indicates a magnetic unit cell of the skyrmion\ncrystal or a single skyrmion. (Reproduced from Ref. [68].)\nnormal toHwith relative angles of 120\u000e, andenis a\nunit vector orthogonal to ezandqnde\fned such that\nallqn\u0001(ez\u0002en) have the same sign. Here M0scales with\nrelative magnitude of the net magnetization in the H-\ndirection. Figure 12 indicates real-space distributions\nofp(r) and\u001a(r) calculated for the skyrmion-crystal\nstate with various directions of H. The obtained\nresults suggest that each skyrmion texture locally\ncarries an electric quadrupole moment under Hk[001]\nor an electric dipole moment along the in-plane ( k[001])\nand out-of-plane ( k[111]) directions under Hk[110] and\nHk[111], respectively. Such a local coupling between\nthe electric dipole and the skyrmion spin texture\nstrongly suggests that skyrmions in an insulator can\nbe driven by a spatial gradient of the external electric\n\feld. Note that the total charge within each skyrmion\nis always zero, which implies nondissipative nature of\nthe electric-\feld-induced skyrmion dynamics.\nA small angle neutron scattering experiment\nunder an applied electric \feld for bulk sample of\nCu2OSeO 3indeed reported that orientation of qvec-\ntors in the skyrmion-crystal state slightly rotates\naround theH-direction in a clockwise or counter-\nclockwise manner depending on the sign of electric\n\feld [78, 79]. This experiment clearly demonstrates\npossible manipulation of magnetic skyrmions by ap-\nplying electric \felds in insulators.Dynamical magnetoelectric phenomena of multiferroic skyrmions 10\n2.3. Magnetoelectric coupling\nBecause of the cubic crystal symmetry, the local\nelectric polarization pifor theith crystallographic\nunit cell is described using the local magnetization\ncomponentsmi= (mia;mib;mic) as\npi=0\n@pia\npib\npic1\nA=\u00150\n@mibmic\nmicmia\nmiamib1\nA; (5)\nwherea,bandcare the Cartesian coordinates in the\ncubic setting [80, 81]. Although the two equations,\nEq. (3) and Eq. (5), seem to be di\u000berent from each\nother at \frst glance, it was con\frmed that these\ntwo expressions give equivalent spatial distribution\nof the local polarizations. The magnetoelectric-\ncoupling constant \u0015is a material parameter, which\nis microscopically related with the relativistic spin-\norbit interaction and the metal-ligand hybridization.\nIts value is common for all the magnetic phases as\nfar as the same compound is concerned. Namely\nthe value is identical in the helical, the skyrmion-\ncrystal and the ferrimagnetic phases in Cu 2OSeO 3,\nand can be evaluated as \u0015=5:64\u000210\u000027\u0016Cm from\nthe experimentally measured P[001]=16\u0016C/m2in\nthe ferrimagnetic phase under Hk[110] at 5 K. The\nreason why we choose the ferrimagnetic phase is\nthat all the tetrahedra with three-up and one-down\nspins give uniform contributions to the ferroelectric\npolarizationPand the net magnetization M. Thus\nthe contributions from each tetrahedron, piand\nmi, can be easily evaluated from the experimentally\nmeasuredPandMaspi=P/Nandmi=M/Nwhere\nNis the number of the tetrahedra in the unit volume.\nIn this way, the ferrimagnetic phase provides us a\nunique opportunity to evaluate the coupling constant\n\u0015.\n3. Spin model and phase diagrams\n3.1. Microscopic spin model\nIn 1980, Bak and Jensen proposed that magnetism on\nthe chiral cubic crystal structure can be described by\nthe following continuum spin model as long as magnetic\norders with su\u000eciently slow spatial and temporal\nvariations are concerned [82]:\nH=Z\ndr\u0014J\n2a(rm)2+D\na2m\u0001(r\u0002m)\n\u0000g\u0016B\u00160\na3H\u0001m\n+A1\na3(m4\nx+m4\ny+m4\nz)\n\u0000A2\n2a[(rxmx)2+ (rymy)2+ (rzmz)2]\u0015\n:\n(6)This model describes competitions among the ferro-\nmagnetic exchange interaction (the \frst term), the\nDzyaloshinskii-Moriya interaction (the second term),\nand the Zeeman coupling to an external magnetic\n\feldH(the third term). Two types of magnetic\nanisotropies allowed by the cubic crystal symmetry\n(the fourth and the \ffth terms) are also incorporated,\nbut they turn out to play only a minor role if we con-\nsider realistic small values of A1andA2. In order to\ntreat this continuum spin model numerically, it is con-\nvenient to divide the space into cubic meshes, which\ngives a classical Heisenberg model on the cubic lat-\ntice [35, 36, 83]. The Hamiltonian is given by,\nH=\u0000JX\ni;^\rmi\u0001mi+^\r\n\u0000DX\ni;^\r(mi\u0002mi+^\r\u0001^\r)\n\u0000g\u0016B\u00160H\u0001X\nimi\n+A1X\ni[(mx\ni)4+ (my\ni)4+ (mz\ni)4]\n\u0000A2X\ni(mx\nimx\ni+^x+my\nimy\ni+^y+mz\nimz\ni+^z): (7)\nThe index ^\rruns over ^x,^y, and ^zfor the three-\ndimensional case, while over ^xand ^yfor the two-\ndimensional case. The magnetic system of Cu 2OSeO 3\nis a network of tetrahedra composed of four Cu2+\n(S=1/2) ions, on which three-up and one-down type\ncollinear spin arrangement is realized below Tc\u001858\nK [63, 64, 65]. This four-spin assembly as a magnetic\nunit can be treated as a classical magnetization\nvectormiwhose norm mis unity. We choose the\nratioD=J=0.09, which gives a skyrmion diameter\nof\u001899 sites for the skyrmion-crystal phase. If we\nassume that the distance between adjacent tetrahedra\nis\u00185\u0017A, this number corresponds to the skyrmion\ndiameter of\u001850 nm in agreement with the observation\nin the Lorentz transmission electron microscopy for\nCu2OSeO 3[60]. For the slowly varying spin textures,\nthe spins are nearly decoupled from the background\nlattice structure. It justi\fes the theoretical treatment\nbased on the spin model on the cubic lattice for\nsimplicity without considering the complicated crystal\nstructure of real materials.\n3.2. Theoretical phase diagrams\nAlthough microscopic studies based on the \frst-\nprinciples calculations [74, 84, 85, 86] and electron\nspin resonance (ESR) experiments [87] have predicted\ncomplicated magnetic system with signi\fcant quantum\nnature, the above-introduced simple classical spin\nmodel has turned out to describe magnetic properties\nof Cu 2OSeO 3well. Ground-state phase diagrams ofDynamical magnetoelectric phenomena of multiferroic skyrmions 11\nFigure 13. (color online). (a) Theoretical phase diagram of the lattice spin model given by Eq. (7) with D=J=0.09 andA1=A2=0\nfor two dimensions as a function of magnetic \feld HzatT=0. Calculated magnetic-\feld dependence of [110] component of the\nnet magnetization M[110] and that of [001] component of the ferroelectric polarization P[001] under H= (0;0;Hz)(k[110]) are\nplotted. (b) Those for three dimensions. In the middle panel, M[110] data for the ground-state magnetic con\fgurations obtained by\nnumerically minimizing the energy are shown by closed circles, while the analytically obtained behavior of Mz=cos\u0012with Eq. (14)\nand Eq. (15) is shown by a solid line. They show perfect coincidence. (c) Schematic illustration of the longitudinal conical spin\nstructure.Dynamical magnetoelectric phenomena of multiferroic skyrmions 12\nthe lattice spin model given by Eq. (7) well reproduce\nthe experimental phase diagrams of Cu 2OSeO 3as\nwell as the B20 compounds. Figure 13(a) displays a\ntheoretical phase diagram for two dimensions. We \fnd\nthat the skyrmion-crystal phase emerges in the range\n1:875\u000210\u00003<jg\u0016B\u00160Hz=Jj<6:3\u000210\u00003sandwiched\nby the helical and ferromagnetic phases. This phase\nevolution is in agreement with an experimental result\nfor thin-\flm samples shown in Fig. 6(b) where the\nhelical, skyrmion-crystal, and ferrimagnetic phases\nsuccessively appear as a magnetic \feld increases\nat low temperatures. For the relevant exchange\nparameter J=3 meV for Cu 2OSeO 3, these critical\n\felds, respectively, correspond to 486 Oe and 1632 Oe,\nwhich coincide well with the experimental values of 500\nOe and 1800 Oe indicated by the phase diagram shown\nin Fig. 6.\nThe net magnetization Mand the ferroelectric\npolarization Pare given by sums of the local\ncontributions as,\nM=g\u0016B\nNVNX\ni=1mi; (8)\nand\nP=1\nNVNX\ni=1pi: (9)\nHere the index iruns over the Cu-ion tetrahedra with\nfour spin pair, Nis the number of the tetrahedra, and\nV(=1.76\u000210\u000028m3) is the volume per tetrahedron.\nThe calculated [110] component of Mand the [001]\ncomponent of PunderHk[110] are plotted in the lower\npanels as functions of H.\nIt might seem to be surprising that the purely two-\ndimensional spin model can reproduce the experimen-\ntally observed phase evolutions in thin slab of mate-\nrials with \fnite thickness. This is probably because\nthe two-dimensional model captures well a physical\nmechanism for enhanced stability of skyrmion crystal\ndue to the destabilization of conical state in thin-\flm\nsamples as discussed in Sec. 1.2. However, because\nphase transitions take place only at zero temperature\nfor the two-dimensional model, consideration of the\nthree-dimensionality is required when we study phys-\nical properties of thin-\flm samples at \fnite tempera-\ntures.\nOn the other hand, a ground-state phase diagram\nof the spin model given by Eq. (7) for three dimensions\nis displayed in Fig. 13(b). In this case, the skyrmion-\ncrystal phase is absent at low temperatures, and\nonly a phase transition from the longitudinal conical\nphase to the ferromagnetic phase is reproduced at\njg\u0016B\u00160Hz=Jj= 8:0\u000210\u00003. This result again agrees\nwith the experimental result for bulk samples shown\nin Fig. 6(a) where the skyrmion-crystal phase does not\nappear at low temperatures.The critical magnetic \feld jg\u0016B\u00160Hz=Jj= 8:0\u0002\n10\u00003corresponds to 2073 Oe if we assume J=3 meV\nandm=1, which is again in good agreement with\nthe experimental value of 1900 Oe. Note that there\nappears a multiple qconical phase in the experimental\nphase diagram at low-\feld region owing to degeneracy\nof the spiral qvectors and higher-order magnetic\nanisotropies, whereas it is absent in the present\ntheoretical phase diagram because we neglect magnetic\nanisotropies for simplicity. It should also be mentioned\nthat a recent Monte-Carlo study demonstrated that\nthis three-dimensional lattice spin model can reproduce\nwhole experimental phase diagram for bulk samples\nof Cu 2OSeO 3and MnSi in plane of magnetic \feld\nBand temperature Twith small skyrmion-crystal\nphase near the paramagnetic{conical phase boundary\nat \fnite temperatures [83].\nIn the longitudinal conical state, the ferromagnet-\nically ordered planes are stacked along the Hdirection\naccompanied by rotation of the in-plane magnetization\ncomponents around Hwith a uniform turn angle \u001e\nupon propagating along the Hdirection as shown in\nFig. 13(c). In addition this state has a net magneti-\nzation parallel to H. With the turn angle \u001eand the\nmagnetization per site mz=mcos\u0012, the magnetization\nvector on the ith plane and that on the ( i+ 1)th plane\nare written, respectively, as\nmi=m(sin\u0012cos\u001e0;sin\u0012sin\u001e0;cos\u0012); (10)\nand\nmi+1=m(sin\u0012cos(\u001e+\u001e0);sin\u0012sin(\u001e+\u001e0);cos\u0012):(11)\nThen the energy per magnetization is given using \u0012and\n\u001eas,\nE[\u0012;\u001e] =\u0000Jm2(sin2\u0012cos\u001e+ cos2\u0012)\n\u0000Dm2sin2\u0012sin\u001e\n\u0000g\u0016B\u00160Hzmcos\u0012: (12)\nThe expressions of \u0012and\u001ecan be derived from the\nsaddle-point equations of the energy with respect to \u0012\nand\u001e:\n@E[\u0012;\u001e]\n@\u0012= 0;@E[\u0012;\u001e]\n@\u001e= 0: (13)\nThey are given by\n\u001e = tan\u00001\u0012D\nJ\u0013\n; (14)\ncos\u0012=g\u0016B\u00160Hz\n2m[J(cos\u001e\u00001) +Dsin\u001e]: (15)\nNote that if the magnitude of the external His small\nenough, the propagation vector qfor the conical state\nis not necessarily directed along Hin reality, but could\nbe pinned in a certain direction due to weak magnetic\nanisotropies. However it is assumed here that the\nvectorqis always parallel to Hfor simplicity.Dynamical magnetoelectric phenomena of multiferroic skyrmions 13\nFigure 14. (color online). Short rectangular pulses of magnetic\n\feld and electric \feld used in the numerical simulations for\ncalculating the dynamical susceptibilities.\nIn the lower two panels of Fig. 13(b), calculated\nH-dependence of [110] component of the net magne-\ntizationM[110] and [001] component of the ferroelec-\ntric polarization P[001] underHk[110] are plotted for\nthe three-dimensional case. Here the M[110]data indi-\ncated by closed circles are calculated for the ground-\nstate magnetic con\fgurations obtained by numerically\nminimizing the energy, while the analytically obtained\nbehavior of Mz=cos\u0012with Eq. (14) and Eq. (15) is in-\ndicated by a solid line. They show perfect coincidence.\n4. Electromagnetism in Multiferroics\n4.1. Dynamical susceptibilities\nDynamical properties of resonantly oscillating mag-\nnetizations and magnetically induced polarizations of\nmultiferroic skyrmions are captured by the following\ndynamical susceptibilities:\nDynamical dielectric susceptibilities:\n\u001fee\n\u000b\f(!) =\u0001P!\n\u000b\n\u000f0E!\n\f(16)\nDynamical magnetic susceptibilities:\n\u001fmm\n\u000b\f(!) =\u0001M!\n\u000b\n\u00160H!\n\f(17)\nDynamical magnetoelectric susceptibilities:\n\u001fem\n\u000b\f(!) =\u0001P!\n\u000bp\u000f0\u00160H!\n\f=c\u0001P!\n\u000b\nH!\n\f(18)\nDynamical electromagnetic susceptibilities:\n\u001fme\n\u000b\f(!) =r\u00160\n\u000f0\u0001M!\n\u000b\nE!\n\f=c\u00160\u0001M!\n\u000b\nE!\n\f: (19)\nThe latter two susceptibilities describe cross-correlation\nresponses, that is, responses of polarization Pto the\nac magnetic \feld H!and responses of magnetization\nMto the ac electric \feld E!. Here the indices \u000band\n\frun over the Cartesian coordinates.These dynamical susceptibilities are calculated nu-\nmerically using the Landau-Lifshitz-Gilbert equation:\ndmi\ndt=\u0000mi\u0002Be\u000b\ni+\u000bG\nmmi\u0002dmi\ndt; (20)\nwhere the local magnetization miis de\fned as mi=\n\u0000Si=\u0016hwithSibeing the spin. Here the \frst\nterm depicts the gyrotropic motion of magnetization\nmiunder the e\u000bective magnetic \feld Be\u000b\ni, while\nthe second term describes the phenomenologically\nintroduced damping e\u000bect with \u000bGbeing the Gilbert-\ndamping coe\u000ecient. After the linearization, the\nequation leads,\ndmi\ndt=1\n1 +\u000b2\nGh\n\u0000mi\u0002Be\u000b\ni\u0000\u000bG\nmmi\u0002(mi\u0002Be\u000b\ni)i\n:(21)\nThe e\u000bective magnetic \feld Be\u000b\niacting on the\nmagnetization miis calculated from the derivative of\nthe Hamiltonian with respect to mias,\nBe\u000b\ni=\u0000@H=@mi: (22)\nwith\nH=H0+H0(t): (23)\nHere the \frst term H0depicts the magnetic system of\nthe chiral-lattice magnet, which is given by,\nH0=\u0000JX\n<i;j>mi\u0001(mi+^x+mi+^y)\n\u0000DX\ni;^\rmi\u0002mi+^\r\u0001^\r\n\u0000g\u0016B\u00160H\u0001X\nimi; (24)\nwhereH=(0, 0,Hz)(kz), and ^\rruns over the\nCartesian coordinates ^x,^yand ^z. The second term\nH0(t) represents a short rectangular pulse of magnetic\n\feld or that of electric \feld as a perturbation whose\ntime width is \u0001 t:\nH0(t) =\u0000g\u0016B\u00160X\ni\u0001H(t)\u0001mi; (25)\nor\nH0(t) =\u0000X\ni\u0001E(t)\u0001pi: (26)\nTime pro\fles of \u0001 H(t) and \u0001E(t) are shown in\nFig. 14. An advantage of utilizing short rectangular\npulses is that the !-dependence in the Fourier\ncomponentsH!andE!shows up only in higher-order\nterms with respect to !\u0001tas,\nE!\n\u000b=Z1\n\u00001\u0001E(t)ei!tdt=E0\u0001t+O[(!\u0001t)2]; (27)\nH!\n\u000b=Z1\n\u00001\u0001H(t)ei!tdt=H0\u0001t+O[(!\u0001t)2]:(28)\nSince the leading terms of E!\n\u000bandH!\n\u000bare!-\nindependent, one only needs to calculate \u0001 M!\n\u000band\n\u0001P!\n\u000bto evaluate the dynamical susceptibilities givenDynamical magnetoelectric phenomena of multiferroic skyrmions 14\nFigure 15. (color online). (a) Frequency dependence of\nimaginary part of the dynamical magnetic susceptibilities,\nIm\u001fmm(!), in the skyrmion-crystal phase with several values of\nmagnetic \feld Hzfor in-plane ac magnetic \felds H!?Hdc. (b)\nThat for out-of-plane ac magnetic \felds H!kHdc. The static\nmagnetic \feld Hdc=(0, 0,Hz) is applied normal to the two-\ndimensional plane. The angular frequency !is normalized by the\nferromagnetic-exchange coupling J, and!=J=0.01 corresponds\nto\u00181 GHz when J=0.4 meV for example. Insets show resonance\nfrequencies !Ras functions of Hz. (Reproduced from Ref. [88].)\nby Eqs. (16)-(19). In the simulations, one should \frst\ntrace the time evolution of the net magnetization,\nM(t) =g\u0016B\nNVX\nimi(t); (29)\nand that of the ferroelectric polarization,\nP(t) =1\nNVX\nipi(t); (30)\nafter application of a short pulse of H0(t) att=0. Then\none performs the Fourier transformation of \u0001 M(t)(=\nM(t)\u0000M(t= 0)) and \u0001P(t)(=P(t)\u0000P(t= 0)) to\nobtain \u0001M!\n\u000band \u0001P!\n\u000b.\n4.2. Spin-wave modes of skyrmion crystal: Theory\nIt was theoretically revealed that the skyrmion-crystal\nstate has peculiar spin-wave modes with frequencies\nof a few gigahertz [88, 89, 90, 91, 92, 93, 94].\nShown in Figs. 15(a) and (b) are calculated frequency\ndependence of imaginary part of the dynamical\nmagnetic susceptibility Im \u001fmm(!) for the in-plane ac\nmagnetic \feld H!?Hdcand that for the out-of-plane\nac magnetic \feld H!kHdc, respectively [88]. The\ncalculations are performed by numerically solving the\nLandau-Lifshitz-Gilbert equation with the lattice spin\nmodel given by Eq. (24) in two dimensions where the\nstatic magnetic \feld Hdcis applied normal to the\nplane. We \fnd that each spectrum in Fig. 15(a) has a\ncouple of resonance peaks, while that in Fig. 15(b) has\nonly one resonance peak.\nIn order to identify these resonant modes, numer-\nical simulations have been performed to trace the spa-\ntiotemporal dynamics of the local magnetizations miand the local scalar spin chiralities Cifor the skyrmion-\ncrystal state by applying a stationary oscillating ac\nmagnetic \feld H!whose frequency coincides with the\nresonance frequency of the corresponding mode [88].\nHere the local scalar spin chirality is de\fned as,\nCi=mi\u0001(mi+^x\u0002mi+^y) +mi\u0001(mi\u0000^x\u0002mi\u0000^y):(31)\nIt is found that for every resonance mode discussed\nhere, all of the skyrmions exhibit uniformly the same\nmotion so that one only needs to focus on the dynamics\nof one skyrmion in the skyrmion crystal to identify the\nmicrowave-active eigenmodes.\nIn Fig. 16, snapshots of the simulated spatiotem-\nporal dynamics of a skyrmion in the collectively oscil-\nlating skyrmion crystal for each mode are displayed. As\nshown in Figs. 16(a) and (b), two resonant modes un-\nderH!?Hdcare rotational modes where distribution\nof the out-of-plane magnetization components circu-\nlates around each skyrmion core. Interestingly such ro-\ntational motions of skyrmions are driven even by a lin-\nearly polarized H!. The rotation sense of the skyrmion\ncirculation is opposite between these two modes. It\nis counterclockwise with respect to the Hdcdirection\nfor the lower-frequency mode, while clockwise for the\nhigher-frequency mode. The rotation sense becomes\nopposite when one reverses the sign of Hdcor the sign\nof the skyrmion number Q. In contrast, it is not af-\nfected by the sign of the Dzyaloshinskii-Moriya param-\neter or swirling direction of the magnetizations. We\n\fnd that intensity of the lower-lying mode is much\nstronger than the higher-lying mode because the rota-\ntion sense of the lower-lying mode matches the preces-\nsional direction of magnetizations determined by the\nsign ofHdc. On the other hand, the single resonant\nmode underH!kHdcturns out to be a breathing mode\nwhere all the skyrmions in the skyrmion crystal ex-\npand and shrink in an oscillatory manner as shown in\nFig. 16(c).\nThe spin-wave excitations activated by the in-\nplane ac magnetic \feld strongly depend on circular\npolarization of H!or that of an irradiated microwave\nbecause of their rotational habits [88]. Numerical sim-\nulation found that irradiation of the left-handed circu-\nlarly polarized microwave with resonant frequency sig-\nni\fcantly enhances the magnetization oscillation of the\nlower-lying counterclockwise rotational mode, whereas\nthe right-handed circularly polarized microwave with\nthe same frequency cannot activate the magnetiza-\ntion oscillation so much. Numerical simulations also\ndemonstrated that melting of the skyrmion crystal\ncan be realized when the counterclockwise rotational\nspin-wave mode is intensely excited in the skyrmion-\ncrystal phase with the left-handed circularly polarized\nmicrowave. The melting occurs as the radius of the\nskyrmion-core rotation exceeds the lattice constant of\nthe skyrmion crystal.Dynamical magnetoelectric phenomena of multiferroic skyrmions 15\nFigure 16. (color online). Simulated spatiotemporal dynamics of local magnetizations (left panels) and local scalar spin chiralities\nCi(right panels) for three di\u000berent spin-wave modes of the skyrmion crystal. Temporal waveforms of the applied ac magnetic \felds\nare shown in the uppermost panels where inverted triangles indicate times at which we observe the snapshots shown here. (a) [(b)]\nLower-energy [Higher-energy] rotational mode activated by the ac magnetic \feld H!normal to the static magnetic \feld Hdc(H!?\nHdc). Distributions of the mzicomponents and the spin chiralities Cicirculate around the skyrmion core in a counterclockwise\n(clockwise) fashion. (c) Breathing mode activated the ac magnetic \feld H!parallel to the static magnetic \feld Hdc(H!kHdc).\nAreas of the skyrmions expand and shrink in an oscillatory manner. (Reproduced from Ref. [88].)\n4.3. Spin-wave modes of skyrmion crystal:\nExperiment\nSince the magnon excitations are characterized by typi-\ncal resonance frequencies ranging from gigahertz to ter-\nahertz, the measurement of absorption and/or trans-\nmission spectra with linearly polarized electromagnetic\nwaves in this energy region is the most straightfor-\nward way to experimentally identify them. However\nthe dominant contribution from the Drude response\nof conduction electrons often prevents the detection\nof pure magnetization dynamics in metallic systems.\nTherefore the employment of insulating materials is\nideal for this purpose because of the absence of the\nDrude contribution. Indeed the predicted skyrmion\nresonant modes were detected by microwave absorption\nand transmission measurements for Cu 2OSeO 3[95, 96].\nThe setup of the experiment in Ref. [95] is shown in\nFig. 17(a).Figure 17(b) indicates measured absorption spec-\ntra for bulk samples of Cu 2OSeO 3at 57.5 K (just below\nTc\u001858 K) for various magnitudes of Hdcwith a mi-\ncrowave polarization of H!?Hdc. The helical spin or-\nder is realized for Hdc\u0014400 Oe except for the region\n140 Oe\u0014Hdc\u0014320 Oe where the skyrmion-crystal\nphase takes place. While the helical spin state shows a\nmagnetic resonance at around 1.5-1.7 GHz, the emer-\ngence of new absorption mode around 1 GHz is clearly\nobserved in the latter Hdc-range. In Fig. 17(c), inten-\nsity of the new absorption mode is indicated by colors\nin plane of TandHdc. This mode has been con\frmed\nto appear only in the skyrmion-crystal phase, and can\nbe assigned to the counterclockwise rotational mode\nof the skyrmion crystal according to the theoretically\nsuggested selection rules with respect to the microwave\npolarization. Note that the existence of the clockwise\nrotational mode at higher resonance frequency was also\npredicted for the skyrmion-crystal state, while its ex-Dynamical magnetoelectric phenomena of multiferroic skyrmions 16\nFigure 17. (color online). (a) Setup of the microwave absorption experiment (left panel), the microstrip line used to detect the\nmagnetic resonant modes of the skyrmion crystal (middle panel), and the electromagnetic-\feld distribution in the present setup\nusing the microstrip line viewed along the microwave propagation direction (right panel). (b) Experimentally measured microwave\nabsorption spectra under various magnitudes of static magnetic \feld Hdcat 57.5 K for bulk samples of Cu 2OSeO 3and (c) temperature\nversus magnetic \feld phase diagram with background color indicating the absorption intensity of the skyrmion resonant modes for\nmicrowave-polarization con\fguration of H!?Hdcwith which clockwise and counterclockwise rotational modes are activated in the\nskyrmion-crystal phase. (d),(e) Corresponding pro\fles for microwave-polarization con\fguration of H!kHdcwith which a breathing\nmode is activated in the skyrmion-crystal phase. (Reproduced from Ref. [95].)\nperimental identi\fcation is yet to be performed because\nof its weak absorption intensity.\nLikewise the corresponding Hdc-dependence of\nthe absorption spectra for the microwave polarization\nH!kHdcis shown in Fig. 17(d). Whereas the\nconventional ferromagnetic resonance is active only\ntoH!(?Hdc) and this selection rule holds also for\nthe helical state, emergence of a sharp absorption\npeak is observed at 1.3-1.5 GHz for the region 50\nOe\u0014Hdc\u0014150 Oe. The color plot of the\nspectral intensity in the H-Tplane indicates that\nthis mode uniquely appears in the skyrmion-crystal\nphase [Fig. 17(e)]. Comparison with the theoreticallyproposed selection rules suggests that this resonant\nabsorption corresponds to the breathing mode of the\nskyrmion crystal.\n5. Microwave magnetoelectric phenomena\n5.1. Electromagnetic waves in multiferroics\nDynamical responses of multiferroic materials with net\nmagnetization Mand ferroelectric polarization Pto\nan irradiated electromagnetic wave can be discussed\nstarting from Maxwell's equations,\nrotE=\u0000@B\n@t; (32)Dynamical magnetoelectric phenomena of multiferroic skyrmions 17\nrotB=@D\n@t; (33)\nwhere\nB=\u00160(^\u00161H+M); (34)\nD=\u000f0^\u000f1E+P; (35)\nP=\u000f0^\u001feeE+ ^\u001femp\u000f0\u00160H; (36)\nM=\u00160^\u001fmmH+ ^\u001fmer\u000f0\n\u00160E: (37)\nNote that because of the multiferroic nature, the\nmagnetoelectric susceptibilities ^ \u001femand ^\u001fmebecome\n\fnite, and there appear a contribution proportional to\nHin the expression of the ferroelectric polarization P\nand that proportional to Ein the expression of the net\nmagnetization M. Inserting the following expressions,\nE=E!exp[i(K!\u0001r\u0000!t)]; (38)\nH=H!exp[i(K!\u0001r\u0000!t)]; (39)\ninto Eq. (32) and Eq. (33), one obtains the following\nequations for Fourier components,\n!B!=K!\u0002E!; (40)\n\u0000!D!=K!\u0002H!; (41)\nwhere\nB!=[^\u00161+ ^\u001fmm(!)]\u00160H!+ ^\u001fme(!)p\u000f0\u00160E!;(42)\nD!=[^\u000f1+ ^\u001fee(!)]\u000f0E!+ ^\u001fem(!)p\u000f0\u00160H!: (43)\nSolving these simultaneous equations in terms of the\nwave vectorK!and considering the relation,\nK!=!N(!)=c; (44)\none obtains an expression of the complex refractive\nindex\nN(!) =n(!) +i\u0014(!): (45)\nThis quantity gives us information about how the\nmaterial responds to an irradiated electromagnetic\nwave. In particular, its imaginary part \u0014(!) is called\nextinction coe\u000ecient and describes to what extent\nthe material absorbs the electromagnetic wave. More\nconcretely, the absorption coe\u000ecient \u000b(!) is related to\n\u0014(!) as,\n\u000b(!) =2!\u0014(!)\nc=2!\ncImN(!); (46)\nand the absorption intensity I(!) is given by,\nI(!) =I0exp[\u0000\u000b(!)d]; (47)\nwithdbeing the sample thickness.\n5.2. Nonreciprocal directional dichroism: Theory\nIn the multiferroic materials with magnetically induced\nelectric polarizations, one can expect dynamical cou-\npling of magnetic and electric dipoles. For exam-\nple, resonant oscillations of magnetizations in multi-\nferroics are coupled to those of electric polarizations\nFigure 18. (color online). Schematic illustration of the\nnonreciprocal directional dichroism of electromagnetic waves\ninduced by the skyrmion-crystal state. An electromagnetic\nwave irradiating in a certain direction to a material is strongly\nabsorbed, while that in the opposite direction is not absorbed so\nmuch.\nowing to the magnetoelectric coupling, and thus can\nbe activated not only magnetically by an ac magnetic\n\feldH!but also electrically by an ac electric \feld\nE![97, 98, 99]. Such simultaneous magnetic and elec-\ntric activities of magnons (so-called electromagnons)\ncause intriguing dynamical magnetoelectric phenom-\nena in the microwave frequency regime [80, 112]. Since\nthe magnetic skyrmions in Cu 2OSeO 3are accompanied\nby electric dipoles, the resonant oscillation of skyrmion\ncrystal active to H!can be also activated by E!. The\ntheoretical calculation indeed predicted that the rota-\ntional modes and the breathing mode of skyrmion crys-\ntal underHdck[110] has \fnite electric susceptibility for\nE!k[110](kHdc) andE!k[001](?Hdc), respectively.\nFor such hybridized electromagnon modes, inter-\nference between the H!-activity and the E!-activity\noften leads to unique optical and/or microwave phe-\nnomena called directional dichroism [80]. This is a\nkind of \"one-way window\" e\u000bect where reversal of in-\ncident direction of an electromagnetic wave gives dif-\nferent absorption or emission spectrum (see schematic\n\fgure in Fig. 18). Since the \frst experimental demon-\nstration by Rikken in 1997 [100], directional dichro-\nism in absorption and emission has been reported for\nvarious frequency range from x-ray to visible-light to\nterahertz [101, 102, 103, 104, 105, 106, 107, 108, 109].\nThe directional dichroism can be considered as a direct\nexpansion of the linear magnetoelectric e\u000bect ( Pi=\n\u000bijHjandMi=\u000bjiEj) into the dynamical regime\nand is allowed when \u000bxycomponent is nonzero for the\nincident direction of an electromagnetic wave K!kz.Dynamical magnetoelectric phenomena of multiferroic skyrmions 18\nFigure 19. (color online). Calculated spectra of (a) imaginary\npart of the dynamical magnetic susceptibility Im \u001fmm\n\u000b\f(!)Cand\n(b) imaginary part of the dynamical dielectric susceptibility\n\u001fee\n\u000b\f(!) for the skyrmion-crystal phase under the external\nmagnetic \feld Hk[110]. Im\u001fmm\nyyin (a) and Im \u001fee\nzzin (b) have\nresonance peaks at the same frequencies, indicating that both\nthe in-plane ac magnetic \feld H!kyand the out-of-plane ac\nelectric \feld E!kzactivate common eigenmodes. On the other\nhand, Im\u001fmm\nzzin (a) and Im \u001fee\nyyin (b) also have a resonance\npeak at the same frequency, indicating that both the out-of-\nplane ac magnetic \feld H!kzand the in-plane ac electric \feld\nE!kyactivate a common eigenmode. For the de\fnition of the\nCartesian coordinates x,yandz, see the inset. (Reproduced\nfrom Ref. [80].)\nThis condition is always satis\fed when the relationship\n(P\u0002M)kK!holds, while the magnitude of direc-\ntional dichroism essentially depends on nature of the\nabsorption at the target frequency.\nWhen one applies a static magnetic \feld\nHdck[110] to Cu 2OSeO 3, the ferroelectric polarization\nPk[001] as well as the net magnetization Mk[110] are\ninduced, which are orthogonal to each other as shown\nin Fig. 7(c) or Fig. 9(b). With this special con\fgura-\ntion ofP?M, bothH!andE!components of a mi-\ncrowave propagating parallel or antiparallel to P\u0002M\ncan excite common oscillation modes.\nSuch a simultaneous activity of the collective\nmodes can be seen via comparison between the dynam-\nical magnetic susceptibilities \u001fmm\n\u000b\f(!) and the dynami-cal dielectric susceptibilities \u001fee\n\u000b\f(!). Figures 19(a) and\n(b) show calculated frequency dependence of Im \u001fmm\n\u000b\f\nand that of Im \u001fee\n\u000b\f, respectively, for the skyrmion-\ncrystal state under Hk[110]. We \fnd that resonances\nactive to ac magnetic \felds seen as peaks in the spec-\ntrum of Im \u001fmm\n\u000b\fcan be found also in the spectrum of\nIm\u001fee\n\u000b\f. More concretely, the spectrum of Im \u001fmm\nyywith\nyk[001] in Fig. 19(a) has a couple of resonance peaks,\nand the spectrum of Im \u001fee\nzzwithzk[110] in Fig. 19(b)\nalso has peaks at the same frequencies. This indi-\ncates that the in-plane ac magnetic \feld H!?Hdcand\nthe out-of-plane ac electric \feld E!kHdcactivate the\nsame eigenmodes. On the other hand, both Im \u001fmm\nzzin\nFig. 19(a) and Im \u001fee\nyyin Fig. 19(b) have a single peak\nat the same frequency, indicating that the out-of-plane\nac magnetic \feld H!kHdcand the in-plane ac electric\n\feldE!?Hdcactivate the same eigenmode.\nIn Figs. 20(a) and (b), snapshots of the simulated\ncoupled dynamics of local magnetizations mi(left\npanels) and local polarizations pi(right panels)\nare shown for the counterclockwise rotational mode\nactivated by H!?HdcorE!kHdcand the breathing\nmode activated by H!kHdcorE!?Hdc, respectively.\nBehaviors of the net magnetization Mand the\nferroelectric polarization Pfor these modes are shown\nin the bottom panels. In the rotational mode, the angle\nbetweenPandM, which is originally 90\u000e, becomes\nlarger and smaller oscillatorily. On the other hand,\nin the breathing mode, simultaneous elongation and\nshrinkage ofPandMoccur in an oscillatory manner,\nwhile keeping the angle between them 90\u000e.\nThese simultaneous electric and magnetic ac-\ntivities of the resonant modes cause peculiar dy-\nnamical magnetoelectric phenomena of skyrmions in\nthe microwave-frequency regime [80]. We \frst dis-\ncuss the case with a linearly polarized microwave\nwithH!k[001](?M) andE!k[110](?P) as shown in\nFig. 21(a). Starting from Maxwell's equations, one can\nderive a relation for electromagnetic waves:\nH!kK!\u0002E!: (48)\nThis equation indicates that the relative relationship\nbetweenH!- andE!-directions of an electromagnetic\nwave is determined by the sign of the wave vector\nK!or the propagation direction. This relative\nrelationship will be reversed upon the sign reversal\nofK!. Accordingly the H!andE!components\nof a microwave propagating in [ \u0016110] (+x) direction\ncooperatively induce an oscillation of the angle between\nPandMso as to intensely activate the rotation\nmode [see bottom panels of Fig. 21(a)]. In turn,\nwhen the microwave is propagating in the opposite\ndirection, that is, [1 \u001610] (\u0000x) direction, its H!andE!\ncomponents work destructively to activate the rotation\nmode. Consequently the microwave with K!k\u0000x\nstrongly excites the spin wave resonance, and thereby isDynamical magnetoelectric phenomena of multiferroic skyrmions 19\nFigure 20. (color online). Spatiotemporal dynamics\nof magnetizations mi(left panels) and polarizations pi\n(right panels) for electromagnon excitations in the skyrmion-\ncrystal under the static magnetic \feld Hk[110](kz). (a)\nCounterclockwise rotational mode activated by in-plane ac\nmagnetic \feld H!?[110] or by out-of-plane ac electric \feld\nE!k[110]. (b) Breathing mode activated by out-of-plane\nac magnetic \feld H!k[110] or by in-plane ac electric \feld\nE!?[110]. Dynamical behaviors of net magnetization Mand\nferroelectric polarization Pare shown for each mode in the\nlowest panel. The angle between MandPbecome larger and\nsmaller in an oscillatory manner in the rotational mode, while\nin the breathing mode, elongation and shrinkage of MandPin\nlength occur synchronously, keeping the angle at 90\u000e.\nstrongly absorbed, while the microwave with K!k+x\nexcites the spin wave resonance only weakly, and\nthus is weakly absorbed. In this way, the microwave\nabsorption intensity becomes di\u000berent depending on\nthe incident direction.\nOne can also expect that interference between\nthe magnetic and electric activation processes of the\nbreathing mode also gives rise to the nonreciprocal di-\nrectional dichroism of microwave when the microwave\npolarization is H!k[110](kM) andE!k[001](kP) as\nFigure 21. (color online). Microwave polarization con\fg-\nurations with which the nonreciprocal directional dichroism\noccurs in the skyrmion-crystal phase with net magnetization\nMk[110](kz) and ferroelectric polarization Pk[001](ky). (a)\nH!k[001](ky) and E!k[001](kz). The H!andE!components\ncontribute in an additive way to the excitation of rotational\nmodes for a microwave propagating in the negative ( \u0000xor [1\u001610])\ndirection, while in a subtractive way for a microwave propagating\nin the positive (+ xor [\u0016110]) direction. Consequently the absorp-\ntion intensity of the former microwave with sgn( ReK!) =\u00001 be-\ncomes larger than that of the latter one with sgn( ReK!) = +1.\n(b)H!k[110](kz) and E!k[001](kz). In this case, a microwave\npropagating in the + xdirection is absorbed intensely through\nstrongly exciting the breathing mode as compared to that prop-\nagating in the\u0000xdirection. (Reproduced from Ref. [80].)\nshown in Fig. 21(b). In this case, the H!andE!\ncomponents of a microwave propagating in the + xdi-\nrection cooperatively induce length oscillations of P\nandMso as to intensely activate the breathing mode,\nwhereas those of a microwave propagating in the \u0000x\ndirection do not [see bottom panels of Fig. 21(b)]. Con-\nsequently, the microwave with K!k+xis strongly ab-\nsorbed, whereas the microwave with K!k\u0000xis weakly\nabsorbed. Again the absorption intensity becomes dif-\nferent depending on the incident direction of the mi-\ncrowave.\nFor quantitative discussion on these e\u000bects, oneDynamical magnetoelectric phenomena of multiferroic skyrmions 20\nFigure 22. (color online). (a),(b) Calculated microwave absorption spectra for the skyrmion-crystal phase realized in a (two-\ndimensional) thin-\flm specimen of Cu 2OSeO 3under the external magnetic \feld Hdck[110]. Here \u000b+and\u000b\u0000are the microwave\nabsorption coe\u000ecients for positive and negative incident directions, that is, sgn(Re K!) = +1 and sgn(Re K!) =\u00001, respectively.\nThe di\u000berence \u0001 \u000brepresents the magnitude of the directional dichroism. The spectra in (a) are obtained for the microwave\npolarization of H!k[001](ky) and E!k[110](kz) where the rotational modes are excited. On the other hand, the spectra in\n(b) are obtained for the microwave polarization of H!k[110](kz) and E!k[001](ky) where the breathing mode is excited. (c)\nCalculated microwave absorption spectra \u000b+and\u000b\u0000for the longitudinal conical phase in a (three-dimensional) bulk specimen of\nCu2OSeO 3under Hdck[110] for H!k[001](ky) and E!k[110](kz). (Reproduced from Ref. [80].)Dynamical magnetoelectric phenomena of multiferroic skyrmions 21\nderives expressions of a complex refractive index N(!)\nby solving the simultaneous equations (40) and (41) for\ntwo di\u000berent cases of microwave polarizations:\nCase (i):H!k[001](ky) andE!k[110](kz)\nN(!)\u0018q\n[\u000f1zz+\u001feezz(!)][\u00161yy+\u001fmmyy(!)]\n\u0000sgn(ReK!)[\u001fme\nyz(!) +\u001fem\nzy(!)]=2; (49)\nCase (ii):H!k[110](kz) andE!k[001](ky)\nN(!)\u0018q\n[\u000f1yy+\u001feeyy(!)][\u00161zz+\u001fmmzz(!)]\n+ sgn(ReK!)[\u001fme\nzy(!) +\u001fem\nyz(!)]=2: (50)\nIn these expressions, we \fnd that there appears a\nterm containing the sign of K!. Since the absorp-\ntion coe\u000ecient of electromagnetic waves is given by\n\u000b(!) = (2!=c)ImN(!), the coe\u000ecient \u000b(!) depends\non the sign of K!or the incident direction of electro-\nmagnetic wave. Magnitude of the nonreciprocal direc-\ntional dichroism or change of the absorption coe\u000ecient\naccompanied by reversal of the microwave incident di-\nrection, that is, \u0001 \u000b(!) =\u000b+(!)\u0000\u000b\u0000(!), is given by:\nCase (i):H!k[001](ky) andE!k[110](kz)\n\u0001\u000b(!) =\u0000Im[\u001fme\nyz(!) +\u001fem\nzy(!)]=2; (51)\nCase (ii):H!k[110](kz) andE!k[001](ky)\n\u0001\u000b(!) = Im[\u001fme\nzy(!) +\u001fem\nyz(!)]=2: (52)\nHere\u000b+(!) and\u000b\u0000(!) are absorption coe\u000ecients for\nan electromagnetic wave propagating in the positive\nand negative xdirections, respectively. We \fnd that\nthe magnitude of nonreciprocal directional dichroism\nis governed by magnetoelectric susceptibilities \u001fem\n\u000b\f(!)\nand\u001fme\n\u000b\f(!):\nFigures 22(a) and (b) display calculated spectra\nof microwave absorption coe\u000ecients \u000b+(!) and\n\u000b\u0000(!) as well as their di\u000berence \u0001 \u000b(!) for the\nskyrmion-crystal state in a (two-dimensional) thin-\n\flm sample of Cu 2OSeO 3underHdck[110] for several\nvalues ofHdc. In Fig. 22(a)Cthere appear two\npeaks in the spectra, which correspond to the low-\nenergy counterclockwise and the high-energy clockwise\nrotational modes, respectively, for the microwave\npolarization of H!k[001] (ky) andE!k[110] (kz). The\ndi\u000berence of absorption coe\u000ecients \u0001 \u000b(!) becomes\nlarger as the magnetic \feld Hdcincreases, and\neventually reaches a maximum value of 0.25 cm\u00001,\nwhich corresponds to the relative di\u000berence of\n\u0001\u000b/\u000bave=2(\u000b+\u0000\u000b\u0000)/(\u000b++\u000b\u0000)\u001820 %. On the other\nhand, in Fig. 22(b), a single resonance peak, which\ncorresponds to the breathing mode appears for the\nmicrowave polarization con\fguration of H!k[110](k\nz) andE!k[001](ky). The di\u000berence of absorptioncoe\u000ecients \u0001 \u000b(!) increases as the magnetic \feld Hdc\nincreases, and reaches 0.14 cm\u00001at maximum which\ncorresponds to the relative di\u000berence of \u001810 %.\nThe microwave directional dichroism can be\nexpected also in other magnetic phases. Figure 22(c)\ndisplay calculated spectra of \u000b+(!) and\u000b\u0000(!) as\nwell as their di\u000berence \u0001 \u000b(!) for the longitudinal\nconical state in a (three-dimensional) bulk sample of\nCu2OSeO 3underHdck[110] for several values of Hdc.\nThe magnitude of \u0001 \u000b(!) increases as Hdcincreases,\nand reaches relative di\u000berence of \u001830 %. In Fig. 23(a)\nand (b), calculated resonant frequency fR(upper\npanels) and magnitude of directional dichroism \u0001 \u000b=\u000b\u0000\n(lower panels) are plotted as functions of dc magnetic\n\feldHdcfor the two-dimensional case (thin-\flm\nsample) and the three-dimensional case (bulk sample)\nfor the microwave polarization H!k[001] (ky) and\nE!k[110] (kz) underHdck[110]. In the calculation, an\nisotropic dielectric tensor with \u000f1\nzz=\u000f1\nyy=\u000f1=8 is used\naccording to the dielectric-measurement data [110,\n111], while \u00161\nzz=\u00161\nyy=1 is used for permeability. The\nvalue ofJis set to be J=1 meV.\nNote that the value of exchange parameter\nin Cu 2OSeO 3isJ\u00183 meV in reality, but here\nwe usedJ=1 meV to reproduce ciritcal \felds\nat a \fnite temperature in order to compare the\ncalculation and the experiment done at T=57.5\nK. For an exact numerical treatment, we should\nperform a \fnite-temperature calculation by taking\naccount of thermal-\ructuation e\u000bects using the\nstochastic Landau-Lifshitz-Gilbert equation with a\nrealistic exchange parameter of J=3 meV. The simple\nreduction of the Jvalue adopted here to mimic\nthe thermal e\u000bects is a rather crude approximation.\nAs a result, the calculation tends to overestimate\nthe magnitude of directional dichroism at \fnite\ntemperatures signi\fcantly. However, even with this\napproximation, not only qualitative behaviors of\nphysical phenomena but also experimentally measured\nresonance frequnecies and magnetic-\feld strengths can\nbe reproduced quantitatively. Also, the predicted\nlarge microwave directional dichroism of \u001820-30 % is\nexpected to be observed at low temperatures.\n5.3. Nonreciprocal directional dichroism: Experiment\nThe above theoretical analysis suggests that signi\fcant\ndirectional dichroism can be expected for the skyrmion\nresonant modes in Cu 2OSeO 3under the experimental\ncon\fguration with Hdck[110] (eventually Pk[001]) and\nK!k[1\u001610] since the condition P\u0002MkK!is satis\fed\nhere. Figure 24(b) shows the nonreciprocal absorption\nspectra \u0001\u000b(!) =\u000b+(!)\u0000\u000b\u0000(!) measured for the\nskyrmion-crystal phase in bulk Cu 2OSeO 3atT=57\nK [112]. Two resonant modes observed at 1.0 GHz and\n1.7 GHz correspond to the counterclockwise rotationalDynamical magnetoelectric phenomena of multiferroic skyrmions 22\nFigure 23. (color online). Calculated Hdcdependence of the resonant frequency fR(upper panels) and the magnitude of\nnonreciprocal directional dichroism \u0001 \u000b=\u000b\u0000(lower panels) for (a) two-dimensional system (thin-\flm case) and (b) three-dimensional\nsystem (bulk case) under Hdck[110] for the microwave polarization H!k[001](?Hdc) and E!k[110](kHdc).\nmode and the breathing mode of the skyrmion crystal,\nrespectively. Note that for the experimental setup\n[see Fig. 24(a)] with electromagnetic-\feld distribution\nshown in Fig. 24(b), both the rotational mode active to\nH!?Hdcand the breathing mode active to H!kHdc\nare simultaneously excited [see Fig. 24(c)]. Both modes\nshow large directional dichroism up to \u00183% but with\nopposite signs [see Fig. 24(d)]. Here the reversal of Hdc,\nwhich reverses the sign of Mbut does not change the\nsign ofP, changes the sign of directional dichroism,\nwhich is consistent with the symmetry requirement of\nthis phenomena. The experimentally and theoretically\nobtainedHdc-dependence of the magnetic resonance\nfrequency as well as the magnitude of directional\ndichroism for bulk Cu 2OSeO 3are summarized in\nFig. 25, which agree well with each other [112].\nThe experimental observation of the large directional\ndichroism inversely proves that the present skyrmion\nresonant modes have \fnite electric activity coupled\nwithE!. The above results demonstrate that ultra-\nfast control of skyrmions by ac electric \felds up to\nGHz frequency range is indeed possible in insulating\nmaterials, and the employment of resonant structures\nin the electric susceptibility as presented here may\nbe useful to improve the e\u000eciency of the electric\nmanipulation of magnetic skyrmions.5.4. Microwave magnetochiral e\u000bect\nIn addition to the Voigt geometry with K!?Hdc\nas discussed above, one can expect the microwave\ndirectional dichroism also for the Faraday geometry\nwithK!kHdc[113, 114]. The directional dichroism\nrealized with this con\fguration is called magnetochiral\ne\u000bect where the absorption or transmission intensity\nof an electromagnetic wave propagating parallel or\nantiparallel to the external magnetic \feld Hdcdi\u000bers\ndepending on its propagation direction.\nConsidering the emergence of \fnite Pk[001](kc)\nunderHdck[110] and the absence of P(P=0) under\nHdck[010](kb) as shown, respectively, in Fig. 9(b)\nand Fig. 26(a), one expects the emergence of\noscillating ferroelectric polarization \u0001 P!k[001](kc)\ninduced by the oscillation of net magnetization M\nwith \u0001M!k[100](ka) when the ac magnetic \feld\nH!k[100](ka) is applied to the Cu 2OSeO 3sample\nunderHdck[010](kb) [see Figs. 26(b) and (c)].\nConsequently the interference between the H!-\nactive and the E!-active processes of this coupled\noscillation of \u0001 M!and \u0001P!, and thereby the di-\nrectional dichroism occur for microwave con\fguration\nofK!k[010](kb),H!k[100](ka) andE!k[001](kc) as\nshown in Fig. 26(d). Namely the H!andE!compo-\nnents of an microwave propagating in [010] or + bdi-Dynamical magnetoelectric phenomena of multiferroic skyrmions 23\nFigure 24. (color online). (a) Experimental setup of the measurement of microwave directional dichroism. Microwaves can\npropagate along the coplanar waveguide in both directions, and the di\u000berence between \u0001 S21and \u0001S12is evaluated as nonreciprocal\nabsorption spectra. (b) Electromagnetic-\feld distribution in the present setup using a coplanar waveguide, viewed from the direction\nof microwave propagation. Here the con\fguration of Pk[001], HdckMk[110], and K!k[1\u001610] is taken for bulk Cu 2OSeO 3, and\nthus the relationship P\u0002MkK!is satis\fed. (c) Microwave absorption spectra for the skyrmion-crystal state at Hdc= 300 Oe.\nA result of the two-peak \ftting is also shown by broken lines in addition to the raw data (red line). The broken lines represent each\ncomponent, and the blue solid line represents a sum of them. (d) The corresponding nonreciprocal absorption spectra measured for\nHdc=\u0006300 Oe. (Reproduced from Ref. [112].)\nrection cooperatively excite the above-mentioned elec-\ntromagnon mode and hence is absorbed signi\fcantly,\nwhile those of an oppositely propagating microwave de-\nstructively contribute to the electromagnon excitation\nand hence is absorbed only weakly.\nThe microwave magnetochiral e\u000bect with this\ncon\fguration is expected not only for the skyrmion-\ncrystal phase but also for other magnetically ordered\nphases. The e\u000bect in the conical phase and the \feld-\npolarized ferromagnetic phase in the bulk Cu 2OSeO 3\nsample at low temperatures was experimentally\nobserved [113] and theoretically predicted [114]\nindividually around the same time.\n6. Summary and Perspectives\nIn this article, we have overviewed recent theoretical\nand experimental studies on the multiferroic properties\nand the dynamical magnetoelectric phenomena of\nmagnetic skyrmions in a chiral-lattice magnetic\ninsulator Cu 2OSeO 3. We have \frst discussed that thenoncollinear skyrmion spin textures in this insulating\nmagnet induce electric polarizations via the so-called\nspin-dependent metal-ligand hybridization mechanism,\nand the system attains multiferroic nature. Resulting\nmagnetoelectric coupling, that is, the coupling\nbetween magnetizations and polarizations enables us\nto manipulate magnetic skyrmions by application of\nelectric \felds instead of injection of electric currents.\nHere, an important future issue is an establishment\nof a method to create skyrmions by electric \felds.\nThis method will be microscopically distinct from\nthat based on spin-transfer torques from spin-polarized\nelectric currents in metallic systems, and can be a\nunique technique for future skyrmion-based storage\ndevices without energy losses due to Joule heating.\nWe have then argued that multiferroic skyrmions\nshow coupled oscillation modes of magnetizations\nand polarizations, so-called electromagnon excitations,\nwhich are both magnetically and electrically active.\nInterference between these electric and magnetic\nactivation processes leads to peculiar magnetoelectricDynamical magnetoelectric phenomena of multiferroic skyrmions 24\nFigure 25. (color online). Comparison between experiment and theory of the skyrmion magnetoelectric resonance. (a)-(c) Hdc\ndependence of (a) ac magnetic susceptibility for Hdck[110], (b) resonance frequency fR, and (c) normalized magnitude of microwave\ndirectional dichroism. In (a), the electric polarization pro\fle is also presented. Broken line in (b) is a guide to the eyes. (d) and\n(e) Calculated magnetic-\feld dependence of (d) the resonance frequencies fRand (e) the magnitudes of the directional dichroism\nnormalized by the absorption coe\u000ecient for T=0. The thick vertical lines in (a)-(c) and the broken vertical lines in (d) and (e)\nrepresent the boundaries between di\u000berent magnetic phases: from low to high \felds, the magnetic phases are helical, skyrmion-crystal,\nconical and collinear phases. (Reproduced from Ref. [112].)Dynamical magnetoelectric phenomena of multiferroic skyrmions 25\nFigure 26. (color online). (a)-(c) In the presence of net\nmagnetization MkHunder Hk[010], oscillating magnetization\ncomponent \u0001 M!(k[100]) is accompanied by the oscillating\npolarization component \u0001 P!(k[001]). (d) Con\fguration of\nmicrowave H!andE!components, for which the magnetochiral\ndichroism occurs under Hk[010]: K!k\u0006H,H!k[100] and\nE!k[001]. (Reproduced from Ref. [114].)\ne\u000bects of skyrmions in the microwave frequency regime.\nSigni\fcant dynamical magnetoelectric e\u000bects such\nas directional dichroism are rare for any frequency\nranges. In particular there have been few reports on\ntheir observations in the gigahertz regime. This is\nbecause usual multiferroic materials based on simple\nshort-period spin structures with antiferromagnetic\ninteractions tend to have rather high resonance\nfrequencies (typically at the terahertz regime) due\nto large spin-wave gaps. In turn, skyrmions and\nother topological spin textures with long-period spin\nmodulations induced by the Dzyaloshinskii{Moriya\ninteractions and ferromagnetic interactions tend to\nhave small spin-wave gaps and speci\fc low-lying\nresonant modes, which o\u000bers a unique opportunity to\nrealize novel microwave functionalities.\nFor technical applications of these e\u000bects to\nfuture microwave devices [115], it is important to\nachieve enhanced cross-correlation responses or larger\nferroelectric polarization. For this purpose, search\nfor new insulating materials which exhibit skyrmions\nis necessary [116]. Materials containing ions with\nstronger spin-orbit interactions other than Cu2+is one\nof the promising research directions. In addition, it isalso promising to explore skyrmion states which induce\nlarge ferroelectric polarizations via a mechanism other\nthan the spin-dependent hybridization mechanism such\nas the inverse Dzyaloshinskii-Moriya mechanism [7].\nThe insulating skyrmionic material also o\u000bers a\nunique opportunity to investigate skyrmion motion\ndriven by magnon currents. It has been theoretically\npredicted that magnon currents can drive translational\nmotion, subsequent Hall motion, and rotational motion\nof skyrmions via the spin-transfer torques [117, 118,\n119, 120, 121]. In particular, thermally-induced\ndi\u000busive \rows of magnons are expected to induce\nthe skyrmion motion in the presence of temperature\ngradient [117, 118, 119]. In the metallic magnets, a\nthermal gradient necessarily induces di\u000busive currents\nof conduction electrons which \row from a higher-\ntemperature side to a lower-temperature side, and\ncontribute to the skyrmion motion. Therefore it is\ndi\u000ecult to identify a contribution purely from magnon\ncurrents to the skyrmion motion in metallic magnets.\nIn turn, insulating magnets are appropriate for the\nresearch because of the absence of conduction electrons\nand low-lying charge excitations. The skyrmion-\nhosting insulating magnets o\u000ber precious playgrounds\nfor research into several interesting issues such as\noptical responses and manipulations of skyrmions [122,\n123, 124], skyrmion-based magnonic crystals [125, 126],\nspin currents [127], spin motive forces [128], spin\nSeebeck e\u000bects [119, 129] in the skyrmion phases,\nelastic responses of skyrmions [130], E-\feld driven\nmotion and manipulations of skyrmions [78, 79,\n131], and critical and dynamical behaviors of phase\ntransitions [132, 133, 134, 135].\nSo far, magnetic skyrmions had been observed\nonly in materials with chiral cubic P 213symme-\ntry such as B20 alloys and Cu 2OSeO 3. However,\nit was theoretically predicted that non-chiral but\npolar magnets with C nvsymmetry can also host\nskyrmions stabilized by the Dzyaloshinskii{Moriya\ninteractions [19]. Indeed, realization of magnetic\nskyrmions was recently discovered in polar mag-\nnet GaV 4S8with rhombohedral C 3vsymmetry [116].\nIn addition to these non-centrosymmetric ferromag-\nnets, observations of skyrmions have been reported\neven for centrosymmetric ferromagnetic insulators with\nuniaxial anisotropy such as Y 3Fe5O12[137, 136],\nRFeO 3[137, 136], BaFe 11:79Sc0:16Mg0:05O19[138],\nLa0:5Ba0:5MnO 3[139], La 2\u00002xSr1+2xMn2O7[140].\nCrystal structures of these materials have spatial in-\nversion symmetry, and thus the Dzyaloshinskii{Moriya\ninteraction is not active. Instead, interplay be-\ntween magnetic dipole{dipole interaction and mag-\nnetic anisotropies play important roles for the real-\nization of magnetic skyrmions. Furthermore, it was\nrecently discovered that surfaces and interfaces of fer-Dynamical magnetoelectric phenomena of multiferroic skyrmions 26\nromagnetic monolayers host atomically small magnetic\nskyrmions [141, 142, 143]. There, the space-inversion\nsymmetry is broken, and thus the Dzyaloshinskii{\nMoriya interaction is active. Now the number of known\nskyrmion-hosting magnetic systems is rapidly increas-\ning. Magnetoelectric dynamics in these systems are\nissues of interest and should be clari\fed in future stud-\nies.\n7. Appendix\nDerivation of the expression of N(!)\nIn this Appendix, we solve Maxwell's equations to\nderive the expression of N(!) given by Eq. (49) for the\nfollowing polarization con\fguration of electromagnetic\nwave as an example: H!(ky) = (0;H!\ny;0),E!(kz) =\n(0;0;E!\nz), andK!(kx) = (K!;0;0).\nInserting Eq. (42) into Eq. (32), we obtain\n!\u00160^~\u001fmmH!+!p\u000f0\u00160^\u001fmeE!=K!\u0002E!\n= (0;\u0000K!E!\nz;0);(53)\nwhere ^~\u001fmm= ^\u00161+ ^\u001fmm.\nAlso inserting Eq. (43) into Eq. (33), we obtain\n\u0000!\u000f0^~\u001feeE!\u0000!p\u000f0\u00160^\u001femH!=K!\u0002H!\n= (0;0;\u0000K!H!\ny);(54)\nwhere ^~\u001fee= ^\u000f1+ ^\u001fee.\nNow we have a set of equations,\n!\u00160H!\ny~\u001fmm\nyy+!p\u000f0\u00160E!\nz\u001fme\nyz=\u0000K!E!\nz; (55)\n!\u000f0E!\nz~\u001fee\nzz+!p\u000f0\u00160H!\ny\u001fem\nzy=\u0000K!H!\ny: (56)\nAfter multiplyingc\n!=1\n!p\u000f0\u00160to both sides of the\nequations and using N=cK!\n!, the equations lead,\n0\n@\u001fme\nyz+ sgn(K!)Nq\n\u00160\n\u000f0~\u001fmm\nyyq\n\u000f0\n\u00160~\u001fee\nzz\u001fem\nzy+ sgn(K!)N1\nA\u0012E!\nz\nH!\ny\u0013\n= 0:\n(57)\nThese equations have nontrivial solutions when the\ndeterminant of the matrix is zero, that is,\n\u001fme\nyz\u001fem\nzy+ sgn(K!)N(\u001fme\nyz+\u001fem\nzy) +N2\u0000~\u001fee\nzz~\u001fmm\nyy= 0:\n(58)\nNeglecting the \frst term of the left-hand side because\n\u001fme\nyzand\u001fem\nzyare small, the equation leads\nN2+ sgn(K!)N(\u001fme\nyz+\u001fem\nzy)\u0000~\u001fee\nzz~\u001fmm\nyy= 0:\nSolving this quadratic equation, we obtain\nN=\u0000sgn(K!)(\u001fme\nyz+\u001fem\nzy)\u0006q\n(\u001fmeyz+\u001femzy)2+ 4~\u001feezz~\u001fmmyy\n2:\n(59)After neglecting the second-order terms with respect\nto\u001fme\nyzand\u001fem\nzyagain, we eventually arrive at\nN(!) =q\n~\u001feezz~\u001fmmyy\u0000sgn(K!)(\u001fme\nyz+\u001fem\nzy)=2: (60)\n8. 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Kubetzka, and R.\nWiesendanger, Science 341, 636 (2013)."    },    {        "title": "1203.6095v1.A_Generic_Property_of_Exact_Magnetic_Lagrangians.pdf",        "content": "arXiv:1203.6095v1  [math.DS]  27 Mar 2012A Generic Property of Exact Magnetic Lagrangians\nM´ ario Jorge Dias Carneiro and Alexandre Rocha.\nOctober 26, 2018\nAbstract\nWe prove that for the set of Exact Magnetic Lagrangians the pr operty “There\nexist finitely many static classes for every cohomology clas s” is generic. We also\nprove some dynamical consequences of this property.\n1 Introduction\nLetMbe a closed manifold equipped with an Riemannian metric g=/an}bracketle{t.,./an}bracketri}ht. A\nLagrangian L:TM→Ris called Exact Magnetic Lagrangian if\nL(x,v) =/bardblv/bardbl2\n2+/an}bracketle{tη,v/an}bracketri}ht\nfor some non-closed 1-form η.\nThis type of Lagrangian fits into Mather’s theory, as developed by R . Ma˜ n´ e and\nA. Fathi, about Tonelli Lagrangians, namely, it is fiberwise convex an d superlinear.\nWe refer the reader to the references Fathi in [6], Contreras and Iturriaga in [4] for\nexpositions of this theory.\nLetM(L) be the set of action minimizing measures. Recall that M(L) is the set\nofµBorel probability measures in TMwhich are invariant under the Euler-Lagrange\nflowϕtgenerated by Land minimizes the action, that is for all invariant probability ν\ninTM: /integraldisplay\nTMLdµ≤/integraldisplay\nTMLdν.\nThe setM(L) is a simplex whose extremal points are the ergodic minimizing\nmeasures.\nSince the Euler Lagrange flow generated by Ldoes not change by adding a closed\none form ζ, we also consider the action minimizing measures M(L−ζ). The minimal\n1action value, depends only on the cohomology class c= [ζ]∈H1(M,R) of the closed\none form, so it is denoted by −α(c). It is known that α(c) is the energy level that\ncontains the Mather set for the cohomology class c:\n/tildewiderMc=/uniondisplay\nµ∈M(L−ζ)supp(µ).\n/tildewiderMcis a compact invariant set which is a graph over a compact subset McofM, the\nprojected Mather set (see [11]). Mcis laminated by curves, which are global (or time\nindependent) minimizers. Mather also proved that the function c/ma√sto→α(c) is convex\nand superlinear.\nIn general,/tildewiderMcis contained in another compact invariant set, which also a graph\nwhose projection is laminated by global minimizers: the Aubry set for the cohomology\nclass c, denoted by/tildewideAc. Ma˜ n´ e proved that /tildewideAcis chain recurrent and it is a challenging\nquestion to describe the dynamics of the Euler-Lagrange flow rest ricted to/tildewideAc. The\ndefinition of Aubry set and some its properties are given in Section 3.\nOf course this question only makes sense if it is posed for generic Lag rangians,\nsince many pathological examples can be constructed. The notion o f genericity in the\ncontext of Lagrangian systems is provided by Ma˜ n´ e in [9]. The idea is to make special\nperturbations by adding a potential: L(x,v)+Ψ(x), for Ψ∈C∞(M).\nApropertyis genericinthesenseofMa˜ n´ eifitisvalidforallLagrangians L(x,v)+\nΦ(x) with Φ contained in a residual subset O.\nIn this setting, G. Contreras and P. Bernard proved in the work A Generic Prop-\nerty of Families of Lagrangian Systems (see [1]) that generically, in the sense of Ma˜ n´ e,\nfor all cohomology class cthere is only a finite number of minimizing measures. This\ntheorem is a consequence of an abstract result which is useful in diff erent situations.\nIn general, when we are dealing with an specific class of Lagrangians, perturba-\ntions by adding a potential are not allowed. However, due to the abs tract nature of\nBernard-Contreras proof it may be addapted to the specific case like the one treated\nhere.\nThe objective of this paper is to prove the genericity of finitely many minimizing\nmeasures for Exact Magnetic Lagrangians and apply it to the dynam ics of the Aubry\nset.\nLet us consider Γ1(M) the set of smooth 1-forms in Mendowed with the metric\nd(ω1,ω2) =/summationdisplay\nk∈Narctan(/bardblω1−ω2/bardblk)\n2k, (1)\n2denoting by /bardblω/bardblktheCk-norm of the 1-form ω.With this metric Γ1(M) is a Frechet\nspace, it means that Γ1(M) is a locally convex topological vector space whose topology\nisdefined by atranslation-invariant metric, andthat Γ1(M) iscomplete forthismetric.\nThe main result of this paper is the following:\nTheorem 1 LetAbe a finitedimensionalconvexfamilyof ExactMagnetic Lagra ngians.\nThen there exists a residual subset OofΓ1(M)such that,\nω∈ O,L∈A⇒dimM(L+ω)≤dimA.\nHence there exist at most 1+dimAergodic minimizing measures of L+ω.\nCorollary 2 LetLbe a Exact Magnetic Lagrangian. Then there exists a residual\nsubsetOofΓ1(M)such that for all c∈H1(M,R)and for all ω∈ O,there are at\nmost1+dimH1(M,R)ergodic minimizing measures of L+ω−c.\nThe last part of this work is dedicated to prove some consequences about the\ndynamics. For instance, using the work of Contreras and Paterna in, [5] we obtain\nconnecting orbits between the elements of the Aubry set that con tain the support of\nminimizing measures (the so called “static classes”).\n2 Adapting the abstract setting of Bernard and\nContreras\nAs it was pointed out previously, the proof of Theorem 1 is an applicat ion of the\nwork of Contreras and Bernard. Here we state their result.\nAssume that we are given\n(i)Three topological vector spaces E,F,G.\n(ii)A continuous linear map π:F→G.\n(iii)A bilinear map /an}bracketle{t,/an}bracketri}ht:E×G→R.\n(iv)Two metrizable convex compact subsets H⊂FandK⊂Gsuch that π(H)⊂K.\nSuppose that\n1. The restriction of the map given by (iii), /an}bracketle{t,/an}bracketri}ht|E×Kis continuous.\n32. The compact Kis separated by E.This means that, if µandνare two different\npoints of K,then there exists a point ωinEsuch that /an}bracketle{tω,µ−ν/an}bracketri}ht /ne}ationslash= 0.\n3.Eis a Frechet space.\nNote then that Ehas the Baireproperty, that is any residual subset of Eis dense.\nWe shall denote by H∗the set of affine and continuous functions defined on H.\nGiven¯L∈H∗denote by\nMH/parenleftbig¯L/parenrightbig\n= argmin ¯L\nthe set of points α∈Hwhich minimizes ¯L|H,and byMK/parenleftbig¯L/parenrightbig\nthe image π/parenleftbig\nMH/parenleftbig¯L/parenrightbig/parenrightbig\n.\nThese are compacts convex subsets of HandK.\nUnder these conditions we have:\nTheorem 3 (G. Contreras and P. Bernard) For every finite dimensional affine\nsubspace AofH∗, there exists a residual subset O(A)⊂Esuch that, for all ω∈ O(A)\nand¯L∈A,we have\ndimMK/parenleftbig¯L+ω/parenrightbig\n≤dimA\nInorder toapply thistheorem, we needto define theaboveobject s inanadequate\nsetting as follows:\nLetCbe the set of continuous functions f:TM→Rwith linear growth, that is\n/bardblf/bardblℓin= sup\nθ∈TM|f(θ)|\n1+|θ|<+∞ (2)\nendowed with the norm /bardbl./bardblℓin.\nWe define:\n•E= Γ1(M) endowed with the metric ddefined in (1).\n•F=C∗is the vector space of continuous linear functionals µ:C→Rprovided with\nthe weak- ⋆topology:\nlim\nnµn=µ⇔lim\nnµn(f) =µ(f),∀f∈C.\n•Gis the vector space of continuous linear functionals µ: Γ0(M)→R,where Γ0(M)\nis the space of continuous 1-forms on M. Note that the Riemannian metric g=/an}bracketle{t.,./an}bracketri}ht\nallows us to represent any continuous 1-form as /an}bracketle{tX,./an}bracketri}ht,for some C0vector field X.We\nendowGwith the weak- ⋆topology:\nlim\nnµn=µ⇔lim\nnµn(ω) =µ(ω),∀ω∈Γ0(M).\n4•The continuous linear π:F→Gis given by\nπ(µ) =µ|Γ0(M).\n•For a given natural number N, let\nBN={(x,v)∈TM:|v| ≤N}.\nLet us denote by M1\nNthe set of the probability measures µinTMsuch that supp µ⊂\nBN.DefineKN=π(M1\nN)⊂G, the restriction of the probabilities in M1\nNto Γ0(M).\nClaim 1. KNis metrizable.\nProof:SinceGis the dual of Γ0(M), we define a norm in Gas follows\n/bardblµ/bardblG= sup\n/bardblω/bardblℓin≤1{|µ(ω)|}.\nIfµ∈KN,\n/bardblµ/bardblG= sup\n/bardblω/bardblℓin≤1/braceleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nTMωdµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracerightbigg\n≤sup\n/bardblω/bardblℓin≤1/braceleftbigg/integraldisplay\nTM∩BN|ω|dµ/bracerightbigg\n= sup\n/bardblω/bardblℓin≤1/braceleftbigg/integraldisplay\nBN|ω(x,v)|\n1+N(1+N)dµ/bracerightbigg\n≤sup\n/bardblω/bardblℓin≤1/braceleftbigg/integraldisplay\nBN|ω(x,v)|\n1+|v|(1+N)dµ/bracerightbigg\n≤(N+1) sup\n/bardblω/bardblℓin≤1/braceleftbigg/integraldisplay\nTM/bardblω/bardblℓindµ/bracerightbigg\n≤N+1.\nThis shows that KN⊂BG,whereBGis the ball of radius N+ 1 inG= Γ0(M)∗.\nThen, by following classical theorem of Analysis, it is enough show tha t Γ0(M) is a\nseparable vector space.\nTheorem 4 LetEa Banach’s space. Then Eis separable if, and only if, the unit ball\nBE∗⊂E∗in the weak- ⋆topology is metrizable.\nThe separability of Γ0(M) follows from the lemma below and of the duality\nbetween 1-forms and vector fields provided by the Riemannian metr ic.\nLemma 5 The space X0(M)of continuous vector fields in a compact manifold Mis\nseparable.\nProof:By compactness of M,we can consider a number finite local trivializations\nˆUi⊂TM→Ui×Rnof the tangent bundle TMand by compactness of Ui,X0/parenleftbig\nUi/parenrightbig\n=\nC0/parenleftbig\nUi,Rn/parenrightbig\nis separable. Let {fi\nn}be a dense subset in X0/parenleftbig\nUi/parenrightbig\nand{αi}a partition\n5of unity subordinate to the open cover {Ui}.It is enough show that {/summationtext\niαifi\nn}is\ndense in X0(M).Letg∈X0(M) and consider gi=αig. Theng=/summationtextαig=/summationtextgi,\nsuppgi⊂Ui⊂Ui.Givenǫ >0 there exists ni∈Nsuch that\n/vextenddouble/vextenddoublefi\nni−gi/vextenddouble/vextenddouble<ǫ\n2i.\nThen\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\niαifi\nni−g/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\niαifi\nni−/summationdisplay\niαig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤/summationdisplay\nisup\nUi/vextendsingle/vextendsinglefi\nni−gi/vextendsingle/vextendsingle\n=/summationdisplay\ni/vextenddouble/vextenddoublefi\nni−gi/vextenddouble/vextenddouble</summationdisplay\niǫ\n2i< ǫ.\nLet us consider ( Xn) a dense sequence in X0(M) andωn=/an}bracketle{tXn,·/an}bracketri}ht ∈Γ0(M).Let\nω=/an}bracketle{tX,·/an}bracketri}ht ∈Γ0(M) andUεbe a ball in Γ0(M) centered at ω., of radius ε >0. Then\nthere exists a Xn∈Vε(X),whereVε(X) is the ball in X0(M) of radius εand center\nX.It follows that\n/bardblωn−ω/bardblℓin= sup\n(x,v)∈TM|(ωn−ω)(x,v)|\n1+|v|= sup\n(x,v)∈TM|/an}bracketle{t(Xn−X)(x),v/an}bracketri}ht|\n1+|v|\n≤sup\n(x,v)∈TM|(Xn−X)(x)||v|\n1+|v|≤sup\nx∈M|(Xn−X)(x)|< ε.\nThis shows that ωn∈ Uεand Γ0(M) is separable, so KNis metrizable. This finishes\nthe proof of the Claim 1.\nObserve that KNis compact and convex since KN=π(M1\nN), πis a continous\nmap and M1\nNis a compact subset of probability measures in TM.\n•The bilinear mapping /an}bracketle{t,/an}bracketri}ht:E×G→Ris given by integration:\n/an}bracketle{tω,µ/an}bracketri}ht=/integraldisplay\nTMωdµ.\nNote that here we apply the Hahn-Banach Theorem for extends th e functional µand\nthat the above integral does not depend on the extension of µto a signed measure on\nTMgiven by Riesz representation Theorem. Moreover,\n/an}bracketle{t,/an}bracketri}ht:E×KN→R\nis continuous. In fact, if ωn→ωandµn→µwith (ωn)⊂Eand (µn)⊂KN,then\nlim\nn/integraldisplay\nTMηdµn=/integraldisplay\nTMηdµ,∀η∈E,\n6andd(ωn,ω)→0 implies that given ǫ >0,there exists n0∈Nsuch that\n∀n≥n0,/bardblωn−ω/bardblℓin<ǫ\n(N+1).\nSinceµn,µ∈KN,we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nTMωndµn−/integraldisplay\nTMωdµn/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplay\nBN|ωn−ω|dµn\n=/integraldisplay\nBN|ωn−ω|\n1+N(1+N)dµn\n≤(1+N)/integraldisplay\nBN|ωn−ω|\n1+|v|dµn\n≤(1+N)/integraldisplay\nBN/bardblωn−ω/bardblℓindµn< ǫ\nWhenn→ ∞,/vextendsingle/vextendsingle/vextendsingle/vextendsinglelim\nn/integraldisplay\nTMωndµn−/integraldisplay\nTMωdµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ǫ,∀ǫ >0.\nTherefore\nlim\nn/an}bracketle{tωn,µn/an}bracketri}ht= lim\nn/integraldisplay\nTMωndµn=/integraldisplay\nTMωdµ=/an}bracketle{tω,µ/an}bracketri}ht.\n•KNisseparated by E.This followsfromtheduality andapproximationofcontinuous\nvector fields by smooth ones and the fact that KNis separated by Γ0(M),that is: if\nµ,ν∈KN, µ/ne}ationslash=ν, then there exists a ω0∈Γ0(M) such that µ(ω0)/ne}ationslash=ν(ω0) or\n/integraldisplay\nTMω0dµ/ne}ationslash=/integraldisplay\nTMω0dν.\nThe next ingredient regarding the steps followed by Bernard and Co ntreras is the\nproof of injectivity of the map π:M(L)→G.\nRecall that M(L) the set of minimizing measures for Land/tildewiderM0=/uniontext\nµ∈M(L)suppµ\nis the Mather set.\nLemma 6 LetLbe a Exact Magnetic Lagrangian. If µandνare two distincts mini-\nmizing measures, then there exists a 1-form ωinΓ0(M)such that\n/integraldisplay\nTMωdµ/ne}ationslash=/integraldisplay\nTMωdν\nProof:Ifµ/ne}ationslash=ν,there exists Ain the Borel sigma algebra such that µ(A)/ne}ationslash=ν(A).\nWe can suppose Ais a closed set and A⊂supp(µ)∪supp(ν).The energy function\nforLis given by E(x,v) =1\n2/bardblv/bardbl2and since\nsupp(µ)∪supp(ν)⊂E−1(α(0)) =/braceleftbig\n(x,v)∈TM:/bardblv/bardbl2= 2α(0)/bracerightbig\n,\n7we haveA⊂E−1(α(0)).Moreover, A⊂/tildewiderM0,where/tildewiderM0is the Mather set. By graph\nproperty Ais a graph on π(A) and we can write\nA=/braceleftbig\n(x,v) :x∈π(A) andv=π−1(x)/bracerightbig\nwhereπ−1is Lipschitz on the projected Mather set. Let\nX(x) =/braceleftBigg\nπ−1(x),ifx∈π(A)\n0,ifx /∈π(A)\nand consider fn:M→[0,1] sequence of smooth bump functions\nfn(x) =/braceleftBigg\n1,ifx∈π(A)\n0,ifx /∈Bn(π(A))\nwhereBn(π(A)) is a neighborhood of the compact π(A) :\nBn(π(A)) =/braceleftbigg\nx∈M:d(x,a)<1\nn,for some a∈π(A)/bracerightbigg\n.\nLet us consider Xa continuous extension of X|π(A)onM.Then the vector field Xn=\nfnX∈X0(M),converges pointwise to X(x) and\n|/an}bracketle{tXn(x),v/an}bracketri}ht|=/vextendsingle/vextendsingle/angbracketleftbig\nfnX(x),v/angbracketrightbig/vextendsingle/vextendsingle≤/vextendsingle/vextendsinglefnX(x)/vextendsingle/vextendsingle|v| ≤/vextendsingle/vextendsingleX(x)/vextendsingle/vextendsingle|v|.\nBy Dominated Convergence Theorem\n/integraldisplay\nTM/an}bracketle{tXn(x),v/an}bracketri}htdµ→/integraldisplay\nTM/an}bracketle{tX(x),v/an}bracketri}htdµ,\nand/integraldisplay\nTM/an}bracketle{tXn(x),v/an}bracketri}htdν→/integraldisplay\nTM/an}bracketle{tX(x),v/an}bracketri}htdν.\nSuppose that for all ω∈Γ0(M),\n/integraldisplay\nTMωdµ=/integraldisplay\nTMωdν.\nThen we have/integraldisplay\nTM/an}bracketle{tXn(x),v/an}bracketri}htdµ=/integraldisplay\nTM/an}bracketle{tXn(x),v/an}bracketri}htdν.\n8Therefore\n/integraldisplay\nTM/an}bracketle{tX(x),v/an}bracketri}htdµ=/integraldisplay\nTM/an}bracketle{tX(x),v/an}bracketri}htdν\n/integraldisplay\nA/an}bracketle{tX(x),v/an}bracketri}htdµ=/integraldisplay\nA/an}bracketle{tX(x),v/an}bracketri}htdν\n/integraldisplay\nA/an}bracketle{tX(x),X(x)/an}bracketri}htdµ=/integraldisplay\nA/an}bracketle{tX(x),X(x)/an}bracketri}htdν\n/integraldisplay\nA2α(0)dµ=/integraldisplay\nA2α(0)dν\nα(0)µ(A) =α(0)ν(A),\nHenceµ(A) =ν(A) because α(0)>0 (See G. Paternain and M. Paternain in [13]) .\nThis finishes the proof.\nThe final step is entirely analogous to Lemma 9 of [1] and we repeat it h ere only\nfor the sake of completeness. Ma˜ n´ e introduced a special type o f probability measures,\nthe holonimic measures which is useful to prove genericity results. A C1curveγ:I⊂\nR→Mof period T >0 define an element µγ∈Fby\nµγ(f) =1\nT/integraldisplayT\n0f(γ(s),˙γ(s))ds\nfor eachf∈C.Let\nΘ =/braceleftbig\nµγ:γ∈C1(R,M) periodic of integral period/bracerightbig\n⊂F.\nThe setHof holonomic probabilities is the closure of Θ in F.One can see His convex\n(see Ma˜ n´ e [9]). The elements µofHsatisfyµ(1) = 1.We define the compact HN⊂F\nas the set of holonomic probability measures which are supported in BN.Therefore we\nhaveπ(HN)⊂KN.\nThe each Tonelli Lagrangian Lit is associated an element ¯L∈H∗\nNas follows\nµ/ma√sto→/integraldisplay\nTMLdµ,µ∈HN.\nRecalling that we have defined MHN(L) as the set of measures µ∈HNwhich minimize\nthe action/integraltext\nLdµonHN.\nLemma 7 IfLis a Exact Magnetic Lagrangian then there exists N∈Nsuch that\ndimMKN(L) = dimM(L).\n9Proof:Ma˜ n´ e proves in [9] that M(L)⊂ H. The Mather set /tildewiderM0is compact, therefore\nM(L)⊂HNfor some N∈N.Ma˜ n´ e also proves in [9] that minimizing measures are\nalso all the minimizers of action functional AL(µ) =/integraltext\nLdµon the set of holonomic\nmeasures, therefore M(L) =MHN(L). By previous Lemma the map π:M(L)→G\nis injective, so that\ndimπ(MHN(L)) = dim π(M(L)) = dim M(L)\nProof:(of Theorem 1) Given n∈Napply Theorem 3 and obtain a residual subset\nOn(A)⊂E= Γ1(M) such that\nL∈A,ω∈ On(A)⇒dimMKn(L+ω)≤dimA.\nLetO(A) =/intersectiontext\nnOn(A).By the Baire property O(A) is residual. We have that\nL∈A,ω∈ O(A),n∈N⇒dimMKn(L+ω)≤dimA.\nThen by previous Lemma, dim M(L+ω)≤dimAfor allL∈Aand allω∈ O(A).\nThis finishes the proof.\n3 Some Dynamical Consequences\nAs it was pointed out in the Introduction, the Mather set /tildewiderMcassociated to a\ncohomology class cis contained in another compact invariant set called the Aubry set\n/tildewideAc. It is also a graph over a compact subset of the manifold Mand it is contained\nin the same energy level α(c) as/tildewiderMc. Moreover,/tildewideAcis chain recurrent set. All these\nproperties are proven in [4], see also [6].\nIn order to state the dynamical consequences of our Theorem 1, we need to intro-\nduce the Aubry set and the concept of static classes for a genera l Tonelli Lagrangian.\nLet us consider the actionon a curve γ: [0,T]→Mdefined by\nAL−c+k(γ) =/integraldisplayT\n0[L(γ,˙γ)−η(γ)˙(γ)+k]dt\nwherekis a real number and ηis a representative of the class c.The energy level α(c),\nnamely Ma˜ n´ e’s critical value of the Lagrangian L−c,may be characterized in several\nways.α(c) is defined by Ma˜ n´ e as the infimum of the numbers ksuch that the action\nAL−c+k(γ) is nonnegative for all closed curve γ: [0,T]→M.\n10Recall that, for a given real number kthe action potential Φ L−c+k:M×M→R\nis defined by\nΦL−c+k(x,y) = infAL−c+k(γ)\ninfimum taken over the curves γjoiningxthey.\nMa˜ n´ e proved that\n−α(c) = inf\nµ∈M(L)/integraldisplay\nTM(L−η)dµ,\nwhereηis a representative of the class cand that α(c) is the smallest number such\nthat the action potential is finite, in other words, if k < α(c), then Φ L−c+k(x,y) =−∞\nand fork≥α(c), ΦL−c+k(x,y)∈R.\nObserve that by Tonelli’s Therorem (See for example in [4]), for fixed t >0, there\nalways exists a minimizing extremal curve connecting xtoyin timet. The potential\ncalculates the global (or time independent) infimum of the action. Th is value may not\nbe realized by a curve.\nThe potential Φ L−c+α(c)is not symmetric in general but\nδM(x,y) = ΦL−c+α(c)(x,y)+ΦL−c+α(c)(y,x)\nis a pseudo-metric. A curve γ:R→Mis calledsemistatic if minimizes action between\nany of its points:\nAL−c+α(c)/parenleftbig\nγ|[a,b]/parenrightbig\n= ΦL−c+α(c)(γ(a),γ(b)),\nandγis called staticif is semistatic and δM(γ(a),γ(b)) = 0 for all a,b∈R.\nForexample, theorbitscontainedintheMatherset /tildewiderMcprojectontostaticcurves.\nThe Aubry set /tildewideAcis the set of the points ( x,v)∈TMsuch that the projection\nγ(t) =π◦ϕt(x,v) is a static curve, where ϕtis the Euler-Lagrange flow. We just saw\nthat the Mather set /tildewiderMcis contained in the Aubry set /tildewideAc.\nDenoting the projected Aubry set by Ac, the function δM|Ac×Ac:Ac×Ac→R\nis called Mather semi-distance. We define the quotient Aubry set (AM,δM) to be\nthe metric space by identifying two points x,y∈ Acif their semi-distance δM(x,y)\nvanishes. When we consider δMon the quotient space AMwe will call it the Mather\ndistance and the elements of AMare called static classes forL−c. Observe that the\nstatic classes are disjoint subsets of the energy level set α(c) and a static curve is in\nthe same static class.\nThen we have the following corollary of the Theorem 1:\n11Corollary 8 LetLbe a Exact Magnetic Lagrangian. Then there exists a residual\nsubsetOofΓ1(M)such that for all c∈H1(M,R)and for all ω∈ O, the Lagrangian\nL+ω−chas at most 1+dimH1(M,R)static classes.\nProof:Itsufficestoshowthateachstaticclasssupportsatleastoneerg odicminimizing\nmeasure. In fact, let Λ be a static class for L+ω−cand (p,v)∈/tildewideAcwithp∈Λ.For\nT >0 we define a Borel probability measure µTonTMby\nµT(f) =1\nT/integraldisplayT\n0f(ϕs(p,v))ds\nAll these probability measures have their supports contained in /tildewideActhat is a compact\nsubset, consequently, we can extract a sequence µTnweakly convergent to µ:\nµ(f) = lim\nT→∞1\nTn/integraldisplayTn\n0f(ϕs(p,v))ds,\nwhich is a ergodic minimizing measure whose support is contained in Λ (Se e [6] for\ndetails).\nNow we present some dynamical consequences assuming that the L agrangian L\nhas finitely many static classes. In this manner, by previous corollar y, the properties\npresented here are generic on set of Exact Magnetic for all cohom ology class.\nThe projected Aubryset Acischain recurrent andthestaticclasses areconnected\nso they are the connected components of Ac. Moreover the static classes are the chain\ntransitivecomponents of Acandweobtainthefollowingcycle property: Iftwo supports\nof ergodic minimizing measures are contained in a static class, then th ere exists a cycle\nconsisting of static curves in the same static class connecting them .\nContreras and Paternain prove in [5] that between two static clas ses there exists\na chain of static classes connected by heteroclinic semistatic orbits . More precisely\nthey show\nTheorem 9 Suppose that the number of static classes is finite. Then give n two static\nclassesΛkandΛl,there exist classes Λ1= Λk,Λ2,...,Λn= Λlandθ1,θ2,...,θn−1∈TM\nsuch that for all i= 1,...,n−1we have that γi(t) =π◦ϕt(θi)are semistatic curves,\nα(θi)⊂Λiandω(θi)⊂Λi+1.\nAnother important property, demonstrated by P. Bernard in [2], is the semi-\ncontinuity of the Aubry set\nH1(M,R)∋c/ma√sto→/tildewideAc,\nwhenAMis finite. In order to be more precise he showed the following Theorem\n12Theorem 10 LetLkbe a sequence of Tonelli Lagrangians converging to L.Then given\na neighborhood Uof/tildewideA0inTM,there exists k0such that/tildewideA0(Lk)⊂Ufor eachk≥k0,\nwhere/tildewideA0(Lk)is the Aubry set for the Lagrangian Lk.\nIn fact Bernard showed that this Theorem is true with a weaker hyp othesis than\nAMbe finite, namely coincidence hypothesis (See [2]).\n4 Example\nIn this section we present an example of a Exact Magnetic Lagrangia n on flat\ntorusT2whose quotient Aubry set AMis a Cantor set, therefore not every Exact\nMagnetic Lagrangian has finitely many static classes.\nLetL:TT2→Rbe a Exact Magnetic Lagrangian defined by\nL(x,y,v1,v2) =/bardbl(v1,v2)/bardbl2\n2+/an}bracketle{t(0,f(x)),(v1,v2)/an}bracketri}ht,\nwherefis aC2nonpositive and periodic function whose set of minimum points Γ min\nis a Cantor set and f|Γminis a negative constant.\nIn this case the system of Euler-Lagrange is given by\n/braceleftBigg\n˙x=v\n˙v=−f′(x)Jv\nwhereJis the canonical sympletic matrix.\nLemma 11 The Ma˜ n´ e’s critical values of Lisα(0) =f(a)2/2,wherea∈Γmin.\nMoreover, the closed curves γadefined by γa(t) = (a,−f(a)t),are static curves.\nProof:Given any curve β(t) = (x(t),y(t)) onT2,we have\nL/parenleftBig\nβ,˙β/parenrightBig\n=˙x2+ ˙y2\n2+f(x) ˙y(t) =(˙y+f(x))2+ ˙x2\n2−f(x)2\n2≥ −f(a)2\n2.(3)\nThen\nAL+f(a)2/2(β) =/integraldisplayT\n0/parenleftBigg\nL/parenleftBig\nβ,˙β/parenrightBig\n+f(a)2\n2/parenrightBigg\ndt≥0,\nand we obtain α(0)≤f(a)2\n2.Observe that if 0 < k≤f(a)2\n2,the closed curve given\nbyγk(t) =/parenleftBig\na,√\n2kt/parenrightBig\n,wherea∈Γmin,is Euler-Lagrange solution and its energy is\nE=k. Moreover,\nAL+k(γk) =/integraldisplayT\n0(L(γk,˙γk)+k)dt=/integraldisplayT\n0/parenleftBig\n2k+f(a)√\n2k/parenrightBig\ndt.\n13Therefore\nAL+k(γk)<0 ifk <f(a)2\n2andAL+k(γk) = 0 ifk=f(a)2\n2.\nThis shows that α(0) =f(a)2\n2and the curve γais semistatic, i.e., realizes the action\npotential. Since γais a semistatic closed curve, it is static curve.\nTo complete the example, it suffices to show that the application Ψ : Γ min→ AM,\ngiven by Ψ( a) = [(a,0)],where [(a,0)] is a representative of the static class conteining\nthe curve γa,is a Lipschitz bijection. In fact, since the action potential Φ L+α(0)is\nLipschitz, the distance δMon quotient Aubry set AMalso is Lipschitz.\nIn order to show the surjectivity of Ψ it is enough to show that the p rojected\nAubry set A0is exactly the union of the closed curves γawitha∈Γmin.Suppose that\nthere exists p∈ A0such that π(p)/∈Γmin,whereπis the canonical projection of T2on\nR/Z.Then there exists a neighborhood Vpofpsuch that f(x)> f(a) for alla∈Γmin\nandx∈Vp. Letγbe a piece, contained in Vp,of the static curve passing through p.\nThe inequality 3 implies AL+α(0)(γ)>0.Moreover, it follows by inequality 3 which\nthe action L+α(0) of any curve is nonnegative, so Φ L+α(0)(x,y)≥0 for allx,y∈T2.\nThen\nAL+α(0)(γ) = ΦL+α(0)(γ(0),γ(T)) =−ΦL+α(0)(γ(0),γ(T))≤0.\nThis is a contradiction.\nIf Ψ is not injective there exists b∈Γmin,b/ne}ationslash=asuch that ( b,0)∈[(a,0)].Since\neach static class is connected (See G. Contreras and G. Paternain in [5], Proposition\n3.4) and b∈π([(a,0)]) we have that π([(a,0)])⊂R/Zis connected so it is an interval.\nBy total disconnectedness of Γ min, there exists q∈π([(a,0)])−Γmin.The contradiction\nfollows of the inequality 3 by same argument above.\nReferences\n[1] Bernard, P., Contreras, G., A generic property of families of Lagrangian systems,\nAnnals of Mathematics, Pages 1099-1108 from Volume 167 (2008).\n[2] Bernard, P., On the Conley Decomposition of Mather sets , Rev. Mat. Iberoameri-\ncana, vol. 26, no. 1, pp. 115–132 (2010).\n14[3] Carneiro, M. J., Lopes, A., On the minimal action function of autonomous la-\ngrangians associated to magnetic fields , Annales de l’I. H. P., section C, tome 16,\nN.6, 667-690, (1999).\n[4] Contreras, G., Iturriaga, R., Global Minimizers of Autonomous Lagrangians ,\nCIMAT, M´ exico, (2000).\n[5] Contreras, G., Paternain, P., Connecting orbits between static classes for generic\nLagrangian systems, Topology, 41 645-666, (2002).\n[6] Fathi, A., Weak KAM Theorem in Lagrangian Dynamics, Workshop and School\non Conservative Dynamics, (2006).\n[7] Fathi, A., Figalli A., Rifford L., On the Hausdorff dimension of the Mather quo-\ntient.Comm. Pure Appl. Math. 62, no. 4, 445-500, (2009).\n[8] Ma˜ n´ e, R., On the minimizing measure of Lagrangian dynamical systems, Nonlin-\nearity, 5, N.3, 623-638, (1992).\n[9] Ma˜ n´ e, R., Generic properties and problems of minimizing measure of La grangian\ndynamical systems, Nonlinearity, 9, N.2, 273-310, (1996).\n[10] Ma˜ n´ e, R., Global Variational Methods in Conservative Dynamics , IMPA, (1993).\n[11] Mather, J. N., Action minimizing invariant measures for positive definite La-\ngrangian Systems , Math. Zeitschrift, 207, 169-207, (1991).\n[12] Mather, J. N., Variational construction of connectings orbits, Ann. Ins. Fourier\n(Grenoble) 43, no.5, 1349-1386, (1993).\n[13] Paternain, M., Paternain, G., Critical Values of autonomous Lagrangian systems,\nComment. Math. Helvetici, 7 481, (1997).\n15"    },    {        "title": "2311.05852v2.Exchange_stiffness_proportional_to_power_of_magnetization_in_permalloy_co_doped_with_Mo_and_Cu.pdf",        "content": "1 \n Exchange stiffness proportional to power  of magnetization  \nin permalloy co -doped with Mo and Cu  \n \nShiho Nakamura, Nobuyuki Umetsu, Michael Quinsat, and Masaki Kado  \nInstitute of Memory Technology Research and Development, Kioxia Corporation  \n3-13-1, Moriya -cho, Kanagawa -ku, Yokohama, 221 -0022, Japan  \n \nAbstract  \nThe exchange stiffness of magnetic materials is one of the essential parameters governing magnetic \ntexture and its dynamics in magnetic devices. The effect of single -element doping on exchange \nstiffness has been investigated for several doping elements for permalloy (NiFe alloy), a soft magnetic \nmaterial whose soft magnetic properties can be controlled by doping. H owever, the impact of more \npractical multi -element doping on the exchange stiffness of permalloy is unknown. This study \ninvestigates the typical magnetic properties, including exchange stiffness, of permalloy systematically \nco-doped with Mo and Cu using br oadband ferromagnetic resonance spectroscopy. We find that the \nexchange stiffness, which decreases with increasing doping levels, is proportional to a power of \nmagnetization, whic h also decreases with increasing doping levels. The magnetization , Ms, depend ence \nof the exchange stiffness constant, A, of all the investigated samples, irrespective of the doping level s \nof each element, lies on a single curve expressed as A∝Msn with exponent n close to 2 . This empirical \npower -law relationship provides a guideline for predicting unknown exchange stiffness in non -\nmagnetic element -doped permalloy systems.  \n \nKeywords: permalloy , ferromagnetic resonance,  exchange stiffness , g-factor, Gilbert damping constant  2 \n 1. Introduction  \nExchange stiffness represents the amount of energy difference associated with a spatial change in \nspins (magnetic moments) in a non -uniformly magnetized ferromagnet , reflect ing the strength of the \ninteraction between spins  in the material  [1-3]. It determine s magnetic textures, including magnetic \ndomain  [1], domain wall width  [3-7], magnetic skyrmions  [8,9], and vortexes  [10,11], as well as their \ndynamics  [12-16], together with magnetization and magnetic anisotropy . Therefore, it is one of the \nessential  parameter s to simulate and then predict the performance of various magnetic devices such as \nmagnetic memories and logics . Despite its indispensability, there is far less information compared to \nthe other basic parameters of magnetic materials , magnetization and magnetic anisotropy. One of the \nreasons for this is the lack of simple methods to measure exchange stiffness  constant A. An efficient \nway to measure A is to excite spin waves and measure their spectra. Excitation methods include neutron \nscattering [17,18], Brillouin light scattering (BLS) using photons  in the visible r ange  [19-22], and \nferromagnetic resonance (FMR) using microwaves of fixed [2 3,24] or b roadband frequencies [ 22, 25-\n27]. In particular, broadband FMR with microstrip lines or coplanar waveguides, which have become \nreadily available due to recent advances in high-frequency technology, is a relatively accessible and \nsensitive method that can also measure other parameters including magnetic damping constant.   \nPermalloy , based on NiFe alloys , is a well -known typical soft magnetic material  [28-31]. In \nparticular, NiFe  alloys in combination with several doping elements , such as Mo and Cu , are widely \nused in practical applications  due to their ultra-high magnetic permeability [ 29-31]. Changes in \nfundamental physical p roperties of permalloy , including m agnetization and exchange stiffness , with \ndoping , have been investigated for single -element doping such as Pt, Cu , V, and Ta by means of spin-\nwave measurements using FMR and BLS methods , and ab initio  calculations  [22,25, 27,32-33]. \nHowever, to the best of our knowledge, there are no reports on the exchange stiffness of multi -element \ndoped systems, which are closer to practical materials.   3 \n In this study, we report on the effect of Mo and Cu co -doping on the exchange stiffness of permalloy \nalong with the effect on other typical magnetic properties  including magnetization, g-factor, and \nGilbert damping constant . The composition ratio of NiFe (Py) is 80:20 at . % and t he doping level of \nMo ranges from 0 to 9.4 at . % in Mo/Py, and that of Cu from 0 to 11.7 at. %  in Cu/Py. Broadband FMR \nmethod  is used to evaluate the parameters . We find the dependence of exchange stiffness on saturation \nmagnetization , Ms, exhibits a  relationship,  A∝Msn, with the exponent n is 2.24  ± 0.04 , for all \ncombinations of Mo and Cu doping levels investigated. We also investigate the magnetization \ndependence of exchange stiffness in single -element doped systems from literatures, and f ind that the \npower  law relationship seems to be a general trend for non-magnetic element doped permall oy systems .  \n \n2. Experimental  \nThe investigated Py-Mo-Cu samples were formed b y alternating deposition of sub-atomic layers \nof the constituent elements in order to systematically control the composition. A magnetron sputter \ndeposition system with base pressure less than 3 ×10-6 Pa was used for deposition, and Si wafers with \na thermal oxidation layer were used as substrates. The film stacking procedures were [Fe  (0.11 nm)  / \nNi (0.35 nm)  / Cu (n × 0.02 nm, n = 0~3) / Mo (m × 0.01 nm, m = 0~4)] × 110 cycles and [Fe  \n(0.11 nm)  / Ni (0.35 nm)  / Cu (4× 0.02 nm)]  × 110 cycles. All the films were capped with Ta (3 \nnm) and SiO 2 (10 nm) to protect surface oxidation. The thickness es of the obtained Py-Mo-Cu film \nranged from 49.7 nm to 62.5 nm, depending on the doping level. The composition of the films was \ndetermined by X -ray florescence (XRF) measurements with fundamental parameter analysis  [34]. A \ntypical XRF spectrum  of the film is shown in Fig. 1. The average  Py concentration in the 21 samples \nwas 80.2 at. % for Ni and 19.8 at. % for Fe, with a standard deviation of 0.7 at. % for both Ni and Fe  \nin the  21 samples . An excellent linear relationship between the measured doping concentratio n and \nthe number of sub -atomic layers of dopant was confirmed. Table 1 summarize s the film composition. 4 \n Here, linear fit values were used as nominal values for Cu and Mo concentrations in Cu/Py and Mo/Py, \nrespectively. The difference between these nominal values and measured values is negligible, about \n±0.1 at. % in  average and a maximum of 0.4 at. % for highest concentration of Cu. Therefore, we \nused the nominal values in all the manuscript.  The X -ray diffraction patterns for each film showed a \nsingle diffraction peak corresponding to (111) reflections  of face-centered cubic NiFe alloy , \nsuggesting that the elements are well alloyed and the films are polycrystal line with <111> preferred \ncrystal orientation. The lattice constant of undoped Py was 0.3546 nm, which is in good agreement \nwith that of the bulk  value  of Py [17]. The presence of Cu dopants hardly change d the lattice constant, \nas reported for a single Cu doping [ 22]. Whereas t he addition of 9.4 % Mo increase d it to 0.3581 n m, \nbut the increase rate is as small as 1 %.  \nTypical m agnetization curves measured using a vibrating sample magnetometer (VSM) are shown \nin Fig.  2. The coercive force for the magnetic field applied along easy -axis was less than 1 mT  for all \nthe compo sitions. Whereas for the hard -axis, the anisotropy field Hk increased  from 1.7  mT to 11.7  \nFigure 1. X -ray florescence spectra of Py co -doped with 9.4 at. % Mo/Py and \n9.4 at. % Cu/Py (Py85Mo8Cu7). The thick black curves represent the \nmeasurement results and the thin colored curves represent each component \nanalyzed.  BG i ndicates  the background signal.  \n 5 \n mT, depending on the doping levels , particularly affected by Mo doping  as shown in Table 1 . This \nuniaxial anisotropy must be caused by the induced magnetic anisotropy during deposition and by the \nnon-zero-magnetostriction due to doping [ 30,35].  \nFMR and perpendicularly standing spin wave ( PSSW ) were measured using the Broadband FMR \nmethod using  a coplanar waveguide with microwave frequency range up to 40 GHz . The external  field \nof up to 1 T was applied in the plane of the film perpendicular to  the microwave field.  Typical \nmeasurements of both FMR and PSSW modes  are shown in the insets in Fig.  3 with the PSSW mode \nappearing at lower field. The spectr a were fit with a sum of symmetrical and antisymmetrical \nLorent zian derivatives in Equation (1) in Ref. 25. The field dependence of the extracted FMR and \nPSSW resonance frequencies are shown in Fig.  3, together with fits to the equation  for films with in -\nplane uniaxial anisotropy ,  \n𝑓=𝛾𝜇0\n2𝜋[(𝐻res+𝐻k(𝜑)𝑎+2𝐴\n𝜇0𝑀s(𝑝𝜋\n𝑑)2\n)×(𝐻res+𝐻k(𝜑)𝑏+𝑀s+2𝐴\n𝜇0𝑀s(𝑝𝜋\n𝑑)2\n)]1\n2\n,(1)  \nFigure 2. Magnetization curves of Py co -doped with 2.3 at. % Mo/Py and 5.9 at. % \nCu/Py (Py92Mo2Cu6) for easy - and hard -axis. 6 \n where  f is the frequency,  𝐻res is the magnetic resonance field of either the FMR ( p = 0) or the PSSW \nmode ( p = 1), γ is the gyromagnetic ratio, 𝜇0 is the permeability of free space , 𝐻k(𝜑)𝑎 and 𝐻k(𝜑)𝑏 are \nthe in -plane anisotropy field s expressed as 𝐻k(𝜑)𝑎=−𝐻kcos(𝜋−2𝜑) and 𝐻k(𝜑)𝑏=𝐻kcos2𝜑, 𝜑 is \nthe azimuthal angle  of the applied field from the easy axis , and d is the thickness of the film  [25,36-\n38]. When 𝜑  is zero, 𝐻k(𝜑)𝑎  = 𝐻k(𝜑)𝑏  = 𝐻k . For  𝐻k , we use d the v alues obtained f rom VSM \nmeasurements. From such fits, we extracted Ms and γ from the FMR mode and then,  extracted A from \nthe PSSW mode. γ gives the g-factor  g from the relation  𝛾=2𝜋𝑔𝜇B/ℎ. The determined values of un -\ndoped Py were  𝜇0Ms = 0.995  ± 0.028  T, A =11.7 ± 1.2  pJ/m, and g = 2.09  ± 0.01 , which  are consistent \nwith the literatures  [25,28,37,39]. Further, i n order to derive Gilbert damping co nstant  𝛼 , the \nfrequency -dependent FMR linewidth ∆𝐻F of the films were fit using  \n∆𝐻F=∆𝐻0+4𝜋𝛼\n𝛾𝑓, (2)  \nFigure 3. Frequency versus resonance magnetic field of the FMR mode (closed circles) and \nPSSW mode (open circles) for Py co -doped with 4.7 at. % Mo/Py and 2.9 at. % Cu/Py \n(Py93Mo4Cu3).  The solid lines are fits to the Kittel equation. Inset shows the resonance curv e \nof the same sample for frequency of 24 GHz.  \n 7 \n wher e ∆𝐻0  is the inhomogeneous broadening [4 0,41]. The parameters extracted from the fits are \nlisted in Table 1.    \n \n \n \n3. Results   Table 1. Extracted values for μ0Ms, g-factor, A, and α of Py -Mo-Cu alloys  from FMR measurements . \n___________________________________________________________________________________________  \nCu/Py  Mo/Py  Composition      Hk         μ0Ms         g-factor          A           α \n(at. %)  (at. %)               (10-4 T)        (T)                         (pJ/m)        (10-3) \n____________________________________________________________________________________________  \n0      0     Py100        17 ± 2     0.995 ± 0.0 18    2.09 0 ± 0.01 2   11.72 ± 1. 03    7.1 ± 0.2  \n0     2.3    Py98Mo2       20 ± 2     0.833 ± 0.0 15    2.091 ± 0.01 1    7.91 ± 0.88    8.3 ± 0.2  \n0     4.7    Py96Mo4       26 ± 2     0.680 ± 0.01 2    2.091 ± 0.01 0    4.82 ± 0. 65    9.9 ± 0.2  \n0     7.0    Py93Mo7       39 ± 2     0.533 ± 0.01 0    2.095 ± 0.0 09    2.70 ± 0. 45   13.2 ± 0.2  \n0     9.4    Py91Mo9       62 ± 2     0.393 ± 0.0 08    2.100 ± 0.0 09    1.63 ± 0. 38   18.0 ± 0.3  \n2.9     0     Py97Cu3       23 ± 2     0.943 ± 0.0 18    2.09 4 ± 0.01 2   10.74 ± 1. 06    7.7 ± 0.3  \n2.9    2.3    Py95Mo2Cu3    27 ± 2     0.785 ± 0.0 15    2.094 ± 0.01 1    6.75 ± 0. 78   9.0 ± 0.2  \n2.9    4.7    Py93Mo4Cu3    33 ± 2     0.632 ± 0.01 2    2.095 ± 0.0 10    4.11 ± 0. 57   10.8 ± 0.2  \n2.9    7.0    Py91Mo6Cu3    50 ± 2     0.490 ± 0.01 0    2.097 ± 0.009     2.47 ± 0.4 3   15.1 ± 0.4  \n2.9    9.4    Py89Mo8Cu3    85 ± 2     0.352 ± 0.0 07    2.103 ± 0.0 08    1.44 ± 0. 30   18.4 ± 0.5  \n5.9     0     Py94Cu6        25 ± 2     0.894 ± 0.0 17    2.095 ± 0.0 11    9.36 ± 0.98    7.8 ± 0.2  \n5.9    2.3    Py92Mo2Cu 6    31 ± 2     0.737 ± 0.0 14    2.096 ± 0.01 1    5.92 ± 0. 74    9.1 ± 0.2  \n5.9    4.7    Py9 1Mo4Cu5    43 ± 2     0.593 ± 0.01 1    2.094 ± 0.01 0    3.63 ± 0. 59   12.0 ± 0.3  \n5.9    7.0    Py89Mo6Cu5    57  ± 2     0.446 ± 0.0 09    2.100  ± 0.0 09    2.04 ± 0. 38   15.7 ± 0.4  \n5.9    9.4    Py87Mo8Cu 5    95 ± 2     0.309 ± 0.0 07    2.105 ± 0.0 09    1.20 ± 0.2 9   19.6 ± 0.6  \n8.8     0     Py92Cu8        31 ± 2     0.852 ± 0.0 16    2.094 ± 0.0 12    8.31 ± 0.96    7.7 ± 0.2  \n8.8    2.3    Py90Mo2Cu8    38 ± 2     0.693 ± 0.0 13    2.096 ± 0.0 10    5.15 ± 0.7 7   10.2 ± 0.2  \n8.8    4.7    Py88Mo4Cu8    51 ± 2     0.547 ± 0.01 0    2.100  ± 0.0 09    3.01 ± 0.5 7   13.0 ± 0.3  \n8.8    7.0    Py86Mo6Cu8    68 ± 2     0.407 ± 0.0 09    2.104 ± 0.0 09    1.67 ± 0. 40   16.5 ± 0.4  \n8.8    9.4    Py85Mo8Cu7   117 ± 3     0.273 ± 0.0 07    2.109 ± 0.0 09    0.97 ± 0.3 3   20.9 ± 0.5  \n11.7     0     Py90Cu10      29 ± 2     0.807 ± 0.0 15    2.098  ± 0.01 1    7.25 ± 0.91    8.1 ± 0.2  \n____________________________________________________________________________________________  \n \n 8 \n Mo and Cu doping level  dependences of magnetization and exchange stiffness in the c o-doped Py \nsystem are shown in Fig.  4 (a) and (b). Although b oth magnetization and exchange stiffness show a \ndecrease with increasing doping levels, the former is very monoton ic. The magnetization shows a \nlinear  relation w ith dop ing in the composition range studied.  The Cu (Mo) doping level dependence s \nat different Mo (Cu) doping levels are almost parallel with each other, indicating that the Mo and Cu \naffect the magnetization independently. The rates of magnetization reduction , per each doping level \nexpressed in at. % with respect to total alloy , are 6.4 and 1 .5 % for Mo and Cu, respectively , which are \ngreater  than the 1 % reduction expected if the NiFe atoms were simply replaced by non -magnetic \natoms . These reduction rates are similar to the results o btained from density functional theory \ncalculations for single -element doped of Mo or Cu, in which Mo addition lowers the Ni and Fe site \nmoments and Mo polarizes antiferromagnetically, while Cu addition lowers the Ni site moment  [42]. \nSince Mo doping and Cu doping can be treated as independent of each other for magnetization, w e can \ndetermine the magnetization of co -doped Py  by summing the reduction in magnetization due to Mo \nand Cu.  \nThe exchange stiffness of co-doped Py decreases more steeply than the decrease in magnetization , \nsimilar to that reported for single -element  dopin g Py systems [22,27]. The reduction is more \npronounced with Mo doping compared to Cu doping. The slope of the curve s showing the Mo (Cu) \ndependence of exchange stiffness at each Cu (Mo) doping level varies somewhat depending on the Cu \n(Mo) doping level . It indicates that the two dopants, Mo and Cu, affect the exchange stiffness in weak \ncorrelation with each other, unlike the behavior of magnetization . Here, to clarify the relationship \nbetween the magnetization and the exchange stiffness, the exchange st iffness is plotted as a function \nof magnetization in Fig. 5 for all composition s examined . Surprisingly, t he marks representing  the A-\nMs relation ship all lie on a single curve  expressed in a power , regardless of doping  levels of ea ch 9 \n element . The power exponent is found to be 2.24  ± 0.04 . In order to examine  the dependence in more  \nFigure 4. Doping level dependence of magnetization (a), exchange stiffness (b), g-factor (c), and \ndamping constant (d) for Py co -doped with Mo and Cu. Black -filled marks indicate the dependence \non Mo concentration at constant Cu concentration, with closed circles,  closed triangles, closed large \nsquares, closed diamonds, and closed small squares corresponding to Cu doping levels of 0, 2.9, 5.9, \n8.8, and 11.7 at . % in Cu/Py, respectively. White -filled marks indicate the dependence on Cu \nconcentration at constant Mo c oncentration, with open circles, open triangles, open large squares, open \ndiamonds, and open small squares corresponding to the Mo doping levels of 0, 2.3, 4.7, 7.0, and 9.4 \nat. % in Mo/Py, respectively. The dotted lines in (a) show the results of the line ar approximation.  Error \nbars for Figs. 4 (a), (b) and (d) (summarized in Table 1) are omitted. On the other hand, for Fig. 4 (c), \nwhere the estimated errors are large, an error bar is marked at a representative point to indicate \nuncertainty. All plots in Fig. 4 ( c) have similar error bars with an average of ±0.010.  10 \n detail, power -law fitting is performed for each Cu doping level. The results are shown in the first half \nof Table 2. The exponents of the power law for each Cu doping level are consistent within the error, \nindicating that they can still be approximated by a single curve represented by a power. We will discuss \nthis relationship in the next section .  \nThe dependence s of g-factor and damping constant  𝛼 on Mo and Cu doping levels in co-doped \nsystem s are shown in F ig. 4 (c) and (d). Note here that the error bar in Fig.  4 (c) is large. Unlike the \ndecreasing trends in the magnetization and the exchange stiffness mentioned above , doping appears  to \nslightly increase the g-factor , albeit within the margin of error , and it increases damping , especially for \nMo doping . If doping increases the g-factor above 2, it implies an increase in the orbital magnetic  \nFigure 5. Exchange stiffness as a function of magnetization obtained from Py co -doped with \nMo and Cu. Color of the symbol indicates Mo doping level and shape indicates Cu doping \nlevel. The soli d line is a fit to t he power low and the broken line is a fit to square of \nmagnetization.   \n 11 \n moment . The increase in orbital moment leads to an increase in the damping through the interaction  \nbetween spin -orbit interaction (SOI) and spin angular momentum [ 43, 44]. Thus, one would expect the \nbehavior of g-factor to be similar to that of damping. What is characteristic of the damping behavior \nis that the increase in damping is kept very small for Cu doping compared to Mo doping.  It is the \nelectrons near Fermi energy that contribute to the damping [ 40]. Since the Fermi energy of dopant Cu \nis occupied by s -electrons which is less susceptible to the SOI, and SOI itself is smaller in lighter Cu \nthan heavier Mo, the damping is considered to be less susceptible  to Cu doping.   \n \n4. Discussion  \nIn the previous section , we have shown from the experimental results that for two -element doped \nPy, the exchange stiffness  can be expressed as a power of the magnetization . Based on the classical \nHeisenberg model  considering the nearest neighbor spins , its H amiltonian  can be written as   \n𝐻=−𝐽∑𝑆𝑖⃗⃗⃗ ∙𝑆𝑗⃗⃗⃗ \n𝑖,𝑗, (3) \nwith 𝑆𝑖⃗⃗⃗  being a spin for atom i and J is the exchange constant between the nearest neighbor spins. \nAssuming long wave -length deviation s and continuum limit,  the energy necessary to produce the \ndeviation from the uniform ly aligned mode in three -dimensional lattice is derived as follows [ 2,3,5],  \n∆𝐻∝𝐽𝑎2𝑀s2(|∇𝑀𝑥|2+|∇𝑀𝑦|2+|∇𝑀𝑧|2) (4) \nThe prefactor on the right -hand  side corresponds to the exchange  stiffness, thus A∝Ja2Ms2. If the \nchange in the lattice constant a is almost negligible, as in the present case, A∝JMs2. Strictly speaking, \nthis model applies to localized electron system s. In itinerant systems  such as in this study , the \ninteractions not only with the nearest -neighbor  atoms but also with the atoms further away can affect \nthe exchange stiffness  [45]. Nevertheless , the trend  of the exchange stiffness such as dopant \nconcentration dependence can still be explained  by the interactions betwe en the nearest  neighbor s [33].  12 \n For impurity -doped systems, the relation  A∝Ja2Ms2 was first suggested by Ref. 22 in order to \nexplain the experimentally obtained steeper decrease in the exchange stiffness than the magnetization \nwith increasing Cu concentration in Py-Cu system. We have  investigate d the magnetization \ndependence of the exchange stiffness for some single -element dop ed Py  systems , based on the \ndisclosed values of A and Ms in literature s for doped Py  [22,25,33]. The results are summarized in the \nsecond half of Table 2. It shows that the A - Ms relationship  can be fitted  approximately  by power  and \nthe exponents ranged from 2 to just over 3 . The power law therefore seems to be a general trend in Py \nsystems with non-magnetic single -element doping . Interestingly, our results show that the power law \nholds even for co -doped systems .  \nTable 2.  Exponents derived from fitting the Ms dependence of A to the power law and the coefficients of \ndetermination of the fitting.  \n \nComposition  Method  n in \nA = c×Msn Coefficient of \nDetermination  \nR2 Reference  \n(Ni80Fe20) -Mo-Cu broadband \nFMR  2.24 ± 0.04  0.997  This study  \n(Ni80Fe20) -Mo-Cu0%   2.28 ± 0.06  0.999   \n(Ni80Fe20) -Mo-Cu2.9%   2.29 ± 0.11  0.996   \n(Ni80Fe20) -Mo-Cu5.9%   2.20 ± 0.10  0.997   \n(Ni80Fe20) -Mo-Cu8.8%   2.19 ± 0.10  0.997   \n(Ni80Fe20) -Pt broadband \nFMR  2.31 ± 0.42  0.881  25 \n(Ni80Fe20) -Au  1.94 ± 0.18  0.974   \n(Ni80Fe20) -Ag  2.31 ± 0.16  0.988   \n(Ni80Fe20) -Cu broadband \nFMR  3.39 ± 0.49  0.962  22 \n(Ni81Fe19) -V ab initio  \ncalculations  2.76 ± 0.04  1.000  33 \n(Ni81Fe19) -Au  2.16 ± 0.02  0.999   \n 13 \n The exponents of the power law obtained from the doped Py were not 2 but close to 2, predicted \nby the simple Heisenberg model. Because it is the itinerant electron system, deviations from the \nprediction will occur and we need to modify the model. In addition, J defined in Equation (3) can also \nvary depending on the doping elements and their amounts, resulting in the deviation of the expon ents. \nHowever, since the magnetization dependence of the exchange stiffness of co -doped system exhibits a \nsingle curve despite the presence of two doping elements, the change in J may not be so pronounced. \nThe result obtained in this study indicates that m agnetization is a dominant factor for exchange stiffness \nin the doped Py system. We also need to take into considerations the temperature dependence of the \nmagnetization and the exchange stiffness [ 4,5,46-48]. In Fig . 5, small deviations from the fitting curve  \nare observed at small exchange stiffness and small magnetization  region . The decrease in the exchange \nstiffness may have resulted in a lower Curie temperature , which in turn may have resulted in a smaller \nmagnetization.  \nThis study has dealt with non -magnetic dopants. For magnetic dopants, th e power law is not \nsuitable. It has been reported that the doping of Gd, one of the magnetic elements, decreases the \nexchange stiffness whereas it increases the magnetization [33]. The power law would apply to non -\nmagnetic doping elements  and, moreover, to a range of doping levels that do not significant ly affect \nthe exchange interaction itself, which is also a useful range in practical permall oys.  \n \n5. Conclusion s \nIn this study, the effect of Mo and Cu co -doping on the exchange stiffness of Py was investigated \nusing broadband FMR  spectroscopy, along with the effect of co -doping on other typical magnetic \nproperties. Mo and Cu co -doping reduce s the magnetization of Py linearly with respect to each dop ing \nlevels  and affects magnetization independently of each other, irrespective of co -doping. The rates of \nmagnetization reduction per doping level were 6.4 % and 1.5 % for Mo and Cu, respectively , which \nare greater than 1% for simply substituting non -magnetic element . For the exchange stiffness, the two 14 \n dopants lead to a more rapid decrease in the exchange stiffness with a weak correlation with each other. \nIn contrast to these decreasing trends, the g-factor appears to have increased slightly, although within \nthe margin of error, and damping was increased, especially with Mo doping. We found  that the \nmagnetization dependence of exchange stiffness for all the co-doped Py samples lie on a single curve \npresented by  A∝Msn, with the exponent n of 2.24  ± 0.04 . The relationship that the exchange stiffness \nis proportional to the power of magnetization, is also found in the previously reported non -magnetic \nsingle -element doped Py systems. The result obtained in the present study that the exchange stiffness \ncan be expressed by a single power law  even in the presence of two doping elements indicates that \nmagnetization is the dominant factor in exchange stiffness.  The feature that the exchange stiffness  is \nproportional to the power of magnetization remains phenomenological but will provide guidance for \nthe unknown exchange stiffness of non -magnetic element doped systems , facilitating the prediction of \nthe performance of magnetic devices using those syste ms. \n \nAcknowledgment  \nWe thank T. Kondo for useful  discussions.   \n  15 \n References  \n[1] A. Hubert and R. Schäfer, Magnetic Domains, Springer -Verlag, 1998.  \n[2] C. A. F. Vaz, J. A. C. Bland, and G. Lauhoff, Rep. Prog. Phys. 71  (2008)  056501.  \n[3] S. 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Phys. 53  (2020)  39LT01 . "    },    {        "title": "2005.07049v2.Magnetic_field_dependent__t_Hooft_determinant_extended_Nambu__Jona_Lasinio_Model.pdf",        "content": "Magnetic field dependent ’t Hooft determinant extended Nambu–Jona-Lasinio model\nJoão Moreira,1,\u0003Pedro Costa,1,yand Tulio E. Restrepo2, 1,z\n1CFisUC - Center for Physics of the University of Coimbra,\nDepartment of Physics, Faculty of Sciences and Technology,\nUniversity of Coimbra, 3004-516 Coimbra, Portugal\n2Departamento de Física, Universidade Federal de Santa Catarina, Florianópolis, SC 88040-900, Brazil\nWe study the implications of recent lattice QCD results for the magnetic field dependence of the\nquarks dynamical masses on the ’t Hooft determinant extended Nambu–Jona-Lasinio model in the\nlight and strange quark sectors ( up,downandstrange). The parameter space is constrained at\nvanishing magnetic field, using the quarks dynamical masses and the meson spectra, whereas at\nnonvanishing magnetic field strength the dependence of the dynamical masses of two of the quark\nflavors is used to fit a magnetic field dependence on the model couplings, both the four-fermion\nNambu–Jona-Lasinio interaction and the six-fermion ’t Hooft flavor determinant. We found that\nthis procedure reproduces the inverse magnetic catalysis, and the strength of the scalar coupling\ndecreases with the magnetic field, while the strength of the six-fermion ’t Hooft flavor determinant\nincreases with the magnetic field.\nPACS numbers: 11.30.Qc,11.30.Rd,12.39.Fe,14.65.-q,21.65.Qr,75.90.+w\nI. INTRODUCTION\nThe effect of strong magnetic fields in strongly inter-\nacting matter plays a very important role in several phys-\nical contexts such as in heavy ion collisions [1–3], in as-\ntronomic compact object like magnetars [4, 5], and in\nthe first phases of the early Universe [6, 7]. Quantum\nchromodynamics (QCD) is the theory of the strong inter-\naction between quarks and gluons whose phase diagram\nhave been widely studied by lattice QCD (LQCD) simu-\nlations and in the context of effective models also in the\npresence of magnetic fields (see [8] for a review).\nAt zero temperature, and in the presence of magnetic\nfields, LQCD and the vast majority of effective models\npredict magnetic catalysis, which is the increment of the\nchiral order parameter, the light quark condensate, as\nthe magnetic field, B, increases. This phenomenon is\noriginated by the dominant contribution of the lowest\nLandau-level [9]. Nevertheless, at finite temperature and\nmagnetic field, apart from a few exceptions, for example\nRef. [10],withouttheinsertionofadditionalmechanisms,\nthe most effective models fail to predict inverse mag-\nneticcatalysis(IMC),contradictingLQCDresults, where\nthe pseudocritical temperature for the chiral symmetry\nrestorationdecreasesas Bincreases[11–13]. Possiblerea-\nsonsforthisdiscrepancyhavebeengiveninRefs. [14–17].\nParticularly, in Ref. [16], it was argued that IMC comes\nfrom the rearrangement of the Polyakov loop induced\nby the coupling of magnetic field with the sea quarks.\nThiskindofbackreactionofthePolyakovloopwasimple-\nmented in the entangled Polyakov–Nambu–Jona-Lasinio\n(EPNJL) model [18], where the scalar coupling of the\nNambu–Jona-Lasinio (NJL) model [19] is a function of\n\u0003jmoreira@uc.pt\nypcosta@uc.pt\nztulio.restrepo@posgrad.ufsc.brthe Polyakov loop. Even though this model by itself\nalso fails to reproduce IMC, in Ref. [20], it was shown\nthat if the pure-gauge critical temperature T0is fitted\nto reproduce LQCD data [11], then the EPNJL model\nreproduces IMC. Later, it was argued that IMC can be\nreproduced by mimicking asymptotic freedom, which is\nabsent in many effective models and it is one of the most\nimportant characteristics of QCD. For instance, in Ref.\n[21], the scalar coupling G(B)of the SU(3) version of the\nNJL model, was fitted to reproduce the LQCD pseud-\nocritical temperatures for the chiral transitions, Tc(B).\nIn Refs. [22, 23] the scalar coupling is also made ex-\nplicitly temperature dependent by fitting it to reproduce\nthe LQCD quark condensate [11]. More recently, in Ref.\n[24], the constituent quark masses were calculated as a\nfunction of the magnetic field, B, using LQCD simula-\ntions and then the coupling G(B)was set to reproduce\nthose constituent quark masses. It is important to re-\nmark that this new way to set G(B)was made within\nthe SU(2) version of the Polyakov–Nambu–Jona-Lasinio\nmodel using the proper time formalism.\nIn the present paper we aim to apply the method pro-\nposed in [24] to the more laborious SU(3) version of the\n’t Hooft extended NJL model (which will henceforth be\nreferred to as the NJL model), where now, due to the ’t\nHooft six fermions interactions that appear in the model,\nwe have to set two couplings, the scalar coupling, G(B),\nand the six fermions coupling, \u0014(B)1. Also, in this appli-\ncation we consider the more general nondegenerate case\nmu6=md6=ms. We are particularly interested in the be-\nhavior of\u0014, since in previous applications of the ’t Hooft\n1It is interesting to note that, motivated by phenomenology ar-\nguments, a temperature [25] and a density dependence [26, 27]\nof\u0014, in the form of decreasing exponentials, were already pro-\nposed in order to achieve an effective restoration of axial U A(1)\nsymmetry.arXiv:2005.07049v2  [hep-ph]  21 Jul 20202\nextended NJL model, this coupling was kept independent\nof the magnetic field.\nFor the fit of the parameters we follow the procedure:\nfirst we fit the six parameters of the model, \u0003,G,\u0014,mu,\nmdandmsatT=B= 0; then, atB6= 0the couplings\nGand\u0014are set to reproduce MdandMsmagnetized\nconstituent quark masses respectively, calculated in [24],\nwhile the other parameters are kept Bindependent. The\nother constituent quark mass, Mu, is an output in our\nprocedure.\nThe present paper is organized as follows. In Sec. II\nwe present the SU(3) ’t Hooft extended NJL model. In\nSec. III we fit the parameters of the model to reproduce\nLQCD results [24] and we present the respective analysis\nofourresults. Finally,inSec. IVwedrawourconclusions\nand final remarks.\nII. THE MODEL\nA. ’t Hooft extended NJL model\nThe SU(3) version of the NJL model is given by the\nLagrangian density [28],\nLNJL= f\u0002=D\u0016\u0000^mc\u0003\n f+Lsym+Ldet\n(1)\nin which the quark sector includes scalar-pseudoscalar\nand ’t Hooft six fermions interactions that models the\naxial UA(1) symmetry breaking, with LsymandLdetbe-\ning [28]\nLsym=Gh\u0000\n f\u0015a f\u00012+\u0000\n fi\r5\u0015a f\u00012i\n;\nLdet=\u0014\b\ndetf\u0002\n f(1 +\r5) f\u0003\n+detf\u0002\n f(1\u0000\r5) f\u0003\t\n; (2)\nwhere fare the quark fields with f=u;d;s,mc=\ndiagf(mu;md;ms)is the quark current mass matrix, \u0015a\nare the Gell-Mann matrices and Gand\u0014are coupling\nconstants.\nIn the mean field approximation de effective quarks\nmasses are given by the gap equations\n8\n<\n:Mu=mu\u0000Gh u ui\u0000\u0014h d dih s si\nMd=md\u0000Gh d di\u0000\u0014h u uih s si\nMs=ms\u0000Gh s si\u0000\u0014h u uih d di(3)\nwhere the condensates are\nh f fi=\u00004MfZd4p\n(2\u0019)41\np2\n4+p2+M2\nf(4)\nB. Inclusion of temperature, chemical potential\nand background magnetic field\nThe inclusion of the effect of a finite magnetic field at\na Lagrangian level is done by replacing the Lagrangian(1) by\nL=LNJL\u00001\n4F\u0016\u0017F\u0016\u0017(5)\nwhereF\u0016\u0017is the electromagnetic field tensor. The cou-\npling between the magnetic field Band the quarks is\nnow inside the covariant derivative D\u0016=@\u0016\u0000iqfA\u0016\nEM,\nwhereqfis the quark electric charge, AEM\n\u0016=\u000e\u00162x1Bis\na constant magnetic field, pointing in the zdirection and\nF\u0016\u0017=@\u0016AEM\n\u0017\u0000@\u0017AEM\n\u0016.\nAt the mean field level, the extension to take into ac-\ncount the medium effects of finite temperature and/or\nchemical potential can be done in the usual way by re-\nplacing the p4integration by a summation over Matsub-\nara frequencies,\np4!\u0019T(2n+ 1)\u0000i\u0016\nZ\ndp4!2\u0019T+1X\nn=\u00001: (6)\nTheinclusionoftheeffectofafinitemagneticfieldcanbe\nviewed as the substitution of the integration over trans-\nverse momentum, with respect to the direction of the\nmagnetic field by a summation over Landau levels (de-\nnoted by the index m) averaged over the spin related\nindex,s,\nZd2p?\n(2\u0019)2!2\u0019jqjB\n(2\u0019)21\n2X\ns=\u00001;+1+1X\nm=0;\np2\n?!(2m+ 1\u0000s)jqjB: (7)\nHere we have taken the direction of z-axis as to coincide\nwith that of the magnetic field such that\u0000 !B=B^z.\nThe medium part as well as the nonmagnetic field de-\npendent at a vanishing chemical potential (the standard\nvacuum part) are regularized using a three-dimensional\ncutoff on the spatial part of the momentum integrals\nwhereas the magnetic field dependent contribution at a\nvanishing temperature and chemical potential is done as\nin [29–31]. This is achieved by first performing the full\nsum over the Landau levels in the vacuum part, which\nenables the separation of the magnetic field dependent\ncontributionfromthestandard, nonmagneticfielddepen-\ndent, contribution. The former is then evaluated using\ndimensional regularization as in [29].\nIII. FITTING LQCD RESULTS\nAs stated previously, the main purpose of this paper is\nto explore the consequences of the magnetic field depen-\ndence of the dynamical masses of the light sector quarks\n[Mf(B); f2fu;d;sg] as reported in [24] (at a vanishing\ntemperature and chemical potential) in the framework of\nthe NJL model (for convenience these values are listed in\nTable I). We chose to split this procedure in two steps:3\nTable I. Dynamical mass of the light sector quarks ( Mf) as a function of the magnetic field strength ( B) [24] as well as the\nerrors in their estimations ( \u001bMf).\neB[GeV2]Mu[GeV]\u001bMu[MeV]Md[GeV]\u001bMd[MeV]Ms[GeV]\u001bMs[MeV]\n0.0 0.3115047 8.900894 0.3116843 8.852091 0.5500066 15.88578\n0.1 0.3272839 17.06525 0.2933519 14.70449 0.5246419 23.01390\n0.2 0.3341793 20.73042 0.2799108 17.68701 0.4999245 27.28394\n0.3 0.3348220 19.81968 0.2695186 17.65552 0.4776900 26.89854\n0.4 0.3301029 18.31746 0.2611673 17.98711 0.4568607 28.97203\n0.5 0.3194990 20.62144 0.2581306 19.89098 0.4369001 34.67607\n0.6 0.3037266 28.12296 0.2595464 24.61544 0.4163851 40.79873\n0.7 0.2859020 34.44003 0.2607701 29.56910 0.4029641 44.11870\n1. fix the values of G,\u0014and\u0003and the current masses\n(mf; f2fu;d;sg) at a vanishing magnetic field\nstrength;\n2. usethevaluesoftwoofthedynamicalquarkmasses\nat finite magnetic field strength to fit the cou-\npling strengths of the interactions thus introduc-\ning a magnetic field strength dependence on them\n[G(B)and\u0014(B))] while keeping \u0003andmffixed.\nA. Fits at vanishing magnetic field\nThere are 6 degrees of freedom in our fit: current\nmasses, coupling strengths and cutoff. Three of these\nshouldbefixedusingthedynamicalmasses(throughEqs.\n3) as the main idea behind this paper relies on taking at\nface value the LQCD values for Mf. Let us think of\nthe coupling strengths and cutoff as being fixed by these\nconditions. That leaves us with the choice of the three\ncurrent masses.\nNaively one could expect to be able to fit these using\nthe mass of some of the lightest pseudoscalar mesons. As\nthe NJL model, by construction, relies on the relevance\nof chiral and axial symmetry breaking (spontaneous and\nexplicitinthecaseoftheformerandexplicit, throughthe\n’t Hooft determinant, in the case of the latter) using the\npion, kaonandetaprimemesonmassesseemstheobvious\nchoice. Some other reasonability criteria, such as the size\nof the cutoff and the sign of the coupling strengths (or\nmore precisely its consequences in the obtained meson\nspectra), will however come into play.\nLet us start by fixing the charged pion and kaon to\ntheir physical values fitting muandmdand leaving ms\nas a free parameter. In Figs. 1 the results of the fit are\npresented. A negative coupling constant for the ’t Hooft\ndeterminant is only obtained for strange quark current\nmass above a critical value of ms>0:180 GeV .\nAs can be seen in Figs. 2(a) at a positive \u0014(ms<\n0:180 GeV ) one of the neutral light pseudoscalars be-\ncomes lighter than the pion. Below a critical value of\nthe strange quark current mass ( ms<0:163 GeV ) it be-\ncomes massless and, as can be seen in Fig. 2(b), gains a\nfinite decay width. As can be seen in Fig. 2(c), the de-\ncay width of the \u00110meson vanishes for the choice of mscorresponding to vanishing \u0014[see Fig. 1(b)] which also\nresultsindegenerate \u00190and\u0011. Thesepositive \u0014scenarios\nare unphysical and avoiding them results in a maximum\nvalue of the cutoff \u0003<0:414 GeV as can be seen in Fig.\n3.\nA simultaneous fit of M\u0019,MKandM\u00110could not be\nachieved in the preformed scan [which already goes into\nunreasonably low values of the cutoff as can be seen in\nFig. 1(c)]. It should be noted, however, that raising the\nvalueMKenables us to reach the physical value of M\u00110.\nIn Fig. 4 one can see the dependence of the cutoff, eta\nmeson mass and eta prime decay width on the choice of\nmass for the kaon, while keeping the pion and eta prime\nmeson masses at their experimental values. In Tables II\nand III some parameter sets resulting in cutoffs of 0:500,\n0:600,0:700 GeV are presented as well as a set which\nreproduces the physical values of the masses of pion, eta\nprime and eta mesons.\nB. Fitting the magnetic field dependence\nFor the second point, pertaining to the magnetic field\ndependence, we kept the cutoff ( \u0003) and the current\nmasses (mf) fixed while fitting the couplings strengths,\nwhich thus gain a magnetic field dependence [ G(B)and\n\u0014(B)], using two of the dynamical masses. The results\npresented here were obtained by fitting the downand\nstrangequarks dynamical mass. The magnetic field de-\npendence of the dynamical mass of the upquark can\ntherefore be used as one of the criteria to check the ad-\nequacy of the several scenarios. An attempt to use the\nmassoftheupanddownquarksasinputsinthisfit(leav-\ning the dynamical mass of the strange quark as output)\nwas not successful.\nThe results of the magnetic field dependent fit of the\ncouplings can be seen in the Figs. 5 and Table IV. The\nsplit between MuandMdis smaller for our parameter\nsets [see Fig. 5(a)] and starts to deviate markedly for\nlarger fields. This larger deviation appears however to\nbe related to regularization effects as it occurs for larger\nmagnetic field strengths when we move to higher cutoffs.\nThese regularization effects are also patent when consid-\nering the dependence of the couplings [see Figs. 5(b) and4\n0.05 0.10 0.15 0.20ms020406080G\nMπ±=0.140 GeV\nMK±=0.494 GeV\n(a)\n0.05 0.10 0.150.20ms\n-500-400-300-200-1000100κ\nMπ±=0.140 GeV\nMK±=0.494 GeV (b)\n0.05 0.10 0.15 0.20ms0.00.51.01.5Λ\nMπ±=0.140 GeV\nMK±=0.494 GeV (c)\nFigure 1. Coupling strengths and cutoff dependence on the choice of a strange quark current mass and a fit to reproduce the\nphysical masses of the charged pion and kaon.\n0.05 0.10 0.15 0.20ms0.00.20.40.60.81.0\nMπ±=0.140 GeV\nMK±=0.494 GeVMη'\nMη\nMπ0Mη'Exp\nMηExp\n(a)\n0.05 0.10 0.15 0.20ms0.00.10.20.30.40.50.6Γη\nMπ±=0.140 GeV\nMK±=0.494 GeV (b)\n0.05 0.10 0.15 0.20ms0.00000.00050.00100.00150.00200.00250.0030Γη'\nMπ±=0.140 GeV\nMK±=0.494 GeV (c)\nFigure 2. Pseudoscalar meson masses dependence on the choice of strange quark current mass in Fig. 2(a). The lightest\nsolution drops below that of the pion for \u0014 >0(which happens for this fit for ms<0:180 GeV ) and goes to zero below the\ncritical value of ms= 0:163 GeV . The imaginary part of the pole of the propagator (for a given meson Xthe pole is located\natMX\u0000{\n2\u0000X, with \u0000Xcorresponding to the decay width) is presented for the \u0011meson in Fig. 2(b) (finite for the massless\nsolution) and for the \u00110in Fig. 2(c). It is noteworthy that the decay with for the \u00110vanishes for \u0014= 0, when it is degenerate\nwith\u00190.\n5(c)]. There is a clear onset of a deviation from the be-\nhavior that is observed at lower magnetic field strengths\nwhich occurs at smaller field strengths for smaller cutoff.\nFor the sets with larger cutoff (sets candd) the behavior\nis approximately linear with a decreasing positive Gas a\nfunction of the magnetic field and an increasing contri-\nbution coming from the ’t Hooft determinant interaction\nas we see a negative \u0014increasing in absolute value. This\nincrease in the ’t Hooft term relevance is possibly con-\nnected to an increase in the relevance of UA(1)due to the\nquark spin interaction with electromagnetic fields [32].\nRecent LQCD simulations [11, 12] have shed light into\nan interesting interplay of temperature and magnetic\nfield concerning chiral symmetry. On the one hand they\npoint to a decrease in the critical temperature for chi-ral (partial) restoration with increasing magnetic field\nstrength, while on the other hand, when looking at the\nchange in the renormalized chiral condensate due to the\nmagnetic field, the LQCD estimates point to an increas-\ning condensate with magnetic field strength for tem-\nperatures well below the critical temperature for chiral\nrestoration, a decrease well above said temperature and\na nonmonotonic behavior close to that temperature (an\nincrease followed by a decrease): the inverse magnetic\ncatalysis phenomenon.\nThis variation of the magnetic field dependence of the\nrenormalized chiral condensate change at different tem-\nperatures is depicted in Fig. 6 (LQCD data taken from\n[12]) and 7 (the results obtained with the parameter sets\nfrom Table II). The quantity displayed in Figs. 6(a) and5\n0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Λ\n-100-80-60-40-2002040κ\nMπ±=0.140 GeV\nMK±=0.494 GeV\n(a)\nFigure 3. Dependence of the coupling strength of the ’t determinant interaction on the cutoff imposing both the pion and kaon\nmasses on the physical values.\n0.54 0.56 0.58 0.60MK±0.20.40.60.81.01.21.4Λ\nMπ±=0.140 GeV\nMη'=0.958 GeV\n(a)\n0.54 0.56 0.58 0.60MK±0.450.500.550.60Mη\nMπ±=0.140 GeV\nMη'=0.958 GeV (b)\n0.54 0.56 0.58 0.60MK±0.050.100.150.200.25Γη'\nMπ±=0.140 GeV\nMη'=0.958 GeV (c)\nFigure 4. From left to right the dependence of the cutoff, mass of the eta meson and decay width of the eta prime on the choice\nof kaon mass while keeping the pion and eta prime meson masses fixed to their experimental values (all in GeV).\nTable II. Parameter sets obtained fitting the dynamical masses of the quarks to the values reported in [24], the mass of the\npion (M\u0019\u0006= 0:140 GeV ) and the eta prime mesons ( M\u0011= 0:958 GeV ) at a vanishing magnetic field strength. Sets a,bandc\nwere chosen so as to have a cutoff of 0:500,0:600and0:700 GeV respectively. Set dis chosen so as to reproduce the eta meson\nmassM\u0011= 0:548 GeV .\nmu[MeV]md[MeV]ms[MeV]G\u0002\nGeV\u00002\u0003\n\u0014\u0002\nGeV\u00005\u0003\n\u0003 [GeV]\na) 7.06242 7.19017 196.403 14.2933 -409.947 0.500000\nb) 5.89826 6.01442 180.450 9.47574 -149.340 0.600000\nc) 4.99322 5.09877 165.328 6.77511 -63.7862 0.700000\nd) 4.75221 4.85464 160.820 6.17046 -50.1181 0.731313\nTable III. Kaon and eta meson masses along with the eta prime decay width for the sets listed in Table II.\nMK\u0006[GeV] \u0000\u00110[MeV] M\u0011[GeV]\na) 0.543936 0.205968 0.512422\nb) 0.557196 0.222571 0.530023\nc) 0.568573 0.224593 0.544170\nd) 0.571768 0.223918 0.5480006\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7eB 0.10.20.30.40.50.6Mi\nMπ±=0.140 GeV\nMη'=0.958 GeV\nMs=MsLQCD\nMd=MdLQCDMuLQCD\nMu\n(a)\n0.1 0.2 0.3 0.4 0.5 0.6 0.7eB\n-5051015G\nMπ±=0.140 GeV\nMη'=0.958 GeV (b)\n0.1 0.2 0.3 0.4 0.5 0.6 0.7eB\n-1000-800-600-400-2000κ\nMπ±=0.140 GeV\nMη'=0.958 GeV (c)\nFigure 5. Results of the magnetic field strength dependent fit of the couplings: G(B)and\u0014(B). In 5(a) the magnetic field\ndependence of the dynamical masses of the quarks [24]. The fit is done by imposing the reproduction of the LQCD results\nforMdandMs, as such,Muis an output. In increasing thickness the full lines labeled Mucorrespond to \u0003 = 0:500 GeV ,\n0:600 GeV and0:700 GeV (setsa,bandcin Table II) whereas the thick dashed line corresponds to the choice of cutoff which\nreproduces the eta meson mass M\u0011= 0:548 GeV (setdin Table II). All sets reproduce the pion and the eta prime masses.\nTable IV. Values of the coupling strengths of the model at sample values for the magnetic field fitted as to reproduce the\ndynamical masses of the down and strange quarks (as reported in [24]) and using the current quark masses and regularization\ncutoff from Table II.\na) b) c) d)\neB G \u0014 G \u0014 G \u0014 G \u0014\n[GeV]\u0002\nGeV\u00002\u0003 \u0002\nGeV\u00005\u0003 \u0002\nGeV\u00002\u0003 \u0002\nGeV\u00005\u0003 \u0002\nGeV\u00002\u0003 \u0002\nGeV\u00005\u0003 \u0002\nGeV\u00002\u0003 \u0002\nGeV\u00005\u0003\n0.000 14.2933 -409.947 9.47574 -149.34 6.77511 -63.7862 6.17046 -50.1181\n0.025 13.9768 -425.237 9.30076 -155.053 6.66863 -66.2936 6.07794 -52.1048\n0.050 13.6360 -438.726 9.11346 -160.544 6.55517 -68.8060 5.97949 -54.1126\n0.075 13.2740 -450.005 8.91565 -165.649 6.43581 -71.2560 5.87602 -56.0895\n0.100 12.8933 -459.012 8.70884 -170.285 6.31147 -73.6004 5.76833 -58.0012\n0.125 12.4960 -465.883 8.49437 -174.415 6.18300 -75.8110 5.65714 -59.8236\n0.150 12.0840 -470.836 8.27354 -178.029 6.05121 -77.8674 5.54317 -61.5385\n0.175 11.6593 -474.099 8.04770 -181.123 5.91694 -79.7513 5.42716 -63.1290\n0.200 11.2244 -475.857 7.81835 -183.686 5.78113 -81.4428 5.30991 -64.5773\n0.225 10.7980 -474.748 7.59499 -185.209 5.64925 -82.7222 5.19612 -65.7105\n0.250 10.3624 -472.860 7.36897 -186.329 5.51640 -83.8301 5.08159 -66.7121\n0.275 9.91500 -470.722 7.13923 -187.186 5.38198 -84.8125 4.96583 -67.6159\n0.300 9.45341 -468.778 6.90474 -187.905 5.24545 -85.7123 4.84837 -68.4533\n0.325 8.98913 -465.858 6.67073 -188.101 5.10956 -86.3739 4.73150 -69.1042\n0.350 8.50145 -464.395 6.42799 -188.553 4.96940 -87.0983 4.61110 -69.7986\n0.375 7.98110 -465.306 6.17265 -189.561 4.82289 -88.0050 4.48542 -70.6284\n0.400 7.41725 -469.525 5.90025 -191.437 4.66770 -89.2195 4.35249 -71.6908\n0.425 6.78212 -479.527 5.59935 -194.981 4.49788 -91.0640 4.20734 -73.2350\n0.450 6.07885 -494.515 5.27189 -199.917 4.31445 -93.4357 4.05079 -75.1837\n0.475 5.29586 -515.254 4.91364 -206.442 4.11522 -96.4163 3.88102 -77.6013\n0.500 4.41941 -542.789 4.51957 -214.790 3.89762 -100.101 3.69584 -80.5623\n0.525 3.34654 -585.110 4.04499 -227.351 3.63731 -105.472 3.47463 -84.8363\n0.550 2.16839 -636.099 3.53317 -241.747 3.35846 -111.524 3.23796 -89.6340\n0.575 0.907604 -695.043 2.99558 -257.393 3.06752 -117.988 2.99134 -94.7446\n0.600 -0.403001 -760.236 2.44764 -273.513 2.77300 -124.521 2.74199 -99.8967\n0.625 -1.71787 -828.289 1.90952 -289.085 2.48582 -130.687 2.49918 -104.747\n0.650 -2.97697 -893.873 1.40666 -302.800 2.21958 -135.947 2.27440 -108.874\n0.675 -4.10572 -950.085 0.969434 -313.073 1.99032 -139.679 2.08117 -111.791\n0.700 -5.01754 -989.212 0.631820 -318.137 1.81575 -141.211 1.93440 -112.9787\nΔ(Σu+Σd)/2\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7eB -0.20.00.20.40.60.8\nT=0.000\nT=0.130T=0.148\nT=0.153\nT=0.163T=0.176\n(a)\nFigure 6. Renormalized chiral condensate change with magnetic field at different temperatures as given by LQCD calculations\n[12] displaying the behavior known as inverse magnetic catalysis. The chiral condensate dependence on the magnetic field\nstrength is qualitatively different for temperatures depending on its relation to the chiral transition temperature, T\u001f\nc. For\ntemperatures well below T\u001f\ncit increases monotonically with B(slope decreases with increasing temperature) whereas for\ntemperatures well above T\u001f\ncit decreases monotonically (slope increases with temperature). Close to the transition temperature\nit exhibits a nonmonotonic behavior with a local maxima.\n7 is given by:\n\u0006i(B;T) =\n2mi\nM2\u0019F2\u0019\u0000\nh i(B;T) i(B;T)i\u0000h i(0;0) i(0;0)i\u0001\n+ 1;\n\u0001\u0006i(B;T) = \u0006i(B;T)\u0000\u0006i(0;T): (8)\nHereF\u0019is the pion decay constant ( F\u0019= 86 MeV for\nthe LQCD data and F\u0019= 77:088,87:7928,96:8062and\n99:3428 MeV for setsa,b,canddfrom Table II), M\u0019\nthe pion mass ( M\u0019= 135 GeV for the LQCD data and\nM\u0019= 140 GeV for our sets) and iis the flavor of the\nquark.\nAs can be seen in Figs. 7 the behavior is well repro-\nduced qualitatively by our parameter sets. It should be\nnoted however that while on the LQCD data the non-\nmonotonic behavior (the bump) is more marked just be-\nlow the pseudocritical transition temperature for partial\nchiral restoration in the light sector ( T\u001flc= 0:158 GeV\nfor the LQCD data), in our case this behavior occurs for\nlower temperatures. The pseudocritical temperatures for\nour sets (determined as the inflexion points of the av-\nerage light quark condensates) are T\u001flc= 0:170,0:175,\n0:180and0:182 GeV for setsa,b,canddfrom Table II\nrespectively. The marked bumpoccurs in ours sets for a\ntemperature around half the pseudocritical temperature\nor even slightly lower. There is also a small decrease for\nlowermagneticfieldswhichisnotobservedinLQCDdata\npoints and which is more pronounced at larger cutoff.\nThedecreaseinthepseudocriticaltemperatureforpar-\ntial chiral defined by the inflection point in the average\nlight quark condensate\nd2\ndT21\n2\u0000\nh u ui+h d di\u0001\f\f\f\f\nT=T\u001flc= 0 (9)is depicted in Fig. 8(b) where one can see that the tem-\nperature is larger and its behavior becomes more linear\nfor larger cutoffs. The decrease of the critical tempera-\nture with increasing magnetic field strength is however\nmuch more pronounced in our model calculations. As\none can see in Fig. 8(b) by eB= 0:7 GeV2the critical\ntemperature has dropped to a value which is approxi-\nmately half of its value for vanishing field while, for the\nsame magnetic field strength, LQCD show a reduction of\nonly\u001810%approximately. It should also be noted that\nwhile for larger cutoff an approximate linear response to\nmagnetic field is obtained in our model, for the LQCD\nsimulation the slope changes with magnetic field, hinting\nat an inflexion point.\nOne should stress that both the magnetic field depen-\ndence of the pseudocritical temperature for partial chiral\nrestoration as well as the temperature dependence of the\nmagnetic field change induced renormalized chiral con-\ndensate are easily modified by considering further exten-\nsions of the model such as the inclusion of a Polyakov\npotential which mimics gluon dynamics and would mod-\nify the finite temperature aspects of the model. Also the\nidea that extensions of the model to include higher order\ninteraction terms, for instance as in [33–37], or go be-\nyond mean field corrections as in [38–41], could possibly\nfix these issues and deserves to be explored.\nIV. CONCLUSIONS\nIn this work we have studied the implications of the\nmagneticfielddependenceofthedynamicalquarkmasses\non the up,downand strangesectors (as reported in\n[24])byusingthe’tHooftextendedNambu–Jona-Lasinio\nmodel. Our study can be broken down into two separate8\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7eB -1.5-1.0-0.50.00.51.01.5Δ(Σu+Σd)/2\nMπ±=0.140 GeV\nMη'=0.958 GeV\nΛ=0.500 GeVT=0.000\nT=0.080\nT=0.120T=0.180T=0.240\nTcrit\n(a)\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7eB -1.5-1.0-0.50.00.51.01.5Δ(Σu+Σd)/2\nMπ±=0.140 GeV\nMη'=0.958 GeV\nΛ=0.600 GeVT=0.000\nT=0.080\nT=0.120T=0.180T=0.240\nTcrit (b)\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7eB -1.5-1.0-0.50.00.51.01.5Δ(Σu+Σd)/2\nMπ±=0.140 GeV\nMη'=0.958 GeV\nΛ=0.700 GeV\nT=0.000\nT=0.090\nT=0.120\nT=0.160T=0.240\nTcrit\n(c)\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7eB -1.5-1.0-0.50.00.51.01.5Δ(Σu+Σd)/2\nMπ±=0.140 GeV\nMη'=0.958 GeV\nMη=0.548 GeV\nT=0.000\nT=0.090\nT=0.120\nT=0.160T=0.240\nTcrit (d)\nFigure7. Renormalizedchiralcondensatechangewithmagneticfieldatdifferenttemperaturesasgivenbyourmodelcalculations\nfor sets from Table II.\n0.0 0.2 0.4 0.6 0.8eB 0.130.140.150.160.17Tcχl\n(a)\n0.1 0.2 0.3 0.4 0.5 0.6 0.7eB 0.000.050.100.15Tcχl\nMπ±=0.140 GeV\nMη'=0.958 GeV (b)\nFigure 8. Pseudocritical temperature for the partial restoration of chiral symmetry in the light sector. On the left-hand side the\nLQCD results reported in [11] whereas on the right-hand side the results obtained with our model are presented. On the latter,\nthe full lines in increasing order of thickness correspond to: \u0003 = 0:500 GeV ,0:600 GeV and0:700 GeV . The thick dashed line\ncorresponds to set din Table II which reproduces M\u0011= 0:548. All sets reproduce M\u0019= 0:140 GeV andM\u00110= 0:958 GeV .9\nsteps:\n\u000fat vanishing magnetic field we developed four pa-\nrameter sets, all of which reproduce the pion and\nthe eta prime physical meson masses as well as the\nvacuum dynamical masses of the quarks; for three\nof them we imposed a choice of cutoff whereas for\nthe remaining set we chose to reproduce the eta\nmeson mass.\n\u000fat vanishing magnetic field we used the variation in\nthedownandstrangequarks dynamical masses to\nfit the coupling strengths of the model interactions\n[G(B)and\u0014(B)for the NJL four-quark interaction\nand the ’t Hooft flavor determinant six-quark inter-\naction respectively] while keeping the other param-\neters frozen at their vacuum value. Then, we stud-\nied the inverse magnetic catalysis phenomenology\nhaving arrived at an acceptable qualitative agree-\nment with LQCD data both for the decrease in\nthe pseudocritical temperature for the partial chi-\nral symmetry restoration in the upanddownsector\nwith increasing magnetic field as well as the differ-\nent magnetic field dependent behaviors at different\ntemperatures.\nWe found for the four-fermion coupling, G(B), an overall\nsimilar behavior to the ones reported in previous works.\nRegarding the six-fermion ’t Hooft coupling, \u0014, our re-\nsults showed that the absolute value of \u0014increases as B\nincreases. To our knowledge this is the first work with\na six-fermion ’t Hooft flavor determinant running withthe magnetic field. We also verified that the obtained\nparameter sets, as well as their magnetic field dependen-\ncies, show promising results which can be easily applied\nto further studies, for instance, the Polyakov loop dy-\nnamics (see [42–53]). This same procedure can also be\napplied to any new LQCD data as well as for different\nregularization procedures.\nACKNOWLEDGMENTS\nThis work was supported by a research grant under\nProject No. PTDC/FIS-NUC/29912/2017, funded by\nnational funds through FCT (Fundação para a Ciên-\ncia e a Tecnologia, I.P, Portugal) and cofinanced by the\nEuropean Regional Development Fund (ERDF) through\nthe Portuguese Operational Program for Competitive-\nness and Internationalization, COMPETE 2020, by na-\ntional funds from FCT, within the Projects No. UID/\n04564/2019 and No. UID/04564/2020. This study was\nfinanced in part by Coordenação de Aperfeiçoamento de\nPessoal de Nível Superior-(CAPES-Brazil)-Finance Code\n001. We would like to thank G. Endrödi for his availabil-\nity in supplying the data points from the LQCD study\nreported in [24]. 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Michael Bartram1,2,#, Meng Li1,#, Liangyang Liu1, Zhiming Xu1, Yongchao Wang3, Mengqian Che1, Hao Li4, Yang Wu4,5, Yong Xu1,6,7,8, Jinsong Zhang1,6,7, Shuo Yang1,6,7 and Luyi Yang1,2,6,7,* 1State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China 2Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada 3Beijing Innovation Center for Future Chips, Tsinghua University, Beijing 100084, China 4Tsinghua-Foxconn Nanotechnology Research Center, Department of Physics, Tsinghua University, Beijing 100084, China 5Beijing Univ Chem Technol, Coll Sci, Beijing, 100029, China 6Frontier Science Center for Quantum Information, Beijing 100084, China 7Collaborative Innovation Center of Quantum Matter, Beijing 100084, China 8RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan #These authors contributed equally. *e-mail: luyi-yang@mail.tsinghua.edu.cn  Abstract Atomically thin van der Waals magnetic materials have not only provided a fertile playground to explore basic physics in the two-dimensional (2D) limit but also created vast opportunities for novel ultrafast functional devices. Here we systematically investigate ultrafast magnetization dynamics and spin wave dynamics in few-layer topological antiferromagnetic MnBi2Te4 crystals as a function of layer number, temperature, and magnetic field. We find laser-induced (de)magnetization processes can be used to accurately track the distinct magnetic states in different magnetic field regimes, including showing clear odd-even layer number effects. In addition, strongly field-dependent antiferromagnetic magnon modes with tens of gigahertz frequencies are optically generated and directly observed in the time domain. Remarkably, we find that magnetization and magnon dynamics can be observed in not only the time-resolved magneto-optical Kerr effect but also the time resolved reflectivity, indicating strong correlation between the magnetic state and electronic structure. These measurements present the first  2 comprehensive overview of ultrafast spin dynamics in this novel 2D antiferromagnet, paving the way for potential applications in 2D antiferromagnetic spintronics and magnonics as well as further studies of ultrafast control of both magnetization and topological quantum states.  Keywords: 2D topological antiferromagnet, ultrafast magnetization dynamics, ultrafast magnon dynamics, spintronics, magnonics, ultrafast spectroscopy  1. Introduction Ultrafast pump-probe techniques have proved to be a powerful tool for understanding the dynamics of magnetic materials on short timescales, such as demagnetization, excitation of collective modes [1]. Femtosecond laser pulses can also be used to control magnetic order, including exciting transient magnetic states inaccessible in the static regime [1-3]. Although most ultrafast research has been performed in ferromagnets, these techniques have also demonstrated their power in the probing and manipulation of spins in antiferromagnets [2], which are otherwise hard to detect (especially for small samples) due to their vanishing net magnetic moment.   The discovery of 2D magnetic materials about 6 years ago [4-11] has provided new opportunities for both fundamental studies of spin physics in reduced dimensions and for the creation of novel ultrafast communication, logic and memory devices based on 2D van der Waals magnets and their heterostructures. Unfortunately, due to their small volume the magnetic order in atomically thin magnetic materials cannot be probed by conventional tools such as neutron scattering. Using optical techniques based on magneto-optical effects has turned out to be a very effective alternative and has been widely applied to determine magnetic states in 2D magnetic materials [6,7,11]. However, to date most work has focused on the static magnetic order; ultrafast spin dynamics – which are both fundamentally important and crucial for ultrafast device applications – have received much less attention and remain largely unexplored.    As the first intrinsic magnetic topological insulator [16-22], atomically thin MnBi2Te4 crystals bridge three fascinating realms in condensed matter physics: magnetism, topology and 2D  3 materials. Below a critical temperature of ~24 K, MnBi2Te4 forms an A-type antiferromagnetic order, with intralayer ferromagnetic coupling and antiferromagnetic coupling between adjacent layers. The properties of few-layer MnBi2Te4 have been probed by magneto-optical circular dichroism [23,24] and magnetic force microscopy [25], wherein the much stronger intralayer coupling makes each layer act as a single unit, allowing few-layer MnBi2Te4 to be modeled as a finite one-dimensional antiferromagnetic chain. MnBi2Te4 further exhibits quantum anomalous Hall and axion insulator phases for odd- and even-layer samples respectively [21,22], demonstrating the connection between magnetism and topology [26-29]. Ultrafast magnetization dynamics have been reported in bulk MnBi2Te4 [30], and a recent magneto-Raman study reported the detection of magnon modes in bilayer MnBi2Te4 [31]. Recent theoretical and experimental studies have also demonstrated optical axion electrodynamics and optical control of the antiferromagnetic order in even layer MnBi2Te4 [32,33]. However, systematic studies of ultrafast spin dynamics in atomically thin crystals of MnBi2Te4 (or in 2D magnetic materials in general) are still at an early stage [14,15], as these studies are technically demanding, involving ultrafast time-resolved spectroscopy combined with high-spatial-resolution microscopy, cryogenic temperatures and high magnetic fields. Sufficient understanding of ultrafast (de)magnetization mechanisms, timescales and magnon dynamics is critical for the development of 2D ultrafast data processing and storage devices.  Here we explore, for the first time, ultrafast dynamics of magnetization and magnons in 4- to 8-septuple layer (SL) thick MnBi2Te4 samples via measurements of both the time-resolved magneto-optical Kerr effect (MOKE) and the time-resolved reflectivity. While time-resolved MOKE is well-established probe of magnetization dynamics, we also observe a large magnetization-dependent reflectivity signal in our samples, implying an unusually strong connection between electronic states and magnetic order in this material. The magnetic state of few-layer MnBi2Te4 under an applied magnetic field differs distinctly depending on whether the number of layers is even or odd, due to an odd layer of numbers creating a necessary asymmetry between AFM sublattices (e.g., N+1 up, N down for 2N+1 layers) [23,24]. We have observed laser-induced (de)magnetization processes which are highly sensitive to the magnetic state, which reflect these odd-even layer effects and also allow us to accurately track magnetic transitions across varying applied magnetic fields. In fact, this  4 use of pump-probe techniques for probing the static magnetic state turns out to be a more effective tool compared to a conventional static method, which could be of use in studying other 2D magnets. In addition, by applying an in-plane external field we directly observe collective spin wave excitations in real time and show that their frequency and damping time also depend on both the even-or-odd layer number and the external magnetic field. The magnetization and recovery timescales and magnon frequencies and lifetimes revealed in this work provide critical parameters for further designing 2D ultrafast spintronic/magnonic devices based on few-layer MnBi2Te4 and heterostructures. Our results also highlight some apparent limitations of the current theoretical models, suggesting some need for improvements in the understanding of the magnetic interactions. Since topological quantum states in MnBi2Te4 are closely linked to the magnetic states, it is important to have a clear picture of the magnetic behavior. By presenting a comprehensive overview of pump-induced magnetization dynamics, we can therefore provide a basis for future experiments investigating ultrafast manipulation of topological states.  2. Methods MnBi2Te4 crystals were prepared by mechanical exfoliation as described previously [22,34]. The layer thickness was determined by optical contrast, atomic force microscopy and time-resolved reflectivity measurements of the interlayer breathing mode [35].   To probe the nonequilibrium magnetization and spin wave dynamics, we performed two-color pump-probe reflectivity and MOKE measurements with pulsed lasers in two different configurations (see Supplementary Note 1 for details). In the first case, the magnetic field was applied perpendicular to the sample plane (Faraday geometry, Fig. 1a). In the second case, the magnetic field was almost in the sample plane, but with a small tilting angle of ~ 7.5 degrees (close to the Voigt geometry, Fig. 5a). In both cases, the pump and probe beams were at normal incidence, where the probe is sensitive to the change of the out-of-plane magnetization.  3. Ultrafast magnetization dynamics  5 We start with experiments performed in the Faraday geometry (shown schematically in Fig. 1a), where the external field is out-of-plane. Previous reports have studied the magnetic states at varying fields in such a configuration using static MOKE measurements [23,24]. As in those reports, our static MOKE measurements (shown in Fig. 1c,d for 4- and 5-layer samples at 3 K) show large steps at the transition from antiferromagnetic (AFM) order to canted antiferromagnetic (cAFM) order, as indicated by the lower edge of the shaded regions (~2 T and ~3.5 T for the 4- and 5-layer sample respectively). At a higher field, the cAFM order eventually coalesces in ferromagnetic (FM) order, with all spins aligned to the external field, indicated by the upper edge of the shaded regions (~3.3 T and ~6 T for the 4- and 5-layer samples). In static MOKE measurements, the cAFM-FM transition is marked by the signal flattening out, a change which can often be challenging to clearly identify [23,24]. As we will show, this transition can be more clearly distinguished by time-resolved measurements. We also note that like previous studies, we observe an anomalous hysteresis loop in the AFM region of even-layered samples. The exact origin of this loop remains under debate, which could result from uncompensated magnetization [23] or from optical axion electrodynamic response [32] as recently observed experimentally [33]. We do not attempt to address it here as it is not the focus of this work but note that it consistently appears in both our static MOKE and time-resolved data.  Figure 1b shows the time-resolved Kerr rotation for a 4-layer region of the sample at 3 K with an external field of 7 T, which is typical of the signals in this configuration. The signal is negative (corresponding to a decrease in magnetization) and shows four distinct components. First, there is a step-like feature at t = 0 (t0), followed by a slow rise, with the signal reaching the maximum (negative) value after about ~1 ns. Part of the signal then recovers on a timescale of a few nanoseconds, with the remainder lasting on a scale which we estimate to be ~100 ns, much longer than the interval of the adjacent pump pulses (12.5 ns), causing a large background offset (even before t0) due to accumulation from previous pump pulses. We also note that this time-resolved response shows a strong non-linearity with respect to the pump and probe beam powers due to laser heating effects (we show a clear example of this for the second configuration in Supplementary Figure 6). We therefore have minimized the powers of the pump and probe beams as much as possible while maintaining acceptable signal quality.  6  This two-step demagnetization signal is very similar to typical cases of ultrafast demagnetization in rare-earth ferromagnets (e.g., Gd) [36] or ferromagnetic half-metals or insulators (e.g., La1-xSrxMnO3, CoCr2S4, CrI3) [37,38]. Reports on bulk MnBi2Te4 samples have also shown similar results [30]. Such cases are generally described by the microscopic three-temperature model [36], which we discuss in more detail in Supplementary Note 2. The pump deposits energy into electronic states, which become very hot before rapidly cooling back down within a few ps as they thermalize with the lattice. During this hot electron phase, rapid demagnetization occurs, giving the step-like feature. In the next phase, the magnetization slowly shifts towards the appropriate equilibrium value for the new (increased) temperature of the system, leading to a slow rise time. We note that the ~ns scale rise time in these measurements is relatively long compared to that of many other materials (e.g., Ni, Gd) [1,36,37], but comparable to that of half-metallic ferromagnets (e.g., La1-xSrxMnO3) [37]. The long demagnetization time in MnBi2Te4 may be because the Mn d bands are far away from the Fermi level [18] and therefore the Elliot-Yafet type spin relaxation is limited (see Supplementary Note 2 for further discussion). Finally, the signal will slowly decay as the entire system cools back to the ambient temperature via thermal diffusion. In our case there are two distinct decay times, one relatively short (~ a few ns) and one very long (on the order of 100 ns), which can be explained as originating from the out-of-plane diffusion (into the SiO2 substrate) and in-plane diffusion (within the sample) respectively. For comparison, a very rough estimate of the in-plane diffusion timescale can be given by d2/Dip, where d is the pump spot diameter and Dip is the in-plane thermal diffusivity. Using a diffusivity of ~0.073 µm2 ns-1 (from specific heat and in-plane thermal conductivity values in [39,40]) and a pump spot diameter of 2 µm gives a similar timescale of ~50 ns.  A simple measure of the field-dependent behavior of the signal can be obtained via the long-lived background, which we measure using the value of the signal just before t0 (corresponding to ~12 ns after the previous pulse, where all faster components have already decayed away). Sweeping the external field back and forth gives results shown in Fig. 1 e,f for 4SL and 5SL samples at 3 K, which are antisymmetric about the magnetic field and show obviously layer-dependent patterns, including large peaks at the AFM-cAFM transition, and  7 a noticeable change in the signal at the cAFM-FM transition. These features match extremely well with the features in the static MOKE signals and are notable in that they are easier to identify, particularly in the case of the cAFM-FM transition. We additionally include static MOKE data and background sweeps for 6 to 8 SL in Supplementary Figure 1, with similar results. By taking temperature dependent background sweeps for the 5-layer sample, shown in Supplementary Figure 2, we can also see how the transition fields evolve with temperature. These results demonstrate the efficacy of pump-probe measurements as a tool for probing the static magnetic order of few-layer MnBi2Te4, which could also be widely applied to the studies of other atomically thin magnetic materials.   Based on our three-temperature model, this long-lived background corresponds to an elevated sample temperature from pumping, meaning we are effectively measuring the change in 𝜃! with temperature. To verify this, we have also measured the static MOKE signals at 10 K, where we plot the difference between the 10K and 3K data in Fig. 1g,h. This plot shows clear similarities to the background sweep, suggesting that thermal effects are indeed the main driver of that signal. We also note that these signals are independent of the pump polarization, further supporting the conclusion of thermally driven transient magnetization dynamics [1].  Since typically, the polar Kerr rotation is proportional to the out-of-plane magnetization, we also calculate Δ𝑀\"/Δ𝑇 from our theoretical model (discussed in detail in Supplementary Note 3) for comparison, shown in Figure 2a and 2b. The results are qualitatively very similar, with peaks near the AFM-cAFM transitions and a distinct cusp at the cAFM-FM transition. In the AFM region, Δ𝑀\"/Δ𝑇  in the 4-layer region starts from zero (at zero field) and rises as the field increases, in contrast to the 5-layer region, in which it starts at a negative value and decreases in amplitude towards zero as the field increases. This odd-even layer effect exactly mirrors what is seen in the data, aside from a small offset in the 4-layer region due to the anomalous hysteresis loop. One notable difference is that the cAFM-FM transition is predicted to be much larger by theory compared to what we observe in the experiment. We show in Figure 2c transition fields for a variety of thicknesses extracted from our data (crosses) compared to theoretical calculations (dots), where while good agreement can be found for the AFM-cAFM transition points (𝐻#$%&), the cAFM-FM transitions (𝐻%&)  8 observed in the data show an unexpected rapid decrease for thinner samples. This issue has not been noted in previous studies [23,24], where the focus was on the (more obvious) AFM-cAFM transition points.  Full time- and field- dependent data for 7- and 8-layer samples are shown in Figure 3. The overall behavior roughly just tracks that of the background sweeps, with peaks in amplitude near the AFM-cAFM transition points and a small additional peak at the transition between two cAFM configurations [23] in the 8-layer sample. One difference of note is that in the even-layered AFM region at low field, where the background is positive (due to magnetization increasing with temperature in this region, as discussed above), the initial step-like feature remains negative, meaning it still represents a decrease in magnetization. We can qualitatively understand this as interactions in the hot electron phase causing large demagnetizations in a small fraction of spins, compared to the later interactions causing smaller but more widespread shifts towards the new equilibrium (which has a larger magnetization, in the case of the even-layered sample). As a uniquely AFM behavior, such a feature is not well captured by the standard three-temperature model, which is designed to describe ferromagnetic systems. We also note that the timescales change somewhat as a function of field, though within the FM region they appear (nearly) constant. However, detailed quantitative analysis of changes in dynamics for different applied fields proves difficult, likely due to the influence of the various transition points. Around the AFM-cAFM transitions, we were sometimes able to observe small oscillatory signals (see Supplementary Figure 3), which are linked to the magnon modes discussed later. Measuring a set of different thicknesses all in the FM state suggested a trend of increasing rise (and fall) times for thicker samples (shown in Supplementary Figure 4), which implies a significant contribution to the underlying interactions from the interface.  Interestingly, in addition to the time-resolved Kerr signal, we can also observe a large time-resolved reflectivity change at low temperature, shown in Figure 4. Like the Kerr data, we observe clear features near the transition points, with the noticeable difference of being (approximately) symmetric with respect to the external field (b,c). We have verified that the signal also shows a sudden drop in amplitude above the Neel temperature (a inset). Such pronounced magnetorefractive effects do not exist in many other magnetic materials,  9 suggesting that the electronic structure in MnBi2Te4 is unusually sensitive to the magnetic order. To further support this idea, we have performed first-principles DFT calculations (Supplementary Note 5) which indeed show a noticeable difference in calculated reflectivity between FM and AFM states. Surprisingly, a large signal can be observed in an even-layered sample with no external field, where the net magnetic moment (and any changes induced by heating) should be very small (ideally zero). The fact that it is relatively large in the zero-field even-layered samples suggests that these reflectivity changes cannot simply be thought of in terms of changes in net magnetic moment (as is conventionally done in MOKE), and might instead be related to changes in the Neel vector [2] or some more novel effect from the axion insulator state [32,33]. Separate reports have also indicated measurable shifts in the static reflectivity as bulk samples are cooled below the transition temperature [41], lending further experimental support to our observations.  4. Coherent spin wave dynamics Next, we discuss the measurements in the (near) Voigt geometry, where the external field direction is set at a small ~7.5-degree angle to the sample plane, as shown in Fig 3a. In this configuration, any non-zero applied field causes the magnetic moments to begin tilting, forming a cAFM state, with the spins eventually aligning with the field, giving FM order. The tilt of the magnetic field slightly out of the sample plane gives us better control of the zero-field magnetic state and leads to magnon modes which cause modulations in the out-of-plane magnetization, which is typically necessary for detection.  Figure 5b displays a set of time-resolved reflectivity data for various applied magnetic fields in a 5-layer sample at 4K, where prominent oscillations with tens of GHz frequency starting right after t0 emerge at around 3 T before again vanishing near 7 T. We note that observing magnons with time-resolved reflectivity measurements is unusual, though it has also been done in other materials such as EuTe [42] (via magnetization-induced bandgap shifts) and CrSBr [15] (via coupling between magnons and excitons). As discussed above, we have observed a strong magnetorefractive effect in this system, so the fact that the magnon signal also appears is not surprising. In fact, the oscillations are present in both reflectivity and Kerr angle, but the signal to noise ratio turns out to be much better in the reflectivity  10 measurements (shown in Supplementary Figure 5). As before, the sample is highly sensitive to laser heating, which can be demonstrated here by observing that increasing the power causes a noticeable shift in the oscillation frequency (Supplementary Figure 6). We emphasize that time-resolved techniques are capable of detecting dynamics in a wide frequency range from sub-GHz to a few THz, making them particularly suitable for probing the ~20-50 GHz frequency magnons observed here. By contrast, Raman spectroscopy is limited by the low energy cutoff (usually around 3 cm-1 ≈ 90 GHz) and traditional ferromagnetic resonance techniques are limited to lower frequencies (typically below ~40 GHz).  To show the full field dependence of this oscillatory signal, we plot the FFT amplitude across field and frequency in Fig. 5c. The frequency clearly decreases towards zero as the field approaches ~6.8 T, after which the signal vanishes. As before, we observe a long-lived (thermal) component of the time-resolved signals, which can be used to probe the magnetic state. In the background sweep of the time-resolved Kerr rotation (shown in Fig 5d) a large cusp can be seen at the same ~6.8 T field value where the magnons vanish, which we identify as the cAFM-FM transition field. A comparison of background sweeps across different thickness is shown in Supplementary Figure 7, along with theoretical calculations of Δ𝑀\"/Δ𝑇. Surprisingly, such a calculation does not resemble the observed data, which may be explained by including sensitivity to the Neel vector, similar to what was done in studies of the anomalous central loop in even-layered samples [32,33]. We have additionally measured the 5-layer sample at a selection of higher temperatures (shown in Supplementary Figure 8), wherein the magnon frequencies decrease with increasing temperature and the field where the magnon signal drops to zero (which decreases at higher temperature) consistently matches with the cusp in the corresponding background sweep. For a theoretical comparison, we plot the field dependence of magnon frequencies calculated via a linearized Landau-Lifshitz-Gilbert (LLG) equation (detailed in Supplementary Note 4) in Fig. 5e, wherein the lowest frequency mode just below the cAFM-FM transition (highlighted region) shows an obvious similarity to our measurements, also dropping to zero frequency at the transition from cAFM to FM order.   11 In addition to the 5-layer sample, we also observe similar oscillations in other regions of the sample (varying between 4 and 8 layers thick), which we show in Fig. 6 (all at 4 K), along with parameters obtained from fitting to a damped sinusoid plus a slow exponentially decaying background. The FFT amplitude is shown as a function of field and frequency, with the fitted oscillation frequencies marked with green dots (a-e), alongside the initial amplitude of the oscillation from the fit (f,g) and the oscillation lifetime (h,i). A clear difference between the even- and odd-layered samples can be observed. The even-layered samples show visible oscillations down to generally lower fields than in the odd-layered ones, and redshift slightly at the lowest fields. For all the even-layered samples the lifetime also steadily increases with decreasing field, whereas in the odd-layered samples the lifetime reaches a maximum and then decreases. The increase of spin wave damping towards the saturation field is likely due to the inhomogeneous broadening of local magnetic fields from doping or strain, in line with previous work in bilayer CrI3 [14]. For comparison, we calculate the damping parameter 𝛼=1/(𝑓'𝜏) from the frequency 𝑓' and lifetime 𝜏 (shown in Supplementary Figure 9) where we find that our damping parameter is somewhat larger (0.1 – 10) than what was observed observed in CrI3 (<0.4).  Extrapolating our measurements towards zero field indicates a finite spin wave gap which slightly decreases from ~60 GHz (~0.25 meV) for thicker samples down to ~40 GHz (~0.17 meV) for the thinnest region, consistent with that observed in bilayer MnBi2Te4 by magneto-Raman measurements (~0.2 meV) [31]. In the theoretical calculation, the spin-wave gap would be expected to be layer-independent for even-layered samples, given by 𝛾/𝐾(𝐾+2𝐽), with slightly reduced gap for thinner odd-layered samples (down to the obvious result of 𝛾𝐾 for monolayer), where 𝛾 is the gyromagnetic ratio. We note that as mentioned before, the thinner samples show a reduced cAFM-FM transition field not well captured by our model, which may be related to why the frequency decreases despite theoretically being layer-independent. Using our estimated values of 𝐽=2.3 T and 𝐾=0.35𝐽 with the gyromagnetic ratio of a free electron (()*~28 GHz T-1) gives value of 58 GHz for the even-layered spin-wave gap, which matches well with the observed data on the thicker regions. For a bulk sample, eliminating the influence of the surface via periodic boundary conditions gives a slightly increased gap of 𝛾/𝐾(𝐾+4𝐽) = 79 GHz (0.33 meV).  12 This is somewhat smaller than the value of 0.4-0.5 meV obtained from inelastic neutron scattering measurements on bulk samples [43], but this may be simply due to sample-dependent variations in the anisotropy, as in some previously reported MOKE measurements [23,24] the observed AFM-cAFM transition field has been higher, suggesting a higher anisotropy level of 𝐾\t~\t0.6\t𝐽 (alongside a slightly higher 𝐽 value of 2.55 T), which would in turn lead to a higher bulk spin-wave gap of about 116 GHz (0.48 meV).    In principle, an N-layer system holds N magnon modes, but as mentioned above, the only mode observed here appears to be the lowest order mode, which is an “out-of-phase” mode wherein each spin precesses directly out of phase with its neighbors (see Supplementary Note 4 for further description of this mode). The signal is only visible above an onset field of a few tesla (which is smaller in even-layered samples) and vanishes again above the cAFM-FM transition. Note that the previous report on magnons in bilayer MnBi2Te4 using Raman spectroscopy observed the higher order of the two modes [31], in which the spins all precess in phase. To try to explain the limited field range in which magnons can be observed, we have performed theoretical calculations (shown in detail in Supplementary Note 4) which suggest that modeling the effect of the pump pulse as suddenly changing the effective exchange coupling gives a qualitatively similar result, with no excitation occurring in the FM region and a falling amplitude at low fields, with the odd-layered samples falling off more rapidly. A similar light-induced change in exchange coupling was also proposed for the excitation of the out-of-phase magnon mode in EuTe [42]. Note that another angle that can be taken is to consider the detection sensitivity. In a typical time-resolved MOKE experiment, the signal is taken to be proportional to modulations in the net out-of-plane magnetization 𝑀\" caused by precession of the spins. As mentioned previously, this was part of the motivation for tilting the field slightly out-of-plane, since a purely in-plane field would result in this magnon mode having no effect on 𝑀\" for even-layered samples. Even with the field tilted out-of-plane, this remains true in the FM region, which serves as an alternate explanation for the lack of signal above 𝐻%&. However, the effects on 𝑀\" are nonzero throughout the cAFM state for both odd- and even-layered samples, even increasing somewhat at lower fields, so this cannot serve as an explanation of the low-field cutoff observed in our data. In addition, the detection of magnons via time- 13 resolved reflectivity makes it difficult to draw clear conclusions in this way, as it is unclear whether signal amplitude should even be related to changes in 𝑀\".  5. Conclusion In summary, we have observed magnetization and magnon dynamics in atomically thin MnBi2Te4 across a range of thicknesses. We have found that the magnetization-dependent signals are observed not only in time-resolved MOKE but also in time-resolved reflectivity, indicating strong coupling between the magnetic state and electronic structure. The time-resolved MOKE (and reflectivity) signals can accurately track different magnetic regimes, which could also be applied to the studies of other 2D van der Waals antiferromagnets. Here, this revealed a lower-than-expected transition into FM order for thin samples, which is difficult to detect via static MOKE measurements. We have also demonstrated direct optical excitation and detection of coherent magnons, whose frequencies can be tuned by external magnetic fields. Our work helps establish the magnetic response of MnBi2Te4 to femtosecond pulses in the few-layer limit, giving a basis for future work on optical control of spins or topological states [1-3,32,33,44,45]. The magnon frequencies in few-layer MnBi2Te4 are compatible with those of existing qubit technologies, showing that it has the potential for magnonic applications [46,47] in 2D devices. Many of the magnon dynamics discussed here are widely applicable to layered antiferromagnetic systems, and similar observations have been reported in previously mentioned materials like CrI3 [14] and CrSBr [15]. Some techniques used in these studies, such as voltage gating the sample or measuring propagation of magnons through the sample, could be applicable to future studies of MnBi2Te4. We hope this work can also help inform studies of other layered antiferromagnets in the future, which may benefit from the techniques and analysis used in our experiments.    Competing interests The authors declare no competing interests.    14 Acknowledgements Sample preparation, ultrafast optical measurements and calculations were all carried out at Tsinghua University. The work was supported by the National Key R&D Program of China (Grant Nos. 2020YFA0308800 and 2021YFA1400100), the National Natural Science Foundation of China (Grant No. 12074212). Y.Wu was supported by the National Natural Science Foundation of China (Grant No. 21975140 and 51991343) and Fundamental Research Funds for the Central Universities (Buctrc202212). F.M.B. was also supported by funds from the University of Toronto.  Author contributions Luyi Yang conceived and supervised the project.  F. Michael Bartram built the static MOKE and time-resolved MOKE and reflectivity experiments and performed all the optical measurements with help from Liangyang Liu and Mengqian Che supervised by Luyi Yang.  F. Michael Bartram carried out all the data analysis. Meng Li and F. Michael Bartram performed the theoretical calculations supervised by Suo Yang and Luyi Yang.  Zhiming Xu. performed the DFT calculations supervised by Yong Xu.  Hao Li and Yang Wu grew the samples.  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An example of a typical signal at 3 K is shown in b, where we observed 4 distinct features: (i) a sudden change immediately after the pump, (ii) a slow ns-scale rise, (iii) a few-ns scale decay, and (iv) a much larger and longer-lived signal that decays on a scale of ~100 ns. This latter feature can be easily measured by the signal just before the pump (~12ns after the previous pump pulse), where we can observe a striking field-dependence that varies with sample thicknesses (shown in e and f for 4 and 5 SL respectively). We compare this to static MOKE measurements done with a 633 nm HeNe laser (c,d) on the same sample regions (also at 3 K). In g,h we show the difference between static MOKE measurements at 10 K and 3 K, which show very similar features to those seen in the time-resolved measurements, suggesting a thermally driven mechanism. Arrows in c, e, g indicate the field sweep direction.  19  Figure 2. (Color online) Theoretical calculations of 𝜟𝑴𝒛/𝜟𝑻 and transition fields. The transition between magnetic states and the temperature dependent magnetization is calculate using our theoretical model described in Supplementary Note 3, with exchange and anisotropy parameters 𝐽=2.3 T and 𝐾=0.35\t𝐽, and theoretical temperature values 𝑇=2 and 𝑇=4 for the 𝛥𝑀\"/𝛥𝑇 plots. The difference in z-direction magnetization between the two temperatures is shown for 4- and 5-layer samples in a and b. In panel c, the AFM-cAFM transition fields (𝐻#$%&) and cAFM-FM transition fields (𝐻%&) extracted from the real data (crosses) are compared to the same transition fields calculated from the theory (dots). The comparison reveals a dramatic difference in the cAFM-FM transitions, especially for the thinnest samples. \n 20  Figure 3. (Color online) Field dependence of time-resolved Kerr measurements. We show the time-resolved data after subtracting the background signal across the full field range (swept from negative to positive) for 8 SL (a) and 7 SL (b) sample regions. The AFM-cAFM transitions are indicated with solid lines, and a secondary transition in the 8 SL sample between different cAFM configurations is indicated with a dotted line. Individual signals at 1 T and 7 T, where both samples are in the AFM and cAFM states respectively, are plotted in c and d. In the cAFM state, both regions have similar signals representing a decrease in Mz from heating. In the AFM state, dMz/dT becomes positive for an even-layered sample, giving a signal of opposite sign, whereas for the odd-layered sample it remains negative. The initial step-like component is also smaller, and remains negative in the even-layered sample, in contrast to the other components.  \n 21  Figure 4. (Color online) Magnetic state dependence of time-resolved reflectivity. The time-resolved reflectivity (shown for a 6-layer region of the sample at 3 K with no applied field in a) shows the same general features seen in the Kerr measurements. Note that this large signal vanishes above 𝑇' (inset in a), and the background signals (b and c for 6- and 7-layer samples respectively) show clear cusps near the transition points (where, as before, the cAFM region is indicated by grey shading), indicating it is genuinely connected to the magnetic state. It differs notably in that the signal is (nearly) symmetric about the field, compared to the antisymmetric Kerr. Additionally, the even-layer sample has a large signal with zero applied field, where both the time-resolved Kerr signal and the expected change in net magnetization with temperature from calculations are very small.    22  Figure 5. (Color online) Magnon oscillations in 5-SL MnBi2Te4. The external field is applied at a small (≈7.5°) angle with respect to the sample plane, as shown in a, which causes the spins to begin tilting for all non-zero field strengths, eventually transitioning to (nearly in-plane) FM order at high field. Transient reflectivity measurements at varying fields are shown in b (curves offset for clarity), where starting at around 3 T we can observe a distinct oscillation which decreases in frequency with increasing applied field before vanishing close to 7 T. The field dependence of this mode can be seen more clearly in the FFT amplitude (c, central frequencies from fitting shown with green dots), dropping to zero frequency at ~6.8 T (dashed line). Like the out-of-plane field case, we observe a long-lived component of the Δ𝜃( signal, which can be measured as a function of field (d, with arrows indicating sweep direction) to reveal the magnetic state. A cusp at 6.8 T (dotted line, the same field where the magnons vanished) can be observed, which we attribute to the cAFM-FM transition.  Comparing our observations to magnon modes calculated from theory (green lines in e), they appear to match well with the portion of the lowest-frequency mode just below the cAFM-FM transition field (yellow highlight in e). \n 23  Figure 6. (Color online) Field- and thickness-dependence of magnons in MnBi2Te4. In a-e we show the FFT amplitude of the transient reflectivity for various thicknesses (all at 4 K) along with central frequencies from fits (green dots), where a distinct difference can be observed between the even (a-c) and odd (d,e) layered sample regions. We further plot how the amplitude (f,g) and lifetime (h,i) of the magnon mode, obtained from least-squares fitting, vary across magnetic field for each of the sample regions, where again distinct differences can be observed between the even (f,h) and odd (g,i) layered regions.   \n"    },    {        "title": "2003.12700v1.Spin_transport_and_dynamic_properties_of_two_dimensional_spin_momentum_locked_states.pdf",        "content": "Spin transport and dynamic properties of two-dimensional spin-momentum locked\nstates\nPing Tang1;2, Xiufeng Han1, and Shufeng Zhang2\u0003\n1Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, University of Chinese Academy of Sciences,\nChinese Academy of Sciences, Beijing, 100190, China and\n2Department of Physics, University of Arizona, Tucson, AZ 85721, USA\nMaterials with spin-momentum locked surface or interface states provide an interesting playground\nfor studying physics and application of charge-spin current conversion. To characterize their non-\nequilibrium magnetic and transport properties in the presence of a time-dependent external magnetic\n\feld and a spin injection from a contact, we introduce three macroscopic variables: a vectorial helical\nmagnetization, a scaler helical magnetization, and the conventional magnetization. We derive a set\nof closed dynamic equations for these variables by using the spinor Boltzmann approach with the\ncollision terms consistent with the symmetry of spin-momentum locked states. By solving the\ndynamic equations, we predict several intriguing magnetic and transport phenomena which are\nexperimentally accessible, including magnetic resonant response to an AC applied magnetic \feld,\ncharge-spin conversion, and spin current induced by the dynamics of helical magnetization.\nI. INTRODUCTION\nThe electronic states with spin-momentum locking\n(SML) into mutually perpendicular directions occur at\nthe surface of a topological insulator or at the interface\nwith a strong Rashba interfacial spin-orbit coupling [1{\n3]. These SML states are two-dimensional itinerant mag-\nnetic states without spontaneous magnetic moment in\nthe absence of the magnetic \feld and there is no mag-\nnetic hysteresis response to the magnetic \feld. Yet, these\nmaterials have displayed profound magnetic phenomena.\nFor example, an applied electric current can induce a\nnon-equilibrium spin density known as the Edelstein ef-\nfect (EE) [4{6], and reciprocally, a spin current injection\nproduced by, e.g., spin pumping [7, 8], to such electronic\nsurface states yields an electric charge current termed\nas the inverse Edelstein e\u000bect (IEE) [9{14]. These ob-\nserved EE and IEE phenomena have drawn considerable\ninterest due to their potential applications for spin-based\nelectronic devices [15].\nIn this paper, we theoretically study the time-\ndependent magnetic and spin transport properties of such\nSML systems in the presence of the external spin injec-\ntion and magnetic \feld. In the conventional magnetic\nmaterials, the time-dependent magnetization dynamics is\ndescribed by the Landau-Lifshitz-Gilbert equation with\nan additional Slonczewski spin transfer torque [16, 17]\nwhen a spatially varying spin current is present. For\nthe SML systems, there is no net magnetization with-\nout a magnetic \feld in the equilibrium even though the\nspin state of an electron with a given momentum is well-\nde\fned, i.e., the spin is ordered perpendicularly to the\nmomentum. Such unique ordered spin con\fguration in\nthe momentum space may be described by three macro-\nscopic variables: a vectorial helical magnetization (VHM)\n\u0018\u0011<^p\u0002\u001b>and a scalar helical magnetization (SHM)\n\u0011\u0011<^p\u0001\u001b>, in addition to the conventional magne-tizationm\u0011<\u001b>, where ^p=p=p is the direction of\nmomentum and \u001bis the Pauli matrix. Clearly, VHM and\nSHM characterize the relative orientation of the momen-\ntum and spin. In equilibrium, these macroscopic vari-\nables take simple values for the SML system, \u0018eq6= 0,\n\u0011eq= 0 andmeq= 0.\nThe main objective of the paper is to determine these\nvariables in the presence of a driving force such as a time-\ndependent magnetic \feld or a spin current injected from a\nnearby contacting metallic layer, and more interestingly,\nto establish the relations between these variables and\ntransport properties that can be measured experimen-\ntally. Due to the special band structure of SML systems,\nwe \fnd that the spin transport and dynamic properties\ndisplay a number of unique characteristics and the above\nde\fned three macroscopic variables provide a convenient\nway to understand and explain the non-equilibrium pro-\ncesses. This paper is organized as follows. In Sec. II,\nwe present our model and introduce a spinor form of the\nBoltzmann equation. In Sec. III, we derive the dynamic\nequations of these three macroscopic variables from the\nspinor Boltzmann equation, with appropriate simpli\fca-\ntions. In Sec. IV, we solve the equations in several cases\nthat are experimentally accessible. In Sec. IV, we con-\nsider the e\u000bect of the time-dependent motion of the VHM\non the spin pumping. Finally, we conclude the paper in\nSec. V.\nII. MODEL HAMILTONIAN AND SPINOR\nBOLTZMANN EQUATION\nWe consider a spin-orbit coupled two-dimensional band\nstructure with a simple dispersion relation given by\n^\"p=\"0\np+\u000b(^z\u0002p)\u0001\u001b (1)arXiv:2003.12700v1  [cond-mat.mes-hall]  28 Mar 20202\nFIG. 1. (a) The schematics of the SML surface in contact\nwith a normal metal layer. On the other side of NM, a fer-\nromagnetic layer is served as a either spin injection source\nor spin detection probe with an attached normal metal. The\nrole of the normal metal is to avoid the magnetic coupling\nbetween the spin source (the ferromagnetic layer) and SML\nlayer while to allow spin current to \row through the entire\nlayers. Inversely, the dynamic resonant states of SML excited\nby an AC magnetic \feld can pump a spin current into the\nnormal metal that will be detected by the magnetic probe at\nthe top. (b) A simple picture of spin directions at the Fermi\ncircle of the SML states.\nwhere\"0\npis the spin-independent part of the electron dis-\npersion which we will take zero for the surface state of\na topological insulator and \"0\np=p2=2m\u0003for a Rashba\nband, where m\u0003is the e\u000bective mass, and \u000bis the spin-\norbit coupling constant. For a given momentum p, the\neigenvalues take \"p\u0006=\"0\np\u0006\u000bp with the corresponding\nspin eiegnstates \u001fp\u0006satisfying ( \np\u0001\u001b)\u001fp\u0006=\u0006\u001fp\u0006,\nwhere \np\u0011(^z\u0002^p) is the unit vector representing the\nquantization axis of the spin for a given momentum p.\nThe above model describes a simple one-electron SML\nstate at equilibrium. We now turn on an AC magnetic\n\feldhex(t) and a DC electric \feld Eex, in addition to\na possible spin injection from a contacting normal metal\n(NM), as shown in Fig. 1. The electron Hamiltonian with\na given momentum then reads\n^Hp= ^\"p+hex(t)\u0001\u001b+eEex\u0001r (2)\nTo calculate the spin transport properties in response\nto the magnetic and electric \felds, we introduce a spinor\nform of the semiclassical Boltzmann distribution function\nfor the SML band,\n^f(p;t) =^f0(p) +fc(p;t) +fs(p;t)\u0001\u001b (3)\nwhere ^f0is the equilibrium Fermi distribution function\nand we take the distribution at T= 0, i.e., ^f0(p) =\n\u0002(\"F\u0000^\"p) with\"Fthe Fermi energy. In Eq. (3), the\nnon-equilibrium distribution function is separated into\nthe spin-independent ( fc) and spin-dependent ( fs) parts.\nThe generalized spinor Boltzmann equation is,\n@^f\n@t+eEex\u0001^v \n\u0000@^f0\n@^\"p!\n+ih\n^Hp;^fi\n= \n@^f\n@t!\ncol(4)where ^v=@^\"p=@pis the velocity and the anticommuta-\ntor is implied for the product between Pauli matrices in\nthis paper.\nAt this point, we want to emphasize that the pres-\nence of the commutator in Eq. (4) between ^Hpand the\nspinor distribution function allows the electron to occupy\nthe states that are notthe spin eigenstates at equilib-\nrium; this term represents the procession of the nonequi-\nlibrium electron spin around the e\u000bective magnetic \feld\n(the spin-orbit \feld \u000b(^z\u0002p) and the external magnetic\n\feldhex). Thus, we do not assume that fsis parallel to\nthe\np. Recall that in a ferromagnetic metal, the spin\ndependent distribution function fsis always con\fned to\nthe states parallel or antiparallel to the local magnetiza-\ntionM, i.e.,fs/Mand thus the Boltzmann equation in\nthe conventional ferromagnet has only two components\n(spin up and down relative to the local magnetization).\nThe transverse component of spin accumulation or spin\ncurrent injected to a ferromagnet is absorbed by the fer-\nromagnet within a few atomic distance due to the strong\nexchange interaction between the transverse spin and lo-\ncal magnetization; the absorption of the transverse spin\ncurrent is considered as the manifestation of spin transfer\ntorque. In the present case, we argue that the spin orbit\ncoupling responsible for the dephasing of injected spins\nis much weaker compared to that of the transverse spin\nin conventional ferromagnets since a typical spin orbit\ncoupling in a Rashba Hamiltonian or TI, which is about\ntens of meV [6, 13, 14], is much smaller than the ex-\nchange interaction in ferromagnets (of the order of eV);\nwe will return to this point later when we model relax-\nation times. Thus, we do not specify the direction of fs\nand instead we will determine the dynamic equations rel-\nevant to it from the above spinor Boltzmann equation.\nThe resulting commutator in Eq. (4) for an arbitrary fs\nis\nh\n^Hp;^fi\n= 2ih\u0010\n\u000b^z\u0002p+hex\u0011\n\u0002fsi\n\u0001\u001b (5)\nThe collision term on the right-hand side of Eq. (4)\nhas two contributions: one is the internal spin and mo-\nmentum relaxations of the SML system, and the other\nis the interfacial scattering with the attached NM layer.\nWe may parameterize these processes below,\n \n@^f\n@t!\ncol=\u0000fc+fL\u0001\u001b\n\u001cp\u0000fT\u0001\u001b\n\u001c\u001e\n+X\np0^\u0000pp0h\n^g(p0;t)\u0000^f(p;t)i\n(6)\nwhere we have introduced two characteristic relaxation\ntimes for the SML: the \frst term is caused by the mo-\nmentum scattering of electrons, which is responsible for\nthe relaxation of the longitudinal part of spin-dependent\ndistribution function (relative to the spin-orbit \feld di-\nrection), i.e., fL\u0011(fs\u0001\np)\npwith a relaxation time3\n\u001cp, and the second term is the spin dephasing for the\ntransverse part of the spin-dependent distribution func-\ntionfT\u0011fs\u0000(fs\u0001\np)\npwith the relaxation time \u001c\u001e.\nThe last term in Eq. (6) represents the contribution from\nthe interfacial scattering between the SML and NM layers\nwith a transition rate ^\u0000pp0and ^g(p0;t) being the spinor\ndistribution function of the NM layer at the interface.\nIII. DYNAMIC EQUATIONS FOR\nMACROSCOPIC VARIABLES\nThe Boltzmann equation, Eq. (4), along with Eqs. (5)\nand (6), remains mathematically complicated since the\ncollision terms make the Boltzmann equation an inte-\ngral equation. To further reduce the mathematical com-\nplication, in this Section, we propose to generate the\nspin-di\u000busion-like equations from the Boltzmann equa-\ntion such that resulting simpler equations can be directly\nused for experimental analysis. The following approx-\nimations are made. First, we assume the distribution\nfunction of the NM layer ^ g(p0;t) can be described by a\ntime-independent single parameter, i.e., the spin chemi-\ncal potential, de\fned as \u0016s= (1=eNF)P\npTr\u001b(\u001b^g) with\nNFits density of state at the Fermi level, which acts as a\nbias injecting the spin current into the SML layer. Note\nthat the spin chemical potential in the NM layer may\ncome from the spin injection of the source layer such as\nthe spin pumping of ferromagnetic layer, see Fig. 1. In\nprinciple, one should self-consistently determine the dis-\ntribution function ^ g(p0;t) or the spin chemical potential\n\u0016s; this will involve the Boltzmann equation and bound-\nary conditions for ^ g(p0;t) at the interface of the NM and\nspin source layers. For the purpose of deriving the closed\nform of macroscopic dynamic equations, the presence of\nthe spin chemical potential near the SML layer is su\u000e-\ncient. The second approximation is to assume that the\nelectron transition across the interface conserves the spin\nand the interface transition rate ^\u0000pp0is independent of\nthe momentum p0of electrons in the NM layer; this is the\nassumption frequently used for modeling electron tunnel-\ning or di\u000busion across a rough interface. It is, however,\nthat ^\u0000pp0does depend on the momentum pof electrons\nin the SML layer even for the rough interface since the\nelectron spin in the SML layer is coupled to its momen-\ntum. The symmetry of the SML states demands ^\u0000pp0to\nhave the following form,\n^\u0000pp0=\u00121\n\u001cc\u0013\n+\u00121\n\u001cs\u0013\n\np\u0001\u001b (7)\nwhere 1=\u001ccand 1=\u001cscharacterize the sum and di\u000berence\nof the transition rate across the interface for two spin\nsubbands \\\u0006\" of the SML, respectively. In the case of\na single subband at the Fermi Level, i.e., a topological\ninsulator band, 1 =\u001cc= 1=\u001cs[18], while in the case of a\nRashba band we assume that the transition rates for twosubbands are the same in the limit \"F\u001d\u000bpFand thus\n1=\u001cs= 0.\nFinally, we assume the momentum relaxation time is\nmuch faster than the transverse spin relaxation time\n(\u001cp=\u001c\u001e\u001c1), as we have discussed above. Physically, \u001cpis\ndue to the impurity or defect scattering, while \u001c\u001einvolves\ninelastic or interband scattering. More quantitatively, \u001c\u001e\nscales inversely with the strength of spin orbit coupling\nwhich is the order of several tens of meV for the TI or\nRashba systems, and thus \u001c\u001eis about picoseconds while\nthe momentum relaxation time is typically a few fem-\ntoseconds [10]. The separation between these two time\nscales are critically important since we are able to treat\nthe dynamics for the longitudinal and transverse spins\ndi\u000berently. In fact, we will limit our dynamic description\nbetween these two time scales such that the longitudi-\nnal spin reaches steady states instantly. Equivalently,\nwe take the limit @fc=@t= 0 and@fL=@t= 0 in the\nBoltzmann equation, and we focus the time dependence\nof the distribution function on the \\ slow\" dynamics of\nthe transverse spin part fT.\nWith above simpli\fcations, we can now explicitly es-\ntablish the dynamic equations for three macroscopic vari-\nables by inserting the Boltzmann equation Eq. (4) along\nwith the explicit relations of Eqs. (5), (6) and (7) into\nthe de\fnitions of the VHM, \u0018(t) =P\npTr\u001b(^p\u0002\u001b^f);the\nSHM,\u0011(t) =P\npTr\u001b(^p\u0001\u001b^f);and the conventional mag-\nnetization,m(t) =P\npTr\u001b(\u001b^f). After a tedious but\nstraightforward algebra, we obtain,\n@m\n@t=!0h\n^z\u0002\u0018\u0000\u0011^zi\n+ 2hex\u0002m\u0000m\u0000mp\n\u001c\n+ginth\u0000\n\u0016s\u0001^z\u0001^z+1\n2\u0000^z\u0002\u0016s\u0001\n\u0002^zi\n(8)\n@\u0018\n@t=!0^z\u0002h\nm\u0000mpi\n+ 2\u0011hex\u0000\u0018\u0000\u0018eq\n\u001c(9)\n@\u0011\n@t=!0^z\u0001m\u00002hex\u0001\u0018\u0000\u0011\n\u001c(10)\nwhere!0= 2\u000bpFwith pFthe Fermi momentum, gint=\neNF=\u001ccis an interface spin conductance, 1 =\u001c= 1=\u001c\u001e+\n1=\u001ccandmp=P\npTr\u001b[(fL\u0001\u001b)\u001b] reads\nmp=eNF\u001cp\n2(\u001cp+\u001cc)^z\u0002(\u0016s\u0002^z) +\u001ccqEE^z\u0002Eex (11)\nwhereqEEis the EE coe\u000ecient,\nqEE=8\n<\n:\u000be\u001cpm\u0003\n2\u0019(\u001cp+\u001cc)for Rashba,\n\u0000sgn(\"F)epF\u001cp\n4\u0019(\u001cp+\u001cc)for TI(12)\nwhere sgn(\"F) = +1 for \"F>0 and sgn(\"F) =\u00001 for\n\"F<0. Eq. (8)-(10) along with Eq. (11) are our main\nresults and they can be broadly used for capturing the\ndynamics of the SML states in the presence of exter-\nnal \felds and spin current injection. We shall point out4\nthat the di\u000berence between the total magnetization m\nand themp: the latter describes the shift of momentum\ncenter due to the presence of an electric \feld and spin\ncurrent injection. Since we assume a fast relaxation for\nthe electron momentum, mpis treated as a steady state\nsolution. Microscopically, mpis comprised of the states\nwith the longitudinal spins (relative to the spin-orbit \fled\ndirection \np), while themincludes both transverse and\nlongitudinal spin components. It is the transverse com-\nponent ofmthat gives arise to the time-dependent mo-\ntion. When a spin current injected from the contact,\nits longitudinal spin component induces a spin accumu-\nlation (mp) and thereby converts to an electric current,\ni.e., the inverse Edelstein e\u000bect, while its transverse spin\ncomponent is equivalent to the spin transfer torque which\ndrives the magnetization dynamics. One might compare\nthis picture with the conventional spin injection to a fer-\nromagnet where the transverse spin current leads to the\nspin transfer torque on a macros-spin while the longitu-\ndinal component generates a magnetoresistance (or spin\naccumulation).\nIV. APPLICATIONS OF THE DYNAMIC\nEQUATIONS\nThese equations may be considered as an extension of\nthe Landau-Lifshitz-Gilbert-Slonzcewski (LLGS) to the\nSML systems. For a conventional itinerant ferromagnet,\nthe LLG equation involves only one macroscopic vari-\nable in magnetization; we have three coupled equations\nfor three helix-dependent magnetizations \u0018(t),m(t) and\n\u0011(t). Therefore, it may be more appropriate to compare\nEq. (8)-(10) with the LLGS for antiferromagnets in which\nthe dynamics of the staggered magnetic moment is always\ncoupled to the magnetization because a time-dependent\nchange of the staggered moment is only possible when\nthe magnetic moment of each sublattice does not exactly\ncompensate for each other. In the present case, our VHM\nand SHM are coupled to the in-plane and out-of-plane\ndirection of the conventional magnetization through the\nspin-orbit coupling, respectively. In this Section, we solve\nthese equations in two simple cases that can be readily\ntested experimentally.\nA. Magnetic resonance\nAs in the cases of ferromagnets and antiferromag-\nnets, the dynamic equations contain characteristic res-\nonant states. To see this, we ignore all the relaxation\nterms and then take simple time dependent solutions,\ni.e.,\u0018(t),m(t) and\u0011(t)/exp(\u0000i!t). By placing them\ninto Eqs.(8)-(10), we immediately obtain three degener-\nate resonant modes at frequency !=!0. There are two\ntransverse modes consisting of the left-handed and right-\nFIG. 2. Microscopic dynamics of the electron spin in mo-\nmentum space for three degenerate resonant modes, in which\neach spin precesses along the local spin-orbit \feld. (a) The\ntwo in-plane modes include the right-handed and left-handed\nprecession of mutually coupled VHM \u0018and the in-plane com-\nponent ofm. (b) The out-of-plane mode represents the cou-\npled oscillation of SHM \u0011and the out-of-plane component of\nm. The small black arrow represents the electron spin and\n\\\u0006\" is the sign of spin projection along z direction.\nhanded precessions of the VHM and in-plane component\nof magnetization, and one longitudinal mode represent-\ning the oscillation of SHM and longitudinal component\nof magnetization. In Fig. 2, we show the correspond-\ning microscopic spin dynamics in momentum space for\nthe three modes. We shall emphasize that the resonant\nfrequency is controlled by the spin-orbit coupling or the\nspin-momentum locking parameter \u000b; this is rather dif-\nferent from the conventional magnetic systems where the\nresonant frequency is determined either by the anisotropy\n\feld for a ferromagnet, or by the geometric mean of the\nexchange coupling and the anisotropy \feld for an anti-\nferromagnet.\nThe above resonant states can be excited by apply-5\ning a time-dependent magnetic \feld, similar to the fer-\nromagnetic resonance (FMR). To excite an in-plane res-\nonant mode, one applies a combined out-of plane DC\nand a small in-plane AC magnetic \feld hex(t) =h0^z+\n\u000ehac(cos!t^x+ sin!t^y). By placing the magnetic \feld\ninto Eq.(8)-(10) with no electric \feld Eex= 0 and spin\ninjection\u0016s= 0 and by de\fning the in-plane VHM sus-\nceptibility, \u000e\u0018=\u001fin\u000ehace\u0000i!t, where\u000e\u0018=\u000e\u0018x\u0000i\u0018y, we\nobtain\n\u001fin(!) =\u00124h0\u0018eq\n1 +!2\n0\u001c2\u0013(1\u0000i!\u001c)\n!2\u0000!2\n0\u00002h0!+ 2i!=\u001c(13)\nwhere we ignored the second-order term in 1 =\u001candh0,\nand the magnetization is related to the VHM through\nm=\u0000i!0\u001c\n1\u0000i!\u001c+ 2ih0\u001c\u000e\u0018 (14)\nwherem=mx\u0000imy. Note that the DC magnetic \feld\nh0is necessary to excite the in-plane resonant mode since\nthe resonant amplitude of VHM is proportional to h0.\nClearly, as shown in Eq. (13), the DC magnetic \feld ap-\nplied along ^zdirection can also lift the degeneracy of two\nin-plane processional modes, i.e., !\u0006=h0\u0006!0, where\n+(\u0000) correspond to the right (left)-handed precessions.\nSimilarly, if one applies a small AC \feld along the ^z\naxis,hex(t) =hace\u0000i!t^z, one is able to excite the dynam-\nics of SHM \u0011(t). The out-of-plane susceptibility of the\nSHM\u001fout(!)\u0011\u0011=hacreads,\n\u001fout(!) =2(1=\u001c\u0000i!)\u0018eq\n!2\u0000!2\n0+i\u000f(15)\nwith the magnetization in this case\nmz=\u0000!0\u001c\n1\u0000i!\u001c\u0011 (16)\nIn contrast to the in-plane modes, the out-of plane mode\ncan be directly by an AC magnetic \feld along the z di-\nrection.\nB. E\u000bects of spin injection\nAs we mentioned earlier, the spin current injection\nfrom a contacting conductor has two e\u000bects. One is to ac-\ncumulate spins in the SML states, and consequently the\nspin accumulation leads to a charge current via spin or-\nbit coupling, or the IEE e\u000bect. The other e\u000bect is a spin\ntransfer torque exerted on the magnetization if the spin\ncurrent is not parallel to the direction of local spin-orbit\n\feld ( \np). To quantitatively include both e\u000bects, we set\nhex= 0 and \frst consider the steady state solution, i.e.,\n@\u0018=@t=@\u0011=@t =@m=@t= 0. From Eq. (8)-(10), wehave\nm=mp+\u001cgint\n1 +!2\n0\u001c2h\u0000\n\u0016s\u0001^z\u0001^z+1\n2\u0000^z\u0002\u0016s\u0001\n\u0002^zi\n;(17)\n\u000e\u0018=!0\u001c2gint\n2(1 +!2\n0\u001c2)^z\u0002\u0016s; (18)\n\u0011=!0\u001c2gint\n1 +!2\n0\u001c2^z\u0001\u0016s: (19)\nThe deviation of the VHM and SHM from the equilibrium\nvalues implies that the spin and momentum are no longer\nlocked into completely perpendicular directions due to\nthe spin transfer torque; this leads to an additional mag-\nnetization beyond the simple momentum-shift-relevant\nmagnetization ( mp), i.e., the second term in Eq.(17).\nThe spin current injected into the SML layer is corre-\nlated with the spin chemical potential at the interface.\nIn the steady state, the spin current across the interface\nis given by,\njs=X\npp0Tr\u001bn\n\u001b^\u0000pp0h\n^g(p0)\u0000^f(p)io\n=eNF\n2(\u001cp+\u001cc)^z\u0002(\u0016s\u0002^z)\u0000qEE^z\u0002Eex\n+\u0010\n1\u0000\u0010\u0011\nginth\u0000\n\u0016s\u0001^z\u0001^z+1\n2\u0000^z\u0002\u0016s\u0001\n\u0002^zi\n(20)\nwhere the \frst term is the conventional spin current and\nthe second term is the electric current-driven EE e\u000bect,\nand the last term represents the spin transfer torque from\nthe spin injection whose spin component is not parallel\nto (\np) with a back\row factor \u0010=\u001c\u001c\u00001\nc(1 +!2\n0\u001c2)\u00001.\nSimilarly, the electric current in the SML layer also gains\na contribution from the spin transfer torque,\nje=\u001beEex+\u0012\n\u0015IEE+e\u001c! 0gint\n1 +!2\n0\u001c2\u0013\n^z\u0002\u0016s (21)\nwhere\u001beis the electric conductivity and \u0015IEE =\n\u0000(2e=~)qEEis the conventional IEE coe\u000ecient. Besides\nthe IEE term, there is an additional contribution to the\nelectric current, which is irrelevant to the momentum\nshift and caused by the spin torque. Note that here we\nchoose the electric \feld and spin accumulation in the NM\nas the driving forces and hence there exists an Onsager\nreciprocal relation \u0000(2e=~)qEE=\u0015IEE.\nC. Spin pumping\nSpin pumping is the reciprocal e\u000bect of spin current\ninjection, similar to the non-magnetic metal-ferromagnet\nbilayer system, in which the time-dependent magnetiza-\ntion generates an outgoing spin current to the contacting\nnon-magnetic metal. In this Section, we formulate the\nspin pumping current due to the dynamics of the VHM6\nand SHM. The pumping spin current across the interface\ncan be de\fned as,\njpump\ns =X\npp0Tr\u001bf\u001b^\u0000pp0^f(p;t)g (22)\nwhere we do not include the \\\row back\" of the spin cur-\nrent by the induced spin accumulation in the contact\nlayer since it is the second e\u000bect of dynamics of VHM\nand SHM. By utilization Eq. (3) and (7), one can im-\nmediately identify that the spin pumping in Eq. (22) is\nproportional to the total magnetization m(t). Provided\nthat the SML is in resonant modes and expressing the\nmagnetization in terms of the time-derivatives of VHM\nand SHM, we \fnd\njpump\ns =\u00001\n\u001cc!0\u001a\n^z\u0002@\u0018\n@t\u00001\n!0\u001c^z\u0002\u0012@\u0018\n@t\u0002^z\u0013\u001b\n+1\n\u001cc!0\u0012\n1\u0000i\n!0\u001c\u0013@\u0011\n@t^z (23)\nwhere the dynamics of VHM and SHM pump out an in-\nplane and an out-of-plane polarized spin current, respec-\ntively.\nWe want to emphasize the di\u000berences between the con-\nventional spin pumping in ferromagnets and the above\nformula. In our model, the dynamics of the VHM and\nSHM are induced by the AC magnetic \feld and we have\nused the approximation which is valid up to the \frst-\norder in the magnetic \feld. Thus, the spin current pump-\ning contains only the AC component. In the conventional\npumping, the magnetization dynamics could be gener-\nated by various methods and the the spin pumping for-\nmula is written beyond the linear response. As a result,\nthe conventional spin pumping contains both AC spin\ncurrent and a higher order DC component of the spin\ncurrent.\nV. DISCUSSION AND SUMMARY\nWe have considered the magnetic and spin transport\nproperties in the presence of the time-dependent mag-\nnetic \feld and spin current injection from a contact. We\nwant to comment on the di\u000berences of the SML spin\ntransport compared to other materials.\nUp until now, we have not included any interaction\namong spins, and thus it is more appropriate to classify\nthe SML as a paramagnetic materials with a momentum\ndependent magnetic \feld on each particle. The mag-\nnetic resonant frequency is given by the strength of spin-\norbit magnetic \feld \u000bpF, similar to the external mag-\nnetic \feld as the paramagnetic resonance. On the other\nhand, the spins of electrons of the SML is perfectly or-\ndered in a helical state in the momentum space, similar\nto the ferromagnetic or antiferromagnetic spins ordered\nin real space. Although the spin ordering in the SML isnot driven by the exchange interaction, there are some\nshared spin transport properties such as spin dephasing\nand spin pumping.\nIn conventional ferromagnetic systems, there are two\ndi\u000berent degree of freedom for the magnetization dy-\nnamics and spin transport. The magnetization density\nconsists all electrons of occupied states while the spin\ntransport is con\fned to the electron near or at the Fermi\nlevel. Thus the dynamics involves the time-dependence of\nthe magnetization and of the conduction electrons. The\nmagnetization dynamics are considered much slower than\nthat of the conduction electrons, even though they are\nstrongly coupled. In the SML system, both magnetiza-\ntion and transport are governed by the states near the\nFermi level.\nA key observation of the SML is the di\u000berent spin re-\nlaxations for the longitudinal and transverse components.\nFor conventional ferromagnetic metals, the spins have\nmuch shorter transverse relaxation time (or dephasing\ntime) than the longitudinal spins, and thus the magneti-\nzation dynamics modeled by the LLGS do not address the\nconduction electrons, but the much slow dynamics of the\nlocal magnetization. While for the SML system, there is\nno spontaneous local magnetization. On the other hand,\nthe transverse spins relax much slower than the longitu-\ndinal spins, and thus our dynamic equations address the\ndynamics of the conduction electrons.\nThe VHM and SHM provide a useful tool to visualize\nthe spin orientation of the SML. When the VHM devi-\nates from the ground state, the in-plane component of\nthe VHM indicates the degree of the spin tilting away\nfrom the perfect perpendicular locking between the mo-\nmentum of the spin.\nIn summary, we introduce three macroscopic variables\nfor the SML and establish their equations of motion.\nAmong other things, we have discussed the magnetic res-\nonant frequency, spin injections associated with the EE\nand IEE, and we propose a spin pumping formalism.\nVI. ACKNOWLEDGES\nThis work was partially supported by US National Sci-\nence Foundation under Grant No. ECCS-1708180, the\nNational Natural Science Foundation of China(NSFC,\nGrant No.11434014) and China Scholarship Council\n(CSC, [2017]3109).\n\u0003zhangshu@email.arizona.edu\n[1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[2] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and\nR. A. Duine, Nat. Materials 14, 871 (2015).7\n[3] A. Soumyanarayanan, N. Reyren, A. Fert, and\nC. 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Segawa, Y. Ando, and E. Saitoh, Phys. Rev. Lett.\n113, 196601 (2014).\n[13] J.-C. Rojas-S\u0013 anchez, S. Oyarz\u0013 un, Y. Fu, A. Marty,\nC. Vergnaud, S. Gambarelli, L. Vila, M. Jamet, Y. Oht-\nsubo, A. Taleb-Ibrahimi, P. Le F\u0012 evre, F. Bertran,\nN. Reyren, J.-M. George, and A. Fert, Phys. Rev. Lett.\n116, 096602 (2016).\n[14] W. Zhang, M. B. Jung\reisch, W. J. Jiang, J. E. Pear-\nson, and A. Ho\u000bmann, Journal of Applied Physics 117,\n17C727 (2015).\n[15] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[16] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[17] J. C. Slonczewski, Journal of Magnetism and Magnetic\nMaterials 159, L1 (1996).\n[18] S. Zhang and A. Fert, Phys. Rev. B 94, 184423 (2016)."    },    {        "title": "2210.07357v2.Influence_of_Ferromagnetic_Interlayer_Exchange_Coupling_on_Current_induced_Magnetization_Switching_and_Dzyaloshinskii_Moriya_Interaction_in_Co_Pt_Co_Multilayer_System.pdf",        "content": "Influence of Ferromagnetic Interlayer Exchange\nCoupling on Current-induced Magnetization\nSwitching and Dzyaloshinskii-Moriya Interaction in\nCo/Pt/Co Multilayer System\nKrzysztof Grochot1,2,*, Piotr Ogrodnik3,**, Jakub Mojsiejuk1, Piotr Mazalski4, Urszula\nGuzowska4, Witold Skowro ´nski1, and Tomasz Stobiecki1,2\n1Institute of Electronics, AGH University of Science and Technology\n2Faculty of Physics and Computer Science, AGH University of Science and Technology\n3Faculty of Physics, Warsaw University of Technology, Warsaw, Poland\n4Faculty of Physics, University of Bialystok, Bialystok, Poland\n*grochot@agh.edu.pl\n**piotr.ogrodnik@pw.edu.pl\nABSTRACT\nThis paper investigates the relationship among interlayer exchange coupling (IEC), Dzyaloshinskii-Moriya interaction (DMI),\nand multilevel magnetization switching within a Co/Pt/Co heterostructure, where varying Pt thicknesses enable control over\nthe coupling strength. Employing Brillouin Light Scattering to quantify the effective DMI, we explore its potential role in\nmagnetization dynamics and multilevel magnetization switching. Experimental findings show four distinct resistance states\nunder an external magnetic field and spin Hall effect related spin current. We explain this phenomenon based on the asymmetry\nbetween Pt/Co and Co/Pt interfaces and the interlayer coupling, which, in turn, influences DMI and subsequently impacts the\nmagnetization dynamics. Numerical simulations, including macrospin, 1D domain wall, and simple spin wave models, further\nsupport the experimental observations of multilevel switching and help uncover the underlying mechanisms. Our proposed\nexplanation, supported by magnetic domain observation using polar-magnetooptical Kerr microscopy, offers insights into both\nthe spatial distribution of magnetization and its dynamics for different IECs, thereby shedding light on its interplay with DMI,\nwhich may lead to potential applications in storage devices.\nIntroduction\nAmong magnetization switching methods, spin-orbit torque current-induced magnetization switching (SOT-CIMS) in metallic\nmultilayers and magnetic tunnel junctions offers a very short switching time (less than 1 ns) with no breakdown risk of\nthe tunnel barrier used in typical spin-transfer torque (STT)-based memory cells.1–5Until recently, SOT-CIMS has been\nobserved in a variety of heavy metal (HM)-based layered systems, such as simple HM/ferromagnet (FM) bilayers1, 6, 7, or\nHM/FM/antiferromagnet (AFM)8–12and FM/HM/FM trilayers.13–22Heavy metal is a source of spin current due to its strong\nspin-orbit interactions, which cause spin current generation when a charge current flows, as claimed in Refs.23 –26. Spin current\nmay be injected into the FM layer as a result of the spin accumulation gradient at the interface and exerts effective SOT fields,\nfield-like ( HFL) and damping-like ( HDL), which can flip the magnetization of the FM layer.27–33In systems where the FM layer\nis characterized by perpendicular magnetic anisotropy, the magnetization dynamics is also driven, in addition to the SHE, by a\nRashba-Edelstein effect originating from the inversion symmetry breaking at the HM/FM interfaces.26, 34–38To the diversity of\nmagnetic effects present in FM/HM multilayers, one should add the interface-dependent Dzyaloshinskii-Moriya interaction\n(DMI), which affects both the dynamics and switching of the magnetization, as well as the domain structure at the remanent\nstate.39–41\nIn a bilayer HM/FM system, the magnetization state can be determined using anisotropic or spin-Hall magnetoresistance,\nand in a binary system two resistance stated are written using CIMS42. The spin current generated in the HM accumulates at\nboth HM interfaces; however, it can only act on a single FM layer, potentially causing the reversal of its magnetization. In\ncontrast, in trilayer FM/HM/FM systems, the spin current has a different polarization at both interfaces. The energy efficiency\nof the magnetization reversal in such trilayer systems may be slightly higher than that of the bilayer ones. Another advantage of\ntrilayers is that they provide the possibility of four stable resistance states, making them attractive for potential use in low-powerarXiv:2210.07357v2  [cond-mat.mes-hall]  20 Dec 2023consumption and high-density memory design43and bioinspired neuromorphic computations.16, 44\nIn this work, we present a detailed study of multilevel switching via SOT-CIMS in the Co/Pt/Co system with different Pt\nthicknesses. As shown in our previous work38, the thickness of Pt varies along the wedge shape of the sample, resulting in a\ndifferent efficiency of the spin current generation, interlayer exchange coupling (IEC) and the effective magnetic anisotropy of\nthe two layers. For a thin Pt layer between 1and2nm, the transition of effective anisotropy was observed from in-plane to\nperpendicular. We also showed a significant difference in the atomic structure of the lower Co/Pt and upper Pt/Co interfaces,\nwhich consequently affects the amount of spin current flowing into both Co layers and thus, the switching mechanisms of their\nmagnetizations. We also discuss the role of co-existing IEC and DMI in multilevel magnetization switching. The analysis is\nsupported by a number of experimental techniques, such as polar-magnetooptical Kerr microscopy (p-MOKE) for domain\nobservations and Brillouin Light Scattering (BLS) to quantitatively estimate the strength of DMI. We show that the IEC can\ninfluence DMI through modifying the magnetization distribution at HM/FM interfaces. To gain insight into the physical\nmechanisms of the switching and enhance our understanding, we employ macrospin modeling as well as simple models of\ndomain-wall (DW) and spin-wave (SW) dynamics. Additionally, we provide a qualitative explanation for the magnetization\nswitching mechanism in the investigated trilayers. Finally, we demonstrate the dependence of the critical switching current on\nthe magnitude of the IEC in the presence of DMI.\nResults and Discussion\nDzyaloshinskii-Moriya Interaction\nIn FM/HM/FM system, apart from IEC, which has been studied before29, 38, the DMI plays an important role. Pt with its\nsubstantial spin-orbit coupling, not only induces DMI but also modulates the strength of ferromagnetic IEC as its thickness\nvaries. As both of these components are present, it is crucial to understand their contributions and relationship within the context\nof switching mechanism. To study the influence of the IEC on the magnitude of the DMI, we carried out BLS measurements\nin the Damon-Eshbach (DE) configuration on the continuous layer at several points along the Pt wedge. We showed in our\nprevious paper38that the ferromagnetic IEC decreases inversely with Pt thickness.\nFigure 1. BLS measurement to quantify DMI. (a) BLS spectra of Stokes and anti-Stokes peaks for different Pt thicknesses\nfrom regions I-III, where the applied magnetic fields are 3.8 kOe or 6.6 kOe. (b) Extracted values of the ∆f,Deff, and HDMIas\na function of Pt spacer thickness. The solid line represents the fitted theoretical ∆fcalculated using experimentally derived\nDMI constants, where the DMI fields at the top and bottom Co/Pt interfaces are equal ( χ=0). Dashed lines: asymmetric DMI\nfields ( χ̸=0). Upper (lower) dashed line corresponds to a 15% higher DMI field at the top (bottom) Co/Pt interface while\nmaintaining their average at the value used for the solid line (see Spin Wave model in Methods section for details).\nFig.1(a) shows example spectra along with the Gaussian fit for several selected Pt thicknesses from regions I to III (see\nFig.S1 in Supplementary Materials for the BLS spectra from a wider range of HM thickness, spectra in region IV were not\nmeasured due to equipment limitations). Larger Pt thicknesses exhibit lower intensity peaks and broader peak widths. This\ncorrelates with a significant increase in damping at the border between regions II and III and an increase in anisotropy (see\n2/14Fig.S2 in the Supplementary Materials). For larger Pt thicknesses, it was necessary to apply a stronger in-plane magnetic field,\nbecause of the shifting of peaks toward lower frequencies when the Pt thickness increases. The frequency difference ( ∆f) was\nobtained by comparing the frequencies of the Stokes peaks ( fs) anti-Stokes peaks ( fas). In Fig.1(b), the relationship ∆f(tPt)is\nshown by black dots, and the uncertainty, indicated by bars, arises due to the uncertainty of the function fitting to the BLS\nspectra.\nBased on the obtained ∆fvalues, the effective DMI constant ( Deff) was calculated as45:\nDeff=∆fπMs\n2γk(1)\nwhere: Msis the saturation magnetization determined in our previous work38,γis the gyromagnetic ratio of 1.76×1011T−1s−1\nand k = 11.81 µm−1is the wave vector.\nKnowing Defffor each studied Pt thickness, it was possible to calculate the field HDMI, according to the formula45:\nHDMI=Deff\nµ0Ms∆(2)\nwhere ∆=p\n(A/Keff)is the DW width. We used the values of Kefffor a given thickness of Pt from our previous work38, while\nA = 16 pJ/m is the exchange stiffness45. The calculated ∆is in the range of 4 to 9 nm. This range agrees with data commonly\nreported in the literature45, 46.\nIn order to verify the result for the Deffconstant, we recalculated ∆fbased on experimentally derived values of Deffusing\nthe SW model presented in Methods section. In our previous research38, we showed that in the considered thicknesses of Pt,\nthere are large variations in IEC as well as anisotropies in the top and bottom Co layers. Such a procedure provided insight\ninto the reason for the dependence of Deff(dPt)and its reliability in a trilayer system. As shown in Fig.1(b), the calculated\n∆fagrees well with the experimental dependence. The result was obtained for parametrized anisotropy constants and IEC\nas in Ref.38. Importantly, our calculations showed that the dependence holds even if anisotropies and IEC are fixed with a\nconstant value, while DMI fields are the only parameters varying with Pt thickness. This means that the ∆f(tPt)dependence\noriginates only from the change of DMI fields at interfaces, and not from the variations of other parameters. Therefore, the\nexperimentally derived Deffis reliable. Moreover, the calculations indicate that the DMI fields at both interfaces exhibit either\nsimilar magnitudes or, in the presence of asymmetry ( χ=0), the asymmetry is of a nature such that the average of these two\nDMI fields closely approximates Deff.\nIn Fig.1(b), the value of Deffis inversely correlated with the IEC, which, in turn, varies with the thickness of the Pt spacer\n(the dependence of IEC on Pt thickness is shown in Fig.9(f)). This correlation suggests that changes in the IEC may be\nassociated with adjustments in the DMI. Since the DMI arises from the exchange interaction between adjacent ferromagnetic\nspins, it implies that the IEC may influence this type of interaction. As the Pt thickness increases, the IEC interaction tends to\nweaken, potentially allowing the DMI coupling to play a more prominent role.\nWe also investigated how in-plane ( Hx) and out-of-plane ( Hz) magnetic field pulses affect the expansion of bubble-like\ndomains in a Co/Pt/Co films induced at tPt= 2.2 nm (Fig.2(a)). We have shown, like others47that the DMI causes asymmetric\nDW motion, as the external Hxfield modifies the energy of Néel-type DWs (N-DWs) differently depending on their core\nmagnetization direction. By applying a Hzsaturation field, then tuning the HxandHzfields and generating successive field\npulses, we controlled the asymmetry of domain growth and determined the chirality of Co/Pt/Co. Our results show that the\nN-DWs have anticlockwise (ACW) spin configuration, indicating a positive DMI constant ( Deff> 0). A differential p-MOKE\nimage of a continuous Co(1)/Pt(1.38-1.72)/Co(1) sample, subjected to pulsed HxandHz, provides compelling evidence of the\nsubstantial impact of IEC magnitude on the size of the observed domain structure (see Fig.2(b)). The investigated Pt thicknesses\nfall within the range where the most significant reduction in coupling with increasing Pt thickness is observed.\nIn the range of the Pt thickness from 1.38 to 1.55 nm (Fig.2(b)), a remarkably fine-grained domain structure is observed,\nwhich progressively enlarges as the IEC decreases. At approximately 1.62 nm thickness, the structure transitions to a single\nbubble form. Furthermore, the influence of IEC on the shape of the domain wall within the single bubble domain structure was\nalso observed (see Fig.2(c)). For high coupling ( tPt= 1.67 nm), the domain wall exhibits a highly jagged character, while it\nbecomes progressively smoother as the IEC strength decreases.\nCurrent-induced magnetization switching\nAfter determining the IEC and DMI dependence on Pt thickness, we now turn to the switching mechnism in the Co/Pt/Co\nsystem. We measured the anomalous Hall effect (AHE) to observe magnetization switching between two stable high and low\nresistance states of the current-pulsed loop. CIMS takes place in regions II-IV only, where at least one layer is perpendicularly\nmagnetized in the remanent state. To achieve magnetization saturation, the samples were subjected to a large magnetic field in\n3/14Figure 2. (a) p-MOKE difference image showing the growth of bubble domains under the influence of the in-plane magnetic\nfield ( Hx) and the out-of-plane magnetic field ( Hz) in the Co(1)/Pt(2.2)/Co(1) system. The applied field values and directions\nare illustrated in the accompanying image. The initial position of the bubble domain is indicated by an orange ring. Schematic\nrepresentation of the lateral magnetization profile along the dashed white line is depicted in the image. The orange and red\narrows denote the anticlockwise chirality of the N-DWs (Néel domain walls). (b) p-MOKE difference image of the continuous\nwedge layer of the studied system under the influence of HxandHzmagnetic fields, where the orange dashed lines indicate the\nconstant Pt thickness within the wedge. (c) Example p-MOKE differential images of the p-MOKE bubble domain structure for\nseveral Pt thicknesses located in regions II-IV shows the change in roughness of the bubble domain wall. Magnetic fields are\napplied in opposite directions to (a).\nthe−zdirection. Then, to drive magnetization switching, we applied a sequence of 1ms voltage pulses, with a pulse spacing of\n2ms in the xdirection. The pulse amplitude was swept from 0V to a maximum positive value ( +Vmax), then to the maximum\nnegative value ( −Vmax), and then back to 0V . Simultaneously, we measured the transverse voltage ( Vxy) in the presence of an\nin-plane magnetic field Hx, which is co-linear to the current direction. The measurement setup is presented schematically in\nFig.3(a).\nThe in-plane magnetic field Hxwas changed sequentially after each CIMS loop in the wide range of ±7 kOe . As a result,\nwe obtained a set of CIMS loops in different Hxfor representative Pt thicknesses from regions II to IV , and examples are plotted\nin Fig.3(b-e).\nAs shown in Figs.3(b) and (c), experimentally obtained CIMS loops measured at positive and negative magnetic fields are\nclearly separated in regions II and III. Both stable resistance states of the CIMS loops have a higher resistance for +Hx(blue\nloop) than those measured for −Hx(red loop). When the direction of the magnetic field changes from +x to -x, we observed a\nsmooth transition from the high-resistance loop to the low-resistance loop. For the thicker Pt spacer in Fig.3(d) ( tPt=1.64 nm )\nseparation gap becomes smaller compared to the sample of Pt = 1.36 nm thick (Fig.3(b). In region III ( tPt=2.16nm) (Fig.3(d),\nhowever, the four resistance states can still be observed. In the case of the thickest Pt ( tPt=3.57nm), for which only one Co\nlayer exhibits perpendicular anisotropy, the separation gap disappears. Regardless of the direction of Hxonly two resistance\nstates exist, as in the case of the HM/FM bilayer (not shown here).1, 6–8\nSubsequently, we performed an analysis of the critical current densities ( jc,Pt) required to switch magnetization. For\nthis purpose, the dependence of jc,Ptthrough Pt was plotted as a function of the applied external magnetic field ( Hx). As\ndemonstrated in Fig.3(f) for Hx≪Hk,eff, the experimental dependencies measured in all devices are linear, which remains\nconsistent with Ref.48.\nIn Fig.3(g) we show the jc,Ptdependence on the Pt layer thickness. The critical jc,Ptdecreases linearly in a wide range of Pt\n4/14Figure 3. (a) Device used for CIMS measurements. CIMS loops for devices from regions: (b)-(c) II, (d) III, and (e) IV . Blue\nand red dashed lines indicate the resistance levels of the bottom and top Co layers, respectively. The blue and red solid lines\nindicate CIMS loops for positive and negative magnetic fields, respectively. In accordance with the macrospin model, we\ndenoted the gap between loops by δ. Critical switching current density as a function of the external magnetic field ( Hx) for\nsamples from regions II and III (f), critical current density ( jc,Pt) as a function of Pt thickness (g).\nthickness, from 1.6 to 3 nm, when it reaches its lowest value. However, for the thinnest and thickest Pt layer, jc,Ptdeviates from\nthe linear dependence by slightly dropping and rising, respectively. The highest values of the critical current amplitude required\nfor switching are found for elements with a small thickness of Pt and then decrease linearly slightly to a value of approximately\n0.5 TAm−2for the element with tPt=2.92 nm.\nFigure 4. (a)∆R/RAHEratio as a function of the external magnetic field for elements with different Pt layer thicknesses. (b)\nThe maximum values of ∆R/RAHEratio as a function of IEC.\nOne of the most important components of the current-induced magnetization switching process, from an application point\n5/14of view, is the difference between high and low resistance levels of the current switching loop, denoted ∆R=Rhigh−Rlow. In\nthe case of the devices studied, the amplitude ∆Rdepends on the thickness of the Pt spacer and, therefore, on the magnitude of\nthe IEC. Elements with a thin Pt layer (region II and III) and, thus, with strong coupling, exhibit small values of ∆R, reaching\na∆R/∆RAHEvalue of 0.8(Fig.4(a). The fact that the amplitude ratio does not reach the maximum value ( ∆R/∆RAHE< 1)\nFigure 5. Four stable resistance states for the sample from region II ( tPt= 1.55 nm (a), obtained by manipulating the\nmagnitude of the current pulse (b) and the external magnetic field (c). Resistance levels in (a) correspond to the resistance\nlevels of CIMS loops in (d).\nindicates a domain-specific origin of switching and therefore the magnetic domains persist in remanence. Elements in region\nIV , where the coupling is negligible (Fig.4(b), show ∆R/∆RAHEvalues close to 1, suggesting that the switching is practically\nsingle-domain and only Co layer with perpendicular anisotropy switches.\nTherefore, to investigate multilevel switching, we focused on tPt=1.55nm (region II). For this thickness, we chose two\nvalues of the external magnetic field (+0.5 kOe and -0.5 kOe) at which both loops exhibit a significant amplitude ( ∆R) and\nare completely separated. As shown in Fig.5(a), there are four different resistance states. On the basis of the switching loops,\nwe determined the critical current densities ( jc,Pt) of + 1.29 TAm−2and - 1.29 TAm−2needed to switch the magnetization at\n±0.5kOe. Then, to switch the resistance between four well-separated levels, we applied both current pulses of ±Icamplitude\nand the magnetic field of magnitude of ±0.5kOe (Fig. 5(a). By carefully choosing the combination of signs Hx, shown in the\nFig.5(b), and jc, given in Fig.5(c), we obtained a ladder-shaped waveform of resistance of the system (Fig.5(d). The procedure\nof tuning the switching pulse duration and its amplitude for an arbitrary field allows the system to be set in a single well-defined\nresistance state and therefore to store considerably more information in a single memory cell.\nDomain mechanism of multilevel switching\nWe qualitatively explain the observed CIMS loops in terms of magnetic domains in microstrips of Hall-bar devices. For this\npurpose, selected Hall-bars from each region of Pt thickness were imaged with p-MOKE while the Hxfield was applied. The\nresistance Rxyvs.Hxrepresents the reversal magnetization process by blue loops (Fig.6 a,b). It enabled us to relate the change\nof magnetic domain structure with the resistance level measured during CIMS. Firstly, the magnetization of each Hall bar was\nsaturated with Hzfield to the lowest resistance state, indicated by A and E in Figs.6(a) and (b), respectively. Therefore, it was\npossible to assign images of the domain structure to the corresponding resistances in the current switching loops, as shown in\nFigs.6(a) and (b).\nWe also repeated this procedure for the perpendicular field Hzand in this case the magnetization reversal was performed by\na single-domain wall motion, represented by a rectangular AHE loop (not shown).\nThe p-MOKE images reveal that the resistances within the CIMS loops correspond to a very fine-grained domain structure\n(Fig.6(a) A-D) in the Hall bar with thickness tPt=1.36nm (region II) in the remanence state H=0 (j=0). When current-induced\nSOT switches the magnetization, a number of domains change their state to the opposite. The new distribution of magnetic\ndomains results in an intermediate state (yellow dots B and C) placed between the two extremes marked in Fig.6(a) with letters\nA and D. This transition of magnetic domains can be observed as a change in the gray color level of elements marked A and\nB (or C and D) in Fig.6(a). The smooth shape of the CIMS loops in this region confirms the fine-grained magnetic domain\nswitching mechanism.\nThe opposite behavior occurs in the region IV element with a Pt thickness of 3.57nm, where we observed a complete\nmagnetization reversal driven by a current-induced domain-wall motion. This behavior demonstrates itself as a perfectly\n6/14Figure 6. CIMS loops for (a) tPt=1.36 nm thick element (region II) and (b) tPt=3.57 nm thick element (region IV). The\nCIMS loops (red and green) were obtained in fields Hxof+0.8 kOe and −0.8 kOe in (a) and +3.8 kOe and −3.8 kOe in (b),\nrespectively. The blue triangles indicates the Rxyloops measured in the Hxfield. The letters (A-H) in figures (a) and (b)\nindicate the relevant p-MOKE images labeled with the same letter.\nrectangular shape of the CIMS loops with only two stable resistance states for both directions of the magnetic field (+ Hxand\n-Hx) (Fig.6(b)\nMatching the CIMS resistances involved generating the fine-grained structure visible in Fig.6(a). It was achieved by, first,\nsaturating the sample with a perpendicular field ( Hz), then applying a field Hxof about 10 kOe , and then gradually reducing its\nvalue to approximately 1 kOe . As a result, the field-free resistance of the system is not equal to the high-resistance state of the\nAHE loop due to the uneven distribution of the mzcomponents of the magnetic domains in both Co layers. This condition is\npresented in Fig.6(a), where there is a predominance of domains with a + mzcomponent at remanence. Reapplying a small\nHxfield generates a mxcomponent parallel to the direction of the magnetic field in both Co layers, the existence of which is\nnecessary for the switching of magnetization by the spin-polarized current.\nThe following scenario explains the behavior of CIMS: in the top Co layer, the current-induced SOT damping-like effective\nfield ( HDL∼ −m×ey) acts oppositely on domains with +mzand−mzcomponents, i.e., for positive currents, + HDLforces\n+mzdomains to switch, while −HDLpushes −mzdomains back to the perpendicular direction. On the other hand, the spin\ncurrent flowing into the bottom Co layer has the opposite sign. Therefore, SOT stabilizes the +mzdomains while switching\nthe−mzdomains in this layer (Fig.7(a). Firstly we discuss the effect of the SOT field on an uncoupled and fully symmetric\ntrilayer as depicted in Fig.7(a). When the current pulse reaches a critical amplitude value, each of the Co layers can switch only\npartially. However, we note that, in a fully symmetric and uncoupled case, the SOT would not result in a resistance change.\nThen, the increase of −mzin one layer would be balanced by the increase of +mzin the second layer, which is illustrated in\nFig.7(a) with horizontal arrows pointing in opposite directions. However, thin Pt devices (region II) are far from the symmetric\ncase (Ref.38). In 7(b) we show the scenerio of the coupled and asymmetric case. The top and bottom interfaces differ, and\ntherefore the magnitudes of effective HDLfields acting on each Co are not equal, as shown in our previous paper38. Moreover,\nthe magnetic anisotropies in both layers are different, and a large ferromagnetic IEC is present in this region.38For this reason,\nwhen the lower Co layer switches, the ferromagnetic coupling forces the magnetization of the upper Co layer to switch as well.\nThe switching process results in a higher transversal resistance related to a larger number of domains with magnetization\npointing in the +z, rather than the -z direction, in both Co layers.\nThe described mechanism is consistent with the dependence of the critical current density ( jc,Pt) on the thickness of Pt\npresented in Fig.3(g). jc,Ptdecreases for the Pt thickness, ranging from 1.7to3.0nm (regions II and III). This decrease is due\nto a more efficient SOT for the thicker Pt layer.49However, for the thinnest Pt in region II ( 1.3nm), jc,Ptdrops by about ∆jc=\n0.10 TAm−2). Similarly, for the thickest Pt in region IV ( 3.57nm), the critical current abruptly increases approximately ( ∆jc=\n0.17 TAm−2). The deviations from the linear dependence are correlated with very strong coupling (for the thinnest Pt) and\nnegligible coupling (for the thickest Pt). The switching in the thin Pt case relies on the magnetization reversal in both Co layers.\nThese two layers have different anisotropy fields ( Hk,eff(top) > 0, Hk,eff(bottom)<0).38It means that the bottom layer is more\nsusceptible to torque from the SOT effect. Therefore, when the IEC field is strong enough, it easily overcomes Hk,effin the top\nlayer, allowing it to switch at a lower current (SOT). Then, both Co layers are magnetically stiff and behave somewhat like one\nlayer with the effective anisotropy: Hk,eff(top)> Hk,eff>Hk,eff(bottom).\nFor the intermediate IEC (border of regions II and III), the bottom layer is still more switchable, but the coupling does not\nprovide the top layer with enough torque to switch. Both layers become less magnetically stiff, so more current (more SOT) is\nneeded to switch both of them.\nThe bottom layer magnetization is in-plane when the coupling becomes negligible (region IV). It means that the SOT only\n7/14Figure 7. Mechanism of SOT-CIMS in two cases: With no coupling ( JIEC=0) and symmetric Co/Pt – Pt/Co interfaces (a) and\nin the presence of strong coupling and asymmetric interfaces (top interface with less spin-transparency is marked as solid navy\nblue layer) (b). The red(blue) areas represent magnetic domains with average + mz(-mz) components. The dashed areas together\nwith horizontal arrows indicate the change in domain size under the HDLSOT components (thick red arrows) and\nferromagnetic coupling Hcoupfield (thick green arrows, thin solid orange arrows show the coupling between Co layers). The\nspin current with polarization + ey(-ey) is depicted as red (green) bold points with arrows.\nswitches the top layer with higher anisotropy ( Hk,eff(top)). Therefore, the critical current rises despite the thick Pt and large\nSOT.\nMacrospin and 1D domain wall simulation for multilevel switching\nWe attempted to reproduce the experimental results with the simplest possible model. To this end, we employed two macrospin-\nbased models: the LLGS model and the 1D domain wall model (see Methods for details). Reproduction of the hysteresis gap\nunder field reversal is achieved by a small modification of the resistance model of Kim et al.50For simplicity, we neglected\n∆RSMR xyand∆RAMR xydue to their small contributions, since the term (∆RSMR xy+∆RAMR xy)mxmytends to vanish in macrospin\nsimulations as the mycomponent becomes negligible. Based on the previous work38, we posited that the contributions of the\ntop and bottom FM layers are not equal due to the different intermixing of Co and Pt atoms at both interfaces. This assumption\nis accounted for in the model as an additional resistance asymmetry parameter β, leading to Rxybeing computed as:\nRxy≈R(1)\nxy0+R(2)\nxy0+1\n2κ∆RAHE(m(1)\nz+βm(2)\nz) (3)\nwhere the superscript refers to the top (1)or the bottom (2)FM layer. The parameter βranges from 0 exclusive (asymmetric\ninterfaces) to 1 inclusive for entirely symmetric interfaces. The dimensionless parameter κeffectively corrects the amplitude\nRxyfor the multidomain behavior in regions II through III, which is necessary due to limitations of the macrospin model.\nIn the simplified domain wall motion model, we rephrase the resistance model as the net distance traveled by the domains\nin the two layers, xnet. Assuming a finite strip length, the domain can eventually reach either the left or right edge of the strip. If\nthe domain separates the up ( mz>0) and down ( mz<0) states, then the more the domain moved to the right, respective to the\ncenter marked at x=0, the higher the resistance state Rxywould be, as per Eq.3. Figure 8 presents the simulated results of the\nmacrospin depicted in the left side panels (a, c, e) and the 1D domain wall motion model shown in the right side panels (b, d, f).\nThe absence of a proper hysteresis shape in the 1D domain wall simulation is attributed to the lack of an edge field, which\ntypically slows down the domain as it approaches the sample’s edge. Instead, we simply simulated reaching the edge by the\ncurrent reversal; i.e. in our model the domain always reaches the edge of the sample for the minimum and the maximum current.\nIn Fig.8 two main features of the experimental findings are preserved; first, the gap separation is achieved and it shrinks as the\nPt layer becomes thicker (and the coupling decreases). This shrinkage corresponds to an increase in the value of β, indicating a\nsmaller interfacial asymmetry. Second, the critical currents decrease with the growing thickness of the HM layer, which our\nmodel replicates with an adequate increase in the field- and damping-like torque values.\nImportantly, the simulations demonstrated that DMI does not alter the qualitative outcome. Although DMI influences the\namplitude of the Hxfield, depending on its orientation, it does not sufficiently, by itself, explain the distinct separation of\nthe hysteresis states during the field reversal. However, its presence may slightly reduce the gap between the two loops. We\nconclude that the asymmetry of the two Co/Pt and Pt/Co interfaces is crucial for obtaining the multilevel switching behavior.\nIn summary, the experimentally observed multilevel switching primarily originates from the differences in spin-transparency\nat the interfaces. This feature enables effective IEC-mediated domain structure switching in both layers, as demonstrated by\n8/14MOKE imaging, electrical CIMS measurements and related simulations. Additionally, it is notable that the mechanism of\nIEC-mediated switching aligns with the critical current dependence on Pt thickness. While the IEC can potentially tailor DMI\nif it holds sufficient strength to alter the magnetization distributions at interfaces, its direct impact on the multilevel switching\nitself seems relatively limited. Therefore, both DMI and IEC as well as the asymmetry of the HM/FM interfaces are necessary\nto design a structure with more than two transversal resistance states.\nFigure 8. (a, c, e) CIMS multilevel switching for a range of thicknesses. The external field was the same for all simulations\nwith Hx=±0.8 kOe. An example of δseparation gap was marked in (e). (b, d, f) Multilevel switching for a range of\nthicknesses in the 1D domain wall model.\nMethods\nDevice fabrication\nThe continuous, wedge-shaped FM/HM/FM heterostructure was deposited using a magnetron sputtering technique on the\n20×20 mm2Si/SiO 2substrate at room temperature and under the same conditions as in Ref.38. The sample cross-section\nscheme and the coordinate system used are shown in Fig.9(a). Layers are ordered as follows: Si/ SiO 2/Ti(2)/Co(1)/Pt(0-\n4)/Co(1)/MgO(2)/Ti(2) (thicknesses listed in parentheses are in nanometers). Both the bottom and the top Ti layers function as\nthe buffer and the protection layer, respectively. They do not contribute to the studied phenomena as a result of their partial\noxidation and small spin-orbit coupling.51–53After the deposition process, the sample was characterized by X-ray diffraction.\nWe detected the presence of a face-centered cubic fcc(111) texture at the Pt/Co and Co/Pt interfaces and confirmed the existence\nof an asymmetry between these two interfaces. Details of the structural analysis of the studied samples are described in Ref.38.\nWe performed X-ray reflectivity measurements (XRR) to precisely calibrate the thickness of each layer as a function of the\nposition on the sample wedge. In doing so, we were able to precisely determine the thickness of the layers located at a specific\nposition on the wedge of Pt. The variation in thickness of the Pt layers in the device was less than 0.006 nm, so the Pt thickness\nis constant throughout the device. The sample was nanopatterned by optical laser lithography, ion etching, and lift-off to a\nmatrix of different sizes of Hall bar devices, which were optimized for the measurement techniques used. We used Hall bars of\nsize80x10µm2for current-induced magnetization switching (CIMS) measurements, while resistance and magnetoresistance\nmeasurements were performed on 140 ×20µm2devices using the 4-probe method.\nAnomalous Hall effect and effective anisotropies\nWe measured the AHE for all elements along the Pt wedge. As a result, we obtained a set of AHE resistance loops as a function\nof the external magnetic field applied along the zdirection ( Hz). By analyzing their shapes, we could distinguish four regions of\nPt thickness (marked regions I-IV) in which the AHE loops exhibit a similar shape (Fig.9(b)-(e)). As shown in our previous\nwork38, in region I, the magnetizations of both Co layers are in-plane ( Keff<0) and, as a consequence, AHE depends linearly\n9/14Figure 9. (a) Cross-section through the studied heterostructure. The blue arrows depict the direction of magnetization vectors\nin both ferromagnetic Co layers in specified Pt thickness regions. The dashed lines indicate the border of each region. (b)-(e)\nAHE loops for Hall-bar devices with different thicknesses of the Pt spacing layer. The solid black lines in the inset denote the\nsimulated AHE loops using the model described in Sect. Macrospin models. The depicted diagrams of multilayer cross\nsections for all regions indicate the direction of magnetizations of magnetic layers at remanence. (f) Interlayer coupling derived\nfrom the macrospin simulations of the spin diode FMR spectra.\non the magnetic field Hz. Therefore, it is not possible to distinguish resistance states with AHE during CIMS in this region.\nRegions II and III were characterized by two Co layers magnetized perpendicularly to the sample plane in the remanent state\nand strong IEC (Fig.9(f)) (both Co layers switch simultaneously), and as a result, the AHE hysteresis loops become rectangular,\nas demonstrated in Fig.9(b),(c). Moving from region III to IV , the interlayer exchange coupling (IEC) decreases substantially as\nthe Pt spacing layer thickness.38Consequently, the top Co remains magnetized perpendicularly, whereas the bottom layer tends\nto be magnetized in the plane again.\nMacrospin model\nThe model is based on Landau-Lifshitz-Gilbert-Slonczewski (LLGS) of the following form54–56:\ndm\ndt=−γ0m×Heff+αGm×dm\ndt−γ0HFL(m×ey)−γ0HDL(m×m×ey) (4)\nwhere m=M\nMsis the normalized magnetization vector, with Msas magnetization saturation, αGis the dimensionless Gilbert\ndamping coefficient, γ0is the gyromagnetic ratio, HDLandHFLare damping-like and field-like torque amplitudes respectively,\nandeyis the spin polarization vector in y direction. The Heffis the effective field vector that includes contributions from the\nexternal magnetic field, anisotropy, IEC, and demagnetization energy. For the reproduction of the experimental results, we used\nthe open source package CMTJ57, taking the simulation parameters from Ref.38. Small in-plane components of anisotropy help\nbreak the symmetry under an external field Hx. In macrospin simulations, we take the Gilbert damping of αG=0.05.\n1D domain wall model\nThe equations of domain wall motion are given by45, 58:\n(1+α2)˙X=∆(ΓA+αΓB)\n(1+α2)˙φ=−αΓA+ΓB (5)\nwhere Xandφare coordinates: position and angle, ∆is the domain width, ΓAcontains effective field terms and ΓBcontains\nnon-conservative SHE field:\nΓA=γ0\u0012\n−1\n2Hksin2φ+π\n2Hxsinφ+π\n2HDMIsinφ+2JIEC\nµ0MstFMsin(φ−φ′)\u0013\nΓB=γ0π\n2HSHEcosφ\nwhere φandφ′denote the angles of DWs in the coupled layers58. We solved the equations numerically using the Runge-Kutta\n45 method. The magnitudes of the Dzyaloshinskii-Moriya interaction (DMI) were taken from Fig.1.\n10/14Spin wave model\nTo compare the results with theoretical predictions, we employed a simplified model for the Damon-Eschbach (DE) mode of\nspin waves. As a starting point, we have used a SW model by Kuepferling et al.59. Next, we extended the model to Co/Pt/Co\ntrilayer systems by incorporating the Co interlayer coupling energy density:\nEIEC=−JIECm1(r)·m2(r)\nwhere m1,2(r)is a space- and time-dependent magnetization in both layers, and JIECdenotes IEC coupling amplitude. Such\nan approach does not explicitly account for the coupling due to the dynamic dipolar field produced by the SW modes. It\ncould be done as long as the interlayer coupling through Pt is much larger than that for the dipolar fields acting on each other.\nDespite the simplifications we made, the dipolar fields influence the dynamics of each Co layer separately, similarly as in\nRef.59. Our extension required considering two Co layers. Therefore, we assumed that the magnetizations in the two layers\nare in the form m(1,2)=M(1,2)(r,t) =Ms(δmx,1,δmz), where δm(x,z)are small deviations of the magnetizations saturated in\ntheydirection due to the SW DE mode. Therefore, the dynamical components xandzof magnetization had the following\nform: δm(x,z)(r,t) =δm(x,z)ei(k·r+ωt). Next, we linearized the Landau-Lifschitz(LL) equation:dm(1,2)\ndt=−γ0m(1,2)×Heff,(1,2),\nwhere, similarly as in Eq. 4, Heff,(1,2)includes the external and anisotropy field as well as the micromagnetic spin-wave-induced\nfields related to demagnetization: Hd=−Ms\u0000\nδmx(1,2)P(|k|),0,δmz(1,2)(1−P(|kd|)\u0001\n, where P(|kd|) =1−1−e−|kd|\n|kd|where d\nrepresents the thickness of the Co layers, and kdenotes the wave vector length. 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Matter 29,\n303001, DOI: 10.1088/1361-648X/aa752d (2017).\n59.Kuepferling, M. et al. Measuring interfacial dzyaloshinskii-moriya interaction in ultrathin magnetic films. Rev. Mod. Phys.\n95, 015003, DOI: 10.1103/RevModPhys.95.015003 (2023).\n13/14Acknowledgements\nThis work was supported by the National Science Centre, Poland, Grant No. 2016/23/B/ST3/01430 (SPINORBITRONICS).\nP.M. acknowledges National Science Centre Grant Beethoven 2 (UMO-2016/23/G/ST3/04196). W.S. acknowledges grant no.\n2021/40/Q/ST5/00209 from the National Science Centre, Poland.\nAuthor contributions statement\nK.G. carried out microstructurization and conducted the electrical conductivity, spin Hall and anomalous Hall experiments.\nP.O. performed formal analysis and visualization. J.M. is a software developer of modeling software and has performed\nformal analysis and visualization. P.M. conducted magnetic domain imaging using p-MOKE microscopy measurement and\nBLS measurements. U.G. conducted BLS measurements. W.S. was responsible for the microstructurization, design, and\nprogramming of the measurement methods. T.S. supervised the experimental and theoretical modeling aspects of the project.\nAll authors reviewed the manuscript.\nAdditional information\nCompeting Interests: The authors declare that they have no competing interests.\n14/14"    },    {        "title": "1812.06684v1.Dynamically_generated_magnetic_moment_in_the_Wigner_function_formalism.pdf",        "content": "arXiv:1812.06684v1  [hep-th]  17 Dec 2018Dynamically generated magnetic moment in the Wigner-funct ion formalism\nShijun Mao1,2and Dirk H. Rischke2,3\n1School of Science, Xi’an Jiaotong University, Xi’an, Shaan xi 710049, China\n2Institute for Theoretical Physics, Goethe University,\nMax-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany\n3Interdisciplinary Center for Theoretical Study and Depart ment of Modern Physics,\nUniversity of Science and Technology of China, Hefei, Anhui 230026, China\n(Dated: December 18, 2018)\nWestudyhowthemass andmagnetic momentofthequarksare dyn amicallygenerated in nonequi-\nlibrium quark matter. We derive the equal-time transport an d constraint equations for the quark\nWigner function in a magnetized quark model and solve them in the semi-classical expansion. The\nquark mass and magnetic moment are self-consistently coupl ed to the Wigner function and con-\ntrolled by the kinetic equations. While the quark mass is dyn amically generated at the classical\nlevel, the quark magnetic moment is a pure quantum effect, ind uced by the quark spin interaction\nwith the external magnetic field.\nThe intrinsic magnetic moment of an electron is related to its spin by ms=gµBs, whereµB=e/(2m) is\nBohr’s magneton, with eandmbeing the electron charge and mass, gthe Lande factor, and sthe electron spin\nangular momentum, respectively. Dirac theory predicts g= 2 in the non-relativistic limit, but this result was\nlater challenged by many refined experimental measurements, sho wing a larger gfactor. Schwinger calculated\nthe first-order radiative correction to msfrom the electron-photon interaction [1]. The one-loop contributio n to\nthe fermion self-energy was taken into account in a weak magnetic fi eld, which leads to an anomalous magnetic\nmoment, reflected in a correction to the gfactor of order ∼(g−2)/2 =α/(2π), whereαis the fine-structure\nconstant. Higher-order radiative corrections to ghave subsequently been considered [2, 3], resulting in a series\nin powers of α/π. These corrections are in excellent agreement with experimental d ata. In an external magnetic\nfieldB, the anomalous magnetic moment affects the electron energy in the lowest Landau level by turning the\nmass intom2\neff≃m2+ (g−2)eB/2. In the case of massless quantum electrodynamics (QED), the an omalous\nmagnetic moment cannot be described through Schwinger’s pertur bative approach, and chiral symmetry breaking\nwill dynamically generate an anomalous magnetic moment [4].\nThe anomalous magnetic moment in QED is a fundamental phenomenon in gauge field theory. It should happen\nalso for quarks in quantum chromodynamics (QCD) [5, 6]. Considering its non-Abelian and non-perturbative\nproperties, it becomes much more difficult to directly investigate the quantum fluctuations in QCD, and effective\nmodels without gauge fields, like chiral perturbation theory [7, 8] at the hadron level and the Nambu–Jona-Lasinio\nmodel [9–12] at the quark level, are used to calculate the propertie s of, and spontaneous symmetry breaking in,\nstrong-interaction systems. For instance, U(1)Asymmetry breaking and spontaneous chiral symmetry breaking in\nvacuum and their restoration in medium are investigated for therma l equilibrium systems in an SU(3) linear sigma\nmodel [13] and for non-equilibrium systems in an NJL model [14]. In the chiral limit, the quark magnetic moment\nis closely related to the chiral symmetry of QCD [4–6], which is spontan eously broken in vacuum through the\nchiral condensate /an}b∇acketle{t¯ψψ/an}b∇acket∇i}htor the dynamical quark mass mq. Recent lattice-QCD simulations [15–17] show that the\nbreaking is further enhanced in an external magnetic field. Since th e constituent quark and anti-quark of the chiral\ncondensatehaveoppositespinsandoppositecharges,the pair’sm agneticmomentwillalignwiththemagneticfield,\nleading to a condensate /an}b∇acketle{t¯ψγ1γ2τ3ψ/an}b∇acket∇i}htin the ground state. Therefore, the chiral condensate will inexor ably provide\nthe quasi-particles with both a dynamical mass and a dynamical magn etic moment. The tensor condensate is\ndiscussed at finite temperature in a one-flavor NJL model in the lowe st-Landau-level approximation in a magnetic\nfield in Ref. [18], and the discussion is extended to a two-flavor NJL mo del at finite density in Ref. [19].\nThe only possibility to realize a magnetic field in the laboratory which is co mparable in strength with typical\nQCD energy scales is via high-energy heavy-ion collisions. For heavy- ion collisions at the Relativistic Heavy-Ion\nCollider and the Large Hadron Collider, the magnetic field can reach a m agnitude of eB∼m2\nπ[20–23], however,\nonly for a very short time in the early stage of the collision. Considerin g that the colliding system is initially\nin a state far from equilibrium, one should study the magnetic moment induced by chiral symmetry breaking in\nthe framework of quantum transport theory. One possible way to formulate this theory is the Wigner-function\nformalism [24–28]. In this Letter, we study the space-time depende nt magnetic moment dynamically generated\nin quark matter, by applying equal-time transport theory [27, 28] t o anSU(2) NJL model. We first calculate\nthe temperature dependence of the dynamical magnetic moment in the equilibrium case, and then focus on the\nclassical and quantum kinetic equations for the dynamical quark ma ss and the dynamical magnetic moment.\nThe Lagrangian of the magnetized SU(2) NJL model with a tensor interaction reads [9–12]\nL=¯ψ(iγµDµ−m0)ψ+Gs/bracketleftBig/parenleftbig¯ψψ/parenrightbig2+/parenleftbig¯ψiγ5τψ/parenrightbig2/bracketrightBig\n−Gt\n4/bracketleftBig/parenleftbig¯ψγµγντψ/parenrightbig2+/parenleftbig¯ψiγ5γµγνψ/parenrightbig2/bracketrightBig\n, (1)\nwhere the covariant derivative Dµ=∂µ+iQAµcouples quarks with electric charge Q= diag(Qu,Qd) =\ndiag(2e/3,−e/3) to an external magnetic field B= (0,0,B) pointing in the x3-direction through the potential\nAµ= (0,0,Bx1,0). The coupling constant Gsin the scalar/pseudo-scalar channel controls the spontaneous c hiral2\nsymmetry breaking, which generates a dynamical quark mass, and the coupling constant Gtin the tensor/pseudo-\ntensor channels controls the spin-spin interaction, which leads to a dynamical magnetic moment. Here, m0is the\ncurrent quark mass characterizing the explicit chiral symmetry br eaking. In the following, we focus on the chiral\nlimit withm0= 0. When the magnetic field is turned on, the chiral symmetry SU(2)L⊗SU(2)Ris reduced to\nU(1)L⊗U(1)R. Throughout the paper we use the notation a= (a1,a2,a3) for 3-vectors and aµ= (a0,a) for\n4-vectors.\nThe order parameter for the chiral phase transition is the chiral c ondensate /an}b∇acketle{t¯ψψ/an}b∇acket∇i}htor the dynamical quark mass\nmq=−2Gs/an}b∇acketle{t¯ψψ/an}b∇acket∇i}ht. In a magnetic field, we also introduce a tensor condensate F3=−iGt/an}b∇acketle{t¯ψγ1γ2τ3ψ/an}b∇acket∇i}ht, which plays\nthe role of the dynamical magnetic moment of the quarks. Here we c onsider the dynamical magnetic moment\nalong the direction of the magnetic field. In mean-field approximation , the Lagrangian of the model becomes\nL=¯ψ(iγµDµ−mq−iF3γ1γ2τ3)ψ−m2\nq\n4Gs−F2\n3\n2Gt. (2)\nBy taking the quark propagator in a magnetic field in the Ritus scheme [29–31], the thermodynamical potential\nof the quark system contains a mean-field part and a quasi-quark p art,\nΩ =m2\nq\n4Gs+F2\n3\n2Gt+Ωq, (3)\nΩq=−Nc/summationdisplay\nf,η,n/integraldisplaydp3\n2π|QfB|\n2π[ǫfηn−2Tlng(−ǫfηn)],\nwhereg(x) =/parenleftbig\n1+ex/T/parenrightbig−1is the Fermi-Dirac distribution, ǫfηn=/radicalbigg\np2\n3+/parenleftBig/radicalBig\nm2q+2n|QfB|+ηF3/parenrightBig2\nis the quark\nenergy of flavor f=u,d, and the summation over the discrete Landau energy levels runs ov ern= 0,1,2,...for\nη= + and over n= 1,2,3,...forη=−. The spectrum of the quasi-quarks in Landau levels n >0 exhibits\na Zeeman splitting ( η=±) due to the tensor condensate F3. Therefore, we always use the term “dynamical\nmagnetic moment” for the tensor condensate F3. No splitting is present in the n= 0 mode, since the fermion\nin the lowest Landau level has only one spin projection. The dynamica l quark mass and magnetic moment are\nself-consistently determined by the minimum of the thermodynamic p otential,\n∂Ω\n∂mq= 0,∂Ω\n∂F3= 0. (4)\nBecause of the contact interaction among quarks, the NJL model is non-renormalizable, and it is necessary to\nintroduce a regularization scheme to remove the ultraviolet diverge nces of the momentum integrals. To guarantee\nthe law of causality in a magnetic field, we take a covariant Pauli-Villars r egularization as explained in detail in\nRef. [32]. The two parameters of the model in the chiral limit, namely t he quark coupling constant Gs= 3.52\nGeV−2and the Pauli-Villars mass parameter Λ = 1127 MeV are fixed by fitting t he pion decay constant fπ= 93\nMeV and the chiral condensate /an}b∇acketle{t¯ψψ/an}b∇acket∇i}ht= (−250 MeV)3in vacuum at T=B= 0. The coupling constant Gtin the\ntensor channel is treated as a free parameter.\nLet us first consider the lowest-Landau-level approximation. In t his case, the two gap equations simplify con-\nsiderably and become\nmq\n2Gs+(mq+F3)I0= 0,\nF3\nGt+(mq+F3)I0= 0, (5)\nwith\nI0=−Nc|eB|\n(2π)2/integraldisplaydp3\nǫ3[1−2g(ǫ3)]. (6)\nThe quark energy ǫfηnbecomes flavor-independent in the lowest Landau level with n= 0 andη= +,ǫ3=/radicalbig\np2\n3+(mq+F3)2.\nFrom the two gap equations (5), we readily observe that the dynam ical magnetic moment F3and the dynamical\nquark mass mqare proportional to each other,\nF3\nmq=Gt\n2Gs, (7)\nindependent of temperature, magnetic field, and the regularizatio n scheme used. Once quarks acquire a dynamical\nmass, they should also acquire a dynamical magnetic moment. This eff ect has also been reported in massless QED3\nand in a one-flavorNJL model [4, 5, 18]. The constituent quark and a nti-quark forming the chiral condensate have\nopposite spins and opposite charges, the magnetic moment of the p air is then aligned with the external magnetic\nfield. This leads to a dynamical magnetic moment F3in the ground state. From the view of symmetry, once the\nchiral symmetry is dynamically broken, there is no symmetry protec ting the dynamical magnetic moment, because\na nonvanishing value of the latter breaks exactly the same symmetr y.\nIncluding all Landau levels, the proportionality (7) between F3andmqno longer holds exactly, but is still\napproximately satisfied, see the numerical calculations of the origin al gap equations (4) shown in Fig. 1. With\nincreasingtemperature, the scalarand tensorcondensates con tinuouslymelt and approachzero at the same critical\ntemperature, and F3remains zero in the chirally restored phase, characterized by mq= 0. This proves the original\nidea that the dynamical magnetic moment is induced by chiral symmet ry breaking. With increasing coupling\nstrengthGtin the tensor channel, F3is significantly enhanced but mqchanges only slightly. While there is\nstill an approximate proportionality between F3andmq, the proportionality constant in the full calculation is\nmuch smaller than Gt/(2Gs) in the lowest-Landau-level approximation, see the lower panel of Fig. 1. This is\ndue to the different contributions from the higher Landau levels to mqandF3. The quarks in higher Landau\nlevels participate in the scalar condensate in the same way as the qua rk in the lowest Landau level and therefore\nenhance the dynamical quark mass considerably. However, the qu ark in the lowest Landau level constitutes the\nmajor contribution to the dynamical magnetic moment due to its sing le spin projection. Including higher Landau\nlevels, the dynamical magnetic moment is only slightly changed becaus e of the cancellation between the two spin\nprojections of the quarks in higher Landau levels.\n0 50 100 150050100150200250\n/RParen1/GothicMq/LParen1MeV/RParen1\neB/Equal10mΠ2\nGt/Equal2GsGt/EqualGsGt/Equal0\n0 50 100 15005101520\n/RParen1/CapDigamma3/LParen1MeV/RParen1\n0 50 100 1500.000.020.040.060.08\nT/LParen1MeV/RParen1/CapDigamma3/Slash1/GothicMq\nFIG. 1: The dynamical quark mass, dynamical magnetic moment , and their ratio as functions of temperature in a constant\nmagnetic field eB= 10m2\nπfor different values of the coupling strength Gtin the tensor channel.\nApart from the nonzero coupling Gt, the other necessary condition for a nonvanishing dynamical magn etic\nmoment is a nonzero external magnetic field. When the magnetic field is turned off, the gap equations (4) become\nmq/braceleftBigg\n1+2GsNcNf/summationdisplay\nη/integraldisplayd3p\n(2π)31+ηF3/ǫ⊥\nǫη[g(ǫη)−g(−ǫη)]/bracerightBigg\n= 0,\nF3+GtNcNf/summationdisplay\nη/integraldisplayd3p\n(2π)3F3+ηǫ⊥\nǫη[g(ǫη)−g(−ǫη)] = 0, (8)\nwith quark energy ǫη=/radicalbig\np2\n3+(ǫ⊥+ηF3)2and transverse energy ǫ⊥=/radicalBig\np2\n1+p2\n2+m2q. The solution of the gap\nequations is F3= 0 in both the chiral symmetry broken and restored phases. Phys ically, without magnetic field,\nthe randomly oriented quark spins lead to a vanishing dynamical magn etic moment in the ground state.4\nAs the magnetic field is turned on, a nonzero dynamical magnetic mom ent is induced and increases with\nmagnetic field. Figure 2 shows the dynamical magnetic moment as a fu nction of magnetic field at zero and\nfinite temperature for different values of the tensor coupling Gt. The dynamical magnetic moment is linearly\nproportional to the external magnetic field at zero temperature , analogously to the anomalous magnetic moment\nin Schwinger’s calculation in QED [1]. The linear relation is broken by the t hermal motion of quarks, see the lower\npanel of Fig. 2.\n0 5 10 15 20010203040\n/Slash12/CapDigamma3/LParen1MeV/RParen1\n/LParen1a/RParen1T/Equal0Gt/Equal2GsGt/EqualGsGt/Equal0\n0 5 10 15 200102030\neB/Slash1m/DoublePi2/CapDigamma3/LParen1MeV/RParen1\n/LParen1b/RParen1T/Equal140 MeV\nFIG. 2: The dynamical magnetic moment as a function of magnet ic field at different temperature and different values for\nthe coupling constant Gtin the tensor channel.\nWe now turn to non-equilibrium systems. For systems in a sufficiently s trong magnetic field, like matter created\nin the early stages of relativistic heavy-ion collisions, the calculation in the framework of finite-temperature field\ntheory fails, and we need to treat the dynamical evolution of the sy stem in the framework of transport theory.\nIn the following, we consider the dynamical evolution of the quark ma ss and magnetic moment in an external\nelectromagnetic field by using the Wigner-function formalism applied t o the NJL model with a tensor interaction.\nTo appropriately treat the quantum fluctuations, especially the off -shell effect, order by order, we apply equal-\ntime quantum transport theory, which has been successfully deve loped in QED [26–28]. We will see clearly that\nthe dynamical quark mass is generated at the classical level, but th e magnetic moment arises from quantum\nfluctuations.\nThe covariant quark Wigner function in a gauge field theory is defined as\nW(x,p) =/integraldisplay\nd4yeipy/angbracketleftBig\nψ(x+)eiQ/integraltext1/2\n−1/2dsA(x+sy)y¯ψ(x−)/angbracketrightBig\n, (9)\nwhere the exponential function is the gauge link between the two po intsx−=x−y/2 andx+=x+y/2, which\nguaranteesgaugeinvariance[26], and the symbol /an}b∇acketle{t.../an}b∇acket∇i}htmeans ensembleaverageofthe Wigneroperator. Forexternal\n(classical) gauge fields, the link factor can be moved out of the ense mble average.\nFrom the mean-field Lagrangian (2) in the chiral limit, we obtain the Dir ac equation for the quark field,\n(iγµDµ−mq−iF3γ1γ2τ3)ψ= 0. (10)\nAgain, we consider here the dynamical magnetic moment F3along the direction of the magnetic field.\nUsing the Dirac equation, we derive the generalized Vasak-Gyulassy -Elze equation [26] for the quark Wigner\nfunction for flavor f,\n(γµKµ−M+K3γ1γ2)W= 0, (11)\nwith the operators\nKµ= Πµ+i\n2/planckover2pi1Dµ,\nΠµ=pµ−i/planckover2pi1Qf/integraldisplay1/2\n−1/2dssFµν(x−i/planckover2pi1s∂p)∂ν\np,\nDµ=∂µ−Qf/integraldisplay1/2\n−1/2dsFµν(x−i/planckover2pi1s∂p)∂ν\np, (12)5\nrelated to the electromagnetic interaction,\nM=M1+iM2,\nM1= cos/parenleftbigg/planckover2pi1\n2∂x·∂p/parenrightbigg\nmq(x),\nM2=−sin/parenleftbigg/planckover2pi1\n2∂x·∂p/parenrightbigg\nmq(x), (13)\nrelated to the dynamical quark mass controlled by the scalar intera ction, and\nK3=Fo+iFe,\nFe=−sgn(Qf)cos/parenleftbigg/planckover2pi1\n2∂x·∂p/parenrightbigg\nF3(x),\nFo=−sgn(Qf)sin/parenleftbigg/planckover2pi1\n2∂x·∂p/parenrightbigg\nF3(x), (14)\nrelated to the dynamical magnetic moment controlled by the tensor interaction. We have explicitly exhibited the\n/planckover2pi1-dependence in order to be able to discuss the semi-classical expan sion of the kinetic equation in the following.\nConsidering that the Wigner function defined through Eq. (9) is a 4 ×4 matrix in Dirac space and in general not\na real-valued function, its physical meaning becomes clear only afte r the spinor decomposition [26]\nW=1\n4/parenleftbigg\nF+iγ5P+γµVµ+γµγ5Aµ+1\n2σµνSµν/parenrightbigg\n. (15)\nTo compare the covariant Wigner function W(x,p) defined in 4 −dimensional momentum space with the observ-\nable physics densities such as the number density defined in 3 −dimensional momentum space, we introduce the\nequal-time Wigner function W(x,p) by integrating the covariant Wigner function W(x,p) over the energy p0and\nfurthermore apply the corresponding spinor decomposition,\nW=/integraldisplay\ndp0Wγ0\n=1\n4(f0+γ5f1−iγ0γ5f2+γ0f3+γ5γ0γ·g0+γ0γ·g1−iγ·g2−γ5γ·g3). (16)\nThe physical meaning of the spinor components of the equal-time Wig ner function fi(x,p) andgi(x,p),i=\n0,1,2,3, is discussed in detail in Ref. [27] in QED. For instance, f0is the number density, g0the spin density, and\ng1the number current.\nSince the kinetic equation (11) is a complete equation, when taking th e spinor decomposition (15) it becomes 16\ntransport equations with derivative Dµplus 16 constraint equations with operator Π 0for the spinor components\nF,P,Vµ,Aµ, andSµν. The former controls the dynamical evolution of the 16 component s in phase space, and\nthe latter is the quantum extension of the classical on-shell condit ion [28]. By taking the energy integration of\nthese kinetic equations, we obtain a set of transport equations fo r the spinor components of the equal-time Wigner\nfunction,\n/planckover2pi1\n2(d0f0+d·g1)−m2f3−fog3·e3= 0,\n/planckover2pi1\n2(d0f1+d·g0)+m1f2−feg2·e3= 0,\n/planckover2pi1\n2d0f2+π·g3−m1f1+feg1·e3= 0,\n/planckover2pi1\n2d0f3−π·g2−m2f0−fog0·e3= 0,\n/planckover2pi1\n2(d0g0+df1)−π×g1−m2g3−feg3×e3−fof3e3= 0,\n/planckover2pi1\n2(d0g1+df0)−π×g0+m1g2+fog2×e3−fef2e3= 0,\n/planckover2pi1\n2(d0g2+d×g3)+πf3−m1g1−fog1×e3+fef1e3= 0,\n/planckover2pi1\n2(d0g3−d×g2)−πf2−m2g0−feg0×e3−fof0e3= 0, (17)\nand a set of constraint equations,\nV′\n0+π0f0−π·g1−m1f3+feg3·e3= 0,6\nA′\n0−π0f1+π·g0+m2f2+fog2·e3= 0,\nP′+π0f2+/planckover2pi1\n2d·g3+m2f1+fog1·e3= 0,\nF′+π0f3−/planckover2pi1\n2d·g2−m1f0+feg0·e3= 0,\nA′−π0g0+/planckover2pi1\n2d×g1+πf1+m1g3+fog3×e3−fef3e3= 0,\nV′+π0g1−/planckover2pi1\n2d×g0−πf0−m2g2−feg2×e3−fof2e3= 0,\nS′\n0iei−π0g2+π×g3−/planckover2pi1\n2df3−m2g1−feg1×e3−fof1e3= 0,\nS′\njkǫijkei+2π0g3+2π×g2−/planckover2pi1df2−2m1g0−fog0×e3+fef0e3= 0, (18)\nwhereΓ′(x,p) =/integraltext\ndp0p0Γ(x,p) (Γ =F,P,Vµ,Aµ,Sµν) arethe first-orderenergymoments ofthe covariantWigner\nfunction, e1,e2, ande3are the unit vectors along the Cartesian coordinates x1,x2, andx3in coordinate space,\nand the equal-time operators related to the quark electromagnet ic, scalar, and tensor interactions are the energy\nintegrals of the corresponding covariant operators,\nd0=∂t+Qf/integraldisplay1/2\n−1/2dsE(x+i/planckover2pi1s∇p,t)·∇p,\nd=∇+Qf/integraldisplay1/2\n−1/2dsB(x+i/planckover2pi1s∇p,t)×∇p,\nπ0=i/planckover2pi1Qf/integraldisplay1/2\n−1/2ds sE(x+i/planckover2pi1s∇p,t)·∇p,\nπ=p−i/planckover2pi1Qf/integraldisplay1/2\n−1/2ds sB(x+i/planckover2pi1s∇p,t)×∇p,\nm1= cos/parenleftbigg/planckover2pi1\n2∇·∇p/parenrightbigg\nmq(x),\nm2= sin/parenleftbigg/planckover2pi1\n2∇·∇p/parenrightbigg\nmq(x),\nfe=−sgn(Qf)cos/parenleftbigg/planckover2pi1\n2∇·∇p/parenrightbigg\nF3(x),\nfo= sgn(Qf)sin/parenleftbigg/planckover2pi1\n2∇·∇p/parenrightbigg\nF3(x). (19)\nHere we have replaced the field strength tensor Fµν(x) by the electric and magnetic fields E(x) andB(x). Note\nthat the energy moment/integraltext\ndp0p0W(x,p)γ0in the constraint equations is in general independent of the equal-t ime\nWigner function W(x,p) due to the quantum off-shell effect of particle transport in the me dium [28]. Only in the\nclassical case, any order energy moment can be expressed as/integraltext\ndp0pn\n0W(x,p)γ0=ωn\npW(x,p), n= 0,1,2,..., in\nterms of the quasi-particle energy ωpand the equal-time Wigner function due to the classical on-shell con dition\nδ(p0−ωp).\nUsing the definitions of the scalar and tensor condensates mq=−2Gs/an}b∇acketle{t¯ψψ/an}b∇acket∇i}ht=−2Gs/an}b∇acketle{t¯ψuψu+¯ψdψd/an}b∇acket∇i}htandF3=\n−iGt/an}b∇acketle{t¯ψγ1γ2τ3ψ/an}b∇acket∇i}ht=−iGt/an}b∇acketle{t¯ψuγ1γ2ψu−¯ψdγ1γ2ψd/an}b∇acket∇i}ht, thesequantitiescanbeexpressedintermsoftheWignerfunction ,\nmq(x) =−2Gs/integraldisplayd3p\n(2π)3[f3u(x,p)+f3d(x,p)],\nF3(x) =−Gt/integraldisplayd3p\n(2π)3[g3u(x,p)−g3d(x,p)]·e3. (20)\nThis shows clearly the physics of the spinor components f3andg3: they are the source of the quark mass and the\nquark magnetic moment, respectively, and are then called mass den sity and magnetic-moment density. By solving\nthe kinetic equations (17) and (18), the quark mass and magnetic m oment are self-consistently generated through\nthe dynamical evolution of the quark Wigner function.\nTo see clearly the quantum effect on the equal-time kinetic theory, w e apply the semi-classical ( /planckover2pi1) expansion for\nthe Wigner functions and the equal-time operators,\nW=W(0)+/planckover2pi1W(1)+O(/planckover2pi12),\nW=W(0)+/planckover2pi1W(1)+O(/planckover2pi12),7\nd0=∂t+QfE·∇p+O(/planckover2pi12),\nd=∇+QfB×∇p+O(/planckover2pi12),\nπ0=O(/planckover2pi12),\nπ=p+O(/planckover2pi12),\nm1=mq+O(/planckover2pi1),\nm2=−/planckover2pi1\n2∇mq·∇p+O(/planckover2pi12),\nfe=−sgn(Qf)F3+O(/planckover2pi1),\nfo=−/planckover2pi1\n2sgn(Qf)∇F3·∇p+O(/planckover2pi12). (21)\nBy substituting them into the kinetic equations and comparing order s of/planckover2pi1on both sides, we obtain the transport\nand constraint equations order by order in /planckover2pi1. In the classical limit, i.e., /planckover2pi1= 0, the constraint equations (18)\ndetermine automatically the on-shell energy\np0=χǫη, χ, η =±, (22)\ncorresponding to the four independent quasi-particle solutions wit h positive and negative energies ( χ=±) and\nup and down spin projections ( η=±). In this case we can express the distribution functions as the sum of the\ndistributions for the four quasi-particle modes, fi=/summationtext\nχ,ηfχη\niandgi=/summationtext\nχ,ηgχη\ni. To simplify the notation, we\nhave here and in the following neglected the subscript (0) of the clas sical components f(0)\niandg(0)\ni. The constraint\nequations determine not only the on-shell condition but also give rise to relations among the classical components,\nfχη\n1= sgn(Qf)χηmq\nǫηp3\nǫ⊥fχη\n0,\nfχη\n2= 0,\nfχη\n3=χmq\nǫη/parenleftbigg\n1+ηF3\nǫ⊥/parenrightbigg\nfχη\n0,\ngχη\n0= sgn(Qf)ηmq\nǫ⊥e3fχη\n0,\ngχη\n1=χ1\nǫη/bracketleftbigg\np−ηF3\nǫ⊥(p×e3)×e3/bracketrightbigg\nfχη\n0,\ngχη\n2= sgn(Qf)ηp×e3\nǫ⊥fχη\n0,\ngχη\n3=−sgn(Qf)χη\nǫηǫ⊥/bracketleftbig\np3p−/parenleftbig\nǫ2\n⊥+ηF3ǫ⊥/parenrightbig\ne3/bracketrightbig\nfχη\n0. (23)\nThese relations greatly simplify the calculation of the classical Wigner function. The nonzero tensor condensate\ncouples the spin-related distributions to the number density-relat ed distributions. Therefore, there is only one\nindependent distribution function, the number density f0, and all others can be expressed in terms of f0. Note\nthat the classical limit of the transport equations (17) can reprod uce a part of the classical relations shown in\nEq. (23) but does not give any new relations.\nSubstitutingtheclassicalrelationsbetween f3,g3, andf0intotheexpressions(20)for mqandF3, andconsidering\nthe trivial color degrees of freedom in the NJL model, the non-trivia l quark mass mq(x) and magnetic moment\nF3(x) at the classical level are controlled by the gap equations\n1+2GsNc/summationdisplay\nχ,η/integraldisplayd3p\n(2π)31+ηF3/ǫ⊥\nχǫη(fχη\n0u+fχη\n0d) = 0,\nF3+GtNc/summationdisplay\nχ,η/integraldisplayd3p\n(2π)3F3+ηǫ⊥\nχǫη(fχη\n0u+fχη\n0d) = 0. (24)\nThese two classical gap equations have the same structure as Eq. (8) for systems in thermal equilibrium, the only\ndifference being the non-equilibrium distribution f0(x,p), which is controlled by a classical transport equation and\nwill be discussed below. When replacing f0by the Fermi-Diracdistribution, the gap equations(24) and (8) bec ome\nexactly the same. Remember that F3= 0 is the only solution of the gap equations (8), the same structure , namely\nthe same dynamics of Eqs. (24) and (8) leads to the conclusion that F3= 0 in the classical limit. Physically, the\ndynamical magnetic moment F3is generated by the quark spin, which is a quantum phenomenon and w ill not\nappear at the classical level. F3has a nonzero value only when quantum fluctuations are included. On the other\nhand, the chiral symmetry restoration in medium is a classical phase transition, the space-time dependence of the8\norder parameter mq(x) is solely controlled by\n1+2GsNc/summationdisplay\nχ,η/integraldisplayd3p\n(2π)31\nχǫη(fχη\n0u+fχη\n0d) = 0. (25)\nWhen the tensor condensate vanishes in the classical limit, the spin d ensityg0becomes an independent Wigner\ncomponent, and the constraint equations (18) result in the classic al relations\nfχ\n1=χp·gχ\n0\nǫ,\nfχ\n2= 0,\nfχ\n3=χmq\nǫfχ\n0,\ngχ\n1=χp\nǫfχ\n0,\ngχ\n2=p×gχ\n0\nmq,\ngχ\n3=χǫ2gχ\n0−(p·gχ\n0)p\nmqǫ, (26)\nwith the quark energy ǫ=/radicalBig\nm2q+p2.\nWe now consider the transport equations to linear order in /planckover2pi1. Taking into account the classical solution F3= 0\nandf2= 0, we have\nd0f0+d·g1−∇mq·∇pf3= 0,\nd0f1+d·g0+2mqf(1)\n2+2sgn(Qf)F(1)\n3g2·e3= 0,\np·g(1)\n3−mqf(1)\n1−m(1)\nqf1−sgn(Qf)F(1)\n3g1·e3= 0,\nd0f3−2p·g(1)\n2−∇mq·∇pf0= 0,\nd0g0+df1−2p×g(1)\n1−∇mq·∇pg3+2sgn(Qf)F(1)\n3g3×e3= 0,\nd0g1+df0−2p×g(1)\n0+2mqg(1)\n2+2m(1)\nqg2= 0,\nd0g2+d×g3+2pf(1)\n3−2mqg(1)\n1−2m(1)\nqg1−2sgn(Qf)F(1)\n3f1e3= 0,\nd0g3−d×g2−2pf(1)\n2−∇mq·∇pg0+2sgn(Qf)F(1)\n3g0×e3= 0, (27)\nwheref(1)\ni,g(1)\ni,m(1)\nq, andF(1)\n3are the first-order quantum corrections, and d0anddare at the classical level,\nd0=∂t+QfE·∇pandd=∇+QfB×∇p.\nWith the help of the classical relations (26), a careful but straight forward treatment of Eqs. (27) determines the\nquantum correction from the quark spin density g0to the magnetic moment F(1)\n3,\n2(mqg3+f1p)·e2sgn(Qf)F(1)\n3=/bracketleftBigg\n∇m2\nq\n2·∇pg3−mq(d0g0+df1)+p×(d0g2+d×g3)/bracketrightBigg\n·e1,(28)\nand leads to the transport equations for the two independent clas sical components, the number density f0and\nspin density g0,\n/parenleftBigg\nd0+χp\nǫ·d−χ∇m2\nq·∇p\n2ǫ/parenrightBigg\nfχ\n0= 0,\n/parenleftBigg\nd0+χp\nǫ·d−χ∇m2\nq·∇p\n2ǫ/parenrightBigg\ngχ\n0=Qf\nǫ2[p×(E×gχ\n0)−χǫB×gχ\n0]−1\n2ǫ2m2q/parenleftbig\n∂tm2\nqp+χǫ∇m2\nq/parenrightbig\n×(p×gχ\n0)\n−sgn(Qf)χ\nmqǫ/bracketleftbig\nm2\nqgχ\n0×e3+((p×gχ\n0)·e3)p/bracketrightbig\nF(1)\n3. (29)\nThe quarks obtain a dynamical mass mqfrom the interaction with the medium. When the medium is inhomo-\ngeneous, a mean-field force F=−∇m2\nq/(2ǫ) is exerted on the moving quark, which leads to the third term on\nthe left-hand side of the two transport equations. While in mean-fie ld approximation there is no collision term\non the right-hand side of the transport equation for the number d ensityf0, the quark spin interactions with the\nelectromagnetic field, the space-time dependent quark mass, and the magnetic moment lead to the three kinds of\ncollision terms shown on the right-hand side of the transport equat ion for the quark spin density g0.9\nLet us now consider the limit of a homogeneous medium and a constant magnetic field. In this limit, the\nquantum correction F(1)\n3becomes\nF(1)\n3=|Qf|{p×[(B×∇p)×g3]−mqB×∇pf1}·e1\n2(mqg3+f1p)·e2. (30)\nIt is clear that the quantum correction vanishes when the magnetic field disappears. Moreover, the quantum\ncorrectionF(1)\n3cannot be generated from the inhomogeneous medium, if the electr omagnetic field is turned off.\nSubstituting the kinetic equation (29) for g0into the first-order quantum correction (28) to F(1)\n3, and making use\nof the relations in Eq. (26), we can straightforwardly prove F(1)\n3= 0.\nOur strategy to extract quantum corrections from a general kin etic theory is the following. The classical kinetic\ntheory for quasi-particles arises from the constraint equations a t zeroth order in /planckover2pi1and the transport equations at\nfirst order in /planckover2pi1. The quantum correction induced by the spin of the quasi-particles comes also from the transport\nequations at first order in /planckover2pi1. When we go to higher-order quantum corrections, the particles a re no longer on the\nenergy shell, and the quasi-particle treatment fails. In this case, t he first-order energy moment/integraltext\ndp0p0W(x,p)γ0\nis independent of the zeroth-order energy moment/integraltext\ndp0W(x,p)γ0=W(x,p). Therefore, all 16 spin components\nf(j)\niandg(j)\ni(i= 0,1,2,3,j= 1,2,...) become independent of each other, and their behavior is controlle d by the\nfull set of transport equations (17).\nIn summary, we investigated the dynamically generated quark mass and magnetic moment in the Wigner-\nfunction formalism. We derived the transport and constraint equa tions for the spinor components of the\nequal-time Wigner function in the magnetized NJL model with tensor in teraction. We expanded the kinetic\nequations in the semi-classical expansion and solved them order by o rder. The space-time dependent quark\nmass and magnetic moment are self-consistently coupled to the Wign er function and determined by the ki-\nnetic equations. 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D94, 036007 (2016)."    },    {        "title": "1912.12663v1.Synchronized_excitation_of_magnetization_dynamics_via_spin_waves_in_Bi_YIG_thin_film_by_slot_line_waveguide.pdf",        "content": " 1 Synchronized excitation of magnetization dynamics via spin waves in Bi-YIG thin film by slot line waveguide  Tetsunori Koda1, Sho Muroga2 and Yasushi Endo3,4,5  1 General Education Division, National Institute of Technology, Oshima College, Suo-Oshima 742-2193, Japan 2 Department of Mathematical Science and Electrical-Electronic-Computer Engineering、Graduate School of Engineering Science, Akita University, Akita 010-8502, Japan 3 Department of Electrical Engineering、Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan 4 Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, JAPAN 5 Center for Science and Innovation in Spintronics (CSIS), Organization for Advanced Studies, Tohoku University, Sendai 980-8577, JAPAN     2 ABSTRACT  We have studied magnetization dynamics of single Bi-YIG thin films by means of the high frequency power response induced by a slot line waveguide. Multiple absorption peaks that correspond to excitement states in magnetization dynamics appeared without the ferromagnetic resonance (FMR) condition. The peaks were strongly influenced by a waveguide line width and a distance between the lines. Micromagnetics simulation reveals that each line induces a local magnetization dynamics oscillation and generates spin waves. The spin wave that propagates from one of the lines interferences with the other side of local magnetization dynamics oscillation around the other line, resulting in an amplification of the oscillation when they are in synchronization with each other. This amplification occurs at both sides of the lines by the interference. Thus, the possible mechanism of the excitation in the magnetization dynamics oscillation is the synchronization of mutual magnetization dynamics oscillation via spin waves. This technique resonantly excites the local magnetization dynamics without the FMR condition, which is applicable as a highly coherent spin waves source.     3 When a magnetic field is applied to a ferromagnetic material, the magnetization precesses around the axis of final state direction. The precession motion of magnetization, magnetization dynamics, is a basic phenomenon of magnetism. Especially, near the ferromagnetic resonance (FMR) conditions, the precession motion is effectively excited. Since the precession frequency is of the order of GHz, this phenomenon is a strong interest to both fundamental physics and an engineering points of view. For example, a spin torque nano oscillator shows promising properties for microwave emission [1-3] and reservoir computing [4-5]. At near FMR condition, a response of magnetization precession motion shows strong nonlinearity with respect to the change of an applied magnetic field applicable for highly sensitive magnetic sensors. We have reported that the change in magnetization dynamics was sensitive enough to detect a small change in magnetic field and is capable of being used for biomagnetic field measurements [6]. There are several ways to induce the magnetization dynamics oscillation. Typically, a coplanar waveguide (CPW) consisted of one signal line and two ground lines is used as a high frequency waveguide. A strong magnetic field induces local magnetization dynamics at around the signal line, which has been widely used to study the magnetization dynamics of magnetic films and sub micro magnetic elements [7-10].  The design of the waveguide is a key factor because it controls magnetic field dependence of magnetization dynamics in ferromagnetic materials. In our previous work, the magnetic field dependence of reflected RF power in Bi doped YIG (Bi-YIG) single crystal thin films was measured  4 using an asymmetrical CPW, and multiple peaks corresponding to the excitation of magnetization dynamics were observed [6]. Those height of peak profile were strongly influenced by the width of each line and the distance between the lines. At a critical condition, the peak height changed drastically. This change could help detecting a small change in magnetic field. However, the study has not completely explained the origin of the multiple peaks. In this paper, we report magnetization dynamics in a Bi-YIG single crystal thin film induced by a slot line waveguide to understand the origin of the multiple peaks. The slot line waveguide is a representative planar waveguide including two lines with the same width. We systematically changed the width of the lines and the distance between the lines and measured the high frequency power response. The results showed that the interference between local magnetization dynamics oscillation and spin waves generated around each line was a crucial rule for the excitation of magnetization dynamics of the system. Bi-YIG (111) single crystal thin films with a thickness of about 10 µm were epitaxially grown on gadolinium gallium garnet (GGG) (111) single crystal substrates using liquid phase epitaxy technique. To evaluate the high-frequency response of the samples, Cu slot line waveguides with a thickness of 500 nm were directly fabricated on the Bi-YIG film using a combination of photolithography, Ar ion milling and sputtering techniques. The width and distance between the lines were changed and the length of the sample’s signal line(s) and ground lines was fixed at 300 µm (Fig.1 (a)). Since all the waveguides were fabricated on a single Bi-YIG thin film, the magnetic property was constant. When magnetization dynamics was resonantly excited, the input power was absorbed for the use of the  5 precession motion of magnetization, resulting in the significant reduction of the reflected signal strength. Magnetization dynamics was evaluated by measuring the reflected power from the slot line waveguide with the vector network analyzer (Keysight Model N5230A) while sweeping the magnetic field parallel to the sample plane. The input signal strength was set at 0 dBm for all the measurements and the frequency was fixed at a frequency between 6.0 and 9.0 GHz for each measurement. It should be noted that the short length of lines help high-frequency input signals propagating along the waveguide, although the waveguide impedance was different from its measurement setup impedance. The numerical simulations based on micromagnetics were performed using object oriented micromagnetic framework (OOMMF) [11]. Figure 2 (a) shows the reflected high frequency signal strength of various input frequency according to the distance between the lines. Multiple peaks were clearly detected in all conditions. The peaks shifted towards higher magnetic field with the increase of frequency. This indicates that the peaks are associated with magnetic phenomenon. For the samples with the same line width, the number of multiple peaks increased and shifted towards higher magnetic field with the increase of the distance between the lines. Furthermore, one of the peaks shifted higher magnetic field with increasing the width for the samples with the same line’s distance as shown in Fig.2 (b).  To understand the experimental results and the origin of the peaks for slot lines waveguides, we carried out the micromagnetics simulation on the RF line’s number dependence of magnetization dynamics. The simulation models are shown in Fig. 3 (a). Those names are one-line and two-line models. The dimensions used in the simulation were 200 µm long (l), 320 µm wide (w), and 10 µm  6 thick (z), and one or two lines were arranged. The cell sizes were set at l=50 nm, w=200 µm and z=1 µm. The length of the lines was fixed at 200 µm, and the width and the distance (d) were varied. The Gilbert damping factor  a of the Bi-YIG thin film was set at 0.001. A small RF magnetic field (0.5 Oe) was applied perpendicular locally at the position of the lines with the frequency at 8.0 GHz. A DC magnetic field was applied parallel to the length direction to the film. The simulation results are shown in Fig 3 (b). Only one peak associated with FMR is observed for the one-line model. The FMR conditions are determined by the magnetization properties and line width. The broad FMR peak is due to the inhomogeneous demagnetizing field caused by the limitation of simulation size. On the other hand, multiple peaks were identified for the two-line models. When the distance between the lines are identical, the peak shifted toward higher magnetic field with the increase of line widths. The magnetic field where magnetization reaches its peak shifts with the increase of the distance between the lines. Those results faithfully reproduce the experimental results. The difference of the magnetic fields for the appearance of the peaks between the experimental and the simulation results is caused by the difference of the internal magnetic field owing to the limitation of the simulation size. In the experiments, the slot line waveguides with the length of 300 µm were directly fabricated on the Bi-YIG thin film with the substrate size of 11 mm ´ 11 mm. We simulated the magnetization orientation distribution for x-direction for one-line model as shown in Fig. 3 (c). The DC magnetic field was set at 3.95 kOe, which was the condition for the appearance of the peak of the two-line model in Fig.3 (b). This shows that the line induces local magnetization dynamics. The oscillation of localized magnetization dynamics propagates perpendicular to the line direction via exchange or dipole  7 interaction, and it generates magneto static surface spin waves (MSSW) [12]. This result reveals that the lines also work as spin wave sources. Figure 3 (c) also shows the combined wave of one-line and the 40 µm shifted waves as the green and red lines, respectively. The geometric condition of the combined wave is the same as that of the two-line model. Figure 3 (d) shows the comparison between the combined wave and the spin wave for the two-line model under the same simulation condition. The amplitude of the spin wave for the two-line model is larger than that for the combined wave. This indicates that the local magnetization dynamics oscillation is more amplified in the two-line model. For the two-line model, a creation of spin waves from the lines propagates toward both sides of the lines, and the spin waves reach the other side of the line. The wavelength of MSSW is described as [13-14] 𝑓\"##$=&(𝑓(+𝑓\"2⁄)-−(𝑓\"2⁄)-exp(−2𝑘𝑑4),       (1) where fH=gH0 and fM=4pgMS and d0 is film thickness. Here, g =2.8 MHz Oe-1 is the electron gyromagnetic ratio, k is the wavenumber of spin wave, H0 is the effective internal field, MS=140 emu/cc is the saturation magnetization of YIG [12]. The wave length increases with the increase in the applied magnetic field. Fig. 4 shows the experimental results for the magnetic fields at the magnetization peaks to the calculated dispersion curve of the equation (1) for several input frequencies. The peaks appeared when the wave length of the spin wave was almost the same as the distance of the lines. This result indicates that the spin wave generation is potentially relevant with the excitement of magnetization dynamics.  If there is an interaction between the local magnetization dynamics and the spin waves, the condition  8 of the interference should be influenced by the phase difference of those dynamics. The phase difference can be set by the input RF magnetic fields condition of the lines. The simulation results are shown in Fig.5. The peaks shifted with the change of the phase difference. This strongly indicates the existence of the interaction between the local magnetization dynamics and the spin waves.  The possible mechanism of the excitation of magnetization dynamics is as follows. Each local magnetization dynamics under each line generates the spin wave. This spin waves propagate each other and interact with the other side of local magnetization dynamics. When the phase between the local magnetization dynamics and the spin wave coincides each other, the local magnetization dynamics is amplified resonantly. The amplified magnetization dynamics emits the spin wave with large amplitude. The enhanced spin waves propagate and amplify the local magnetization dynamics under the other side of the line. Thus, the local magnetization dynamics under the both side of lines are resonantly excited via the spin waves. This is a typical mutual synchronization phenomenon and have been reported in different systems [15-20]. The reason why we did not detect the FMR peak in our system was that the wave length of spin waves at FMR condition was much longer than the distance between the lines. Thus, the local magnetization dynamics directly influenced each other via the spin waves. We found that the phase difference of the slot line waveguide was about 180° based on numerical electromagnetic analysis. This phase difference effectively suppressed the excitation of the magnetization dynamics for both lines.  Our experimental results show that the magnetization dynamics can be controlled by using multiple sources of local magnetization dynamics. This finding gives the artificial control of the emission  9 condition of highly coherent spin waves without using the FMR, which is attractive for the applications that use the interference of spin waves such as logic gates devices [21-23] and magnonic crystals [24-26]. In conclusion, we studied magnetization dynamics of single Bi-YIG thin films by measuring the high frequency power response induced by the slot lines waveguide. Multiple peaks corresponding to the excitement of magnetization dynamics appeared. We found that the peaks were strongly influenced by the width and the distance of lines. Micromagnetics simulation successfully reproduced the experimental results and revealed that the spin waves generated from the local magnetization dynamics at each line propagated and interfered with the other side of the local magnetization. At a critical condition, this interference resonantly excites the both side of magnetization dynamics owing to the synchronization via spin waves. This finding gives a way for the excitement of the local magnetization dynamics without the FMR condition, which is applicable as a highly coherent spin waves source.  ACKNOWLEDGMENT We thank GRANOPT Co,Ltd. for their support of Bi-YIG single crystal thin films. This work was supported by JSPS KAKENHI Grant Number JP18K14114.  References [1] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and  10 D. C. Ralph, Nature 425, 380 (2003). [2] W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett. 92, 027201 (2004). [3] D. Grollier, D. Querlioz, and M. D. Stiles, Proc. of the IEEE 104, 2024 (2016). [4] S. Tsunegi, T. Taniguchi, K. Nakajima, S. Miwa, K.Yakushiji, A. Fukushima, S. Yuasa, and H. Kubota, Appl. Phys. Lett. 114, 164101 (2019). [5] T. Kanao, H. Suto, K. Mizushima, H. Goto, T. Tanamoto, and T. Nagasawa, Phys. Rev. Appl. 12, 024052 (2019). [6] T. Koda, S. Muroga, Y . Endo, IEEE Trans. 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Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, Nature 437, 389 (2005). [17] F. B. Mancoff, N. D. Rizzo, B. N. Engel, and S. Tehrani, Nature 437, 393–395 (2005). [18] V . E. Demidov, H. Ulrichs, S. V . Gurevich, S. O. Demokritov, V . S. Tiberkevich, A. N. Slavin, A. Zholud and S. Urazhdin, Nat. Commun. 5, 3179 (2014). [19] A. Houshang, E. Iacocca, P. Dürrenfeld, S. R. Sani, J. Åkerman and R. K. Dumas, Nat. Nanotechnol. 11, 280–286 (2016). [20] A. A. Awad, P. Dürrenfeld, A. Houshang, M. Dvornik, E. Iacocca, R. K. Dumas and J. Åkerman, Nat. Phys. 13, 292 (2017). [21] Chumak, A. V., Serga, A. A. & Hillebrands, B., Nature Commun. 5, 4700 (2014). [22] K. Vogt, F.Y. Fradin, J.E. Pearson, T. Sebastian, S.D. Bader, B. Hillebrands, A. Hoffmann and H. Schultheiss, Nat. Commun. 5, 3727 (2014). [23] T. Schneidera, A. A. Serga, B. Leven, and B. Hillebrands, Appl. Phys. Lett. 92, 022505 (2008). [24] G Gubbiotti, S Tacchi, M Madami, G Carlotti, A O Adeyeye, and M Kostylev, J. Phys. D 43, 264003 (2010). [25] Y . V . Gulyaev, S. A. Nikitov, L. V. Zhivotovskii, A. A. Klimov, P. Tailhades, L. Presmanes, C.  12 Bonningue, C. S. Tsai, S. L. Vysotskii, and Y. A. Filimonov, JETP Lett. 77, 567–570 (2003). [26] M. Krawczyk, and D. Grundler, J. Phys. Cond. Matter 26, 123202 (2014).    13    \n  Fig. 1 Schematic view of measurement set up for magnetization dynamics using a slot line waveguide.    \n 14  \n  Fig. 2 (a) Applied magnetic field dependence of the power of reflected high frequency wave with various frequency of input power. The distance between lines and the width of the line were systematically changed. (b) Slot line waveguide’s width dependence of the reflected power measured at 8.0 GHz.   \n 15  \n \n \n 16 Fig. 3 (a) Schematic view of micromagnetics simulation models. The number of local areas applied RF magnetic field was set at one or two. The dimensions used in the simulation were 200 µm long, 320 µm wide, and 10 µm thick. and one or two lines were arranged. The length of the lines was fixed at 200 µm, and the width (w) and the distance (d) were varied.  (b) Micromagnetic simulation results for the line’s number dependence of magnetization dynamics. W and D stand for the width of lines and the distance between lines, respectively.  (c) Magnetization orientation distribution for x-direction for one-line model at 3.95kOe. Y-axis indicates the z component of normalized magnetization at each position. The green line shows the 40 µm shifted curve of the blue line. The red line shows the combined curve of the blue and green curves. (d) Comparison for simulation results of the spin waves for two-lines model and the combined curve shown in Fig.3 (c).    17  Fig. 4 Lines’ distance dependence of magnetic field for the appearance of magnetization dynamics. The solid lines show the calculated dispersion curves using the equation (1).       \n  Fig. 5 Magnetization dynamics of the two-line model. The input RF magnetic fields have various range of phase difference.   \n"    },    {        "title": "2110.11158v1.Magnetic_hysteresis_of_individual_Janus_particles_with_hemispherical_exchange_biased_caps.pdf",        "content": "Magnetic hysteresis of individual Janus particles with hemispherical exchange\nbiased caps\nS. Philipp,1B. Gross,1M. Reginka,2M. Merkel,2M. Claus,1M. Sulliger,1A. Ehresmann,2and M. Poggio1\n1)Department of Physics, University of Basel, 4056 Basel, Switzerland\n2)Institute of Physics, University of Kassel, 34132 Kassel, Germany\nWe use sensitive dynamic cantilever magnetometry to measure the magnetic hysteresis of individual magnetic\nJanus particles. These particles consist of hemispherical caps of magnetic material deposited on micrometer-\nscale silica spheres. The measurements, combined with corresponding micromagnetic simulations, reveal\nthe magnetic configurations present in these individual curved magnets. In remanence, ferromagnetic Janus\nparticles are found to host a global vortex state with vanishing magnetic moment. In contrast, a remanent\nonion state with significant moment is recovered by imposing an exchange bias to the system via an additional\nantiferromagnetic layer in the cap. A robust remanent magnetic moment is crucial for most applications of\nmagnetic Janus particles, in which an external magnetic field actuates their motion.\nJanus particles (JPs) are nano- or micronsized parti-\ncles that possess two sides, each having different phys-\nical or chemical properties. There is a multitude of\ntypes of JPs1, differing in shape, material, and func-\ntionalization. As a subgroup of micron and sub-micron\nsized magnetic particles, which are discussed as a multi-\nfunctional component in lab-on-chip or micro-total anal-\nysis systems2,3, magnetic JPs, consisting of a hemispher-\nical cap of magnetic material on a non-magnetic spheri-\ncal template, allow not only a controlled transversal mo-\ntion, but also a controlled rotation by rotating external\nmagnetic fields4,5. Such JPs can be mass-produced via\nthe deposition of magnetic layers on an ensemble of sil-\nica spheres. The transversal and rotary motion of these\nparticles can be controlled via external magnetic fields,\nwhich exert magnetic forces and torques6. This ability to\nexternally actuate magnetic JPs has led to applications\nin microfluidics, e.g. as stirring devices7, as microprobes\nfor viscosity changes8, or as cargo transporters in lab-on-\nchip devices9–11. Magnetic JPs have also been proposed\nas an in vivo drug delivery system12.\nAlthough, in general, a transversal controlled motion\ncan be achieved by both superparamagnetic particles or\nparticles with a permanent magnetic moment, a con-\ntrol over the rotational degrees of freedom can only be\nachieved, if the particles possess a sufficiently large per-\nmanent magnetic moment. Streubel et al.13analyzed\nthe remanent magnetic state of magnetic JPs with fer-\nromagnetic (fm) magnetic caps. Their simulations show\nthat permalloy JPs with diameters larger than 140 nm\nhost a global vortex state at remanence. Because this\nflux-closed state has a vanishing net magnetic moment,\nmagnetic JPs hosting such a remanent configuration are\nunsuited for applications involving magnetic actuation.\nThus, for JPs larger than this critical diameter, strate-\ngies to overcome this limitation need to be developed.\nHere, we make use of exchange bias14–16, which, in\na simplified picture imposes a preferred direction on\nthe magnetic moments of the fm layer and is thereby\nable to prevent the formation of a global vortex at\nremanence. We apply an exchange bias to the fm layer\nby adding an antiferromagnetic (afm) layer beneath thefm layer. In order to verify that this addition leads to a\nremanent configuration with large magnetic moment, we\nmeasure the magnetic hysteresis of individual JPs with\nand without this layer. The measurement of individual\nJPs is necessary in order to eliminate the effects of\ninteractions between neighboring JPs. For this task, we\nemploy dynamic cantilever magnetometry (DCM) and\nanalyze the results by comparison to corresponding mi-\ncromagnetic simulations. This technique overcomes the\nlimitation of earlier measurements, that were restricted\nto ensembles of interacting JPs on a substrate17. These\nmeasurements relied on the longitudinal magneto-optical\nKerr effect and magnetic force microscopy. They found\nan onion state with a large remanent magnetization in\nJPs with an afm layer. Nevertheless, given that the\nmeasurements were done on close-packed ensembles of\nJPs, they do not exclude effects due to the interaction\nbetween the particles and, therefore, cannot be used to\ninfer the behavior of isolated JPs.\nWe fabricate the magnetic JPs by coating a self-\nassembled template of 1 .5µm-sized silica spheres with\nthin layers of different materials via sputter-deposition.\nThe non-magnetic silica spheres are arranged on a silica\nsubstrate using entropy minimization18, which allows\nthe formation of hexagonal close-packed monolayers.\nJPs with two different layer stacks, shown in Fig. 1\n(b), are produced. Ferromagnetic JPs (fmJPs) are\nfabricated by depositing a 10 nm-thick Cu buffer layer\ndirectly on the silica spheres, followed by a 10 nm-thick\nlayer of ferromagnetic CoFe. The film is sealed by\na final 10 nm-thick layer of Si. A second type of JP,\nwhich we denote exchange-bias JPs (ebJPs), includes an\nadditional 30 nm-thick afm layer of Ir 17Mn83between\nthe Cu buffer and the fm layer. Layer deposition is\nperformed by sputtering in an external magnetic field\nof 28 kA/m applied in the substrate plane, i.e. in the\nequatorial plane of the JPs, in order to initialize the\nexchange bias by field growth. This fabrication process\nis described in detail in Tomita et al.17. Individual\nJPs are then attached to the apex of a cantilever for\nmagnetic characterization in a last fabrication step, asarXiv:2110.11158v1  [cond-mat.mes-hall]  21 Oct 20212\n-x\nyz\nx\nyz(a)\n(b)(c)\n(d)\nBuffer CuFM CoFeCapping Si\nAFM IrMnθ\nϕ\nϕθeb\neb\nJP\nJP\n500 nm‘‘\n‘\nBuffer CuFM CoFeCapping SifmJPebJPTruncation\nFigure 1. (a) Sketch of a cantilever with a JP attached to\nits tip and definition of the coordinate system. (b) Cross-\nsectional SEM of a JP showing the gradient of the layer thick-\nness. The two investigated layer stacks of the hemispherical\ncap are shown in the insets. (c), (d) Definition of the angles\nsetting the orientation of the unidirectional anisotropy vector\nused to mimic exchange bias effects ( θeb,ϕeb), and the angles\ndefining the orientation of a JP on the cantilever ( θJPand\nϕJP).\nshown in the scanning electron micrographs (SEMs) of\nFig. 2 (a) and (b).\nNote that the values given for thicknesses are nominal\nand that the film thickness gradually reduces towards\nthe equator of the sphere with respect to the top,\nas shown in Fig. 1 (b), because of the deposition\nprocess17. Furthermore, the touching points of the\nnext neighbors in the hexagonal closed packed arrange-\nment of the silica spheres on the substrate template\nimpose a lateral irregularity on the equatorial line of\nthe capping layers. This is best seen in Fig. 2 (a) and (b).\nWe measure the magnetic hysteresis of each an\nindividual fmJP and an individual ebJP via DCM.\nDCM is a technique to investigate individual, nano- to\nmicrometer-sized magnetic specimens, similar to a stan-\ndard vibrating superconducting quantum interference\ndevice (SQUID) magnetometer. The key differences\nare that DCM is sensitive enough to measure much\nsmaller magnetic volumes than a vibrating SQUID\nmagnetometer and that it measures magnetic properties\nwith respect to rotations of the external magnetic\nfield, rather than modulations of its amplitude as in\nmeasurements of magnetic susceptibility. Details on\nthe technique and measurement setup can be found in\nRefs. 19 and 20.A magnetic specimen is attached to the tip of a can-\ntilever, which is driven in a feedback loop at its resonance\nfrequencyfwith a fixed amplitude, actuated by a piezo-\nelectric transducer. A uniform external magnetic field\nHis applied to set the magnetic state of the specimen\nunder investigation. Hcan be rotated within the plane\nperpendicular to the cantilever’s rotation axis ( xz-plane)\nwith a span of 117 .5°and a maximum field amplitude of\nH=±3.5 T. Its orientation is set by the angle θhas\ndefined and indicated in Fig. 1 (a). The magnetic torque\nacting on the sample results in a deflection of the can-\ntilever as well as a shift in its resonance frequency, which\nis given by\n∆f=f−f0=f0\n2k0l2e/parenleftBigg\n∂2Em\n∂θ2c/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nθc=0/parenrightBigg\n. (1)\nf0andk0are the resonance frequency and spring\nconstant of the cantilever at zero applied field, respec-\ntively,leis the effective length of the cantilever, θc\nthe oscillation angle, and Emthe magnetic energy of\nthe specimen. Properties of the cantilevers used in the\nexperiments can be found in section I of the appendix.\nIn the limit of large applied magnetic field, such that the\nZeeman energy dominates over the anisotropy energy,\nall magnetic moments align along H. In this limit ∆ f\nasymptotically approaches a value determined by the\ninvolved anisotropies, their respective directions, the to-\ntal magnetic moment, θh, and the mechanical properties\nof the cantilever19,20. In particular, the determination\nof these asymptotes allows us to extract the direction of\nmagnetic easy and hard axes by measuring the angular\ndependence of the frequency shift at high field ∆ fhf.\nThe maximum positive (minimum negative) ∆ fhf(θh)\nindicates the easy (hard) axis. For these measurements,\nshown in Fig. 2 (c), we apply µ0H= 3.5 T, where we\nexpect to be in the high field limit. See section VI\nand VII of the appendix for more details. We also\nmeasure the magnetic hysteresis of ∆ f(H) by sweeping\nthe applied field Hup and down for θhfixed to the\nvalues determined for the magnetic easy and hard axes,\nrespectively. This procedure reveals signatures of the\nJPs’ magnetic reversal, as shown in Figs. 2 (d), (e) and 3.\nIn order to analyze both these types of measure-\nments, we perform micromagnetic simulations using the\nfinite-element software Nmag21. We calculate ∆ f(H)\nfrom the micromagnetic state for parameters set by\nthe experiment19,22. Matching these simulations to the\nmeasurements gives us a detailed understanding of the\nprogression of the magnetic configurations present in the\nJPs throughout the reversal process. Events, such as the\nnucleation of a magnetic vortex, can be identified and\nassociated with features in ∆ f(H) measured via DCM.\nFig. 2 (a) and (b) shows false color SEMs of the\nmeasured fmJP and ebJP, respectively, each attached to\nthe tip of a cantilever. The orientation of the particles3\n/gid00132/gid00001/gid00048/gid00040/gid00004/gid00028/gid00041/gid00047/gid00036/gid00039/gid00032/gid00049/gid00032/gid00045ebJP fmJP H\n(a) (b)\n(c)\n(d)\n(e)/gid00132/gid00001/gid00048/gid00040\nH || easy \naxisH || hard\naxis\nFigure 2. False color SEMs of the (a) fmJP and (b) ebJP\nattached to the tip of a cantilever, respectively. The coordi-\nnate system is shown on the right. (c) ∆ fhf(θh) measured at\nµ0H= 3.5 T for the fmJP (blue) and ebJP (brown). Black\ncircles indicate θhof the hysteresis measurements, which are\nshown in (d), (e), and Fig. 3.\nin the images can be correlated with the angle θhof the\nmaxima and minima found in the high-field frequency\nshift ∆fhf(θh) in dependence of the magnetic field angle\nθh, shown in Fig. 2 (c). Doing so, we find a magnetic\neasy direction in the equatorial plane of the particles and\na hard direction along the axis of the pole. The 90 °angle\nbetween easy and hard direction is a clear indication\nthat uniaxial anisotropy is the dominant anisotropy in\nthe system. We ascribe the latter to the shape of the\nJPs, because no other strong anisotropies are expected.\nThe field-dependent frequency shift ∆ f(H) for H\naligned along the easy axis, see Fig. 2 (d), shows a\ntypical hysteretic, V-shaped curve, that approaches\na horizontal asymptote for high field magnitudes19.\nThe fmJP shows a symmetric asymptotic behavior\nforµ0H= 3.5 T and −3.5 T (blue curve). Magnetic\nreversal at low fields, µ0Haround ±20 mT, is symmetric\nupon reversal of the field sweep direction, as shown in\nFig. 2 (e). This behavior is expected for a ferromagnetic\nparticle with a magnetic field applied along its easy axis.In contrast, measurements of the ebJP reveal asymmet-\nric asymptotic behavior with ∆ fhfvalues differing by\nabout 0.9 Hz forµ0H=±3.5 T, as seen in the brown\ncurve of Fig. 2 (d). Furthermore, after a full hysteresis\ncycle, we observe a reduction in the difference of ∆ fhfat\n±3.5 T by about 0 .4 Hz, which is evidence for magnetic\ntraining23,24. Measurements of the ebJP also show a\nhighly asymmetric magnetic reversal, which occurs at\nµ0H=−44 mT when sweeping the field down and at\nµ0H= 12 mT when sweeping the field up. All of these\nfindings are characteristic of an exchange bias imposed\non the fm layer by the afm layer.\nIn order to draw conclusions about the magnetic state\nof the JPs, we establish a micromagnetic model for\neach of the two types of JPs over many iterations of\ncomparison to measured ∆ f(H) and variations of the\nparameters for Happlied along the magnetic easy and\nhard axis, respectively. This iterative process, informa-\ntion from literature, and observations from SEMs lead\nto a final set of parameters (see appendix, section I)\nand a model that reproduces the measured ∆ f(H). The\nmodel assumes that the magnetic JPs are made from a\nhemispherical shell with a thickness gradient from the\npole towards the equator, which accounts for the gradual\nreduction of the shell thickness away from the pole, as\nshown in Fig. 1 (b). The hemisphere is also truncated25\nby a latitudinal belt around the equator, reflecting\nobservations from the SEM images in Figs. 2 (a) and\n(b). For simplicity, in the simulations, we do not account\nfor the magnetic film’s irregular edge at the equator and\na possible change in the crystallographic texturing with\nrespect to the particle surface as a function of position\nwithin the cap. The orientation of a JP with respect to\nthe cantilever rotation axis and His set by infering the\norientation from the SEMs and followed by an iterative\ntuning of the angles ( θJP,ϕJP), as defined in Fig. 1 (d),\nto match the measured ∆ f(H).\nFigs. 3 (a) and (b) shows agreement between the\nmeasured and simulated ∆ f(H) curves for the fmJP\nforHapplied along the magnetic easy and hard axis,\nrepectively. Similar curves are shown for the ebJP\nin Figs. 3 (c) and (d). Details on the progression of\nthe magnetization configurations as a function of H,\nas indicated by the simulations, are found in section\nII of the appendix. Here, we focus on the remanent\nmagnetization configurations in both types of particles,\nas shown on the right of Fig. 3.\nFor the fmJP, an onion state is realized after sweeping\nHalong the easy axis while a global vortex state is found\nafter sweeping along the hard axis. In applications, mag-\nnetic JPs are subject to considerable disturbances from\nthe outside, including thermal activation, interactions\nwith other nearby magnetic particles, and alternating\nexternal magnetic fields for actuation. Hence, we can\nexpect a magnetic JP to relax to its ground state\nconfiguration over time. The simulation of the fmJP4\nfmJP\nebJP\nx\ny\nzhard\neasy\n-1\n mz1-1\n1 mzH || easy \naxis\nH || easy \naxisH || hard\naxis\nH || hard\naxis(a)\n(b)\n(c)\n(d)\nFigure 3. Measured and simulated ∆ f(H) of the fmJP for H\napplied along the (a) easy and (b) hard axis, respectively. A\nvisualization of each corresponding simulated remanent mag-\nnetic state is shown on the right. The same set of data for\nthe ebJP is shown in (c) and (d).\nshows that when the magnetization is in the global\nvortex state, its magnetic energy is 101 aJ lower than\nwhen it is in the onion state. The global vortex state\nis therefore energetically more favorable than the onion\nstate, which is also true if compared to any other\nremanent state that we have found in simulations for\nfmJPs, as discussed in appendix, section IV. This\nanalysis suggests that a remanent global vortex state,\nwhich has a vanishing total magnetic moment, is realized\nin fmJPs over time, independent of magnetic history. If\nwe normalize the magnetic moment of this state by the\nsaturation moment, MsV, we find that the global vortex\nstate hosted by the fmJP has a moment value of 0.03,\nprecluding the use of such particles in applications.\nWe establish a similar micromagnetic model for theebJP. To model the effect of an exchange bias imposing\na preferred direction on the magnetic moments in the\nferromagnet, a unidirectional anisotropy is added to\nthe simulation. Note that other influences of the afm\nlayer are not accounted for, especially contributions\nto the coercive field26, rotational anisotropies27, or\ncontributions arising due to its granular structure28.\nFor this reason, neither the asymmetric values of the\nmagnetic reversal fields nor the observed training\neffects are correctly reproduced by the simulation. The\nunidirectional anisotropy, described by a unit vector ˆ ueb\nwith orientation angles ( θeb,ϕeb), as shown in Fig. 1 (c),\nand an anisotropy constant Keb, is expected to lie\nsomewhere in the equatorial plane of the ebJP. The\nspecific orientation of ˆ uebwithin this plane is induced in\nthe afm by the magnetic field applied during deposition.\nNote that for simplicity, Kebis kept constant within\nthe whole volume of the magnetic cap, despite thickness\nvariations of the afm layer with θeb.\nThe knowledge of where ˆ uebpoints within the equa-\ntorial plane is lost after attaching the JP to the can-\ntilever. In the simulations, we choose to align ˆ uebalong\nthe direction in the equatorial plane that coincides with\nthe applied external field in the easy-axis configuration,\neven though it could point along any direction in this\nplane. This assumption that Handˆ uebare collinear in\nthe easy-axis measurements means that our simulations\npredict the maximum possible ∆ ffor any given choice\nof the unidirectional anisotropy constant Keband hence\ngive a lower bound for Keb.\nWe adjust this anisotropy to match the measured\n∆f(H) along both the easy and hard axes and find\ngood agreement for Keb= 22.5 kJ/m3. In particular, the\nmodel reproduces the asymmetry in ∆ fhffor both the\neasy and hard-axis alignments, as shown in Fig. 3 (c)\nand (d). This value of Kebis also consistent with results\nfrom Ref. 28, once the influence of a reduced sample size\nis considered29.\nAt remanence, the simulations show that the ebJP\nhosts an onion state irrespective of its magnetic history,\nas shown in Figs. 3 (c) and (d). To exclude the presence\nof an equilibrium global vortex state, we test this state’s\nstability by initializing the ebJP in a global vortex\nstate at remanence and then relaxing the system to a\nlocal energetic minimum. Following this procedure, the\nsystem relaxes to the onion state. As a result, we can\nexclude the global vortex state as a possible equilibrium\nremanent state in this ebJP.\nFor the simulations of the ebJP we find a total\nmagnetic moment at remanence, normalized by its\nmaximum value of MsV, of 0.89 and 0.71 depending on\nwhether His applied along the hard or easy direction of\nthe external field, respectively. This remanent moment\nrepresents an increase of more than one order of mag-\nnitude compared to the remanent moment of the fmJP.5\nHence, introducing exchange bias to magnetic JPs, if\nstrong enough, succeeds in stabilizing a high-moment\nonion state in remanence.\nTo conclude, micrometer-sized JPs capped with an\nantiferromagnetic/ferromagnetic or purely ferromagnetic\nthin film system have been mass produced through a\nsputter-deposition process. We have investigated the\nmagnetic reversal and remanent magnetic configurations\nof individual specimens of these JPs using DCM and\ncorresponding micromagnetic simulations. Although\nthe fmJPs host a global vortex state in remanence\nwith a vanishing magnetic moment, the addition of an\nantiferromagnetic layer in ebJPs successfully changes the\nremanent configuration to a stable high-moment onion\nstate. Unlike previous measurements on close packed\nparticle arrays, our measurements on individual JPs\nshow that the stability of this high magnetic moment\ntexture in remanence is a property of the individual\nparticles and present in absence of interparticle interac-\ntions.\nWe thank Sascha Martin and his team in the machine\nshop of the Physics Department at the University of\nBasel for help building the measurement system. We\nacknowledge the support of the Canton Aargau and the\nSwiss National Science Foundation under Grant No.\n200020-159893, via the Sinergia Grant Nanoskyrmionics\n(Grant No. CRSII5-171003), and via the National Cen-\ntre for Competence in Research Quantum Science and\nTechnology. Calculations were performed at sciCORE\n(http://scicore.unibas.ch/) scientific computing core\nfacility at University of Basel.\nThe data that support the findings of this study are\navailable from the corresponding author upon reasonable\nrequest.\nREFERENCES\n1Andreas Walther and Axel H. E. M¨ uller. Janus Particles: Synthe-\nsis, Self-Assembly, Physical Properties, and Applications. Chem.\nRev., 113(7):5194–5261, July 2013.\n2D. Issadore, Y. I. Park, H. Shao, C. Min, K. Lee, M. Liong,\nR. Weissleder, and H. Lee. Magnetic sensing technology for\nmolecular analyses. Lab Chip , 14:2385–2397, 2014.\n3Arno Ehresmann, Iris Koch, and Dennis Holzinger. Manipu-\nlation of superparamagnetic beads on patterned exchange-bias\nlayer systems for biosensing applications. Sensors , 15(11):28854–\n28888, 2015.\n4Brandon H. McNaughton, Karen A. Kehbein, Jeffrey N. Anker,\nand Raoul Kopelman. 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Interrelation between polycrystalline structure\nand time-dependent magnetic anisotropies in exchange-biased bi-\nlayers. Physical Review B , 102(14):144421, October 2020. Pub-\nlisher: American Physical Society.\n29J. Nogu´ es, J. Sort, V. Langlais, V. Skumryev, S. Suri˜ nach, J. S.\nMu˜ noz, and M. D. Bar´ o. Exchange bias in nanostructures.\nPhysics Reports , 422(3):65–117, December 2005.Supplementary Information: Magnetic hysteresis of individual Janus particles with hemispherical exchange\nbiased caps\nS. Philipp,1B. Gross,1M. Reginka,2M. Merkel,2M. Claus,1M. Sulliger,1A. Ehresmann,2and M. Poggio1\n1)Department of Physics, University of Basel, 4056 Basel, Switzerland\n2)Institute of Physics, University of Kassel, 34132 Kassel, Germany\nI. CANTILEVER PROPERTIES AND\nSIMULATION DETAILS\nCantilevers are fabricated from undoped Si. They\nare 75 µm-long, 3.5µm-wide, 0.1µm-thick with a mass-\nloaded end and have a 11 µm-wide paddle for optical\nposition detection. The resonance frequency f0of the\nfundamental mechanical mode used for magnetometry\nis on the scale of a few kilohertz. Spring constant k0\nand effective length leare determined using a finite\nelement approximation1. For the cantilever used for\nthe fmJP we find f0= 5285.8 Hz,k0= 249 µN/m\nandle= 75.9µm. For the cantilever of the ebJP\nf0= 5739.3 Hz,k0= 240 µN/m andle= 75.9µm.\nMicromagnetic simulations are performed with the\nfinite-element software package Nmag2. As an approx-\nimated geometry for the JPs a semi-sphere shell with\na thickness gradient from the pole towards the equa-\ntor is used, that is truncated at the equator, as dis-\ncussed in the main text. The exchange constant is set\ntoAex= 30 pJ/m3.\nIn case of the fmJP, opposing to the SEM image in\nFig. 2 (a) in the main text, which suggests a truncation\nof the fm layer by about 250 nm, it needs to be set to\n350 nm or even more to match the high field progression\nof ∆f(H). For the same reason the nominal thickness of\n10 nm of the fm layer needs to be increased to at least\n12 nm at the pole, which is then gradually reduced to\n0 at the equator. These two geometric constraints are\nnecessary to keep Msat a reasonable value below the bulk\nvalue of 1.95 MA/m4. This suggests, that significantly\nmore fm material than anticipated is deposited on the\nregion around the pole of the JPs, which is the most\ndirectly exposed area of the sphere during deposition.\nFor the simulation of the fmJP we set the following\nparameters: saturation magnetization Ms= 1.8 MA/m,\nsilica sphere diameter of 1 .5µm, truncation d= 350 nm,\nparticle orientation ( θJP,ϕJP) = (91 °,2°) and maximum\nallowed mesh cell size 7 .5 nm.\nFor the ebJP, the geometric parameters have to be ad-\njusted less from their nominal values than for the fmJP,\nin order to match between the micromagnetic model to\nthe experiment. This result suggests that the afm layer,\nwhich is deposited before the fm, acts as an adhesive for\nthe fm, and the ebJP is coated more homogeneously than\nthe fmJP.\nFor the simulation of the ebJP we set the follow-\ning parameters: Ms= 1.44 MA/m, fm layer thickness\nof 10 nm at the pole, gradually reduced to 0 at theequator, silica sphere diameter of 1 .5µm,d= 350 nm,\n(θJP,ϕJP) = (85 °,10°), maximum allowed mesh cell size\n7.0 nm, (θeb,ϕeb) = (−90°,0°), unidirectional anisotropy\nconstantKeb= 22.5 kJ/m3.\nFor the generic simulations in sections V and IV we\nhave usedMs= 1.8 MA/m, , fm layer thickness of 10 nm\nat the pole, gradually reduced to 1 nm at the equator,a\nsilica sphere diameter of 500 nm, ( θJP,ϕJP) = (0 °,0°),\nand (θeb,ϕeb) = (−90°,0°). Parameters that are not\nmentioned here are given in the main text.\nII. PROGRESSION OF THE MAGNETIC\nSTATE WITH EXTERNAL FIELD\nA. Ferromagnetic Janus particles\n∆f(H), measured for Hparallel to the magnetic easy\n(blue data) and hard axis (orange data), respectively,\nas shown in Fig. 1 (a), gives direct information on\nhigh field behavior and magnetic reversal of the JPs.\nForHparallel to the magnetic easy axis, an overall\nV-shape suggest Stoner-Wohlfarth like behavior for\nmost of the field range in Fig. 1 (a). As seen in the\nclose-up in Fig. 1 (b), magnetic reversal appears to take\nplace through a few sequential switching events at small\nnegative reverse fields.\nThe simulated ∆ f(H), also shown in Fig. 1 (green\npoints) together with a few exemplary configurations\nof the simulated magnetic state of the JP, can give\nmore insight into what happens during the field sweep.\nStarting from full saturation, most magnetic moments\nstay aligned with the easy direction down to very low\nreverse fields, nicely seen in Fig. 1, configuration 1 at\n3.5 T and configuration 2 at remanence. The latter is\nan onion state. This progression of configurations is\nconsistent with the Stoner-Wohlfarth-like behavior of\nthe experimental ∆ f(H). Magnetic reversal takes place\nthrough the occurrence of a so-called S-state, for which\nthe magnetization follows the curvature of an S. The\nreversal is shown in configurations 3 and 4 in Fig. 1.\nThen, until full saturation is reached in reverse field,\nonly magnetic moments in proximity to the equator of\nthe JP are slightly canted away from the direction of the\nexternal field (and the easy plane). This progression is\nrobust in simulation, even though sometimes, depending\non slight variations of simulation parameters, a vortex\nappears in reverse field instead of the S-state. The\nobservation of several, individual switching events\nduring magnetic reversal in experiment may originate\n1arXiv:2110.11158v1  [cond-mat.mes-hall]  21 Oct 20211\n3(b)\n5\n 6\n 8 7mz1\n0\n-1\nmz1\n0\n-1mx1\n0\n-1\nmz\n1\n 3 2 41\n0\n-1mz1\n0\n-1mx1\n0\n-1mx1\n0\n-1\nmzEasy direction \nzx\ny\nH\nzx\nyHHard direction \n1\n0\n-1\n627(a)\n4\n58Figure 1. Data for the ferromagnetic Janus particle. (a) Measured ∆ f(H) for easy (blue) and hard (orange) alignment of the\nexternal field, as well as the simulated magnetic configurations in green (easy) and red (hard). (b) Close-up of (a) for low fields.\nData with easy orientation is offset by 2 Hz for better visibility. Colored arrows indicate field sweep directions. Numbers in (a)\nand (b) denote the field values for the configurations of the magnetic state.\nin vortex hopping, or switching of different regions in\nthe JP due to variations in material and geometric\nparameters. Magnetic reversal through a vortex rather\nthan an S-state may also explain the big difference of\nthe coercive fields between experiment ( Hc≈32 mT)\nand simulation ( Hc= 6 mT).\nForHparallel to the magnetic hard axis, the exper-\nimental data has an inverted V-shape, and there is no\neasily identifiable sign of magnetic reversal. Yet, for rel-\natively large fields, around 1 .5 T, switching events are\nobserved, and exist up to negative fields of similar mag-\nnitude. Typically, W-shaped curves are observed for\nmeasurements with the field aligned with the hard direc-\ntion, and can be understood in a simple Stoner-Wohlfarth\nmodel, which is discussed in section III. The inverted V-\nshape instead of the W-shape is a peculiarity of the fact\nthat the fm layer of the JP is curved everywhere. The\nangle between the local surface normal and the direction\nof the external magnetic field is different for every polar\ncoordinate of the JP, which leads to a dependence of the\nlocal demagnetizing field on the polar coordinate. In con-\nsequence, the magnitude of the external magnetic field,\nfor which the local magnetic moments start to rotate to-\nwards their local easy direction depends strongly on the\nposition in the magnetic cap. This leads to the observed\ncurve shape of ∆ f(H), a more detailed discussion can befound in section IV. The magnetic progression in simula-\ntion for external field alignment with the hard direction\ncan be summarized as follows: Starting from full sat-\nuration, the magnetic moments start rotating towards\nthe easy plane with decreasing field magnitude due to\nthe competition between shape anisotropy and Zeeman\nenergy. This takes place for different magnitudes of H\ndepending on where a magnetic moment is located in\nthe JP, as discussed earlier. Configuration 5 in Fig. 1\nshows a state for which magnetic moments at the pole\nhave already started to rotate, while magnetic moments\nin proximity to the equator remain aligned with the ex-\nternal field. Superimposed to this rotation, a minimiza-\ntion of the system’s energy by formation of a magnetic\nvortex localized at the pole for around 1 .8 T takes place,\nwhich grows in size with decreasing field, see configura-\ntion 6. In simulation, this is a gradual evolution, and only\nfor fields below about 300 mT jumps in ∆ fdue to vortex\nmovement are observed. This process is in contrast to\nthe discontinuities that occur at around 1 .5 T in the ex-\nperiment, but can be explained by vortex hopping from\npinning site to pinning site. The latter may be present\ndue to fabrication inhomogeneities in the JPs5. For zero\nfield the vortex dominates the magnetic configuration of\nthe JP and has evolved into a global vortex state, as\nshown in configuration 7. Configuration 8 shows the vor-\ntex in reverse field, which has changed polarity, and has\n2jumped to a slightly off-centered position. The latter is\ntoo small to be visible in the figure. Further decreasing\nthe field, the vortex sits centrally in the JP and shrinks in\nsize, and vanishes around −1.82 T. At the same time, the\nmagnetic moments rotate towards the field direction de-\npending on their position in the JP, as described earlier.\nNote, that by slightly changing simulation parameters,\nwe find that features due to vortex entrance and hopping\nmay manifest themselves in ∆ f(H) with strongly differ-\ning magnitude and for different field values. Introducing\nartificial pinning cites in simulation can be used to adjust\nthe vortex hopping to match the observed signals more\nprecisely5, but consumes vast amounts of computational\ntime and should still be understood only as an exemplary\nprogression of the magnetic state.\nB. Exchange-biased Janus particles\nThe progression of the magnetic state for the ebJP\nis very similar to the fmJP, yet, there are crucial\ndifferences. See Fig. 2 for the DCM data, simulation\nresults, and configurations of some magnetic states. For\nthe field oriented in the magnetic easy direction the\nnearly polarized state, shown in Fig. 2 configuration\n1, is similar to that shown in Fig. 1, configuration 1.\nReducing the field down to remanence, as shown in\nFig. 2 configuration 2, we find an onion state just as\nfor the fmJP. Magnetic reversal occurs again through\nan S-state, rather than via vortex formation, as shown\nin configuration 3. However, the reversal is shifted\ntowards negative fields, and occurs for −15 mT for the\ndown sweep, and for −17.5 mT for the up sweep of the\nmagnetic field. This does not match the experimentally\nobserved values, especially for the latter case, for which\nthe switching occurs for positive field. This is no\nsurprise, since the employed model does not account for\nthe contribution of the exchange bias to the coercivity.\nYet, both simulation and experiment show a shift of the\nhysteresis loop towards negative fields as compared to\nthe fmJP.\nWe only observe a single switching event in experiment\nfor the magnetic reversal, which is consistent with the\nbehavior of the S-state in simulation. For the alternative\nmagnetic reversal process through vortex formation, we\nwould expect several switching events due to vortex\nhopping. We find such a situation e.g. for a few reversal\nprocesses without unidirectional anisotropy, where\ngeometrical parameters of the JPs have been varied, see\nsection V. Yet, it is also possible, that a strong pinning\nsite favors the formation of a vortex, and keeps it in\nplace for all field magnitudes up to the reversal point.\nIf the magnetic field is swept from negative saturation up\nto remanence (not shown here), an onion state is present,\nthat has its total magnetic moment pointing opposite to\nthe exchange bias direction. For applications, this is an\nundesirable state. It is energetically less favorable than\nthe state of parallel alignment, and if the energy barrierbetween the two states is overcome by an external\ninfluence, the JP will switch.\nIf the external field is aligned with the hard direc-\ntion, and starts at full saturation, the magnetic moments\nrotate towards the easy plane depending on their posi-\ntion in the JP for decreasing field just as for the fmJP.\nFurther, the same, superimposed vortex formation takes\nplace, starting for 1 .34 T. Again the vortex occupies more\nand more volume of the JP with further decreasing field.\nSuperimposed, a local vortex forms at the pole of the\nebJP for an applied field of 1 .34 T. As for the fmJP, the\nvortex occupies more and more volume of the JP with\nfurther decreasing field. However, upon further reducing\nthe field, the vortex, rather than inhabiting the whole\nJP as a global vortex centered at the pole of the fmJP,\nit prefers to move to the side of the ebJP, as shown in\nconfiguration 4 of Fig. 2. Moving down from the pole to-\nwards the equator, the vortex exits from the JP through\nthe equator for 5 mT, and an onion state is formed at\nremanence, as shown in configuration 5. The orientation\nof the onion state is governed by ˆ ueb. For a small reverse\nfield a domain wall state forms, as shown in configura-\ntion 6. With further decreasing field, the domain wall\nis rotated with respect to the polar axis of the JP. This\nstate seems to be a precursor of the vortex state, and the\nwall is subsequently replaced by the vortex, sitting again\nin the center of the JP, as shown in configuration 7. The\nvortex vanishes for -1 .36 T. Whether such a domain wall\nstate is indeed realized in the ebJPs for reverse fields, or\nif a vortex enters from the equator and moves back to the\ncenter of the JP, as seen for simulations of smaller JPs\n(see section V), remains an open question. The DCM\nsignal shows in both experiment and simulation many ir-\nregularities for the lower field range, which does not allow\nus to draw clear conclusions on the magnetic state present\nin the JPs. Nevertheless, the simulations clearly suggest\nthat an onion state should be realized at remanence, irre-\nspective of the states present during the hysteresis. This\nsituation is markedly different than that of the fmJP and\nis a direct consequence of the presence of exchange bias.\nIII. STONER-WOHLFARTH MODEL FOR\nEASY PLANE TYPE ANISOTROPY\nThe shape of the magnetic material of the fmJPs is,\nat least in a first approximation, rotationally symmetric\naround the pole axis. Further, the thickness gradient of\nthe CoFe layer, as shown in the cross-sectional SEM in\nFig. 1 (b) in the main manuscript, suggest that the mag-\nnetic material is concentrated in proximity to the pole,\nand that there is less material towards the equator. This\nmaterial distribution suggests that a uniaxial anisotropy\nof easy plane type is imposed on the sample by its shape.\nThe most basic approach to describe such a system is\na Stoner-Wohlfarth model, in which a single macro mag-\nnetic moment replaces the ensemble of distributed mag-\n361\n23Hard direction1\n0\n-1\nEasy direction mz1\n0\n-1\nmz1\n0\n-1mx1\n0\n-1\nmz1\n0\n-1\nmz\n(a) (b)1 2 3\n4 5\n1\n0\n-1\nmz1\n0\n-1\nmz7\nzxy\nHzx\ny\nH\n4\n567\nFigure 2. Data for the exchange biased Janus particle. (a) Measured ∆ f(H) for easy (blue) and hard (orange) alignment of\nthe external field, as well as the simulated pendants in green (easy) and red (hard). (b Close-up of (a) for low fields. Data with\neasy orientation is offset by 2 Hz for better visibility. Numbers in (a) and (b) indicate the field values for the configurations of\nthe magnetic state shown in (c) for hard and in (d) for easy alignment.\nnetic moments6. Following Ref. 7, it is straight forward\nto calculate the magnetic hysteresis and connected DCM\nresponse for easy plane anisotropy, where the latter is\nmanifested in a positive, effective demagnetizing factor\nDu, opposing to a negative Dufor easy axis anisotropy.\nThe model is a good approximation for high fields, where\nall magnetic moments are aligned with the external field,\nand essentially behave like a single macro spin. At lower\nfields deviations from the curve of the SW-model indicate\ninhomogeneous spin orientation. We show the magnetic\nhysteresis of the components mx,myandmzof the nor-\nmalized magnetization mand the DCM response ∆ fin\nFig. 3 for several different orientations ( θu,φu) of the\nuniaxial anisotropy axis ˆ u, which is perpendicular to the\neasy plane. The external field Hhas to be fixed in the\nz-direction in the model for technical reasons, which is\nwhyˆ uis varied rather than H, contrary to the situation\nin experiment. Hence, the angles θJP−θhandφJP, de-\nscribing the equivalent situation in experiment, have to\nbe compared to θuandφu, respectively. However, this\ndoes not limit the validity of the model.\nForθu= 0°, for which His perpendicular to the easy\nplane, we find the typical W-shape for ∆ ffor a magnetic\nhard orientation of H7. The columnar arrangement of\nthe components of the magnetization mx,myandmz\ntogether with ∆ fallow to correlate changes in magneticbehavior with features in ∆ f, such as e.g. the transition\nfrom field alignment to the beginning of a rotation of m\ntowards the easy axis.\nForθu= 90°, for which Hlies in the easy plane, the typi-\ncal V-shaped curve for an easy orientation of His found,\njust as expected7. This V-shape is connected to a perfect\nsquare hysteresis of m. Intermediate values of θulead to\nintermediate curve progression, which are a combination\nof the two extrema described above.\nCompared to experiment (see Figs. 2 and 3 in the main\nmanuscript) we find, that while in the case of the exter-\nnal field Hin the easy plane (last panel in the last row\nin Fig. 3) the model generates relatively similar curve\nshapes for ∆ f, this is not true for Hperpendicular to\nthe easy plane (first panel in the last row). The vertical\nasymptotes for h≈±0.5 are missing in experiment. Fur-\nther, the horizontal asymptote at high field is approached\nfrom negative rather than positive values, as it is seen in\nexperiment. A more detailed analysis in the following\nsection will show, that the latter originates in the cur-\nvature of the magnetic shell, which cannot be captured\nby a single spin model. Even though typically a good\nstarting point, the SW model seems to be of rather lim-\nited validity for the description of the relatively specific\ngeometry of a spherical cap.\n4Figure 3. Components of Mnormalized by Ms(first 3 rows) and ∆ fnormalized byf0µ0VM2\ns\n2k0l2e(last row) vs. normalized magnetic\nfieldh=H\nMsDufor different orientations of the anisotropy axis. Valid for uniaxial anisotropy with Du>0 (hereDu= 0.5).θu\nis increased from 0 °in the first column by 30 °per column up to 90 °.φuis changed in the same steps, given by the different,\ncolor-coded graphs within each column (blue for φu= 0 °, orange for 30 °, green for 60 °and red for 90 °). Arrows indicate\nswitching of the magnetization. ∆ fhas been scaled by a factor 0.5 and offset by 0.5 for θu= 0 °, and scaled by 2 for the other\nvalues ofθu.\nIV. SHAPE ANISOTROPY OF A\nTRUNCATED SPHERICAL HALFSHELL\nAs a measure for the strength of the shape anisotropy\npresent in the JPs, and as a parameter used for SW-\nmodeling, the knowledge of the effective demagnetization\nfactorDuof the geometry is of value. If possible, Duis\ndetermined using an analytical expression8, which, how-\never, is not known for the given geometry. Using micro-\nmagnetic simulations as discussed in the main text, we\ncan extract a good approximation to the demagnetiza-\ntion factor of a given geometry, without necessity for an\nanalytical formula. Here, we analyze a generic, truncated\nspherical halfshell as defined in section I for different de-\ngrees of truncation d.\nWe quickly recall the definition of the effective demagne-\ntization factor Du=Dz−Dx, whereDzandDy=Dx\nare the demagnetization factors of a magnetic object thatis rotationally symmetric in the xyplane.−0.5<D u<1\nis true, and Du<0 is valid for prolate and Du>0 for\noblate bodies. Du= 0 is the case for a perfectly spheri-\ncal body. We find a minimum Duof approximately 0.25\nfor the smallest truncation, and Duincreases with trun-\ncation as shown in Fig. 4 (a). Hence, shape anisotropy\ngets stronger as dis increased, which can be roughly un-\nderstood as the transformation from a spherical halfshell\nto a disc. Note, that a spherical halfshell without thick-\nness gradient and without truncation leads to a Dujust\nslightly large than zero. This implies, that pointing in\nthe easy plane is not very much more favorable that any\nother direction for the magnetization averaged over the\nwhole sample, see the orange dots in Fig. 4.\nThe high field frequency shift, ∆ fhf, which is\nof relevance for extracting anisotropy constants from\nexperiment7,9, first increases with truncation, but later\ndecreases again, see Fig. 4 (c). This is owed to the loss\n50.00.51.0Du\n(a)gradient shell\nhalfshell, no gradient\n012V (nm3)1e6\n(b)\ngradient shell\nhalfshell, no gradient x0.5\n0 50 100 150 200 250\nd (nm)024fhf (Hz)\n(c)\ngradient shell\nhalfshell, no gradientFigure 4. (a) Effective demagnetization factor Duof a trun-\ncated spherical halfshell with a gradient shell thickness in de-\npendence of the truncation d(blue) and of a full halfshell\n(orange). (b) Volume and (c) high field frequency shift ∆ fhf\nof the same geometries as in (a).\n0 -1.4e6\nA/m\nFigure 5. Cut through the geometry of the JP at the position\nof thexzplane showing the xcomponent of the demagnetizing\nfield within the magnetic layer for H/bardblˆ xandµ0H= 20 T.\nof magnetic material for increasing truncation as seen in\nFig. 4 (b).\nIn contrast to the SW model the micromagnetic simu-\nlations are able to correctly reproduce the experimental\ncurve shape of ∆ ffor the hard axis orientation, see Fig. 6\n(d). To understand the reason for this, the xcomponent\nof the demagnetizing field Hdemag,x within the magnetic\nlayer is visualized for a cut through the geometry in Fig. 5\nat high applied field in xdirection. It shows a gradual\nchange of the demagnetizing field magnitude with zpo-\nsition, which is a good measure of the preferred orienta-\ntion of a magnetic moment (top part with Hdemag,x≈0\nprefersxorientation, opposing to bottom part with max-\nimumHdemag,x , which needs maximum external field to\nbe aligned in xdirection). This shows, that magnetic\nmoments in proximity to the pole need the smallest field\nmagnitude to be aligned with a field in xdirection. The\nrequired field magnitude gradually increases the closer a\nmagnetic moment is situated to the equator. This ex-\nplains the gradual change of ∆ fwith increasing field inthis orientation, opposing to what is evident in the SW\nmodel, where all magnetic moments rotate in unison.\nV. GENERIC SIMULATIONS OF\nTRUNCATED SPHERICAL HALFSHELLS\nWITH AND WITHOUT EXCHANGE BIAS\nThe speedup of simulations due to a reduced size of\nthe JPs, as defined in section I, allows us to analyze the\ninfluence of simulation parameters such as the trunca-\ntion on the magnetic hysteresis. In Fig. 6 a set of data\nfor JPs with different truncations is shown. Besides the\ndifferences in high field asymptotes as already discussed\nin section IV, the truncation also significantly changes\nthe curvature of ∆ f(H), see Figs. 6 (a) and (d). By ad-\njusting the truncation, this allows to fit the curvature of\na given experimental ∆ f(H) in the simulation.\nThe magnetic state evolves very much as discussed in\nsection II for field alignment in both easy ( H/bardblˆ x) and\nhard ( H/bardblˆ z) orientation. Yet, especially for the for-\nmer we observe some distinct differences for a truncation\nofd > 100 nm: Rather than through the formation of\nan S-state close to remanence, magnetic reversal occurs\nthrough a vortex state, that only appears for small re-\nverse fields. This manifests itself e.g. in the magnitude\nof the total magnetic moment |µ|, which is significantly\nless for the vortex state as compared to the S-state, as\nshown in Fig. 6 (b).\nFor the external magnetic field perpendicular to the\neasy plane, we find that magnetic reversal takes place via\nvortex formation for all values of the truncation. Con-\nsequently, the total magnetic moment at remanence is\nalways small, as seen in Figs. 6 (b) and (e).\nThere is a big difference between the magnetic\nmoments at remanence depending on through which\nprogression remanence has been reached. Hence, in\norder to find the most stable state, it is instructive to\ncompare the energies of the different progressions as\nshown in Figs. 6 (c) and (f). A state with significant\n|µ|at remanence always seems to possess higher energy\nthan the vortex state. Compare, for example, the energy\nat remanence for the JP with 200 nm truncation for\neasy (no vortex) and hard (vortex state) field alignment.\nThe former has about 40 aJ, while the latter is more\nfavorable with only 7 aJ.\nAs discussed in the main text, exchange bias, which\nforces magnetic moments into a certain direction, can\navoid a global vortex state at remanence. We simulate\nthe magnetic hysteresis of reduced-size JPs with global\nunidirectional anisotropy and vary its strength Keb. As\nan example, we pick a JP with a cut of 100 nm, and match\nthe orientation of the unidirectional anisotropy to the\nparallel field direction, which gives the maximum effect\nin the DCM signal.\nIn Fig. 7, we plot the same set of data for the JP\nwith exchange bias as for the purely ferromagnetic JPs\n60123f (Hz)\n(a)\n0.00.51.01.52.02.5| | (fAm3)\n(b) 40\n20\n02040Etotal (aJ)\n(c)\n0 10\n5\n 510\n0H (T)\n2\n1\n0f (Hz)\n(d)\n100\n 50\n 0 50 100\n0H (mT)\n0.000.250.500.751.001.25| | (fAm3)\n(e)\n50\n 25\n 0 25 50\n0H (mT)\n10\n5\n0510Etotal (aJ)\n(f)\n5\n50\n100\n150\n200Figure 6. Frequency shift ∆ f, total magnetic moment µand total energy Etotal for JPs with different truncation as indicated\nin the legend in nm. Top row: External field applied in the easy plane ( H/bardblˆ x). Bottom row: External field applied in the hard\ndirection ( H/bardblˆ z).\nin Fig. 6. The frequency shift for high fields for H/bardblˆ x\ndevelops an increasing asymmetry for increasing strength\nKuof the unidirectional anisotropy, as shown in Fig. 7\n(a). In turn, for perpendicular alignment this asymme-\ntry is not evident, as shown in Fig. 7 (d). Hence, the\nexperimentally observed asymmetry in the asymptotes\ncan serve as good indicator for the strength of the undi-\nrectional anisotropy, given that its orientation is known.\nThe main influence of the unidirectional anisotropy on\nthe progression of the magnetic state with external field\nfor easy field alignment is to shift magnetic reversal in\nmagnetic field magnitude. This can e.g. be seen in |µ|,\nsee Fig. 7 (b). Magnetic reversal takes place around zero\nfield forKeb= 1 kJ/m3, and is shifted to happen around\n50 mT forKeb= 100 kJ/m3. The dominant state dur-\ning reversal remains an onion state for all values of Keb.\nFor hard alignment, a vortex appears in the JP for all\nvalues ofKeb, just as described in section II B. The vor-\ntex is centered in the JP for larger field magnitudes, but\nmoves to the side of the JP, if the field is reduced. De-\npending on Keb, the vortex moves only very little (small\nvalues forKeb), moves significantly to the side of the JP\n(intermediate Keb), or even escapes the JP through the\nequatorial line (large values of Keb). If the latter hap-\npens, the vortex reenters the JP in reverse field from the\nother side of the JP and then moves back to the cen-\nter for increasing reverse field. The larger the Keb, the\nlarger is the total magnetic moment µat remanence, for\nKeb= 100 kJ/m3we find almost full saturation in direc-\ntion of the anisotropy vector, see Fig. 7 (e).A clear trend is observable, if we consider the energy\nthat sets which remanent state is more likely over long\ntime scales, as shown in Figs. 7 (c) and (f): The difference\nin energy between the remanent states gets smaller for\nincreasingKeb, and hence diminishes the relevance of the\nchosen orientation of an applied field to magnetize the JP.\nThe trend for the total magnetic moment is, irrespective\nof the which orientation for the external field is chosen,\nto be larger in magnitude if Kebis increased.\nVI. DCM IN THE HIGH-FIELD LIMIT\nWITH UNIDIRECTIONAL ANISOTROPY\nThe high-field limit in DCM is reached for |H| /greatermuch\n|/summationtext\niKi/(µ0Ms)|, whereKiare the different anisotropy\ncontributions for a given direction. The frequency shift\nof the cantilever resonance is determined in this limit by\nthe competition of the different anisotropy contributions,\nand can be calculated analytically7,9. For the presence\nof unidirectional anisotropy of strength Kebthis is given\nby:\n∆funidir =−f0VKeb\n2k0l2e·\n/parenleftbigg\ncosθu(sinθhsinθebcosφucosφeb+ cosθhcosθeb)\n−sinθeb(sinθucosθhcosφeb+ sinθhsinφusinφeb)\n+ sinθusinθhcosθebcosφu/parenrightbigg(1)\n70123f (Hz)\n(a)\n0.00.51.01.52.02.5| | (fAm3)\n(b)\n10000\n25000\n50000\n100000\n100\n50\n050Etotal (aJ)\n(c)\n10\n 0 10\n0H (T)\n3\n2\n1\n0f (Hz)\n(d)\n100\n 50\n 0 50 100\n0H (mT)\n0.00.51.01.52.02.5| | (fAm3)\n(e)\n100\n 50\n 0 50 100\n0H (mT)\n100\n50\n0Etotal (aJ)\n(f)Figure 7. Frequency shift ∆ f, total magnetic moment µand total energy Etotal for JPs with 100 nm truncation and different\nstrengths of the unidirectional anisotropy as indicated in the legend in J /m3. Top row: External field applied in the easy plane\n(H/bardblˆ x). Bottom row: External field applied in the hard direction ( H/bardblˆ z).\nHere, (θh,φh= 0) define the orientation of the external\nfield, (θu,φu) of the axis of the uniaxial shape anisotropy\nas defined in Ref. 9, and ( θeb,φeb) of the unidirectional\nanisotropy vector. The latter is oriented first, and ro-\ntated by (θu,φu) in a second step to be consistent with\nthe situation in experiment. Cantilever and magnetic\nparameters are as defined before.\nWe evaluate the high-field limit for the sum of shape\nand unidirectional anisotropy for an exemplary situation\nas it may be present for the exchange biased JPs, using\nsimilar parameters as in section IV. However, we increase\nKebsignificantly to magnify its effects. The angles are\nset to be (θu,φu) = (−3°,0°) and (θeb,φeb) = (−90°,0°),\nwhileθhis varied as in experiment. The result is shown\nin Fig. 8 (a), together with the individual contributions\nfrom shape and unidirectional anisotropy. This shows,\nthat the sum of the two contributions may lead to a pe-\nriodicity that deviates slightly from 180 °, which wold be\ngiven for pure uniaxial shape anisotropy. Furthermore,\nthe magnitude of maxima and minima may differ signif-\nicantly. Here we find 1 .9 Hz for the maxima and 2 .4 Hz\nfor the minima, respectively. We have indeed observed\nsuch large asymmetries in experiment for low tempera-\ntures (not shown here), however, the origin is different,\nas will be discussed in the following. Nevertheless, for\nstrong unidirectional anisotropies these findings should\nbe observable in experiment.\nThe SW model, as described in section III, can be used\nto calculate ∆ f(θh) for a fixed field magnitude, as done\nin experiment for 3 .5 T. This allows to compare the re-\nsult of the SW model with the high field limit as dis-\n2\n02f (Hz)\n(a) sum\nshape\nunidir\n0 50 100 150\nh (°)\n2\n02f (Hz)\n(b)3.5 T\nlimitFigure 8. (a) ∆ f(θh) in the high field limit with shape and\nunidirectional anisotropy. ∆ f(θh) in the SW model for 3 .5 T\napplied field magnitude, and in the high field limit from (a).\ncussed above, see Fig. 8 (b). The curve of the high field\nlimit follows a (negative) cosine with 2 θhin the argument.\nIn turn, for the SW model at 3 .5 T, minima are deeper\nand maxima are shallower in ∆ f, respectively. However,\nthere is no deviation from the 180 °periodicity. Further,\nmaxima are wider than minima, which is a consequence\nof the fact that positive and negative asymptotes are ap-\nproached with a different curvature when ramping up the\nexternal field in the SW model, compare curves in the last\n8panel in the first column with those in the last panel in\nthe last column in Fig. 3. In experiment this behavior\nof the maxima and minima is inverted, which is caused\nby the extreme curvature of the magnetic layer JPs, as\ndiscussed in section IV.\nREFERENCES\n1COMSOL AB. Comsol multiphysics ®, 2021.\n2T. Fischbacher, M. Franchin, G. Bordignon, and H. Fan-\ngohr. A systematic approach to multiphysics exten-\nsions of finite-element-based micromagnetic simulations:\nNmag. IEEE Transactions on Magnetics , 43(6):2896–\n2898, 2007.\n3D. V. Berkov, C. T. Boone, and I. N. Krivorotov. Mi-\ncromagnetic simulations of magnetization dynamics in\na nanowire induced by a spin-polarized current injectedvia a point contact. Physical Review B , 83(5):054420,\nFebruary 2011. Publisher: American Physical Society.\n4J. M. D. Coey. Magnetism and Magnetic Materials .\nCambridge University Press, 2010.\n5N. Rossi, B. Gross, F. Dirnberger, D. Bougeard, and\nM. Poggio. Magnetic force sensing using a self-assembled\nnanowire. Nano Letters , 19(2):930–936, 2019.\n6Allan H. Morrish. The Physical Principles of Mag-\nnetism . Wiley, January 2001.\n7B. Gross, D. P. Weber, D. R¨ uffer, A. Buchter, F. He-\nimbach, A. Fontcuberta i Morral, D. Grundler, and\nM. Poggio. Dynamic cantilever magnetometry of in-\ndividual CoFeB nanotubes. Phys. Rev. B , 93(6):064409,\nFebruary 2016.\n8A. Hubert and R. Sch¨ afer. Magnetic Domains: The\nAnalysis of Magnetic Microstructures . 1998.\n9B. Gross, S. Philipp, E. Josten, J. Leliaert, E. Wetter-\nskog, L. Bergstr¨ om, and M. Poggio. Magnetic anisotropy\nof individual maghemite mesocrystals. Phys. Rev. B ,\n103:014402, Jan 2021.\n9"    },    {        "title": "2101.04649v1.Modeling_of_Thermal_Magnetic_Fluctuations_in_Nanoparticle_Enhanced_Magnetic_Resonance_Detection.pdf",        "content": "1 \n MODELING OF THERMAL MAGNETIC FLUCTUATIONS IN NANOPARTICLE ENHANCE D \nMAGNETIC RESONANCE  DETECTION  \n \nTahmid Kais ar1, Md Mahadi Rajib1, Hatem ElBidweihy2, Mladen  Barbic3 and Jayasimha Atulasimha1* \n \n1Department of Mechanical and Nuclear Engineering, Virginia Commonwealth University, Richmond, VA 23284, USA  \n2United States  Naval Academy , Electrical and Computer Engineering Department , Annapolis, Md 21402 , USA  \n3NYU Langone  Health , Tech4Health Institute , New York, NY 10010 , USA  \n* Corresponding author: jatulasimha@vcu.edu  \n \nAbstract  \nWe present a systematic numerical modeling investigation of magnetization dynamics and thermal \nmagnetic moment fluctuations of single magnetic domain nanoparticles in a configuration applicable  to \nenhancing  inductive  magnetic resonance  detection  signal to noise ratio ( SNR ). Previous proposals for \noriented anisotropic single magnetic domain nanoparticle amplification of magnetic flux in MRI coil  \nfocused only on the coil pick-up voltage signal enhancement. Here we extend the analysis to the numerical \nevaluation of  the SNR by modeling the inherent thermal magnetic noise introduced in to the detection coil \nby the insertion of such anisotropic nanoparticle -filled coil core.  We utilize  the Landau -Lifshitz -Gilbert \nequation under the Stoner -Wohlfarth single magnetic domain (macrospin ) assumption to simula te the \nmagnetization dynamics in such nanoparticles due to AC drive field as well as thermal noise . These \nsimulations are used to evaluate the nanoparticle configurations and shape effects on enhanc ing SNR . \nFinally , we explore  the effect of narrow band filtering of the broadband magnetic moment thermal \nfluctuation  noise on the SNR . Our results provide the impetus for relatively simple modifications to existing \nMRI systems for achieving enhanced detection SNR in scanners with mo dest polarizing magnetic fields .2 \n Introduction  \nSensitivity enhancement in magnetic resonance detection continues to be an important challenge due to the \nimportance of NMR and MRI in basic science, medical diagnostics, and materials characterization  [1-5]. \nAlthough many alternative methods of magnetic resonance detection have been developed over the years, \ninductive coil detection of magnetic resonance of precessing proton  nuclear magnetic moments is by far \nthe most common  [6]. The challenge in magnetic resonance detection stems from the low nuclear spin \npolarization at room temperature and laboratory static magne tic fields. An a dditional challenge is the \nfundamental requirement that the detector in magnetic resonance experiment needs to be compatible wit h \nand immune to the large polarizing DC magnetic field while also sufficiently sensitive to weak AC magnetic \nfields generated by the precessing nuclear spins. The inductive coil , operating on the principle of Faraday’s \nlaw of induction , satisfies this requ irement, and  enhancing the inductive coil detection SNR  has been \npursued through various  techniques  [7-9]. However, unlimited increase of the polarizing magnetic field is \ncost prohibitive, and technical challenges often inhibit  the development of mobile MR I units, their access, \nsustainability, and size. Therefore, solutions to achieving sufficient or improved SNR in NMR inductive \ncoil detection in lower magnetic fields and more accessible and compact configurations remains highly \ndesirable  [10]. \n \nSignal amplification by magnetic nanoparticle -filled coil core  \nAn idea has been put forward to increase the magnetic field flux from the sample through the coil by filling \nthe coil with a core of oriented anisotropic single domain magnetic nanoparticles  [11, 12 ], as shown in \nFigure 1. The sample and the inductive coil detector are both in the prototypical MRI environment of a \nlarge DC polarizing magnetic field along the z -axis, B Zdc=µ0HZdc, where µ 0 is the permeability of free space. \nThis field generates a fractional nuclear spin polarization of protons in the sample. Application of RF \nmagnetic fields along the x -axis is subsequently used to tilt the magnetic moment of the sample away from \nthe z -axis and generate precession of the sample magnetization around the z -axis at the proton NMR \nfrequency ω 0=γB Zdc, where γ is the proton nuclear gyromagnetic ratio. This sample moment precession \naround the z -axis generates a time -varying magnetic field B Xac=µ0HXac through the inductive coil detector \nof N turns and sensing area A along the x -axis. By Faraday’s law of induction, an AC signal voltage V at \nfrequency ω 0 generated across the coil terminals is:  \n𝑉=𝑁∙𝐴∙𝜔0∙𝐵𝑋𝑎𝑐                (1) 3 \n It is a well -known  practice in electromagnet design and ambient inductive detectors that a soft \nferromagnetic core within the coil significantly amplifies the magnetic flux through the coil  [13, 14 ]. The \nchallenge, however, in the configuration of NMR detection of Figure 1 is that the presence of the large \npolarizing magnetic field along the z -axis, B Zdc=µ0HZdc, would generally saturate the detection coil core \nmade of a soft ferromagnet along the z -axis and render the AC magnetic field due to proton precession  \nalong the x -axis of the coil ineffective . The solution proposed [11, 12 ] was that the oriented anisotropic \nmagnetic nanoparticles filling the coil core actually have an appreciable magnetic susceptibility along the \nx-axis precisely in the presence of a si gnificant DC magnetic field along the z -axis. The pick -up coil voltage \nis then:  \n𝑉=𝑁∙𝐴∙𝜔0∙𝜇0∙(𝐻𝑋𝑎𝑐+𝑀𝑋𝑎𝑐)            (2) \nWhere M Xac is the magnetization component of the nanoparticle -filled  coil core along the x -direction \n(sensing direction of the coil) due to the magnetic field B Xac=µ0HXac from the precess ing sample nuclear \nspin moment, M Xac=χRTHXac (where χ RT=ΔM Xac/ΔH Xac is defined as reversible transverse susceptibility).  \nTherefore, if the reversible transverse s usceptibility, χ RT, of the magnetic nanoparticle -filled  coil core along \nthe x -axis is significant at the large polarizing DC magnetic field B Zdc along the z -axis, the inductive coil \nsignal voltage will be enhanced. Various theore tical [15-17] and experimental investigations [18-25] of \nreversible transverse susceptibility in oriented magnetic nanostructures indeed reveal that its magnitude can \nbe appreciable and therefore might  provide a viable route for magnetic resonance signal a mplification , as \ndiagrammatically shown in Figure 1.  \nIn this work, we numerically evaluate the coil signal  voltage  by modeling individual nanoparticle magnetic \nmoment dynamics in the Stoner -Wohlfarth uniform magnetization approximation  [26]. More specifically, \nwe evaluate the AC nanoparticle moment along the x -axis in Figure 1 , m Xac, in the presence of a large DC \nmagnetic field BZdc=µ0HZdc along the z -axis and under the driven sample AC magnetic field B Xac=µ0HXac \nalong the x -axis. We assume for simplicity that the total coil core of volume, v C, is composed of “n” number \nof identical oriented single domain magnetic nanoparticles, and that each particle  x-component of the  AC \nmagnetic moment, m Xac, equally contributes to the coherent amplification of the pick-up voltage signal of  \nthe coil  detector . Therefore, the total coil AC voltage due to the magnetic nanoparticle core contribution is: \n𝑉=𝑁∙𝐴∙𝜔∙𝜇0∙𝑛∙𝑚𝑋𝑎𝑐\n𝑣𝐶              (3) \n 4 \n  \nFIG. 1.  Schematic diagram for NMR detection with magnetic nanoparticle filled coil core . \n \nNoise contri bution by magnetic nanoparticle -filled coil core  \nEssential  to the SNR consideration of any NMR experimental arrangement is the evaluation of the noise \nsources in  the signal chain. Since in this work we focus specifically on the magnetic nanoparticle -filled \ninductive coil detector, we will not consider the sample noise,  the a mplifier noise , and the Johnson noise \ncontributions which are addressed in numerous works  [27-31]. Any magnetic material placed inside the \ninductive detection coil  will introduce additional pick -up voltage noise due to intrinsic magnetization \nfluctuations  [32]. In our specific case, diagrammatically shown in Figure 1, these t hermal fluctuations of \nthe coil core magnetization along the x -axis, which we numerically model in detail in this work, generate a \ntotal mean squared coil noise voltage:  \n<𝑉2>=𝑁2∙𝐴2∙𝜔2∙𝜇02∙<𝑀𝑋2>            (4) \nWe assume for simplicity that the total coil core of volume, v C, is composed of n number of identical \noriented single domain magnetic nanoparticles , and that each particle magnetic moment, m, undergoes \nrandom uncorrelated thermal fluctuation. Therefore, the total mean squared coil noise voltage due to the \nmagnetic nanoparticle core is [33, 34 ]: \n5 \n <𝑉2>=𝑁2∙𝐴2∙𝜔2∙𝜇02∙𝑛∙<𝑚𝑋2>\n𝑣𝐶2            (5) \nWe note that the magnetic moment fluctuation phenomena has previously been investigated  in various spin \nsystems, materials , and detection modalities  [35-42]. However, we are not aware of any theoretical, \nnumerical, or experimental study where thermal magnetic moment fluctuations of single domain \nnanoparticles of the configuration of Figure 1 (where the large polarizing magnetic field is applied \nperpen dicular to the nanoparticle hard axis and the coil detection axis) and their contribution to the coil \nnoise voltage has been carried out.  \nIn this article, therefore, in order to assess the signal and the noise of the configuration of the magnetic \nresonanc e coil detector of Figure 1, we simulate  the room temperature magnetization dynamics of  a single \ndomain nanomagnet in the coil core. Importantly, we also model  the magnetic nanoparticle moment thermal \nfluctuation  under  the same conditions . We explore  such single nanomagnet dynamics for the macrospin \nStoner -Wolhfarth model uniform magneti zation oblate and prolate ellipsoid  geometries. This will explain  \nthe optimum nanoparticle orientation  and bias field needed to maxim ize SNR of the experimental \narrange ment of Figure 1. We then extend this analysis to scaling properties of an ensemble of nanomagnets \nand study the effect of applying a band -pass filter  to provide an estimate on the extent to which the insertion  \nof magnetic  particles  in the sensing c oil can enhance the limits of detection of magnetic fields due to proton \nspin resonances in MRI/NMR.  \n \nModelling particle magnetizati on dynamics  in the presence of room temperature thermal noise  \nModelling of the single particle magnetization dynamics was performed by solving the Landau -Lifshitz -\nGilbert (LLG) equation  [43]: \n𝜕𝑚⃗⃗ \n𝛿𝑡=(−𝛾\n1+𝛼2)[𝑚⃗⃗ ×𝐻𝑒𝑓𝑓+𝛼{𝑚⃗⃗ ×(𝑚⃗⃗ ×𝐻⃗⃗ 𝑒𝑓𝑓)}]                                                                                (6) \nIn equation (6 ), 𝛾 is the gyromagnetic ratio (rad/Ts), 𝛼 is the Gilbert damping coefficient and 𝑚⃗⃗  is the \nnormalized magnetization vector, found by normalizing the magnetization vector ( 𝑀⃗⃗ ) with respect to \nsaturation magnetization (M s). The effective field (𝐻⃗⃗ 𝑒𝑓𝑓) was obtained from the derivative of the total  \nenergy (E) of the system with respect to the magnetization ( 𝑀⃗⃗ ): \n𝐻⃗⃗ 𝑒𝑓𝑓=−1\n𝜇0𝛺𝑑𝐸\n𝑑𝑀⃗⃗ +𝐻⃗⃗ 𝑡ℎ𝑒𝑟𝑚𝑎𝑙                                                       (7) 6 \n where 𝜇0 is the permeability of the vacuum and 𝛺 is the volume of the nanomagnet.  \nThe total potential energy in equation (7 ) is given by:  \n𝐸=𝐸𝑠ℎ𝑎𝑝𝑒  𝑎𝑛𝑖𝑠𝑜𝑡𝑟𝑜𝑝𝑦 +𝐸𝑧𝑒𝑒𝑚𝑎𝑛                     (8) \nwhere 𝐸𝑠ℎ𝑎𝑝𝑒  𝑎𝑛𝑖𝑠𝑜𝑡𝑟𝑜𝑝𝑦  is the shape anisotropy due to the prolate or oblate shape and can be calculated \nfrom the following equation:  \n𝐸𝑠ℎ𝑎𝑝𝑒  𝑎𝑛𝑖𝑠𝑜𝑡𝑟𝑜𝑝𝑦 =(𝜇0\n2)𝛺[𝑁𝑑_𝑥𝑥𝑀𝑥2+𝑁𝑑_𝑦𝑦𝑀𝑦2+𝑁𝑑_𝑧𝑧𝑀𝑧2]           (9) \n \nwith N d_xx, N d_yy, and N d_zz representing the demagnetization factors in the respective directions of the \nnanomagnet and 𝐸𝑧𝑒𝑒𝑚𝑎𝑛  is the potential energy of nanomagnet for an external magnetic field ( 𝐻⃗⃗ ), given \nby: \n𝐸𝑧𝑒𝑒𝑚𝑎𝑛=−𝜇0𝛺𝐻⃗⃗ .𝑀⃗⃗           (10) \nThe thermal field 𝐻⃗⃗ 𝑡ℎ𝑒𝑟𝑚𝑎𝑙  is modeled as a random field incorporated  in the manner of [44, 45] \n                                                  𝐻⃗⃗ 𝑡ℎ𝑒𝑟𝑚𝑎𝑙  (t)=√2𝑘𝑇𝛼\n𝜇0𝑀𝑠𝛾𝛺∆𝑡  𝐺 (𝑡)                                                                  (11) \nIn equation (11 ), 𝐺 (𝑡) is a Gaussian distribution with zero mean and unit variance in each Cartesian \ncoordinate axis , k is the Boltzman constant, T is the temperature, 𝑀𝑠 is saturation magnetization and γ is \nGyromagnetic ratio.  ∆t is the time -step used in the numerical solution of equation ( 6) and was chosen to be  \n100 fs . This wa s chosen to be small enough to ensure that all results are independent of the time step . We \nemployed Euler method to solve the differential equations  [44, 45].  \nTable I lists the values of the material properties of the nanomagnet . \nTable 1: Material properties  of Co Fe [46] \nParameters  Material  \nProperty  \nSaturation Magnetization (M s) 1.6x 106 (A/m)  7 \n   \n \n \n \nResults and discussions  \nConsider a prolate ellipsoid  of volume ~5000 nm3 (100nm×10nm×10nm)  shown in Fig ure 2(a) where we \nassume the sample proton spin precession produces a magnetic field along the easy (long) x-axis of the \nnanomagnet while DC bias field is applied along one of the hard (short) axes, viz. the z -axis. When the DC \nbias field is zero there are two deep energy wells at θ =0, 180˚. When the magnetization is in one of these \nstates, the magnetization response to an AC magnetic field along the x -axis (a simplified representation of \nthe signal at the pick -up coil due to proton spin precession  of Figure 1) is very small as the magnetizati on \nis in this deep potential  as seen in Fig ures 3(a)-(b) and Table II . The corresponding magnetization \nfluctuation due to room temperature thermal noise (that is modelled as a random effective magnetic field, \nsee equation 7) is also very small.  \nAs the DC field increases along the z -axis to the point H dc=H c (the DC b ias field is equal to the coercive \nfield H c) the mean magnetization orientation is  at 90˚. However, the  potential well at 90˚ (Fig ure 2(a)) is \ncharacterized by a flat energy profile where the energy is nearly independent of the polar angle θ, around \nθ=90˚. This leads to a large magnetization response along the x -axis to an applied AC magnetic field along \nthe x -axis (Figures 3(a)-(b) and Table II) in the presence of a large  DC magnetic field along the z -axis. \nImportantly, since  the energy profile is flat  in this configuration,  the particle moment  fluctuation due to \nroom tempera ture thermal noise is also high. Nevertheless,  we find the Signal to Noise Ratio (SNR) is \nhighest at H dc=H c. In fa ct, the magnetization response and the SNR ratio are  found to increase \nmonotonically with the applied bias field up to H c (see Table II  based on the select ed simulations shown in \nFigures 3(a)-(b) and all the simulations shown in Supplementary Fig ure S1) and then decrease s as H dc>H c \n(for example at H dc=1.25H c in Table II  and Supplementary Fig ure S1) due to an energy well deepening at \nθ=90˚ for H dc>H c (Figure 2(a)).  We note that the SNR is calculated in the following manner. First, we do \nnot include any thermal noise and perform the LLG simulation to determi ne the magnetization response  \ndue to only the AC field from sample proton spin precession . Then we perform another LLG simulation \nwith no signal and study  the magnetization fluctuation due to the thermal noise only. The ratio of th e rms \nvalues of the response due to the sample precessing field  and that due to noise is defined as the SNR.  Gilbert Damping (α)  0.05 \nGyromagnetic Ratio (γ)  2.2x105(m/As ) 8 \n We note that the AC drive magnetic field along the z -axis has an ampli tude of 800 A/m (10 Oe or 1 mT) \nfor all cases discussed in this work , which is much larger tha n the typical signal due to proton spins which \nmay be several orders of magnitude smaller . However, we chose a higher drive amplitude as we wanted to \nelicit a  reasonable magnetization response  that would be easily visible in the plots and result in reasonable \nSNR ratios for a single nanomagnet. In practise, the number of nanomagnets placed in the d etector coil \ncould be over n~1012 or more resulting in sub nT se nsing capability as discussed later.  Furthermore, the \nproton resonance of 42.5 MHz occurs at a DC field ~ 1T, so the frequency of spin precession would be \ndifferent at different anisotropy fields as the Zeeman (DC bias field) seen by the nanoparticles is t he same \nas that seen by the proton spins in the MRI sample. However, to keep the simulations consistent, we assume \nsignal due to proton resonance as 42.5 MHz for all cases as this would not change our qualitative findings . \nNext , we c onsider a prolate ellipsoid shown in Fig ure 2(b) where we assume the proton spin precession \nproduces a magnetic field along one of the hard (short ) axes, viz. the x-axis of the nanomagnet while a DC \nbias field is applied along the long (easy) z-axis. Initially, the magnetization points downward (θ=180˚) and \nat H dc=0 is in a deep potential well (Fig ure 2(b)). As the magnetic field applied along the +z direction  (θ=0˚)  \nincreases, the energy well around θ=180˚ is flattened.  Thus, for higher field, the magnetization resp onse \nincreases, as does the magnetization fluctuation due to thermal noise as shown in Table II (detailed \nsimulations are all show n in the supplementary Fig ure S2). However, the shallow wells improve the \nmagnetization response to the drive magnetic field more than the increased magnetization fluctuations due \nto thermal noise (as in the prior case) and increase the overall SNR ratio (Table II).  \nHowever, as one approaches H dc=H c the SNR drops significantly. This is because the energy profile is flat \nat θ=180 ˚ but even small perturbations from this angle make the magnetization switch and rotate to the +z -\naxis (θ=0˚). Once it reaches this state, the energy well profile at Hdc=Hc at θ=0˚ is a deeper  than the well at \nHdc=0. The reason is that t he Zeeman energy due to field along +z -axis makes the already deep shape \nanisotropy well even deeper at θ=0˚ reducing both the magnetization response and the magnetization \nfluctuations due to thermal noise , as well as  the SNR ratio. Thus, the best SNR is s een at H dc<H c (see the \nhigh SNR at 0.875H c in Table II) but close to H c.  9 \n  \n FIG. 2. Energy profile for various DC bias magnetic field (H dc) for (a) Prolate: Bias field along m inor axis, \n(b) Prolate: Bias field along m ajor axis and ( c) Oblate:  Bias field along minor axis . \nFinally , we consider the case of an oblate ellipsoid  of volume ~5000 nm3 (40nm× 40nm×6nm), as same as \nthe volume of prolate ellipsoid,  shown in Fig ure 2(c) where we assume the proton spin precession produces \na magnetic field along one of the easy (long)  x-axis of the nanomagnet while DC bias field is applie d along \nthe hard z-axis. The symmetry of the problem is such that at H dc=0 the magnetization is free to rotate in the \nx-y plane as there is no energy barrier to such rotation. As  Hdc increases, the magnetization is still free to \nmove in a cone of the x-y plane at a specific angle to the z -axis that decreases with increasing H dc, finally \ncoinciding with it when H dc=H c. Thus, at a range of DC bias fields (for example from Hdc=0.25H c to \nHdc=0.75 H c) we observe a high SNR > 1.4 when a single nanomagnet is driven by an AC magnetic fie ld. \nThis is due to the combination of high magnetization response given the symmetry and noise limited by \npresence of the DC bias field. The simulations of magnetic response to the AC magnetic field and \nmagnetization fluctuations due to random thermal noise are respectively shown in Fig ures 3(c)-(d) \ncomparing the H dc=0 and H dc=0.625H c cases with all other bias field cases shown in Fig ure S3 of the \nsupplement.  \nIn summary, as far as the SNR is concerned, the prolate ellipsoid with DC bias magnetic field along the \nhard axis (Fig ure 1(a)) is the better choice over the prolate ellipsoid with DC bias mag netic field along the \neasy axis. However, the oblate geometry and configuration shown i n Fig ure 2(c) produce s the highest SNR \n(more than twice higher than the highest SNR for the prolate ellipsoid configuration in Fig ure 2(a), and \nmore than 10 times for the prolate ellipsoid configuration of Figure 2(b)). What makes this oblate \nconfiguration even more attractive to detection coils in MRI/NMR applications is that the high SNR \nperformance is seen over a large range of DC bias field (e.g. Hdc=0.25H c to Hdc=0.75 H c), making it \nattractive for broad range of MR I scanner fields.  \n10 \n                    \nFIG. 3. Magnetization dynamics with (a) 800 A/m , 42.5 MHz field with no thermal noise for prolate \nnanomagnet with bias along minor axis and (b) only t hermal noise at H dc=0 and H dc=H c. Magnetization \ndynamics with (c) 800 A/m , 42.5 MHz field with no thermal noise for oblate nanomagnet with bias along \nminor axis and (d)  only thermal noise at H dc=0 and H dc= 0.625 Hc. (e) SNR vs Hdc/Hc for prolate and oblate \ncases  and (f) zoomed version of SNR  vs Hdc/Hc for prolate nanomagnet with bias along major axis.  \n \n \nTable II: Magnetization oscillations in the single nanomagnet of different geometries for different values \nof DC bias magnetic field  (based on simulations shown in Figure  3 of the main paper and Figures S1, S2, \nS3 of the supplement) .  \n11 \n  Cases  Value of bias field \n(Hdc) RMS normalized \nmagnetization  \n(M/M s) for a \nsinusoidal magnetic \nsignal  of 800 A/m \n(10 Oe) amplitude  RMS normalized \nmagnetization  \n(M/M s) due to \nthermal noise only \n(no signal)      SNR          \n(defined here as \nratio of Column 2 \nand Column 3).  \nProlate applying \nbias field along \nminor axis  0 1.66x10-6 6.45x10-4 .003 \n0.25 Hc 2.82x10-4 .0075  .05 \n0.5 Hc 0.002  0.0103  0.2885  \n0.75 Hc .0128  0.0202  0.63 \nHc 0.1035  0.1464  .7042  \n1.25 Hc 0.009  .0484  0.193  \nProlate applying \nbias field along \nmajor axis  0 9.3x10-4 0.025  0.03 \n0.5 Hc .0017  .034 0.0465  \n0.75 Hc .0032  .05 .0653  \n0.875  Hc 0.0063  .0662  0.09 \nHc .0018  .025 .075 \nOblate applying \nbias field along \nminor axis  0.25 Hc .94 .68 1.4 \n0.5 Hc 0.843  0.582  1.51 \n0.625  Hc 0.76 0.45 1.71 \n0.75 Hc 0.645  0.46 1.44 \n Hc .0934  .0921  0.95 12 \n Finally , we take this  best-case nanoparticle (oblate ellipsoid at H dc=0.625 Hc with SNR=1.71 ) and \ninvestigate  if the SNR can be further improved by applying a narrow band filter around 42.5 MHz. The \nrationale is that the magnetization response driven by the magnetic field due to proton spin precision at 1 \nTesla applied DC field is dominant around 42.5 MHz while the magnetization fluctuatio ns driven by \nthermal noise is broad band as evidence d by the singe -sided amplitude spectrum shown in Fig ure 4(a). \nWhen a  band -pass filter (42-44) MHz  was applied,  the SNR improved to ~8 . \n \nFIG. 4. (a)  Frequency spectrum of  signal + noise, before filtering (b) Frequency spectrum of  signal + noise, \nafter filtering . Both cases for oblate nanomagnet . \n \n \n13 \n Scaling of SNR with coil core nanoparticle number  \nWhile the SNR with a n AC magnetic field of 10 Oe (equivalent of 1mT) shows a SNR ~8 for optimal \nconditions, this was assuming a single nanomagnet. However, for a large number n of nanomagnets  within \nthe core , the magne tization response increases as n  (as more nanomagnets coher ently contribute to more \nmagnetic moment and therefore greater induced voltage in the coil) according to Equation 3 , while the \nmagnetic  noise would only increase as √n  (as the thermal noise induced fluctuations of nanomagnets have \na random phase)  according  to Equation 5 , leading to a SNR increase of n/√n =√n .  \nNow consider, a square detection coil  (for ease of calculation) of 2 cm side . Considering the pitch between \nnanomagnets ~ 200 nm to avoid significant dipole coupling 10 billion nanomagnets can be accommodated \nin a single layer of 2 cm by 2 cm. Additionally, as the single nanoparticle layer thickness is ~ 6 nm the \naverage distance between two such layers  can be ~25 nm. Thus,  400,000 such magnetic nanoparticle layers \ncan be accommodated in 1 cm coil thickness . Consequently, n=40×1014 nanomagnets can be incorporated \ninto the sensing coil leading to an increase in SNR from 8 to  ~5×108. In other words, with a SNR of 5, one \ncould detect an AC magnetic field of 1/(108) mT, i.e. an AC magnetic field of 10 pT. Hence, insertion of \nsuch nanomagnets could allow  detectability to ~ 10 pT  and perhaps even better  depending on the density \nwith which nanoparticles that can be inserted in the detection coil . However, it should be n oted that pinning \nsites, i nhomogeneities, etc can impede  magnetization dynamics  [47], create a phase difference, etc , thus \ndecreasing detectability.  \nThe key point is merely  filling the detection coil with a soft (high permeability) core would not help as the \ncore would be saturated under the high DC bias fields used for MRI. However, by using appropriate \ngeometry nanoparticles that are still responsive to AC fields from prot on precession under strong DC fields, \nthe coil detection sensitivity can be enhanced.   \nConclusion  \nOur numerical investigation of nanoparticle magnetization dynamics and LLG of room temperature thermal \nmoment fluctuations confirm the initial zero temperature proposal for nanoparticle based amplification of \nNMR signal. Such consideration of the thermal fluctuation allows us to predict not just signal amplification \nvalues, but realistic room temperature SNR values. Our analysis suggests  specific  proposal for using \noriented anisotropic  oblate ellipsoid particles as optimal for achieving SNR improvements over \nconventional air -filled MRI coils. Much will depend on the quality of the particles used in the coil core: \nshape uniformity, quality of parti cles orientation within the core, smoothness of the particles and surface \npinning sites (that degrade the effect of magnetization dynamics ), and uniformity of the nanoparticle aspect 14 \n ratio (which determines where the particle has a peak in transverse susce ptibility ). Further consideration \nwould have to be made of the effect of magnetic particles on the field uniformity within the nuclear spin \nsample that is being imaged, since such field distortions will broaden the sample spin resonance and will \nhave to be  address in both the MRI scanner bore designs that incorporate th e nanoparticles within the coils , \nas well as in the pulse sequences that deal with such inhomogeneous broadening. 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Interfaces, 10, 20, 17455 (20 18). \n \n \n 17 \n Supplemental Information  \nMODELING OF THERMAL MAGNETIC FLUCTUATION S IN NANOPARTICLE \nENHANCED MAGNETIC RE SONANCE DETECTION  \nTahmid Kaisar1, Md Mahadi Rajib1, Hatem ElBidweihy2, Mladen  Barbic3 and Jayasimha Atulasimha1* \n1Dept of Mechanical and Nuclear Engineering, Virginia Commonwealth University, Richmond, VA, USA  \n2United States Naval Academy, Electrical and Computer Engineering Department, Annapolis, M D, USA  \n3NYU Langone Health, Tech4Health Institute, New York, NY 10010, USA  \n* Corresponding author: jatulasimha@vcu.edu  \n \nIn the main paper, we had plott ed the normalized magnetization (M/M s) for (i) a sinusoidal magnetic signal \nof 800 A/m (10 Oe) amplitude and (ii) thermal noise . However, these plots were provided for only : \n1. Two select ed cases of DC bias magnetic field for prolate  nanomagnet with bias along minor axis ; and \n2. Two select ed cases of DC bias magnetic field for oblate nanomagnet with bias along minor axis . \n \nHowever, in the  supplement we provide figures for normalized magnetization (M/M s) for (i) a sinusoidal \nmagnetic signal of 800 A/m (10 Oe) amplitude and (ii) thermal noise ; for all cases of DC bias magnetic \nfields for:  \n1. All cases of DC bias magnetic field for prolate  nanomagnet with bias along minor axis  (Fig S1);   \n2. All cases of DC bias magnetic field for prolate  nanomagnet with bias along m ajor axis (Fig S2);  and \n3. All cases of DC bias magnetic field for oblate nanomagnet with bias along minor axis  (Fig S3).  \nThese suppl ementary plots form the basis for Table 2 as well as the Signal to Noise Ratio (SNR) plot Fig 3 \n(e) in the main paper.  \n \n \n \n \n \n \n \n \n \n \n 18 \n                                Prolate Nanomagnet with Bias along Minor Axis  \n \nFig S1.  Normalized magnetization  in a prolate  nanomagnet for different bias fields  along minor -axis:          \nFor s inusoidal magnetic signal of 800 A/m (10 Oe) amplitude without thermal noise (plots on left) and \ncorresponding case with thermal noise  only and no signal (plots on right).  \n19 \n                                  Prolate Nanomagnet with Bias along M ajor Axis  \nFig S 2. Normalized magnetization  in a prolate  nanomagnet for different bias fields along majo r-axis: For \nsinusoidal magnetic signal of 800 A/m (10 Oe) amplitude without thermal noise (plots on left) and \ncorresponding case with thermal noise  only and no signal (plots on right).  \n20 \n                                Oblate Nanomagnet with Bias along Minor Axis   \n \nFig S 3. Normalized magnetization  in an oblate  nanomagnet for different bias fields along minor -axis:          \nFor s inusoidal magnetic signal of 800 A/m (10 Oe) amplitude without thermal noise (plots on left) and \ncorresponding case with thermal noise  only and no signal (plots on right).  \n"    },    {        "title": "1504.06042v1.Magnetization_damping_in_noncollinear_spin_valves_with_antiferromagnetic_interlayer_couplings.pdf",        "content": "arXiv:1504.06042v1  [cond-mat.mes-hall]  23 Apr 2015Magnetizationdamping innoncollinearspinvalveswithant iferromagnetic interlayer couplings\nTakahiro Chiba1, Gerrit E. W. Bauer1,2,3, and Saburo Takahashi1\n1Institute for Materials Research, Tohoku University, Send ai, Miyagi 980-8577, Japan\n2WPI-AIMR, Tohoku University, Sendai, Miyagi 980-8577, Jap an and\n3Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n(Dated: October 29, 2018)\nWe study the magnetic damping in the simplest of synthetic an tiferromagnets, i.e. antiferromagnetically\nexchange-coupled spin valves in which applied magnetic fiel ds tune the magnetic configuration to become\nnoncollinear. We formulate the dynamic exchange of spin cur rents in a noncollinear texture based on the spin-\ndiffusiontheorywithquantum mechanicalboundaryconditionsa ttheferrromagnet|normal-metal interfacesand\nderive the Landau-Lifshitz-Gilbert equations coupled by t he static interlayer non-local and the dynamic ex-\nchange interactions. We predict non-collinearity-induce d additional damping that can be sensitively modulated\nbyanapplied magnetic field. The theoretical results compar e favorablywithpublished experiments.\nI. INTRODUCTION\nAntiferromagnets (AFMs) boast many of the functionali-\ntiesofferromagnets(FM)thatareusefulinspintroniccirc uits\nanddevices: Anisotropicmagnetoresistance(AMR),1tunnel-\ning anisotropicmagnetoresistance(TAMR),2current-induced\nspintransfertorque,3–8andspincurrenttransmission9–11have\nall been found in or with AFMs. This is of interest because\nAFMshaveadditionalfeaturespotentiallyattractivefora ppli-\ncations. InAFMsthetotalmagneticmomentis(almost)com-\npletely compensated on an atomic length scale. The AFM\norder parameter is, hence, robust against perturbations su ch\nas external magnetic fields and do not generate stray fields\nthemselveseither. AspintronictechnologybasedonAFM el-\nementsisthereforeveryattractive.12,13Drawbacksarethedif-\nficulty to controlAFMs by magnetic fields and much higher\n(THz)resonancefrequencies,14–16whicharedifficulttomatch\nwith conventional electronic circuits. Man-made magnetic\nmultilayers in which the layer magnetizations in the ground\nstate isorderedin anantiparallelfashion,17i.e. so-calledsyn-\nthetic antiferromagnets,donot su ffer fromthis drawbackand\nhave therefore been a fruitful laboratory to study and modu-\nlate antiferromagnetic couplings and its consequences,18but\nalso found applications as magnetic field sensors.19Trans-\nport in these multilayers including the giant magnetoresis -\ntance (GMR)20,21are now well understood in terms of spin\nand charge diffusive transport. Current-induced magnetiza-\ntionswitchinginF|N|Fspinvalvesandtunneljunctions,22has\nbeen a game-changer for devices such as magnetic random\naccess memories(MRAM).23A keyparameterof magnetiza-\ntiondynamicsisthemagneticdamping;asmalldampinglow-\nersthethresholdofcurrent-drivenmagnetizationswitchi ng,24\nwhereasalargedampingsuppresses“ringing”oftheswitche d\nmagnetization.25\nMagnetization dynamics in multilayers generates “spin\npumping”, i.e. spin current injection from the ferromagnet\ninto metallic contacts. It is associated with a loss of an-\ngular momentum and an additional interface-related magne-\ntization damping.26,27In spin valves, the additional damp-\ning is suppressed when the two magnetizations precess in-\nphase, while it is enhanced for a phase di fference ofπ(out-\nof-phase).27–30This phenomenon is explained in terms of a\n“dynamic exchange interaction”, i.e. the mutual exchange o fnon-equilibriumspin currents,which shouldbe distinguis hed\nfrom(butcoexistswith)theoscillatingequilibriumexcha nge-\ncoupling mediated by the Ruderman-Kittel-Kasuya-Yosida\n(RKKY) interaction. The equilibriumcoupling is suppresse d\nwhenthespacerthicknessisthickerthantheelasticmean-f ree\npath,31,32while the dynamiccouplingise ffective onthe scale\noftheusuallymuchlargerspin-flipdi ffusionlength.\nAntiparallel spin valves provide a unique opportunity to\nstudy and control the dynamic exchange interaction between\nferromagnets through a metallic interlayer for tunable mag -\nnetic configurations.33,34An originallyantiparallel configura-\ntionisforcedbyrelativelyweakexternalmagneticfieldsin toa\nnon-collinearconfigurationwith a ferromagneticcomponen t.\nFerromagneticresonance(FMR)andBrillouinlightscatter ing\n(BLS) are two useful experimentalmethodsto investigateth e\nnature and magnitude of exchange interactions and magnetic\ndamping in multilayers.35Both methods observe two reso-\nnances, i.e. acoustic (A) and optical (O) modes, which are\ncharacterizedbytheirfrequenciesandlinewidths.36,37\nTimopheev etal.observedaneffectoftheinterlayerRKKY\ncoupling on the FMR and found the linewidth to be a ffected\nby the dynamic exchange coupling in spin valves with one\nlayerfixed by the exchange-biasof an inert AFM substrate.38\nThey measured the FMR spectrum of the free layer by tun-\ning the interlayer coupling (thickness) and reported a broa d-\nening of the linewidth by the dynamic exchange interaction.\nTaniguchi et al.addressed theoretically the enhancement of\nthe Gilbert damping constant due to spin pumping in non-\ncollinear F|N|F trilayer systems, in which one of the magne-\ntizations is excited by FMR while the other is o ff-resonant,\nbutadoptaroleasspinsink.39Thedynamicsofcoupledspin\nvalvesinwhichbothlayermagnetizationsarefreetomoveha s\nbeencomputedby oneof us29and bySkarsvåg et al.33,49but\nonly for collinear (parallel and antiparallel) configurati ons.\nCurrent-induced high-frequency oscillations without app lied\nmagnetic field in ferromagnetically coupled spin valves has\nbeenpredicted.40\nInthepresentpaper,wemodelthemagnetizationdynamics\nof the simplest of synthetic antiferromagnets, i.e. the ant i-\nferromagnetically exchange-coupled spin valve in which th e\n(in-plane) ground state magnetizations are for certain spa cer\nthicknesses ordered in an antiparallel fashion by the RKKY\ninterlayercoupling.41We focusonthecoupledmagnetization2\nmodes in symmetric spin valves in which in contrast to pre-\nvious studies, both magnetizations are free to move. We in-\nclude static magnetic fields in the film plane that deform the\nantiparallelconfigurationintoacantedone. Microwaveswi th\nlongitudinal and transverse polarizations with respect to an\nexternalmagneticfieldthenexciteAandOresonancemodes,\nrespectively.31,42–46We develop the theory for magnetization\ndynamics and damping based on the Landau-Lifshitz-Gilbert\nequationwithmutualpumpingofspincurrentsandspintrans -\nfer torques based on the spin di ffusion model with quantum\nmechanical boundary conditions.27,47,48We confirm28,49that\nthe additional damping of O modes is larger than that of the\nA modes. We report that a noncollinear magnetization con-\nfigurationinducesadditionaldampingtorquesthat to the be st\nof ourknowledgehavenotbeen discussedin magneticmulti-\nlayers before.50The external magnetic field strongly a ffects\nthe dynamics by modulating the phase of the dynamic ex-\nchange interaction. We compute FMR linewidths as a func-\ntion of applied magnetic fields and find good agreement with\nexperimental FMR spectra on spin valves.31,32The dynam-\nics of magnetic multilayers as measured by ac spin trans-\nfer torque excitation30reveals a relative broadening of the O\nmodes linewidths that is well reproduced by our spin valve\nmodel.\nIn Sec. II we present our model for noncollinear spin\nvalves based on spin-di ffusiontheory with quantum mechan-\nical boundary conditions. In Sec. III, we consider the mag-\nnetization dynamics in antiferromagnetically coupled non -\ncollinear spin valves as shown in Fig. 1(b). We derive the\nlinearized magnetization dynamics, resonance frequencie s,\nand lifetimes of the acoustic and optical resonance modes in\nSec. IV. We discuss the role of dynamicspin torqueson non-\ncollinear magnetization configurations in relation to exte rnal\nmagnetic field dependence of the linewidth. In Sec. V, we\ncompare the calculated microwave absorption and linewidth\nwith published experiments. We summarize the results and\nendwiththeconclusionsinSec. VI.\nII. SPINDIFFUSIONTRANSPORTMODEL\nWe consider F1|N|F2 spin valves as shown in Fig. 1(a), in\nwhichthemagnetizations MjoftheferromagnetsF j(j=1,2)\nare coupled by a antiparallel interlayer exchange interact ion\nand tilted towards the direction of an external magnetic fiel d.\nApplied microwaves with transverse polarizations with re-\nspect to an external magnetic field cause dynamics and, via\nspinpumping,spincurrentsandaccumulationsinthenormal -\nmetal (NM) spacer. The longitudinal component of the spin\naccumulation diffuses into and generates spin accumulations\ninF thatwe showtobesmall later,butdisregardinitially. L et\nusdenotethepumpedspincurrent JP\nj,whileJB\njisthediffusion\n(back-flow)spin currentdensity inducedby a spin accumula-NM zy\nx\nF1 F2 0 dN\nJ1PJ2P\nJ2BJ1Bµsθθ\n(c) Acoustic mode (d) Optical mode (a)\nm1 m2\nH\n*OUFSMBZFS\u0001DPVQMJOH \n(b)\n(b) H\nhx hy\nFIG.1. (a) Sketch of the sample withinterlayer exchange-co uplings\nillustrating the spin pumping and backflow currents. (b) Mag netic\nresonance in an antiferromagnetically exchange-coupled s pin valve\nwith a normal-metal (NM) film sandwiched by two ferromagnets\n(F1,F2)subjecttoamicrowave magneticfield h. Themagnetization\nvectors (m1,m2) are tilted by an angle θin a static in-plane mag-\nneticfield Happliedalongthe y-axis. Thevectors mandnrepresent\nthesum anddifference ofthe twolayer magnetizations, respectively.\n(c) and (d): Precession-phase relations for the acoustic an d optical\nmodes.\ntionµsjin NM,bothat theinterfaceF j, with27,51\nJP\nj=Gr\nemj×/planckover2pi1∂tmj, (1a)\nJB\nj=Gr\ne/bracketleftBig/parenleftBig\nmj·µsj/parenrightBig\nmj−µsj/bracketrightBig\n, (1b)\nwheremj=Mj//vextendsingle/vextendsingle/vextendsingleMj/vextendsingle/vextendsingle/vextendsingleis the unit vector along the magnetic\nmoment of F j(j=1,2). The spin current througha FM |NM\ninterface is governed by the complex spin-mixing conduc-\ntance (per unit area of the interface) G↑↓=Gr+iGi.27The\nreal component Grparameterized one vector component of\nthe transverse spin-currentspumped and absorbed by the fer -\nromagnets. The imaginary part Gican be interpreted as an\neffective exchange field between magnetization and spin ac-\ncumulation, which in the absence of spin-orbit interaction is\nusuallymuchsmallerthantherealpart,forconductingaswe ll\nasinsultingmagnets.523\nThediffusionspin-currentdensityin NMreads\nJs,z(z)=−σ\n2e∂zµs(z), (2)\nwhereσ=ρ−1is the electrical conductivity and µs(z)=\nAe−z/λ+Bez/λthe spin accumulationvectorthat is a solution\nofthespindiffusionequation∂2\nzµs=µs/λ2,whereλ=√Dτsf\nis the spin-diffusion length, Dthe diffusion constant, and τsf\nthe spin-flip relaxation time. The vectors AandBare de-termined by the boundary conditions at the F1 |NM (z=0)\nand F2|NM (z=dN) interfaces: Js,z(0)=JP\n1+JB\n1≡Js1and\nJs,z(dN)=−JP\n2−JB\n2≡−Js2. Theresultingspin accumulation\ninN reads\nµs(z)=2eλρ\nsinh/parenleftBigdN\nλ/parenrightBig/bracketleftBigg\nJs1cosh/parenleftBiggz−dN\nλ/parenrightBigg\n+Js2cosh/parenleftbiggz\nλ/parenrightbigg/bracketrightBigg\n,(3)\nwithinterfacespin currents\nJs1=ηS\n1−η2δJP\n1+η2/parenleftBig\nm2·δJP\n1/parenrightBig\n1−η2(m1·m2)2m1×(m1×m2), (4a)\nJs2=−ηS\n1−η2δJP\n2+η2/parenleftBig\nm1·δJP\n2/parenrightBig\n1−η2(m1·m2)2m2×(m2×m1). (4b)\nHere\nδJP\n1=JP\n1+ηm1×(m1×JP\n2), (5a)\nδJP\n2=JP\n2+ηm2×(m2×JP\n1), (5b)\nS=sinh(dN/λ)/grandη=gr/[sinh(dN/λ)+grcosh(dN/λ)]\nare the efficiency of the back flow spin currents, and gr=\n2λρGris dimensionless. The first terms in Eqs. (4a) and (4b)\nrepresent the mutual pumping of spin currents while the sec-\nondtermsmaybeinterpretedasa spincurrentinducedbythe\nnoncollinear magnetization configuration, including the b ack\nflowfromthe NMinterlayer.\nIII. MAGNETIZATIONDYNAMICSWITH DYNAMIC\nSPINTORQUES\nWe consider the magnetic resonance in the non-collinear\nspin valve shown in Fig. 1. The magnetization dynamics are\ndescribedbytheLandau-Lifshitz-Gilbert(LLG)equation,\n∂tm1=−γm1×Heff1+α0m1×∂tm1+τ1,(6a)\n∂tm2=−γm2×Heff2+α0m2×∂tm2−τ2.(6b)\nThe first term in Eqs. (6a) and (6b) represents the torque in-\nducedbytheeffectivemagneticfield\nHeff1(2)=H+h(t)−4πMsm1(2)zˆz+Jex\nMsdFm2(1),(7)\nwhich consists of an in-plane applied magnetic field H,\na microwave field h(t), and the demagnetization field\n−4πMsm1(2)zˆzwith saturation magnetization Ms. The inter-\nlayer exchange field is Jex/(MsdF)m2(1)with areal density of\ntheinterlayerexchangeenergy Jex<0(forantiferromagnetic-\ncoupling) and F layer thickness dF. The second term is the\nGilbert dampingtorque that governsthe relaxationcharact er-\nized byα0itowards an equilibrium direction. The third term,τ1(2)=γ/planckover2pi1/(2eMsdF)Js1(2), is the spin-transfertorque induced\nby the absorption of the transverse spin currents of Eqs. (4a )\nand (4b), andγandα0are the gyromagnetic ratio and the\nGilbert dampingconstant of the isolated ferromagneticfilm s,\nrespectively. SometechnicaldetailsofthecoupledLLGequ a-\ntionsarediscussedinAppendixA.Introducingthetotalmag -\nnetizationdirection m=(m1+m2)/2andthedifferencevector\nn=(m1−m2)/2,theLLG equationscanbewritten\n∂tm=−γm×(H+h)\n+2πγMs(mzm+nzn)׈z\n+α0(m×∂tm+n×∂tn)+τm, (8a)\n∂tn=−γn×/parenleftBigg\nH+h+Jex\nMsdFm/parenrightBigg\n+2πγMs(nzm+mzn)׈z\n+α0(m×∂tn+n×∂tm)+τn,(8b)\nwhere the spin-transfer torques τm=(τ1+τ2)/2 andτn=\n(τ1−τ2)/2become\nτm/αm=m×∂tm+n×∂tn\n+2ηm·(n×∂tn)\n1−ηCm+2ηn·(m×∂tm)\n1+ηCn,(9a)\nτn/αn=m×∂tn+n×∂tm\n−2ηm·(n×∂tm)\n1+ηCm−2ηn·(m×∂tn)\n1−ηCn,(9b)\nandC=m2−n2,while\nαm=α1gr\n1+grcoth(dN/2λ), (10a)\nαn=α1gr\n1+grtanh(dN/2λ), (10b)\nwithα1=γ/planckover2pi12/(4e2λρMsdF).4\nIV. CALCULATIONANDRESULTS\nWe consider the magnetization dynamics excited by lin-\nearly polarized microwaves with a frequency ωand in-plane\nmagnetic field h(t)=(hx,hy,0)eiωtthat is much smaller than\nthe saturation magnetization. For small angle magnetizati on\nprecession the total magnetization and di fference vector may\nbe separated into a static equilibrium and a dynamic com-\nponent as m=m0+δmandn=n0+δn, respectively,\nwherem0=(0,sinθ,0),n0=(cosθ,0,0),C=−cos2θ,\nands=−ˆzsin2θ. The equilibrium (zero torque) conditions\nm0×H=0 andn0×(H+Jex/(MsdF)m0)=0 lead to the\nrelation\nsinθ=H/Hs, (11)\nwhereHs=−Jex/(MsdF)=|Jex|/(MsdF) is the saturation\nfield. TheLLGequationsread\n∂tδm=−γδm×H−γm0×h\n+2πγMs(δmzm0+δnzn0)׈z\n+α0(m0×∂tδm+n0×∂tδn)+δτm,(12a)\n∂tδn=−γδn×H−γn0×h\n+2πγMs(δnzm0+δmzn0)׈z\n−γHs(m0×δn−n0×δm)\n+α0(m0×∂tδn+n0×∂tδm)+δτn,(12b)\nwithlinearizedspin-transfertorques\nδτm/αm=m0×∂tδm+n0×∂tδn\n−ηsin2θ\n1+ηcos2θ∂tδnzm0+ηsin2θ\n1−ηcos2θ∂tδmzn0,(13a)\nδτn/αn=m0×∂tδn+n0×∂tδm\n+ηsin2θ\n1−ηcos2θδmzm0−ηsin2θ\n1+ηcos2θδnzn0,(13b)\nTo leading order in the small precessing components δmand\nδn,theLLG equationsinfrequencyspace become\nδmx=γhxγ(Hs+4πMs)+iω/parenleftBig\nα0+αm(1+η)\n1−ηcos2θ/parenrightBig\nω2−ω2\nA−i∆Aωsin2θ,(14a)\nδny=−γhxγ(Hs+4πMs)+iω/parenleftBig\nα0+αn(1−η)\n1−ηcos2θ/parenrightBig\nω2−ω2\nA−i∆Aωcosθsinθ,\n(14b)\nδmz=−γhxiω\nω2−ω2\nA−i∆Aωsinθ, (14c)\nδnx=−γhy4πγMs+iω/parenleftBig\nα0+αn(1−η)\n1+ηcos2θ/parenrightBig\nω2−ω2\nO−i∆Oωcosθsinθ,(15a)\nδmy=γhy4πγMs+iω/parenleftBig\nα0+αm(1+η)\n1−ηcos2θ/parenrightBig\nω2−ω2\nO−i∆Oωcos2θ, (15b)\nδnz=γhyiω\nω2−ω2\nO−i∆Oωcosθ. (15c)The A modes (δmx,δny,δmz) are excited by hx, while the O\nmodes (δnx,δmy,δnz) couple to hy. The poles inδm(ω)and\nδn(ω)define the resonance frequencies and linewidths that\ndo not depend on the magnetic field since we disregard\nanisotropyandexchange-bias.\nA. Acoustic andOpticalmodes\nAn antiferromagnetically exchange-coupled spin valves\ngenerallyhave non-collinearmagnetizationconfiguration sby\nthepresenceofexternalmagneticfields. For H<Hs(0<θ<\nπ/2),theacousticmode:\nωA=γH/radicalbig\n1+(4πMs/Hs), (16)\n∆A=α0γ/parenleftBig\nHs+4πMs+Hssin2θ/parenrightBig\n+αmγ(Hs+4πMs)+αA(θ)γHs,(17)\nandthe opticalmode:\nωO=γ/radicalBig\n(4πMs/Hs)(H2s−H2), (18)\n∆O=α0γ/parenleftBig\n4πMs+Hscos2θ/parenrightBig\n+αn4πγMs+αO(θ)γHs, (19)\nwhere\nαA(θ)=α1grsin2θ\n1+grtanh(dN/2λ)+2grsin2θ/sinh(dN/λ),(20)\nαO(θ)=α1grcos2θ\n1+grtanh(dN/2λ)+2grcos2θ/sinh(dN/λ).(21)\nThe additional broadeningin ∆Ais proportional toαmand\nαAwhile that in∆Oscales withαnandαOin. Figure 2 (a)\nshowsαmandαnas a function of spacer layer thickness, in-\ndicating thatαnis always larger than αm, and thatαn(αm)\nstrongly increases (decreases) with decreasing N layer thi ck-\nness, especially for dN<λand large gr. Figure 2(b) shows\nthedependenceofαAandαOonthetiltedangleθfordifferent\nvaluesof dN. Asθincreases,αAincreasesfrom0to αmwhile\nαOdecreasesαmto 0. The additional damping can be ex-\nplained by the dynamic exchange. When two magnetizations\nin spin valves precess in-phase, each magnet receives a spin\ncurrent that compensates the pumped one, thereby reducing\nthe interface damping. When the magnetizations precess out\nof phase, theπphase difference between both spin currents\nmeans that the moduli have to be added, thereby enhancing\nthedamping.\nWhen the magnetizations are tilted by an angle θas\nsketched in Fig. 1, we predict an additional damping torque\nexpressedbythesecondtermsofEqs.(4a)and(4b). Figure3\nshows the ratiosαA(θ)/αmandαO(θ)/αnas a function ofθ\nandgrfor different values ofλ/dN, thereby emphasizing the\nadditional damping in the presence of noncollinear magneti -\nzations. In Fig. 3(a,b)with λ/dN=1, i.e. for a spin-di ffusive\ninterlayer,the additionaldampingof both A- and O-modesis\nsignificant in a large region of parameter space. On the other5\n0.1 1 10 100λ/d N012345αm/α1, αn/α1/α1 /α12\n1\n0.5\n0 30 60 90 \nθ (degree)0.00.10.20.30.40.5αA, αOαO αAλ/d N=1 \n2\n4\n10(b)(a)\nαnαmgr=5\nFIG.2. (a)αm(dashed line)andαn(solidline) as a functionof λ/dN\nfor different values of the dimensionless mixing conductance gr. (b)\nαAC(dashed line)andαOP(solidline)as afunction oftiltangle θfor\ngr=5and different values ofλ/dN.\nhand, in Fig. 3(c,d) with λ/dN=10, i.e. for an almost spin-\nballistic interlayer, the additional damping is more impor tant\nfortheA-mode,whiletheO-modeisa ffectedonlyclosetothe\ncollinear magnetization. In the latter case the intrinsic d amp-\ningα0dominates,however.\nB. In-phaseandOut-of-phasemodes\nWhentheappliedmagneticfieldislargerthanthesaturation\nfield (H>Hs), both magnetizations point in the ˆydirection,\nand theδm(A) andδn(O) modes morph into in-phase and\n180◦out-of-phase (antiphase) oscillations of δm1andδm2,\nrespectively. The resonance frequency53and linewidth of the\nin-phasemodefor H>Hs(θ=π/2)are\nωA=γ/radicalbig\nH(H+4πMs), (22)\n∆A=2(α0+αm)γ(H+2πMs), (23)\nwhilethoseoftheout-of-phasemodeare\nωO=γ/radicalbig\n(H−Hs)(H−Hs+4πMs),(24)\n∆O=2(α0+αn)γ(H−Hs+2πMs).(25)\u0011 \u001a\u0011 \u0017\u0011 \u0014\u0011 \u0011 \u001a\u0011 \u0017\u0011 \u0014\u0011 \u0011\u0017\n\u0015\n\u0013\u0019\u0012\u0011 \n\u0011\u0017\n\u0015\n\u0013\u0019\u0012\u0011 \tB\n \tC\n \n\tD\n \tE\nHS\nHS\nθ (degree) θ (degree)\n\u0011 \u001a\u0011 \u0017\u0011 \u0014\u0011 \nθ (degree)\u0011 \u001a\u0011 \u0017\u0011 \u0014\u0011 \nθ (degree)\u0011\u0017\n\u0015\n\u0013\u0019\u0012\u0011 HS\n\u0011\u0017\n\u0015\n\u0013\u0019\u0012\u0011 HS0.1 0.3 0.5 0.7 0.9 \n0.1 0.2 0.3 0.4 0.5 0.6 \n0.55 0.65 0.75 0.85 0.95 \n0.01 0.02 0.03 0.04 0.05 \nFIG. 3. (a,c)αA(θ)/αmand (b,d)αO(θ)/αnas a function ofθandgr\nfor different values ofλ/dN. (a,b) withλ/dN=1, (c,d) withλ/dN=\n10\nFigure 4(a) shows the calculated resonance frequencies of\nthe A andO modesas a functionof an appliedmagneticfield\nHwhile 4(b) displays the linewidths for α1/α0=1, which\nis representative for ferromagnetic metals, such as permal -\nloy (Py) with an intrinsic magnetic damping of the order of\nα0=0.01andacomparableadditionaldamping α1duetospin\npumping. A value gr=4/5 corresponds toλ=20/200nm,\nρ=10/2.5µΩcmfor N=Ru/Cu,54,55Gr=2/1×1015Ω−1m−2\nfor the N|Co(Py) interface56,57, anddF=1nm, for example.\nThecolorsinthefigurerepresentdi fferentrelativelayerthick-\nnessesdN/λ. The linewidth of the A mode in Fig. 4(b) in-\ncreases with increasing H, while that of the O mode starts\nto decrease until a minimum at the saturation field H=Hs.\nFigure 4(c) shows the linewidths for α1/α0=10, which de-\nscribes ferromagnetic materials with low intrinsic dampin g,\nsuch as Heusler alloys58and magnetic garnets.59In this case,\nthe linewidth of the O modeis much largerthanthat of the A\nmode,especiallyforsmall dN/λ.\nIn the limit of dN/λ→0 is easily established experimen-\ntally. The expressions of the linewidth in Eqs. (17) and (19)\narethengreatlysimplifiedto ∆A=γ(Hs+4πMs+Hssin2θ)α0\nand∆O=γ/parenleftBig\n4πMs+Hscos2θ/parenrightBig\nα0+(4πγMs)grα1,and∆A≪\n∆Owhengrα1≫α0. The additional damping, Eq. (10b) re-\nduces toαm→0 andαn→2[γ/planckover2pi1/(4πMsdF)(h/e2)Gr] when\nthemagnetizationsarecollinearandintheballisticspint rans-\nport limit.27In contrast to the acoustic mode, the dynamic\nexchange interaction enhances damping of the optical mode.\n∆O≫∆AhasbeenobservedinPy |Ru|Pytrilayerspinvalves32\nandCo|Cu multilayers30, consistentwiththepresentresults.\nFor spin valves with ferromagnetic metals,\nthe interface backflow spin-current [(1b)] reads\nJB\nj=(Gr/e)/bracketleftBig\nξF/parenleftBig\nmj·µsj/parenrightBig\nmj−µsj/bracketrightBig\n,whereξF=6\n012ω/(4 πγ Ms)ωAωO\n02468∆/(4 πMsα0) dN/λ=\n0.3\n1(a)\n(b) \n(c)\n0 0.5 1 1.5\nH/H s050dN/λ=0.01\n0.1\n0.3\n1∆/(4 πγ Msα0)0.01γ\nA mode  O mode 0.1\nFIG. 4. (a) Resonance frequencies of the A and O modes as a func -\ntion of magnetic field for Hs/(4πMs)=1. (b), (c) Linewidths of the\nA (dashed line) and the O (solid line) modes for Hs/(4πMs)=1,\ngr=5, and different values of dN/λ. (b)α1/α0=1 and (c)\nα1/α0=10.\n1−(G/2Gr)(1−p2)(1−ηF) (0≤ξF≤1),Gis the\nN|F interface conductance per unit area, and pthe conduc-\ntance spin polarization.51Here the spin diffusion efficiency\nis\n1\nηF=1+σF\nGλFtanh(dF/λF)\ncosh(dF/λF), (26)\nwhereσF,λF, anddFare the conductivity, the spin-flip dif-\nfusion length, and the layer thickness of the ferromagnets,\nrespectively. For the material parameters of a typical fer-\nromagnet with dF=1 nm, the resistivity ρF=10µΩcm,\nG=2Gr=1015Ω−1m−2,λF=10nm,and p=0.7,ξF=0.95,\nwhichjustifiesdisregardingthiscontributionfromtheout set.\nV. COMPARISONWITH EXPERIMENTS\nFMRexperimentsyieldtheresonantabsorptionspectraofa\nmicrowavefield ofa ferromagnet. Themicrowaveabsorptiont\ndP/dH ϕ=90 o\n20 o\n0o×5\n×5ϕHh(t)(a)\n5\nH (kOe) 22 4 6 0 0  0.4  0.8  1.2 H/H s\nCo(3)|Ru(1)|Co(3) Co(3)|Ru(1)|Co(3) \nExperiment (Ref. 11) Calculation \n 0  2  4  6  8  0  0.2  0.4  0.6  0.8  1 ∆/(4 πγ Msα₀)H/H s\nA mode \nO mode \n 0  2000  4000  6000 \nField (Oe) (b)\nExperiment (Ref. 10): [Co(1)|Cu(1)] 10 Co(1) Calculation: Co(1)|Cu(1)|Co(1) \nFIG.5. (a)Derivativeofthemicrowaveabsorptionspectrum dP/dH\nat frequencyω/(2π)=9.22 GHz for different anglesϕbetween the\nmicrowavefieldandtheexternalmagneticfieldfor Hs/(4πMs)=0.5,\nω/(4πγMs)=0.35,dN/λ=0.1,dF/λ=0.3α0=α1=0.02, and\ngr=4. The experimental data have been adopted from Ref. 31.\n(b) Computed linewidths of the A and O modes of a Co |Cu|Co spin\nvalve (dashed line) compared with experiments on a Co |Cu multi-\nlayer (solidline).30\npowerP=2/angbracketlefth(t)·∂tm(t)/angbracketrightbecomesinourmodel\nP=1\n4γ2Ms(Hs+4πMs)∆A\n(ω−ωA)2+(∆A/2)2h2\nxsin2θ\n+1\n4γ2Ms(4πMs)∆O\n(ω−ωO)2+(∆O/2)2h2\nycos2θ. (27)\nPdepends sensitively on the character of the resonance, the\npolarization of the microwave, and the strength of the ap-\nplied magnetic field. In Figure 5(a) we plot the normalized\nderivative of the microwave absorption spectra dP/(P0dH)\nat different anglesϕbetween the microwave field h(t) and\nthe external magnetic field H, where P0=γMsh2and\nh(t)=h(sinϕ,cosϕ,0)eiωt. Here we use the experimen-\ntal values Hs=5kOe, 4πMs=10kOe,dN=1nm,\ndF=3nm, and microwave frequency ω/(2π)=9.22GHz\nas found for a symmetric Co |Ru|Co trilayer.31λ=20nm for7\nRu,α0=α1=0.02, andgr=4 is adopted (correspond-\ning toGr=2×1015Ω−1m−2).Whenh(t) is perpendicularto\nH(ϕ=90◦), only the A mode is excited by the transverse\n(δmx,δmz) component. When h(t) is parallel to H(ϕ=0◦),\nthe O mode couples to the microwave field by the longitudi-\nnalδmycomponent. For intermediate angles ( ϕ=20◦), both\nmodes are excited at resonance. We observe that the opti-\ncal mode signal is broader than the acoustic one, as calcu-\nlated. The theoretical resonance linewidths of the A and O\nmodes as well as the absorption power as a function of mi-\ncrowave polarization reproduce the experimental results f or\nCo(3.2nm)|Ru(0.95nm)|Co(3.2nm)well.31\nFigure 5(b) shows the calculated linewidths of A and\nO modes as a function of an applied magnetic field for a\nCo(1nm)|Cu(1nm)|Co(1nm) spin valve. The experimental\nvaluesλ=200nm andρ=2.5µΩcm for Cu,α0=0.01\nand 4πMs=15kOe for Co, and gr=5 (corresponding to\nGr=1015Ω−1m−2) for the interface have been adopted.57\nWe partially reproduce the experimental data for magnetic\nmultilayers; for the weak-field broadenings of the observed\nlinewidthsagreementisevenquantitative. Theremainingd is-\ncrepanciesintheappliedmagneticfielddependencemightre -\nflect exchange-dipolar49and/or multilayer30spin waves be-\nyondourspinvalvemodelinthe macrospinapproximation.\nVI. CONCLUSIONS\nIn summary, we modelled the magnetization dynamics\nin antiferromagnetically exchange-coupled spin valves as a\nmodel for synthetic antiferromagnets. We derivethe Landau -\nLifshitz-Gilbert equations for the coupled magnetization s in-\ncluding the spin transfer torques by spin pumping based on\nthe spin diffusion model with quantum mechanical boundary\nconditions. We obtain analytic expressionsfor the linewid ths\nof magnetic resonance modes for magnetizations canted byapplied magneticfields and achieve goodagreementwith ex-\nperiments. We findthatthelinewidthsstronglydependonthe\ntype of resonance mode (acoustic and optical) as well as the\nstrength of magnetic fields. The magnetic resonance spectra\nreveal complex magnetization dynamics far beyond a simple\nprecessionevenin the linear responseregime. Our calculat ed\nresults compare favorably with experiments, thereby provi ng\ntheimportanceofdynamicspincurrentsinthesedevices. Ou r\nmodel calculation paves the way for the theoretical design o f\nsyntheticAFMmaterialthatisexpectedtoplayaroleinnext -\ngenerationspin-baseddata-storageandinformationtechn olo-\ngies.\nVII. ACKNOWLEDGMENTS\nTheauthorsthanksK.Tanaka,T.Moriyama,T.Ono,T.Ya-\nmamoto, T. Seki, and K. Takanashi for valuable discussions\nand collaborations. This work was supported by Grants-in-\nAidforScientificResearch(GrantNos. 22540346,25247056,\n25220910,268063)fromtheJSPS,FOM(StichtingvoorFun-\ndamenteel Onderzoek der Materie), the ICC-IMR, EU-FET\nGrant InSpin 612759, and DFG Priority Programme 1538\n“Spin-CaloricTransport”(BA 2954 /2).\nAppendixA: CoupledLandau-Lifshitz-Gilbertequationsin\nnoncollinearspinvalves\nBoth magnets and interfaces in our NM |F|NM spin valves\nare assumed to be identical with saturation magnetization Ms\nandGrthe real part of the spin-mixing conductance per unit\narea (vanishing imaginary part). When both magnetizations\nare allowed to precess as sketched in Fig. 1 (a), the LLG\nequationsexpandedtoincludeadditionalspin-pumpandspi n-\ntransfertorquesread\n∂mi\n∂t=−γmi×Heffi+α0imi×∂mi\n∂t\n+αSPi/bracketleftBigg\nmi×∂mi\n∂t−ηmj×∂mj\n∂t+η/parenleftBigg\nmi·mj×∂mj\n∂t/parenrightBigg\nmi/bracketrightBigg\n+αnc\nSPi(ϕ)mi×/parenleftBig\nmi×mj/parenrightBig\n, (A1)\nαnc\nSPi(ϕ)=αSPiη2\n1−η2(mi·mj)2/bracketleftBigg\nmj·mi×∂mi\n∂t+η/parenleftBigg\nmi·mj×∂mj\n∂t/parenrightBigg\n(mj·mi)/bracketrightBigg\n, (A2)\nwhereγandα0iare the gyromagnetic ratio and the Gilbert\ndamping constant of the isolated ferromagnetic films labele d\nbyiand thickness dFi. Asymmetric spin valves due to the\nthickness differencedFisuppress the cancellation of mutual\nspin-pumpinA-mode,whichmaybeadvantagetodetectboth\nmodesintheexperiment. Thee ffectivemagneticfield\nHeffi=Hi+h(t)+Hdii(t)+Hexj(t) (A3)consistsoftheZeemanfield Hi,amicrowavefield h(t),thedy-\nnamic demagnetization field Hdii(t), and interlayer exchange\nfieldHexj(t). The Gilbert damping torque parameterized\nbyα0igoverns the relaxation towards an equilibrium direc-\ntion. The third term in Eq. (A1) represents the mutual spin\npumping-induced damping-like torques in terms of damping\nparameter8\nαSPi=γ/planckover2pi12Gr\n2e2MsdFiηS\n1−η2, (A4)\nwhere\nη=gr\nsinh(dN/λ)+grcosh(dN/λ)(A5)\nandgr=2λρGrisdimensionless. ThefourthterminEq. (A1)\nis the damping Eq. (A2) that depends on the relative angleϕbetween the magnetizations. When mjis fixed along the\nHidirection, i.e. a spin-sink limit, Eq. (A1) reduces to the\ndynamicstiffnessin spinvalveswithoutanelectricalbias.60\nWhen the magnetizations are noncollinear as in Fig. 1, we\nhave to take into account the additional damping torques de-\nscribedbythe secondtermsin Eqs.(4a) and(4b ,). 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In the spin coherent state represe ntation, the master equation becomes\na Fokker-Planck equation, which naturally includes the spi n transfer and quantum fluctuation.\nThe current electron scattering state is correlated to the m agnet quantum states, giving rise to\nquantum correction to the electron transport properties in the usual semiclassical theory. In the\nlarge spin limit, the magnetization dynamics is shown to obe y the Hamilton-Jacobi equation or the\nHamiltonian canonical equations.\nPACS numbers: 75.78.-n, 05.10.Gg, 85.75.-d\nI. INTRODUCTION\nThe theory of open quantum systems, which has been\ngreatly developed in the past several decades, plays a\ncritical role in the understanding and control of the dy-\nnamics of quantum systems that are coupled to the sur-\nrounding environments.1–3The basic issues such as dissi-\npation, decoherence, measurement, and noise source, etc.\nof the systems are usually investigated in the open quan-\ntum system framework. This theory has wide applica-\ntions in the fields including quantum optics,1–3ultracold\natoms,4and quantuminformationand computation.5On\nthe other hand, in micromagnetics6,7and spintronics8,9\nthe magnetization dynamics is commonly treated as clas-\nsical even though the control and dissipation parame-\nters are couched as from quantum sources. While these\nmethods have been highly successful in simulating mag-\nnetization reversal and spin-torque driven magnetization\ndynamics,10thereisthequestionwhethertherearequan-\ntum effects not exhibited by these classical treatments,\nwhen the magnet is in the mesoscopic range, of 103−107\nspins, between the molecular magnets and the macro-\nscopic magnets as defined by Ref. 11. Addressing this\nquestion may not only pave the way for the future tech-\nnology developments, but also broaden our vision and\ndeepen our understanding of the emerging mesoscopic\nquantum world between the well established microscopic\nand macroscopic ones. In this paper, we present a the-\noryofasingledomainmesoscopicmagnetasamemberof\nthe family of open quantum systems for its spin-current-\ndriven dynamics.\nWhen the spin-polarized current passes through the\nferromagnetism (FM) layer, the spin angular momentum\nof the current electrons is transferred to the FM layer\nand thus rotate the magnetization12,13. This so-called\n“spin transfer torque (STT)” effect has now become the\nmost important method to control the magnetization dy-\nnamics in the nano-scale structures, where the conven-\ntional Oersted field generated by the electric current is\nless practical8,9. Numerous magnetoelectronics devices\nhave been proposed and fabricated based on the STT-\ndriven magnetization dynamics.8,9In spite of its greatsuccess and importance, the fundamental physics of STT\nin the standard theory is semiclassical. The magneti-\nzation dynamics is described by the classical Landau-\nLifshitz-Gilbert (LLG) equation10, while the STT terms\nin the LLG equation is obtained from the quantum scat-\ntering ofthe spin currentelectrons by the classicalpoten-\ntial of the magnetization12,13. This semiclassical picture\nisexpected to breakdownasthe magnetis further minia-\nturized to the mesoscopic regime. Furthermore, the STT\nhas been used to stimulate and control the spin waves14,\nor magnons. Therefore, a more sophisticated investiga-\ntion of the STT in the full quantum picture is necessary\nin order to adapt the new developments in the field. In\na previous study, we shown that the continuous scatter-\nings between the quantum macrospin state of a magnet\nandspin-polarizedelectronsin a simple model simulation\nnot only induce the STT effect but also generate quan-\ntum fluctuations due to the quantum disentanglement\nprocess.15In this paper, we will show that the quantum\nmacrospinscatteringmodelweexploitedbeforeisexactly\nsolvable,andwillinvestigatethemagnetizationdynamics\nfrom the full quantum picture by applying the standard\ntheoretical techniques for open quantum systems to this\nmodel.\nII. QUANTUM MACROSPIN MODEL\nIn parallel with the original study of STT in the semi-\nclassical picture12,13, we consider the motion of a single-\ndomain magnet driven by the spin-polarized current.\nHowever, the magnet here is not described by the clas-\nsical magnetization vector M, but is represented by the\nquantum operators of the total spin angular momentum\n/hatwideJ. The spin-polarized electrons are injected along the\nx-direction in sequence independent of one another, and\ninteract with the magnet located at x= 0 through the\nexchange interaction. The model Hamiltonian for each\nelectron interacting with the magnet is15\nH=−1\n2∂2\nx+δ(x)/parenleftig\nλ0/hatwideJ0+λ/hatwides·/hatwideJ/parenrightig\n.(1)2\nwhere the first term is the kinetic term of a single current\nelectron, and the second term is the interaction between\nthe electrons and the magnet; /hatwideJ0and/hatwideJare the unit and\ntotal spin operators of the magnet, and /hatwidesis the electron\nspin operator; the parameters λ0andλare the spin-\nindependent and spin-dependent interaction strength re-\nspectively, which are used in the semiclassical model if\nthe operators /hatwideJ0and/hatwideJare replaced by their mean field\napproximation according to the correspondence princi-\nple. Note that the Hamiltonian is that of a free magnet\nwithout the external magnetic field and anisotropic crys-\ntal field. This treatment will simplify the calculations\nbelow without invalidating of the general conclusions.\nThe STT effect originates from the elementary\nentangle-disentangle processes of the scattering states\nbetween the magnet and the spin-polarized electrons.15\nThese scattering states are deduced from the scattering\nmatrixSof the Hamiltonian (1). Unlike the scatter-\ning matrix in the semiclassical picture, which is defined\nonly in the Hilbert space of the electron, the scattering\nmatrixSin this full quantum picture is defined on the\nlarger Hilbert space including both the magnet and the\nelectron (see Appendix A), which gives the STT directly\nand more informations compared with the semiclassical\ncase.\nIII. QUANTUM DYNAMICS EQUATIONS FOR\nMAGNET\nA. Quantum Master Equation\nConsider the scattering of the magnet spins by an in-\njected electron as uncorrelated. The incoming compos-\nite state of the magnet and the current electron is as-\nsumed to be a product of their respective density ma-\ntricesρJ\ninandρe\nin. After scattering, the outgoing states\nof the whole system, ρout=SρJ\nin⊗ρe\ninS†, as a result of\nthe unitary scattering matrix S, has a degree of entan-\nglement. Next, the surrounding environment decoheres\nthe entangled state into a joint probability distribution\nof the possible magnet states and the corresponding elec-\ntron states. Properties of the resultant magnet state or\nthe electron state are characterized by their respective\nreduced density ρJ\noutor electron ρe\noutfrom tracing over\nthe degrees of freedom of the other component in ρout.\nThecorrelated quantum dynamics of the magnet and the\nelectron injected in sequence in the spin-polarized cur-\nrent drives the time evolution of the magnet state ρJand\nthe magnetization-dependent electron transport proper-\nties in the electron density matrix ρe. While the mean\nmagnetization dynamics is within reach of the semiclas-\nsical picture, the magnetization fluctuation is given only\nby the full quantum treatment followed here without ad-\nditional stochastic assumptions.\nForatheoryofdynamicsofthe ferromagnetasanopen\nsystem, we treat the current electrons as the equivalent\nof the environment. The Kraus operator16of the magnetis defined in terms of the scattering matrix Sas the evo-\nlution operatorofeachencounter with acurrent electron,\nKk,s;k′,s′≡ /an}b∇acketle{tk,s|S|k′,s′/an}b∇acket∇i}ht, (2)\nwith a specific basis set {|k,s/an}b∇acket∇i}ht}of an incoming electron\nstate of wave vector kand spin up or down state s=±.\nWe have adopted the simple model (1) for the dynamics\nof the rigid macro-spinstates {|J,m/an}b∇acket∇i}ht}with the total spin\nnumberJand thezcomponent quantum number mand\nleave the effects of spin waves for future study. Then,\nthe current electron kinetic energy is conserved and the\nKraus operators are non-zero only if kandk′are on the\nsame energy shell, given by\nKk,s;±k,s= (ξ±1\n2)/hatwideJ0+sζ/hatwideJz,Kk,−s;±k,s=ζ/hatwideJs,(3)\nwhere the coefficients ξandζare functions of the basic\nparameters λ0,λandk,J. The first operator Kk,s;±k,s\nwiththesamespinindex scomesfromscatteringwithout\nspin transfer, and the second operator Kk,−s;±k,swith a\nchange insrepresents spin transfer. These Kraus oper-\nators are functions of the macro-spin /hatwideJ0and/hatwideJ(see Ap-\npendix B). In the semiclassical picture, these Kraus op-\nerators will reduce to scalars representing effective fields\nfor the magnetization dynamics.\nIn a scattering event, let the initial state of the cur-\nrent electron be given by the density matrix ρe\nin=/summationtext\ns,s′fs,s′(k)|k,s/an}b∇acket∇i}ht/an}b∇acketle{tk,s′|. This simple form may be ex-\ntended to account for a wave vector distribution or quan-\ntum coherence between different wave vectors, but will\nnotbeexploitedheretokeeptheexpositionsimple. With\ntheaboveKrausoperators,thequantummapofthemag-\nnet state from ρJ\nintoρJ\noutin the scattering is\nρJ\nout=/summationdisplay\n±,s,s′,s′′fs,s′(k)K±k,s′′;k,sρJ\nin(K±k,s′′;k,s′)†.(4)\nIf the spin-polarized current is considered as a sequence\nof electrons injected at equal time interval τ(a measure\nof the inverse current), the continuous evolution of the\nmagnetic density matrix driven by Eq. (4), with a coarse\ngraining of a time scale much larger than τ, yields the\nquantumdynamicsofthemagnetgovernedbythe master\nequation,\n∂\n∂tρJ(t) =1\nτ[T0(t)+S(t)·T(t)], (5)\nwhereS= Tr[σρe\nin] is the Bloch vector of the spin-\npolarized current electrons, σbeing the Pauli matrices.\nThe operators T0andTare polynomial functions of /hatwideJ0\nand/hatwideJ(see Appendix C). T0corresponds to the unpolar-\nized part of the current which causes fluctuations of the\nmagnet motion without a net spin transfer effect. On\nthe other hand, Tis induced by the electron spin polar-\nization, giving rise to both STT and the magnetization\nfluctuations.\nNote that the master equation (5) is an exact solu-\ntion from the model (1) for arbitrary J. Thus, Eq. (5)3\ncan be applied to molecular magnets of small J. This is\nin contrast with the simulation of the quantum stochas-\ntic dynamics of the magnet in a previous study15which\nused the same scattering matrix but required the large\nJcondition to keep the approximation of the magnetic\nquantum state as a spin coherent state after scattering.\nB. Fokker-Planck Equation\nTo facilitate computation in the large Jcase and, more\nimportantly, to study the quantum-classical crossover of\nthe magnetization dynamics, we put the master equa-\ntion (5) in the spin coherent state P-representation1,4\nanalogous to the boson case. The chosen basis set is the\novercomplete and non-orthogonal states {|J,Ω/an}b∇acket∇i}ht}, where\n|J,Ω/an}b∇acket∇i}htis the spin coherent state of total spin Jin the\ndirection of Ω = (Θ ,Φ). The density matrix ρJin this\nrepresentation is ρJ(t)≡/integraltext\ndΩPJ(Ω,t)|J,Ω/an}b∇acket∇i}ht/an}b∇acketle{tJ,Ω|, and\nthe spin operators /hatwideJtake the form of the differential\noperators.17Substituting these expressions of ρJand/hatwideJ\ninto Eq. (5) with some algebraic manipulations, we ob-\ntain the Fokker-Planck equation for PJ(see Appendix\nD),\n∂\n∂tPJ(/hatwidem,t) =−∇·(TPJ)+∇2(DPJ),(6)\nwhere the unit vector /hatwidempoints in the direction of the\nmacrospin Ω = (Θ ,Φ), the drift vector T=A(/hatwidem×S)×\n/hatwidem+B/hatwidem×Scontainsthetwowell-knowntermsofSTT,8,9\nthe diffusion coefficient D=A(1−/hatwidem·S)/(2J+1) orig-\ninates from the quantum fluctuation generated by the\nscattering.15Theparameters AandBarefunctionsofthe\nparameters ξandζin the Kraus operators (3), namely,\nA= (2J+ 1)|ζ|2/τ,B= 2ℑ[ξ∗ζ]/τ(ℑdenoting the\nimaginary part of) which can be determined from the\nbasic parameters of the Hamiltonian (1).\nThe quasi-probability distribution function PJin\nEq. (6) is different from the one considered in the classi-\ncal theory. Its value could be negative in some situations\nbecausePJdescribes the quantum state ofthe magnet as\nfaithfully as the density matrix ρJ. As shown in Eq. (6),\nthe STT terms naturally arise from the open quantum\ndynamics of the magnet in the presence of the contin-\nuous scatterings by the spin-polarized electrons. Unlike\nthe semiclassical picture, where the STT terms are indi-\nrectly obtained from the current electron spin polariza-\ntion after potential scattering, in the quantum case, the\nSTTtermsfollowsdirectlyfromthefullquantumscatter-\ning. Furthermore, the full quantum treatment also gives\nthe quantum fluctuations accompanying the spin trans-\nfer processes, which does not exist in the semiclassical\ntreatment.\nThe diffusion coefficient Dexpression shows depen-\ndence on the relative angle between the magnet and the\nelectron spin, with its maximal (minimal) value when /hatwidem\nandSare anti-parallel(parallel). This coincides with our\nprevious simulation15that the quantum magnetizationfluctuation is first enhanced and then suppressed during\nthe STT-driven magnet switching, which is observed in\na recent experiment.18For the special case where the in-\njected electrons are fully unpolarized ( S=0),Dis still\na non-zero constant which again suggests that the unpo-\nlarized current can cause quantum magnetization fluctu-\nations without net spin transfer. The steady solution of\nEq. (6) is a constant, which means a uniform distribution\nfunction in the spin coherent state space and the magne-\ntizationwillvanishonaverage. Bycontrast,thesemiclas-\nsical theory predicts only the zero spin torque but no dif-\nfusion dynamics for the magnet. This STT-induced mag-\nnetization fluctuation becomes dominant over the ther-\nmal magnetization fluctuation at low temperatures. The\ncrossovertemperature is estimated by comparing the dif-\nfusion coefficient Din Eq. (6) with the one for thermal\nmagnetization fluctuation,19,20\nαgγgkBT\n|M|∼A\n2J+1(1−/hatwidem·S), (7)\nwhereMis the magnetic moment of the magnet, αg\nthe Gilbert damping coefficient, γgthe gyromagnetic ra-\ntio,kBthe Boltzmann constant, and Tthe temperature.\nSince|M|=γgJ/planckover2pi1and|ζ|2∼ O(1/J2) forJ≫1, we\nobtainαgκBT∼/planckover2pi1/Jτ, in agreement with the quantum\nnoise estimate.15\nThePJdistribution as the solution of the Fokker-\nPlanck equation (6) gives the exact quantum dynamics\nof the magnet based on the quantum macrospin model\n(1). The expectation values of any observable physics\nqualities, such as the magnetization and its fluctuations,\ncan be calculated from the PJdistribution. An exam-\nple is demonstrated in subsection D. In the semiclassical\npicture, the STT is defined on the mean field level of\nthe magnetization, and quantum correlation between the\nmagnetization states does not exist. The quantum the-\nory includes the quantum correlationand is applicable to\nany possible exotic quantum states of the magnet in the\nmesoscopic or microscopic regime.\nC. WKB Approximation for Large J\nInthelarge Jregime, wedemonstratethatthesolution\nof Eq. (6) leads to classical behavior. The expressions for\nAandBsuggest that T∼ O(1/J) andD ∼ O(1/J2),\nthus the diffusion term is smaller than the drift term in\nEq. (6) by the order of magnitude O(1/J). Then, the\nWKB approximation is applied for large J.21Substitut-\ningPJ(/hatwidem,t) =e−JW(/hatwidem,t)into Eq. (6) and keeping the\nterms to the leading order of 1 /J, leads to the Hamilton-\nJacobi equation for W(/hatwidem,t),\n∂W\n∂t+T·∇W+JD(∇W)2= 0. (8)\nThus, the STT-driven magnet obeys the canonical dy-\nnamics in the classical limit, and the function Wplays4\ntheroleofaction. Fortheconstantspin-polarizedcurrent\ncase, the corresponding Hamiltonian canonical equations\nin the spherical polar coordinates are\ndΘ\ndt=∂H\n∂PΘ,dPΘ\ndt=−∂H\n∂Θ,\ndΦ\ndt=∂H\n∂PΦ,dPΦ\ndt=−∂H\n∂Φ. (9)\nHere, the Hamiltonian function is explicitly written as\nH=TΘPΘ+TΦ\nsinΘPΦ+JDP2\nΘ+JD\nsin2ΘP2\nΦ.(10)\nwiththegeneralizedmomentumcomponents, PΘ=∂ΘW\nandPΦ=∂ΦW, and the STT components TΘandTΦ.\nThe equations for Θ and Φ in (9) show that two more\nterms, which originate from the diffusion term in Eq. (6),\ncontribute to the magnetic dynamics in addition to the\nSTT terms even in the classical limit. Eq. (9) and (10)\ngive a more complete description of the classical magne-\ntization dynamics than the semiclassical STT theory.\nD. Numerical Example\nHere we demonstrate how to apply the approach de-\nveloped above to the STT-driven quantum dynamics of\na magnet. We consider a magnet with J= 104, and the\ninitial distribution function obeys the Boltzmann distri-\nbution, i.e. PJ(/hatwidem,0) =Ce−E/kBT, where the energy of\nthe magnet in a magnetic field BisE=−M·B,Cis\nthe normalization factor, and the magnetic moment is\nM=γg/planckover2pi1J/hatwidem. We set the temperature as T= 1 K, and\nthe magnitude of the magnetic field B= 0.05 T. The di-\nrection of Bis chosen that maximum value of PJlocates\nat the angle Ω 0= (2.8,1.0). The initial distribution is\nshowintheFig.1(a). Thenweapplyaspincurrentpulse,\nwhich includes 1 .5×105electrons. The Bloch vector of\nthe electron spin is S= (0,0,1), and the wavevector is\nk= 13.6 nm−1. To calculate the scattering matrix, the\nparameters λ0andλin Eq. (1) are estimated for a mag-\nnet with effective potentials ∆ += 1.3 V, ∆ −= 0.1 V\nand thickness d= 3 nm.\nIn the simulations, we have used the method of\ncharacteristics22to solve the Hamilton-Jacobi equation\n(8), which gives the time-evolution of W(/hatwidem,t) and then\nthe distribution function PJ(/hatwidem,t) =e−JW(/hatwidem,t). Direct\nsolution of the Fokker-Planck equation (6) or the mas-\nter equation (5) is also practical but not explored here.\nThe time is measured in tN=Nτ. The distribution\nfunctions PJatt= 0.3tN,0.5tN,tNare shown in Fig. 1\n(b)(c)(d) respectively. In order to keep the normaliza-\ntion ofPJ, it is renormalized after every 500 scatterings.\nNote the different scales for Θ in Fig. 1. We found that\nPJis first expanded and then compressed, and its center\nmoves from (2.8,1.0) to (0.05,4.2), which show the effect\nof spin current on the distribution function./s50/s46/s54 /s50/s46/s55 /s50/s46/s56 /s50/s46/s57/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s40/s97/s41\n/s48/s46/s48/s50/s48/s52/s48/s54/s48\n/s49/s46/s50 /s49/s46/s53 /s49/s46/s56 /s50/s46/s49/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s40/s98/s41\n/s48/s46/s48/s50/s48/s52/s48/s54/s48\n/s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53\n/s40/s99/s41\n/s48/s46/s48/s50/s48/s52/s48/s54/s48\n/s48/s46/s48/s48 /s48/s46/s48/s52 /s48/s46/s48/s56 /s48/s46/s49/s50/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48\n/s40/s100/s41\n/s48/s46/s48/s50/s48/s52/s48/s54/s48\nFIG. 1. (color online). Distribution function PJfor the nano-\nmagnet at four different time t. (a)t= 0; (b) t= 0.3tN; (c)\nt= 0.5tN; (d)t=tN. The simulation parameters are set as:\nJ= 104, ∆+= 1.3 V, ∆ −= 0.1 V,d= 3 nm, N= 1.5×105,\nk= 13.6 nm−1,S= (0,0,1). A 200 ×200 lattice in Θ-Φ plane\nis used in the simulations.\n/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s116/s32/s47/s32/s116\n/s78/s40/s97/s41\n/s32/s32/s74 /s32/s47/s32 /s74\n/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53\n/s32/s32/s32 /s74 /s32/s47/s32 /s74\n/s116/s32/s47/s32/s116\n/s78/s40/s98/s41\nFIG. 2. (color online). Time-evolution of Jµ=x,y,zand its\nfluctuation δJµ=x,y,z(x: solid line; y: dash line; z: dotted\nline.) caused by the spin transfer torque obtained from the\nprobability distribution function (thickline) and thequa ntum\ntrajectory method (thin line).\nThe meanvalue ofthe macrospin /hatwideJandits fluctuations\nare calculated from PJas\nJµ=x,y,z(t) =/integraldisplay\ndΩP(Ω,t)/an}b∇acketle{tJ,Ω|/hatwideJµ|J,Ω/an}b∇acket∇i}ht,\nδJ2\nµ=x,y,z(t) =/integraldisplay\ndΩP(Ω,t)/an}b∇acketle{tJ,Ω|δ/hatwideJ2\nµ|J,Ω/an}b∇acket∇i}ht.\nThe results are shown in Fig. 2, where the mean magne-\ntization is switched by the STT, and the magnetization\nfluctuations are first enhanced and finally suppressed.\nComparison with the results obtained from the quantum\ntrajectory method in Ref. 15 gives reasonable agreement.\nThe approachdeveloped here together with the quantum\ntrajectory method15fits in the toolbox for micromagnet-\nics simulations with the added value of accounting for\nrelevant quantum effects.5\nIV. ELECTRON DENSITY MATRIX AFTER\nSCATTERING\nFinally, we briefly discuss the electron states after the\nscattering,whichcontaintheinformationsabouttheelec-\ntric current and current noise, etc. After tracing over\nthe degrees of freedom for the magnet in the total den-\nsity matrix ρout, the reduced density matrix of the elec-\ntron isρe\nout=TrJ[Sρe\nin⊗ρJ\ninS†]. This expression is\na generalization of the electron potential scattering in\nthe semiclassical picture23. Here, the transformation of\nthe electron state is no longer unitary due to the recoil\nof the magnet. For instance, we assume that one elec-\ntron with wavevector k >0 and spin-polarized vector\nS= (0,0,1) is injected, i.e., ρe\nin=|k,+/an}b∇acket∇i}ht/an}b∇acketle{tk,+|, and take\ntheP-representation for the quantum states of the mag-\nnet. With the scattering matrix S, we obtain\nρe\nout=/summationdisplay\nk′,s′;k′′,s′′|k′,s′/an}b∇acket∇i}ht/an}b∇acketle{tk′′,s′′|\n×/integraldisplay\ndΩP(Ω)/an}b∇acketle{tΩ|(Kk′′,s′′;k,+)†Kk′,s′;k,+|Ω/an}b∇acket∇i}ht.(11)\nFor the model (1), the terms in Eq. (11) are non-zero\nonly ifk′=±kandk′′=±k. The electron scattering\nstate is correlatedwith the quantum state of the magnet.\nThe scattering formalism in the semiclassical picture will\nbe reproduced if the Kraus operators are replaced by the\ncorresponding scattering matrix elements there. As the\nmagnets are miniaturized further24and the quantum de-\nscription becomes necessary, the semiclassical scattering\nformalismforthe electrontransportwill breakdown, and\nEq.(11)anditsgeneralizationsshouldbeexploitedasthe\nnew starting point.\nV. CONCLUSION\nIn conclusion, the STT-driven magnetization dynam-\nics has been investigated by treating the magnet as an\nopen quantum system in the exactly solvable quantum\nmacrospin model. A set of dynamical equations is es-\ntablished and the quantum-classical connection is made.\nThe full quantum picture here provides a unified and\ncomplete description of the magnetization dynamics and\nelectron transport, and further explorations of the quan-\ntum physics in spintronics along this line is expected.\nACKNOWLEDGMENTS\nWe acknowledge the support of this work by the U.\nS. Army Research Office under contract number ARO-\nMURI W911NF-08-2-0032 and NSF ECCS-1202583.Appendix A: Scattering Matrix\nFirst, wecalculatethescatteringmatrix Softhemodel\nHamiltonian\nH=−1\n2∂2\nx+δ(x)/parenleftig\nλ0/hatwideJ0+λ/hatwides·/hatwideJ/parenrightig\n.(A1)\nConsidering that one electron in state |ψe\nin/an}b∇acket∇i}htis injected\nalong thex-direction and the initial quantum state of\nthe magnet is |ψJ\nin/an}b∇acket∇i}ht, then the incoming state |Ψin/an}b∇acket∇i}htof the\nwhole system before scattering is the product state of\n|ψe\nin/an}b∇acket∇i}htand|ψJ\nin/an}b∇acket∇i}ht, i.e.,|Ψin/an}b∇acket∇i}ht=|ψe\nin/an}b∇acket∇i}ht ⊗ |ψJ\nin/an}b∇acket∇i}ht. After scat-\ntering, the outgoing state |Ψout/an}b∇acket∇i}htwill be|Ψout/an}b∇acket∇i}ht=S|Ψin/an}b∇acket∇i}ht.\nThe scattering matrix Sare determined by the boundary\nconditions at x= 0,\n(Ψin+Ψout)|x=0−= (Ψin+Ψout)|x=0+,\n/integraldisplay0+\n0−H(Ψin+Ψout)dx=ε/integraldisplay0+\n0−(Ψin+Ψout)dx,(A2)\nwhereεis the total energy of the whole system.\nThe scattering problem (A2) is simplified by utilizing\nthe symmetries of the model Hamiltonian (A1). First,\nthe kinetic energy of the electron is conserved during the\nscattering process. Thus the scattering matrix elements\nofSisnon-zeroonlyiftheabsolutevaluesoftheincoming\nand outgoing wavevectors of the electron are the same.\nSecond,theoperators( /hatwides+/hatwideJ)2and/hatwidesz+/hatwideJzarecommutative\nwith the Hamiltonian (A1), and their eigenstates |J,µ/an}b∇acket∇i}ht\nare given as\n(/hatwides+/hatwideJ)2|J,µ/an}b∇acket∇i}ht=J(J+1)|J,µ/an}b∇acket∇i}ht,\n(/hatwidesz+/hatwideJz)|J,µ/an}b∇acket∇i}ht=µ|J,µ/an}b∇acket∇i}ht,\nwithJ=J±1\n2andµ=−J,...,J. Choosing the basis\nset{|k;J,µ/an}b∇acket∇i}ht}, the scattering problem (A2) reduced to\na set ofδ-potential scattering equations, and gives the\nscattering matrix Sin this representation15. Then after\narepresentationtransformationwith the helpofClebsch-\nGorden coefficients, we obtain the scattering matrix Sin\nthe basis set {|k,s;J,m/an}b∇acket∇i}ht}, which takes a block form\nS=\n...0 0\n0Sk,µ0\n0 0...\n, (A3)\nand the form of each block Sk,µis\nSk,µ=\nt++\nk,µr++\nk,µt+−\nk,µr+−\nk,µ\nr++\nk,µt++\nk,µr+−\nk,µt+−\nk,µ\nt−+\nk,µr−+\nk,µt−−\nk,µr−−\nk,µ\nr−+\nk,µt−+\nk,µr−−\nk,µt−−\nk,µ\n.(A4)\nHere, the element tss′\nk,µ(rss′\nk,µ) is the transmission (re-\nflection) probability amplitude from |k,s′;J,µ−1\n2s′/an}b∇acket∇i}htto\n|k,s;J,µ−1\n2s/an}b∇acket∇i}ht(| −k,s;J,µ−1\n2s/an}b∇acket∇i}ht). The spin transfer6\nis related to those elements with s/ne}ationslash=s′. The explicit\nexpressions for the matrix elements are\nt++\nk,µ= cosηJ,+e−iηJ,+cos2αJ,µ+cosηJ,−e−iηJ,−sin2αJ,µ,\nr++\nk,µ=−i(sinηJ,+e−iηJ,+cos2αJ,µ+sinηJ,−e−iηJ,−sin2αJ,µ),\nt−−\nk,µ= cosηJ,+e−iηJ,+sin2αJ,µ+cosηJ,−e−iηJ,−cos2αJ,µ,\nr−−\nk,µ=−i(sinηJ,+e−iηJ,+sin2αJ,µ+sinηJ,−e−iηJ,−cos2αJ,µ),\nt−+\nk,µ= (cosηJ,+e−iηJ,+−cosηJ,−e−iηJ,−)sinαJ,µcosαJ,µ,\nr−+\nk,µ=−i(sinηJ,+e−iηJ,+−sinηJ,−e−iηJ,−)sinαJ,µcosαJ,µ,\nt+−\nk,µ= (cosηJ,+e−iηJ,+−cosηJ,−e−iηJ,−)sinαJ,µcosαJ,µ,\nr+−\nk,µ=−i(sinηJ,+e−iηJ,+−sinηJ,−e−iηJ,−)sinαJ,µcosαJ,µ.\nHere, the phase shifts are given as ηJ,±= tan−1∆J,±\nk,\nwith the effective potentials ∆ J,+=1\n2(Jλ0+Jλ),∆J,−=\n1\n2[Jλ0−(J+1)λ]. The Clebsch-Gordan coefficients\ncosαJ,µand sinαJ,µare given as cos αJ,µ=/radicalig\nJ+µ+1\n2\n2J+1,\nsinαJ,µ=/radicalig\nJ−µ+1\n2\n2J+1.\nAppendix B: Kraus Operators\nNow we express the Kraus operators Kk,s;k′,s′in the\nbasis set {|J,m/an}b∇acket∇i}ht}based on the scattering matrix Sob-\ntainedabove. Theblockformof Smeansthat Kk,s;k′,s′is\nnon-zero only if kandk′have the same absolute values.\nFor example, we have\nKk,+;k,+\n=/an}b∇acketle{tk,+|S|k,+/an}b∇acket∇i}ht\n=\ntk,J+1\n2···0···0\n......0......\n0 0t++\nk,m+1\n20 0\n......0......\n0···0···tk,−J+1\n2\n,(B1)\nwhich is a (2 J+1)-dimension diagonal matrix. With the\nClebsch-Gordan coefficients, the diagonal elements are\nrewritten as\nt++\nk,m+1\n2= (ξ+1\n2)+ζm,\nwhere\nξ=J+1\n2J+1cosηJ,+e−iηJ,++J\n2J+1cosηJ,−e−iηJ,−−1\n2\n=−i(J+1\n2J+1sinηJ,+e−iηJ,++J\n2J+1sinηJ,−e−iηJ,−)+1\n2,\nζ=1\n2J+1(cosηJ,+e−iηJ,+−cosηJ,−e−iηJ,−)\n=−i1\n2J+1(sinηJ,+e−iηJ,+−sinηJ,−e−iηJ,−).\nConsidering the matrix form of the angular momentum\noperator /hatwideJzin the basis set {|J,m/an}b∇acket∇i}ht}, the matrix (B1)shows that the Kraus operator Kk,+;k,+is just\nKk,+;k,+= (ξ+1\n2)/hatwideJ0+ζ/hatwideJz,\nwhere/hatwideJ0is the unit matrix.\nSimilarly, the other Kraus operators are written in the\ncompact form as\nKk,s;±k,s= (ξ±1\n2)/hatwideJ0+sζ/hatwideJz,Kk,−s;±k,s=ζ/hatwideJs.\nAppendix C: Quantum Master Equation\nThemasterequation(5)inthemaintextisobtainedby\nsubstituting the Kraus operators (3) into Eq. (4) there.\nThe calculations are straightforward, and yield the ex-\nplicit expressions for the operators T0andTas\nT0≡(|ξ|2−1\n4)ρJ+|ζ|2(/hatwideJzρJ/hatwideJz+/hatwideJ+ρJ/hatwideJ−)+h.c.,\nTx≡2ξζ∗ρJ/hatwideJx+|ζ|2(/hatwideJzρJ/hatwideJ+−/hatwideJ+ρJ/hatwideJz)+h.c.,\nTy≡2ξζ∗ρJ/hatwideJy+i|ζ|2(/hatwideJ+ρJ/hatwideJz−/hatwideJzρJ/hatwideJ+)+h.c.,\nTz≡2ξζ∗ρJ/hatwideJz+|ζ|2(/hatwideJ+ρJ/hatwideJ−−/hatwideJ−ρJ/hatwideJ+)+h.c..\nThe terms containing a single angular momentum oper-\nators inTcan be interpreted as a field-like torque, while\nthe terms including two angular momentum operators\nthe Slonczewski-type torque and the quantum fluctua-\ntions. This becomes clearer in the spin coherent state\nrepresentation.\nAppendix D: The Fokker-Planck Equation\nHere we explain the derivation of the Fokker-Planck\nequation (6) in the paper. In the spin coherent state rep-\nresentation {|J,Ω/an}b∇acket∇i}ht}, the density matrix ρJis expressed\nas17\nρJ=/integraldisplay\ndΩPJ(Ω)|J,Ω/an}b∇acket∇i}ht/an}b∇acketle{tJ,Ω|. (D1)\nUsingS= (α,β,γ), and substituting the expression (D1)\ninto the master equation (5) in the paper, and utilizing\nthe differential forms of the operators17/hatwideJi|J,Ω/an}b∇acket∇i}ht/an}b∇acketle{tJ,Ω|/hatwideJj\n(i,j= 0,+,−,z), we derive the differential equation for\nPJ(Ω) as\n∂PJ\n∂t=1\nsinΘ∂(−TΘPJ)\n∂Θ+1\nsinΘ∂(−TΦPJ)\n∂Φ\n+1\nsinΘ∂\n∂Θ[sinΘ∂(DPJ)\n∂Θ]+1\nsin2Θ∂2(DPJ)\n∂Φ2.\n(D2)\nHere,\nTΘ=A(αcosΘcosΦ+ βcosΘsinΦ −γsinΘ)\n+B(αsinΦ−βcosΦ),7\nTΦ=A(−αsinΦ+βcosΦ)\n+B(αcosΘcosΦ+ βcosΘsinΦ −γsinΘ),\nD=A\n2J+1(1−αsinΘcosΦ −βsinΘsinΦ −γcosΘ),\nwith the coefficients A= (2J+1)|ζ|2\nτ,B=2ℑ(ξ∗ζ)\nτ. Fur-\nther analysis shows that TΘandTΦare the components\nof the spin transfer torque T=A(/hatwidem×S)×/hatwidem+B/hatwidem×S\nin the spherical coordinates, where the unit vector /hatwidemdenotes the direction of the macrospin. Replacement of\nthe differential operators in spherical coordinates by the\ndivergence operator ∇and Laplace operator ∇2reduces\nEq. (D2) to the simple form of the Fokker-Planck equa-\ntion (FPE)\n∂\n∂tPJ(/hatwidem,t) =−∇·(TPJ)+∇2(DPJ).(D3)\n∗Present address: Department of Physics, University of\nHong Kong\n†lsham@ucsd.edu\n1H. J. Carmichael, Statistical methods in quantum optics 1:\nMaster equations and Fokker-Planck equations (Springer,\nBerlin, 1999).\n2H. J. Carmichael, Statistical methods in quantum optics 2:\nNonclassical fields (Springer, Berlin, 2008).\n3H.-P. Breuer and F. Petruccione, The Theory of Open\nQuantum Systems (Oxford University Press, 2002).\n4F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas,\nPhys. Rev. A 6, 2211 (1972).\n5C. 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Krause, States, Effects, And Operations Fundamental\nNotions Of Quantum Theory , edited by A. Bohm, J. Dol-\nlard, and W. Wootters (Springer-Verlag, Berlin, 1983).\n17L. M. Narducci, C. M. Bowden, V. Bluemel, G. P. Gar-\nrazana, and R. A. Tuft, Phys. Rev. A 11, 973 (1975).\n18V. E. Demidov, S. Urazhdin, E. R. J. Edwards, M. D.\nStiles, R. D. McMichael, and S. O. Demokritov, Phys. Rev.\nLett.107, 107204 (2011).\n19W. F. Brown, Phys. Rev. 130, 1677 (1963).\n20Z. Li and S. Zhang, Phys. Rev. B 69, 134416 (2004).\n21H. Risken, The Fokker-Planck Equation (Springer-Verlag,\nBerlin, 1984).\n22R. Courant and D. Hilbert, Methods of Mathematical\nPhysics (Interscience, New York, 1953).\n23J. Foros, A. Brataas, Y. Tserkovnyak, and G. E.W. Bauer,\nPhys. Rev. Lett. 95, 016601 (2005).\n24J. K. W. Yang, Y. Chen, T. Huang, H. Duan, N. Thiya-\ngarajah, H. K. Hui, S. H. Leong, and V. Ng, Nanotechnol-\nogy22, 385301 (2011)."    },    {        "title": "1110.4749v2.Classical_Orbital_Magnetic_Moment_in_a_Dissipative_Stochastic_System.pdf",        "content": "arXiv:1110.4749v2  [cond-mat.stat-mech]  7 Aug 2012Classical Orbital Magnetic Moment in a\nDissipative Stochastic System\nN. Kumar\nRaman Research Institute, Bangalore 560080, India\nAbstract\nWe present an analytical treatment of the dissipative-stoc hastic dy-\nnamics of a charged classical particle confined bi-harmonic ally in a plane\nwith a uniform static magnetic field directed perpendicular to the plane.\nThe stochastic dynamics gives a steady state in the long-tim e limit. We\nhave examined the orbital magnetic effect of introducing a pa rametrized\ndeviation ( η−1) from the second fluctuation-dissipation (II-FD) relatio n\nthat connects the driving noise and the frictional memory ke rnel in the\nstandard Langevin dynamics. The main result obtained here i s that the\nmoving charged particle generates a finite orbital magnetic moment in\nthe steady state, and that the moment shows a crossover from p ara- to\ndia-magnetic sign as the parameter ηis varied. It is zero for η= 1 that\nmakes the steady state correspond to equilibrium, as it shou ld. The mag-\nnitude of the orbital magnetic moment turns out to be a non-mo notonic\nfunction of the applied magnetic field, tending to zero in the limit of an\ninfinitely large as well as an infinitesimally small magnetic field. These\nresults are discussed in the context of the classic Bohr-van Leeuwen theo-\nrem on the absence of classical orbital diamagnetism. Possi ble realization\nis also briefly discussed.\nPACS numbers: 05.40.-a, 05.10.Gg, 75.20.-g.\nThe Bohr-van Leeuwen (BvL) theorem [1]–[3] on the stated absenc e of or-\nbital diamagnetism for a classical system of charged particles in equ ilibrium has\nbeen one of the surprises of physics [4]. The static external magne tic field ex-\nerts a Lorentz force on the moving charged particle, acting at righ t angle to its\ninstantaneous velocity ( v). While such a gyroscopic force does no work on the\nparticle, it does induce an orbital cyclotron motion that subtends a n amperean\ncurrent loop. The magnetic field associated with this current loop is e xpected\nto be non-zero, and directed oppositely to the externally applied ma gnetic field\n– the Lenz’ law. Hence the expectation of a finite orbital diamagnet ic moment\nq/2c(r×v) [5]. Yet, as is known well, the partition function for a classical sys-\ntem in equilibrium turns out to be independent of the applied magnetic fi eld,\nthus giving a zero orbital magnetic moment. And this has been the su rprise\n[4]. (The field independence of the classical partition function derive s simply\nfrom the fact that the classical partition function involves integra tion of the\ncanonical momentum over an infinite range for any given value of the conjugate\ncoordinate, and thus the magnetic vector potential ( A) entering the Hamilto-\nnian through minimal coupling ( p→p−q\ncA) gets eliminated through a trivial\n1shift of the momentum variable. This shift is, however, not allowed fo r a quan-\ntum system because of the canonical non-commutation involved th ere. Hence\nthe stated quantum origin of the orbital magnetic moment in equilibriu m – the\nLandau orbital diamagnetism [6]). A remarkably heuristic real-space explana-\ntion for the vanishing of the classical orbital moment was first sugg ested by\nBohr [1] in terms of a cancellation of the orbital diamagnetic moment o f the\ncompleted amperean orbits (Maxwell cycles) in the bulk interior by th e para-\nmagnetic moment subtended by the incompleted orbits skipping the b ounding\nsurface of the system in the opposite sense. This ‘ edge current ’ has a large\narm-length, or leverage and, therefore, can effectively cancel o ut the bulk dia-\nmagnetic moment. The cancellation has indeed been demonstrated g raphically\nfor a simple planar geometry [7]. This real space-time picture is consis tent with\nthe zero orbital magnetic moment following from an exact analytical solution\nof the classical Langevin dynamics with a white noise describing the mo tion of\nthe charged particle confined harmonically in two dimensions, with a un iform\nstatic magnetic field applied perpendicular to the plane [8]. Here, the s teady\nstate (t→ ∞) orbital moment indeed vanishes for the given potential confine-\nment (owing to the spring constant kof the harmonic potential, providing a soft\nboundary). Interestingly, this null result persists in the limit k→0, provided\nit is taken after taking the limit t→ ∞. This suggests that in this case the\nstochastic particle dynamics has had time enough to sense (i.e., be infl uenced\nby) the confinement ( k). On the other hand, the moment survives to a non-zero\nvalue if the order of the two limits is interchanged. (The effect of the se so-called\nDarwinian limiting processes is also manifest in the case of the quantum version\nof the above Langevin treatment [9]). It is to be noted, however, t hat in the\ncase of quantum Langevin equation the orbital moment tends to ze ro as the\nPlanck constant is formally reduced to zero, i.e., in the classical limit, f or which\nthe noise term reduces to a classical white noise which is consistent w ith a local\nStokes friction constant – in accord with the second fluctuation-d issipation (II-\nFD) theorem [10, 11]. This reasonably suggests to us that it may well be the\nconstraint of the second fluctuation-dissipation relation that for ces the orbital\nmagnetic moment to vanish in the classical case. This is further supp orted by\nour recent numerical simulation [12]. Motivated by these observatio ns, we have\ncarried out an exact analytical calculation of the orbital magnetic m oment of\na charged particle confined bi-harmonically in two dimensions with a unif orm\nstatic magnetic field applied normal to the xy-plane, but now with the proviso\nthat the stochastic driving force (noise) is a sum of two uncorrelat ed noise terms\n– an exponentially correlated term and a delta-correlated term – an d there is a\nparametrized ( η−1) deviation from the II-FD relation. Interestingly now, we\ndo obtain a non-zero orbital magnetic moment in the infinite time limit – in\nthe steady state. Moreover, the sign of the orbital magnetic mom ent turns out\nto show a dia- to para-magnetic crossover as the parameter ηis tuned through\nη= 1, where the moment vanishes. In the following we present an exac t analyt-\nical treatment of this dissipative-stochastic system and discuss t he results that\nfollow.\nConsider the classical dissipative-stochastic dynamics of a particle , of charge\n−eand mass m, which is confined bi-harmonically in the xy-plane in the pres-\nence of a uniform static magnetic field Bapplied perpendicular to the plane.\n2The governing stochastic (Langevin) equations are\nm¨x(t) =−kx(t)−/integraldisplayt\n0/parenleftbiggΓ\ntce−(t−t′)/tc+Γ0δ(t−t′)/parenrightbigg\n˙x(t′)dt′−eB\nc˙y(t)+ξx(t)\n(1a)\nm¨y(t) =−ky(t)−/integraldisplayt\n0/parenleftbiggΓ\ntce−(t−t′)/tc+Γ0δ(t−t′)/parenrightbigg\n˙y(t′)dt′+eB\nc˙x(t)+ξy(t),\n(1b)\nwhereξi(t) is the noise term with /an}bracketle{tξi(t)/an}bracketri}ht= 0, and\n/an}bracketle{tξi(t)ξj(t′)/an}bracketri}ht=δi,jmkBT/parenleftbiggγ\ntce−|t−t′|/tc+2ηγ0δ(t−t′)/parenrightbigg\n, (2)\nand we are interested in the long-time limit t→ ∞. Herei=x,y, and the\nangular brackets /an}bracketle{t···/an}bracketri}htdenote average over realizations of the two un-correlated\nnoiseterms –one adelta-correlated(white) noiseand the otheran exponentially\ncorrelated noise with a correlation time tc. This sum of a white noise and\nan exponentially correlated noise, we believe, is the simplest non-Mar kovian\ngaussian process allowed by Doob’s theorem [13]. Further, ( η−1) parametrizes\ndeviation from the II-FD relation as noted above.\nIt is convenient to introduce here the quantities Ω 0=/radicalbig\nk/m(harmonic os-\ncillator circular frequency),eB\nmc= Ωc(cyclotron circular frequency), andΓ\nm=γ\n(the frictional relaxation frequency). We further define the follo wing dimension-\nless parameters τ=γt(dimensionless time), and ω0=Ω0\nγ,ωc=Ωc\nγ(dimension-\nless circular frequencies).\nFollowing now the ’Landau trick’, the two coupled Langevin equations f or\nthe real displacements x(τ) andy(τ) as functions of the dimensionless time τ,\ncan be conveniently combined into a single Langevin equation for the c omplex\ndisplacement z(τ)≡x(τ)+iy(τ), giving:\n¨z(τ) =−ω2\n0z(τ)−/integraldisplayτ\n0/parenleftbigg1\nτce−(τ−τ′)/τc+γ0\nγδ(τ−τ′)/parenrightbigg\n˙z(τ′)dτ′+iωc˙z(τ)+g(τ)\n(3)\nwith/an}bracketle{tg(τ)/an}bracketri}ht= 0 and /an}bracketle{tg(τ)g∗(τ′)/an}bracketri}ht=2kBT\n(mγ2)/parenleftBig\n1\nτce−(τ−τ′)/τc+(2η)γ0\nγδ(τ−τ′)/parenrightBig\n.\nNote the complex conjugation ( ∗) that we have introduced in /an}bracketle{tg(τ)g∗(τ′)/an}bracketri}ht\nabove, as the same will be needed in subsequent calculations. Also, w e have\nchanged over to the dimensionless time parameter ( τ) , but have retained the\nsame symbols for the dynamical variables without the risk of confus ion. Inas-\nmuch as the particle motion along the uniform magnetic field normal to the\nxy-plane decouples from that in the xy-plane, the present model e qually well\ndescribes a 3-dimensional system. The orbital magnetic moment ca n now be\nre-written as\n/an}bracketle{tM(τ)/an}bracketri}ht=eγ\n2cIm/an}bracketle{tz(τ)˙z∗(τ)/an}bracketri}ht. (4)\nIt is convenient now to introduce the Laplace transform\n˜z(s) =/integraldisplay∞\n0e−sτz(τ)dτ,and˜˙z(s) =s˜z(s), (5)\nwith the initial conditions z(0) = 0 = ˙ z(0) (without loss of generality, as we are\ninterested in the steady state (long-time limit τ→ ∞). We obtain straightfor-\n3wardly\n˜z(s) =˜g(s)(1+τcs)/τc/parenleftBig\ns3+/parenleftBig\n1\nτc+γ0\nγ−iωc/parenrightBig\ns2+/parenleftBig\nω2\n0+1\nτc+γ0\nγτc−iωc\nτc/parenrightBig\ns+ω2\n0\nτc/parenrightBig(6)\nand\n˜˙z(s) =s(1+τcs)˜g(s)/τc/parenleftBig\ns3+/parenleftBig\n1\nτc+γ0\nγ−iωc/parenrightBig\ns2+/parenleftBig\nω2\n0+1\nτc+γ0\nγτc−iωc\nτc/parenrightBig\ns+ω2\n0\nτc/parenrightBig,(7)\nwhere we have used ˜˙z(s) =s˜z(s), with the initial conditions z(0) = ˙z(0) = 0 as\nnoted above. In order to inverse-Laplace transform the expres sions above, it is\nconvenient to introduce the factorized denominator\nD(s) =/parenleftbigg\ns3+/parenleftbigg1\nτc+γ0\nγ−iωc/parenrightbigg\ns2+/parenleftbigg\nω2\n0+1\nτc+γ0\nγτc−iωc\nτc/parenrightbigg\ns+ω2\n0\nτc/parenrightbigg\n≡(s−s1)(s−s2)(s−s3), (8)\nwheresi(i= 1,2,3) are the three roots of the cubic denominator D(s). These\nroots are readily obtained following the Cardano procedure. We can then write\n˜z(s) and˜˙z(s) as partial fractions\n˜z(s) =/summationdisplay\niai\ns−si˜gz(s), (9)\nand\n˜˙z(s) =/summationdisplay\niAi\ns−si˜gz(s), (10)\nwhere{ai}are given as solutions of the set of equations\na1+a2+a3= 0\na1(s2+s3)+a2(s3+s1)+a3(s1+s2) =−1\na1s2s3+a2s3s1+a3s1s2=1\nτc, (11)\nand similarly, {Ai}are given by\nA1+A2+A3= 1\nA1(s2+s3)+A2(s3+s1)+A3(s1+s2) =−1\nτc\nA1s2s3+A2s3s1+A3s1s2= 0, (12)\nIn terms of the above, we now have the inverse Laplace transform s as\nz(τ) =/summationdisplay\niai/integraldisplayτ\n0esi(τ−τ′)/τcg(τ′)dτ′(13a)\nand\n˙z(τ) =/summationdisplay\njAj/integraldisplayτ\n0esj(τ−τ′′)/τcg(τ′′)dτ′′. (13b)\n4With this, the orbital magnetic moment in Eq. (4) turns out to be\n/an}bracketle{tM(τ)/an}bracketri}ht=/parenleftBige\nmc/parenrightBig/parenleftbiggkBT\nγτ3c/parenrightbigg\nIm/summationdisplay\ni,j=1,2,3aiA∗\nj/integraldisplayτ/integraldisplay\n0/bracketleftBig\nesi(τ−τ′)/τces∗\nj(τ−τ′′)/τce−|τ′−τ′′|/τc\n+2ηγ0τc\nγesi(τ−τ′)/τces∗\nj(τ−τ′′)/τcδ(τ′−τ′′)/bracketrightbigg\ndτ′dτ′′, (14)\nwhere Im denotes the imaginary part. Straightforward integratio n gives the\nτ→ ∞(steady-state) limit for the orbital magnetic moment as\nM(∞) =/parenleftBige\nmc/parenrightBig/parenleftbiggkBT\nγτ3c/parenrightbigg\nIm/summationdisplay\ni,j=1,2,3aiA∗\nj/bracketleftBigg/parenleftBigg\n1\n1\nτc+s∗\nj/parenrightBigg\n×/braceleftBigg\n2\nτc(s∗\nj−1\nτc)(si+s∗\nj)+1\n(si−1\nτc)/bracerightBigg\n−2ηγ0\nγτc/parenleftBigg\n1\nsi+s∗\nj/parenrightBigg/bracketrightBigg\n.(15)\nAfter some simplification, the above expression for the orbital mag netic mo-\nmentM(∞) reduces to\nµ≡M(∞)\n(ekBT\nmcγτ c)=−2(η−1)γ0τc\nγIm/summationdisplay\ni,j=1,2,3aiA∗\nj/parenleftBigg\n1\nsi+s∗\nj/parenrightBigg\n(16)\nInFig. 1,wehaveplottedthelimitingsteady-statevalueofthedimen sionless\norbital magnetic moment µagainst the dimensionless applied magnetic field\nβ(=eB\nmcγ), where ηparametrizes the II-FD violation and is varied over the\nrange 0.5–2.0. As we are interested here mainly in the matter of princ iples, we\nhave made a simple choice for the dimensionless parameters involved, namely\n1\nτc=ω0= 1 andγ0\nγ= 0.5 (the strength of the white noise relative to the\nexponentially correlated noise).\nAs we observe the field-induced orbital magnetic moment is clearly no n-zero\nin the steady state. There is a crossover from the paramagnetic t o the diamag-\nnetic sign as the parameter ηis tuned from η= 0.5 toη= 2.0. This crossover\nis a surprise. The magnetic moment is zero for η= 1, which corresponds to\nthe canonical II-FD consistent (equilibrium) state. Hence no violat ion of the\nBvL theorem. The orbital magnetic moment is obviously zero for zer o magnetic\nfield; but, not so obviously it tends to zero for large magnetic field as well. The\nlatter behaviour can, however, be understood from the following, namely that\nthe radius/frequency of the cyclotron orbit tends to zero/infinit y as the applied\nmagnetic field is made infinitely large [4]. The fact that the orbital magn etic\nmoment can be paramagnetic in certain range of the II-FD deviation parame-\nterηis significant in that, unlike diamagnetism, it leads to a positive feedbac k\nfor a collection of charged particles in such a classical system – it can give an\nenhancement of the orbital paramagnetism.\nPhysicalrealizationof such a classicalsystem in the laboratoryis ad mittedly\nsomewhat demanding. One needs to create a dilute (highly non-dege nerate) gas\nof charged particles (e.g., electrons/holes) at sufficiently high temp eratures, and\nconfined on a mesoscopic scale in the presence of a static magnetic fi eld. The\ntemperature has to be high enough so as to wash out the quantum e ffects,\nnamely the discreteness of the quantized level spacings owing to th e mesoscopic\n5Figure 1: Plot of dimensionless magnetic moment µagainst dimensionless mag-\nnetic field βfor four different values of ηthat parametrizes deviation from the\nII-FD relation. It clearly shows a dia- to para-magnetic crossover asηis varied\nthrough η= 1. Also, the moment tends to zero in the limit of zero as well as\nlarge magnetic field β.\nconfinement. Now, the II-FD violating parameter ( η−1) necessarily requires a\nnon-equilibrium steady-state condition. This is the real problem for an experi-\nmental realization. One is tempted to think that such a non-equilibriu m steady\nstate may be induced through a noisy laser excitation, e.g., the Kubo -Anderson\nnon-Markoviannoise [14, 15], of charged particles confined in an opt ical tweezer\n[16]. There is, however, a problem here involving the energy injected by the\nlaser, its dissipation in the system and the associated rise of temper ature. In\nfact, one necessarily needs to have a two-temperature configur ation. Thus, e.g.,\na possible physical realizationof our model can, in principle, havecha rgedparti-\ncles in contact with a thermal reservoir at one temperature, while a nother type\nof neutral particles is in contact with the reservoir at a different te mperature.\nThen particle collisions will ensure a stationary (steady-state) non -equlibrium\nstate with flow of energy between the reservoirs. This then accor ding to our\nmodel calculattion should give a non-zero orbital magnetic moment in an exter-\nnally applied magnetic field [17].\nGiven that classically a static magnetic field does no work on a moving\ncharged particle, our model calculation giving a non-equilibrium stead y-state\nsolution in the presence of a static magnetic field could lead to some ins ightful\nmolecular dynamical (MD) simulations when appropriately thermosta tted [18].\nAlso, it has a significant bearing on the work related to generalized flu ctuation-\ndissipation theorem for steady-state systems [19].\nIn conclusion, we have presented an exact analytical treatment o f a classical\ndissipative-stochastic model system in a uniform static magnetic fie ld, which is\nfound to give afinite orbitalmagnetic moment in the steady state. I nterestingly,\nwefindthatthereisacrossoverfromthediamagnetictotheparam agneticsignof\n6the magnetic moment as function of a parametrized deviation from t he second\nfluctuation-dissipation relation. We think that these results do com plement,\nrather than violate the classic Bohr-van Leeuwen theorem.\nThe author would like to thank A.A. Deshpande and K. V. Kumar for ma ny\ndiscussions. The author would also like to thank the referee for con structive\ncriticisms.\nReferences\n[1] J.H.V. Leeuwen, Journal de Physique 2, 361 (1921).\n[2] N. Bohr, Studies over Metallerners Elektrontheori, Ph.D. thesis , Copen-\nhagen, 1911.\n[3] J.H.V. Vleck, The TheoryofElectric andMagnetic Susceptibilities (O xford\nUniversity Press, London, 1932).\n[4] R.E. Peierls, Surprises in Theoretical Physics (Princeton Univers ity Press,\nPrinceton, 1979). The fact that diamagnetism was known much ear lier,\nindeed from the time of Faraday, its absence for a classical system could be\nviewed as the first macroscopic evidence for the incompleteness of classical\n(statistical) mechanics.\n[5] J.D. Jackson, Classical Electrodynamics (John Wiley, New York, 1 975).\n[6] L.D. Landau, Zeits. f. Physik 64, 629 (1930).\n[7] S.K. Ma, Statistical Mechanics (World Scientific, Singapore, 1985 ) p. 283.\n[8] N. Kumar and A.M. Jayannavar, J. Phys. A 14, 1399 (1981).\n[9] S.DattaguptaandJ.Singh, Phys.Rev.Lett. 79,961(1997); S.Dattagupta,\nN. Kumar and A.M. Jayannavar, Curr. Sci. 8, 863 (2001).\n[10] R. Kubo, Rep. Prog. Phys. 29, 255 (1966).\n[11] V. Balakrishnan, Pramana 12, 301 (1979).\n[12] A. A. Deshpande, K. V. Kumar and N. Kumar (unpublished).\n[13] J.L. Doob, Selected Papers on Noise and Stochastic Processes , ed. N. Wax\n(Dover, New York, 1954) p. 319.\n[14] R. Kubo, J. Phys. Soc. Jpn 9, 935 (1954).\n[15] P.W. Anderson, J. Phys. Soc. Jpn. 9, 316 (1954).\n[16] T. Li et al., Science 328, 1673 (2010); R. Huang et al., Nature Ph ysics, p.1,\n27 March 2011.\n[17] The author would like to thank the referee for suggesting the p ossible phys-\nical realization of our model.\n[18] W.G. Hoover, Phys. Rev. A 31, 1695 (1985); S. Nos, J. Chem. P hys. 81,\n511 (1984).\n7[19] J. Prost, J.-F. Joanny and J.M.R. Parrondo, Phys. Rev. Lett. 103, 090601\n(2009)\n8"    },    {        "title": "2209.04483v1.Magnetization_dynamics_and_reversal_of_two_dimensional_magnets.pdf",        "content": "Magnetization dynamics and reversal of two-dimensional magnets\nEssa M. Ibrahim and Shufeng Zhang\nDepartment of Physics, University of Arizona, 1117 E 4th Street, Tucson, AZ 85721\n(Dated: September 13, 2022)\nMicromagnetics simulation based on the classical Landau-Lifshitz-Gilbert (LLG) equation has\nlong been a powerful method for modeling magnetization dynamics and reversal of three-dimensional\n(3D) magnets. For two dimensional (2D) magnets, the magnetization reversal always accompanies\nthe collapse of the magnetization even at the low temperature due to intrinsic strong spin \ructuation.\nWe propose a micromagnetic theory that explicitly takes into account the rapid demagnetization\nand remagnetization dynamics of 2D magnets during magnetization reversal. We apply the theory\nto a single domain magnet to illustrate fundamental di\u000berences of magnetization trajectories and\nreversal times for 2D and 3D magnets.\nINTRODUCTION\nThe classical Landau-Lifshitz-Gilbert (LLG) equation\nis an essential equation for studying magnetization dy-\nnamics driven by an applied magnetic \feld or spin\ntorques. Almost all of magnetic phenomena related to\nthe magnetic structure and dynamics can be approxi-\nmately understood in terms of the LLG equation. The\nbasic assumption in the LLG equation is the invariace of\nthe magnitude of magnetization which is a constant at a\n\fxed temperature, independent of the magnetic \feld or\nother driving sources such as the spin torque. As long\nas the temperature is well below the Curie temperature\nof the 3D magnet, the amplitude of the magnetization\nis mainly controlled by the exchange interaction which\nis usually several oreders of magntitude larger than the\nmagnetic \feld or anisotropy \feld.\nFor 2D magnets, however, the assumption of a constant\namplitude of the magnetization during the dynamic pro-\ncess completely fails at any temperature. To see this,\nconsidering a uniaxial anisotropic magnet with the mag-\nnetization intially aligned in one of the easy direction\ndenoted as m(0) =m0^z. When a magnetic \feld with\nits magnitude same as the anisotropic \feld applied in\nthe opposite direction of the magneization, the sum of\nthe magnetic \feld and the anisotropy \feld becomes zero.\nConsequently, the energy of the quanta of the long wave-\nlength of spin wave excitation, or magnons, scales as\n\u000fk/k2where kis a wave vector. Since the magnons\nare Bosons, it follows that the total number of magnons\nN/R\nd2k[exp(\u000fk=kBT)\u00001]\u00001diverges at the thermal\nequalibrium, i.e., the external magnetic \feld cancels with\nthe anisotropy \feld and thus the magnon spectrum is\ngapless, leading to completely demagnization at any \f-\nnite temperature. The above argument is consistent with\nthe Wagner-Mermin's theory which excludes the long-\nrange magnetic ordering for Heisenberg model without\nthe anisotropy and external \felds. Therefore, when the\nreversal magnetic \feld reaches the anisotropic \feld, the\nmagnetization disapears in its initial direction and in-\nstead, the magnetization spontaneously appears at the\ndirection of the reversal magnetic \feld. We de\fne suchcollapse of the magnetization in the original direction and\nre-appeear in the direction of the applied \feld as demag-\nntization/remagnetization (DMRM).\nOn the contrary, DMRM does not occur for 3D mag-\nnets because the external magnetic \feld has little e\u000bect\non the magnitude of the magnetization as long as the\nmagnetic \feld is much smaller than the exchange energy\nand thus the magnetization reversal is governed by the\nrotation of the local magnetization. In a single domain,\nthe magnetic \feld must be applied with an angle rela-\ntive to the magnetization to generate the rotation. The\nrotating dynamics is usually modelled by the classical\nLLG with the reversal time determined by the damping\nparameter and the magnetic \feld.\nThe question is how fast is the DMRM process com-\npared with the LLG rotation dynamics? The origin of\nthe DMRM is the change of the magnitude of the mag-\nnetization by the thermal magnons. To see the DMRM\ntime scale, we recall the DMRM experiments in which 3D\nmagnetic \flms are exposed to a short-pulsed laser \feld,\ninducing a fast temperature change to the Curie temper-\nature. It has been found the demagnetization is achieved\nwithin a picosecond, and subsequently, the remagnetiza-\ntion follows after the laser \feld is turned o\u000b. The mag-\nnetization dynamics in these experiments involves both\nlongitudinal magnetization dynamics which is the process\nof reaching the thermal equilibrium of magnons, and the\ntransverse magnetization dynamics which is the process\nof rotation of the order parameter (magntization). The\nformer is several orders of magnitude faster than the lat-\nter.\nFor magnetization reversal of a 2D magnet, both longi-\ntudinal and transverse dynamics are present at any tem-\nperature as the reversal inevitably invlove the quench of\nmagnetizaion due to magnetization instability generated\nby the divesgence of the number of magnons. In the next\nSection we propose our model for calculating the magne-\ntization dynamics of 2D magnets.arXiv:2209.04483v1  [cond-mat.mes-hall]  9 Sep 20222\nDYNAMIC EQUATIONS FOR 2D MAGNETS\nTo be more speci\fc for the dynamic equations we pro-\npose, let's start with a spin Hamiltonian of the 2D mag-\nnet\n^H=\u0000JX\n<i;j>^Si\u0001^Sj\u0000AX\n<i;j>^Sz\ni^Sz\nj\u0000X\niH(t)\u0001^Si(1)\nwhere ^Siand ^Sz\niare respectively the spin and the\nz-component (taken as perpendicular to the two-\ndimensional plane) of the spin operators at lattice site Ri,\nJis the isotropic exchange integral, Ais the anisotropic\nexchange integral, < ij > indicates the sum over near-\nest neighbors, and His the time-dependent external \feld.\nBefore the magnetic \feld is turned on, the magnetization\nis initially in the direction of the ferromagnetic ground\nstate. At su\u000ecient low temperature compared to the\nCurie temperature, we can use the random phase approx-\nimation (RPA) to calculate the average magnetization.\nThe resulting self-consistent equation for the magnetiza-\ntion is [23],\nM=Ms\u0000Zd2k\n(2\u0019)22M\ne\fEk\u00001(2)\nwhereMsis the magnetization at zero temperature, and\nEkis the magnon energy; in the long wave length limit,\nEk=zM(Jk2=2+2A) wherezis the number of nearest-\nneighbor sites. Equation (2) has a straightforward expla-\nnation: the magnetization is subtracted by the number\nof the magnon which is softened by the factor of Mat\nthe \fnite temperature. We note that a) Equation (3)\nis the RPA approximation for spin-1/2, the higher spins\nwould lead a more complicated equation and the RPA is\nconsidered an excellent approximation for temperature\nfar below the Curie temperature, and b) we consider\nthe magnetic anisotropy from the anisotropic exchange\nrather than on-site anisotropy in the form of \u0000A(Sz\ni)2.\nBy using the quadratic dispersion in the energy, we may\nintegrate out d2k, resulting a simple analytical expres-\nsion,\nM=Ms\u00001\n\u0019zJ\u00121\n\fln\f\f\f\fe\f(\u0001+W)\u00001\ne\f\u0001\u00001\f\f\f\f\u0000W\u0013\n(3)\nwhere \u0001 = 8 AM2andW= 8\u0019JM (assuming a square\nlattice,z= 4) are the e\u000bective magnon gap and the\nmagnon bandwidth, respectively.\nAtt= 0, a magnetic \feld H(t) turns on. To de-\ntermine how the magnetization proceeds with the mag-\nnetic \feld, we make the following postulations. First,\nthe longitudinal magnetization is determined by Eq. (2),\nin which the band gap is replaced by the total e\u000bective\n\feld \u0001 = 8 AM2+M\u0001H(t), i.e., the magnon distribu-\ntion reaches the thermal equlibrium much fast than the\nchange of the magnetic \feld H. As we stated earlier,this assumption has been supported by the experiments\non laser-induced magnetization dynamics in which the\ntime scale to reach thermal equilibrium is less than 1\npicosecond. When \u0001 becomes zero or a negative value\nat a timet, the magnetization will immediately drops\nto zero; this is consistent with the Wagner-Mermin the-\norem. Susequently, the magnetization reappears in the\ndirection of the total magnetic \feld. The magnetization\nswitching through this DMRM process is the new physics\nfor the magnetization reversal. When the gap \u0001 >0,\nthe solution of Eq. (3) detrmines the magnitude of the\nmagnetization as a function time. Then our second pos-\ntulation is the transverse dynamic which describes the\nrotation of the magnetization given by the conventional\nLLG equation,\ndM\ndt=\u0000\rM\u0002Heff+\u000bdM\njMj\u0002dM\ndt(4)\nwhere\ris gyromagnetic ratio, \u000bis the damping param-\neter, and Heffthe total e\u000bective magnetic \feld.\nThe above two hypothesis on the longitudinal and tran-\nverse dynamics completely determine the magnetization\ndynamics of 2D magnets. The proposed dynmic equa-\ntions combine the quantum Boson statistics and the clas-\nsical equation of motion. Next, we shall \frst demonstrate\nhow the above dynamics di\u000ber from the conventional 3D\nmagnets in a simplest single domain particle.\nAPPLICATION TO 2D SINGLE DOMAIN\nLet us consider a single domain 2D magnet with the\ninitial magnetization aligned in + ^ z. If the magnetic \feld\nis applied in the\u0000^ zdirection, there will be no transverse\ndynamics without a thermal kick to allow the magneti-\nzation deviating from ^zdirection. For a 3D magnet in\nwhich the dynamics is always rotational, the time to re-\nverse the magnetization is long even the magnetic \feld is\nmuch larger than the anisotropy \feld. For 2D, however,\nas long as the magnetic \feld is larger than anisotropy\n\feld, the switching to the reversel direction is considered\nimmediately via the demagnetization and remagnetiza-\ntion in the direction of the magnetic \feld. More interest-\ning case is the reversal magnetic \feld in a direction not\nparallel to the\u0000^ zdirection. As an example, we consider\nthe magnetic \feld at 45owith respect to\u0000^ z. In this case,\nthe magnetization trajectories display several distinct be-\nhaviors as shown in Fig. (1). For a small magnetic \feld,\nmagnetization dynamics is governed by the rotation with-\nout reversal and the tranjectories are almost identical for\n2D and 3D magnets (except that the amplitude of the\nmagnetization has a small variation in the trajectories).\nWhen the external \feld reaches a critical value Hc,\nthe magnetization trajectories for 2D and 3D magnets\nqualitatively di\u000ber. In 3D, the magnetization processes\nwith spiral rotation towards the \fnal equlibrium position3\nas shown in Fig. 2(h) while for 2D, the magnetization\nstarts rotation for a short period of time before the gap\n\u0001 becomes zero, at which time the demagnetization oc-\ncurs and the magnetization appears at the direction of\nthe \feld. After the remagnetization, the anisotropy \feld\nwill make the magnetization rotates to the \fnal equilib-\nrium direction determined by the competition between\nthe external \feld at 45oand the anisotropy \feld at \u0000^z,\nas shown in Fig. 2(c). Thus, the magnetization reversal\nhas three distinct processes: rotation, DMRM, and the\n\fnal rotation.\nIf one further increases the external magnetic \feld, the\nmagnon energy becomes negative for the initial state of\nthe magnetization. In this case, the swithing immedi-\nately occurs via DMRM without the prior rotation. The\n\fnal rotation characterizes the transverse dynamics that\ndescrbs the rotation of the magnetization from parallel\ntoHto the total \feld direction, as noted earlier.\nAn important measure for magnetic memory elements\nis the switching time which is de\fned as the time tsfor\nthe magnetization crossing the \\equater\" from the initial\nposition. For the 3D magnets, the time scale is limited\nby the product of the damping parameter and the ap-\nplied \feld, typically of the order of sub-nanoseconds to\na few nanoseconds. The switching time could be much\nshorter with the DMRM process of the 2D magnets. In\nFigure x, we show the swicthing time as the function of\nthe applied \feld. If the \feld is too small, both 2D and\n3D are unable to switch the magnetization. At an in-\ntermediate \feld, the 2D magnet switching is much faster\nbecause the magnetization rotation is only a small por-\ntion of the reversal projectory before the DMRM process.\nAt a su\u000eciently large \feld, the 2D magnet switching is\ninstantaneous, only limited by the relaxation mechanism\nof thermal magnons.\nAs the DMRM process depends on the temperature,\nwe show temperature dependence of the switching time\nin Fig.xx.|discussion on the \fgure follows.\nDISCUSSION AND CONCLUSION\nWe have proposed the magnetization dynamic equa-\ntions by considering the longitudinal and transverse re-\nlaxations. The logitudinal process is governed by the\nquantum statistics, i.e., Boson statistics of magnons.\nWe postulate that the equalibrium distribution of the\nmagnons for a given spin Hamiltonian is much faster\nthan the classical transverse dynamics. Our proposed\nquantum-classical magnetization dynamic processes in-\ntroduce a novel demagnetization and remagnetization dy-\nnamics for 2D magnets which is the key physics that\nmakes the magnetization reversal much faster. Our ex-\nample for a single domain gives quantitative di\u000berent be-\nhaviors for 2D and 3D magnets.\nThe calculation scheme we proposed here shall replace\nFIG. 1. Comparison between the magnetization trajectories\nfor the 3D case (Right) and the 2D case (Left) at di\u000berent\nintensities of the external magnetic \feld (from the top to the\nbottomHex= 0:2J;0:3J;0:4J;0:5J). The dotted line repre-\nsents the process of DMRM and the blue line represents the\nexternal magnetic \feld.\nthe convensional micromagnetics which do not consider\nthe DMRM physics. Since the DMRM is fundamen-\ntally present in 2D magnets as shown in Wagner-Menmin\ntheory, any micromagnetic calculations on 2D materials\nmust take into account DMRM dynamics.\nThis work was partially supported by the U.S. National\nScience Foundation under Grant No. ECCS-2011331.\n[1] B. Huang, et al, (2017) Layer-dependent ferromagnetism\nin a van der Waals crystal down to the monolayer limit\nNature 546, 270.\n[2] Gong, C. et al., (2017) Discovery of intrinsic ferromag-\nnetism in two-dimensional van der Waals crystals, Nature\n546, 265{269.\n[3] D. J. O'Hara, et al., (2018) Room Temperature Intrinsic\nFerromagnetism in Epitaxial Manganese Selenide Films\nin the Monolayer Limit, Nano Lett. 18, 3125.\n[4] D. R. Klein, et al., (2018) Probing magnetism in 2D van4\nFIG. 2. The switching time for di\u000berent intensities of external\nmagnetic \feld. Starting from a threshold value of Hex, we\n\fnd that increasing the external \feld continuously reducing\nthe switching time until the point when the \feld is strong\nenough to make the energy gap collapses before the smooth\nswitching process \fnishes, which in turn reduces the switching\ntime even more. And after Hex>2zAM s(T), the DMRM\nprocess happens instantaneously. We can also notice that\nas the temperature decreases, the behaviour reduces to the\nclassical 3D known behaviour. For this graph we used z= 4,\nA= 0:25J,and\u000b= 0:2.\nder Waals crystalline insulators via electron tunneling,\nScience 360, 1218.\n[5] S. Jiang, et al., (2018) Controlling magnetism in 2D CrI 3\nby electrostatic doping, Nature Nanotechnol. 13, 549.\n[6] J. U. Lee, et al., (2016) Ising-Type Magnetic Ordering in\nAtomically Thin FePS 3, Nano Lett. 16, 7433{7438.\n[7] B. Huang et al., (2020) Emergent phenomena and\nproximity e\u000bects in two-dimensional magnets and het-\nerostructures, Nat. Mater. 19, 1276.\n[8] C. Gong and X. Zhang, (2019) Two-dimensional mag-\nnetic crystals and emergent heterostructure devices, Sci-\nence363, 4450.\n[9] Song, T. et al., (2018) Giant tunneling magnetoresis-\ntance in spin-\flter van der Waals heterostructures, Sci-\nence360, 1214{1218.[10] V. Gupta et al., (2020) Manipulation of the van der Waals\nMagnetCr2Ge2Te6by Spin{Orbit Torques, Nano Lett.\n20, 7482.\n[11] D. MacNeill, et al., (2017) Control of spin{orbit torques\nthrough crystal symmetry in WTe 2/ferromagnet bilay-\ners, Nature Phys. 13, 300.\n[12] M. Alghamdi, et al., (2019) Highly E\u000ecient Spin{Orbit\nTorque and Switching of Layered Ferromagnet\nFe3GeTe 2, Nano Lett. 10, 4400.\n[13] X. Wang, et al., (2019) Current-driven magnetization\nswitching in a van der Waals ferromagnet Fe3GeTe 2, Sci-\nence Adv. 5, 8904.\n[14] C. Fang, et al., (2019) Observation of large anomalous\nNernst e\u000bect in 2D layered materials Fe3GeTe 2, Appl.\nPhys. Lett. 115, 212402.\n[15] T. Liu, et al., (2020) Spin caloritronics in a CrBr 3-based\nmagnetic van der Waals heterostructure, Phys. Rev. B\n101, 205407.\n[16] N. Ito, et al., (2019) Spin Seebeck e\u000bect in the layered\nferromagnetic insulators CrSiTe 3andCrGeTe 3, Phys.\nRev. B. 100, 060402(R).\n[17] J. Xu, et al., (2019) Large Anomalous Nernst E\u000bect in a\nvan der Waals Ferromagnet Fe3GeTe 2, Nano Lett. 19,\n8250.\n[18] Stoner E. C. and Wohlfarth E. P. (1948) A mechanism\nof magnetic hysteresis in heterogeneous alloys, Philos.\nTrans. Royal Soc. A 240(826): 599{642.\n[19] C. Tannous and J. Gieraltowski (2008) The\nStoner{Wohlfarth model of ferromagnetism, Eur. J.\nPhys. 29475\n[20] N. D. Mermin and H. Wagner, (1966) Absence of Fer-\nromagnetism or Antiferromagnetism in One- or Two-\nDimensional Isotropic Heisenberg Models, Phys. Rev.\nLett.17, 1133.\n[21] Igor \u0014Zuti\u0013 c, et al., (2004) Spintronics: Fundamentals and\napplications, Rev. Mod. Phys. 76, 323.\n[22] D. L. Cortie, et al., (2020) Two-Dimensional Magnets:\nForgotten History and Recent Progress towards Spin-\ntronic Applications, Adv. Func. Mater. 30, 1901414.\n[23] P. Tang, X. F. Han, and S. Zhang, (2021) Quantum the-\nory of spin-torque driven magnetization switching, Phys.\nRev. B 103, 094442."    },    {        "title": "2311.09900v1.Meridional_Circulations_of_the_Solar_Magnetic_Fields_of_Different_Strength.pdf",        "content": "arXiv:2311.09900v1  [astro-ph.SR]  16 Nov 2023Solar Physics\nDOI: 10.1007/ •••••-•••-•••-••••-•\nMeridional Circulations of the Solar Magnetic Fields\nof Different Strength\nIrina A. Bilenko ••\n©Springer ••••\nAbstract The meridional circulation of the solar magnetic fields in Solar Cycles\n21–24 was considered. Data from both ground-based and space o bservatories\nwere used. Three types of time-latitude distributions of photosph eric magnetic\nfieldsandtheirmeridionalcirculationswereidentifieddependingonth emagnetic\nfield intensity. (i) low-strength magnetic fields. Positive- and negat ive-polarity\nmagneticfieldsweredistributedevenlyacrosslatitudeandtheywea klydepended\non the magnetic fields of active regions and their cycle variation; (ii) m edium-\nstrengthmagneticfields.Forthesepositive-andnegative-polarit ymagneticfields\na wave-like, pole-to-pole, antiphase meridional circulation with a per iod of≈22\nyears was revealed. The velocities of meridional flows were slower at the minima\nof solar activity, when they were at high latitudes in the opposite hem ispheres,\nand maximal at the solar maxima, when the positive- and negative-po larity\nwaves crossed the equator. The meridional circulation of these fie lds reflects the\nsolarglobalmagneticfielddynamicsanddeterminesthesolarpolarfie ldreversal;\n(iii) high-strength (local, active region) magnetic fields. They were d istributed\nsymmetrically in the Northern and Southern hemispheres. The magn etic fields\nof active regions were formed only during the periods when the posit ive- and\nnegative-polarity waves of medium-strength magnetic fields appro ached at low\nlatitudes.Magneticfieldsofbothleadingandfollowingsunspotpolarit ymigrated\nfrom high to low latitudes. The meridional-flow velocities of high-stren gth mag-\nnetic fields were higher at the rising and maxima phases than at the min ima.\nSome of the high-latitude active region magnetic fields were capture d by the\nsecond type meridional circulation flows and transported along with them to\nthe appropriate pole. But the magnetic fields of active regions are n ot the main\nones in the solar polar field reversal. The results indicate that high-s trength\nmagnetic fields were not the main source of weak ones. The butterfl y diagram is\nthe result of a superposition of these three types of magnetic field time-latitude\ndistributions and their meridional circulation. The results suggest t hat different\nI. A Bilenko\nbilenko@sai.msu.ru\nSternberg Astronomical Institute Moscow M.V. Lomonosov St ate University,\nUniversitetsky pr.13, Moscow, 119234, Russia\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 1Irina A. Bilenko\nstrength magnetic fields have different sources of their generatio n and cycle\nevolution.\nKeywords: Magnetic fields, Latitudinal drifts, Meridional flow, Solar Cycle\n1. Introduction\nSolar plasma meridional circulation play a major role in the solar magnet ic\nfield dynamics. Meridional flows globally redistribute the solar plasma, heat,\nangular momentum, and magnetic fields in all layers from convective z one to\nthe solar corona. Meridional flows are important in explaining the sola r dynamo\nand differential rotation. They play a key role in the flux-transport dynamo\nmodels (Sheeley, 2005; Jiang et al., 2014; Hanasoge, 2022). Cycle variations of\nthe meridionalflowhavebeen suggestedto explainthe changesin th e amplitudes\nand lengths of the solar activity cycles (Charbonneau, 2020; Karak, 2023) and\nthey also used for their prediction (Petrovay, 2020). Solar meridional circulation\nis considered to be an axisymmetric flow system, extending from the equator to\nthe poles with ≈20 m s−1at the solar surface.\nVarious methods and traces are used to identify and measure merid ional\nflows. Poleward flow has been detected using the Doppler method. D uvall and\nJr. (1979) found a poleward flow of 20 m s−1approximately constant over the\nlatitude range of 10◦–50◦. Studying giant velocity features about 15◦in latitude\nand 30–60◦in longitude with amplitudes around 40 m s−1on the solar surface,\nHoward( 1979) havealso found a meridional flow towardthe poles of the order of\n20 m s−1. Using doppler velocities Hathaway ( 1996) determined an antisymmet-\nric flow towards the poles of 20 m s−1about the equator in both hemispheres.\nFor bright Ca+-mottles Schroeter and Woehl ( 1975) have found a systematic\nmeridional motion of about 0.1 km s−1for latitudes around 10◦. Topka et al.\n(1982) shown that polar filaments track the boundary latitudes of the un ipolar\nmagnetic regions and drift poleward with the regions at about 10 m s−1. They\nnoted that such a filament motion cannot be explained by diffusion alon e, and\nthat a poleward meridional flow carries magnetic flux of both polaritie s along\nwith it.\nSmall magnetic elements are often used to determine the velocity of merid-\nional flows (Rightmire-Upton, Hathaway, and Kosak, 2012). Komm, Howard,\nand Harvey ( 1993) found poleward meridional flow velocity of the order of\n10 m s−1in each hemisphere for small magnetic elements using high-resolution\nmagnetograms taken from 1978 to 1990 with the NSO Vacuum Telesc ope on\nKitt Peak. The meridional flow was found to change during solar cycle . They\nshowed that meridional flow increased in amplitude from the equator , reached\na maximum of 13.2 m s−1at 39◦, and decreased poleward. They found no\nsignificant hemispheric asymmetry and no equator ward migration. H owever,\nanalyzing the Mt. Wilson magnetograms, Snodgrass and Dailey ( 1996) found\nthat for latitudes near the equator the flow was towards the equa tor.\nMeridional flows were found to change during solar cycles. Hathawa y and\nRightmire ( 2010) using SOHO data showed that the average flow speed of mag-\nnetic features was poleward at all latitudes up to 75◦. They found that the flow\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 2Meridional Circulation of Magnetic Fields\nwas faster at sunspot cycle minimum than at maximum and it was chang ed\nfrom cycle to cycle. In Hathaway and Upton ( 2014) it was also found that the\nmeridional flow speed of magnetic elements was fast at solar cycle min imum\nand slow at maximum. The meridional flow weakening on the poleward sid es of\nthe active region latitudes and it was slower at Solar Cycle 23. They pr oposed\nthat an inflow towards the sunspot zones was superimposed on a mo re general\npoleward meridional flow. Imada et al. ( 2020) found that the average meridional\nflow in Solar Cycle 24 was faster than that in Solar Cycle 23.\nMeridional motions have also been determined using coronal bright p oints\nfrom SOHO/EIT, Yohkoh/SXT, Hinode, and SDO/AIA data (Vrˇ sna k et al.,\n2003; Sudar et al., 2016). Vrˇ snak et al. ( 2003) noted that a velocity pattern of\nbright point motions reflected the large-scaleplasma flows. In a com plex pattern\nof meridional motion, they found that the equator ward flows were dominated\nat low (B <10◦) and high ( B >40◦) latitudes, and a poleward flow at mid-\nlatitudes ( B≈10◦–40◦). Faster flow tracers had equator ward motion and the\nslower ones showed poleward motion.\nHoward and Gilman ( 1986) studying the meridional motions of sunspots and\nsunspot groups found a midlatitude northward flow with a few hundr edths of\na degree per day in each hemisphere. For sunspot groups, a gener ally poleward\nmotions at higher latitudes was determined. Sudar et al. ( 2014) considered the\nlocation measurements of sunspot groups covering 1878–2011. T hey found that\nthe meridional motion of sunspot groups is towards the centre of a ctivity from\nall latitudes and in all solar cycle phases. The range of meridional velo cities was\nfound to be ±10 m s−1.\nHelioseismologymakes it possible to study meridional flows at different layers\nin the solar interior (Gizon, 2004). Chou and Dai ( 2001) studying the subsurface\nmeridionalflowasafunction oflatitudeanddepth from1994to2000 ,havefound\nthat the velocity of meridional flow increased when solar activity dec reased. A\nnew meridional flow component at about 20◦appeared in each hemisphere as\nsolar activity increased. At low latitudes, the new flow changed from poleward\nat solar minimum to equatorward at solar maximum. The velocity of the new\ncomponent increases with depth. In Basu and Antia ( 2003), it was revealed that\nthe meridional flows showed solar activity-related changes. The an tisymmetric\ncomponent of the meridional flow decreased in speed with activity. B asu and\nAntia (2010) have shown that meridional-flow speed increases with depth. For\nSolar Cycle 23, they found that solar meridional flows in the outer 2% of the\nsolar radius was connected with a flow pattern drifting equator war d in parallel\nwiththe activitybelts. Thedifferentflowcomponentswasfoundtoh avedifferent\ntime dependencies, and the dependence was different at different d epths. Zhao\net al. (2013) using the helioseismology observations from the SDO/HMI, found\nthe poleward meridional flow of a speed of 15 m s−1from the photosphere to\nabout 0.91 R ⊙and an equator ward flow of a speed of 10 m s−1between 0.82\nand 0.91 R ⊙. They also found that the meridional flow turned poleward again\nbelow 0.82 R ⊙.\nFrom all of the above it follows that in various studies, different auth ors\ndefined meridional flows of different directions and velocities. In som e studies\nsignificant temporal variations in the meridional flows were found, t hough they\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 3Irina A. Bilenko\nwere not always in complete agreement with each other. The purpos e of this\nstudy is to determine time-latitude variations in meridional flows by an alyzing\nthe dynamics of various strength positive- and negative-polarity p hotospheric\nmagnetic fields. Section 2describes the data used. In Section 3, the time-latitude\ndistributions of the large-scale solar magnetic fields of positive and n egative\npolarity in Cycles 21–24 are investigated. The time-latitude distribut ions of\npositive- and negative-polarity photospheric magnetic fields using t he data with\nhigh spatial resolution are analyzed in Section 4. Variations in the magnitudes of\nthe photosphericmagnetic fields ofdifferent strength areconside red in Section 5.\nMeanlatitudesandvelocitiesofdifferenttype meridionalcirculations arestudied\nin Section 6. The results are discussed in Section 7. The main conclusions are\nlisted in Section 8.\n2. Data\nSynoptic magnetic field data from ground based WSO (Wilcox Solar Obs erva-\ntory, (Scherrer et al., 1977; Duvall et al., 1977; Hoeksema, Wilcox, and Scherrer,\n1983), NSO KPVT (Kitt Peak Vacuum Telescope, Livingston et al. ( 1976a,b);\nJones et al. ( 1992)) and SOLIS/VSM (Synoptic Optical Long-Term Investi-\ngations/Vector Stokes Magnetograph, Keller, Harvey, and Giamp apa (2003)),\nand space based SOHO/MDI (Solar and Heliospheric Observatory/M ichelson\nDoppler Imager, Scherrer et al. ( 1995)) and SDO/HMI (Solar Dynamics Obser-\nvatory/Helioseismic and Magnetic Imager, Scherrer et al. ( 2012); Schou et al.\n(2012)) observatories were used. These observatories were chosen b ecause they\nrepresent the most often used data that overlap the greatest t ime interval of\nobservations. All synoptic maps were used in their original format w ithout any\ninterpolation or changes. Synoptic maps are maps of the magnetic fi eld latitude-\nlongitude distribution, created on the base of daily observations. E ach synoptic\nmap span a full Carrington Rotation (CR, 1 CR = 27.2753 days). The e ntire\ndata set consists of 616 synoptic maps and covers CRs 1642–2258 (May 1976–\nMay 2022). Data on magnetic fields are given in a longitude versus sine -latitude\ngrid.\nWSO provides the longest homogeneous series of observational da ta of the\nlow-resolution large-scale photospheric magnetic fields (Hoeksema and Scherrer,\n1988) since 1976 without major updates of magnetograph. The WSO syn optic\nmaps represent the radial component of the photospheric magne tic field, de-\nrived from observations of the line-of-sight field component by ass uming the\nfield to be approximately radial. WSO aperture size is 3 arc-min, which m eans\napproximately 33 pixels in longitude in the equator. WSO synoptic magn etic-\nfield maps consist of 30 data points in equal steps of sine latitude fro m 70◦S to\n70◦N. Longitude is presented in 5◦intervals. To convert WSO measured data to\nunits of Gauss, a factor of 1.85 is required (Riley et al., 2014). ”F-data” files,\nwhere missing data are interpolated, were applied. WSO coronal mag netic field\nsynoptic maps calculated from large-scale photospheric fields with a potential\nfield radial model with the source-surface location at 2.5 R ⊙(Schatten, Wilcox,\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 4Meridional Circulation of Magnetic Fields\nand Ness, 1969; Altschuler and Newkirk, 1969; Hoeksema, Wilcox, and Scherrer,\n1983) were also used.\nObservations from KPVT (from 1976 to 2003, CRs 1625–2007) and SOLIS\n(from 2003 to 2017, CRs 2007–2127) were used to study the high- resolution\nphotospheric magnetic fields. At the KPVT, the pixel size of images w as 1.0”\nfrom CR 1625 to CR 1853, when a 512-channel Babcock type instru ment was\nused, and 1.14” after CR 1855, when a CCD spectromagnetograph was used.\nSOLIS’s resolution was 1.125” until 2010, and 1.0” after that. Ever y KPVT and\nSOLIS synoptic map consists of 360 ×180pixels of magnetic field strength values\nin Gauss. KPVT and SOLIS CR maps span 0◦–360◦solar longitude and from\n90◦S to 90◦N sine solar latitude.\nMDI instrument on board the SOHO spacecraft was operational fr om 1996\nto 2011. There were data gaps in SOHO for June–October 1998 and Jan-\nuary–February 1999. MDI synoptic map consists of 3600 ×1080pixels. MDI was\nsucceeded by the HMI in 2010, with a short overlapping period. HMI p rovides\nmagneticfielddatawithmuchhigherspatialandtemporalresolution sandbetter\nquality. The HMI synoptic maps have a size of 3600 ×1440 pixels.\nSynoptic data presented by WSO, KPVT, SOLIS, MDI, and HMI are d iffer-\nent (Virtanen and Mursula, 2016,2017,2019). The observatories use different\ninstruments, observation methods, data processing methods, d ifferent spectral\nlines, and different technic used for synoptic maps construction. T heir data have\ndifferent spatial and spectral resolutions. The instrumentation u sed at ground-\nbased observatories changed during the period concerned. The m ethod of polar\nmagnetic field interpolation in synoptic maps used in different observa tories\nalso cause differences between the data sets. Therefore the valu es of magnetic\nfield strength from different observatories differ significantly. For comparison of\nmagnetic field data obtained by different instruments, the scaling co efficients\nwere proposed (Pietarila et al., 2013; Riley et al., 2014).\n3. Distributions and Meridional Flows of Large-Scale\nMagnetic Fields\nButterfly diagrams are the main means of studying solar meridional fl ows and\ncreating various dynamo models. When constructing a butterfly dia gram, the\nlongitude-average total of the magnetic field for all latitudes for e ach CR is\ncalculated. The result is a diagram of the change in the average value of the\nmagnetic field in latitude with time. Information about the changes in t he lon-\ngitudinal distribution of magnetic fields in each CR is lost. Figure 1(a) shows\nthe time-latitude distribution oflongitudinally averagedlarge-scalep hotospheric\nmagnetic fields, the so-called butterfly diagram. Red color indicates magnetic\nfields of positive polarity and green that of negative polarity. Differe nt struc-\ntures of magnetic fields of positive and negative polarity were disting uished at\ndifferent phases of Solar Cycles 21–24. A change in the sign of the ma gnetic\nfield at the North and South poles of the Sun is clearly seen. The merid ional\ncirculation is believed to be directed from the equator to the poles on the\nsurface of the Sun. But if similar diagrams are created separately f or positive-\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 5Irina A. Bilenko\nFigure 1. Distributions of large-scale photospheric magnetic fields (WSO). (a) Butterfly\ndiagram. Time-latitude distributions of longitude-avera ged positive-polarity (b) and nega-\ntive-polarity (c) photospheric magnetic fields. (d) Superp osition of the distributions of positive-\nand negative-polarity magnetic fields shown in panels (b) an d (c). Color indicates the field\nintensity. Red denotes the positive-polarity and green den otes the negative-polarity magnetic\nfields. The maximum and minimum of each cycle are market at the top.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 6Meridional Circulation of Magnetic Fields\nand negative-polarity magnetic fields, then their time-latitude distr ibutions will\nbe completely different (Figure 1(b and c)). In these diagrams, low-strength\npositive- and negative-polarity magnetic fields revealed wave-like po le-to-pole\nmeridional flows. The high-strength active region magnetic fields mig rate from\nhigh latitudes to the equator. Red and green arrows show the direc tions of the\nmeridional wave flows on the photosphere in each cycle (Figure 1(a–d) as well\nas on the corresponding panels of all subsequent figures). The me ridional wave\nflows of positive- and negative-polarity magnetic fields were antipha se in each\ncycle. Each wave spanned a large range of latitudes in a CR. They cro ssed the\nequator at the period of sunspot maximum in each cycle. During perio ds of solar\nactivity minimum, they were at opposite poles. The period of waves wa s equal\nto two solar cycles (approximately 22 years). These waves are a ma nifestation\nof the meridional circulation of the large-scale magnetic fields of pos itive and\nnegative polarity in each cycle.\nFigure1(d) presents the superposition of the distributions of positive- an d\nnegative-polarity magnetic fields shown in Figure 1(b and c). Comparison of\ndiagrams in Figure 1(a and d) shows that they are almost the same. Some small\ndifferences are due to the fact that the average value of total ma gnetic fields for\neachCR isnot equal tothe sum ofthe averagesofpositive-andneg ative-polarity\nmagnetic fields for the same CR.\nMeridional flows of large-scale positive- and negative-polarity magn etic fields\nwith different strength are shown in Figure 2for magnetic fields 0 <|B|≤0.6 G\nand in Figure 3for magnetic fields |B|≥1 G. Note that the color bars in\nFigures1–3aredifferent.MagneticfieldvariationsinFigure 2reflectthedynam-\nics of low- and medium-strength magnetic fields. The distributions of magnetic\nfields in the range of 0 <|B|≤0.6 G were almost the same. Both positive-\nand negative-polarity magnetic fields were distributed evenly from t he South to\nthe North pole. Pole-to-pole meridional waves of positive- and nega tive-polarity\nmagnetic fields appear from approximately |B| ≈0.6 G. They were antiphase\nand antisymmetrical with respect to the equator. At the solar cyc le minima\nthe wave-like meridional flows were shifted to high latitudes in opposit e hemi-\nspheres. The change in their meridional flow direction occurred sys tematically\nin the rising and ascending phases of each solar cycle. During periods of solar\nmaximum, the meridional flows of positive and negative polarity got clo ser and\napproachedthe equator.Thentheycrossedtheequatorandco ntinuedtomigrate\nto the opposite poles. From Figures 1and2it follows that these meridional flows\ndetermine the process of solar polar field reversal.\nAs the magnetic field strength increases, the wave-like meridional fl ows dis-\nappear. The active region (high-strength) magnetic fields remain o nly. Figure 3\npresents the time-latitude distributions of large-scale longitude-a veraged high-\nstrength positive- and negative-polarityphotospheric magnetic fi elds in different\nranges. The magnetic fields of active regions began to dominate initia lly in\nlow Solar Cycles 23 and 24. In high Solar Cycles 21 and 22, the meridiona l\nwave flows were observed up to |B|≈3 G. Whereas in low Solar Cycles 23\nand 24, these meridional flows ceased to be observed at lower magn etic fields,\n|B|≈1 G. Meridional flows of active region magnetic fields both positive and\nnegative (leading and following sunspot) polarity were symmetric with respect\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 7Irina A. Bilenko\nFigure 2. Time-latitude distributions of large-scale positive- and negative-polarity\nlow-strength photospheric magnetic fields in different rang es (WSO). (a) 0 < B≤0.6 G;\n(b)−0.6≤B <0 G; (c) 0 .6< B≤0.8 G; (d) −0.8≤B <−0.6 G; (e) 0 .8< B≤1 G; (f)\n−1≤B <−0.8 G. Designations are the same as in Figure 1.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 8Meridional Circulation of Magnetic Fields\nFigure 3. Time-latitude distributions of large-scale positive- and negative-polarity high-\n-strength photospheric magnetic fields in different ranges ( WSO). (a) 1 < B≤3 G; (b)\n−3≤B <−1 G; (c) 3 < B≤7 G; (d) −7≤B <3 G; (e) B >7 G; (f) B <−7 G.\nDesignations are the same as in Figure 1.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 9Irina A. Bilenko\nFigure 4. Time-latitude distributions of large-scale magnetic field s calculated at the source\nsurface (2.5 R ⊙, WSO). (a) Butterfly diagram. Time-latitude distributions of longitude-aver-\naged positive-polarity (b) and negative-polarity (c) magn etic fields. (d) Superposition of the\ndistributions of positive- and negative-polarity magneti c fields shown in panels (b) and (c).\nTime-latitude distributions of positive-polarity (e) and negative-polarity (f) magnetic fields in\nthe range of 2 <|B|≤7 G. Designations are the same as in Figure 1.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 10Meridional Circulation of Magnetic Fields\nto the equator and they migrated from high to low latitudes in the Nor thern\nand Southern hemispheres in all ranges.\nFigure4shows similar magnetic field meridional circulation diagrams for\ncoronal magnetic fields calculated from large-scale photospheric fi elds (WSO)\nwith a potential radial field model with the source-surface location at 2.5 R ⊙.\nThesamearrowsareplottedonallpanels,asinFigure 1.Itshouldbeemphasized\nthat the magnetic fields of active regions are absent from these dia grams, since\ntheyarebelievedtobemainlyconfinedbelowthesourcesurface.Th ereforelarge-\nscale magnetic field dynamics is only presented. The dynamics of magn etic fields\non the source surface reflects the cycle variations of the solar glo bal magnetic\nfield. Figure 4(a) presents the total distribution of all magnetic fields. In Fig-\nure4(b), the time-latitude distribution of longitude averaged positive-p olarity\nmagnetic fields and in Figure 4(c), that of the negative-polarity are presented.\nSuperposition of positive- and negative-polarity magnetic field distr ibutions is\nshown in Figure 4(d). Latitudinal distributions ofpositive- and negative-polarity\nmagnetic fields in the range of 2 <|B|≤7 G (without polar fields) arepresented\nin Figure 4(e and f). The wave-like, pole-to-pole, antiphase meridional flows o f\nthe positive- and negative-polarity magnetic fields stand out even m ore clearly\nin Figure 4.\nThus, three types of time-latitude distributions and meridional circ ulations of\nlarge-scalemagnetic fields, depending on the field strength, have b een identified.\nThe first one is the distribution of low-strength positive- and negat ive-polarity\nmagnetic fields (0 <|B|≤0.6 G). They distributed uniformly across the solar\ndisk andsymmetricallywith respectto the equator.The secondtyp e is the wave-\nlike, antiphase, pole-to-pole meridional circulation of medium-stren gth positive-\nandnegative-polaritymagneticfieldsintherangeof0 .6<|B|≤3.0G.Thethird\ntype is the well known distribution of strong magnetic fields of active regions\nthat show meridional flows from high latitudes toward the equator f or both\npositive-andnegative-polarity(leadingandfollowingsunspotpolarit y)magnetic\nfields. From Figures 2and3it follows that the time-latitude distributions of\nlow- and high-strength magnetic field are very different from each o ther. This\nindicates the independent formation of low-strength magnetic field s and argues\nin favor of the small-scale dynamo theory. The meridional circulation of the\nmedium-strength photospheric magnetic fields is also very different from that of\nactive regions. This suggests that they have different sources of their generation\nand cycle evolution. The wavy pole-to-pole meridional circulation of m edium-\nstrength magnetic fields indicates that it is these magnetic fields tha t determine\nthe process of solar polar field reversal, and not the magnetic fields of active\nregions.\nThe butterfly diagram is the result of a superposition of magnetic fie lds of\ndifferent strengths and their meridional circulations. The various d etails and\nstructures in the butterfly diagram, for example poleward surges , are the result\nof the domination of magnetic field polarity of one of the meridional cir culation\ntype.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 11Irina A. Bilenko\n4. Distributions and Meridional Flows of High Spatial\nResolution Magnetic Fields\nThe resolution of WSO magnetic fields is 3 arc-min. Therefore, WSO ma gnetic-\nfield data show the dynamics of low-resolution large-scale magnetic fi elds. To\ninvestigate the distribution and meridional circulation of high-resolu tion mag-\nnetic fields, synoptic data from ground based KPVT and SOLIS and s pace\nSOHO/MDI and SDO/HMI observatories were used. Figure 5shows the time-\nlatitude distributions of longitude-averagedpositive- and negative -polarity mag-\nnetic fields at high spatial resolution (KPVT, CRs 1625–2007 and SOL IS, CRs\n2008–2196). Figure 6shows that in the range of 0 <|B|≤25 G, and Figure 7\nshowsthatfor |B|>25G.Figures 8–10showsimilartime-latitudedistributions\nconstructed from the synoptic magnetic maps of SOHO/MDI (CRs 1 911–2104)\nand SDO/HMI (CRs 2105–2267). HMI maps (3600 ×1440 pixels) were recalcu-\nlated to MDI scale (3600 ×1080 pixels). To convert HMI magnetic field data to\nthat of MDI, the conversion factor of 1.01 (Riley et al., 2014) was used.\nUnlike WSO, the fine structure of the magnetic field distribution appe ars\nmore clearly in the diagrams Figures 5and8. Comparison of Figures 1–10\nshows that different data show a very similar distributions and meridio nal flows\nin similar magnetic field ranges, but the intensity of the magnetic fields greatly\ndiffers between the data sets. It should be noted that magnetic fie ld distribu-\ntions were analyzed on the basis of the synoptic maps of different ind ependent\nobservatories. Despite this, the results obtained using instrumen ts with low and\nhigh spatial resolution are the same. Three types of magnetic field d istributions\nand meridional circulations depending on the magnetic field strength were also\nidentified.Thefirsttypearelow-strengthmagneticfields(0 <|B|≤7G,KPVT-\nSOLIS and MDI-HMI). They were almost cycle-independent and dist ributed\nevenly over the solar disk. The second type includes medium-streng th magnetic\nfields (10 <|B|≤25 G, KPVT-SOLIS and 30 <|B|≤70 G, MDI-HMI) High-\nresolution medium-strength magnetic fields also revealed wave-like, antiphase,\npole-to-pole meridional circulation with a period of ≈22 years. They transport\nthe new polarity magnetic field to the poles. The third type consists o f strong\nmagnetic fields ( |B|>100 G, KPVT-SOLIS and |B|>200 G, MDI-HMI).\nHigh-strength magnetic fields show meridional circulation of active r egions from\nhigh to low latitudes in the Northern and Southern hemispheres. Neit her the\nleading nor the following polarity sunspot magnetic fields migrate to th e poles\nin any of the magnetic field ranges.\nFrom Figures 7(a and b) and 10(a and b) it follows that some high-latitude\nactive region magnetic fields, whose polarity coincided with the polarit y of the\nsecond type meridional circulation waves, i.e. medium-strength mag netic field\nflows, indicated by arrows in Figures 7(a and b) and 10(a and b), were picked up\nby the second type pole-to-pole meridional circulation waves and tr ansported to\nthe poles. Thus some active region magnetic fields participate in the s olar pole\nfield reversals, but they are not the main source of the new polarity magnetic\nfields at the solar poles.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 12Meridional Circulation of Magnetic Fields\nFigure 5. Distributions of high-resolution photospheric magnetic fi elds from KPVT (CRs\n1625–2007) and SOLIS (CRs 2008–2196) synoptic maps. (a) But terfly diagram.Time-latitude\ndistributions of longitude-averaged positive-polarity ( b) and negative-polarity (c) magnetic\nfields. (d) Superposition of the distributions of positive- and negative-polarity magnetic fields\nshown in panels (b) and (c). Designations are the same as in Fi gure1. Black vertical line marks\nthe transition from KPVT to SOLIS in CR 2007.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 13Irina A. Bilenko\nFigure 6. Time-latitude distributions of high-resolution longitud e-averaged low-strength and\nmedium-strength positive- and negative-polarity photosp heric magnetic fields in different\nranges (KPVT, CRs 1625–2007 and SOLIS, CRs 2008–2196). (a) 0 < B≤7 G; (b)\n−7≤B <0 G; (c) 10 < B≤15 G; (d) −15≤B <−10 G; (e) 15 < B≤25 G; (f)\n−25≤B <−15 G. Designations are the same as in Figure 5.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 14Meridional Circulation of Magnetic Fields\nFigure 7. Time-latitude distributions of high-resolution longitud e-averaged high-strength\npositive- and negative-polarity photospheric magnetic fie lds in different ranges (KPVT, CRs\n1625–2007 and SOLIS, CRs 2008–2196). (a) 25 < B≤100 G; (b) −100≤B <−25 G; (c)\n100< B≤500 G; (d) −500≤B <−100 G; (e) B >500 G; (f) B <−500 G. Designations\nare the same as in Figure 5.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 15Irina A. Bilenko\nFigure 8. Distributions of high-resolution photospheric magnetic fi elds from SOHO/MDI\n(CRs 1911–2104) and SDO/HMI (CRs 2105–2267) synoptic maps. (a) Butterfly diagram.\nTime-latitude distributions of longitude-averaged posit ive-polarity (b) and negative-polarity\n(c) magnetic fields. (d) Superposition of the distributions of positive- and negative-polarity\nmagnetic fields shown in panels (b) and (c). Black vertical li ne marks the transition from\nSOHO/MDI to SDO/HMI in CR 2105. Designations are the same as i n Figure 1.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 16Meridional Circulation of Magnetic Fields\nFigure 9. Time-latitude distributions of high-resolution longitud e-averaged low-strength and\nmedium-strength positive- and negative-polarity photosp heric magnetic fields (SOHO/MDI,\nCRs 1911–2104 and SDO/HMI, CRs 2105–2267). (a) 0 < B≤7 G; (b) −7≤B <0 G;\n(c) 30< B≤50 G; (d) −50≤B <−30 G; (e) 50 < B≤70 G; (f) −70≤B <−50 G.\nDesignations are the same as in Figure 8.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 17Irina A. Bilenko\nFigure 10. Time-latitude distributions of high-resolution longitud e-averaged high-strength\npositive- and negative-polarity photospheric magnetic fie lds in different ranges (SOHO/MDI,\nCRs 1911–2104 and SDO/HMI, CRs 2105–2267). (a) 100 < B ≤200 G; (b)\n−200≤B <−100 G; (c) 200 < B≤500 G; (d) −500≤B <−200 G; (e) B >500 G;\n(f)B <−500 G. Designations are the same as in Figure 8.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 18Meridional Circulation of Magnetic Fields\n5. Variations in the Magnetic Field Magnitudes of Different\nType Meridional Circulations\nFigure11shows the variations in the mean strength of magnetic fields in each\nCR for the selected magnetic-field ranges according to WSO (green ), KPVT-\nSOLIS (blue), and MDI-HMI (red) data. Variations in the mean of low -strength\nmagnetic fields (the first type) are shown in Figure 11(a) Variations in WSO\n(0<|B|≤0.6 G) data were insignificant in amplitude and did not coincide\nwith the solar cycle variations of active region magnetic fields. They a re minimal\nat cycles maxima. The mean magnetic field values of KPVT-SOLIS data (0<|\nB|≤7 G) increased slightly toward the maximum and decreased toward th e\nminimum of each solar cycle. MDI-HMI low-strength magnetic fields did not\nreveal solar cycle variations at all.\nThe mean magnetic field values of the second type meridional circulat ion,\ni.e. the medium-strength wave-like, pole-to-pole positive- and nega tive-polarity\nmagnetic field flows, are presented in Figure 11(b). The selected magnetic field\nranges correspond to 0 .6<|B|≤1.0 G for WSO, 15 <|B|≤25 G for KPVT-\nSOLIS, and 50 <|B|≤70 G for MDI-HMI data. For the second type meridional\ncirculation, the magnitudes of magnetic fields remained approximate ly at the\nsame level at all solar cycle phases during Solar Cycles 21–24.\nIn Figure 11(c) the variations in the mean magnetic field of the third type\nmeridional circulation (meridional flows of high-strength positive- a nd negative-\npolarity active region magnetic fields) are shown. The selected magn etic field\nranges correspond to |B|>7 G for WSO, |B|>100 for KPVT-SOLIS, and\n|B|>200 G for and MDI-HMI. The solar cycle variations are clearly visible.\nThe mean values of magnetic fields in these ranges increased to the m aximum\nand decreased to the minimum of solar activity in each cycle.\nThemean ofmagnetic fieldscalculated onthe baseofhigh-resolution synoptic\nmaps, turn out to be low, since weak magnetic fields occupy larger ar eas in the\nsynoptic maps and their contribution to the total magnetic field is hig her. It\nshould be noted that the MDI and HMI data in the range of 0 <|B|≤7.0 G\ndid not agree well with each other despite the using of conversion co efficient.\nThe values of magnetic fields according to the HMI data were much low er than\nthose according to the MDI data. There is no such difference in othe r magnetic\nfield ranges. This may indicate the nonlinearity of the relationship bet ween the\nMDI and HMI magnetic field data.\n6. Mean Latitudes and Velocities of Different Type Meridional\nCirculations\nFigures12–14show the variations in the mean latitudes and velocities of dif-\nferent type meridional circulations in Solar Cycles 21–24. The mean la titudes\nwere calculated for each CR from the corresponding time-latitude d istributions\nof magnetic fields. Variations in the mean latitudes of low-strength p ositive- and\nnegative-polarity magnetic fields are shown in Figures 12(a),13(a), and 14(a).\nTheir mean latitudes were near the equator. Positive- and negative -polarity\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 19Irina A. Bilenko\nFigure 11. Variations in the mean magnetic field strength. (a) 0 <|B|≤0.6 G (WSO);\n0<|B|≤7 G (KPVT-SOLIS and MDI-HMI); (b) 0 .6<|B|≤1.0 G (WSO); 10 <|B|≤25 G\n(KPVT-SOLIS); 30 <|B|≤70 G (MDI-HMI); (c) |B|>7 G (WSO); |B|>100 G\n(KPVT-SOLIS); |B|>200 G (MDI-HMI). Green denotes WSO data (right yaxes), blue\ndenotes KPVT-SOLIS data, and red denotes MDI-HMI data. Dots represent CR-averaged\ndata and thick lines represent seven-CR-averaged data. Sol ar cycles maxima and minima are\nmarked at the top. The black vertical lines mark the transiti on from KPVT to SOLIS and\nfrom MDI to HMI.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 20Meridional Circulation of Magnetic Fields\nFigure 12. Mean latitudes and velocities of different type meridional fl ows (WSO). Variations\nin the mean latitudes of low-strength positive- and negativ e-polarity magnetic fields in the\nrange of 0 <|B|≤0.6 G (a). (b) Variations in the mean latitudes of medium-stren gth\npositive- and negative-polarity magnetic fields in the rang e of 0.8<|B|≤1.0 G. (c) Velocities\nof medium-strength positive-polarity (red) and negative- polarity (blue) magnetic field flows.\nVariations in the mean latitudes of high-strength positive -polarity (d) and negative-polarity (e)\nmagnetic fields in the range of |B|>7 G. (f) Velocities of the high-strength positive-polarity\nmagnetic field flows inthe North(green) and South (lilac)hem ispheres.In (a)–(e) redindicates\nmagnetic fields of positive polarity, and blue indicates tha t of negative polarity. Thin lines show\nthe values averaged for each CR. Thick lines indicate the val ues averaged over 31 CRs.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 21Irina A. Bilenko\nFigure 13. Mean latitudes and velocities of different type meridional fl ows (KPVT-SOLIS).\nVariations in the mean latitudes of low-strength positive- and negative-polarity magnetic fields\nin the range of 0 <|B|≤7 G (a). (b) Variations in the mean latitudes of medium-stren gth\npositive- and negative-polarity magnetic fields in the rang e of 15<|B|≤25 G. (c) Velocities\nof medium-strength positive-polarity (red) and negative- polarity (blue) magnetic field flows.\nVariations in the mean latitudes of high-strength positive -polarity (d) and negative-polarity (e)\nmagnetic fields in the range of |B|>500 G. (f) Velocities of the high-strength positive-polari ty\nmagnetic field flows in the North (green) and South (lilac) hem ispheres. Designations are the\nsame as in Figure 12.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 22Meridional Circulation of Magnetic Fields\nFigure 14. Mean latitudes and velocities of different type meridional fl ows (MDI-HMI).\nVariations in the mean latitudes of low-strength positive- and negative-polarity magnetic fields\nin the range of 0 <|B|≤7 G (a). (b) Variations in the mean latitudes of medium-stren gth\npositive- and negative-polarity magnetic fields in the rang e of 30<|B|≤50 G. (c) Velocities\nof medium-strength positive-polarity (red) and negative- polarity (blue) magnetic field flows.\nVariations in the mean latitudes of high-strength positive -polarity (d) and negative-polarity (e)\nmagnetic fields in the range of |B|>500 G. (f) Velocities of the high-strength positive-polari ty\nmagnetic field flows in the North (green) and South (lilac) hem ispheres. Designations are the\nsame as in Figure 12.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 23Irina A. Bilenko\nmagnetic fields varied in anti phase, but their variations had no any c ycle\ndependence.\nVariations in the mean latitudes of the second type meridional circula tion\nflows are shown in Figures 12(b),13(b), and 14(b). Wave-like flows were located\nat low latitudes during solar activity maxima. The averagelatitude of t he merid-\nional magnetic flows of both polarities increased with a decrease in so lar activity\nto the minimum of each cycle in the northern and southern hemispher es. Waves\nof positive and negative polarity remained at high latitudes in each hem isphere\nuntil the rising phase of the next cycle. It should be noted, that wa ves of the\nsecond type meridional flows occupied a wide range of latitudes in eac h CR.\nTherefore, variations in their mean latitudes indicate the general d irection of\nthe wave migration.\nIn Figures 12(c),13(c), and 14(c) the velocity of the second type meridional\nflows in degrees per CR are shown. Velocities were higher when meridio nal the\nflows were at low latitudes and they crossed the equator migrating f rom the\nNorth (South) to the South (North) hemisphere during solar activ ity maximum.\nVelocities decreased to minimal values during the periods when the flo ws were at\nhighlatitudesduringsolaractivityminimum.Inthepolarregions,them eridional\nflows seem to turn around and their velocity decreases to zero.\nFigures12(d, e),13(d, e), and 14(d, e) show variations in the mean latitudes\nof the third type meridional circulation, i.e. the meridional circulation of high-\nstrength positive- and negative-polarity magnetic fields. Their mer idional flows\nreflect the well known solar cycle dynamics of active regions. They a ppeared at\nhigh latitudes during the rising phases of solar activity and drifted to wards low\nlatitudes. In Figures 12(f),13(f), and14(f) the velocities (degree per CR) of the\nthird type meridionalcirculationflowsarepresented. The latitudes ofmeridional\nflows of high-strength positive- and negative-polarity magnetic fie lds are almost\nidentical. When their latitudinal distributions are superimposed, the y coincide\nwith each other with great accuracy. Therefore the velocities of p ositive-polarity\nmagnetic fields in the North and South hemispheres are presented. The maximal\nvelocitieswereattherisingsolarcyclephasesinallcycles.Withcyclea ctivitythe\nvelocities decreased. Comparison of meridional circulation of medium -strength\nmagnetic fields with that of high-strength shows that active region s did not form\nwhen the second type meridional flows were at the highest latitudes in each\ncycle. Active regions began to form when the latitudes of antiphase meridional\nflow waves of medium-strength magnetic fields shifted to lower latitu des as they\ndrifted to the opposite hemispheres. The formation of active regio ns stopped\nwhen the wave meridional flows of medium-strength positive- and ne gative-\npolarity magnetic fields migrated away from the equator and approa ched the\nopposite poles.\nThe results indicate that both low-resolution large-scale (WSO) and high-\nresolution (KPVT-SOLIS and MDI-HMI) magnetic field data show simila r solar\ncyclechangesin the mean latitude variationsofdifferent strengthm agnetic fields\nand the velocities of their meridional circulations.\nIt should be noted, that the WSO mean latitudes were shifted to the South\nhemisphere in all ranges. It seems to be the instrumental effect. T here were no\nsuch shift neither in KPVT-SOLIS nor in MDI-HMI data.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 24Meridional Circulation of Magnetic Fields\n7. Discussion\nThree different types of time-latitude distributions and meridional c irculations\nof solar magnetic fields were revealed. They depend on the strengt h of the\nphotospheric magnetic fields. Both low-resolution large-scalemagn etic field data\n(WSO) and high-resolution data (KPVT-SOLIS and MDI-HMI) show t he same\nthree types of photospheric magnetic field dynamics, but the magn etic field\nstrength values are different due to different instruments. The fir st type in-\ncludes low-strengthmagnetic fields. Low-strengthmagnetic fields aredistributed\nthroughoutthesolarsurface,buttheirpropertiesandspatial-t emporalvariations\nare not yet well known. Currently, it is unclear what mechanism unde rlies the\norigin and dynamics of weak magnetic fields. It is unclear whether the y are\nthe remnants of the active region magnetic fields or the result of a t urbulent\nsmall-scale dynamo. In the latter case, their distribution and variat ions in the\nstrength of their magnetic field should not coincide with that in the ma gnetic\nfields of active regions. The small-scale dynamo does not depend on t he large-\nscaledynamo.Theresultsshowthattheyweredistributedevenlya crossthesolar\ndisk, their time-latitude distribution and the mean value of their magn etic field\nstrength did not change during solar cycles from the minimum to maxim um of\nsolaractivity.Theirbehaviorisalmostcycle-independent.Thisis con sistentwith\nthe results obtained from daily magnetograms by (Kleint et al., 2010; Buehler,\nLagg, and Solanki, 2013; Lites, Centeno, and McIntosh, 2014; Jin and Wang,\n2015a,b). Thus, low-strength magnetic fields show little variation over spac e and\ntime that was not coincide with that of active regions, indicating that they are\npredominantly governed by a process that is independent of the ac tive region\nsolar cycle. The results indicate that low-strength magnetic fields a re not a\nproduct of the decay of active region magnetic fields. This support s the notion\nthat low-strength magnetic fields caused by a turbulent small-scale dynamo.\nThe second type meridional circulation is demonstrated by medium-s trength\nmagnetic fields. Positive- and negative-polarity magnetic fields reve al wavy an-\ntiphase meridional flows from pole to pole with a period of approximate ly 22\nyears. They cross the equator during the maximum of solar activity . Cycle evo-\nlution of coronal holes (CHs) revealed some temporal and spatial r egularities\n(Bilenko, 2002; Bilenko and Tavastsherna, 2016) that match well the wave-\nlike pole-to-pole meridional circulation of medium-strength magnetic fields. In\nBilenko ( 2002) the pole-to-pole antiphase meridional drift was revealed for CHs\nassociated with positive- and negative-polarity photospheric magn etic fields. In\nBilenko and Tavastsherna ( 2016) we found two different type waves of non-polar\nCHs. The first are short waves that trace the poleward movement of unipolar\nmagnetic fields from approximately 35◦latitude to the corresponding pole in\neach hemisphere. The second type of non-polar CH waves forms tw o antiphase\nsinusoidal branches associated with the positive- and negative-po larity photo-\nspheric magnetic fields with a period of ≈268 CRs (22 years). These CH waves\ncompletely coincide with the meridional flows of the medium-strength positive-\nand negative-polarity photospheric magnetic fields revealed in Sect ions3and4.\nThe study of CHs by other authors confirmed our results. Pevtso v and Abra-\nmenko (2010) described the observational evidence of a transport of one CH\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 25Irina A. Bilenko\nacross the equator from one solar polar region to the other. Huan g, Lin, and Lee\n(2017) found that during the rising phase, the opposite directed open ma gnetic\nfluxes (CH regions), showed pole-to-pole trans-equatorial migra tions in opposite\ndirections. Maghradze et al. ( 2022) also found sinusoidal cyclic migration of the\ncenters of CH activity with opposite directed magnetic fluxes from p ole to pole\nacross the solar equator.\nCHs are considered to be good traces of the solar large-scale, glob al magnetic\nfield cycle evolution (Stix, 1977; Bagashvili et al., 2017; Bilenko and Tavast-\nsherna,2017,2020). Thus, medium-strength magnetic fields reflect the solar\nglobal magnetic field cycle evolution. Cycle changes in the latitude loca tion of\nthe CH waves coincide with that of the axisymmetric component of th e solar\ndipole (g0\n1) (Bilenko and Tavastsherna, 2016). When both CH branches were\nabove≈ ±35◦latitude, the zonal structure was observed. When two branches\nof positive- and negative-polarity magnetic fields and CHs move below ≈ ±35◦\nlatitude, the zonal structure of the GMF changes to sectorial on e (Bilenko and\nTavastsherna, 2016).This moment coincides with the beginning ofthe formation\nofhigh-strength active regionmagnetic fields. Apparently, the fo rmation ofhigh-\nstrength magnetic fields is somehow associated with the second typ e wave-like\nmeridional circulation, i.e. medium-strength magnetic field dynamics.\nIt is important to note, that Komm ( 2022) determined the direction and\namplitude of cross-equatorial flows below the solar surface. They found that the\ncross-equatorial flow was mainly southward during Solar Cycle 23 an d mainly\nnorthwardduring SolarCycle 24. At depths less than 7 Mm, the aver agevelocity\nwas -1.1 ±0.2 m s−1in Solar Cycle 23 and +1.3 ±0.1 m s−1in Solar Cycle\n24. At the beginning of Solar Cycle 25, the cross-equatorial flow ch anged sign\nagain.Theyconcluded,thatthesubsurfacecross-equatorialfl owwasnonzeroand\ncaused by the inflows associated with active regions located close to the equator\nand it was directed to the hemisphere with the greater amount of flu x. But,\napparently, these subsurface meridional flows were associated w ith the second\ntype meridional circulationofmedium-strength magnetic fields. The second type\nmeridional circulation can be associated with processes occurring a t the base of\nthe convective zone. Jones ( 2005), for example, noted that CHs may be rooted\nas deep as the base of the convection zone.\nThe third type is the meridional circulation of high-strength magnet ic fields,\ni.e. fields of active regions. Both positive- and negative-polarity high -strength\nmagnetic fields migrate from high latitudes towards the equator in ea ch cycle.\nNeither the magnetic fields of the leading sunspot polarity nor the fo llowing\nones migrate to the poles. It is interesting to note that using the Ko daikanal\nObservatory archives for 1906–1987 and the Mt. Wilson Observat ory archives\nfor 1917–1985, Sivaraman et al. ( 2010) found that the latitudinal drifts (or the\nmeridional flows) of spot groups classified into three categories of area: 0–5,\n5–10, and >10 millionths of the solar hemisphere were directed equator ward in\nboththe NorthandSouth hemispheres.Theequatorwarddriftve locityincreases\nfrom almost zero at 35◦latitude in both hemispheres reaches maximum around\n20◦latitude and slows down towards zero near the equator Sivaraman e t al.\n(2010). Consequently, both the active regions themselves and their mag netic\nfields migrate from latitudes of 35◦to the equator, and not to the poles. Some\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 26Meridional Circulation of Magnetic Fields\nof the high-latitude active region magnetic fields were captured by t he second\ntype wave-like flows in their drift to the appropriate pole and transp orted to the\npoles. However, the magnetic fields of active regions are not the ma in ones in\nthe process of the solar polar field reversal.\nSo, the observed magnetic-field cycle dynamics is a superposition of tree\ndifferent type time-latitude distributions and meridional circulations . Therefore,\nthe solar cycle is governed by different types of dynamo. Benevolen skaya (1998)\nproposed a new dynamo model to explain the solar magnetic cycle tha t consists\noftwomainperiodiccomponents:alow-frequencycomponent(Hale ’s22yrcycle)\nandahigh-frequencycomponent(quasi-biennialcycle). Theesse nceofthe model\nis that two dynamo sources are at different levels. The first is locate d near the\nbottom of the convection zone, and the other is near the top. It is possible that\nthe assumption of the joint operation of different types of dynamo s at different\nlevels in the convective zone and near the surface will make it possible to better\nexplain the observed distributions of magnetic fields of different str engths and\ntheir solar cycle dynamics.\n8. Conclusions\nMeridional circulation of the solar magnetic fields were analyzed using synoptic\nmagnetic field data from ground based WSO, NSO KPVT, and SOLIS/V SM\nand space based SOHO/MDI and SDO/HMI observatories for Solar C ycles\n21–24. It have been found that cycle variations of low-, middle-, an d hight-\nstrength magnetic fields significantly different. Depending on the int ensity of\nthe photospheric magnetic fields, three types of time-latitude dist ributions and\nmeridional circulations were identified for both low and high spatial re solution\ndata.\nThefirsttype includes low-strengthmagneticfields (0 <|B|≤0.6G forWSO\nand 0<|B|≤7 G for KPVT-SOLIS and MDI-HMI). They were distributed\nevenly across latitude and their average strength weakly depende d on cycle\nvariations of the active region magnetic fields. These fields are believ ed to be\ndetermined by a small-scale dynamo.\nThe second is medium-strength magnetic field meridional circulation ( 0.6<|\nB|≤1 G for WSO, 10 <|B|≤25 G for KPVT-SOLIS, and 30 <|B|≤70 G\nfor MDI-HMI). Fields of positive and negative polarity revealed a wav e-like an-\ntiphase meridional circulation from pole to pole with a period of approx imately\n22 years. The velocity of meridional flows were slower at the minima of solar\nactivity, when they were at high latitudes in the opposite hemisphere s, and max-\nimal at the solar cycle maxima, when the positive- and negative-polar ity waves\ncrossed the equator. The flows were more pronounced in large-sc ale magnetic\nfield data and in magnetic fields calculated at the source surface. Th e meridional\ncirculation of these fields reflects the solar global magnetic field dyn amics and\ndetermines the solar polar field reversal.\nThe third type meridional circulation includes high-strength magnet ic fields,\nthe fields of active regions ( |B|>7 G for WSO, |B|>100 G for KPVT-\nSOLIS,and |B|>200G for MDI-HMI). Magnetic fieldsofpositiveand negative\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 27Irina A. Bilenko\npolarity were distributed symmetrically in both hemispheres with resp ect to the\nequator. Active region magnetic fields of both leading and following su nspot\npolarity migrate from high to low latitudes in all magnetic field ranges. T he\nvelocities of these meridional flows were higher at the rising and maxim a phases\nthan at the minima.\nThemagneticfieldsofactiveregionsarenotthe mainsourceofmagn eticfields\nthat determine the solar polar field reversal. The magnetic fields of s ome high-\nlatitude active regions picked up by the meridional wave-like flows (me ridional\ncirculation of the second type) of the corresponding polarity durin g the rising\nsolar cycle phases and transported along with the flows to the poles .\nThe butterfly diagram is just the result of a superposition of cyclica lly chang-\ning meridional flows of different strength positive- and negative-po larity mag-\nnetic fields. The various details and structures in the butterfly diag ram, for\nexamplepolewardsurges,aretheresultofthedominationofoneof thepolarities.\nAcknowledgements Wilcox Solar Observatory data used in this study was obtaine d via\nthe web site http://wso.stanford.edu at 2023:04:28 12:04:38 PDT courtesy of J.T. Hoeksema.\nThe Wilcox Solar Observatory is currently supported by NASA .\nNSO/Kitt Peak magnetic data used here are produced cooperat ively by NSF/NOAO,\nNASA/GSFC, and NOAA/SEL.\nThis work utilizes SOLIS data obtained by the NSO Integrated Synoptic Program (NISP),\nmanaged by the National Solar Observatory, which is operate d by the Association of Universi-\nties for Research in Astronomy (AURA), Inc. under a cooperat ive agreement with the National\nScience Foundation.\nSOHO/MDI data were also used. 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L29, 6 pp. DOI.\nSOLA: Bilenko.tex; 17 November 2023; 1:49; p. 31"    },    {        "title": "1702.08394v4.Magnetization_reversal_by_superconducting_current_in___varphi_0__Josephson_junctions.pdf",        "content": "arXiv:1702.08394v4  [cond-mat.supr-con]  17 Apr 2017Magnetization reversal by superconducting current in ϕ0Josephson junctions\nYu. M. Shukrinov,1,2I. R. Rahmonov,1,3K. Sengupta,4and A. Buzdin5\n1)BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n2)Dubna State University, Dubna, 141980, Russia\n3)Umarov Physical Technical Institute, TAS, Dushanbe, 73406 3, Tajikistan\n4)Theoretical Physics Department, Indian Association for th e Cultivation of Science, Jadavpur, Kolkata 700032,\nIndia\n5)University Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence C edex, France\n(Dated: 4 September 2018)\nWe study magnetization reversal in a ϕ0Josephson junction with direct coupling between magnetic moment\nand Josephson current. Our simulations of magnetic moment dynam ics show that by applying an electric\ncurrent pulse, we can realize the full magnetization reversal. We pr opose different protocols of full magneti-\nzation reversal based on the variation of the Josephson junction and pulse parameters, particularly, electric\ncurrent pulse amplitude, damping of magnetization and spin-orbit int eraction. We discuss experiments which\ncan probe the magnetization reversal in ϕ0-junctions.\nKeywords: Superconducting electronics, ϕ0-junction, magnetization reversal, spin-orbit interaction\nSpintronics, which deals with an active control of spin\ndynamicsinsolidstatesystems,isoneofthemostrapidly\ndeveloping field of condensed matter physics1. An im-\nportant place in this field is occupied by superconduct-\ning spintronics dealing with the Josephson junctions (JJ)\ncoupled to magnetic systems2. The possibility of achiev-\ning electric control over the magnetic properties of the\nmagnet via Josephson current and its counterpart, i.e.,\nachieving magnetic control over Josephson current, re-\ncently attracted a lot ofattention3–5. Spin-orbit coupling\nplaysamajorroleinachievingsuchcontrol. Forexample,\nin superconductor/ferromagnet/superconductor(S/F/S)\nJJs, its presence in a ferromagnet without inversion sym-\nmetryprovidesamechanismforadirect (linear)coupling\nbetween the magnetic moment and the superconducting\ncurrent. In such junctions, called hereafter ϕ0-junction,\ntime reversal symmetry is broken, and the current–phase\nrelation is given by I=Icsin(ϕ−ϕ0), where the phase\nshiftϕ0is proportional to the magnetic moment per-\npendicular to the gradient of the asymmetric spin-orbit\npotential, and also to the applied current.6–8. Thus such\nJJs allow one to manipulate the internal magnetic mo-\nment by Josephson current6,9. The static properties of\nS/F/Sstructures are well studied both theoretically and\nexperimentally; however, the magnetic dynamics of these\nsystems has not been studied in detail beyond a few the-\noretical works6,9–14.\nThe spin dynamics associated with such ϕ0-junctions\nwas studied theoretically in Ref. 9. The authors con-\nsidered a S/F/S ϕ 0-junction in a low frequency regime\nwhich allowed usage of quasi-static approach to study\nmagnetizationdynamics. It wasdemonstrated that a DC\nsuperconductingcurrentproducesastrongorientationef-\nfect on the magnetic moment of the ferromagnetic layer.\nThus application of a DC voltage to the ϕ0-junction is\nexpected to lead to current oscillations and consequently\nmagnetic precession. This precession can be monitored\nby the appearance of higher harmonics in current-phase\nrelation; in addition, it also leads to the appearance ofa DC component of the current which increases near a\nferromagnetic resonance9. It is then expected that the\npresence of external radiation in such a system would\nlead to several phenomena such as appearance of half-\ninteger steps in the current-voltage (I-V) characteristics\nof the junction and generation of an additional magnetic\nprecession with frequency of external radiation9.\nIn this paper we study the magnetization reversal in\nϕ0-junction with direct coupling between magnetic mo-\nment and Josephson current and explore the possibility\nofelectricallycontrollablemagnetizationreversalinthese\njunctions. We carry out investigations of the magneti-\nzation dynamics for two types of applied current pulse:\nrectangular and Gaussian forms. An exact numerical\nsimulation of the dynamics of magnetic moment of the\nferromagnetic layer in the presence of such pulses allows\nus to demonstrate complete magnetization reversal in\nthese systems. Such reversal occurs for specific param-\neters of the junction and the pulse. We chart out these\nparameters and suggest a possible way for determination\nof spin-orbit coupling parameter in these systems. We\ndiscuss the experiment which can test our theory.\nIn order to study the dynamics of the S/F/S system,\nwe use the method developed in Ref. 9. We assume that\nthe gradient of the spin-orbit potential is along the easy\naxis of magnetization taken to be along ˆ z. The total\nenergy of this system can be written as\nEtot=−Φ0\n2πϕI+Es(ϕ,ϕ0)+EM(ϕ0),(1)\nwhereϕis the phase difference between the supercon-\nductors across the junction, Iis the external current,\nEs(ϕ,ϕ0) =EJ[1−cos(ϕ−ϕ0)], andEJ= Φ0Ic/2πis\nthe Josephson energy. Here Φ 0is the flux quantum, Icis\nthe critical current, ϕ0=lυsoMy/(υFM0),υFis Fermi\nvelocity, l= 4hL//planckover2pi1υF,Lis the length of Flayer,his\ntheexchangefieldofthe Flayer,EM=−KVM2\nz/(2M2\n0),\nthe parameter υso/υFcharacterizesa relative strength of\nspin-orbit interaction, Kis the anisotropic constant, and\nVis the volume of the Flayer.2\nThe magnetization dynamics is described by the\nLandau-Lifshitz-Gilbert equation3(see also Supplemen-\ntary Material) which can be written in the dimensionless\nform as\ndmx\ndt=1\n1+α2/braceleftbig\n−mymz+Grmzsin(ϕ−rmy)\n−α/bracketleftbig\nmxm2\nz+Grmxmysin(ϕ−rmy)/bracketrightbig/bracerightbig\n,\ndmy\ndt=1\n1+α2/braceleftbig\nmxmz\n−α/bracketleftbig\nmym2\nz−Gr(m2\nz+m2\nx)sin(ϕ−rmy)/bracketrightbig/bracerightbig\n,\ndmz\ndt=1\n1+α2/braceleftbig\n−Grmxsin(ϕ−rmy)\n−α/bracketleftbig\nGrmymzsin(ϕ−rmy)−mz(m2\nx+m2\ny)/bracketrightbig/bracerightbig\n,(2)\nwhereαis a phenomenological Gilbert damping con-\nstant,r=lυso/υF, andG=EJ/(KV). Themx,y,z=\nMx,y,z/M0satisfy the constraint/summationtext\nα=x,y,zm2\nα(t) = 1. In\nthis system of equations time is normalized to the in-\nverse ferromagnetic resonance frequency ωF=γK/M 0:\n(t→tωF),γis the gyromagnetic ratio, and M0=/ba∇dblM/ba∇dbl.\nIn what follows, we obtain time dependence of magne-\ntizationmx,y,z(t), phase difference ϕ(t) and normalized\nsuperconducting current Is(t)≡Is(t)/Ic= sin(ϕ(t)−\nrmy(t)) via numerical solution of Eq.(2).\nLet us first investigate an effect of superconducting\ncurrent on the dynamics of magnetic momentum. Our\nmain goal is to search for cases related to the possibility\nof the full reversal of the magnetic moment by supercon-\nducting current. In Ref.9 the authors have observed a\nperiodic reversal, realized in short time interval. But, as\nwe see in Fig. 1, during a longtime interval the character\nofmzdynamics changes crucially. At long times, /vector mbe-\ncomes parallel to y-axis, as seen from Fig.1(b)) demon-\nstrating dynamics of my. The situation is reminiscent\nof Kapitza pendulum (a pendulum whose point of sus-\npension vibrates) where the external sinusoidal force can\ninvert the stability position of the pendulum.15Detailed\nfeatures of Kapitza pendulum manifestation will be pre-\nsented elsewhere.\ntmz\n050100150200-1-0.500.51(a)\nr=0.1,G=500 π,\nα=0.1, ω=5 mz\n0 0.5-101\ntmy\n010203000.20.40.60.81(b)\ntmy\n0 0.501\nFIG. 1. (a) Dynamics of mzin case of ωJ= 5,G= 500π,r=\n0.1,α= 0.1. The inset shows the character of time depen-\ndence in the beginning of the time interval; (b) The same as\nin (a) for my.\nThe question we put here is the following: is it pos-\nsible to revers the magnetization by the electric current\npulse and then preserve this reversed state. The answer\nmay be found by solving the system of equations (2) to-\ngether with Josephson relation dϕ/dt=V, written inthe dimensionless form. It was demonstrated in Ref.13\nthat using a specific time dependence of the bias voltage,\napplied to the weak link leads to the reversal of the mag-\nnetic moment of the nanomagnet. The authors showed\nthe reversal of nanomagnet by linearly decreasing bias\nvoltageV= 1.5−0.00075t(see Fig.3 in Ref.13). The\nmagnetization reversal, in this case, was accompanied by\ncomplex dynamical behavior of the phase and continued\nduring a sufficiently long time interval.\nIncontrast,inthepresentworkweinvestigatethemag-\nnetization reversal in the system described by the equa-\ntions (2) under the influence of the electric current pulse\nof rectangular and Gaussian forms. The effect of rectan-\ngular electric current pulse are modeled by Ipulse=As\nin the ∆ttime interval ( t0−∆t\n2,t0+∆t\n2) andIpulse= 0\nin other cases. The form of the current pulse is shown in\nthe inset to Fig.2(a).\nHere we consider the JJ with low capacitance C (\nR2C/LJ<<1, where LJis the inductance of the JJ and\nRis its resistance), i.e., we do not take into account the\ndisplacement current. So, the electric current through\nJJs is\nIpulse=wdϕ\ndt+sin(ϕ−rmy) (3)\nwherew=VF\nIcR=ωF\nωR,VF=/planckover2pi1ωF\n2e,Ic- critical current, R-\nresistance of JJ, ωR=2eIcR\n/planckover2pi1- characteristic frequency.\nWe solved the system of equations ( 2) together with\nequation ( 3) and describe the dynamics of the system.\nTime dependence of the electric current is determined\nthrough time dependence of phase difference ϕand mag-\nnetization components mx,my,mz.\nWe first study the effect ofthe rectangularpulse shown\nin the inset to Fig.2(a). It is found that the reversal of\nmagnetic moment can indeed be realized at optimal val-\nues of JJ ( G,r) and pulse ( As,∆,t0) parameters . An\nexample of the transition dynamics for such reversal of\nmzwith residual oscillation is demonstrated in Fig. 2(a);\nthe corresponding parameter values are shown in the fig-\nure.\nDynamics of the magnetic moment components, the\nphase difference and superconducting current is illus-\ntrated in Fig.2(b). We see that in the transition region\nthe phase difference changes from 0 to 2 πand, corre-\nspondingly, the superconducting current changes its di-\nrection twice. This is followed by damped oscillation of\nthe superconducting current. There are some character-\nistic time points in Fig.2(b), indicated by vertical dashed\nlines. Line 1 correspondsto a phase difference of π/2 and\nindicate maximum of superconducting current Is. The\nline 1′which corresponds to the maximum of my, and\nmz= 0 has a small shift from line 1. This fact demon-\nstrates that, in general, the characteristic features of mx\nandmytime dependence do not coincide with the fea-\ntures on the Is(t), i.e., there is a delayin reaction ofmag-\nnetic moment to the changes of superconducting current.\nAnother characteristic point corresponds to the ϕ=π.\nAt this time line 2 crosses points Is= 0,my= 0, and3\ntmi, Is\nϕ\n222426283032-1-0.500.51\n0123456mz\nIs\nmyϕ\nπ/2π3π/21’23\n1\nOn Off(b)\nFIG. 2. Transition dynamics of the magnetization compo-\nnentmzfor a system with rectangular current pulse shown\nin the inset; (b) Dynamics of magnetization components to-\ngether with the phase difference ϕand superconducting cur-\nrentIs. Arrows indicate the beginning and end of the electric\ncurrent pulse. Vertical dashed lines indicate the common fe a-\ntures while the horizontal ones mark the corresponding valu es\nof the phase difference.\nminimum of mz. At time moment when ϕ= 3π/2 line\n3 crosses minimum of Is. When pulse is switched off,\nthe superconducting current starts to flow through the\nresistance demonstrating damped oscillations and caus-\ningresidualoscillationsofmagneticmomentcomponents.\nNote also, that the time at which the current pulse ends\n(t= 28) is actually does not manifest itself immediately\nin themy(and not shown here mx) dynamics. They\ndemonstrate continuous transition to the damped oscil-\nlating behavior.\nFig. 2(b) provides us with a direct way of de-\ntermining the spin-orbit coupling strength in the\njunction via estimation of r. For this, we note\nthat the ϕ(t) =ϕ00+/integraltextt\n0V(t′)dt′can be deter-\nmined, up to a initial time-independent constant\nϕ00, in terms of the voltage V(t) across the junc-\ntion. Moreover, the maxima and minima of Isoccurs\nat times tmaxandtmin(see Fig. 2(b)) for which\nsin/bracketleftBig\nϕ00+/integraltexttmax[tmin]\n0V(t′)dt′−rmy(tmax[tmin])/bracketrightBig\n=+[−]1.Eliminating ϕ0from these equations, one gets\nsin1\n2/bracketleftbigg/integraldisplaytmin\ntmaxV(t′)dt′+r[my(tmax)−my(tmin)]/bracketrightbigg\n= 1\n(4)\nwhich allows us, in principal, to determine rin terms\nof the magnetization myat the position of maxima and\nminima of the supercurrent and the voltage Vacross the\njunction. We stress that for the experimental realization\nof proposed method one would need to resolve the value\nofthe magnetizationat the time difference ofthe orderof\n10−10-10−9c. Atthepresentstagethestudyofthemag-\nnetization dynamics with such a resolution is extremely\nchallenging. To determine the spin-orbit coupling con-\nstantrexperimentally it may be more convenient to vary\nthe parameters of the current pulse I(t) and study the\nthreshold of the magnetic moment switching.\nThe dynamics of the system in the form of magneti-\nzation trajectories in the planes my−mxandmz−mx\nduring a transition time interval at the same parameters\nof the pulse and JJ at α= 0 is presented in Fig.3. We\nmxmy\n0 0.5 1-1-0.500.51\nB\nA’\nCA\nQ(a)\nmxmz\n0 0.5 1-1-0.500.51\nB\nA’QA\nC(c)\nFIG. 3. Trajectories of magnetization components in the\nplanesmy−mxin the transition region: (a) during electric\npulse action (between points AandC), (b)after switchingthe\npulse off; In (c) and (d) the same is shown for the my−mz\nplane. Parameters of the pulse and the JJ are the same as in\nFig.2(a) at α= 0.\nsee that magnetic moment makes a spiral rotation ap-\nproaching the state with mz=−1 after switching off the\nelectric current pulse. The figures show clearly the spe-\ncific features of the dynamics around points B,A′and\nQand damped oscillations of the magnetization compo-\nnents (see Fig.3(b) and Fig.3(d)). The cusps at point B\nin Fig.3(a) corresponds just to the change from an in-\ncreasing of absolute value of mxto its decreasing and,\nopposite, at point A′in Fig.3(c). The behavior of mag-\nnetic system happens to be sensitive to the parameters4\nof the electric current pulse and JJ. In the Supplement\nwe show three additional protocols of the magnetization\nreversal by variation of As,Gandr.\nIt is interesting to compare the effect of rectangular\npulse with the Gaussian one of the form\nIpulse=As1\nσ√\n2πexp/parenleftbigg\n−(t−t0)2\n2σ2/parenrightbigg\n.(5)\nwhereσdenotes the full width at half-maximum of the\npulse and Ais its maximum amplitude at t=t0. In this\ncase we also solve numerically the system of equations\n(2) together with equation (3) using (5). An example\nof magnetic moment reversal in this case is presented in\nFigure 4, which shows the transition dynamics of mzfor\nthe parameters r= 0.1,G= 10,As= 5,σ= 2 at\nsmall dissipation α= 0.01. We see that the magneti-\ntmz\n0 50 100 150-1-0.500.51\nα=0.01\nG=10,r=0.1tIsignal\n152025303500.51t0=25\nσ=2As=5\nFIG. 4. Demonstration of transition dynamics of mzfor a\nGaussian electric current pulse (shown in the inset).\nzation reversal occurs more smoothly in compare with a\nrectangular case.\nWe also note that very important role in the reversal\nphenomena belongs to the effect of damping. It’s de-\nscribed by term with αin the system of equations (2),\nwhereαis a damping parameter. The examples of the\nmagnetization reversal at G= 50,r= 0.1 and different\nvalues of αare presented in Fig.5. We see that dissi-\ntmz\n0204060-1-0.500.51\nα=0.07(b)\nFIG. 5. Magnetization dynamics under rectangular pulse\nsignal in the system at different values of the dissipation pa -\nrameter α.\npation can bring the magnetic system to full reversal,even if at α= 0 the system does not demonstrate re-\nversal. Naturally, the magnetic moment, after reversal,\nshows some residual oscillations as well. We stress that\nthe full magnetization reversal is realized in some fixed\nintervals of dissipation parameter. As expected, the vari-\nation of phase difference by πreflects the maxima in the\ntime dependence of the superconducting current. Fig.6\ndemonstrates this fact. The presented data shows that\ntIs\nϕ\n0 20 40-1-0.500.51\n05101520ϕ\nIs5ππ\n6π2πG=50,\nr=0.1,\nAs=2,\ndt=10,\nt0=25\nα=0.07\nFIG. 6. Transition dynamics of the phase difference and the\nsuperconducting current for the case presented in Fig.5(b) .\nthe total change of phase difference consists of 6 π, which\ncorresponds to the six extrema in the dependence Is(t).\nAfterthefullmagnetizationreversalisrealized,thephase\ndifference shows the oscillations only.\nOne of the important aspect of the results that we ob-\ntain here is the achievement of a relatively short switch-\ning time interval for magnetization reversal. As we have\nseen in Figs. 2(a) and 4, the time taken for such reversal\nisωFt≃100 which translate to 10−8seconds for typical\nωF≃10GHz. We note that this amounts to a switching\ntime which is 1 /20thof that obtained in Ref. 13.\nExperimental verification of our work would involve\nmeasurement of mz(t) in aϕ0junction subjected to a\ncurrent pulse. For appropriate pulse and junction pa-\nrameters as outlined in Figs. 4 and 5, we predict obser-\nvation of reversal of mzat late times ωFt≥50. More-\nover, measurement of myat times tmaxandtminwhereIs\nreaches maximum and minimum values and the voltage\nV(t) acrossthe junction between these times would allow\nfor experimental determination of rvia Eq. 4.\nAs a ferromagnet we propose to use a very thin F\nlayer on dielectric substrate. Its presence produces the\nRashba-type spin-orbit interaction and the strength of\nthis interaction will be large in metal with large atomic\nnumber Z. The appropriate candidate is a permalloy\ndoped with Pt.16InPtthe spin-orbit interaction play\na very important role in electronic band formation and\nthe parameter υso/υF, which characterizes the relative\nstrength of the spin-orbit interaction is υso/υF∼1. On\nthe other hand, the Ptdoping of permalloy up to 10 %\ndid not influenced significantly its magnetic properties16\nand then we may expect to reach υso/υFin this case 0.1\nalso. If the length of Flayer is of the order of the mag-5\nnetic decaying length /planckover2pi1υF/h, i.e.l∼1, we have r∼0.1.\nAnother suitable candidate may be a Pt/Cobilayer, fer-\nromagnetwithoutinversionsymmetrylikeMnSiorFeGe.\nIf the magnetic moment is oriented in plane of the F\nlayer,thanthespin-orbitinteractionshouldgeneratea ϕ0\nJosephson junction6with finite ground phase difference.\nThe measurement of this phase difference (similar to the\nexperiments in Ref.17) may serve as an independent way\nfor the parameter revaluation. The parameter Ghas\nbeen evaluated in Ref.9 for weak magnetic anisotropy of\npermalloy K∼4×10−5K·˚A−3(see Ref.18) and S/F/S\njunction with l∼1 andTc∼10K asG∼100. For\nstronger anisotropy we may expect G∼1.\nIn summary, we have studied the magnetization rever-\nsal inϕ0-junction with direct coupling between magnetic\nmoment and Josephson current. By adding the electric\ncurrent pulse, we have simulated the dynamics of mag-\nnetic moment components and demonstrate the full mag-\nnetization reversal at some parameters of the system and\nexternal signal. Particularly, time interval for magneti-\nzation reversal can be decreased by changing the ampli-\ntude of the signal and spin-orbit coupling. The observed\nfeatures might find an application in different fields of\nsuperconducting spintronics. They can be considered as\na fundamental basis for memory elements, also.\nSee supplementary material for demonstration of dif-\nferent protocols of the magnetization reversal by vari-\nation of Josephson junction and electric current pulse\nparameters.\nAcknowledgment : The authors thank I. Bobkova and\nA. Bobkov for helpful discussion. The reported study\nwas funded by the RFBR research project 16–52–\n45011India, 15–29–01217, the the DST-RFBR grant\nINT/RUS/RFBR/P-249, and the French ANR projects\n”SUPERTRONICS”.\n1I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323\n(2004).\n2Jacob Linder and W. A. Jason Robinson, Nature Physics, 11,\n307 (2015).\n3A. I. Buzdin, Rev. Mod. Phys., 77, 935 (2005).\n4F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. ,\n77, 1321 (2005).\n5A. A. Golubov, M. Y. Kupriyanov, and E. Ilichev, Rev. Mod.\nPhys. ,76, 411 (2004).\n6A. Buzdin, Phys. Rev. Lett., 101, 107005 (2008).\n7I. V. Krive, A. M. Kadigrobov, R. I. Shekhter, and M. Jonson,\nPhys. Rev. B 71, 214516 (2005).\n8A. A. Reynoso, G. Usaj, C. A. Balseiro, D. Feinberg, and M.\nAvignon, Phys. Rev. Lett. 101, 107001 (2008).\n9F. Konschelle, A. Buzdin, Phys. Rev. Lett ., 102, 017001 (2009).\n10X. Waintal and P. W. Brouwer, Phys. Rev., B 65, 054407 (2002).\n11V. Braude and Ya. M. Blanter, Phys. Rev. Lett., 100, 207001\n(2008).\n12J. Linder and T. Yokoyama, Phys. Rev., B 83, 012501 (2011).\n13Liufei Cai and E. M. Chudnovsky, Phys. Rev., B 82, 104429\n(2010).\n14Eugene M. Chudnovsky, Phys. Rev., B 93, 144422 (2016).\n15P. L. Kapitza, Soviet Phys. JETP, B21, 588-592 (1951); Usp.\nFiz. Nauk, B44, 7-15 (1951).\n16A. Hrabec, F. J. T. Goncalves, C. S. Spencer, E. Arenholz, A. T .\nNDiaye, R. L. Stamps and Christopher H. Marrows, Phys. Rev.\nB93, 014432 (2016).17D. B. Szombati, S. Nadj-Perge, D. Car, S. R. Plissard, E. P.\nA. M. Bakkers and L. P. Kouwenhoven, Nture Physics 12, 568\n(2016).\n18A. Yu. Rusanov, M. Hesselberth, J. Aarts, A. I. Buzdin, Phys.\nRev. Lett. 93, 057002 (2004).6\nSupplementary Material to “Magnetization reversal by\nsuperconducting current in ϕ0Josephson junctions”\nI. GEOMETRY AND EQUATIONS\nGeometry of the considered ϕ0-junction9is presented\nin Fig. 7. The ferromagnetic easy-axis is directed along\nthe z-axis, which is also the direction n of the gradient of\nthe spin-orbit potential. The magnetization component\nmyis coupled with Josephson current through the phase\nshift term ϕ0∼(− →n[− →m− →∇Ψ]), where Ψ is the supercon-\nducting order parameter (− →∇Ψ is along the x-axis in the\nsystem considered here).\nSyz,n\nx, IsM\nF S\nFIG. 7. Geometry of the considered ϕ0-junction.\nIn order to study the dynamics of the S/F/S system,\nwe use the method developed in Ref. 9. We assume that\nthe gradient of the spin-orbit potential is along the easy\naxis of magnetization taken to be along ˆ z. The total\nenergy of this system is determined by the expression\n(S1) in the main text.\nThe magnetization dynamics is described by the\nLandau-Lifshitz-Gilbert equation3\ndM\ndt=−γM×Heff+α\nM0/parenleftbigg\nM×dM\ndt/parenrightbigg\n(6)\nwhereγis the gyromagnetic ratio, αis a phenomeno-\nlogical Gilbert damping constant, and M0=/ba∇dblM/ba∇dbl. The\neffective field experienced by the magnetization Mis de-\ntermined by Heff=−1\nV∂Etot\n∂M, so\nHeff=K\nM0/bracketleftbigg\nGrsin/parenleftbigg\nϕ−rMy\nM0/parenrightbigg\n/hatwidey+Mz\nM0/hatwidez/bracketrightbigg\n(7)\nwherer=lυso/υF, andG=EJ/(KV).\nUsing (6) and (7), we obtain the system of equations\n(2) in main text, which describes the dynamics of the\nSFS structure.II. MAGNETIZATION REVERSAL UNDER ELECTRIC\nCURRENT PULSE\nMagnetic system is very sensitive to the parameters of\nthe electric current pulse and Josephson junction. Here\nwe show three additional protocols of the magnetization\nreversal by variation of As,Gandr.\nA. Effect of As-variation\nFigure 8 demonstrates the magnetization reversal by\nchanging pulse parameter As. We see that change of\nV1V2\n26.52727.5-1.01-1-0.99-0.98\n1.38\n1.41.44\n1.42\ntmz\n0 50 100 150-1-0.500.51\n1.3\n1.4G=10,r=0.1,\n∆t=6\n1.5DifferentAs\nFIG. 8. Magnetization reversal by changing pulse parameter\nAs. The number near curve shows value of As.\npulse amplitude As= 1.3 toAs= 1.4 reverses magnetic\nmoment. At As= 1.5 this feature is still conserved, but\ndisappears at larger values.\nB. Effect of G-variation\nFigure 9 demonstrates the magnetization reversal by\nchanging Josephson junction parameter G.\nC. Effect of r-variation\nFigure 10 demonstrates the magnetization reversal by\nchanging Josephson junction parameter of spin-orbital\ncoupling r. Figure 10 demonstrates the magnetization\nreversal by changing Josephson junction parameter of\nspin-orbital coupling r.We see that there is a possibil-\nity of magnetization reversal around G= 10. In this\ncase a decrease of spin-orbit parameter may lead to the\nmagnetization reversal also. The magnetization reversal7\ntmz\n0 50 100 150-1-0.500.51\n7\n10DifferentG\nr=0.1,\nAs=1.4,∆t=6\nFIG. 9. Magnetization reversal by changing Josephson junc-\ntion parameter G. The number near curve shows value of\nG.tmz\n0 50 100 150-1-0.500.51\n0.10.5\nDifferentr\nG=10,\nAs=1.4,∆t=60.3\nFIG. 10. Magnetization reversal by changing Josephson junc -\ntion parameter of spin-orbital coupling r. The number near\ncurve shows value of G.\ndepends on the other parameters of the system and, nat-\nurally, the minimal value ofparameter rdepends ontheir\nvalues. Inparticularcasepresentedhereitisaround0.05."    },    {        "title": "1210.4036v1.Optimal_Control_of_Stochastic_Magnetization_Dynamics_by_Spin_Current.pdf",        "content": "arXiv:1210.4036v1  [cond-mat.mes-hall]  15 Oct 2012Optimal Control of Stochastic Magnetization Dynamics by Sp in Current\nYong Wang and Fu-Chun Zhang\nDepartment of Physics, The University of Hong Kong, Hong Kon g SAR, China\nFluctuation-induced stochastic magnetization dynamics p lays an important role in magnetic\nrecording and writing. Here we propose that the magnetizati on dynamics can be optimally con-\ntrolled by the spin current to minimize or maximize the Freid lin-Wentzell action functional of the\nsystem hence to increase or decrease the happening probabil ity of the rare event. We apply this\nmethodtostudythermal-drivenmagnetization switching pr oblem and todemonstrate theadvantage\nof the optimal control strategy.\nOne of the key issues in the modern magnetoelectron-\nics is the reliable control of the magnetization dynam-\nics. Compared with the conventional Oersted field gen-\nerated by the electric current, the spin transfer torque\n(STT)[1, 2] acting on the magnet by the spin current has\nbeen proven to be a more efficient manipulation method\nas the magnetic structures are miniaturized towards the\nnanoscale[3, 4]. Various devices based on this principle\nhas been realized, such as the STT-magnetoresistive ran-\ndom access memory, spin-torque oscillators, spin logic\netc[5]. Although the effect of STT on the magnetization\ndynamics has been extensively studied, its effect on the\nmagnetization fluctuation, which can have a prominent\nimpact on the performance of the spintronics devices[6–\n10], has not received enough attentions. Recently, it was\ndemonstratedexperimentallythatspincurrentcaneither\nsuppress or enhance the magnetization fluctuations[11],\nwhich coincides with the theoretical studies[12, 13]. Fur-\nthermore, it raises the interest to control the magneti-\nzation fluctuation by spin current and different control\nstrategies have been proposed[14, 15]. However, there\nstill exists the controversywhether the effective damping\nshould be decreased or increased in order to suppress the\nmagnetization fluctuation[14, 15], and the choice of con-\ntrol strategies is somewhat arbitrary without solid the-\noretical foundation[15]. Thus deeper understanding the\neffect of spin current on the stochastic magnetization dy-\nnamics and searching for better control strategy are de-\nmanded for further applications of the spin current in\nspintronics devices. In this letter, we will analyze how\nthe spin current can affect the stochastic magnetization\ndynamics basedon the largedeviation principle, and how\nto optimize the spin current pulses to control the mag-\nnetization fluctuation. The application of the idea to\nthe problem of thermal-driven magnetization switching\nin the presence of STT will be demonstrated as an ex-\nample.\nThe thermal-driven magnetization dynamics under\nSTT is usually given by the stochastic LLG equation[12]\nin the case that the quantum effect is negligible[13]. For\ngiven initial magnetization configuration, the fluctuating\nmagnetic field due to the thermal noise will make the\ntrajectories of the magnetzation configuration randomly\ndeviate from its deterministic trajectory when there isno thermal noise. When the noise amplitude is small,\nmost of these random trajectories are around the deter-\nministic trajectory, and only few trajectories can deviate\nfrom the deterministic trajectory largely. Thus the mag-\nnetization dynamics at long-time scale can be regarded\nas the combination of the quasi-deterministic motion for\nmost of the time and some large deviation trajectories\nhappened rarely and randomly. Nevertheless, it is be-\ncause the stochastic magnetization dynamics is mainly\ngoverned by these rare trajectories that the control of\nthe stochastic dynamics is reduced to the control of these\nrare trajectories.\nFor simple, we will considerthe dynamics ofthe single-\ndomain magnet with magnetization vector M=Msm\nand volume V, whereMsdenotes the constant magneti-\nzation magnitude and mdenotes the unit direction vec-\ntor. The case for multidomain magnetic structures is\nmore complicated but can still be treated in the simi-\nlar way. Generally, the happening probabilities P[m(t)]\nof the rare magnetization trajectories {m(t)}in stochas-\ntic magnetization dynamics satisfy the large deviation\nprinciple[16–19], i.e. taking the exponential form\nP[m(t)]≍e−S[m(t),Is(t)]/ǫ, (1)\nwhere the Freidlin-Wentzell (FW) action functional\nS[m(t),Is(t)] depends on the applied spin current Is(t),\nand the parameter ǫreflects the noise amplitude. In or-\nder to increase(or decrease) the happening probabilityof\ncertain magnetization trajectory {m(t)}, one can adjust\ntheappliedspincurrent Is(t)todecrease(orincrease)the\nFW action functional. Specially, the maximal (or min-\nimal) happening probability for {m(t)}is achieved by\nminimizing (or maximizing) the FW action functional.\nUsually, certain constrait conditions for the spin current\nshould be set up, such as the total consumed energy dur-\ning the control should be constant, etc. Then the optimal\nspin current Is\nopt(t) to control the happening probability\nof the given magnetization trajectory {m(t)}is given by\nthe variational problem[20]\nδ{S[m(t),Is(t)]+λF[Is(t)]}= 0, (2)\nwhere we have set the constraint conditions F[Is(t)] = 0\nfor the spin current, and λis the Lagrange multiplier.2\nThe dynamics of the single-domain magnet are driven\nby the deterministic effective magnetic field given by the\nmicromagnetic energy density, the STT due to the spin\ncurrent, and the stochastic magnetic field arising from\nthe thermal fluctuation. Explicitly, the effective mag-\nnetic field is Heff=−∇mE(m)/Ms, with the micro-\nmagnetic energy density E(m). The spin current Isis\ndefined by its amplitude aJand spin-polarization vector\nP, and we will only consider the adiabatic STT[1, 2] al-\nthough the non-adiabatic STT can be important in some\ncases[21]. The fluctuating magnetic field is assume as√ǫ˙W, wherethe amplitude ǫ=2αKBT\nγMsVis proportionalto\ntemperature TandGilbertdampingcoefficient α, and˙W\nis the Gaussian white noise processsatisfying /an}b∇acketle{t˙W(t)/an}b∇acket∇i}ht= 0\nand/an}b∇acketle{t˙Wi(t)˙Wj(t′)/an}b∇acket∇i}ht=δijδ(t−t′). Herewehaveintroduced\nthe Boltzmann constant kBand the gyromagnetic ratio\nγ. Then the stochastic LLG equation can be written in\nthe compact form[19]\n˙m=b(m)+√ǫσ(m)˙W, (3)\nwhere the vector band the matrix σare defined as\nb(m) =σHeff−aJγ′KSP,\nσ(m) =γ′(KA+αKS).\nHere we have γ′=γ\n1+α2, and the elements of the anti-\nsymmetric matrix KAand symmetric matrix KSare[19]\n(KA)µν=ǫµνρmρ,(KS)µν=δµν−mµmν.In this case,\nthe expression of the FW action functional S[m(t),Is(t)]\nwill be[18, 19]\nS[m(t),Is(t)] =1\n2/integraldisplaytf\nti|σ−1(˙m−b)|2dt,(4)\nwheretiandtfare the initial and final time for the tra-\njectory{m}. Equations (2) and (4) are the theoretical\nfoundations for us to optimally control the targeted tra-\njectory{m}of the sinlge-domain magnet with spin cur-\nrentIs.\nFor the above optimal control problem, the usual situ-\nation is that many targeted trajectories exist. For exam-\nple, the magnetmayswitch fromone stableconfiguration\nto anotherviadifferent paths. Amongthese targetedtra-\njectories, the one which minimizes the FW action func-\ntionalS[m(t),Is] for fixed spin current Iswould be the\nmost probable trajectory, which is the so-called optimal\npath[20]. The optimal path provides the best prediction\nfor the random and rare trajectories happened in prac-\ntice, and the correspondinghappening probabilityis P∼\ne−V[mopt,Is]/ǫ, whereV[mopt,Is] = min{S[m(t),Is]}is\nthe quasi-potential[18, 19]. Thus, the optimal control\nproblem is further reduced to find the optimal spin cur-\nrentIs\noptto minimize or maximize the quasi-potential in\norder to increase or decrease the happening probability\nof the optimal path.\nIn principle, the double optimization problem formu-\nlated above can be solved by various optimization al-\ngorithms. One special but important case is that thespin current Isis weak and can be treated perturba-\ntively in the magnetization dynamics. Then the optimal\npath{mopt}can be assumed to be independent on the\nspin current, and the quasi-potential V[mopt,Is] can be\napproximated to the first order of aJasV[mopt,Is] =\nV0[mopt]+∆V[mopt,Is], where\nV0=1\n2/integraldisplaytf\nti|σ−1(˙mopt−b0)|2dt, (5)\n∆V=/integraldisplaytf\ntiaJγ′/an}b∇acketle{tσ−1(˙mopt−b0),σ−1KSP/an}b∇acket∇i}htdt.(6)\nFor convenience, we have introduced the vector b0≡\nσHeff, and notice that σ,KS,Heffare all functions of\nmopt. Thus, the presence of a weak spin current Isgives\nthe change of the initial quasi-potential V0in the amount\nof∆V, andthehappeningprobabilityoftheoptimalpath\n{mopt(t)}has been exponentially changed by the factor\ne−∆V/ǫ. The optimal spin current Is\noptis obtained by\nminimizing or maximizing ∆ V, i.e.\nδ{∆V[mopt(t),Is(t)]+λF[Is(t)]}= 0.(7)\nThermal-driven magnetization switching is a typical\nexample of the stochastic dynamics described above, and\nthe concept of “effective temperature” has been intro-\nduced to the N´ eel-Brown law to take account into the\neffect of spin current on the magnetization switching\nprobability[7–9, 12]. This fact can be easily verified from\nEq. (5) and (6) for a weak spin current with constant\namplitude, where ∆ Vis now proportional to aJand the\nmagnetization switching probability is formally written\nasP∼e−(1−aJ\nac)V0/ǫ. Here,acis the critical value for the\nspin current to switch the magnet, and is given by\nac=α/integraltexttf\nti|σ−1KSHeff|2dt\n/integraltexttf\nti/an}b∇acketle{tσ−1KSHeff,σ−1KSP/an}b∇acket∇i}htdt.(8)\nTo get Eq. (8), we have used the fact that in the ab-\nsence of STT, the optimal path {mopt}satisfies the\nequation[18]\n˙mopt=γ′(KA−αKS)Heff. (9)\nFrom Eq. (8), it is seen that the value of acis dependent\non the choice of the spin polarization vector Pof the\nspin current. Especially, if Pis always in anti-parallel\nwith the effective magnetic field Heff, thenacwill be\nnegative and the magnetization switching probability P\nwill be decreased exponentially. From Eq. (3), one might\nregard that the Gilbert damping should be effectively\nenhanced by the spin current in order to stabilize the\nmagnet[15]. One the other hand, Pshould be parallel\nwithHeffin order to increase P. Furthermore, Eq. (8)\ngives a rough estimation of the critical spin current mag-\nnitude as ac∼αmax{|Heff|}, which is proportional to\ntheGilbertdampingcoefficientandthemaximaleffective\nfield[12].3\nIf the spin current amplitude is assumed to be time-\ndependent, the optimal pulse shape aopt\nJ(t) to control the\noptimal path {mopt}can be obtained from Eq. (7). One\nnatural constraint condition would be that the total dis-\nsipation energy of the spin current pulse should be con-\nstant, i.e./integraltexttf\ntia2\nJ(t)dt=E. Then the general solution\naopt\nJ(t) of Eq. (7) is given by\naopt\nJ(t)−1\nλαγ′2/an}b∇acketle{tσ−1KSHeff,σ−1KSPopt/an}b∇acket∇i}ht= 0.(10)\nand the Lagrange multiplier λis determined from the\nconstraint condition as\nλ=±αγ′2[1\nE/integraldisplaytf\nti/an}b∇acketle{tσ−1KSHeff,σ−1KSPopt/an}b∇acket∇i}ht2dt]1\n2.(11)\nThis spin current pulse gives the change of the quasi-\npotential as ∆ V=−2λE. According to Eq. (10), the\nsign ofλis postivie (or negative) when the optimal spin\npolarization vector Poptis parallel (or antiparallel) with\nthe effective magnetic field Heff, which will enhance (or\nsuppress) the happening probability of the optimal path\nmopt(t) by the factor e2λE/ǫ.\nBased on the theoritical analysis above, we use the\nstochastic LLG equation to simulate the optimal control\nof the thermal-driven magnetization switching by spin\ncurrent numerically. Considering a ferromagnetism film\nwith easy axis along z-axis, and demagnetization field\ndirection along x-axis, then the energy density E(m) is\ngiven as[12, 15, 19]\nE(m) =−1\n2HaMsm2\nz+2πM2\nsm2\nx.(12)\nHere, we assume that no external magnetic field is ap-\nplied, the anisotropic field is Ha= 0.05 T, and the de-\nmagnetization field is 4 πMs= 1.2 T. Besides, we set the\nGilbert damping coefficient α= 0.03, the temperature\nT= 300 K, and the magnet volume V= 1500 nm3. The\ninitialdirectionofthemagnetisassumedas m= (0,0,1),\nand the stochastic LLG equation (3) is simulated with\nthe 4th-order Runge-Kutta method, where the time step\nis set as 1 ps. We also got the amplitude of the critical\nspin current numerically as ac= 0.024 at zero tempera-\nture. Notice that we have chosen a relatively small mag-\nnet otherwise the switching time will be too long to be\nnumerically simulated.\nThe energy profile E(m) has two saddle points, i.e.\n(0,1,0)and (0 ,−1,0), thus there exist twocorresponding\noptimal paths for the magnet to switch from the initial\nstable point (0 ,0,1) to the other (0 ,0,−1). Fig. 1(a)\nshows the optimal path {m+\nopt(t)}from (0,0,1) to\n(0,1,0), which is obained from Eq. (9). As one can see,\nthere are some oscillations around the stable point before\nthe magnet arrives at the saddle point, and this process\ntakes about 2 ns. For comparision, Fig. 1(b) gives one\nrandom switching trajectory of the magnet simulated by\nthe stochastic LLG equation (3) with no spin current.It is found that the main feature of the real switching\ntrajectories are indeed qualitatively captured by the op-\ntimal path, although there are some quanlitative differ-\nences because the noise amplitude is not small enough\nin our simulations. We can also see that the magnet\nrandomly moves around the stable point for most of the\ntime (about 1249 ns in this sample trajectory), and the\nswitching event happens rather rarely.\nThe optimal spin current pulse Is,+\nopt(t) to enhance\nthe happening probability of the optimal switching path\n{m+\nopt(t)}are given in Fig. 1(c) and (d). The absolute\nvalue of aopt\nJ(t) is determined by the constraint condi-\ntion, and its shape shows that the best control strategy\nis to allocate the spin current when the magnet has de-\nviated from the stable point significantly. The optimal\nspin polarization direction P+\noptis set to be in parallel\nwith the effective magnetic field Heff. In contrast, the\noptimal spin current pulse to suppress the optimal path\n{m+\nopt(t)}should have an opposite spin polarization di-\nrection. Forthe energyprofile(12) consideredhere, Heff\nhas noy-component, and its x-component can be quite\nlarge even for a small mopt\nxdue to the large demagneti-\nzation field.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s109/s43 /s111/s112/s116\n/s84/s105/s109/s101/s32/s91/s110/s115/s93/s32/s109\n/s120\n/s32/s109\n/s121 \n/s32/s109\n/s122/s40/s97/s41\n/s49/s50/s52/s54 /s49/s50/s52/s55 /s49/s50/s52/s56 /s49/s50/s52/s57 /s49/s50/s53/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s40/s98/s41\n/s32/s32\n/s84/s105/s109/s101/s32/s91/s110/s115/s93/s109\n/s32/s109\n/s120\n/s32/s109\n/s121 \n/s32/s109\n/s122\n/s32/s32\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s32\n/s51 /s50/s40/s99/s41\n/s84/s105/s109/s101/s32/s91/s110/s115/s93/s97/s111/s112/s116 /s74\n/s32/s32\n/s49\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s80/s43 /s111/s112/s116\n/s84/s105/s109/s101/s32/s91/s110/s115/s93/s40/s100/s41\n/s32/s32\n/s32/s80\n/s120\n/s32/s80\n/s121 \n/s32/s80\n/s122\nFIG. 1. (Color online) (a) optimal path {m+\nopt(t)}for the\nmagnet to switch from (0 ,0,1) to (0,1,0); (b) a sample mag-\nnetization switching trajectory simulated by the stochast ic\nLLG equation; (c) the amplitude of the optimal spin current\npulseIs,+\noptcorresponding to {m+\nopt(t)}; (d) the spin polariza-\ntion vector of the optimal spin current pulse Is,+\noptto enhance\nthe happening probability of {m+\nopt(t)}.\nAs a test, we calculated the mean switching time\n(MST) of the magnet under different control strategies,\nwhere 105random trajectories are generated for each\ncase. First, the MST without spin current is about\n567 ns. Then a constant spin current with aJ= 0.05ac\nwill increase the MST to be about 913 ns if P=\n(0,0,−1), and decrease the MST to be about 353 ns if\nP= (0,0,1). While for the optimal control, the cor-4\nresponding spin current pulse Is\noptis applied only if the\nmagnet randomly arrives at the critical position mthat\nmz=mc. We have chosen mc= 0.89,0.81,0.65 seper-\nately, and the initial point of the corresponding spin cur-\nrent pulses are denoted as 1 ,2,3 in Fig. 1(c). The ampli-\ntudesaopt\nJare given such that each pulse will contain the\nsame energy as a 2-ns constant pulse with aJ= 0.05ac.\nThus, the number of pulses used directly reflects the en-\nergy consumption during the control process.\nTABLE I. Mean switching times τ(unit : ns) and number of\nspin current pulses nwhen the switching probability is either\nsuppressed (s) or enhanced (e). mc: the critical values to\nactivate the control spin current pulse. Subscript a: the\noptimal spin current pulses Is\noptis applied; subscript b: the\nspin polarization vector is fixed as (0 ,0,−1) or (0,0,1) for\nsuppressing or enhancing the switching respectively.\nmcτs\na/ns\naτs\nb/ns\nbτe\na/ne\naτe\nb/ne\nb\n0.89517/430 1067/892 28/26 292/249\n0.81739/263 1073/390 32/13 297/110\n0.65515/32 903/57 67/4 347/22\nThe resulting MSTs τand mean pulse numbers nare\nshown in Table I, for both the cases to suppress and en-\nhance the switchingprobability. Unfortunately, wefound\nthattheoptimalpulsesdidn’t increasetheMSTs τs\nawhen\nmc= 0.89 and 0 .65. Formc= 0.81, the situation is bet-\nter since the MST τs\nahas been increased to 739 ns with\n263 pulses, but it still has no obvious advantage com-\npared with the constant control current case. The reason\nof the failure is that the noise amplitude is not small\nenough and the real switching paths m(t) can deviate\nfrom the optimal path significantly so that the control\npulse will in fact increase their happening probabilities.\nTo eliminate this effect, we have set the spin-polarization\ndirectionas P= (0,0,−1),andtheresultsindeedbecome\nmuch better, asshownby τs\nbandns\nbin TableI. Especially\nwhenmc= 0.65, the MST τs\nbis 903 ns which is close to\nthe constant spin current case, but the number of pulses\nns\nbhas been greatly decreased to 57. This shows that\nthe optimal control theory is helpful to find the way to\nsave energy even when the noise amplitude is not small\nenough.\nThe advantage of the optimal control strategy is fully\nverified for the case to increase the switching probability.\nInTableI,wefoundthattheMST τe\naisdecreasedto28ns\nwith only26pulsesfor mc= 0.89. Ifmc= 0.65, theMST\nτe\na= 66nsislarger,but merely4pulsesinaverageisused\nhere. For comprison, we also list the results τe\nbandne\nb\nifPis set as (0 ,0,1) for the spin current pulses, which\nare only a little better than the constant spin current\ncase. Obviously, the optimal control strategy successfuly\nincreased the switching probability with greatly reduced\nenergy consumption.\nIn conclusion, we have discussed how to use the spincurrent to optimally control the stochastic magnetiza-\ntion dynamics based on the largedeviation principle, and\nshown the advantage of the proposed control strategy by\napplying it to the thermal-driven magnetization switch-\ning problem. The happening probabilities of the rare\nevents, which are the dominant part of the stochastic\nmagnetization dynamics, are controlled by changing the\ncorresponding quasi-potential with optimized spin cur-\nrent pulse. Further generalization of the idea to the cases\nincluding the multi-domain magnetization configuration,\nquantum fluctuations etc. are necessary for more practi-\ncal applications .\nY. W. thanks Dr. G. D. Chaves-O’Flynn for helpful\ndiscussions. This work is supported by the Hong Kong\nUniversity Grant Council (AoE/P-04/08).\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater.\n320, 1190 (2008).\n[4] A. Brataas, A.D. Kent, and H. Ohno, Nature Materials\n11, 372 (2012).\n[5] J. W. Lau and J. M. Shaw, J. Phys. D: Appl. Phys. 44,\n303001 (2011).\n[6] D. Weller and A. Moser, IEEE Trans. Magn. 35, 4423\n(1999).\n[7] S. Urazhdin, N. O. Birge, W. P. Pratt, Jr., and J. Bass,\nPhys. Rev. Lett. 91, 146803 (2003).\n[8] R. H. Koch, J. A. Katine, and J. Z. Sun, Phys. Rev. Lett.\n92, 088302 (2004).\n[9] I. N. Krivorotov, N. C. Emley, A.G. F. Garcia, J.C.\nSankey, S. I. Kiselev, D. C. Ralph, and R. A. Buhrman,\nPhys. Rev. Lett. 93, 166603 (2004).\n[10] X. Cheng, C. T. Boone, J. Zhu, and I. N. Krivorotov,\nPhys. Rev. Lett. 105, 047202 (2010).\n[11] V. E. Demidov, S. Urazhdin, E. R. J. Edwards, M. D.\nStiles, R. D. McMichael, and S. O. Demokritov, Phys.\nRev. Lett. 107, 107204 (2011).\n[12] Z. Li and S. Zhang, Phys. Rev. B 69, 134416 (2004).\n[13] Y. Wang and L. J. Sham, Phys. Rev. B 85, 092403\n(2012).\n[14] M. Covington, U. S. Patent No. 7,042,685 (9 May 2006).\n[15] S. Bandopadhyay, A. Brataas, and G.E.W. Bauer, Appl.\nPhys. Lett. 98, 083110 (2011).\n[16] M. Freidlin and A. Wentzell, Random Perturbations of\nDynamical Systems (Springer-Verlag, New York, 1998).\n[17] H. Touchette, Phys. Rep. 478, 1 (2009).\n[18] R. V. Kohn, M. G. Reznikoff, and E. Vanden-Eijnden, J.\nNonlinear Sci. 15, 223 (2005).\n[19] G. D. Chaves-O’Flynn, D. L. Stein, A. D. Kent, and E.\nVanden-Eijnden, J. Appl. Phys. 109, 07C918 (2011).\n[20] V. N. Smelyanskiy and M. I. Dykman, Phys. Rev. E. 55,\n2516 (1997).\n[21] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004)."    },    {        "title": "2106.14858v3.Stability_of_a_Magnetically_Levitated_Nanomagnet_in_Vacuum__Effects_of_Gas_and_Magnetization_Damping.pdf",        "content": "Stability of a Magnetically Levitated Nanomagnet in Vacuum: E\u000bects of Gas and\nMagnetization Damping\nKatja Kustura,1, 2Vanessa Wachter,3, 4Adri\u0013 an E. Rubio L\u0013 opez,1, 2and Cosimo C. Rusconi5, 6\n1Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria.\n2Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria.\n3Max Planck Institute for the Science of Light, Staudtstra\u0019e 2, 91058 Erlangen, Germany\n4Department of Physics, University of Erlangen-N urnberg, Staudtstra\u0019e 7, 91058 Erlangen, Germany\n5Max-Planck-Institut f ur Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany.\n6Munich Center for Quantum Science and Technology,\nSchellingstrasse 4, D-80799 M unchen, Germany.\n(Dated: June 1, 2022)\nIn the absence of dissipation a non-rotating magnetic nanoparticle can be stably levitated in a\nstatic magnetic \feld as a consequence of the spin origin of its magnetization. Here we study the\ne\u000bects of dissipation on the stability of the system, considering the interaction with the background\ngas and the intrinsic Gilbert damping of magnetization dynamics. At large applied magnetic \felds\nwe identify magnetization switching induced by Gilbert damping as the key limiting factor for\nstable levitation. At low applied magnetic \felds and for small particle dimensions magnetization\nswitching is prevented due to the strong coupling of rotation and magnetization dynamics, and\nthe stability is mainly limited by the gas-induced dissipation. In the latter case, high vacuum\nshould be su\u000ecient to extend stable levitation over experimentally relevant timescales. Our results\ndemonstrate the possibility to experimentally observe the phenomenon of quantum spin stabilized\nmagnetic levitation.\nI. INTRODUCTION\nThe Einstein{de Haas [1, 2] and Barnett e\u000bects [3] are\nmacroscopic manifestations of the internal angular mo-\nmentum origin of magnetization: a change in the mag-\nnetization causes a change in the mechanical rotation\nand conversely. Because of the reduced moment of in-\nertia of levitated nano- to microscale particles, these ef-\nfects play a dominant role in the dynamics of such sys-\ntems [4{10]. This o\u000bers the possibility to harness these\ne\u000bects for a variety of applications such as precise magne-\ntometry [11{16], inertial sensing [17, 18], coherent spin-\nmechanical control [19, 20], and spin-mechanical cool-\ning [21, 22] among others. Notable in this context is\nthe possibility to stably levitate a ferromagnetic parti-\ncle in a static magnetic \feld in vacuum [23, 24]. Stable\nlevitation is enabled by the internal angular momentum\norigin of the magnetization which, even in the absence of\nmechanical rotation, provides the required angular mo-\nmentum to gyroscopically stabilize the system. Such a\nphenomenon, which we refer to as quantum spin stabi-\nlized levitation to distinguish it from the rotational stabi-\nlization of magnetic tops [25{27], relies on the conserva-\ntive interchange between internal and mechanical angular\nmomentum. Omnipresent dissipation, however, exerts\nadditional non-conservative torques on the system which\nmight alter the delicate gyroscopic stability [26, 28]. It\nthus remains to be determined if stable levitation can\nbe observed under realistic conditions, where dissipative\ne\u000bects cannot be neglected.\nIn this article, we address this question. Speci\f-\ncally, we consider the dynamics of a levitated magnetic\nnanoparticle (nanomagnet hereafter) in a static magnetic\n\feld in the presence of dissipation originating both fromthe collisions with the background gas and from the\nintrinsic damping of magnetization dynamics (Gilbert\ndamping) [29, 30], which are generally considered to be\nthe dominant sources of dissipation for levitated nano-\nmagnets [8, 13, 31{33]. Con\fned dynamics can be ob-\nserved only when the time over which the nanomagnet is\nlevitated is longer than the period of center-of-mass os-\ncillations in the magnetic trap. When this is the case, we\nde\fne the system to be metastable . We demonstrate that\nthe system can be metastable in experimentally feasible\nconditions, with the levitation time and the mechanism\nbehind the instability depending on the parameter regime\nof the system. In particular, we show that at weak ap-\nplied magnetic \felds and for small particle dimensions\n(to be precisely de\fned below) levitation time can be\nsigni\fcantly extended in high vacuum (i.e. pressures be-\nlow 10\u00003mbar). Our results evidence the potential of\nunambiguous experimental observation of quantum spin\nstabilized magnetic levitation.\nWe emphasize that our analysis is particularly timely.\nPresently there is a growing interest in levitating and con-\ntrolling magnetic systems in vacuum [9, 34, 35]. Current\nexperimental e\u000borts focus on levitation of charged para-\nmagnetic ensembles in a Paul trap [19, 36, 37], diamag-\nnetic particles in magneto-gravitational traps [38{40], or\nferromagnets above a superconductor [14, 20, 41]. Lev-\nitating ferromagnetic particles in a static magnetic trap\no\u000bers a viable alternative, with the possibility of reaching\nlarger mechanical trapping frequencies.\nThe article is organized as follows. In Sec. II we in-\ntroduce the model of the nanomagnet, and we de\fne\ntwo relevant regimes for metastability, namely the atom\nphase and the Einstein{de Haas phase. In Sec. III and\nIV we analyze the dynamics in the atom phase and thearXiv:2106.14858v3  [cond-mat.mes-hall]  31 May 20222\nFigure 1. (a) Illustration of a spheroidal nanomagnet levi-\ntated in an external \feld B(r) and surrounded by a gas at the\ntemperature Tand the pressure P. (b) Linear stability dia-\ngram of a non-rotating nanomagnet in the absence of dissipa-\ntion, assuming a= 2b. Blue and red regions denote the stable\natom and Einstein{de Haas phase, respectively; hatched area\nis the unstable region. Dashed lines show the critical values\nof the bias \feld which de\fne the two phases. In particular,\nBEdH,1\u00115\u0016=[4\r2\n0(a2+b2)M],BEdH,2\u00113 [\u0016B02=(4\r0M)]1=3,\nandBatom = 2kaV=\u0016. Numerical values of physical parame-\nters used to generate panel (b) are given in Table I.\nEinstein{de Haas phase, respectively. We discuss our re-\nsults in Sec. V. Conclusions and outlook are provided in\nSec. VI. Our work is complemented by three appendices\nwhere we de\fne the transformation between the body-\n\fxed and laboratory reference frames (App. A), analyze\nthe e\u000bect of thermal \ructuations (App. B), and provide\nadditional \fgures (App. C).\nII. DESCRIPTION OF THE SYSTEM\nWe consider a single domain nanomagnet levitated in\na static1magnetic \feld B(r) as shown schematically in\nFig. 1(a). We model the nanomagnet as a spheroidal\nrigid body of mass density \u001aMand semi-axes lengths a;b\n(a > b ), having uniaxial magnetocrystalline anisotropy,\nwith the anisotropy axis assumed to be along the major\nsemi-axisa[42]. Additionally, we assume that the mag-\nnetic response of the nanomagnet is approximated by a\npoint dipole with magnetic moment \u0016of constant mag-\nnitude\u0016\u0011j\u0016j, as it is often justi\fed for single domain\nparticles [42, 43]. Let us remark that such a simpli\fed\nmodel has been considered before to study the classical\ndynamics of nanomagnets in a viscous medium [31, 44{\n49], as well as to study the quantum dynamics of mag-\nnetic nanoparticles in vacuum [5, 13, 50, 51]. Since the\nmodel has been successful in describing the dynamics of\nsingle-domain nanomagnets, we adopt it here to inves-\ntigate the stability in a magnetic trap. In particular,\nour study has three main di\u000berences as compared with\n1We denote a \feld static if it does not have explicit time depen-\ndence, namely if @B(r)=@t= 0.Table I. Physical parameters of the model and the values used\nthroughout the article. We calculate the magnitude of the\nmagnetic moment as \u0016=\u001a\u0016V, where\u001a\u0016=\u001aM\u0016B=(50amu),\nwith\u0016Bthe Bohr magneton and amu the atomic mass unit.\nParameter Description Value [units]\n\u001aM mass density 104[kg m\u00003]\na;b semi-axes see main text [m]\n\u001a\u0016 magnetization 2 :2\u0002106[J T\u00001m\u00003]\nka anisotropy constant 105[J m\u00003]\n\r0 gyromagnetic ratio 1 :76\u00021011[rad s\u00001T\u00001]\nB0 \feld bias see main text [T]\nB0\feld gradient 104[T m\u00001]\nB00\feld curvature 106[T m\u00002]\n\u0011 Gilbert damping 10\u00002[n. u.]\nT temperature 10\u00001[K]\nP pressure 10\u00002[mbar]\nM molar mass 29 [g mol\u00001]\n\u000bc re\rection coe\u000ecient 1 [n. u.]\nprevious work. (i) We consider a particle levitated in\nhigh vacuum, where the mean free path of the gas parti-\ncles is larger than the nanomagnet dimensions (Knudsen\nregime [52]). This leads to gas damping which is gen-\nerally di\u000berent from the case of dense viscous medium\nmostly considered in the literature. (ii) We consider\ncenter-of-mass motion and its coupling to the rotational\nand magnetic degrees of freedom, while previous work\nmostly focuses on coupling between rotation and mag-\nnetization only (with the notable exception of [48]). (iii)\nWe are primarily interested in the center-of-mass con\fne-\nment of the particle, and not in its magnetic response.\nWithin this model the relevant degrees of freedom of\nthe system are the center-of-mass position r, the linear\nmomentum p, the mechanical angular momentum L, the\norientation of the nanomagnet in space \n, and the mag-\nnetic moment \u0016. The orientation of the nanomagnet\nis speci\fed by the body-\fxed reference frame Oe1e2e3,\nwhich is obtained from the laboratory frame Oexeyez\naccording to ( e1;e2;e3)T=R(\n)(ex;ey;ez)T, where\n\n= (\u000b;\f;\r )Tare the Euler angles and R(\n) is the\nrotational matrix. We provide the expression for R(\n)\nin App. A. The body-\fxed reference frame is chosen such\nthate3coincides with the anisotropy axis. The magnetic\nmoment\u0016is related to the internal angular momentum F\naccording to the gyromagnetic relation \u0016=\r0F, where\n\r0is the gyromagnetic ratio of the material2.\n2The total internal angular momentum Fis a sum of the individ-\nual atomic angular momenta (spin and orbital), which contribute\nto the atomic magnetic moment. For a single domain magnetic\nparticle, it is customary to assume that Fcan be described as\na vector of constant magnitude, jFj=\u0016=\r0(macrospin approxi-\nmation) [43].3\nA. Equations of Motion\nWe describe the dynamics of the nanomagnet in the\nmagnetic trap with a set of stochastic di\u000berential equa-\ntions which model both the deterministic dissipative evo-\nlution of the system and the random \ructuations due to\nthe environment. In the following it is convenient to de-\n\fne dimensionless variables: the center-of-mass variables\n~r\u0011r=a,~p\u0011\r0ap=\u0016, the mechanical angular momen-\ntum`\u0011\r0L=\u0016, the magnetic moment m\u0011\u0016=\u0016, and\nthe magnetic \feld b(~r)\u0011B(a~r)=B0, whereB0denotes\nthe minimum of the \feld intensity in a magnetic trap,\nwhich we hereafter refer to as the bias \feld. Note that\nwe choose to normalize the position r, the magnetic mo-\nment\u0016and the magnetic \feld B(r) with respect to the\nparticle size a, the magnetic moment magnitude \u0016, and\nthe bias \feld B0, respectively. The scaling factor for an-\ngular momentum, \u0016=\r0, and linear momentum, \u0016=(a\r0),\nfollow as a consequence of the gyromagnetic relation.\nThe dynamics of the nanomagnet in the laboratory\nframe are given by the equations of motion\n_~r=!I~p; (1)\n_e3=!\u0002e3; (2)\n_~p=!Lr~r[m\u0001b(~r)]\u0000\u0000cm~p+\u001ep(t); (3)\n_`=!Lm\u0002b(~r)\u0000_m\u0000\u0000rot`+\u0018l(t); (4)\n_m=m\n1 +\u00112\u0002[!e\u000b\u0000\u0011m\u0002(!+!e\u000b+\u0011!\u0002m)\n+\u0010b(t)]: (5)\nHere we de\fne the relevant system frequencies: !I\u0011\n\u0016=(\r0Ma2) is the Einstein{de Haas frequency, with M\nthe mass of the nanomagnet, !L\u0011\r0B0is the Larmor\nfrequency,!A\u0011kaV\r0=\u0016is the anisotropy frequency,\nwithVthe volume of the nanomagnet and kathe ma-\nterial dependent anisotropy constant [43], !\u0011I\u00001L\nis the angular velocity, with Ithe tensor of inertia,\nand!e\u000b\u00112!A(m\u0001e3)e3+!Lb(~r). Dissipation is\nparametrized by the dimensionless Gilbert damping pa-\nrameter\u0011[29, 53], and the center-of-mass and rotational\nfriction tensors \u0000 cmand \u0000 rot, respectively [32]. The e\u000bect\nof stochastic thermal \ructuations is represented by the\nrandom variables \u001ep(t) and\u0018l(t) which describe, respec-\ntively, the \ructuating force and torque exerted by the\nsurrounding gas, and by \u0010b(t) which describes the ran-\ndom magnetic \feld accounting for thermal \ructuations\nin magnetization dynamics [54]. We assume Gaussian\nwhite noise, namely, for X(t)\u0011(\u001ep(t);\u0018l(t);\u0010b(t))Twe\nhavehXi(t)i= 0 andhXi(t)Xj(t0)i\u0018\u000eij\u000e(t\u0000t0).\nEquations (1-4) describe the center-of-mass and rota-\ntional dynamics of a rigid body in the presence of dis-\nsipation and noise induced by the background gas [32].\nThe expressions for \u0000 cmand \u0000 rotdepend on the parti-\ncle shape { here we take the expressions derived in [32]for a cylindrical particle3{, and on the ratio of the sur-\nface and the bulk temperature of the particle, which\nwe assume to be equal to the gas temperature T. Fur-\nthermore, they account for two di\u000berent scattering pro-\ncesses, namely the specular and the di\u000busive re\rection\nof the gas from the particle, which is described by a\nphenomenological interpolation coe\u000ecient \u000bc. The or-\nder of magnitude of the di\u000berent components of \u0000 cmand\n\u0000rotis generally well approximated by the dissipation\nrate \u0000\u0011(2Pab=M )[2\u0019M=(NAkBT)]1=2, wherePand\nMare, respectively, the gas pressure and molar mass,\nkBis the Boltzmann constant and NAis the Avogadro\nnumber. The magnetization dynamics given by Eq. (5)\nis the Landau-Lifshitz-Gilbert equation in the laboratory\nframe [8, 57], with the e\u000bective magnetic \feld !e\u000b=\r0.\nWe remark that Eqs. (1-5) describe the classical dynam-\nics of a levitated nanomagnet where the e\u000bect of the\nquantum spin origin of magnetization, namely the gy-\nromagnetic relation, is taken into account phenomeno-\nlogically by Eq. (5). This is equivalent to the equations\nof motion obtained from a quantum Hamiltonian in the\nmean-\feld approximation [24].\nLet us discuss the e\u000bect of thermal \ructuations on\nthe dynamics of the nanomagnet at subkelvin temper-\natures and in high vacuum. These conditions are com-\nmon in recent experiments with levitated particles [58{\n60]. The thermal \ructuations of magnetization dy-\nnamics, captured by the last term in Eq. (5), lead\nto thermally activated transition of the magnetic mo-\nment between the two stable orientations along the\nanisotropy axis [54, 61]. Such process can be quan-\nti\fed by the N\u0013 eel relaxation time, which is given by\n\u001cN\u0019(\u0019=! A)p\nkBT=(kaV)ekaV=(kBT). Thermal acti-\nvation can be neglected when \u001cNis larger than other\ntimescales of magnetization dynamics, namely the pre-\ncession timescale given by \u001cL\u00111=j!e\u000bj, and the Gilbert\ndamping timescale given by \u001cG\u00111=(\u0011j!e\u000bj). Con-\nsidering for simplicity j!e\u000bj\u00182!A, for a particle size\na= 2b= 1 nm and temperature T= 1 K, and the\nvalues of the remaining parameters as in Table I, the ra-\ntio of the timescales is of the order \u001cN=\u001cL\u0018103, and\nit is signi\fcantly increased for larger particle sizes and\nat smaller temperatures. We remark that, for the val-\nues considered in this article, \u001cNis much larger than the\nlongest dynamical timescale in Eqs. (1-5) which is associ-\nated with the motion along ex. Thermal activation of the\nmagnetic moment can therefore be safely neglected. The\nstochastic e\u000bects ascribed to the background gas, cap-\ntured by the last terms in Eqs. (3-4), are expected to be\nimportant at high temperatures (namely, a regime where\nMkBT\r2\n0a2=\u00162&1 [32]). At subkelvin temperatures and\nin high vacuum these \ructuations are weak and, con-\nsequently, they do not destroy the deterministic e\u000bects\n3The expressions for \u0000 cmand \u0000 rotfor a cylindrical particle\ncapture the order of magnitude of the dissipation rates for a\nspheroidal particle [55, 56].4\ncaptured by the remaining terms in Eqs. (1-5) [33]. In-\ndeed, for the values of parameters given in Table I and\nfora= 2b,MkBT\r2\n0a2=\u00162\u00190:8T=(a[nm]). For sub-\nkelvin temperatures and particle sizes a>1 nm, thermal\n\ructuations due to the background gas can therefore be\nsafely neglected.\nIn the following we thus neglect stochastic e\u000bects by\nsetting\u001ep=\u0018l=\u0010b= 0, and we consider only the de-\nterministic part of Eqs. (1-5) as an appropriate model\nfor the dynamics [8, 33, 54]. In App. B we carry out\nthe analysis of the dynamics including the e\u000bects of gas\n\ructuations in equations (1-5), and we show that the\nresults presented in the main text remain qualitatively\nvalid even in the presence of thermal noise. For the mag-\nnetic \feld B(r) we hereafter consider a Io\u000be-Pritchard\nmagnetic trap, given by\nB(r) =ex\u0014\nB0+B00\n2\u0012\nx2\u0000y2+z2\n2\u0013\u0015\n\u0000ey\u0012\nB0y+B00\n2xy\u0013\n+ez\u0012\nB0z\u0000B00\n2xz\u0013\n;(6)\nwhereB0;B0andB00are, respectively, the \feld bias, gra-\ndient and curvature [62]. We remark that this is not a\nfundamental choice, and di\u000berent magnetic traps, pro-\nvided they have a non-zero bias \feld, should result in\nsimilar qualitative behavior.\nB. Initial conditions\nThe initial conditions for the dynamics in Eqs. (1-5),\nnamely at time t= 0, depend on the initial state of the\nsystem, which is determined by the preparation of the\nnanomagnet in the magnetic trap. In our analysis, we\nconsider the nanomagnet to be prepared in the thermal\nstate of an auxiliary loading potential at the temperature\nT. Subsequently, we assume to switch o\u000b the loading\npotential at t= 0, while at the same time switching\non the Io\u000be-Pritchard magnetic trap. The choice of the\nauxiliary potential is determined by two features: (i) it\nallows us to simply parametrize the initial conditions by a\nsingle parameter, namely the temperature T, and (ii) it is\nan adequate approximation of general trapping schemes\nused to trap magnetic particles.\nRegarding point (i), we assume that the particle is lev-\nitated in a harmonic trap, in the presence of an external\nmagnetic \feld applied along ex. This loading scheme\nprovides, on the one hand, trapping of the center-of-mass\ndegrees of freedom, with trapping frequencies denoted by\n!i(i=x;y;z ). On the other hand, the magnetic moment\nin this case is polarized along ex. The Hamiltonian of the\nsystem in such a con\fguration reads Haux=p2=(2M) +P\ni=x;y;zM!2\nir2\ni=2+LI\u00001L=2\u0000kaVe2\n3;x\u0000\u0016xBaux, where\nBauxdenotes the magnitude of the external magnetic\n\feld, which we for simplicity set to Baux=B0in all our\nsimulations. At t= 0 the particle is released in the mag-\nnetic trap given by Eq. (6). For the degrees of freedomx\u0011(~r;~p;`;mx)T, we take as the initial displacement\nfrom the equilibrium the corresponding standard devia-\ntion in a thermal state of Haux. More precisely, xi(0) =\nxi;e+ (hx2\nii\u0000hxii2)1=2, wherexi;edenotes the equilib-\nrium value, andhxk\nii\u0011Z\u00001R\ndxxk\niexp[\u0000Haux=(kBT)],\nwithk= 1;2 and the partition function Z. For the Eu-\nler angles \nwe use \n 1(0)\u0011cos\u00001[\u0000p\nhcos2\n1i] and\n\ni(0)\u0011cos\u00001[p\nhcos2\nii] (i= 2;3). The initial condi-\ntions for e3follow from \nusing the transformation given\nin App. A.\nRegarding point (ii), the initial conditions obtained in\nthis way describe a trapped particle prepared in a ther-\nmal equilibrium in the presence of an external loading\npotential where the center of mass is decoupled from the\nmagnetization and the rotational dynamics. It is outside\nthe scope of this article to study in detail a particular\nloading scheme. However, we point out that an auxil-\niary potential given by Hauxcan be obtained, for exam-\nple, by trapping the nanomagnet using a Paul trap as\ndemonstrated in recent experiments [19, 21, 37, 63{70].\nIn particular, trapping of a ferromagnetic particle has\nbeen demonstrated in a Paul trap at P= 10\u00002mbar,\nwith center-of-mass trapping frequency of up to 1 MHz,\nand alignment of the particle along the direction of an\napplied \feld [19]. We note that particles are shown to\nremain trapped even when the magnetic \feld is varied\nover many orders of magnitudes or switched o\u000b. We re-\nmark further that alignment of elongated particles can\nbe achieved using a quadrupole Paul trap even in the\nabsence of magnetic \feld [55, 71].\nC. Linear stability\nIn the absence of thermal \ructuations, an equilibrium\nsolution of Eqs. (1-5) is given by ~re=~pe=`e= 0 and\ne3;e=me=\u0000ex. This corresponds to the con\fguration\nin which the nanomagnet is \fxed at the trap center, with\nthe magnetic moment along the anisotropy axis and anti-\naligned to the bias \feld B0. Linear stability analysis of\nEqs. (1-5) shows that the system is unstable, as expected\nfor a gyroscopic system in the presence of dissipation [28].\nHowever, when the nanomagnet is metastable, it is still\npossible for it to levitate for an extended time before\nbeing eventually lost from the trap, as in the case of a\nclassical magnetic top [25{27]. As we show in the fol-\nlowing sections, the dynamics of the system, and thus its\nmetastability, strongly depend on the applied bias \feld\nB0. We identify two relevant regimes: (i) strong-\feld\nregime, de\fned by bias \feld values B0> B atom, and\n(ii) weak-\feld regime, de\fned by B0< B atom, where\nBatom\u00112kaV=\u0016. This di\u000berence is reminiscent of the\ntwo di\u000berent stable regions which arise as a function of\nB0in the linear stability diagram in the absence of dis-\nsipation [see Fig. 1(b)] [23, 24]. In Sec. III and Sec. IV\nwe investigate the possibility of metastable levitation by\nsolving numerically Eqs. (1-5) in the strong-\feld and\nweak-\feld regime, respectively.5\nIII. DYNAMICS IN THE STRONG-FIELD\nREGIME: ATOM PHASE\nThe strong-\feld regime, according to the de\fnition\ngiven in Sec. II C, corresponds to the blue region in the\nlinear stability diagram in the absence of dissipation,\nshown in Fig. 1(b). This region is named atom phase\nin [23, 24], and we hereafter refer to the strong-\feld\nregime as the atom phase. This parameter regime corre-\nsponds to the condition !L\u001d!A;!I. In this regime, the\ncoupling of the magnetic moment \u0016and the anisotropy\naxise3is negligible, and, to \frst approximation, the\nnanomagnet undergoes a free Larmor precession about\nthe local magnetic \feld. In the absence of dissipation,\nthis stabilizes the system in full analogy to magnetic trap-\nping of neutral atoms [72, 73].\nIn Fig. 2(a-c) we show the numerical solution of\nEqs. (1-5) for nanomagnet dimensions a= 2b= 20\nnm and the bias \feld B0= 200 mT. As evidenced by\nFig. 2(a), the magnetization mxof the particle changes\ndirection. During this change, the mechanical angu-\nlar momentum lxchanges accordingly in the manifesta-\ntion of the Einstein{de Haas e\u000bect, such that the to-\ntal angular momentum m+`is conserved4. The dy-\nnamics observed in Fig. 2(a) is indicative of Gilbert-\ndamping-induced magnetization switching, a well-known\nphenomenon in which the projection of the magnetic mo-\nment along the e\u000bective magnetic \feld !e\u000b=\r0changes\nsign [30]. This is expected to happen when the applied\nbias \feldB0is larger than the e\u000bective magnetic \feld\nassociated with the anisotropy, given by \u0018!A=\r0. Mag-\nnetization switching displaces the system from its equi-\nlibrium position on a timescale which is much shorter\nthan the period of center-of-mass oscillations, estimated\nfrom [24] to be \u001ccm\u00181\u0016s. The nanomagnet thus shows\nno signature of con\fnement [see Fig. 2(b)].\nThe timescale of levitation in the atom phase is given\nby the timescale of magnetization switching, which we\nestimate as follows. As evidenced by Fig. 2(a-b), the\ndynamics of the center of mass and the anisotropy axis\nare approximately constant during switching, such that\n!e\u000b\u0019!e\u000b(t= 0). Under this approximation and as-\nsuming\u0011\u001c1, the magnetic moment projection mk\u0011\n!e\u000b\u0001m=j!e\u000bjevolves as\n_mk\u0019\u0011[!L+ 2!Amk](1\u0000m2\nk): (7)\nAccording to Eq. (7) the component mkexhibits switch-\ning ifmk(t= 0)&\u00001 and!L=2!A>1 [30], both of\nwhich are ful\flled in the atom phase. Integrating Eq. (7)\nwe obtain the switching time \u001c[de\fned as mk(\u001c)\u00110],\nwhich can be well approximated by\n\u001c\u0019ln\u0000\n1 +jmk(t= 0)j\u0001\n2\u0011(!L+ 2!A)\u0000ln\u0000\n1\u0000jmk(t= 0)j\u0001\n2\u0011(!L\u00002!A):(8)\n4We always \fnd the transfer of angular momentum to the center\nof mass angular momentum r\u0002pto be negligible.\nFigure 2. Dynamics in the atom phase. (a) Dynamics of\nthe magnetic moment component mx, the mechanical angular\nmomentum component lx, and the anisotropy axis component\ne3;xfor nanomagnet dimensions a= 2b= 20 nm and the bias\n\feldB0= 200 mT. For the initial conditions we consider\ntrapping frequencies !x= 2\u0019\u00022 kHz and!y=!z= 2\u0019\u000250\nkHz. Unless otherwise stated, for the remaining parameters\nthe numerical values are given in Table I. (b) Center-of-mass\ndynamics for the same case considered in (a). (c) Dynamics of\nthe magnetic moment component mk. Line denoted by circle\ncorresponds to the case considered in (a). Each remaining\nline di\u000bers by a single parameter, as denoted by the legend.\nDotted vertical lines show Eq. (8). (d) Switching time given\nby Eq. (8) as a function of the bias \feld B0and the major\nsemi-axisa. In the region left of the thick dashed line the\ndeviation from the exact value is more than 5%. Hatched\narea is the unstable region in the linear stability diagram in\nFig. 1.(b).\nThe estimation Eq. (8) is in excellent agreement with\nthe numerical results for di\u000berent parameter values [see\nFig. 2(c)].\nMagnetization switching characterizes the dynamics of\nthe system in the entire atom phase. In particular, in\nFig. 2(d) we analyze the validity of Eq. (8) for di\u000berent\nvalues of the bias \feld B0and the major semi-axis a, as-\nsumingb=a=2. The thick dashed line shows the region\nwhere Eq. (8) di\u000bers from the exact switching time, as\nestimated from the full dynamics of the system, by 5%;\nleft of this line the deviation becomes increasingly more\nsigni\fcant, with Eq. (8) predicting up to 20% larger val-\nues close to the stability border (namely, for bias \feld\nclose toBatom = 90 mT). We believe that the signi\f-\ncant deviation close to the border of the atom phase is\ndue to the non-negligible coupling to the anisotropy axis,6\nFigure 3. Dynamics in the Einstein{de Haas phase. (a) Motion of the system in the ey-ezplane until time t= 5\u0016s for\nnanomagnet dimensions a= 2b= 2 nm and the bias \feld B0= 0:5 mT. For the initial conditions we consider trapping\nfrequencies !x= 2\u0019\u00022 kHz and !y=!z= 2\u0019\u00021 MHz. For the remaining parameters the numerical values are given\nin Table I. (b) Dynamics of the projection mkand (c) dynamics of the anisotropy axis component e3;x, for the same case\nconsidered in (a). (d) Dynamics of the center-of-mass component ryand (e) dynamics of the magnetic moment component\nmxon a longer timescale, for the same values of parameters as in (a). (f) Escape time t?as a function of gas pressure P, for\ndi\u000berent con\fgurations in the Einstein{de Haas phase. Circles correspond to the case considered in (a). Each remaining case\ndi\u000bers by parameters indicated by the legend. (g) Escape time t?as a function of the major semi-axis a, with the values of\nthe remaining parameters as in (a). Dashed vertical line denotes the upper limit of the Einstein{de Haas phase, given by the\ncritical \feld BEdH,1 [see Fig. 1(b)].\nwhich results in additional mechanisms not captured by\nthe simple model Eq. (7). In fact, it is known that cou-\npling between magnetization and mechanical degrees of\nfreedom might have an impact on the switching dynam-\nics [74]. As demonstrated by Fig. 2(d), the switching\ntime is always shorter than the center-of-mass oscillation\nperiod\u001ccm, and thus no metastability can be observed in\nthe atom phase.\nLet us note that the conclusions we draw in Fig. 2\nremain valid if one varies the anisotropy constant ka,\nGilbert damping parameter \u0011, and the temperature T,\nas we show in App. C. Finally, we note that the dis-\nsipation due to the background gas has negligible ef-\nfects. In particular, for the values assumed in Fig. 2(a-b)\nthe timescale of the gas-induced dissipation is given by\n1=\u0000 = 440\u0016s.\nIV. DYNAMICS IN THE WEAK-FIELD\nREGIME: EINSTEIN{DE HAAS PHASE\nWe now focus on the regime of weak bias \feld, cor-\nresponding to the condition !L\u001c!A. In this regime\nmagnetization switching does not occur, and the dynam-\nics critically depend on the particle size. In the follow-\ning we focus on the regime of small particle dimensions,i.e.!L\u001c!I, which, as we will show, is bene\fcial for\nmetastability. In the absence of dissipation, this regime\ncorresponds to the Einstein{de Haas phase [red region\nin Fig. 1(b)] [23, 24]. The hierarchy of energy scales in\nthe Einstein{de Haas phase (namely, !L\u001c!A;!I) man-\nifests in two ways: (i) the anisotropy is strong enough to\ne\u000bectively \\lock\" the direction of the magnetic moment \u0016\nalong the anisotropy axis e3(!A\u001d!L), and (ii) accord-\ning to the Einstein{de Haas e\u000bect, the frequency at which\nthe nanomagnet would rotate if \u0016switched direction is\nsigni\fcantly increased at small dimensions ( !I\u001d!L),\nsuch that switching can be prevented due to energy con-\nservation [4]. In the absence of dissipation, the combina-\ntion of these two e\u000bects stabilizes the system.\nIn Fig. 3(a-c) we show the numerical solution of\nEqs. (1-5) for nanomagnet dimensions a= 2b= 2 nm\nand the bias \feld B0= 0:5 mT. The nanomagnet is\nmetastable, as evidenced by the con\fned center-of-mass\nmotion shown in Fig. 3(a). In Fig. 3(b-c) we show the\ndynamics of the magnetic moment component mkand\nthe anisotropy axis component e3;x, respectively, which\nindicates that no magnetization switching occurs in this\nregime. We remark that the absence of switching can-\nnot be simply explained on the basis of Eqs. (7-8). In\nfact, the simple model of magnetization switching, given7\nby Eq. (7), assumes that the dynamics of the rotation\nand the center-of-mass motion happen on a much longer\ntimescale than the timescale of magnetization dynam-\nics. However, in this case rotation and magnetization\ndynamics occur on a comparable timescale, as evidenced\nby Fig. 3(b-c). The weak-\feld condition alone ( !L\u001c!A)\nis thus not su\u000ecient to correctly explain the absence of\nswitching, and the role of particle size ( !L\u001c!I) needs\nto be considered.\nLet us analyze the role of Gilbert damping in this case.\nSince in the Einstein{de Haas phase mk\u00181, we de\fne\nm\u0011e3+\u000em, where\u000emrepresents the deviation of m\nfrom the anisotropy axis e3, and we assumej\u000emj\u001cje3j\n[see Fig. 3(b)]. This allows us to simplify Eq. (5) as\n\u000e_m\u0019!e\u000b\u0002\u000em\u0000\u0011[2!A+!3e3\u0001(m+`)]\u000em;(9)\nwhere!3\u0011\u0016=(\r0I3), withI3the principal moment of\ninertia along e3. As evidenced by Eq. (9), the only e\u000bect\nof Gilbert damping is to align mande3on a timescale\ngiven by\u001c0\u00111=[\u0011(2!A+!3)], irrespective of the dy-\nnamics of e3. For the values of parameters considered in\nFig. 3(a-c), \u001c0= 5 ns, and it is much shorter than the\ntimescale of center-of-mass dynamics, given by \u001ccm\u00181\n\u0016s. For all practical purposes, the magnetization in the\nEinstein{de Haas phase can be considered frozen along\nthe anisotropy axis. The nanomagnet in the presence of\nGilbert damping is therefore equivalent to a hard magnet\n(i. e.ka!1 ) [24].\nThe main mechanism behind the instability in the\nEinstein{de Haas phase is thus gas-induced dissipation.\nIn Fig. 3(d-e) we plot the dynamics of the center-of-\nmass component ryand the magnetic moment compo-\nnentmxon a longer timescale, for two di\u000berent values of\nthe pressure P. The e\u000bect of gas-induced dissipation is\nto dampen the center-of-mass motion to the equilibrium\nposition, while the magnetic moment moves away from\nthe equilibrium. Both processes happen on a timescale\ngiven by the dissipation rate \u0000. When ex=mx\u00190, the\nsystem becomes unstable and ultimately leaves the trap\n[see arrow in Fig. 3(d)]. We de\fne the escape time t?as\nthe time at which the particle position is y(t?)\u00115y(0),\nand we show it in Fig. 3(f) as a function of pressure Pfor\ndi\u000berent con\fgurations in the Einstein{de Haas phase,\nand forb=a=2. Fig. 3(f) con\frms that the dissipation\na\u000bects the system on a timescale which scales as \u00181=P.\nThe metastability of the nanomagnet in the Einstein{de\nHaas phase is therefore limited solely by the gas-induced\ndissipation, which can be signi\fcantly reduced in high\nvacuum. Finally, in Fig. 3(g) we analyze the e\u000bect of\nparticle size on metastability. Speci\fcally, we show the\nescape time t?as a function of the major semi-axis aat\nthe bias \feld B0= 0:5 mT, forb=a=2. The escape time\nis signi\fcantly reduced at increased particle sizes. This\ncon\frms the advantage of the Einstein{de Haas phase to\nobserve metastability, even in the presence of dissipation.V. DISCUSSION\nIn deriving the results discussed in the preceding sec-\ntions, we assumed (i) a single-magnetic-domain nanopar-\nticle with uniaxial anisotropy and constant magnetiza-\ntion, with the values of the physical parameters summa-\nrized in Table I, (ii) deterministic dynamics, i. e. the\nabsence of thermal \ructuations, (iii) that gravity can be\nneglected, and (iv) a non-rotating nanomagnet. Let us\njustify the validity of these assumptions.\nWe \frst discuss the values of the parameters given in\nTable I, which are used in our analysis. The material pa-\nrameters, such as \u001aM,\u001a\u0016,kaand\u0011, are consistent with,\nfor example, cobalt [75{78]. We remark that the uniax-\nial anisotropy considered in our model represents a good\ndescription even for materials which do not have an in-\ntrinsic magnetocrystalline uniaxial anisotropy, provided\nthat they have a dominant contribution from the uniaxial\nshape anisotropy. This is the case, for example, for fer-\nromagnetic particles with a prolate shape [75]. We point\nout that the values used here do not correspond to a spe-\nci\fc material, but instead they describe a general order\nof magnitude corresponding to common magnetic materi-\nals. Indeed, our results are general and can be particular-\nized to speci\fc materials by replacing the above generic\nvalues with exact numbers. As we show in App. C, the re-\nsults and conclusions presented here remain unchanged\neven when di\u000berent values of the parameters are con-\nsidered. The values used for the \feld gradient B0and\nthe curvature B00have been obtained in magnetic mi-\ncrotraps [62, 79{82]. The values of the gas pressure P\nand the temperature Tare experimentally feasible, with\nnumerous recent experiments reaching pressure values as\nlow asP= 10\u00006mbar [58, 68, 70, 83{85]. All the values\nassumed in our analysis are therefore consistent with cur-\nrently available technologies in levitated optomechanics.\nThermal \ructuations can be neglected at cryogenic\nconditions (as we argue in Sec. II A), as their e\u000bect is\nweak enough not to destroy the deterministic e\u000bects cap-\ntured by Eqs. (1-5). In particular, thermal activation of\nthe magnetization, as quanti\fed by the N\u0013 eel relaxation\ntime, can be safely neglected due to the large value of\nthe uniaxial anisotropy even for the smallest particles\nconsidered. As for the mechanical thermal \ructuations,\nwe con\frm that they do not modify the deterministic\ndynamics in App. B, where we simulate the associated\nstochastic dynamics.\nGravity, assumed to be along ex, can be safely ne-\nglected, since the gravity-induced displacement of the\ntrap center from the origin is much smaller than the\nlength scale over which the Io\u000be-Pritchard \feld signi\f-\ncantly changes [24]. Speci\fcally, the gravitational poten-\ntialMgx shifts the trap center from the origin r= 0\nalong exby an amount rg\u0011Mg= (\u0016B00), wheregis\nthe gravitational acceleration. On the other hand, the\ncharacteristic length scales of the Io\u000be-Pritchard \feld\nare given by \u0001 r0\u0011p\nB0=B00for the variation along\nex, and \u0001r0\u0011B0=B00for the variation o\u000b-axis. When-8\neverrg\u001c\u0001r0;\u0001r0, gravity has a negligible role in the\nmetastable dynamics of the system. In the parameter\nregime considered in this article, this is always the case.\nWe note that the condition to neglect gravity is the same\nas for a magnetically trapped atom, since both Mand\u0016\nscale with the volume.\nFinally, we remark that the analysis presented here\nis carried out for the case of a non-rotating nanomag-\nnet5. The same qualitative behavior is obtained even in\nthe presence of mechanical rotation (namely, considering\na more general equilibrium con\fguration with `e6= 0).\nThe analysis of dynamics in the presence of rotation is\nprovided in App. C. In particular, the dynamics in the\nEinstein{de Haas phase remains largely una\u000bected, pro-\nvided that the total angular momentum of the system is\nnot zero. In the atom phase, mechanical rotation leads to\ndi\u000berences in the switching time \u001c, as generally expected\nin the presence of magneto-mechanical coupling [74, 88].\nVI. CONCLUSION\nIn conclusion, we analyzed how the stability of a nano-\nmagnet levitated in a static magnetic \feld is a\u000bected by\nthe most relevant sources of dissipation. We \fnd that in\nthe strong-\feld regime (atom phase) the system is un-\nstable due to the Gilbert-damping-induced magnetiza-\ntion switching, which occurs on a much faster timescale\nthan the center-of-mass oscillations, thereby preventing\nthe observation of levitation. On the other hand, the sys-\ntem is metastable in the weak-\feld regime and for small\nparticle dimensions (Einstein{de Haas phase). In this\nregime, the con\fnement of the nanomagnet in a mag-\nnetic trap is limited only by the gas-induced dissipation.\nOur results suggest that the timescale of stable levitation\ncan reach and even exceed several hundreds of periods of\ncenter-of-mass oscillations in high vacuum. These \fnd-\nings indicate the possibility of observing the phenomenon\nof quantum spin stabilized magnetic levitation, which we\nhope will encourage further experimental research.\nThe analysis presented in this article is relevant for\nthe community of levitated magnetic systems. Speci\f-\ncally, we give precise conditions for the observation of\nthe phenomenon of quantum spin stabilized levitation\nunder experimentally feasible conditions. Levitating a\nmagnet in a time-independent gradient trap represents a\nnew direction in the currently growing \feld of magnetic\nlevitation of micro- and nanoparticles, which is interest-\ning for two reasons. First, the experimental observation\nof stable magnetic levitation of a non-rotating nanomag-\nnet would represent a direct observation of the quantum\nnature of magnetization. Second, the observation of such\n5Rotational cooling might be needed to unambiguously identify\nthe internal spin as the source of stabilization. Subkelvin cooling\nof a nanorotor has been recently achieved [86, 87], and cooling\nto\u0016K temperatures should be possible [56].phenomenon would be a step towards controlling and us-\ning the rich physics of magnetically levitated nanomag-\nnets, with applications in magnetometry and in tests of\nfundamental forces [9, 11, 34, 35].\nACKNOWLEDGMENTS\nWe thank G. E. W. Bauer, J. J. Garc\u0013 \u0010a-Ripoll, O.\nRomero-Isart, and B. A. Stickler for helpful discussions.\nWe are grateful to O. Romero-Isart, B. A. Stickler and\nS. Viola Kusminskiy for comments on an early ver-\nsion of the manuscript. C.C.R. acknowledges funding\nfrom ERC Advanced Grant QENOCOBA under the EU\nHorizon 2020 program (Grant Agreement No. 742102).\nV.W. acknowledges funding from the Max Planck So-\nciety and from the Deutsche Forschungsgemeinschaft\n(DFG, German Research Foundation) through Project-\nID 429529648-TRR 306 QuCoLiMa (\"Quantum Cooper-\nativity of Light and Matter\"). A.E.R.L. thanks the AMS\nfor the \fnancial support.\nAppendix A: Rotation to the body frame\nIn this appendix we de\fne the transformation ma-\ntrix between the body-\fxed and the laboratory reference\nframes according to the ZYZ Euler angle convention,\nwith the Euler angles denoted as \n= (\u000b;\f;\r )T. We\nde\fne the transformation between the laboratory frame\nOexeyezand the body frame Oe1e2e3as follows,\n0\n@e1\ne2\ne31\nA=R(\n)0\n@ex\ney\nez1\nA; (A1)\nwhere\nR(\n)\u0011Rz(\u000b)Ry(\f)Rz(\r) =0\n@cos\rsin\r0\n\u0000sin\rcos\r0\n0 0 11\nA\n0\n@cos\f0\u0000sin\f\n0 1 0\n\u0000sin\f0 cos\f1\nA0\n@cos\u000bsin\u000b0\n\u0000sin\u000bcos\u000b0\n0 0 11\nA:(A2)\nAccordingly, the components vj(j= 1;2;3) of a vector\nvin the body frame Oe1e2e3and the components v\u0017\n(\u0017=x;y;z ) of the same vector in the laboratory frame\nOexeyezare related as\n0\n@v1\nv2\nv31\nA=RT(\n)0\n@vx\nvy\nvz1\nA: (A3)\nThe angular velocity of a rotating particle !can be writ-\nten in terms of the Euler angles as != _\u000bez+_\fe0\ny+ _\re3,\nwhere ( e0\nx;e0\ny;e0\nz)T=Rz(\u000b)(ex;ey;ez)Tdenotes the\nframeOe0\nxe0\nye0\nzobtained after the \frst rotation of the9\nlaboratory frame Oexeyezin the ZYZ convention. By\nusing (A1) and (A2), we can rewrite angular velocity in\nterms of the body frame coordinates,\n!= _\u000b2\n4R(\n)\u000010\n@e1\ne2\ne31\nA3\n5\n3+_\f2\n4R(\r)\u000010\n@e1\ne2\ne31\nA3\n5\n2+ _\re3;\n(A4)\nwhich is compactly written as ( !1;!2;!3)T=A(\n)_\n,\nwith\nA(\n) =0\n@\u0000cos\rsin\fsin\r0\nsin\fsin\rcos\r0\ncos\f 0 11\nA: (A5)\nAppendix B: Dynamics in the presence of thermal\n\ructuations\nIn this appendix we consider the dynamics of a lev-\nitated nanomagnet in the presence of stochastic forces\nand torques induced by the surrounding gas. The dy-\nnamics of the system are described by the following set\nof stochastic di\u000berential equations (SDE),\nd~r=!I~pdt; (B1)\nde3=!\u0002e3dt; (B2)\nd~p= [!Lr~r[m\u0001b(~r)]\u0000\u0000cm~p] dt+p\nDcmdWp;(B3)\nd`= [!Lm\u0002b(~r)\u0000_m\u0000\u0000rot`] dt+p\nDrotdWl;\n(B4)\ndm=m\n1 +\u00112\u0002[!e\u000b\u0000\u0011m\u0002(!+!e\u000b+\u0011!\u0002m)]dt;\n(B5)\nwhere we model the thermal \ructuations as uncorrelated\nGaussian noise represented by a six-dimensional vector\nof independent Wiener increments (d Wp;dWl)T. The\ncorresponding di\u000busion rate is described by the tensors\nDcmandDrotwhich, in agreement with the \ructuation-\ndissipation theorem, are related to the corresponding dis-\nsipation tensors \u0000 cmand \u0000 rotasDcm\u00112\u0000cm\u001f;D rot\u0011\n2\u0000rot\u001f, where\u001f\u0011MkBT\r2\n0a2=\u00162.\nIn the following we numerically integrate Eqs. (B1-B5)\nusing the stochastic Euler method implemented in the\nstochastic di\u000berential equations package in MATLAB. As\nthe e\u000bect of thermal noise is more prominent for small\nparticles at weak \felds, we focus on the Einstein-de Haas\nregime considered in Sec. IV. We show that even in this\ncase the e\u000bect of thermal \ructuations leads to dynamics\nwhich are qualitatively very close to the results obtained\nin Sec. IV. In Fig. 4 we present the results of the stochas-\ntic integrator by averaging the solution of 100 di\u000berent\ntrajectories calculated using the same parameters consid-\nered in Fig. 3(a-c). The resulting average dynamics agree\nqualitatively with the results obtained by integrating the\ncorresponding set of deterministic equations Eqs. (1-5)\nFigure 4. Stochastic dynamics of a nanomagnet for the same\nparameter regime as considered in Fig. 3. (a) Average motion\nof the system in the y-zplane until time t= 5\u0016s. (b) Dy-\nnamics of center of mass along the ey(top) and ez(bottom)\ndirections. (c) Dynamics of the anisotropy axis component\ne3;x. (d) Numerical error as function of time. The simulations\nshow the results of the average of 100 di\u000berent realizations of\nthe system dynamics. In panels (b-d) the solid dark lines are\nthe average trajectories, while the shaded area represents the\nstandard deviation.\n[cfr. Fig. 3(a-c)]. The main e\u000bect of thermal excitations\nis to shift the center of oscillations of the particle's de-\ngrees of freedom around the value given by the thermal\n\ructuations. This is more evident for the dynamics of\ne3[cfr. Fig. 4(c) and Fig. 3(c)]. We thus conclude that\nthe deterministic equations Eqs. (1-5) considered in the\nmain text correctly capture the metastable behavior of\nthe system. We emphasize that the results presented in\nthis section include only the noise due to the surround-\ning gas. Should one be interested in simulating the ef-\nfect of the \ructuations of the magnetic moment, the Eu-\nler method used here is not appropriate, and the Heun\nmethod should be used instead [89].\nLet us conclude with a technical note on the numerical\nsimulations. In the presence of dissipation and thermal\n\ructuations the only conserved quantity of the system is\nthe magnitude of the magnetic moment ( jmj= 1). We\nthus use the deviation 1 \u0000jmj2as a measure of the numer-\nical error in both the stochastic and deterministic sim-\nulations presented in this article. For the deterministic\nsimulations the error stays much smaller than any other\nphysical degree of freedom of the system during the whole\nsimulation time. The simulation of the stochastic dynam-\nics shows a larger numerical error [see Fig. 4(d)], which\ncan be partially reduced by taking a smaller time-step\nsize. We note that, for the value of magnetic anisotropy\ngiven in Table I, the system of SDE is sti\u000b. This, together\nwith the requirement imposed on the time-step size by10\nthe numerical error, ultimately limits the maximum time\nwe can simulate to a few microseconds. However, this is\nsu\u000ecient to validate the agreement between the SDE and\nthe deterministic simulations presented in the article.\nAppendix C: Additional \fgures\nIn this appendix we provide additional \fgures.\n1. Dynamics in the atom phase\nIn Fig. 5 we analyze magnetization dynamics in the\natom phase as a function of di\u000berent system parame-\nters. In Fig. 5(a) we show how magnetization switching\nchanges as the anisotropy constant kais varied. We con-\nsider the bias \feld B0= 1100 mT, which is larger than\nthe value considered in the main text. This is done to en-\nsure thatB0>B atom for all anisotropy values. Fig. 5(a)\ndemonstrates that the switching time \u001c, given by Eq. (8),\nis an excellent approximation for the dynamics across a\nwide range of values for the anisotropy constant ka. The\nlarger discrepancy between Eq. (8) and the line showing\nthe case with ka= 106J/m3is explained by the prox-\nimity of this point to the unstable region (in this case\ngiven by the critical \feld Batom = 900 mT), and better\nagreement is recovered at larger bias \feld values.\nIn Fig. 5(b) we analyze the validity of Eq. (8) for dif-\nferent values of the Gilbert damping parameter \u0011and the\ntemperature T. The thick dashed line shows the region\nwhere Eq. (8) di\u000bers from the exact switching time by\n5%; below this line the deviation becomes increasingly\nmore signi\fcant. As evidenced by Fig. 5(b), \u001cshows lit-\ntle dependence on T; its order of magnitude remains con-\nstant over a wide range of cryogenic temperatures. On\nthe other hand, the dependence on \u0011is more pronounced.\nIn fact, reducing the Gilbert parameter signi\fcantly de-\nlays the switching time, leading to levitation times as\nlong as\u00181\u0016s.\nAdditionally, we point out that \u001cdepends on the \feldgradientB0and curvature B00through the initial con-\nditionmk(t= 0). In particular, magnetization switch-\ning can be delayed by decreasing B0, as this reduces\nthe initial misalignment of the magnetization and the\nanisotropy axis (i. e. jmk(t= 0)j!1).\n2. Dynamics in the presence of rotation\nIn Fig. 6 we consider a more general equilibrium con-\n\fguration, namely a nanomagnet initially rotating such\nthat in the equilibrium point Le=\u0000I3!Sex, with!S>0\ndenoting the rotation in the clockwise direction. This\nequilibrium point is linearly stable in the absence of dis-\nsipation [23, 24], with additional stability of the system\nprovided by the mechanical rotation, analogously to the\nclassical magnetic top [25{27].\nIn Fig. 6(a) we analyze how magnetization switching\nin the atom phase changes in the presence of rotation for\ndi\u000berent values of parameters. The rotation has a slight\ne\u000bect on the switching time \u001c, shifting it forwards (back-\nwards) in case of a clockwise (counterclockwise) rotation.\nThis is generally expected in the presence of magneto-\nmechanical coupling [74, 88].\nIn Fig. 6(b) we show the motion in the y-zplane in\nthe Einstein{de Haas phase for both directions of rota-\ntion. This can be compared with Fig. 3(a). 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Mech. , P09008\n(2014)."    },    {        "title": "2003.07997v1.On_the_saturation_mechanism_of_the_fluctuation_dynamo_at____text_Pr___mathrm_M____ge_1_.pdf",        "content": "On the saturation mechanism of the fluctuation dynamo at Pr M\u00151\nAmit Seta,1, 2,\u0003Paul J. Bushby,2Anvar Shukurov,2and Toby S. Wood2\n1Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia\n2School of Mathematics, Statistics and Physics, Newcastle University, Newcastle Upon Tyne, NE1 7RU, UK\n(Dated: March 19, 2020)\nThe presence of magnetic fields in many astrophysical objects is due to dynamo action, whereby a part\nof the kinetic energy is converted into magnetic energy. A turbulent dynamo that produces magnetic field\nstructures on the same scale as the turbulent flow is known as the fluctuation dynamo. We use numerical\nsimulations to explore the nonlinear, statistically steady state of the fluctuation dynamo in driven turbulence.\nWe demonstrate that as the magnetic field growth saturates, its amplification and diffusion are both affected by\nthe back-reaction of the Lorentz force upon the flow. The amplification of the magnetic field is reduced due\nto stronger alignment between the velocity field, magnetic field, and electric current density. Furthermore, we\nconfirm that the amplification decreases due to a weaker stretching of the magnetic field lines. The enhancement\nin diffusion relative to the field line stretching is quantified by a decrease in the computed local value of the\nmagnetic Reynolds number. Using the Minkowski functionals, we quantify the shape of the magnetic structures\nproduced by the dynamo as magnetic filaments and ribbons in both kinematic and saturated dynamos and derive\nthe scalings of the typical length, width, and thickness of the magnetic structures with the magnetic Reynolds\nnumber. We show that all three of these magnetic length scales increase as the dynamo saturates. The magnetic\nintermittency, strong in the kinematic dynamo (where the magnetic field strength grows exponentially) persists\nin the statistically steady state, but intense magnetic filaments and ribbons are more volume-filling.\nI. INTRODUCTION\nMagnetic fields are observed in a variety of astrophysical\nobjects, including stars, galaxies and galaxy clusters, where\nthey play an important role in various physical processes.\nBased on length and time scales, astrophysical magnetic fields\ncan be divided into two types: the large-scale or mean field,\nwhich is coherent over scales comparable to the size of the\nsystem, and the small-scale or fluctuating field, whose corre-\nlation length is of the order of the driving scale of the underly-\ning turbulent flow. The driving scale of turbulence, l0, is of the\norder of 0:1 kpc in spiral galaxies [1–3], and 10 kpc in galaxy\nclusters [4, 5]. The fluctuating magnetic field is believed to\nevolve over the eddy turnover timescale, which is consider-\nably shorter than the corresponding evolution timescale for\nthe large-scale field (which is typically of the order of 108yr\nin spiral galaxies, comparable to the rotation period). For spi-\nral galaxies, the mean and fluctuating fields have comparable\nmagnitudes and thus both kinds of fields are equally important\nfor the galactic dynamics [6]. There are a number of reviews\ncovering the theoretical, numerical, and observational aspects\nof the subject [7–12].\nThe evolution and maintenance of magnetic fields is gen-\nerally explained by dynamo action, a process by which ki-\nnetic energy is converted to magnetic energy. Astrophysical\nflows leading to dynamo action are typically turbulent; such\nflows may be driven by convection in stars, supernovae in\ngalaxies, and merger shocks, motion of galaxies and AGN\noutflows in galaxy clusters. Magnetic field amplification by\nturbulent motions has also been observed in laboratory exper-\niments [13]. Depending upon the magnetic fields that they\nproduce, such dynamos are generally categorized as either\n\u0003amit.seta@anu.edu.aumean-field or fluctuation (or “small-scale”) dynamos. Mean-\nfield dynamos produce large-scale magnetic fields, whereas\nthe fluctuation dynamo generates the small-scale component\nof the field via random stretching of field lines by the turbulent\nvelocity [14, 15] (as conceptually explained by the stretch-\ntwist-fold mechanism [16, 17]). Fluctuation dynamo action\nplays a crucial role not only in spiral galaxies [7, 10, 18–\n21], elliptical galaxies [22, 23] and galaxy clusters [24–27],\nbut also in stars such as the Sun [28–31], making it a gen-\neral type of astrophysical process. Fluctuation dynamos nat-\nurally produce intermittent magnetic fields [32–34], charac-\nterised by the presence of intense, localised field structures.\nIn the galactic context, a better understanding of these struc-\ntures is needed for cosmic ray propagation studies [35, 36]\nand in the galaxy cluster context for the interpretation of radio\nobservations [37]. The initial stages of magnetic field growth,\nwhen the Lorentz force is negligible, have been thoroughly\nstudied [9, 32], so here we focus on the nonlinear states of\nthe fluctuation dynamo, for which it is possible to consider\nrelatively simple idealised flows (i.e., homogeneous, isotropic\nturbulence). A mean-field dynamo would require additional\nphysics, such as rotation, velocity shear and density stratifi-\ncation; such effects can be safely ignored over the length and\ntime scales that will be of interest here.\nIn a fluctuation dynamo, the root mean square (rms) mag-\nnetic field grows exponentially if the magnetic Reynolds num-\nber Re M(quantifying the efficiency of inductive effects com-\npared to magnetic diffusion) exceeds its critical value Re(crit)\nM,\nwhich depends on the properties of the flow. When the mag-\nnetic energy is low in comparison to the turbulent kinetic en-\nergy, the flow dynamics are not influenced by the magnetic\nfield (the kinematic stage). For an isotropic, incompress-\nible, mirror–symmetric, homogeneous and Gaussian random\nvelocity field, which is also \u000e–correlated in time, it can be\nshown that the magnetic field power spectrum Mkin the kine-\nmatic stage follows a power-law (at low wave numbers) witharXiv:2003.07997v1  [astro-ph.GA]  18 Mar 20202\nslope 32[9, 14]. However, an exponentially growing mag-\nnetic field also leads to the exponential growth of the Lorentz\nforce, which eventually makes the problem nonlinear. This\nslows down the growth and finally leads to the saturation\nof the dynamo (the saturated stage). The nonlinear prob-\nlem is mostly studied via numerical simulations, in which the\nNavier-Stokes and induction equations are solved simultane-\nously [e.g., 26, 28, 30, 33, 38–47]. Our aim in this paper is to\nexplore the saturation mechanism of the fluctuation dynamo\nand to characterize the magnetic structures it generates.\nFor fluctuation dynamos driven by homogeneous and\nisotropic turbulence, the following three quantities are pre-\nscribed: the driving scale of the turbulent flow l0, the fluid vis-\ncosity\u0017, and the magnetic resistivity \u0011. Based on the magnetic\nPrandtl number Pr M(defined to be the ratio of viscosity to re-\nsistivity, Pr M=\u0017\u0011), fluctuation dynamos can be divided into\nsmall and large Pr Mcases. Pr Mis greater than unity ( \u0011 < \u0017 )\nfor hot diffuse plasma (interstellar and intergalactic medium)\nand Pr Mis much smaller than unity ( \u0011 > \u0017 ) for dense plasma\n(planets, stars and liquid metal dynamo experiments). The\ncritical magnetic Reynolds number Re(crit)\nM, which is a thresh-\nold for dynamo action to occur, increases with decreasing Pr M\n[48–52]. We focus upon the Pr M\u00151regime, fixing the un-\nderlying flow (i.e., fixing Re) and then varying Re Min order\nto study the sensitivity of the magnetic structures of nonlinear\ndynamo states to the magnetic Reynolds number.\nThis paper is structured as follows. In Section II, we in-\ntroduce the basic equations and describe the numerical setup\nand provide parameters of the simulations. In Section III, we\ndiscuss magnetic field intermittency and in Section IV, we ex-\namine possible saturation mechanisms. Then, in Section V,\nwe use Minkowski functionals to quantify the magnetic field\nstructures (as a function of the magnetic Reynolds number)\nin both the kinematic and nonlinear regimes. Finally, in Sec-\ntion VI, we conclude with a discussion and propose some fu-\nture directions of research.\nII. BASIC EQUATIONS AND NUMERICAL MODELLING\nTo study the fluctuation dynamo action in a turbulent flow\ndriven by a prescribed random force, we solve the equations\nof magnetohydrodynamics, using the Pencil code [53]. The\ncomputational domain is a triply-periodic cubic box of non-\ndimensional width L=2\u0019, with 2563or5123grid points.\nThe equations are solved with sixth-order finite differences in\nspace and a third-order Runge–Kutta scheme for the temporal\nevolution. The governing equations are\n@\u001a\n@t+r\u0001¹\u001auº=0; (1)\n@b\n@t=r\u0002¹u\u0002bº+\u0011r2b; (2)\n@u\n@t+¹u\u0001rºu=\u0000rp\n\u001a+j\u0002b\nc\u001a\n+\u0017\u0012\nr2u+1\n3r¹r\u0001 uº+2S\u0001rln\u001a\u0013\n+F; (3)where uis the velocity field, bis the magnetic field, \u001ais the\nfluid density, pis the pressure, \u0011is the magnetic diffusivity,\nj=¹c4\u0019ºr\u0002 bis the electric current density, cis the speed\nof light,\u0017is the viscosity, Sij=1\n2\u0010\nui;j+uj;i\u00002\n3\u000eijr\u0001u\u0011\nis\nthe rate-of-strain tensor, and Fis the forcing function (defined\nbelow). We use an isothermal equation of state, p=c2\ns\u001a,\nwhere the constant csis the sound speed. Eq. (2) is solved\nin terms of the magnetic vector potential to ensure that the\nmagnetic field remains divergence free.\nWe drive the flow with a mirror-symmetric and \u000e-correlated\nin time forcing [39] of the form\nF¹x;tº=RefNFk¹tºexp»ik¹tº\u0001x+i\u001e¹tº¼g; (4)\nwhere kis the wave vector, xis the position vector and\n\u0000\u0019 < \u001e\u0014\u0019is a random phase. To ensure that the forcing\nis nearly\u000e–correlated in time, kand\u001eare changed at each\ntime step\u000et. Also, to ensure that the time-integrated force is\nindependent of the chosen time step \u000et, the normalization is\nN=F0cs¹jkjcs\u000etº12, where F0is the non–dimensional forc-\ning amplitude chosen such that the maximum Mach number is\nsmall enough ( urmscs.0:1) to avoid strong compressibility.\nWe select many random wave vectors k, each of magnitude k\n(a multiple of 2\u0019Lto make sure that the flow is periodic) in a\ngiven range. Then we select an arbitrary unit vector e(neither\nparallel nor anti-parallel to k) and set\nFk=k\u0002e\njk\u0002ej: (5)\nThe form of Eq. (5) ensures that the forcing is solenoidal, i.e.\nr\u0001F=0by construction. The average wave number at which\nthe flow is driven is denoted by kF. Even when the flow is\nperiodic, kFneed not be a multiple of 2\u0019L. Physically, 2\u0019kF\nrepresents the driving scale of the turbulent flow, l0, in the\nsystem.\nThe turbulent plasma is characterized by the hydrodynamic\nReynolds number Re and magnetic Reynolds number Re M,\ndefined in terms of the rms velocity urmsand the forcing scale\nkF[54], as\nRe=urms\n\u00172\u0019\nkF;ReM=urms\n\u00112\u0019\nkF: (6)\nWe use non-dimensional units with lengths in units of the do-\nmain size L=2\u0019, speed in units of the isothermal sound speed\ncs, time in units of the eddy turnover time t0=2\u0019urmskF,\ndensity in units of the initial density \u001a0and the magnetic field\nin units of\u00004\u0019\u001a0c2\ns\u000112. Initially, the density is constant ev-\nerywhere and u=0, whilst there is a weak random, seed\nmagnetic field with zero net flux across the domain.\nFor the first set of simulations, with parameters given in\nTable I, the turbulent motions are driven at the wave num-\nbers2\u0019Land2¹2\u0019Lºat equal intensities, which implies that\nkF\u00191:5¹2\u0019Lº. The magnetic field grows for Re M\u0015Re(crit)\nM,\nwith Re(crit)\nM\u0019220for Pr M=1[39]. The evolution of the\nrms velocity field, urms, and magnetic field, brms, is shown\nin Fig. 1 for Re M=1122 . The flow speed is controlled by\nthe forcing function and thus remains nearly constant. The3\nTABLE I. Summary of fluctuation dynamo simulations in a numeri-\ncal domain of size¹L=2\u0019º3with 2563mesh points. In all cases, the\nforcing scale kFis approximately equal to 1:5¹2\u0019Lº, the forcing am-\nplitude F0=0:02, the magnetic Prandtl number Pr M=1and the rms\nvelocity in the saturated state is urmscs\u00190:11. For each simula-\ntion, we quote the Reynolds number, the magnetic Reynolds number,\nthe rms magnetic field in the saturated state brms, the ratio of mag-\nnetic to kinetic energy in the saturated state \"M\"K=b2rmsu2rms,\nthe correlation length of the velocity and magnetic field in the kine-\nmatic stage lukinandlbkin, and similarly in the saturated stage lusat\nandlbsat.\n\u0011;\u0017 ReM;Rebrms\"M\"Klukinlbkinlusatlbsat\n10\u000210\u00004449 0:033 0:08 3:14 1:82 3:77 1:95\n5\u000210\u00004898 0:042 0:14 3:20 1:26 3:45 1:76\n4\u000210\u000041122 0:048 0:20 3:01 0:94 3:64 1:76\n3\u000210\u000041496 0:049 0:21 3:01 0:88 3:39 1:57\n2:5\u000210\u000041796 0:054 0:25 2:95 0:75 3:58 1:57\n2\u000210\u000042244 0:055 0:26 2:95 0:69 3:33 1:56\n0 10 20 30 40 50 60\nt/t010−510−410−310−210−1urms, brms\nurms\nbrms\ne0.4(t/t0)\nkinematic\ntransition\nsaturated\nFIG. 1. Root mean square (rms) velocity field urms(red) and mag-\nnetic field brms(blue) as functions of normalized time tt0(where\nt0=2\u0019urmskFis the eddy turnover time) for Re =ReM=1122 .\nDuring the kinematic stage (area shaded in light red), the black\ndashed line corresponds to the exponential growth. As the magnetic\nfield grows, the dynamo passes through a transitional stage (area\nshaded in light green), before reaching a statistically steady saturated\nstate (area shaded in light blue).\nmagnetic field first decays until it reaches an eigenstate of\nthe induction equation. Then it grows exponentially in the\nkinematic stage at the growth rate of 0:4urmskF2\u0019in dimen-\nsional units. As it becomes stronger, the Lorentz force affects\nthe flow and slows down the exponential increase. Finally,\nwhen the magnetic field becomes strong enough, the dynamo\nreaches a statistically steady state in the saturated stage. The\nexponential growth and then saturation of the magnetic field\noccurs in all of the runs shown in Table I.\nThe shell-averaged (one-dimensional) power spectra, for\nvarious stages of the magnetic field evolution, are shown in\n100101102\nk10−2\n10−6\n10−10\n10−14Ek, MkEk,kin\nMk,kin\nEk,sat\nMk,sat\nk3/2\nk−5/3FIG. 2. The shell–averaged (one–dimensional) kinetic Ek(dashed)\nand magnetic Mk(solid) energy spectra in the kinematic (red) and\nsaturated (blue) stages for Re =ReM=1122 . The kinetic energy\nspectrum is close to the Kolmogorov spectrum, Ek/k\u000053(dotted,\nblack) in the main part of the wave number range. The magnetic\nspectrum is initially of the form Mk/k32(dashed, black) at smaller\nwave numbers. As the magnetic field saturates, its power shifts to\nsmaller wave numbers and the magnetic spectrum flattens.\nFig. 2. At all times, the kinetic energy spectrum is close to the\nKolmogorov spectrum, Ek/k\u000053, in the range 3\u0014kL2\u0019\u0014\n20(flow is driven at k=2\u0019Landk=2¹2\u0019Lº), which sug-\ngest that the velocity field is turbulent in nature. The mag-\nnetic spectrum in the kinematic stage has a broad maximum\nat large wave numbers and its slope agrees with the Kazant-\nsev model, Mk/k32, in the range 2\u0014kL2\u0019\u001410with\nmaximum power at approximately kL2\u0019=10. Kazant-\nsev’s theory assumes that the turbulent flow is \u000e-correlated\nin time. Whilst we have used a \u000e-correlated forcing in the\nNavier-Stokes equation (term Fin Eq. (3)), the flow that it\ndrives is not \u000e-correlated, especially at high Re. However, it\nis known that the slope of the spectrum in the kinematic stage\nremains the same even when the flow has a finite but small\ncorrelation time [55, 56], which explains why we recover the\nKazantsev result in these simulations. As the magnetic field\ngrows, the spectral maximum shifts to smaller wave numbers\nand the spectrum becomes much flatter with a broad maxi-\nmum in the range 2\u0014kL2\u0019\u00145.\nIII. MAGNETIC INTERMITTENCY\nIntermittency in a random field can manifest itself via heavy\ntails in its probability distribution function (PDF) and leads to\nan increased kurtosis in comparison with the Gaussian distri-\nbution. For the random velocity field uwith zero mean, the\nkurtosis is defined by\nK¹uº=hu4i\nhu2i2; (7)4\n−2−1 0 1 2\nux/urms10−310−210−1100PDF\nReM= 1122,kin\nReM= 2244,kin\nReM= 1122,sat\nReM= 2244,sat\nGaussian\nFIG. 3. The PDF of the normalized velocity field component\nuxurmsfor Re M=1122 and Re M=2244 in the kinematic (dashed)\nand saturated (solid) stages for the value of Re Mgiven in the legend.\nThe PDF of the single component of the velocity field is roughly\nGaussian (dashed, black) in both the stages for both Re M. Here only\nuxurmsis shown but similar behaviour is exhibited by all three ve-\nlocity components.\nwith angular brackets denoting the volume average. A useful\ndiagnostic of the spatial structure is the correlation length of\nthe field, lu, which is calculated from the power spectrum Ek\nas\nlu=\u0019\n2¯1\n02\u0019k\u00001Ekdk\n¯1\n0Ekdk: (8)\nHere, using such tools, we discuss the spatial intermittency of\nthe velocity and magnetic fields in nonlinear fluctuation dy-\nnamos.\nFig. 3 shows the PDF of a single component of the velocity\nfield uxurmsin the kinematic and saturated dynamo stages for\nReM=1122 and Re M=2244 . The PDF is nearly Gaussian\nin both the kinematic and saturated stages. This is generally\ntrue for homogenous turbulence [57]. The velocity PDFs re-\nmain Gaussian even in the case of supersonic turbulence with\na compressible forcing [e.g., Fig. A1. in 58]. For all cases\nof Table I, the kurtosis of the velocity field is very close to\nK=3, which is the value for a Gaussian distribution. The\ncorrelation length of the velocity field lu, also given in Ta-\nble I, is about half of the periodic domain size L=2\u0019, as can\nalso be seen from Fig. 4. It decreases slightly as Re increases\nand is slightly larger in the saturated stage than in the kine-\nmatic stage for all Re M. The velocity field thus becomes more\nvolume filling as the magnetic field saturates. This is directly\nattributable to the dynamical effects of the magnetic fields.\nEven though the velocity field statistics are nearly Gaus-\nsian, the magnetic field in both the kinematic and saturated\nstages is spatially intermittent and strongly non-Gaussian.\nThis can be seen from the PDFs of a normalized component\nof the magnetic field bxbrmsin Fig. 5. The distribution is\nfar from a Gaussian one and has long, heavy tails. The non-linearity truncates the most extreme relative magnetic field\nstrengths abovejbxjbrms\u00193. The magnetic field intermit-\ntency is further demonstrated in Fig. 6 which shows the PDF\nofbbrmsfor Re M=1122 and Re M=2244 in the kine-\nmatic and saturated stages. The PDF of the kinematic mag-\nnetic field strength follows a lognormal distribution and it has\nheavier tails in comparison to that of the saturated magnetic\nfield. Thus the magnetic field is intermittent in both the kine-\nmatic and the saturated stages, but the level of intermittency\ndecreases as the field saturates. It should be noted that this\nconclusion is consistent with that of Schekochihin et al. [33]\n(Fig. 6 in this paper is similar to their Fig. 27), who studied a\nclosely related system. This confirms that this finding is ro-\nbust to small variations in the model setup and parameters.\nMagnetic intermittency can also be quantified by measur-\ning the quantity Qnl=h¹bbrmsºnli¹1nlºand its rate of change\nasnlchanges (for example, QnlQnl\u00001). Higher Qnland\nQnlQnl\u00001is a signature of a larger degree of intermittency.\nFig. 7 shows QnlandQnlQnl\u00001for the magnetic field in\nthe kinematic and saturated stages for Re M=1122 fornl=\n1;2;3;\u0001\u0001\u0001;50.Qnland its rate of change are higher for the\nkinematic stage as compared to the saturated stage. This fur-\nther demonstrates that the magnetic field in the saturated stage\nis less intermittent than that in the kinematic stage. We further\ncompare both terms with the corresponding Gaussian versions\nobtained by randomizing phases in Fourier space [keeping the\nexact same magnetic field spectrum but destroying intermit-\ntent structures, as done in 35, 36, 59]. QnlandQnlQnl\u00001\nare higher for the dynamo generated field in comparison to its\nrandomized Gaussian versions in both the kinematic and satu-\nrated stages. Thus, the dynamo generated field is always spa-\ntially intermittent and the degree of intermittency decreases as\nthe field saturates due to nonlinearity.\nThe two-dimensional vector plots of the magnetic fields in\nFig. 8 also show larger structures in the saturated stage. This\ncan be further seen in Fig. 9, which shows the isosurfaces of\nmagnetic fields in the kinematic and saturated stages. The\nkurtosis of the kinematic magnetic field for Re M=1122 is\n5:29but is 3:32in the saturated stage. This also suggests\nthat the magnetic field in the kinematic stage is more intermit-\ntent than the saturated stage. The magnetic field correlation\nlength lbis calculated using Eq. (8) by replacing Ekwith Mk,\nthe magnetic field power spectrum. The magnetic field corre-\nlation length in the kinematic lbkinand saturated lbsatstages\nis given in Table I. The magnetic field correlation length de-\ncreases as Re Mincreases, both for the kinematic and saturated\nstages (see Section V for further details). Thus, the magnetic\nfield intermittency increases during both kinematic and satu-\nrated dynamo stages as Re Mincreases. It is also clear that\nlbsat>lbkinfor all Re Mwhich confirms again that the mag-\nnetic field in the kinematic stage is less volume filling. The\nincrease in the correlation length due to magnetic field satu-\nration is true regardless of the choice of Re Mand agrees with\nprevious numerical studies [26, 40].5\n0π/2π 3π/2 2π\nx0π/2π3π/22πy(a)\n−2−1012\nuz/urms\n0π/2π 3π/2 2π\nx0π/2π3π/22πy(b)\n−2−1012\nuz/urms\nFIG. 4. A 2D cut in the xy-plane with vectors ¹uxurms;uyurmsºand colour showing the magnitude of uzurmsin the kinematic (a) and\nsaturated (b) stages with Re M=2244 . The velocity field in both the stages looks qualitatively the same. The structures span approximately\nhalf of the domain.\n−3−2−1 0 1 2 3\nbx/brms10−310−210−1100PDFReM= 1122,kin\nReM= 2244,kin\nReM= 1122,sat\nReM= 2244,sat\nGaussian\nFIG. 5. The PDF of the normalized magnetic field component\nbxbrmsfor Re M=1122 and Re M=2244 in the kinematic (dashed)\nand saturated (solid) stages for the values of Re Mgiven in the legend.\nThe magnetic field for both Re Min both stages is far from a Gaussian\n(dashed, black). It has heavy tails, which is a sign of intermittency.\nHere only bxbrmsis shown but similar behaviour can be observed\nin all three magnetic field components.\nIV . SATURATION OF THE FLUCTUATION DYNAMO\nSeveral mechanisms have previously been considered to ex-\nplain the saturation of the fluctuation dynamo, including a re-\nduction in magnetic field line stretching due to the suppres-\nsion of the Lagrangian chaos in the velocity field [60, 61],\nchanges in the mutual alignment of the velocity and magnetic\nfield lines [43], the folded structure of magnetic fields and\nenergy equipartition between magnetic and velocity fields for\n0 1 2 3 4 5 6 7\nb/brms10−410−310−210−1100PDFReM= 1122,kin\nReM= 2244,kin\nReM= 1122,sat\nReM= 2244,sat\nLognormalFIG. 6. The PDF of the normalized magnetic field strength bbrms\nfor Re M=1122 and Re M=2244 in the kinematic (dashed) and\nsaturated (solid) stages for the values of Re Mgiven in the legend. The\nPDF of the magnetic field in the kinematic state follows a lognormal\ndistribution (dashed, black). The magnetic field is more intermittent\nin the kinematic stage than in the saturated stage.\nPrM\u001d1[33, 62], enhancement in diffusion due to additional\nnonlinear velocity drift [63, 64] and selective dissipation of\nthe turbulent kinetic energy [65, 66]. From the induction\nequation (2), there are two type of processes that could lead to\nthe saturation: a decrease in the induction term ¹r\u0002¹u\u0002bºº\nor an increase in the dissipation term\u0000\u0011r2b\u0001. We explore each\nscenario here.6\n10 20 30 40 50\nnl12345678Qnl\n(a)kin\nkin (Randomized)\nsat\nsat (Randomized)\n10 20 30 40 50\nnl1.001.051.101.151.201.25Qnl/Qnl−1\n(b)kin\nkin (Randomized)\nsat\nsat (Randomized)\nFIG. 7. Qnl=h¹bbrmsºnli¹1nlº(a) and QnlQnl\u00001(b) as function of nlfor the kinematic (red, solid) and saturated (blue, solid) stages for\nReM=1122 . The corresponding quantities for randomized fields which have almost Gaussian statistics (dashed) are also plotted. The dynamo\ngenerated magnetic field is always intermittent with the degree of intermittency being higher in the kinematic stage.\n0π/2π 3π/2 2π\nx0π/2π3π/22πy(a)\n−4−3−2−101234\nbz/brms\n0π/2π 3π/2 2π\nx0π/2π3π/22πy(b)\n−4−3−2−101234\nbz/brms\nFIG. 8. As Fig. 4 but for the magnetic field. The magnetic field in the kinematic stage (a) is intermittent with random magnetic structures. In\nthe saturated stage (b), the field remains intermittent but the structures are larger.\nA. Alignment of velocity field, magnetic field and electric\ncurrent density\nWe first examine how the induction term is affected when\nthe field becomes stronger. The rms magnitude of both the ve-\nlocity and magnetic fields are statistically steady, as shown in\nFig. 1. Thus, we consider the alignment of the magnetic field\nwith the velocity field as a possible mechanism for the satu-\nration. Such an alignment has been studied in the context of\nconvectively driven fluctuation dynamos [43, 67], MHD tur-\nbulence in the presence of a strong guide field [68] and decay-\ning isotropic MHD turbulence [69]. For the numerical sim-\nulations described in Table I, we calculate the angle between\nthe velocity uand magnetic field b, and between the currentdensity jandb,\ncos¹\u0012ºu;b=u\u0001b\njujjbj;and cos¹\u0012ºj;b=j\u0001b\njjjjbj; (9)\nrespectively. An increase in the level of alignment between\nuandbimplies a decrease in the effectiveness of magnetic\ninduction. On the other hand, an increase in the level of align-\nment between jandbleads to a decrease in the Lorentz force,\ni.e., the field becomes more force-free.\nFig. 10 and Fig. 11 show the probability density func-\ntions of the cosines in the kinematic and saturated stages for\nReM=1122 and Re M=1496 . Since both angles are sym-\nmetric about b=0, we show PDFs of the absolute value of\ntheir cosines. For both values of Re M, the cosine of the angle7\n(a)\n(b)\nFIG. 9. Isosurfaces of b2b2rms=4(blue) and b2b2rms=5(yellow) for the magnetic fields in the kinematic (a) and saturated stages (b) for\nReM=2244 . The structures in the saturated stage are larger in size as compared to that in the kinematic stage.\nbetween the velocity and magnetic field, jcos¹\u0012ºu;bj, tends to\nbe larger in the saturated stage than in the kinematic stage.\nThe better alignment between uandbdecreases the induction\ntermr\u0002¹u\u0002bºand thus reduces the amplification of the mag-\nnetic field. To put this another way, the enhanced alignment\nbetween uandbimplies a decrease in the energy transfer from\nthe flow to the magnetic field (which is a process that has been\nstudied in some detail in the context of shell models of mag-\nnetohydrodynamic turbulence [70–73]). However, there is a\nsignificant fraction of the volume where the two fields are not\naligned and so the amplification is not completely suppressed.\nThis minimum level of amplification is required to balance the\nmagnetic diffusion. The cosine of the angle between the cur-\nrent density and magnetic field cos¹\u0012ºj;bis also statistically\nlarger by magnitude in the saturated stage. Thus, the field be-\ncomes closer to a force–free form as it saturates. This also\nimplies that the morphology of magnetic field changes on sat-\nuration, which motivates us to study the morphology of mag-\nnetic structures in Section V. Overall, because of the enhanced\nlocal alignment between the velocity and magnetic field, the\nfield amplification rate decreases. At the same time, due to\nthe increase in the local alignment between the current den-\nsity and magnetic field, the field becomes more force–free.\nSimilar broad conclusions apply when we consider con-\nditional PDFs that focus exclusively upon the regions of\nstronger field (higher bbrmsin Fig. 10 and Fig. 11). However,\nthe level of alignment between the velocity and magnetic field\nis higher in the strong field regions in both the kinematic and\nsaturated stages. This suggests that the strong field regions\nrequire a larger reduction in amplification by alignment. The\ndistribution of cos¹\u0012ºj;bin the kinematic stage shows some de-\npendence upon the field strength but in the saturated stage thedifference is less pronounced. In the kinematic stage, align-\nment is weakest in the relatively strong field regions, suggest-\ning that in the strong field regions, not only because of its\nhigher strength (as the Lorentz force is proportional to the\nstrength of the field) but also because of the lower level of\nalignment, the field produces a stronger back reaction on the\nflow.\nAnother important question is whether the alignment be-\ntween the velocity and magnetic fields and the magnetic field\nand current density occur in the same spatial region. To an-\nswer this, we show the cross-correlation between the two an-\ngles in Fig. 12 which suggests that the velocity, magnetic field\nand current density are always nearly aligned to each other at\nsame spatial positions. It is difficult to see any further differ-\nence between the kinematic and saturated stages in Fig. 12a\nand Fig. 12b. Fig. 12c and Fig. 12d show the same correlation\nbut only for strong field regions, bbrms>1:5. In Fig. 12c,\nthe kinematic stage shows higher correlation in regions with\nhigh cos¹\u0012ºu;band low cos¹\u0012ºj;b, which is absent in the satu-\nrated stage. The larger misalignment of jandb, especially in\nthe strong field regions, enhances the work done on the mag-\nnetic field by the flow. This promotes growth of the magnetic\nfield. Once the field saturates, the larger correlation at high\ncos¹\u0012ºu;band low cos¹\u0012ºj;bdisappears in Fig. 12d. This im-\nplies a statistical decrease in the back-reaction of the magnetic\nfield on the flow as the field saturates.\nTo summarize, the alignment between the velocity and\nmagnetic field vectors and the magnetic field and current den-\nsity vectors is statistically enhanced as the dynamo saturates.\nThe alignment does not completely inhibit the amplification,\nso there is always some field generated to balance the resistive\ndecay. This in turn also implies that the back reaction of the8\n0.0 0.2 0.4 0.6 0.8 1.0\n|cos(θ)u,b|0.81.01.21.41.61.8PDF\n(a) whole domain ,kin\nwhole domain ,sat\nb/brms>0.5,kin\nb/brms>0.5,sat\nb/brms>1.0,kin\nb/brms>1.0,sat\nb/brms>1.5,kin\nb/brms>1.5,sat\n0.0 0.2 0.4 0.6 0.8 1.0\n|cos(θ)j,b|1.01.41.8PDF\n(b)(b)whole domain ,kin\nwhole domain ,sat\nb/brms>0.5,kin\nb/brms>0.5,sat\nb/brms>1.0,kin\nb/brms>1.0,sat\nb/brms>1.5,kin\nb/brms>1.5,sat\nFIG. 10. The total and conditional probability distribution functions of the cosines of the angles between uandb,cos¹\u0012ºu;b(a) and between j\nandb,cos¹\u0012ºj;b(b) for Re M=1122 in the kinematic (red) and saturated (blue) states. The magnetic field in the saturated stage is more aligned\nwith the velocity field (reducing the induction effects) as compared to the kinematic stage. The magnetic field also becomes better aligned\nwith the electric current density, reducing the back reaction on the velocity field.\n0.0 0.2 0.4 0.6 0.8 1.0\n|cos(θ)u,b|0.81.01.21.41.61.8PDF\n(a) whole domain ,kin\nwhole domain ,sat\nb/brms>0.5,kin\nb/brms>0.5,sat\nb/brms>1.0,kin\nb/brms>1.0,sat\nb/brms>1.5,kin\nb/brms>1.5,sat\n0.0 0.2 0.4 0.6 0.8 1.0\n|cos(θ)j,b|1.01.41.82.2PDF\n(b)(b)whole domain ,kin\nwhole domain ,sat\nb/brms>0.5,kin\nb/brms>0.5,sat\nb/brms>1.0,kin\nb/brms>1.0,sat\nb/brms>1.5,kin\nb/brms>1.5,sat\nFIG. 11. As Fig. 10 but for Re M=1496 .\nLorentz force always remains significant.\nB. Magnetic field stretching\nTo explore another mechanism by which magnetic field\namplification can be suppressed, we consider the stretching\nof the magnetic field lines by the turbulent velocity. For\nthis, we consider the alignment of the magnetic field with\nthe eigenvectors of the rate of strain tensor. Neglecting the\nrather weak divergence of the flow, the symmetric 3\u00023ma-\ntrixSij=1\n2\u0000ui;j+uj;i\u0001is calculated at each point in the do-\nmain using sixth-order finite differences, and its eigenvalues\nand eigenvectors are calculated. The eigenvalues are arranged\nin an increasing order, \u00151< \u0015 2< \u0015 3. The corresponding\neigenvectors are e1;e2;e3. The sum of the eigenvalues is closeto zero since the flow is nearly incompressible. \u00151is always\nnegative and the vector e1corresponds to the direction of lo-\ncal compression of magnetic field, \u00153is always positive and\nthe vector e3corresponds to the direction of local stretching,\nwhereas\u00152can be obtained from \u00151+\u00152+\u00153\u00190. The direc-\ntione2(sometimes referred to as the ‘null’ direction [33, 74])\ncan correspond to either local stretching or compression de-\npending on the sign of \u00152. We then quantify the alignment\nwith the magnetic field bof the vectors e1ande3by consider-\ning\ncos¹\u0012ºe1;b=e1\u0001b\nje1jjbjand cos¹\u0012ºe3;b=e3\u0001b\nje3jjbj: (10)\nFig. 13 shows the PDF of the cosines in the kinematic and\nsaturated stages for Re M=1796 . In most of the volume, the\ndirection of the magnetic field is perpendicular to the direction9\n0.0 0.2 0.4 0.6 0.8 1.0\n|cos(θ)u,b|0.00.20.40.60.81.0|cos(θ)j,b|(a)\n6000700080009000100001100012000\nCounts\n0.0 0.2 0.4 0.6 0.8 1.0\n|cos(θ)u,b|0.00.20.40.60.81.0|cos(θ)j,b|(b)\n600080001000012000140001600018000\nCounts\n0.0 0.2 0.4 0.6 0.8 1.0\n|cos(θ)u,b|0.00.20.40.60.81.0|cos(θ)j,b|(c)\n600800100012001400\nCounts\n0.0 0.2 0.4 0.6 0.8 1.0\n|cos(θ)u,b|0.00.20.40.60.81.0|cos(θ)j,b|(d)\n50010001500200025003000\nCounts\nFIG. 12. The cross correlation of cos¹\u0012ºu;bandcos¹\u0012ºj;bin the kinematic (a,c) and saturated (b,d) stages for Re M=1122 . Panels (a) and\n(b) refer to the whole domain and the difference between them is not significant. Panels (c) and (d) refers to only the strong field regions\n(bbrms\u00151:5). The yellow patch close to low cos¹\u0012ºj;band high cos¹\u0012ºu;bin the kinematic stage vanishes for the saturated stage. The peak\nin the count is always at high cos¹\u0012ºu;band high cos¹\u0012ºj;b, which implies significant alignment between magnetic field, velocity field and\ncurrent density.\nof the local compression (Fig. 13a), which leads to the ampli-\nfication of magnetic field, and this trend is slightly stronger\nin the kinematic stage. The PDF of the angle between the\ndirection of local stretching and the magnetic field cos¹\u0012ºe3;b\nhas maxima at cos¹\u0012ºe3;b=0andcos¹\u0012ºe3;b=1in the kine-\nmatic stage. In the saturated stage, however, all angles are\nnearly equiprobable. This change in behaviour is more pro-\nnounced in the strong field regions, bbrms\u00151. In Fig. 14a,\nwe also show the PDF of cos¹\u0012ºe2;b,cos¹\u0012ºe1;bandcos¹\u0012ºe3;b.\nThe forms of the PDF for cos¹\u0012ºe2;bare different from that of\ncos¹\u0012ºe1;bandcos¹\u0012ºe3;bin the kinematic and saturated stages.\nThe magnetic field is less aligned to the direction e2in the\nkinematic stage as compared to the saturated stage and its ef-\nfect, locally on the magnetic field, is decided by the sign of\nthe eigenvalue \u00152(dashed lines in Fig. 14b). Fig. 14b shows\nthe PDF of all three eigenvalues in the kinematic and satu-\nrated stages. All three eigenvalues are statistically lower inmagnitude in the saturated stage as compared to the kinematic\nstage. However, as can be seen in Fig. 13 and Fig. 14a, the\ndifference between the PDFs in the kinematic and saturated\nstages, whilst statistically significant, is not very strong. This\nsuggest that a small reduction in the local stretching and com-\npression of magnetic field contributes towards the saturation\nof the fluctuation dynamo.\nBefore concluding this section, we note that some of these\nconclusions are similar to those reached independently in the\nPhD thesis of Denis St-Onge [74], albeit for a different model\nsetup.\nC. Local magnetic energy balance\nWe now directly consider the equation for magnetic energy\nevolution and calculate its local growth and dissipation terms.10\n0.00 0.25 0.50 0.75 1.00\n|cos(θ)e1,b|0.91.01.1PDF(a)\nwhole domain ,kin\nwhole domain ,sat\nb/brms<1,kin\nb/brms<1,sat\nb/brms≥1,kin\nb/brms≥1,sat\n0.00 0.25 0.50 0.75 1.00\n|cos(θ)e3,b|0.960.981.001.021.04PDF(b)\nwhole domain ,kin\nwhole domain ,sat\nb/brms<1,kin\nb/brms<1,sat\nb/brms≥1,kin\nb/brms≥1,sat\nFIG. 13. The total and conditional PDFs of the cosine of the angle between the direction of local field line compression and the magnetic field\ncos¹\u0012ºe1;b(a) and between the direction of local field line stretching and the magnetic field cos¹\u0012ºe3;b(b) for Re M=1796 in the kinematic\n(red) and saturated (blue) stages.\n0.00 0.25 0.50 0.75 1.00\n|cos(θ)ei,b|0.91.01.11.2PDF(a)|cos(θ)e1,b|,kin\n|cos(θ)e1,b|,sat\n|cos(θ)e2,b|,kin\n|cos(θ)e2,b|,sat\n|cos(θ)e3,b|,kin\n|cos(θ)e3,b|,sat\n−60−40−20 0 20 40 60\nλi/urms0.000.010.020.030.040.050.060.07PDF(b)λ1/urms,kin\nλ1/urms,sat\nλ2/urms,kin\nλ2/urms,sat\nλ3/urms,kin\nλ3/urms,sat\nFIG. 14. The PDFs of the cosine of the angle between three three eigenvectors ( e1;e2;ande3) with the local magnetic field direction (a) and\nthree three eigenvalues ( \u00151;\u00152;and\u00153) normalized by urms(b) for Re M=1796 in the kinematic (red) and saturated (blue) stages.\nFor an incompressible flow in a periodic domain, the magnetic\nenergy evolution equation can be written as [75]\ndEM\ndt=¹\nVbibjSijdV\u0000\u0011¹\nV¹r\u0002 bº2dV; (11)\nwhere EM=1\n2¯\nVb2dVand summation over repeated indices\nis understood. The term contributing to the energy growth,\nbibjSij, is calculated at each point in the volume as follows.\nFirst, we project the magnetic field vector bon to each of the\neigenvectors of the rate of strain tensor, e1;e2;e3. Let these\nbeb1;b2;b3, and then the local growth term bibjSij=\u00151b2\n1+\n\u00152b2\n2+\u00153b2\n3at each position. This term can be positive or\nnegative (\u00151<0and\u00153>0). A negative local growth term\nleads to a decrease in the magnetic energy, whilst a positive\nvalue leads to an increase. The term contributing to the decay\nin energy is calculated by computing ¹r\u0002 bº2(\u0011=constant )at each point in space.\nFig. 15 and Fig. 16 show the total and conditional PDFs of\nthe local growth and dissipation terms in the kinematic and\nsaturated stages for Re M=1122 and Re M=1796 respec-\ntively. Fig. 15a and Fig. 16a show that the local growth term\ndecreases on saturation and this is equally true of the strong\nand weak field regions. This confirms that the stretching of\nthe magnetic field line reduces, which in turn decreases the\namplification. Numerically, this can be quantified by calcu-\nlating the skewness of the local growth term distribution in\nthe kinematic and saturated stages (solid red and blue lines in\nFig. 15a and Fig. 16a). The skewness is defined for a quan-\ntityXash¹X\u0000hXiº3ih¹X\u0000hXiº2i32, whereh\u0001\u0001\u0001i refers\nto the mean. The skewness of the local growth term distribu-\ntion in the kinematic (solid red line in Fig. 15a) and saturated\n(solid blue line in Fig. 15a) stage for Re M=1122 are0:4and11\n−80−60−40−20 0 20 40 60 80\nλ1(b1/brms)2+λ2(b2/brms)2+λ3(b3/brms)210−510−410−310−210−1100PDF(a) whole domain ,kin\nwhole domain ,sat\nb/brms<1,kin\nb/brms<1,sat\nb/brms≥1,kin\nb/brms≥1,sat\n0 10 20 30 40 50 60 70 80\n(∇×b/brms)210−510−410−310−210−1100PDF(b) whole domain ,kin\nwhole domain ,sat\nb/brms<1,kin\nb/brms<1,sat\nb/brms≥1,kin\nb/brms≥1,sat\nFIG. 15. The total and conditional PDFs of the local growth term \u00151¹b1brmsº2+\u00152¹b2brmsº2+\u00153¹b3brmsº2(a) and the local dissipation\nterm¹r\u0002bº2(b) in the kinematic (red) and saturated (blue) stages for Re M=1122 . The skewness of the local growth term distribution (solid\nred line in (a)) is 0:4in the kinematic stage and 0:1in the saturated stage (solid blue line), so the tendency of this term to promote growth\ndecreases on saturation, as could be expected. The local dissipation term (b) also decreases statistically as the field saturates. This conclusions\nhold in both the weak and strong field regions, except for the local dissipation term, which increases in the weak field regions.\n−80−60−40−20 0 20 40 60 80\nλ1(b1/brms)2+λ2(b2/brms)2+λ3(b3/brms)210−510−410−310−210−1100PDF(a) whole domain ,kin\nwhole domain ,sat\nb/brms<1,kin\nb/brms<1,sat\nb/brms≥1,kin\nb/brms≥1,sat\n0 10 20 30 40 50 60 70 80\n(∇×b/brms)210−510−410−310−210−1100PDF(b) whole domain ,kin\nwhole domain ,sat\nb/brms<1,kin\nb/brms<1,sat\nb/brms≥1,kin\nb/brms≥1,sat\nFIG. 16. As Fig. 15 but for Re M=1796 . The skewness of the local growth term distribution (solid red line in (a)) is 0:9in the kinematic stage\nand0:4in the saturated stage (solid blue line). The conclusions remain the same here as for Re M=1122 in Fig. 15.\n0:1respectively. The corresponding values for Re M=1796\n(Fig. 16a) in the kinematic and saturated stages are 0:9and\n0:4respectively. The local growth term always has a posi-\ntive skewness implying continuous magnetic field generation.\nThe skewness decreases on saturation, where the growth is\nonly required to compensate the dissipation. The dissipation\nterm also exhibits an overall decrease on saturation as shown\nin Fig. 15b and Fig. 16b, but its behaviour differs in the strong\nand weak field regions, where the dissipation increases in the\nlatter regions.\nTo calculate the overall decrease or increase in the mag-\nnetic energy at each point in the domain, we calculate the local\nmagnetic Reynolds number. This helps us to explore the be-\nhaviour of the diffusion term\u0000\u0011r2b\u0001in the induction equation(Eq. (2)) as the dynamo saturates. Both terms in Eq. (11) are\ncalculated at each point in the volume, and the local magnetic\nReynolds number is derived at each position as\n¹ReMºloc=bibjSij\n\u0011¹r\u0002 bº2; (12)\nproviding a measure of the local dynamo efficiency. The lo-\ncal magnetic Reynolds number can be positive or negative,\nsignifying the locally increasing or decreasing magnetic field\nstrength, respectively. Fig. 17 and Fig. 18 show the total\nand conditional PDFs of the local magnetic Reynolds num-\nber in the kinematic and saturated stages for Re M=1122 and\nReM=1796 .¹ReMºlocvaries from values much less than to\nthose much greater than Re(crit)\nMin both the kinematic and sat-12\n−10000−5000 0 5000 10000\n(Re M)loc10−510−410−3PDFwhole domain ,kin\nwhole domain ,sat\nb/brms<1,kin\nb/brms<1,sat\nb/brms≥1,kin\nb/brms≥1,sat\nReM= 1122\nRe(crit)\nM= 220\nFIG. 17. The total and conditional PDFs of the local magnetic\nReynolds number ¹ReMºlocin the kinematic (red) and saturated\n(blue) stages with Re M=1122 . The purple dashed line shows\nthe critical magnetic Reynolds number Re(crit)\nM=220and the black\ndashed line shows Re Mfor this run.\n−10000−5000 0 5000 10000\n(Re M)loc10−510−410−3PDFwhole domain ,kin\nwhole domain ,sat\nb/brms<1,kin\nb/brms<1,sat\nb/brms≥1,kin\nb/brms≥1,sat\nReM= 1796\nRe(crit)\nM= 220\nFIG. 18. As Fig. 17 but for Re M=1796 .\nurated stages. Thus, magnetic field grows and decays in dif-\nferent parts of the volume but remains in a statistically steady\nstate overall in the saturated stage. On saturation, both Fig. 17\nand Fig. 18 show that ¹ReMºlocdecreases statistically. The\nmean of¹ReMºlocfor Re M=1122 in the kinematic stage is\n808and that in the saturated stage is 595. Thus, the mean\nvalue of the local magnetic Reynolds number over the entire\ndomain decreases on saturation (to a value close to but not\nexactly equal to the critical value, Re(crit)\nM\u0019220). This ef-\nfectively implies a relative enhancement in the local diffusion\nin comparison to the local stretching, which also contributes\ntowards the saturation of the fluctuation dynamo.\nTo summarize, the fluctuation dynamo saturates due to both\nreduction in stretching and altered diffusion. The alignment\nbetween the velocity and magnetic fields increases as the field\nsaturates, signifying reduced amplification. Furthermore, theTABLE II. Four Minkowski functionals (MF) V0;V1;V2andV3, their\ngeometrical interpretation and definitions in three dimensions. dVis\nthe volume element, dSis the surface element, and \u00141and\u00142are the\nprinciple curvatures of the surface of a structure.\nMF Geometric interpretation Expression\nV0 V olume±\ndV\nV1 Surface area ¹16º°\ndS\nV2Integral mean curvature ¹16\u0019º°\n¹\u00141+\u00142ºdS\nV3 Euler characteristic ¹14\u0019º°\n¹\u00141\u00142ºdS\ncurrent density and magnetic field are also statistically bet-\nter aligned in the saturated stage, which implies a trend to-\nwards a force–free field. The local growth term statistically\ndecreases (the skewness of the distribution, though remain-\ning positive, decreases on saturation), which implies that the\nreduced magnetic field stretching reduces the amplification,\nwhich contributes towards the saturation of the fluctuation dy-\nnamo. The local magnetic Reynolds number, though varying\nover a wide range from values much less than to much higher\nthan the critical value, decreases on average. This further im-\nplies relative enhancement in the local dissipation compared\nto the local stretching, which also contributes towards the sat-\nuration of the fluctuation dynamo.\nV . MORPHOLOGY OF MAGNETIC STRUCTURES\nAs shown in Section III, magnetic field generated by a fluc-\ntuation dynamo is intermittent as it is concentrated in fila-\nments, sheets and ribbons (Fig. 8 and Fig. 9). To character-\nize the magnetic structures, we use the Minkowski function-\nals [76]. Minkowski functionals have been used in studying\nmorphology of structures in a number of numerical simula-\ntions [34, 77–82] and observations [83–86]. The morphol-\nogy of a d–dimensional structure can be described by d+1\nMinkowski functionals. In three dimensions, there are four\nMinkowski functionals, as described in Table II. We calculate\nthe Minkowski functionals using Crofton’s formulae [87, 88]\nand then calculate the representative length scales ( l1;l2;l3) of\nmagnetic structures (defined by isosurfaces at a fixed value of\nthe magnetic field strength, e.g., see Fig. 9) as [77, 89]\nl1=V0\n2V1;l2=2V1\n\u0019V2;l3=3V2\n4V3: (13)\nWe associate the smallest of these length scales with the thick-\nnessTof the structures, the next largest with the width Wand\nthe largest length scale with the length L, i.e., if l1\u0014l2\u0014l3,\nthen T=l1;W=l2andL=l3. The thickness, width and\nlength can be further used to obtain dimensionless measures\nof the structure shape: planarity pand filamentarity f, given\nby\np=W\u0000T\nW+T;f=L\u0000W\nL+W: (14)\nBy definition, 0\u0014p\u00141and0\u0014f\u00141;p=0andf=1for\na perfect filament, p=0andf=0for a sphere, and p=113\nTABLE III. Parameters of various runs for the nonlinear fluctuation\ndynamo in a numerical domain size of ¹2\u0019º3with 5123mesh points.\nIn all cases, the forcing scale is approximately L5, the forcing am-\nplitude is F0\u00190:02and the hydrodynamic viscosity is \u0017=4\u000210\u00004.\nThe magnetic diffusivity \u0011, the rms velocity in the saturated stage\nurms, the Reynolds number Re, the magnetic Reynolds number Re M,\nthe magnetic Prandtl number Pr Mand the critical magnetic Reynolds\nnumber Re(crit)\nM¹\u0019220Pr\u000012\nMºare given.\n\u0011 urms Re Re MPrMRe(crit)\nM\n4\u000210\u000040:11 346 346 1 :00 220\n3\u000210\u000040:11 346 461 1 :33 191\n2\u000210\u000040:10 314 628 2 :00 156\n1\u000210\u000040:09 283 1131 4 :00 110\n7:5\u000210\u000050:09 283 1508 5 :33 95\n5\u000210\u000050:09 283 2261 8 :00 78\nandf=0for a sheet. The planarity and filamentarity are not\nsensitive to the size of the structures but quantify the shape. It\nis useful to remember that, unlike the Minkowski functionals,\npandfare not additive.\nTo explore the morphology of magnetic structrures for a\nrange of Re Mvalues, we use simulations with parameters\ngiven in Table III. We keep Re about the same for all runs,\nvary Re M(making sure Pr M\u00151), and choose kF\u00195¹2\u0019Lº,\nso there is a sufficient number of magnetic correlation cells\nwithin the volume (with 53velocity correlation cells).\nFig. 19a shows the thickness, width and length of magnetic\nstructures obtained by averaging over 30values of magnetic\nfield strengths ranging from bbrms=2:5to4. The lower\nlimit of the magnetic field strength is chosen to ensure that\nthe structures represent the tail of the PDF (e.g., see Fig. 6),\nwhilst the upper limit is chosen to ensure a sufficient number\nof points within each structure. The computed values of pla-\nnarity and filamentarity also remain roughly constant within\nthis selected range of magnetic field strengths. For the kine-\nmatic stage, we expect that the largest length scale Lwill\nbe independent of Re M. This is because the length of the\nstructures is controlled by the correlation length of the flow\nsince the magnetic correlation function of the fastest grow-\ning dynamo mode decreases exponentially after that scale\n[32]. As seen in Fig. 19a, the length remains roughly con-\nstant but then increases slightly after Re M\u0019600and again\nremains roughly constant. This variation is likely to be due\nto the decrease in the Reynolds number Re (Table III). The\nother two scales ( WandT) decrease as Re\u00000:5\nM. This scal-\ning can be obtained by balancing the rate of magnetic dis-\nsipation with the local shearing rate [90], \u0011W2'urmsl0,\nwhere\u0011is the magnetic resistivity, urmsis the rms turbulent\nvelocity and l0is the driving scale of the turbulence. This\ngives W'l0¹\u0011urmsl0º12=l0Re\u00000:5\nM. This means that the\nshape of the magnetic structures becomes more filamentary\n(L\u001dW\u0019T) and ribbon-like ( T.W\u001cL) as Re Min-\ncreases, but the filamentarity is always larger than the pla-\nnarity, so the filaments dominate among the magnetic struc-\ntures [91]. The differences in Re Mscalings with the previ-ous work [34] is probably due to the following reasons. First,\nthey have a prescribed velocity field with forcing at a range\nof scales, whereas we force the flow at two scales ( k=4and\nk=6) and then let it evolve via the Navier-Stokes equation.\nSecond, our simulations are at a higher resolution\u00005123\u0001as\ncompared to theirs\u00001283\u0001and thus magnetic structures, es-\npecially at higher Re M, are better resolved in our case. Last\nand most importantly, they consider values of Re Mwhich are\nboth lower and higher than Re(crit)\nM, whereas we only con-\nsider Re M>Re(crit)\nM. This is because we strongly believe\nthat those two regimes\u0010\nReM<Re(crit)\nMand Re M\u0015Re(crit)\nM\u0011\nare\nphysically different and must not be considered together to\ncharacterize the length scales of magnetic structures as func-\ntions of Re M.\nAll three scales are larger in the saturated stage than in the\nkinematic stage. Thus, the magnetic structures become larger\nas the magnetic field saturates. This is also the reason that the\nmagnetic field correlation length scale increases as the field\nsaturates (as shown in Table I). The increase in the length (the\nlargest length scale) of magnetic structures on saturation is\nconsistent with the finding by Schekochihin et al. [33]. The\nReMscaling for all three scales is roughly the same for both\nthe kinematic and saturated stages.\nFig. 19b shows the planarity and filamentarity of magnetic\nstructures as functions of Re M. The filamentarity is always\nhigher than the planarity and thus the magnetic structures are\nmore like filaments in both the kinematic and saturated stages.\nThe dependence of these morphological measures on Re Mis\nthe same for the kinematic and saturated stages.\nVI. CONCLUSIONS AND DISCUSSION\nIt is important to understand the saturated state of the fluc-\ntuation dynamo because the saturated state seeds the mean\nfield dynamo, controls the small-scale magnetic field struc-\nture and decides the magnetic field length scales in the system\nwhere the mean field dynamo is absent (for example, ellipti-\ncal galaxies). Moreover, it is crucial to understand the physics\nof the saturation mechanism because numerical simulations,\nat present, are at much lower values of Re Mthan their esti-\nmated values (Re M\u00191018for spiral galaxies, Re M\u00191022for\nelliptical galaxies and Re M\u00191029for galaxy clusters).\nUsing numerical simulations of driven nearly incompress-\nible turbulence, we have explored the saturation mechanism\nof the fluctuation dynamo. We find that the dynamo saturates\nbecause both the amplification and diffusion are affected by\nthe action of the Lorentz force on the flow. Most previously\nsuggested mechanisms hinted at changes in either of those two\nand thus required significant changes in the properties of the\nvelocity and magnetic fields from the kinematic stage. For\nexample, if only the enhancement in diffusion is responsi-\nble, it would require the effective Re Min the saturated state\nto reduce from hugely supercritical levels to values close to\nRe(crit)\nM('102–103[14]). And, if only the decrease in am-\nplification is responsible for saturating the dynamo, it would\nrequire a drastic decrease in the Lyapunov exponents (which14\n103 4×1026×1022×103\nReM10−1T, W, L\n(a)ReM−0.5\nT,kin\nW,kin\nL,kin\nT,sat\nW,sat\nL,sat\n500 1000 1500 2000\nReM0.20.40.60.81.0p,f\n(b)p,kin\np,sat\nf,kin\nf,sat\nFIG. 19. (a) Average length ( L), thickness ( T) and width ( W) of magnetic structures in the kinematic (dashed, color) and saturated stages\n(solid, color) of the nonlinear fluctuation dynamo as functions of Re M. The width and thickness of the magnetic structures both decrease as\nRe\u00000:5\nM. The Re Mdependence of the size of the structures is approximately the same in both the kinematic and saturated stages. (b) Planarity\n(p) and filamentarity ( f) of the magnetic structures, as functions of Re M, for the kinematic (dashed, color) and saturated (solid, color) stages.\nAs Re Mincreases, the filamentarity increases and the planarity decreases but they seem to approach an asymptotic value after Re M\u00191200 .\nThe trend with respect to Re Mis the same for both the kinematic and saturated stages.\nare a measure of chaotic properties of the flow) [60]. We sug-\ngest that both occur and thus such a dramatic change is not\nnecessary. We confirm that the amplification decreases by re-\nduction in the stretching of magnetic field lines. The local\nmagnetic Reynolds number ¹ReMºloc, which is suggested as\na measure of the local magnetic diffusion, decreases slightly.\nThis confirms that the local diffusion of magnetic field relative\nto field line stretching is enhanced, which is also responsible\nfor saturating the dynamo.\nThe fluctuation dynamo-generated magnetic field is spa-\ntially intermittent. So, we studied the morphology of the mag-\nnetic structures in the kinematic and saturated stages. In both\ncases, the largest length scale is roughly independent of Re M\nand the other two scales decrease as Re\u00000:5\nM. We find that the\nstructures are of a larger size (all three length scales increase)\nin the saturated stage as compared to the kinematic stage. This\nagrees with the results in Table I, where we find that the cor-\nrelation length is higher for the saturated magnetic field. This\nalso aligns with the conclusion in the Section III (also shown\nin [33]) that the magnetic field is less intermittent in the sat-\nurated stage as compared to the kinematic stage. However,\nthe Re Mdependence is the same for both the stages and thus\nthe overall shape of magnetic structures produced by the fluc-\ntuation dynamo is not affected by the Lorentz force to any\nsignificant extent (all three length scales increase but in a very\nsimilar way).\nThe study explores physical effects over a range of Re Mfor\nPrM\u00151. However, for Pr M>1at very high Re M(&103), the\nfields might be unstable to fast magnetic reconnection [12].\nThis might change the morphology of magnetic fields, locally\naffect velocity fields and thus might alter the saturated state of\nthe fluctuation dynamo. However, the effect of fast, stochastic\nmagnetic reconnection on the dynamo is not very well under-stood yet [92] and would require high-resolution numerical\nsimulations over a number of very high Re Mvalues to study\nthe effect of fast magnetic reconnection on the fluctuation dy-\nnamo saturation mechanism.\nThe study can be extended in several ways. An immedi-\nate extension would be to repeat the entire analysis for dy-\nnamos in a stratified medium [42, 46, 47, 93], which is more\nrelevant for young galaxies and star-forming gas clouds. We\nhave performed the analysis for Pr M\u00151which is of rele-\nvance to fluctuation dynamo in the interstellar and intergalac-\ntic medium but this should be extended to the Pr M<1regime\nwhich is important for stars, planets and liquid metal exper-\niments [94, 95]. We have adopted the MHD approximation\nbut plasma effects might also play an important role. It would\nalso be interesting to compare our results with those of the\nplasma dynamo [96, 97] and see how the relationship between\nvelocity and magnetic fields and the magnetic field structure\nchange when plasma effects are considered. Plasma effects\nmight be particularly important for the weakly collisional gas\nin galaxy clusters. We aim to consider such problems in our\nfuture work.\nACKNOWLEDGMENTS\nWe thank Kandaswamy Subramanian and Christoph Feder-\nrath for useful discussions and comments on the paper. We\nacknowledge financial support of the STFC (ST/N000900/1,\nProject 2) and the Leverhulme Trust (RPG-2014-427). We\nthank one of the referees for highlighting Denis St-Onge’s\nPhD thesis [74], which independently reports results similar\nto those found in Section IV B. In fact, Fig. 14 was added fol-\nlowing the initial review so as to facilitate a direct comparison15\nbetween the two studies.\n[1] B. M. Gaensler, M. Haverkorn, L. Staveley-Smith, J. M.\nDickey, N. M. 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The criteria to es-\ntablish the planar dynamics reveal that the resulting non-\nrelativistic Hamiltonian has a simplified expression without\nmaking approximations, and some terms have crucial im-\nportance for system dynamics.\n1 Introduction\nThe Aharonov-Bohm effect [1] has been an usual framework\nfor demonstrating the importance of potentials in quantum\nmechanics. After its experimental verification [2, 3] sev-\neral other analogs effects were being proposed along the last\ndecades. For example, in Ref. [4], it was shown that a par-\nticle with a magnetic moment moving in an electric field\nacquires a quantum phase. This phase has been observed in\na neutron interferometer [5] and in a neutral atomic Ram-\nsey interferometer [6]. In Refs. [7, 8] it was verified that a\nneutral particle with an electric dipole moment which moves\nin a magnetic field acquires a topological phase. The exper-\nimental confirmation of this phase was established in Ref.\n[9]. In Ref. [10] was proposed a unified and fully relativis-\ntic treatment of the interaction of the electric and magnetic\ndipole moments of a particle with the electromagnetic field.\nThe essence of this study reveals that new force on dipoles\nare obtained using the non-Abelian nature of this interaction,\nand new experiments analogous to the Aharonov-Bohm ef-\nfect to test this interaction are proposed. Since this interac-\ntion is a consequence of a nonminimal coupling is also in-\nteresting to analyze the consequences of this interaction in\nother contexts. For instance, it may be of interest to study\nscattering and bound states of neutral fermions in external\nelectromagnetic fields and access other physical quantities\nae-mail: edilbertoo@gmail.comsuch as energy bound states [11, 12] and scattering [13, 14].\nThe dual results of the magnetic dipole interaction for the\nelectric dipole interaction for the system considered in Ref.\n[10] has been established in Ref. [15], where the phase shift\nin the interference of a magnetic or electric dipole due to\nthe electromagnetic field is obtained relativistically and non-\nrelativistically.\nIn this work, we consider the same system addressed\nin Ref. [15] but now assuming that the dynamics is purely\nplanar, and derive its equation of motion. This system is a\ngeneralization, for example, from that studied in Ref. [12],\nwhere only the effects of an electric field has been consid-\nered. An interesting feature of this system is that even in\nthe absence of electric field it admits bound and scattering\nstates, which does not occur in the previous work. As an\napplication, we consider the problem of bound state for the\ncase of a magnetic dipole moment interacting with electric\nand magnetic fields generated by an infinitely long charged\nsolenoid, carrying a magnetic field. In our treatment, we\nconsider the self-adjoint extension method [16], which is\nappropriate to address any system endowed with a singu-\nlar Hamiltonian (due to localized fields sources or quan-\ntum confinement) [17–26]. We determine the energy spec-\ntrum and wave functions by applying boundary conditions\nallowed by the system.\n2 The planar Pauli equation\nWe begin with the Dirac equation in (3+1)dimensions [15]\nwhich governs the nonrelativistic dynamics of a neutral par-\nticle that possesses magnetic dipole moment, in the presence\nof electric and magnetic fields (¯h=c=1)\n\u0014\nigm¶m\u0000M+m\n2smnFmn\u0015\nY=0; (1)arXiv:1408.6578v2  [hep-th]  28 Sep 20142\nwhere mis the magnetic dipole moment, Fmnis the electro-\nmagnetic tensor whose components are given by\u0000\nF0i;Fi j\u0001\n=\u0000\n\u0000Ei;\u0000ei jkBk\u0001\n, and\u0000\ns0j;si j\u0001\n=\u0000\niaj;\u0000ei jkSk\u0001\n, where Sk\nis the spin operator, are the components of the operator smn=\ni[gm;gn]=2, which are given in terms of the Dirac matrices.\nWith this notation, it is possible to show that the spin is cou-\npled to the electromagnetic field tensor through the term\n1\n2smnFmn=\u0000SSS\u0001B+iaaa\u0001E (2)\nwhere EandBare the electric and magnetic field strengths.\nThis result is explicitly calculated in the following represen-\ntation of the g-matrices:\ng0=\u00121 0\n0\u00001\u0013\n;ggg=\u00120sss\n\u0000sss0\u0013\n;\naaa=g0ggg=\u00120sss\nsss0\u0013\n;SSS=\u0012sss0\n0sss\u0013\n:\nwithsss= (s1;s1;s3)being the Pauli matrices. Equation (1)\ncan be written as\nˆHDY=EY; (3)\nwhere the operator\nˆHD=bM+aaa\u0001p+m(\u0000bSSS\u0001B+iggg\u0001E) (4)\nis the Dirac Hamiltonian. The non-relativistic limit of Eq.\n(3) was established in Ref. [15], and the relevant equation is\nfound to be:\nˆHy=Ey; (5)\nwhere yis a two-component spinor, with\nˆH=1\n2M[p\u0000(mmm\u0002E)]2\u00001\n2Mm2E2+1\n2Mm(ÑÑÑ\u0001E)\u0000(mmm\u0001B);\n(6)\nwhere mmm=msss. Our goal is to analyze the physical implica-\ntions of the Hamiltonian (6), when we assume that the sys-\ntem dynamics is now planar. This is established as follows.\nBy detaching the third component of Eq. (6), we get\nˆH=1\n2M2\nå\ni=1[pi\u0000(mmm\u0002E)i]2\u00001\n2Mm2E2\n+1\n2M[p3\u0000(mmm\u0002E)3]2\n+1\n2M2\nå\ni=1m(ÑÑÑ\u0001E)i+1\n2Mm(ÑÑÑ\u0001E)3;\n\u00002\nå\ni=1(mmm\u0001B)i\u0000(mmm\u0001B)3;(i=1;2): (7)\nIf we assume that dynamic is planar, the above Hamiltonian\nprovides an important result, namely, the [p3\u0000(mmm\u0002E)3]2\nterm leads exactly to the quantity m2E2=2M. The planar caseis accessed by requiring that pz=z=0 together with the\nimposition of the fields should not have the third direction.\nThis question can also be understood when we look at the\nsymmetry under ztranslations, which allows us to access\nthe solutions of the planar Dirac equation. This type of sim-\nplification is in fact manifested only when we assume that\nthe particle moves in the plane. Thus, since Am= (F;A),\nwe write the electric and magnetic fields, respectively, as\nE=\u0000ˆx¶xF(x;y)\u0000ˆy¶yF(x;y); (8)\nB=ˆz(¶xAy\u0000¶yAx); (9)\nFields EandBabove are now intrinsically two-dimensional.\nNote that the square of Eq. (8) gives exactly E2\n1+E2\n2for the\nplanar case. Also, the restriction imposed on the potential\nAreveals that the (mmm\u0001B)iterm in Eq. (7) is now identically\nzero. Now, we can show that the quantity (mmm\u0002E)3is given\nbym(s1E2\u0000s2E1), and the third term of Eq. (7) results\n[p3\u0000(mmm\u0002E)3]2!m2(s1E2\u0000s2E1)2=m2E2: (10)\nThus, we now can write Eq. (7) as\nˆH=1\n2M2\nå\ni=1n\n[pi\u0000(mmm\u0002E)i]2+m(ÑÑÑ\u0001E)io\n\u0000(mmm\u0001B)3;\n(11)\nwhere the magnetic interaction term (mmm\u0001B)3gives the only\nexplicit dependence of the spin. In Ref. [15] it was assumed\nthat the charge density r=ÑÑÑ\u0001Eand also the m2E2=2Mterm\n(for thermal neutrons) are negligible. In fact, this approxi-\nmation can only be performed, if we are only interested in\nthe study of phase shift. However, when we want to study\nthe dynamics of the system, such as the scattering and bound\nstates problems, all terms of the equation motion of must be\ntaken into account. This, for example, has been addressed\nby Hagen [27] to show that there is an exact equivalence be-\ntween the AB and AC effects for spin-1 =2 particles. In these\neffects, the ÑÑÑ\u0001EandÑÑÑ\u0002Abeing proportional to a delta\nfunction, such terms must now contribute to the dynamics\nof the system, and can not be neglected. For this reason,\nsince we are dealing with Aharonov-Bohm-like system for\nspin-1/2 particles, the m(ÑÑÑ\u0001E)iand(mmm\u0001B)3terms in Eq.\n(11), may not be negligible. Let’s clarify this issue. Con-\nsider an infinitely long solenoid, carrying a magnetic field\nB, and with a charge density ldistributed uniformly about\nit along the z-axis. The electric field and magnetic flux tube\n(in cylindrical coordinates) generated by this configuration\nare known to be\nE=2lˆr\nr;ÑÑÑ\u0001E=2ld(r)\nr; (12)\nB=ÑÑÑ\u0002A=fd(r)\nrˆz; (13)3\nwhere fis the magnetic flux inside the tube, and the vector\npotential in the Coulomb gauge is\nA=f\nrˆjˆjˆj; (14)\nWe see that the fields EandBare proportional to a dfunc-\ntion. By using Eqs. (12) and (13), Pauli equation (5) is now\nwritten as\nˆHy=Ey; (15)\nwith\nˆH=1\n2M2\nå\ni=1\u0012\npi\u0000fEszˆjˆjˆj\nr\u00132\n+(fE\u0000fBsz)d(r)\nr: (16)\nwhere fE=2mlandfB=2Mmf. From Eq. (15), we can\nsee that yis an eigenfunction of sz, whose eigenvalues des-\nignate by s=\u00061, that is, szy=\u0006y=sy. Thus, since ˆH\ncommutes with the operators ˆJz=\u0000i¶j+sz=2, where ˆJzis\nthe total angular momentum operator in the z-direction, we\nseek solutions of the form\ny(r;j) =\u0014fm(r)eimj\ngm(r)ei(m+1)j\u0015\n; (17)\nwith m+1=2=\u00061=2;\u00063=2;:::; (m2Z). Inserting (17)\ninto Eq. (15), we can extract the radial equation for fm(r)\nH fm(r) =k2fm(r); (18)\nwhere\nH=H0+(fE\u0000sfB)d(r)\nr; (19)\nand\nH0=\u0000d2\ndr2\u00001\nrd\ndr+(m\u0000sfE)2\nr2: (20)\nNote that, even in the absence of electric field, bound and\nscattering states are possible. This does not occur, for ex-\nample, in the system studied in Ref. [12], where the particle\ninteracts only with an electric field. Moreover, if a magnetic\nfield is present, the physical system changes completely. We\nwill see later that this fact directly influences on the expres-\nsion for the self-adjoint extension parameter and, hence, on\nthe boundary conditions allowed by the operator H0. In other\nwords, this has direct implications in the dynamics of the\nsystem. This can be seen more easily by studying the signal\noffE\u0000sfBin Eq. (19), where several possible combinations\noffE,fBandsgives us the possibilities for the existence of\nbound and scattering states. As a result of these combina-\ntions, we have\nfE\u0000sfB<0;scattering and bound states ; (21)\nfE\u0000sfB>0;scattering states : (22)\nThe case fE=sfBis not of interest here because it cancels\nthe term that explicitly depends on the spin.3 Physical regularization and the bound states problem\nIn this section, we study the dynamics of the system in all\nspace, including the r=0 region. We consider the prob-\nlem of bound states. To this end, we use the self-adjoint ex-\ntension method in the treatment. As is well known, if the\nHamiltonian has a singularity point, as is the case of the\nHamiltonian in Eq. (19), we must verify that it is self-adjoint\nin the region of interest. Even though H†\n0=H0, their do-\nmains could be different. This is the crucial point in our\nstudy. The operator H0, with domain D(H0), is self-adjoint\nifD(H†\n0) =D(H0)andH†\n0=H0. However, for this to be\nestablished, we must find the deficiency subspaces,\nN+=n\ny2D(H†\n0);H†\n0y=z+y;Imz+>0o\n; (23)\nN\u0000=fy2D(H†\n0);H†\n0y=z\u0000y;Imz\u0000<0g; (24)\nwith dimensions n+andn\u0000, respectively, called deficiency\nindices of H0[16]. We also know of this theory that a nec-\nessary and sufficient condition for H0being essentially self-\nadjoint is that its deficiency indices n+=n\u0000=0. On the\nother hand, if n+=n\u0000\u00151 the operator H0has an infi-\nnite number of self-adjoint extensions parametrized by the\nunitary operators U:N+!N\u0000. With these ideas in mind,\nwe now decompose the Hilbert space H=L2(R2)with\nrespect to the angular momentum H=Hr\nHj, where\nHr=L2(R+;rdr)andHj=L2(S1;dj), with S1denoting\nthe unit sphere in R2. The operator\u0000¶2\njis known to be es-\nsentially self-adjoint in L2(S1;dj). By using the unitary op-\nerator [18]\nV:L2(R+;rdr)!L2(R+;dr); (25)\ngiven by\n(VQ)(r) =r1=2Q(r); (26)\nthe operator H0reads\nH0\n0=VH 0V\u00001=\u00001\n2M\u001ad2\ndr2+1\nr2\u0014\n(m\u0000sfE)2\u00001\n4\u0015\u001b\n;\n(27)\nwhich is essentially self-adjoint for (m\u0000sfE)\u00151, while\nfor(m\u0000sfE)<1, it admits a one-parameter family of self-\nadjoint extensions [16]. To characterize this family, we fol-\nlow the recipe based in boundary conditions given in Ref.\n[28]. Basically, the boundary condition is a match of the log-\narithmic derivatives of the zero-energy solutions for Eq. (18)\nand the solutions for the problem H0plus self-adjoint ex-\ntension. Then, following [28], we temporarily forget the d-\nfunction potential and find the boundary conditions allowed\nforH0. Next, we substitute the problem in Eq. (18) by\nH0fz(r) =k2fz(r); (28)4\nplus self-adjoint extensions. Here, fzis labeled by the pa-\nrameter zof the self-adjoint extension, which is related to\nthe behavior of the wave function at the origin. In order for\ntheH0to be a self-adjoint operator in Hr, its domain of def-\ninition has to be extended by the deficiency subspace, which\nis spanned by the solutions of the eigenvalue equation\nH†\n0f\u0006(r) =\u0006ik2\n0f\u0006(r); (29)\nwhere k2\n02Ris introduced for dimensional reasons. Since\nH†\n0=H0, the only square integrable functions which are so-\nlutions of Eq. (29) are the modified Bessel functions of sec-\nond kind\nf\u0006(r) =Km\u0000sfE(p\n\u0007ik0r); (30)\nwith Imp\u0006i>0. By studying Eq. (30), we verify that it\nis square integrable only in the range m\u0000sfE2(\u00001;1).\nIn this interval, nevertheless, the Hamiltonian (20) is not\nself-adjoint. The dimension of such deficiency subspace is\n(n+;n\u0000) = ( 1;1). So, we have two situations for m\u0000sfE,\ni.e.,\n\u00001<m\u0000sfE<0;\n(31)\n0<m\u0000sfE<1:\nTo address both cases of Eq. (31), we write Eq. (30) as\nf\u0006(r) =Kjm\u0000sfEj(p\n\u0007ik0r): (32)\nEquation (32) allows us to identify the domain of H†\n0as\nD(H†\n0) =D(H0)\bN+\bN\u0000: (33)\nSo, to extend the domain D(H0)and make it equal to D(H†\n0)\nand therefore to turn H0self-adjoint, we get\nD(H0;V) =D(H†\n0) =D(H0)\bN+\bN\u0000: (34)\nEquation (34) establishes the following result. For each value\nofz, we have a possible domain for D(H0;z), but are the\nphysical parameters of the problem that will select a partic-\nular value of it. The Hilbert space is now specified by [16]\nfz(r) =fm(r)\n+Ch\nKjm\u0000sfEj(p\n\u0000ik0r)+eizKjm\u0000sfEj(p\nik0r)i\n;(35)\nwhere fm(r), with fm(0) = ˙fm(0) =0 (˙f\u0011d f=dr), is the\nregular wave function and the parameter z2[0;2p)repre-\nsents a choice for the boundary condition. For each z, we\nhave a possible domain for H0and the physical situation is\nthe factor that will determine the value of z[20, 29–34].\nThus, to find a fitting for zcompatible with the physical sit-\nuation, a physically motivated form for the magnetic fieldis preferable for the regularization of the d-function. This is\naccomplished by replacing (14) with [35]\neA=8\n><\n>:f\nrˆjˆjˆj;r>a\n0;r<a:(36)\nWith this modification, the delta function in Eq. (19) is now\nregularized as d(r\u0000a)=a. A remarkable feature of this regu-\nlarization is that, although the functional structure of d(r)=r\nandd(r\u0000a)=aare quite different, we are free to use any\nform of potential once that the specific details of the reg-\nularization model can be shown to be irrelevant provided\nthat only the contribution is independent of angle and has no\nd-function contribution at the origin [35]. It should also be\nmentioned that the d(r\u0000a)=apotential is one dimensional\nand well defined contrary to the two dimensional d(r)=r.\nNow, we are in the position to determine a fitting value\nforz. To do so, we consider the zero-energy solutions for\nf0(r)with the regularization, and for fz;0(r)without the d\nfunction, respectively, i.e.,\n(\n\u00001\nrd\ndr\u0012\nrd\ndr\u0013\n+(m\u0000sfE)2\nr2\n+ (fE\u0000sfB)d(r\u0000a)\na)\nf0(r) =0;(37)\n(\n\u00001\nrd\ndr\u0012\nrd\ndr\u0013\n+(m\u0000sfE)2\nr2)\nfz;0(r) =0: (38)\nThe value of zis determined by the boundary condition\nlim\na!0+a˙f0(r)\nf0(r)\f\f\f\nr=a=lim\na!0+a˙fz;0(r)\nfz;0(r)\f\f\f\nr=a: (39)\nBy integrating Eq. (37) from 0 to aand noting that the be-\nhavior of f0asa!0 isf0\u0018rjm\u0000sfEj, the left-hand side of\nEq. (39) is found to be\nlim\na!0+a˙f0(r)\nf0(r)\f\f\f\nr=a=fE\u0000sfB: (40)\nTo calculate the right-hand side of Eq. (39), we need to use\nthe asymptotic behavior for Kn(z)in the limit z!0, given\nby\nKn(z)\u0018p\n2sin(pn)\u0014z\u0000n\n2\u0000nG(1\u0000n)\u0000zn\n2nG(1+n)\u0015\n: (41)\nThe substitution of Eq. (41) into Eq. (35) leads to\nlim\na!0+a˙fz;0(r)\nfz;0(r)\f\f\f\nr=a=lim\na!0+˙Wz(r)\nWz(r)\f\f\f\nr=a; (42)5\nwith\nWz(r) =\"\u0000p\u0000ik0r\u0001\u0000jm\u0000sfEj\n2\u0000jm\u0000sfEjG(\u0000)\u0000\u0000p\u0000ik0r\u0001jm\u0000sfEj\n2jm\u0000sfEjG(+)#\n+eiz\"\u0000p\nik0r\u0001\u0000jm\u0000sfEj\n2\u0000jm\u0000sfEjG(\u0000)\u0000p\nik0r\u0001jm\u0000sfEj\n2jm\u0000sfEjG(+)#\n; (43)\nandG(\u0006)=G(1\u0006jm\u0000sfEj)was defined for purposes of\nsimplification. Inserting (40) and (42) in (39), we obtain\nlim\na!0+˙Wz(r)\nWz(r)\f\f\f\nr=a=fE\u0000sfB; (44)\nwhich gives us the parameter zin terms of the physics of\nthe problem, i.e., the correct behavior of the wave functions\nwhen r!0.\nAs promised above, let us now derive determine the bound\nstates for H0. In order systems have a bound state, its energy\nmust be negative, so that in Eq. (28), kis a pure imaginary,\ni.e.,k=ik, with k=p\u00002ME, where E<0 is the bound\nstate energy. Then, with the substitution k!ik, we have\n(\n1\nrd\ndr\u0012\nrd\ndr\u0013\n\u0000\"\n(m\u0000sfE)2\nr2+k2#)\nfz(r) =0; (45)\nThe above equation is the modified Bessel equation whose\ngeneral solution is given by\nfz(r) =Kjm\u0000sfEj\u0010\nrp\n\u00002ME\u0011\n: (46)\nSince these solutions belong to D(Hz;0), it is of the form\n(35) for some zselected from the physics of the problem.\nSo, we substitute (46) into (35) and use (41) to calculate the\nleft-hand side of Eq. (39). After these manipulations, we find\nthe relation\njm\u0000sfEjh\na2jm\u0000sfEjG(\u0000)(\u0000ME b)jm\u0000sfEj+2jm\u0000sfEjG(+)i\na2jm\u0000sfEjG(\u0000)(\u0000ME b)jm\u0000sfEj\u00002jm\u0000sfEjG(+)\n=fE\u0000sfB: (47)\nSolving the above equation for E, we find the energy spec-\ntrum\nE=\u00002\nMa2\n\u0002\u0014\u0012fE\u0000sfB+jm\u0000sfEj\nfE\u0000sfB\u0000jm\u0000sfEj\u0013G(1+jm\u0000sfEj)\nG(1\u0000jm\u0000sfEj)\u0015 1\njm\u0000sfEj:(48)\nNotice that there is no arbitrary parameter in the above equa-\ntion. Also, to ensure that the energy is a real quantity, we\nmust establish that\n\u0012fE\u0000sfB+jm\u0000sfEj\nfE\u0000sfB\u0000jm\u0000sfEj\u0013G(1+jm\u0000sfEj)\nG(1\u0000jm\u0000sfEj)>0: (49)\nThis inequality is satisfied if\njfE\u0000sfBj\u0015jm\u0000sfEj: (50)Because of the condition that jm\u0000sfEj<1, it is sufficient to\nconsiderjfE\u0000sfBj\u00151. A necessary condition for a dfunc-\ntion to generate an attractive potential, which is able to sup-\nport bound states, is that the coupling constant (fE\u0000sfB)\nmust be negative. Thus, the existence of bound states re-\nquires\nfE\u0000sfB\u0014\u00001: (51)\nSo, it seems that we must have\nsfB>fE; (52)\nin such way that the flux and the spin must be parallel, and\nconsequently, a minimum value for jfBjandjfEjis estab-\nlished.\n4 Conclusions\nWe have analyzed the planar quantum dynamics of a mag-\nnetic dipole moment in the presence of electric and magnetic\nfields. We have shown that the initial Hamiltonian system\n(Eq. (6)) reduces to a planar form (Eq. (11)) without making\nany approximations. As an application, we have considered\nthe bound state problem for the case of a magnetic dipole\nmoment interacting with electric and magnetic fields gen-\nerated by an infinitely long solenoid, carrying a magnetic\nfield, and with a charge density distributed uniformly about\nit along the z-axis. The self-adjoint extension approach was\nused to determine the bound states of the particle in terms\nof the physics of the problem, in a very consistent way and\nwithout any arbitrary parameter.\n5 Acknowledgments\nThe author would like to thank R. Casana and F. M. Andrade\nfor the critical reading of the manuscript and for helpful dis-\ncussions. This work was supported by CNPq (Grants No.\n482015/2013-6 (Universal), No.306068/2013-3 (PQ)) and\nFAPEMA (Grant No. 00845/13 (Universal)).\nReferences\n1. Y . Aharonov, D. Bohm, Phys. Rev. 115(3), 485 (1959).\nDOI 10.1103/PhysRev.115.485\n2. R.G. Chambers, Phys. Rev. Lett. 5, 3 (1960). DOI 10.\n1103/PhysRevLett.5.3\n3. M. Peshkin, H.J. 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Taimanov†\n1 Introduction\nThe motion of a charge in a magnetic field on a configuration space Mnis\ndescribed by the Euler–Lagrange equations for the Lagrangian fu nction\nL(q,˙q) =1\n2gik˙qi˙qk+Ai˙qi\nwhere thefirst termisthekynetic energycalculated by using theRie mannian\nmetricgikonMnand the second term describes the interaction of a charge\nwith the magnetic field Fwhich is a closed two-form FonMnsuch that\nF=dA. IfFis non-exact, then the one-form Ais defined only locally. If\nF= 0, then we get the Lagrangian function for geodesics on Mnand for\nthis reason solutions q(t) for the Euler–Lagrange equations with a general\nmagnetic field are called magnetic geodesics.\nTrajectories lying on the energy level\nE=1\n2gik˙qi˙qk= const\nsatisfy to the Euler–Lagrange equations for the reduced Lagran gian function\nLE(q,˙q) =√\n2E/radicalbig\ngik˙qi˙qk+Ai˙qi.\n∗The work was supported by RSF (grant 14-11-00441).\n†Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia, an d Department of\nMechanics and Mathematics, Novosibirsk State University, 630090 Novosibirsk, Russia\ne-mail: taimanov@math.nsc.ru.\n1In this article we consider the magnetic geodesic flow on the two-tor us\nT2=R2/(2πZ)2,\nwhere the coordinates xandyare defined modulo 2 π, with the flat Rieman-\nnian metric\nds2=dx2+dy2\nand with the magnetic field given by the 2-form\nF= cosxdx∧dy.\nThis form is exact:\nF=dAwithA= sinxdy.\nTherewith we\n1.completelyintegratetheflowand, inparticular, describealltrajec tories\nin terms of elliptic functions (Theorem 1);\n2.show that for all contractible periodic magnetic geodesics the redu ced\naction\nSE(γ) =/integraldisplay\nγLEdt\nis positive:\nSE>0;\n(Theorem 2);\n3.forE <1\n2find the minimizers of the action SEextended to submani-\nfoldsΣofT2as follows:\nSE(Σ,f) =√\n2Elength(∂f(Σ))+/integraldisplay\nΣf∗(F),\nwheref: Σ→T2is the embedding (Theorem 31);\n1This extension of SEfor the space of films SEwas introduced in [25, 26]. Theorem 3\ndescribes the minimal films for the case E <1\n2, i.e. when SEattains negative values.\n24.explicitly describe all contractible periodic magnetic geodesics and, in\nparticular, show that they exist only for E <1\n2and simple (not iterates\nofothers)contractible periodicmagneticgeodesics formtwo S1-families\n(Theorem 4).\nThe initial intention for writing this article was to supply the study of\nperiodic problem for magnetic geodesics with a non-trivial explicit exa mple\nof a magnetic geodesic flow with interesting properties.\nThe study of the periodic problem for magnetic geodesics was initiate d\nby Novikov in early 1980s [18, 19, 20] in the framework of qualitative study\nof certain Hamiltonian systems from classical mechanics. It appear s that an\napplication of the classical Morse theory to proving the existence o f periodic\nmagnetic geodesics meets many obstacles which can not be overgon e by clas-\nsical methods [20]. We gave a survey of that as well as of the first re sults in\nthis direction in our survey [24].\nFor dealing with these difficulties new ideas methods arising from sym-\nplectic geometry, dynamical systems and fixed points theory were proposed\n(see, for instance, [1, 5, 9, 10, 11, 12, 13, 14, 15, 16, 17, 21, 22 , 23, 25, 26]).\nOneofthemconsists inaproposaltoextendthereduced actionfu nctional\nSEfor closed curves in an oriented two-manifold to the space of films, a nd\nthentoestablish theexistence ofa minimal filmwhose boundarycomp onents\nwould be locally minimal periodic magnetic geodesics. That was done in\n[25, 26] where the existence of nontrivial periodic magnetic geodes ics was\nestablished for strong magnetic fields.\nLet us assume that a magnetic field Fis exact. Then it is easy to satisfy\nthe strength condition from [25, 26] by multiplying Fby a sufficiently large\nconstant λor by considering only low energy levels E. In fact, the ratioλ√\nEhas to be sufficiently large, or, in other words, E < E 0whereE0is some\nconstant. However if E > E 1withE1a constant, then SEis the length\nfunctional for a Finsler metric and the periodic problem can be studie d by\nthe classical Morse theory. It was established in [11] that E0=E1=C,\nthe constant Cis the Mane critical level and on this level the existence of a\nperiodic magnetic geodesic is derived from the Aubry–Mather theor y. Hence\nthe existence of a periodic magnetic geodesic was established for all energy\nlevels for exact magnetic fields on oriented closed two-manifolds.\nRecently the existence of infinitely many periodic magnetic geodesics was\nestablished for exact magnetic fields on two-manifolds for almost all subcrit-\nical energy levels, i.e. for almost all E < C[2, 3]. For other recent results\n3which clarify the periodic problem on two-manifolds we refer to [4, 6, 8 ].\nTo finish the introduction we are left to make two\nRemarks. 1) Given an exact magnetic field, if the functional SEattains\nnegativevaluesoncertainclosedcontractiblecurves, thenthewh olemanifold\nof one-point curves Mncan be overthrown into domain {SE<0}and hence\neveryk-cyclefrom Mngeneratesanontrivialcyclefrom Hk+1(P0(Mn),{SE≤\n0}) whereP0is the space of contractible closed curves in Mn(“the principle\nof throwing out cycles” ) [23, 27]. It would be interesting to understand how\nthe contractible periodic magnetic geodesics given by Theorem 4 fit in to this\npicture and how to compute H∗(P0(Mn),{SE≤0}) by using them.\n2)Theexistence ofaminimal filmprovedin[25,26]doesnotsayanyt hing\nabout the existence of contractible periodic magnetic geodesics. A priori if\nwe have a nonselfintersecting closed curve γwithSE(γ)<0, then it is not\nnecessarythatthereexistsaminimizerof SEintheclassofnonselfintersecting\nclosed curves with SE<0. Theorems 2 and 3 say that the flow in study\nsupplies an explicit example of such a situation. Actually this article aris es\nas an answer to the question by A. Abbondandolo on such an example asked\nby him after my talk in Bochum in May 2015.\n2 The flow and the first integrals\nThe motion of a charge in T2in the magnetic field Fis defined by the\nLagrangian function\nL(q,˙q) =1\n2gik˙qi˙qk−Ai˙qi=˙x2+ ˙y2\n2+sinx˙y,\nwhereq= (x,y)∈T2.\nThe corresponding Lagrangian system takes the form\n¨x= cosx˙y,\n¨y=−cosx˙x.(1)\nSince the Lagrangian function is independent on y, we have\nProposition 1 The system (1) has two functionally independent first inte-\ngrals:\n1) the energy\nE=˙x2+ ˙y2\n2,\n42) the momentum corresponding to y:\np:=py= ˙y+sinx.\nBy this Proposition, we have\n˙y=p−sinx,˙x=/radicalbig\n2E−(p−sinx)2,\nand therefore these equations are integrable in quadratures:\nt=/integraldisplayx\nx0dv/radicalbig\n2E−(p−sinv)2,\ny(t) =y0+pt−/integraldisplayt\n0sinx(τ)dτ,(2)\nwhere the constants pandEare defined by the initial data:\nx0=x(0), y0=y(0),˙x(0),˙y(0).\nTo findx(t) we have to invert the the integral in the first equation from (2.\nTo do that we make the substitution\nz= sinx\nand derive\nt=/integraldisplaysinx\nsinx0dz\nw\nwhere\nw2= (1−z2)(2E−(p−z)2). (3)\nThis elliptic curve is reduced to the Legendre normal form by the sta ndard\nprocedure (see §5) andx(t) is expressed in terms of the Jacobi elliptic func-\ntions.\nTheorem 1\nsinx(t) = sn/parenleftbiggt+D\nC,k/parenrightbigg\n,\ny(t) =y0+pt−/integraldisplayt\n0sn/parenleftbiggτ+D\nC,k/parenrightbigg\ndτ,\nwhereD=/integraltextsinx0\n0dz\nwand the constants kandDare determined by the re-\nduction of the elliptic curve (3) to the Legendre normal form (see (14) and\n(15)).\n5Remark. This flow is a simplest example of a magnetic analog of the\ngeodesic flow on a surface of revolution. The general family is given b y pairs\nconsistsing of a metric g(x)(dx2+dy2) and a two-form h′(x)dx∧dywhere\nthe functions gandhare both periodic. In this case the additional conser-\nvation law needed for complete integrability takes the form of the “C lairaut\nintegral”:\n∂L\n∂˙y=˙y\ng(x)+h(x).\nWethinkthatinthisclassonecanfindotherinterestingexamplesofm agnetic\ngeodesic flows.\n3 The variational principle for closed trajec-\ntories\nThe Lagrangian is not homogeneous in velocities and therefore the r estric-\ntions of the system on different energy levels are not trajectorica lly isomor-\nphic as in the case of the geodesic flow. In fact, the restriction of t he system\non the energy level is described by the Lagrangian\nLE=√\n2E/radicalbig\ngik˙qi˙qk+Ai˙qi=√\n2E/radicalbig\n˙x2+ ˙y2+sinx˙y.\nIndeed, the Lagrangian LEis homogeneous of first order in velocities and the\ncorresponding trajectories are defined up to parameterizations . The Euler–\nLagrange equations takes the form\nd\ndt/parenleftBigg√\n2Egik˙qk\n/radicalbig\ngik˙qi˙qk/parenrightBigg\n=∂LE\n∂qi\nand forE=1\n2gik˙qi˙qkboth left- and -right-hand sides reduces to the left- and\nright-hand sides for the Euler–Lagrange equation for L.\nThe following statement is easily checked by straightforward compu ta-\ntions\nProposition 2 The Lagrangian function LE(q,˙q)is positive for all ˙q/ne}ationslash= 0if\nand only if E >1\n2.\nIfE=1\n2, thenLEvanishes at {˙x= 0,˙y >0,sinx=−1}and{˙x= 0,˙y <\n0,sinx= 1}.\nLEtakes negative values for every E <1\n2.\n6Remark. The particular level E=1\n2may betreated asthe Mane critical\nlevel of the system. For magnetic geodesic flows the Mane critical le velC\nis defined for magnetic fields Fon Riemannian manifolds Msuch that the\npullback π∗FofFonto the universal covering /tildewiderMofMis exact. In this case\nit is defined as\nC= inf\nθsup\nMH(q,θ)\nwhereH(q,p) =1\n2|p|2is the Hamiltonian function of the system and dθ=\nπ∗F. In our case we have\nC= inf\nθsup\nR21\n2(θ2\n1+θ2\n2)\nwhereθ=θ1dx+θ2dy. Sincedθ= cosxdx∧dy, we have θ1=fx,θ2=\nfy+ sinx, wheredf=fxdx+fydyandf:R2→Ris a smooth function.\nThe restriction of fyonto the line sin x= 1 is 2π-periodic and/integraltext2π\n0fydy=\n0. Therefore on this line fyachieves its minimum at which fy= 0 and\n(fy+ sinx)2= 1 . That implies that supR2H(q,θ)≥1\n2. This lower bound\nbecomes exact at f≡0 and therefore C=1\n2.\nClosed trajectories of the flow lying on the energy level Eare extremals\nof the functional\nSE(γ) =/integraldisplay\nγLEdt\ndefined on the space of closed curves. However this functional is n ot always\nbounded from below in contrast to the length functional studied in t he clas-\nsical Morse theory and moreover for non-exact magnetic fields, i.e . when\nthe closed 2-form is not globally represented as a differential dAof a 1-form,\nthis functional is multi-valued. These differences with the Morse the y were\ndiscussed in detail by Novikov [20] (see also [24]).\nIn our case the magnetic field Fis exact however, by Proposition 2, SE\nis not bounded from below for small values of E.\nThe variational study of the periodic problem is hindered also by the f act\nthat fit is not known does SEsatisfies the Palais–Smale type conditions. A\npriori the deformation decreasing SEmay diverge even when SEapproaches\na critical level.\nIn [25, 26] for studying periodic magnetic geodesics in two-dimension al\nmanifolds we introduced the spaces of films which are embeddings f: Σ2→\nM2of oriented two-manifolds Σ2with boundaries into a two-dimensional\n7closed manifold M2endowed with a Riemannian metric and with a magnetic\nfieldFwhich is not necessarily exact. For such films\nf: Σ2→M2\nthere is defined an action functional\nSE(Σ,f) =√\n2Elength(f(∂Σ))+/integraldisplay\nΣ2f∗(F),\nwhich for exact magnetic fields reduces to SE=/integraltext\nf(∂Σ2)LEdt. It was proved\nin [25, 26] that\nif\n1.Fis exact, or Fis non-exact and/integraltext\nM2F‘>0(as we assume without\nloss of generality),\n2. there is a film with SE<0,2\nthen there exists a film Σon which SEattains its minimal value among films\nand the boundary of Σconsists of closed magnetic trajectories which are\nlocally minimal for SE.\n4 Contractible periodic magnetic geodesics\nBy (2),\ny(t) =y(0)+/integraldisplay(p−z)dz/radicalbig\n(1−z2)(2E−(p−z)2), z= sinx(t).\nTherefore for the closed trajectory γ(t) of the flow we have\n∆y=/integraldisplay\nγ˙ydt=/integraldisplay\nγ(p−z)dz/radicalbig\n(1−z2)(2E−((p−z)2)\nwhere ∆yis the increment of yalong the pullback of the trajectory onto the\nuniversal covering R2→T2. If a closed trajectory is contractible, then\n∆y= 0.\n2Such a field is called strongifFis exact or oscillating ifFis non-exact.\n8Let us compute\nSE(γ) =/integraldisplay\nγLEdt=/integraldisplay\nγ(√\n2E/radicalbig\n˙x2+ ˙y2+sinx˙y)dt=\n=/integraldisplay\nγ(√\n2E/radicalbig\n˙x2+ ˙y2+sinx˙y)dt=/integraldisplay\nγ(√\n2E/radicalbig\n˙x2+ ˙y2+(p−˙y)˙y)dt=\n=/integraldisplay\nγ(√\n2E/radicalbig\n˙x2+ ˙y2−˙y2))dt+p/integraldisplay\nγ˙ydt.\nSince√\n2E/radicalbig\n˙x2+ ˙y2= ˙x2+ ˙y2,pis constant, and/integraltext\nγ˙ydt= ∆y, we have\nSE(γ) =/integraldisplay\nγ˙x2dt+p∆y≥0. (4)\nProposition 3 Given the energy level E, ifγis a nontrivial (different from\na one-point contour) closed magnetic geodesic with ˙x≡0, then it is one of\nthe following orbits:\nγ±±=/braceleftBigg\nx(t) =±π\n2, y(t) =±√\n2Et,0≤t≤π/radicalbigg\n2\nE/bracerightBigg\n,\nand no one of these orbits is contractible.\nThe proof of this Proposition immediately follows from (1). Together\nwith (4) this Proposition implies\nTheorem 2 For a contractible periodic magnetic geodesic γwhich is differ-\nent from a one-point contour and lies on the energy level E, we have\nSE(γ) =/integraldisplay\nγ˙x2dt >0.\nLet us consider a film Π formed by the embedding of the cylinder\nΠ =/braceleftbiggπ\n2≤x≤3π\n2,0≤y≤2π/bracerightbigg\nintoT2. Its image is the closure of the domain on which F <0. The\nboundary of Π is formed by a pair of closed trajectories:\n∂Π =γ−+∪γ+−\n9and it is easy to check that\nSE(Π) =√\n2Elength(∂Π)+/integraldisplay2π\n0dy/integraldisplay3π/2\nπ/2cosxdx= 4π(√\n2E−1).\nTheorem 3 For every E <1\n2the functional SEon the space of films attains\nits minimal value on Π.\nProof. By the results of [25] mentioned in §3,SEattains its minimum\non a film Σ whose boundary components are local minima of SE.\nLet all the boundary components are contractible. This is possible if Σ\nconsists in components diffeomorphic to discs with holes. The bounda ry of\nevery hole is a contractible magnetic geodesic γand, by Theorem 2, SE>0\non such a contour. If we glue the hole by a disc, we obtain a new film /tildewideΣ with\nSE(/tildewideΣ) =SE(Σ)−SE(γ),\ni.e. decrease the value of SEwhich contradicts to the definition of Σ as a\nglobal minimum.\nHence∂Σcontainsanon-contractiblecomponent γ1and, since ∂Σrealizes\na trivial class in 1-homologies, it has to contain another non-contra ctible\ncomponent γ2. Therefore\nSE(Σ)≥length(γ1)+length( γ2)+/integraldisplay\nΨF.\nHowever the lengths of noncontractible contours are at least√\n2E2π, i.e.,\nthe length of γ+−and ofγ−+which are minimal non-contractible geodesics\nonT2. Therefore\nSE(Σ)≥4π+/integraldisplay\nΣF\nand the right-hand side achieves its minimum on the film Π. Q.E.D.\nNow let us describe all nontrivial (different from a point) contractib le\nclosed trajectories.\nBy (2), every periodic in xtrajectory is obtained by the inversion of the\nintegral /integraldisplaydz/radicalbig\n(1−z2)(2E−(p−z)2), z= sinx, (5)\n10wherezgoesalongtheboundedrealoval, i.e. thecontour, ontheRiemannia n\nsurface\nw2=P(z) = (1−z2)(2E−(p−z)2),\nwhich covers the interval Isuch that P(z)≥0 on the interval and P(z)\nvanishes at its ends. It is clear that\nI⊂[−1,1].\nMoreover it is clear that all roots of P(z) are different otherwise the integral\n(5) diverges and does not correspond to an x-periodic solution.\nOn every periodic trajectory there exist a pair of points q1,q2such that\n˙x(q1) = ˙x(q2) = 0,˙y(q1)>0,˙y(q2)<0. (6)\nThe condition ˙ x= 0 is equivalent to\n2E−(p−z)2= 0. (7)\nIfIcontains only one root of this equation then at the corresponding p oints\non anx-periodic trajectory ˙ y=p−zhas the same values and therefore there\nare no points q1andq2meeting (6). In this case the x-periodic trajectory\ndoes not close up and there is a nontrivial translation period in y. Hence for\na periodic trajectory the roots z1< z2of (7) lie inside I:\n−1< z1< z2<1.\nWe are left to check the last condition that the translation period in y\nvanishes. By (2), it is equal to\n∆y= 2/integraldisplayz2\nz1(p−z)dz/radicalbig\n(1−z2)(2E−(p−z)2)=\n= 2/integraldisplayz2\nz1(p−z)dz/radicalbig\n(1−z2)(z−z1)(z2−z), z1+z2= 2p.\nWe have\nz1=p−a, z2=p+a, a >0,\nand by substitution u=p−zwe derive that\n∆y= 2/integraldisplaya\n−audu/radicalbig\n(1−(p−u)2)(a2−u2).\n11By comparing the values of the integrand at ±u, we infer that\n∆y\n\n>0 forp <0\n<0 forp >0\n= 0 for p= 0.\nLet us summarize these facts in the following\nTheorem 4 1. ForE≥1\n2there are no contractible closed orbits.\n2. For every Esuch that 0< E <1\n2, there exist two S1-families of simple\nperiodic magnetic geodesics. These families are invariant with respect\nto translations by x:x→x+const, and obtained by the inversion of\nthe integral\nt=/integraldisplaydz/radicalbig\n(1−z2)(2E−z2),−√\n2E≤z= sinx(t)≤√\n2E,\nand by solving the equation for dynamics in y:\n˙y=−sinx.\nAll other nontrivial contractible periodic magnetic geode sics are iterates\nof these simple closed magnetic geodesics.\n3. These families lie in the domains separated by the contour s on which\nsinx=±1. In particular, no one contractible closed orbit intersect s\nthese contours.\n4. These families degenerate to the pair of contours {sinx= 0}formed by\none-point closed curves as E→0.\n5. For these simple periodic magnetic geodesics\nSE= 2/integraldisplaya\n−a/radicalbig\n2E−sin2xdx, a = arcsin√\n2E.\nStatement 4 is quite evident from the physical point of view: for ver y\nsmall energies closed orbits are trapped near critical points of the magnetic\nfield, i.e. of the function fsuch that F=fdx∧dy.\nStatement 5 is derived by straightforward computations from The orem 2.\n125 Appendix: The Legendre normal form of\nan elliptic curve and elliptic integrals\nIn this section we recall some facts on the reduction of an elliptic cur ve to\nthe Legendre normal form (for more details see, for instance, [7]) . This is\nnecessary for deriving Theorem 1.\nLet\nP(z) = (z−a1)(z−a2)(z−a3)(z−a4),\nbe a polynomial with four different real zeroes a1,...,a 4. We recall how to\ntransform the Riemann surface (elliptic curve)\nw2=P(z) (8)\nto the Legendre form\nη2= (1−ξ2)(1−k2ξ2). (9)\nWe enumerate the zeroes as follows:\na3< a1< a2< a4\nand decompose P(x) into a product P(z) =Q1(z)Q2(z) of two quadratic\npolynomials of the form\nQ1(z) = (z−a1)(z−a2), Q2(z) = (z−a3)(z−a4).\nLet us consider two cases:\n1)Q1(z) =z2−a2\n1,Q2(z) =z2−a2\n3. Then the transformation\nξ=z\na1, η=w\na2\n1a2\n3\nreduces the equation (8) to the form (9) with k2=a2\n1\na2\n3.\n2) If the case 1) does not hold, then there exist λ1andλ2such that\nQ1(z)−λ1Q2(z) = (1−λ1)(z−µ)2, Q 1(z)−λ2Q2(z) = (1−λ2)(z−ν)2.\nTheseconstants λ1,2aredeterminedastheeigenvaluesofthepairofquadratic\nforms defined by Q1andQ2, i.e., as the zeroes of the equation\ndet/parenleftbigg1−λ−a1+a2\n2+λa3+a4\n2\n−a1+a2\n2+λa3+a4\n2a1a2−λa3a4/parenrightbigg\n= 0.\n13Therewith we have\nQ1(z) =B1(z−µ)2+C1(z−ν)2, Q2(z) =B2(z−µ)2+C2(z−ν)2.(10)\nThe constants B1,B2,C1,C2are trivially computed as\nBj=Qj(ν)\n(µ−ν)2, Cj=Qj(µ)\n(µ−ν)2, j= 1,2, (11)\nand a substitution of them into the equations B1+C1=B2+C2= 1 leads\nto the following relations\n2(µν+a1a2) = (µ+ν)(a1+a2),2(µν+a3a4) = (µ+ν)(a3+a4).(12)\nSinceµandνaredifferent fromthezeroes aj,j= 1,...,4, thelatterrelations\nare rewritten as\nν−a1\nν−a2=−µ−a1\nµ−a2,ν−a3\nν−a4=−µ−a3\nµ−a4. (13)\nThe solutions µandνto (13) are obtained as the zeroes of the quadratic\nequation:\nλ2−Aλ+B= 0\nwhere, by (12), we have\nA= 2a1a2−a3a4\na1+a2−a3−a4, B=a1a2(a3+a4)−a3a4(a1+a2)\na1+a2−a3−a4.\nIt is easy to check that µandνhave to correspond to different real ovals of\n(8). This means, without loss of generality, that\na1< ν < a 2\nand\nµ < a3orµ > a4.\nBy (10), we have\nw2= (B1(z−µ)2+C1(z−ν)2)(B2(z−µ)2+C2(z−ν)2) =\n=B1(z−µ)2/parenleftbigg\n1+C1\nB1(z−ν)2\n(z−µ)2/parenrightbigg\nB2(z−µ)2/parenleftbigg\n1+C2\nB2(z−ν)2\n(z−µ)2/parenrightbigg\n=\n14=B1B2(z−µ)4(1−ξ2)(1−k2ξ2)\nfor\nξ=/radicalbigg\n−C1\nB1z−ν\nz−µ, k2=B1C2\nB2C1.\nWe are left to put\nη=w√B1B2(z−µ)2\nto reduce the curve (8) to the Legendre normal form (9). By (11 ) and (13),\nwe have/radicalbig\nB1B2=/radicalbig\nP(ν)\n(µ−ν)2>0,\nk2=/parenleftbiggν−a1\nµ−a1/parenrightbigg2/parenleftbiggµ−a3\nν−a3/parenrightbigg2\n, (14)\nλ=/radicalbigg\n−B1\nC1=/radicalBigg\n(ν−a1)(ν−a2)\n(µ−a1)(a2−µ),\n1\nλa1−ν\na1−µ=ξ(a1) =±1.\nSinceλis defined up to a sign, let us put\nλ=a1−ν\nµ−a1\nto achieve\nξ(a1) =−1.\nTherewith we have\nξ=1\nλz−ν\nz−µ=µ−a1\na1−νz−ν\nz−µ,\nξ(a2) = 1, ξ(a3) =−1\nk, ξ(a4) =1\nkwherek >0,\nand, since the real ovals {(z,w),P(z)≥0}are mapped into real ovals by the\ntransformation ( z,w)→(ξ,η), we conclude that\nk2<1.\nBy\ndξ\ndz=ν−µ\nλ(z−µ)2,\n15we have\n/integraldisplaydz\nw=C/integraldisplaydξ\nη,\n/integraldisplayzdz\nw=C/parenleftBigg\nµ/integraldisplaydξ\nη+µ−ν\nλ/integraldisplaydξ/parenleftbig\nξ−1\nλ/parenrightbig\nη/parenrightBigg\n,\nC=λ(ν−µ)/radicalbig\nP(ν)>0.(15)\nFinally, let us introduce the Jacobi function sn( t,k):\nu=/integraldisplayτ\n0dξ/radicalbig\n1−ξ2/radicalbig\n1−k2ξ2=/integraldisplayθ\n0dϕ/radicalbig\n1−k2sin2ϕ\nwhereξ= sinϕ,τ= sinθ, and we put\nτ= sn(u,k).\nThis function is periodic:\nsn(u+4K,k) = sn(u,k)\nwhere\nK=/integraldisplayπ/2\n0dϕ/radicalbig\n1−k2sin2ϕ.\nReferences\n[1] Abbondandolo, A.: Lectures on the free period Lagrangian actio n\nfunctional. J. Fixed Point Theory Appl. 13(2013), 397–430.\n[2] Abbondandolo, A., Macarini, L., and Paternain, G.P.: On the ex-\nistence of three closed magnetic geodesics for subcritical energie s.\nComment. Math. Helv. 90:1 (2015), 155–193.\n[3] Abbondandolo, A., Macarini, L., Mazucchelli, M., and Paternain,\nG.P.: Infinitely many periodic orbits of exact magnetic flows on sur-\nfaces for almost every subcritical energy level. arxiv.org:1404.764 1.\n16[4] Asselle, L., and Benedetti, G.: Infinitely many periodic orbits of\nnon-exact oscillating magnetic fields on surfaces with genus at least\ntwoforalmosteverylowenergylevel. Calc.Var.andPDE,published\nonline 17 February, 2015.\n[5] Bahri, A., andTaimanov, I.A.: Periodicorbitsinmagneticfieldsand\nRiccicurvatureofLagrangiansystems. Trans.Amer.Math.Soc .350\n(1998), 2697–2717.\n[6] Bangert, V., and Long, Y.: The existence of two closed geodesics on\nevery Finsler 2-sphere. Math. Annalen 346(2010), 335–366.\n[7] Bateman, H., and Erd´ elyi. Higher Transcendental Functions.II .\nMcGraw-Hill, 1955.\n[8] Benedetti, G., andZehmisch, K.: On theexistence of periodic orbit s\nformagneticsystems onthetwo-sphere. J.Mod.Dyn. 9(2015), 141–\n146.\n[9] Benedetti, G.: Magnetic Katok examples on the two-sphere.\narXiv:1507.05341.\n[10] Branding, V., and Hanisch, F.: Magnetic geodesics via the heat\nflow. arXiv:1411.6848.\n[11] Contreras, G., Macarini, L., and Paternain, G.P.: Periodic orbits\nfor exact magnetic flows on surfaces. Int. Math. Res. Not. 2004 , no.\n8, 361–387.\n[12] Contreras, G.: The Palais-Smale condition on contact type ener gy\nlevels for convex Lagrangian systems. Calc. Var. Partial Different ial\nEquations 27(2006), 321–395.\n[13] Frauenfelder, U., and Schlenk, F.: Hamiltonian dynamics on conve x\nsymplectic manifolds. Israel J. Math. 159(2007), 1–56.\n[14] Ginzburg, V.L.: New generalizations of Poincare’s geometric theo -\nrem. Funct. Anal. Appl. 21(1987), 100106.\n[15] Ginzburg, V.L.: On the existence and non-existence of closed tr a-\njectories for some Hamiltonian flows. Math. Z. 223(1996), 397–409.\n17[16] Ginzburg V., and G¨ urel, B.: Periodic orbits of twisted geodesic\nflows and the Weinstein-Moser theorem. Comment. Math. Helv. l84\n(2009), 865–907.\n[17] Kozlov, V.V.: Calculus of variations in the large and classical me-\nchanics. Russian Math. Surveys 40:2 (1985), 37–71.\n[18] Novikov, S.P., and Shmel’tser, I.: Periodic solutions of Kirchhoff’s\nequations for the free motion of a rigid body in a fluid and the\nextended theory of Lyusternik–Shnirel’man–Morse (LSM).I. Func t.\nAnal. Appl. 15:3 (1981), 197–207.\n[19] Novikov, S.P.: Variational methods and periodic solutions of\nKirchhoff-type equations.II. Funct. Anal. Appl. 15:4 (1981), 263–\n274.\n[20] Novikov, S.P.: The Hamiltonian formalism and a many-valued ana-\nlogue of Morse theory. Russian Math. Surveys 37:5 (1982), 1–56.\n[21] Novikov, S.P., and Taimanov, I.A.: Periodic extremals of multival-\nued or not everywhere positive functionals. Sov. Math. Dokl. 29\n(1984), 18–20.\n[22] Schneider, M.: Closed magnetic geodesics on S2. J. Differential\nGeom.87:2 (2011), 343–388.\n[23] Taimanov, I.A.: The principle of throwing out cycles in Morse-\nNovikov theory. Sov. Math. Dokl. 27(1983), 43–46.\n[24] Taimanov, I.A.: Closed extremals on two-dimensional manifolds.\nRussian Math. Surveys 47:2 (1992), 163–211.\n[25] Taimanov, I.A.: Non-self-intersecting closed extremals of multiv al-\nued or not-everywhere-positive functionals. Math. USSR-Izv. 38\n(1992), 359–374.\n[26] Taimanov, I.A.: Closed non-self-intersecting extremals of multiv al-\nued functionals. Siberian Math. J. 33(1992), 686–692.\n[27] Taimanov, I.A.: Periodic magnetic geodesics onalmost every ener gy\nlevel via variational methods. Regular and Chaotic Dynamics 15\n(2010), 598-605.\n18"    },    {        "title": "2009.06187v1.Voltage_driven_Magnetization_Switching_via_Dirac_Magnetic_Anisotropy_and_Spin__orbit_Torque_in_Topological_insulator_based_Magnetic_Heterostructures.pdf",        "content": "arXiv:2009.06187v1  [cond-mat.mtrl-sci]  14 Sep 2020Voltage-driven Magnetization Switching via Dirac Magneti c Anisotropy and Spin–orbit Torque in\nTopological-insulator-based Magnetic Heterostructures\nTakahiro Chiba1and Takashi Komine2\n1National Institute of Technology, Fukushima College, 30 Na gao,\nKamiarakawa, Taira, Iwaki, Fukushima 970-8034, Japan\n2Graduate School of Science and Engineering, Ibaraki Univer sity,\n4-12-1 Nakanarusawa, Hitachi, Ibaraki 316-8511, Japan\n(Dated: September 15, 2020)\nElectric-field control of magnetization dynamics is fundam entally and technologically important for future\nspintronic devices. Here, based on electric-field control o f both magnetic anisotropy and spin–orbit torque,\ntwo distinct methods are presented for switching the magnet ization in topological insulator (TI) /magnetic-TI\nhybrid systems. The magnetic anisotropy energy in magnetic TIs is formulated analytically as a function of the\nFermi energy, and it is confirmed that the out-of-plane magne tization is always favored for the partially occupied\nsurface band. Also proposed is a transistor-like device wit h the functionality of a nonvolatile magnetic memory\nthat uses voltage-driven writing and the (quantum) anomalo us Hall effect for readout. For the magnetization\nreversal, by using parameters of Cr-doped (Bi 1−xSbx)2Te3, the estimated source-drain current density and gate\nvoltage are of the orders of 104–105A/cm2and 0.1 V , respectively, below 20 K and the writing requires n o\nexternal magnetic field. Also discussed is the possibility o f magnetization switching by the proposed method in\nTI/ferromagnetic-insulator bilayers with the magnetic proxi mity effect.\nI. INTRODUCTION\nElectrical control of magnetism is essential for the next ge n-\neration of spintronic technologies, such as nonvolatile ma g-\nnetic memory, high-speed logic, and low-power data trans-\nmission [ 1]. In these technologies or devices, the mag-\nnetization direction of a nanomagnet is controlled by an\nelectrically driven torque rather than an external magnetic\nfield. A representative torque is the current-induced spin– orbit\ntorque (SOT) [ 2] in heavy-metal/ferromagnet heterostruc-\ntures, wherein the spin Hall e ffect in the heavy metal [ 3,4]\nand/or the Rashba–Edelstein e ffect (also known as the inverse\nspin-galvanic effect) at the interface [ 5] play crucial roles in\ngenerating the torque. Recently, several experiments have re-\nported a giant SOT e fficiency in topological insulator (TI)-\nbased magnetic heterostructures such as both TI /magnetic-TI\n[6] and TI/ferromagnetic-metal hybrid systems [ 7,8]. A TI\nhas a metallic surface state in which the spin and momen-\ntum are strongly correlated (known as spin–momentum lock-\ning) because of a strong spin–orbit interaction in the bulk\nstate [ 9,10], which is expected to lead to the giant SOT\n[11]. Indeed, magnetization reversal by SOT has been pro-\nposed theoretically [ 12–15] and demonstrated experimentally\nin magnetic TIs [ 6,16,17] as well as TI/ferromagnet bilay-\ners [18–23]. Remarkably, the critical current density required\nfor switching is of the order of 105A/cm2, which is much\nsmaller than the corresponding values (106–108A/cm2) for\nheavy-metal/ferromagnet heterostructures [ 3–5]. In particu-\nlar, the magnetization switching of magnetic TIs is more e ffi-\ncient: Yasuda et al. [17] succeeded in reducing the switching\ncurrent density by means of a current pulse injected paralle l\nto a bias magnetic field, whereas Fan et al. [16] realized mag-\nnetization reversal by means of a scanning gate voltage with a\nsmall constant current and in-plane magnetic field.\nAnother important method for controlling magnetic proper-\nties is the electric-field e ffect in magnets, such as controllingferromagnetism in dilute magnetic semiconductors [ 24], ma-\nnipulating magnetic moments in multiferroic materials [ 25],\nand changing the magnetic anisotropy in an ultrathin film of\nferromagnetic metal [ 26–28]. In particular, voltage control\nof magnetic anisotropy (VCMA) in ferromagnets promises\nenergy-efficient reversal of magnetization by means of what\nis known as voltage torque, which has been demonstrated\nby using a pulsed voltage under a constant-bias magnetic\nfield in a magnetic tunnel junction [ 29]. This approach is\nbased on the clocking scheme in which one first sets the fer-\nromagnet to an initial stable state under the application of\nan external bias and then inputs the signal voltage pulse to\ndetermine the final state. Recent experiments using heavy-\nmetal/ferromagnet/oxide heterostructures have demonstrated\nthat the critical current for SOT-driven switching of perpe n-\ndicular magnetization can be modulated by an electric field\nvia VCMA [ 30,31]. By contrast, the electric-field e ffect in\na magnetic TI [ 32,33] and a TI/ferromagnetic-insulator (FI)\nbilayer [ 34,35] has been investigated to date in terms of the\nvoltage-torque-driven magnetization dynamics. Note that Se-\nmenov et al. [34] demonstrated magnetization rotation be-\ntween the in-plane and out-of-plane directions by VCMA at\nthe TI/FI interface. Therefore, it becomes highly desirable to\ncontrol the magnetic anisotropy and SOT simultaneously by\nmeans of the electric field in TI-based magnetic heterostruc -\ntures, which may lead to magnetization switching that is mor e\nenergetically efficient.\nIn this paper, inspired by the SOT and VCMA approaches\nfor magnetization control, we combine them and present\ntwo distinct clocking methods for magnetization switching in\nTI/magnetic-TI hybrid systems. First, we model the current-\ninduced SOT and magnetic anisotropy energy (MAE) in TI-\nbased magnetic heterostructures as a function of the Fermi\nenergy to determine a stable magnetization direction at the\nelectrostatic equilibrium. Then we propose a transistor-l ike\ndevice with the functionality of a nonvolatile magnetic mem -\nory that uses (i) VCMA writing that requires no external mag-2\nnetic field and (ii) readout based on the anomalous Hall e ffect.\nFor the magnetization reversal, we estimate the source-dra in\ncurrent density and gate voltage. Finally, we show the switc h-\ning phase diagram for the input pulse width and voltages as a\nguide to realizing the proposed method of magnetization re-\nversal. We also discuss the possibility of using the propose d\nmethod for magnetization reversal in TI /FI bilayers with mag-\nnetic proximity [ 36–38] at the interface.\nII. MODEL\nWe begin this section by deriving the current-induced SOT\nin TI-based magnetic heterostructures by using the current -\nspin correspondence of two-dimensional (2D) Dirac electro ns\non the TI surface. Next, in the same system we formulate the\nMAE analytically to determine a stable magnetization direc -\ntion at the electrostatic equilibrium and to reveal the cont rolla-\nbility of the VCMA e ffect. To model the SOT and VCMA, we\nconsider 2D massless Dirac electrons on the TI surface, whic h\nis exchange coupled to the homogeneous localized moment of\na magnetic TI (or an attached FI as discussed in Sec. IV B ).\nWhen the surface electrons interact with the localized mo-\nment, they have an exchange interaction that can be modeled\nby a constant spin splitting ∆along the magnetization direc-\ntion with unit vector m=M/Ms(in which Mis the mag-\nnetization vector with the saturation magnetization Ms) [39].\nThen, the following 2D Dirac Hamiltonian provides a simple\nmodel for the electronic structure of the TI surface state:\nHk=/planckover2pi1vFˆσ·(k׈z)+∆ˆσ·m, (1)\nwhere h=2π/planckover2pi1is the Planck constant, vFis the Fermi ve-\nlocity of the Dirac electrons, ˆ σis the Pauli matrix operator\nfor the spin, and∆is the exchange interaction. For simplic-\nity, we ignore here the particle–hole asymmetry in the surfa ce\nbands. Introducing the polar angle θand azimuthal angle ϕ\nform=(cosϕsinθ,sinϕsinθ,cosθ), the energy dispersion\nof the Hamiltonian ( 1) can be expressed as\nEks=s/radicalBig\n(/planckover2pi1vFk)2+∆2−2/planckover2pi1vFk∆sinθsin(ϕ−ϕk), (2)\nwhere s=±corresponds to the upper and lower bands, and\ncosθk=∆ cosθ/|Eks|and tanϕk=ky/kxare the polar and\nazimuthal angles of the spinors on the Bloch sphere, respec-\ntively.\nA. Current–induced spin–orbit torque\nWe begin by discussing a current-induced SOT to the mag-\nnetization in TI-based magnetic heterostructures [ 12,14,15,\n40–43]. The SOT stems from the exchange interaction be-\ntween the magnetization and the electrically induced noneq ui-\nlibrium spin polarization µ(in units of m−2) [7,11], which can\nbe described by\nTSO=−γ∆µ\nMsd×m, (3)(a)\n(c)/g39 = 50 meV\n30 meV\n10 meV(d)\n/g39 = 10 meV\n30 meV\n50 meV(b)\n/g39 = 50 meV\n30 meV\n10 meV/g39 = 50 meV\n30 meV\n10 meV/g21/g39 \nFIG. 1. Current-induced nonequilibrium spin polarization ∆µ/d\nscaled by thickness dof ferromagnet as a function of EFfor different\nvalues of∆(with mz=1): (a) x-component∆µx/d; (b) y-component\n∆µy/d; (c)∆µx/dand (d)∆µy/datEF=95 meV (corresponding\ncarrier density∼1012cm−2) as functions of mzfor different values\nof∆. In these graphs, we use vF=4.0×105ms−1,d=10 nm, and\nEx=0.1 V/µm. The details of the calculations are given in the text.\nwhereγis the gyromagnetic ratio and dis the thickness of\nthe ferromagnetic layer (magnetic TI). In short, the SOT is\nobtained by calculating the electrically induced spin pola riza-\ntion on the TI surface.\nAs a characteristic feature of the Dirac Hamiltonian ( 1), the\nspin operator ˆ σis directly proportional to the velocity oper-\nator ˆv=∂Hk/(/planckover2pi1∂k)=vFˆz׈σdue to the spin-momentum\nlock. In this sense, we can identify the nonequilibrium spin\npolarization µwith the electric current Jon the TI surface,\nnamely\nµ=−1\nevFˆz×J, (4)\nwhere−e(e>0) is the electron charge. In the following,\nwe useµandJto denote the quantum statistical expectation\nvalues of ˆ σandˆj=−eˆv, respectively. Here, we empha-\nsize that the nonequilibrium spin polarization involves on ly\nin-plane spin components. Hence, the in-plane component of\nthe spin susceptibility corresponds to the electric conduc tivity\nvia Eq. ( 4). In the framework of Boltzmann transport the-\nory and the Kubo formula, previous studies [ 41,42,44–47]\nhave calculated the longitudinal and transverse (anomalou s\nHall) conductivities on magnetized TI surfaces by assuming\na short-range impurity potential with Gaussian correlatio ns\n/angbracketleftˆV(r1)ˆV(r2)/angbracketrightimp=nV2\n0δ(r1−r2) in which nis the impurity\nconcentration, V0is the scattering potential, and /angbracketleft···/angbracketright impindi-\ncates an ensemble average over randomly distributed impuri -\nties. For an electric field Ealong the TI surface, the driving3\nsheet current can be written as J=σLE+σAHˆz×Ewith [ 42]\nσL=e2\n2hEFτ\n/planckover2pi11−ξ2m2\nz\n1+3ξ2m2z,\nσAH=−4e2\nhξmz1+ξ2m2\nz/parenleftBig\n1+3ξ2m2z/parenrightBig2,(5)\nwhereτ=4/planckover2pi1(/planckover2pi1vF)2//parenleftBig\nnV2\n0EF/parenrightBig\nis the transport relaxation time\nof massless Dirac electrons within the Born approximation,\nξ= ∆/EFwith the Fermi energy EFmeasured from the\noriginal band-teaching (Dirac) point, and mzdenotes the z-\ncomponent of m. Note thatσAHin Eq. ( 5) is independent\nof the impurity parameters but diagrammatically contains t he\nside-jump and skew scattering contributions as well as the i n-\ntrinsic one associated with the Berry curvature of the surfa ce\nbands [ 44,46]. According to Eq. ( 4), the current-induced spin\npolarization for E=Exˆxis therefore\nµ=−1\nevF(−σAHE+σLˆz×E)≡µxmzˆx+µyˆy, (6)\nwhere\nµx=−4eEx\nhvF∆EF/parenleftBig\nE2\nF+∆2m2\nz/parenrightBig\n/parenleftBig\nE2\nF+3∆2m2z/parenrightBig2, (7)\nµy=−eEx\n2hvFEFτ\n/planckover2pi1E2\nF−∆2m2\nz\nE2\nF+3∆2m2z. (8)\nEquations ( 7) and ( 8) are substantially equivalent to the\ncurrent-induced nonequilibrium spin density that Ndiaye et al.\ncalculated directly by using the Kubo–Streda formula in-\nvolving the spin vertex correction [ 15]. Note thatµxorigi-\nnates from the magnetoelectric coupling (the so-called Che rn–\nSimons term) [ 12,39] that is proportional to the anomalous\nHall conductivity [see Eq. ( 6)]. Meanwhile,µystems from the\nRashba–Edelstein e ffect due to the spin-momentum locking\non the TI surface [ 40].\nFrom Eqs. ( 3) and ( 6), we finally obtain the form of SOT\narising from the TI surface [ 15,42] [see the Appendix for the\ncurrent expression of SOT], namely\nTSO=γ∆µx\nMsdmzm׈x+γ∆µy\nMsdm׈y. (9)\nThe first term contributes as a damping-like (DL) torque but\none that is quite di fferent from that of the spin Hall e ffect\nin traditional heavy-metal /ferromagnet heterostructures [ 3,4].\nIn fact, for the in-plane magnetization configuration ( mz=0),\nthis DL torque vanishes because of the absence of the mag-\nnetoelectric coupling via the anomalous Hall e ffect, whereas\nthe SOT driven by the spin Hall e ffect acts on the magneti-\nzation. Meanwhile, despite its origin, the second term acts\nas only a field-like (FL) torque. This feature is also di ffer-\nent from that of the Rashba–Edelstein e ffect in the usual 2D\nferromagnetic Rashba systems in which there might be both\nFL and DL contributions [ 49,50]. Figure 1(a) and (b) show\ntheEFdependence of the x- and y-components, respectively,of∆µ(in units of Jm−2) for different values of the surface\nband gap. For this calculation, ∆is used within the values\nreported experimentally in magnetically doped [ 51] and FI-\nattached [ 37,52] TIs. We also adopt n=1012cm−2and\nV0=0.2 keVÅ2as impurity parameters based on an analy-\nsis of the transport properties of a TI surface [ 53]. These im-\npurity parameters can reproduce the experimentally observ ed\nlongitudinal resistance ( ∼10 kΩ) in magnetic TIs. Remark-\nably, as seen in Fig. 1(a), even when the Fermi level is inside\nthe surface band gap ( EF<|∆|), the x-component survives as\n[12,41]\nµx=−eEx\n2hvF=−1\nevFσQAHEx, (10)\nwhereσQAH=e2/(2h)sgn( mz) characterizes the quantum\nanomalous Hall effect on the magnetized TI surface [ 48], re-\nflecting the topological nature of 2D massive Dirac electron s.\nBy contrast, because of the Rashba–Edelstein e ffect, the y-\ncomponent shown in Fig. 1(b) survives in only the metallic\nsurface states ( EF≥|∆|). Figure 1(c) and (d) show the mzde-\npendence of the x- and y-components, respectively, of ∆µfor\ndifferent values of the surface band gap. In these plots, we in-\nclude mzin the x-component of∆µ. Reflecting the anomalous\nHall effect on the magnetized TI surface, the x-component is\nodd upon magnetization reversal, whereas the y-component is\neven in magnetization reversal because it is proportional t oσL\nvia the Rashba–Edelstein e ffect.\nB. Dirac magnetic anisotropy\nHere, to evaluate the VCMA e ffect in TI-based magnetic\nheterostructures, we investigate the MAE associated with t he\nexchange interaction in Eq. ( 1). The MAE is defined as the\ndifference in the sums over occupied states of energy disper-\nsions ( 2) withθ=0 as the reference state [ 54], namely\nUMAE=occ./summationdisplay\nksEks(θ)−occ./summationdisplay\nksEks(θ=0). (11)\nExpanding Eq. ( 11) aroundθ≈0 leads to UMAE≈Kusin2θ,\nwhere the uniaxial magnetic anisotropy constant Ku(in units\nof Jm−2) is given by\nKu=−occ./summationdisplay\nkss(/planckover2pi1vFk)2∆2sin2(ϕ−ϕk)\n2/bracketleftBig\n(/planckover2pi1vFk)2+∆2/bracketrightBig3/2. (12)\nThe sign of Kuspecifies the type of MAE, namely perpen-\ndicular magnetic anisotropy (PMA, Ku>0) or easy-plane\nmagnetic anisotropy ( Ku<0). For the partially occupied en-\nergy bands, we have Ku>0; i.e., PMA is always favored by\nthe magnetization coupled with Dirac electrons on the TI sur -\nface. A qualitative understanding of the characteristic PM A\nis given by a gain of electronic free energy associated with\nthe exchange interaction between the Dirac electrons and lo -\ncalized moment. When the magnetization is along the out-\nof-plane direction, a surface band gap (2 ∆mz) emerges in the4\nmassless Dirac dispersion, which reduces the electron grou p\nvelocity (kinetic energy). Meanwhile, for the in-plane mag ne-\ntization orientation, the exchange interaction merely shi fts the\nsurface band in the k-space. In terms of the exchange inter-\naction maximizing the energy gain of the Dirac electron sys-\ntem, the case possessing the surface band gap is expected to\nbe more favorable with lower electronic free energy than tha t\nwith the shifted surface bands by an in-plane magnetization ,\nwhose scenario can be interpreted as being analogous to the\nPeierls transition in electron–lattice coupled systems [ 51].\nTo integrate Eq. ( 12), we assume hereinafter that the low-\nenergy Dirac Hamiltonian ( 1) is a valid description for k≤kc\nwith a momentum cut kc=/radicalbig\n∆2c−∆2/(/planckover2pi1vF) [55] in which 2∆c\nis the bulk band gap of TIs induced by the band inversion due\nto the spin–orbit interaction. We also define the Fermi wave\nvector kF=/radicalBig\nE2\nF−∆2/(/planckover2pi1vF), as shown in Fig. 2(a). Without\nloss of generality, we assume the case in which EFcrosses\nthe upper surface band, namely EF>∆. Then, we obtain the\nmagnetic anisotropy constant by integrating over the relev ant\nenergy range as\nKu=∆2\n8π(/planckover2pi1vF)2/bracketleftBigg\n∆c−EF−/parenleftBigg1\nEF−1\n∆c/parenrightBigg\n∆2/bracketrightBigg\n. (13)\nUp to the lowest order of ∆, Eq. ( 13) takes the simplest\nanalytic from of Ku= ∆2kc/(8π/planckover2pi1vF), which corresponds\nto the out-of-plane MAE for EF≤ |∆|derived earlier by\nTserkovnyak et al. [55]. Figure 2(b) shows the EFdepen-\ndence of Kufor different values of the bulk and surface band\ngaps. In this plot, ∆c=150 meV and∆c=100 meV corre-\nspond to the bulk band gaps for Bi 2−xSbxTe3−ySey(BSTS) and\n(Bi1−xSbx)2Te3(BST) [ 10], respectively. Because we assume\nthat the surface states have energy dispersions with partic le–\nhole symmetry, the magnetic anisotropy constant retains th e\nform of Eq. ( 13) in the case in which EFcrosses the lower\nsurface band, namely EF<−∆. Hence, Kuis maximum with\nthe form of Ku|EF=|∆|forEF≤|∆|and decreases apart from\nthe energy level of ±∆. The reason is that there are simply\nfewer active electrons for EF<−∆, while for EF>∆the en-\nergy decrease in the lower band is compensated partially by\nthe upper-band energy increase in conductive electrons.\nAt the end of this section, we compare the formulated\nMAE with a recent experiment employing Cr-doped BST\nthin films which is sandwiched by two di fferent dielectrics\nand hence has the Dirac electron systems on each interface\n[16]. The magnetic anisotropy field due to Eq. ( 13) is de-\nfined by BK=2Ku/(Msd). For the Cr-doped BST films\nwith 7 quintuple-layer ( d≈7 nm), a calculated result is\nBK=117 mT ( Ku/d=0.5 kJm−3) while an experiment\nreports BK≈570 mT without an electric gate in Ref. 16.\nIn this calculation, we choose the parameters for Cr-doped\nBST:∆c=100 meV ,∆ = 30 meV , vF=4.0×105ms−1,\nMs=8.5×103Am−1, and an electron/hole carrier density\n1.0/0.2×1012cm−2(corresponding to|EF|=98/51 meV) for\nthe each interface [ 16]. The difference of BKmight come from\ndisregarding the realistic particle–hole asymmetry induc ed by\nthe higher–order k-term of the energy dispersion which makes\na more sharp electronic density of states in the surface va-Eks\nkxEF\n/g21/g39 c\n/g21/g39 kFkcconduction band\nvalence band(a)(b)\n/g39 = 50 meV\n/g39 = 30 meV\n/g50/g21/g39 \nFIG. 2. (a) Schematic of massless (dashed line) and massive ( solid\nline) surface state dispersions at ky=0 in which EFdenotes the Fermi\nenergy measured from the Dirac point ( Eks=0) of the original mass-\nless surface bands, 2 ∆is the surface band gap due to an exchange\ninteraction, and 2∆cis the bulk band gap. kFandkccorrespond to the\nFermi wave vector and cuto ffwave vector, respectively. (b) Scaled\nmagnetic anisotropy energy Ku/das a function of EFfor different\nvalues of∆cand∆. Red and blue lines are for ∆c=150 meV and\n∆c=100 meV , respectively. In this plot, we use vF=4.0×105ms−1\nandd=10 nm.\nNBHOFUJD\u00015*NFUBM\nEJFMFDUSJDVGxz\ny\ndVS\nmdD\nlx\n5* \nFIG. 3. Schematic geometry (side view) of field-e ffect transistor\n(FET)-like device comprising a magnetic topological insul ator (TI)\nfilm (with thickness d) sandwiched by a nonmagnetic TI and a di-\nelectric attached to a top electric gate VGin which the Dirac electron\nsystem should appear on the top surface of the magnetic TI [ 17].dD\nis the thickness of the dielectric and lxis the length of the conduction\nchannel. VSis the voltage difference between the source and drain\nelectrodes. Current flows on the x–yplane depicted by a yellow line\nthat corresponds to the TI surface state. The arrows denote t he initial\n(red) and final (blue) magnetization directions in the magne tization\nreversal.\nlence band and enhances the hole–mediated Dirac PMA up to\nKu/d=8 kJm−3[34]. In this respect, our simple Dirac model\ncould not reflect the detail of the realistic surface band but\ncould capture the permissible magnitude of BK. Therefore,\nthe above comparison implies that the interfacial Dirac PMA\ngives a significant contribution to the magnetic anisotropy in\ndilute magnetic TIs.5\nNYN[\nNZTPVSDF\u0001\t7 4\nHBUF\u0001\t7 (\n(a) (b) (c)\n\u0012\n\u0013\u0014\u0015\nNBHOFUJD\u00015* NBHOFUJD\u00015* U4\nNYNZN[\nNYNZN[\ntmx,m y,m z\nVV\n\u0013\u0014 \u0012 \u0015\nFIG. 4. (a) Time evolution of each component of magnetizatio n with pulsed source voltage ( VS). The numerical calculation is performed with\nstatic VG=0.32 V , VS=1.5 V , and tS=1 ns. (b), (c) Corresponding magnetization switching traje ctories during duration of (b) 1 →2 and (c)\n3→4 in left upper panel. The vertical arrows denote the initial (red) and final (blue) magnetization directions in the magne tization reversal.\nThe green arrow on the surface of the magnetic TI (black cube) indicates the direction of the current-induced spin polari zation: ˆµ=µ/|µ|.\nThe spin–orbit torque (SOT) is active from 1(3) to the black s tar (⋆).\nIII. MAGNETIZATION SWITCHING\nTo demonstrate magnetization switching via SOT and\nVCMA, we propose a field-e ffect transistor (FET)-like de-\nvice with a magnetic TI film as a conduction channel layer\n[16] in which source-drain ( VS) and gate ( VG) voltages are\napplied, as shown in Fig. 3. To investigate the macroscopic\ndynamics of the magnetization in the device, we solve the\nLandau–Lifshitz–Gilbert (LLG) equation including the SOT\n(9), namely\ndm\ndt=−γm×Beff+αeffm×dm\ndt+TSO(VG), (14)\nwhere Beffis an effective magnetic field obtained by finite m\nfunctional derivatives of the total energy UM, namely Beff=\n−δUM/(Msδm), andαeffdenotes the effective Gilbert damping\nconstant. As discussed in Sec. II B, the interfacial Dirac PMA\ngives a significant contribution to the magnetic anisotropy in\nin dilute magnetic TIs. Because of the thin magnetic TI in\nFig.3, we assume that UMconsists of the MAE ( 13) and the\nmagnetostatic energy that generates a demagnetization fiel d,\ni.e.,\nUM=1\ndKu(VG)/parenleftBig\n1−m2\nz/parenrightBig\n+1\n2µ0M2\nsm2\nz, (15)\nwhereµ0is the permeability of free space. As discussed in\nSec. II, both MAE and SOT depend on the position of EF,\nwhich can be controlled electrically via (see the Appendix f or\ndetails)\nEF(VG)=/planckover2pi1vF/radicalBigg\n4π/parenleftBigg\nnint+∆2\n4π(/planckover2pi1vF)2+ǫ\nedDVG/parenrightBigg\n, (16)\nwhereǫis the permittivity of a dielectric of thickness dDand\nnintis the intrinsic carrier density at VG=0. In this study,for the dielectric layer with dD=20 nm, we adopt a typical\ninsulator, namely Al 2O3(for which the relative permittivity is\nǫ/ǫ0=9.7) [16,56].\nBecause the ferromagnetic Curie temperature ( Tc) of Cr-\ndoped BST is less than 35 K [ 57], we first consider magneti-\nzation switching at zero temperature. Influence of finite tem -\nperatures on the magnetization switching will be discussed in\nSec. IV C . The equilibrium magnetization direction with nei-\ntherVSnorVGis determined by minimizing Eq. ( 15) regard-\ning the polar angle θ. However, for simplicity, we assume that\nθ≈0◦at the electrostatic equilibrium. This assumption is per-\nmissible because hereinafter we consider a magnetic TI such\nas Cr-doped BST with a very small Ms=8.5×103Am−1\n[16], neglecting the demagnetizing field e ffect from the sec-\nond term in Eq. ( 15) at VG=0. For numerical simula-\ntion, Eq. ( 14) is solved by setting θ=1◦andϕ=0◦as\nthe initial condition for m(t=0). In the simulation, we\nchoose the parameters for Cr-doped BST as ∆c=100 meV ,\n∆= 30 meV , vF=4.0×105ms−1,d=7 nm, n=1012cm−2,\nV0=0.2 keVÅ2[53],nint≈0 cm−2(corresponding to\nEF(VG=0)=∆= 30 meV) [ 56,57],γ=1.76×1011T−1s−1,\nandαeff=0.1. Note that a large enhanced damping from 0.03\nto 0.12 due to the strong spin–orbit interaction of TIs has be en\nreported in TI/ferromagnetic-metal bilayers [ 58]. In addition,\nno modulation ofαeffis assumed during the duration of VGbe-\ncause we consider a thicker magnetic TI film ( d=7 nm) than\nthe ferromagnetic metal used in the magnetic tunnel junctio n\n[59].\nA. Switching via a source-drain current pulse JS\nWe investigate the magnetization switching via a source-\ndrain current pulse [ 17]. Under a static gate voltage [ 16], we\napply a step-like voltage pulse of width tSto the source elec-\ntrode, as shown in Fig. 4(a) (see the upper panel). The applied6\n(a) (b)\nNYN[\nNZ(c)\u0015 \u0012\n\u0013 \u0014\nNBHOFUJD\u00015* NBHOFUJD\u00015* U(\nNYNZN[\nNYNZN[\ntmx,m y,m z\nVV\n\u0013 \u0014 \u0012 \u0015TPVSDF\u0001\t7 4\nHBUF\u0001\t7 (\nFIG. 5. (a) Time evolution of each component of magnetizatio n with pulsed gate voltage VG. The numerical calculation is performed with\nVG=0.38 V , VS=0.3 V , and tG=6.7 ns. (b), (c) Corresponding magnetization switching traje ctories during (b) 1 →2 and (c) 2→3 in the\nleft upper panel. The red, blue, and green arrows and the blac k star have the same meanings as in Fig. 4.\ngate voltage can reduce the energy barrier for the magneti-\nzation reversal due to the VCMA e ffect in Eq. ( 13). The re-\nsulting time evolution of m(t) is shown in Fig. 4(a) with the\nconstant gate voltage turned on at t=0. The lower panel\nshows clearly that mzchanges its sign by the pulsed VSin-\nputs, demonstrating the out-of-plane magnetization switc h-\ning. Furthermore, applying subsequent pulses switches the\nmagnetization direction faithfully, and the change is inde pen-\ndent of the pulse’s sign. The estimated switching time be-\ntween points 1 and 2 (3 and 4) is ∼18 ns. In this simu-\nlation with static VG=0.32 V and VS=1.5 V , the mag-\nnetic anisotropy field BKand effective field due to the DL(FL)\nSOT BDL(FL)= ∆µx(y)/(Msd) are evaluated for mz=1 as\nBK=21 mT, BDL=6.0 mT, and BFL=17 mT, respec-\ntively. Note that the calculated magnetic anisotropy field i s\nBK=136 mT at VG=0. The estimated current density\n(corresponding to BDL(FL) ) is JS=1.9×105A/cm2(see\nSec. IV A for details), which is consistent with those of TI-\nbased magnetic heterostructures [ 6,17–20,22]. Figure 4(b)\nand (c) show the magnetization switching trajectories duri ng\nthe durations shown by numbers (1–4) in the left upper panel\nof Fig. 4(a). An equilibrium magnetization almost along ±ˆz\nis rotated steeply around the current-induced spin polariz ation\n(µ) by the pulsed SOT until the black star ( ⋆), after which\nBKgradually stabilizes the magnetization with oscillations\naround the easy axis.\nB. Switching via a pulsed gate voltage VG\nWe also investigate the magnetization switching via a\npulsed gate voltage. Under a constant source-drain bias, we\napply a step-like voltage pulse of width tGto the gate elec-\ntrode, as shown in Fig. 5(a) (see the upper panel). The source-\ndrain bias induces the spin polarization that takes the role of a\nconstant-bias magnetic field [ 29], while the pulsed gate volt-\nage reduces the energy barrier for the magnetization rever-\nsal via the VCMA e ffect during its duration. The resultingtime evolution is shown in Fig. 5(a) with the source-drain bias\nturned on at t=0. Figure 5(b) and (c) show the correspond-\ning magnetization switching trajectories in which an equil ib-\nrium magnetization at an initial state (1 or 3) is rotated ste eply\naround the bias effective magnetic field (proportional to µ)\nby the SOT until the black star, after which BK(=136 mT\natVG=0) aligns the magnetization direction with the easy\naxis by the Gilbert damping. As shown, the z-component\nof the magnetization changes sign by the pulsed VGinputs,\ndemonstrating the out-of-plane magnetization switching. The\nestimated switching time from the initial to final states is a l-\nmost the same as the pulse width tG∼6.7 ns. In this simu-\nlation with static VG=0.38 V and VS=0.3 V , we evaluate\nBK=3.0 mT, BDL=1.2 mT, and BFL=3.6 mT for mz=1.\nWe emphasize that the current density corresponding to the\nSOT is JS=4.1×104A/cm2, which is smaller than that of the\npulsed SOT method discussed above and those of TI-based\nmagnetic heterostructures reported to date [ 22].\nC. Switching phase diagram\nWe conclude this study with a guide for realizing the pro-\nposed magnetization switching methods. In particular, exp er-\nimenters may be interested in how the final-state solution of\nmzdepends on the input pulse width and voltages. In the fol-\nlowing plots, we use the parameters for Cr-doped BST. Fig-\nure6(a) shows the phase diagram of mzfor a source-drain\ncurrent pulse as a function of both tSandVG. The diagram\nis calculated up to VG≈0.39 V to which the Fermi level\nreaches the bottom of a bulk conduction band. The final-state\nsolution of mzoscillates rapidly depending on tSrather than\nVG, whereas the diagram has a threshold VGat the vicinity of\n0.32 V at which the SOT competes with the anisotropy field.\nIn Fig. 6(b), we also show the phase diagram of mzfor a\npulsed gate voltage as a function of both tGandVS. Clearly,\nthe final-state solution of mzoscillates depending on both tG\nandVS, whereas switching tends to succeed in the short pulse7\nN[(a)\n(b)N[\nFIG. 6. (a) Final-state diagram of mzatVS=1.5 V as function of\npulse duration time tSand gate voltage VG. (b) Final-state diagram of\nmzatVG=0.39 V as function of pulse duration time tGand source-\ndrain voltage VS.\nregion of sub-nanosecond order. Consequently, switching w ill\nbe achieved in the wide pulse duration between the nano-\nand sub-microsecond scales. In practice, the proposed de-\nvice would be mounted by combination with semiconductor\ndevices such as CMOS whereas the switching speed of VLSI\n(very large-scale integration) is not so fast at present. He nce,\ncontrol with a pulse width of a few nano-second is considered\nrealistic. Figure 6(b) shows that the controllability of the tG\npulse is better than the tSpulse because of the width of the\npulse. From the viewpoint of the speed, it can be expected\nthat the tGpulse has better compatibility with VLSI.\nIV . DISCUSSION\nA. Source-drain current and Hall voltage: FET and memory\noperations\nTo evaluate the magnitude of current density ( JS) realiz-\ning the magnetization switching in Sec. III, we calculate thesource-drain current flowing on the TI surface in Fig. 7(a). We\nalso calculate the magnitude of the output Hall voltage ( VH)\nto read out a direction of the out-of-plane magnetization by\nits sign [ 60]. According to Eq. ( 5) and Ex=VS/lx, the corre-\nsponding quantities can be written as\nJS=σL(VG)\ndVS\nlx, (17)\nVH=σAH(VG)\ndVS\nlxly, (18)\nwhich are plotted in Fig. 7(b) and (c), respectively. In these\nplots, we use the same parameters as for Cr-doped BST in\nSec. III. As a reminder, note again that nint≈0 cm−2(cor-\nresponding to EF(VG=0)= ∆ = 30 meV) is assumed\nfor the electrostatic equilibrium. Figure 7(d) shows the FET\noperation (on/off) of the proposed device. Clearly, we can\nswitch the source-drain current by a reasonable gate volt-\nage compared with modern FET devices. Therefore, com-\nbining this FET operation with the proposed magnetization-\nswitching method promises an FET with the functionality of\na nonvolatile magnetic memory [ 61]. The bit stored in this\ndevice is read out by measuring VHand determining its sign.\nSo, if EF(VG=0) is tuned within the surface band gap by the\nelement substitution [ 56], the quantum anomalous Hall e ffect\nmight allow the readout process to be free from energy dissi-\npation due to Joule heating.\nB. TI/FI bilayers\nWe discuss the possibility of magnetization switching in\nTI/FI bilayers with the magnetic proximity e ffect at the in-\nterface [ 36–38]. A device corresponding to Fig. 3is proposed\nby replacing the dielectric with an FI and the TI /magnetic-TI\nbilayer with a TI. In this case, it is easily shown that an ex-\nchange interaction between interface Dirac electrons and l o-\ncalized moments of the FI appears in the same form as that\nof the Dirac PMA given by Eq. ( 13) [34,55]. Then, the mag-\nnetization dynamics can be analyzed by using the LLG equa-\ntion ( 14) involving crystalline magnetic anisotropies ( KCMA)\nof the FI. Switching methods similar to those discussed in\nSec. IIIwill be achieved for a magnetically almost-isotropic FI\nwith small net magnetization (desirably KCMAd/Ku≪1 and\nMs/lessorsimilar105Am−1), reducing the demagnetizing field e ffect. The\npotential candidates for the FI layer are 2D van der Waals fer -\nromagnetic semiconductors [ 52,62] and rare-earth iron gar-\nnets [ 63–65] near their compensation point, where the intrin-\nsic magnetic anisotropy and magnetostatic field become infe -\nrior to those of the Dirac PMA. However, the compensation\npoints of the candidates are at a finite temperature, therefo re\nthe thermal excitation of the TI bulk state and the e ffective\nmagnetic field due to the thermal fluctuation of localized mo-\nments might affect the switching probability, which is beyond\nthe scope of the present investigation. An alternative migh t\nbe to use an insulating antiferromagnet with the A-type lay-\nered structure [ 66] that has no magnetostatic field because of\nthe tiny net magnetization. Note that a recent experiment re -8\n(b) (a)\nNBHOFUJD\u00015*lxly\nVS\nxzyJS\n(c)VH\n(d)\nPO P⒎ EF/g21/g39 74\u0001\t7\n \n   \u0012\u000f\u0016\n  \u0001\u0011\u000f\u00147(\u0001\t7\n \n  \u0011\u000f\u0011\u0011\u0012\n  \u0011\u000f\u0011\u0012\n  \u0011\u000f\u0011\u0016\n  \u0011\u000f\u0012\n  \u0011\u000f\u0013\n  \u0011\u000f\u0014\n  \u0011\u000f\u0014\u0019VJ\n7(\u0001\t7\n \n  \u0011\u000f\u0011\u0011\u0012\n  \u0011\u000f\u0011\u0012\n  \u0011\u000f\u0011\u0016\n  \u0011\u000f\u0012\n  \u0011\u000f\u0013\n  \u0011\u000f\u0014\n  \u0011\u000f\u0014\u0019\nVV\n\tN7\n \nVJ\nEF\nFIG. 7. (a) Top view of device proposed in Fig. 3(here the gate\nand dielectric layers are hidden), where lxandlyare the lengths of\nthe source-drain and Hall directions, respectively. (b) So urce-drain\ncurrent density JSof proposed device versus VSfor different values\nofVG. (c) Corresponding Hall voltage |VH|versus VSfor different\nvalues of VG. (d) Transfer characteristics obtained for di fferent VS,\nindicating the FET operation (on /off) of the proposed device. Insets\nshow the band pictures corresponding to the on /offstates. In these\nplots, we assume|mz|=1,d=7 nm, and lx=ly=10µm.\nports that the TI surface states on Bi 2Se3produce a PMA in\nthe attached soft ferrimagnet Y 3Fe5O12[67].\nHerebefore, we have focused only on the all-insulating sys-\ntems (i.e., TI and FI), whereas TI /ferromagnetic-metal(FM)\nbilayer systems are important for spintronic applications and\nexperiments. It is necessary to pay attentions for the direc t\napplication of our model to the TI /FM bilayer represented by\na pair of Bi 2Se3and Py because of the following two reasons.\nFirst, our 2D model cannot capture the three-dimensional\n(3D) nature of the transport in the TI /FM bilayer. Indeed, in\nBi2Se3/Py bilayers, most of the electric current shunt through\nthe Py layer and conductive bulk states of the Bi 2Se3layer,\nwhich reduces the portion of the current interacting with th e\nTI interface state. From this viewpoint, Fischer et al. [68]\nshow that in the FM layer spin-di ffusion transport perpendic-\nular to the interface plays a crucial role to generate the DL\ntorque. In contrast, based on a 3D tight-binding model of the\nTI/FM bilayer, Ghosh et al. [43] demonstrate that a large DL\nSOT is generated by the Berry curvature of the TI interface\nstate rather than the spin Hall e ffect of the bulk states. Sec-\nondly, orbital hybridization between the 3 dtransition metal\nand TI deforms the TI surface states, which shifts the Dirac\npoint to the lower energy and generates Rashba-like metal-\nlic bands across EF[69,70]. Besides, the hexagonal warping\neffect might be important for Bi 2Se3with a relatively large\nEFdue to its crystal symmetry. According to Li et al. [71],the Berry curvature for hexagonal warping bands involves no t\nonly out-of-plane magnetization components but also those of\nthe in-plane, which implies that the in-plane magnetizatio n\ncan contribute to the DL SOT. Note that the hexagonal warp-\ning term is important under threefold-rotational symmetry as\nthe Bi 2Se3crystal structure while it becomes small in bulk\ninsulating TIs (our focus) such as BSTS and Cr-doped BST\ndue to reduction of the symmetry by the elemental substitu-\ntion [ 72].\nC. Influence of finite temperatures\nFor the experimental probe of our proposal, one may be\ninterested in how finite temperatures a ffect the magnetization\nswitching. In our model, there are mainly three temperature\neffects: (i) temperature ( T)–dependence of physical quantities\nof TIs (σL(AH) andKu), (ii) the thermal excitation of the TI\nbulk states at finite temperatures, and (iii) a random magnet ic\nfield due to the thermal fluctuation of localized moments, po-\ntentially leading to a switching error. The cases (i) and (ii )\nattribute to electronic properties while the case (iii) is i n usual\ntreated by magnetization dynamics.\nRegarding the case (i), at low temperatures that satisfies\nkBT≪∆<EF(kBis the Boltzmann constant), we can re-\ngard the Fermi–Dirac distribution function f(E)≈Θ(E−EF)\nas the step function and then ignore T–dependence ofσL(AH)\nandKu. Indeed, when we set ∆= 30 meV and 30 meV ≤\nEF≤100 meV for the system, the above condition is satisfied\nbelow 30 K ( kBT≈2.6 meV) that is a temperature less than\nTcof magnetic TIs or FIs discussed in Sec. IV B . Regarding\nthe case (ii), based on the model of Ref. 53, we investigate the\ncontribution of the bulk states in terms of the electron dens ity\nby using\nRatio=nbdTI\nns, (19)\nwhere nbis the bulk electron density of the TI with thickness\ndTIandnsis the surface electron density. Here, we assume\nthat the bulk state of a TI thin film is a quantum well sys-\ntem along the zdirection, namely Eks,ℓ=/planckover2pi12/(2m∗)/parenleftBig\nk2\nx+k2\ny/parenrightBig\n+\nEz,ℓ(dTI)+∆ c+s∆bwith Ez,ℓ(dTI)=/planckover2pi12/(2m∗)ℓ2π2/d2, where\nℓis an integer, m∗is the effective electron mass, and ∆bis\nthe exchange energy in the bulk of magnetic TIs with spin in-\ndexs=±. The corresponding electron density is given by\nnb/s=/integraltext∞\n−∞dED b/s(E)f(E), where Db/s(E)=/summationtext\nksδ/parenleftbigE−Eb/s/parenrightbig\nis the bulk ( Eb=Eks,ℓ)/surface ( Es=Eksin Eq. ( 2)) den-\nsity of states. The computed result is shown in Fig. 8(a).\nConsequently, we find that Eq. ( 19) is less than 10 % below\n30 K for a Cr-doped BST thin film with EF=95 meV (cor-\nresponding carrier density ∼1012cm−2),∆b=30 meV , and\nm∗/m0=0.15 (m0is the free-electron mass) [ 73], which im-\nplies that the thermal excitation of the TI bulk states is neg li-\ngible below 30 K.\nRegarding the case (iii), we perform similar simulations\nincluding the random magnetic field at T=20 K with an9\nN [(a)\n(b)dTI  = 10 nm\n9\n83BUJP \u0012\u0011 \u000e\u0013 \n\u0012\u0011 \u000e\u0015 \n\u0012\u0011 \u000e\u0017 7\u0012\u0011 \nFIG. 8. (a) Ratio=nbdTI/nsas a function of Tfor different magnetic\nTI thickness dTI. (b) Final-state diagram of at T=20 K as function\nof pulse duration time tGwith VG=0.39 V and source-drain voltage\nVS. The influence of the thermal fluctuation emerges in a longer tG\nregion with lager VS.\nisotropic 3D Gaussian distribution [ 29]\nhth=/radicalBigg\n2kBTαeff\nVMsγ/parenleftBig\n1+α2\neff/parenrightBig\n∆t, (20)\nwhereV≈ lxlyd=10−18m3is the volume of the magnet,\n∆t=0.1 ps, and parameters for Cr-doped BST are used. First,\nwe investigate the e ffect of Eq. ( 20) on the pulsed JSinduced\nmagnetization switching and find that the random magnetic\nfield does not affect the switching diagram of Fig. 6(a) up\nto 20 K. We also test the case of the pulsed VGdriven mag-\nnetization switching and show the result in Fig. 8(b) for the\nextended input parameter ranges. As seen, the influence of\nEq. ( 20) emerges in the longer pulse region with lager VS, re-\nflecting randomness of the thermal fluctuation, whereas we\nconfirm that stable magnetization switching is achieved in t he\nwide parameter range. Note that the thermal stability of mag -\nnetKuV/(kBT)≈5.2×105atEF=30 meV ( VG=0) and\nT=20 K satisfies the required condition ( >60) for a non-\nvolatile memory.V . SUMMARY\nIn summary, we have presented two distinct methods for\nmagnetization switching by using electric-field control of the\nSOT and MAE in TI /magnetic-TI hybrid systems. We for-\nmulated analytically the uniaxial magnetic anisotropy in m ag-\nnetic TIs as a function of the Fermi energy and showed that the\nout-of-plane magnetization is always favored for the parti ally\noccupied surface band. We further proposed a transistor-li ke\ndevice with the functionality of a nonvolatile magnetic mem -\nory adopting (i) the VCMA writing method that requires no\nexternal magnetic field and (ii) read-out based on the anoma-\nlous Hall effect. For the magnetization reversal, by using pa-\nrameters of Cr-doped BST, the estimated source-drain curre nt\ndensity and gate voltage were of the orders of 104–105A/cm2\nand 0.1 V , respectively, below 20 K. As a conclusion of this\nstudy, we showed the switching phase diagram for the input\npulse width and voltages as a guide for realizing the propose d\nmagnetization-reversal method. We also discussed the pos-\nsibility of magnetization switching by the proposed method\nin TI/FI bilayers with the magnetic proximity e ffect. Simi-\nlar magnetization switching may be achieved by the FI layer\nwith 2D van der Waals ferromagnetic semiconductors or rare-\nearth iron garnets near their compensation point. However,\nthe compensation points of the FIs are at a finite temperature ,\nso the thermal excitation of the TI bulk state and the e ffective\nmagnetic field due to the thermal fluctuation of localized mo-\nments might affect the switching probability, which is beyond\nthe scope of the present investigation. Simultaneous contr ol\nof the magnetic anisotropy and SOT by an electric gate may\nlead to low-power memory and logic devices utilizing TIs.\nVI. ACKNOWLEDGMENTS\nThe authors thank Yohei Kota, Koji Kobayashi, Seiji Mi-\ntani, Jun’ichi Ieda, and Alejandro O. Leon for valuable dis-\ncussions. This work was supported by Grants-in-Aid for Sci-\nentific Research (Grant No. 20K15163 and No. 20H02196)\nfrom the JSPS.\nAppendix A: Current-expression of SOT\nWe rewrite the SOT in terms of a current density flow-\ning on the magnetized TI surface. According to Eq. ( 6) in\nthe main text, the current-induced spin polarization invol ving\nJS=σLE/d=(σL/d)(VS/lx)ˆx(in units of Am−2) is given by\nµ=−d\nevF(−θAHJS+ˆz×JS), (A1)\nwhere\nθAH≡σAH\nσL=8/planckover2pi1\nEFτ∆mzEF/parenleftBig\nE2\nF+∆2m2\nz/parenrightBig\n/parenleftBig\nE2\nF−∆2m2z/parenrightBig/parenleftBig\nE2\nF+3∆2m2z/parenrightBig (A2)10\nis an anomalous Hall angle. Inserting Eq. ( A1) into Eq. ( 3) in\nthe main text, we obstinate the current-expression of SOT\nTSO=γ∆\nevFMsm×θAHJS−γ∆\nevFMsm×(ˆz×JS). (A3)\nRecalling that the first term is responsible for the DL-torqu e\nassociated with a magnetoelectric coupling, one may expect\nthat a giant current-induced SOT is obtained by Eq. ( A2) in\nthe case of EF≈∆mz. However, JSthen becomes nearly\nzero because the relation between θAHandJSis tradeoff[see\nEq. ( 5) in the main text], and therefore in Eq. 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Jalil, 2 Takashi Fujita 1,2  \n1Data Storage Institute, Agency for Science, Technol ogy and Research (A*STAR) \nDSI Building, 5 Engineering Drive 1, Singapore 1176 08 \n \n2Information Storage Materials Laboratory, Electrica l and Computer Engineering Department, National \nUniversity of Singapore, 4 Engineering Drive 3, Sin gapore 117576 \n \nAbstracts    \nA unified theoretical treatment is presented to des cribe the physics of electron dynamics in \nsemiconductor and graphene systems. Electron spin’s  fast alignment with the Zeeman magnetic field \n(physical or effective) is treated as a form of adi abatic spin evolution which necessarily generates a  \nmonopole in magnetic space. One could transform thi s monopole into the physical and intuitive \ntopological magnetic fields in the useful momentum (K) or real spaces (R).  The physics of electron \ndynamics related to spin Hall, torque, oscillations  and other technologically useful spinor effects ca n be \ninferred from the topological magnetic fields in sp intronic, graphene and other SU(2) systems.  \n \nPACS: 03.65.Vf, 73.63.-b, 73.43.-f \nContact:  \nDr  S. G. Tan  \nDivision of Spintronic, Media and Interface \nData Storage Institute  \n(Agency for Science and Technology Research) \nDSI Building  5 Engineering Drive 1 \n(Off Kent Ridge Crescent, NUS) \nSingapore 117608 \nDID: 65-6874 8410 \nMobile:  65-9446 8752 \n 2 \n INTRODUCTION \n    Spintronics 1,2 is an interesting area of research that straddles the border between fundamental physics and \ntechnology, offering an almost unique opportunity t o translate physics into real applications. One exa mple is the \ngiant magnetoresistance 3,4  (GMR) effects which have found applications in the  form of a multilayer spin valve 5,6  \nrecording heads for reading magnetic data stored on  the magnetic media. Another phenomenon, the spin t ransfer \ntorque 7-9, is currently under intense investigation for curr ent-induced magnetization switching, noise control in spin \nvalve, and sustained spin torque oscillations 10 .  Micromagnetic studies of magnetization configura tions have improved \nthe design of magnetic media and read-heads for rec ording purposes.  Modern interest in micromagnetics  consider \nthe dynamics of both itinerant and local electron s pin, providing new insights into anomalous Hall, as  well as spin \ntransfer with respect to the itinerant, local spin dynamics, respectively. Of interest recently is the  spin orbital (SO) \neffect 11-14, particularly the Rashba and the Dresselhaus in se miconductor materials. The SO effect has direct \nimplication to the spin and the momentum dynamics o f electrons, leading to recent interest that spans fundamental, \ndevice and engineering physics, and subjects that r ange from spin Hall 15-17  to spin current 18  transistor.  In fact, SO \neffect is highly relevant to spintronics, ranging f rom the well-known anisotropic magnetoresistance, t he anisotropy \nenergy of local moment density, the keenly studied spin Hall and spin current in semiconductor spintro nics, to more \nsubtle implications like spin torque, spin dynamics , spin oscillations, and Zitterbewegung. \nA SO system can be viewed as one which provides an effective Zeeman magnetic (b) field which varies in  the \nmomentum (K) space. As such, one can draw an analog y between a SO system with a locally varying b fiel d system, \nin which a conduction electron experiences the vary ing b field in real space (R).  In the event that t he electron spin \nevolves and aligns adiabatically to the Zeeman b fi eld in their respective K or R spaces, under the th eoretical \nframework of gauge and symmetry 19,20 , the two systems will be analogous in that electro n spin evolving adiabatically \nin both systems “see” a Dirac monopole in the Zeema n field space (B).  Since monopole in B space has n o direct \nbearing on the spin or orbital dynamics of electron s, it would be essential to transform the B space m onopole to \nsome topological magnetic fields (curvature) in the  more useful space of K or R under which the equati on of \nmotion 15,21  can be constructed to describe the electron’s orbi tal dynamics. A similar SU (2) system which resembl es \nthe SO is the special carbon system of monolayer an d bilayer graphene.  But the spinor of these system s does not \nrepresent the spin state of conducting particle in the carbon system.  Instead, the spinor describes t he pseudo-spin \nwhich consists of a linear combination of waves due  to different sub-lattice sites. This article would  be devoted to \ndiscussing the physics of monopole fields 22  originating from spinor dynamics in spintronics an d graphene.  The 3 \n monopole field in these so-called SU(2) systems can  be viewed as a mathematical object which can lead to \ninstructive description of the electron’s orbital d ynamics 23,24  or motion.  Here we will present a thorough descri ption \nof the Dirac gauge potential arising from spinor dy namics (fast alignment with b fields) in the strong  Zeeman field \n(adiabatic) limit.  The strong Zeeman effect has di rect relevance to both the SO or graphene systems a nd the local \nmagnetic system; the former would relate to the tra nsformed topological magnetic fields in K space, th e latter to R \nspace. In local micromagnetic systems which have be en studied intensively in the magnetic media for ha rd disk \ndrives, or domain wall spintronics, one needs to in vestigate the topological magnetic field in real sp aces which will \nnot be discussed in this article.  \n \nTHEORY  \n    When an electron propagates in the SU(2) system , its spin precesses about the effective b field. T his mechanism \nhas been studied in great details in spintronics wh ere precession of spin due to the Rashba or the Dre sselhaus effects \nleads to spin current 25,26   when a finite dimension (boundary condition) is i mposed on the system.  But here we will \nconsider a system where the effective Zeeman b fiel d is infinitely strong, such that in this limit ele ctron spin relaxes \nto the field. The alignment of the electron spin to  the local field means that the electron assumes th e low-energy spin \neigenstate of the system, with no admixture from th e other spin eigenstate. Such system is known as ad iabatic \nwhere spin is constantly aligned to the local field  and there is no probability of the spin assuming i ts other eigenstate.  \nOne can now apply a continuous unitary transformati on to the Hamiltonian such that the spin reference axis (z) in \nthe rotated frame coincides with the local b field direction. The Hamiltonian of such system, when rea d in this rotated \nframe reveals a momentum term that appears modified . In the language of symmetry, the Hamiltonian has been \ntransformed and corresponding transformation of the  wave-function would be required to ensure the inva riance of \nthe Lagrangian.   \n \nI.  Adiabatic Gauge and Path Integral (Monopole in B sp ace) \nAs is well known, the adiabatic change of an eigens tate (eg. spin state) in parameter space (eg. b fie ld) gives rise to \na geometric phase known as the Berry’s phase 27,28 , which by requirement of symmetry transforms the H amiltonian \nby simply modifying the momenta with a set of gauge  potentials which are Abelian by nature; the genera l expression \nfor the Dirac gauge potential is:   4 \n /g1827/g4634/g3015/g3404/g1328\n/uni0032/g1857/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016/g3015/g4667/g2034/g1486/g3015\n/g2034/g1866 /g3556 \n                                                                (1) \nwhere N indicates the type of space, while n is the  parameter in N space.  In the above, the eigenstat e at any point \nin time has the same angular orientation /g4666/g2016/uni002C/g1486/g4667 as those of the parameters.  In the event where th e parameter is \noriented differently than the evolving state, e.g. spin is perpendicular to the momentum in the SO sys tem at any one \ntime, transformation would be required to find the gauge field in the momentum space.  Explicitly, Eq. ( 1) is /g1827/g4634/g3015/g3404\n/g1328\n/g2870/g3032/g4672/uni0031/g3398/g3041/g3301\n/g3041/g4673/g3436/g3105/g1486\n/g3105/g3041/g3299/uni002C/g3105/g1486\n/g3105/g3041/g3300/uni002C/uni0030/g3440.  To understand the acquisition of geometric phase  in an adiabatic spinor system, we will first \nexamine the evolution of an eigenstate from initial  to final state as described by the path integral o f spatial \npropagators as below  \n/g2032/g4666/g1876/g3041/g2878/g2869/uni002C/g1872/g3041/g2878/g2869/g4667/g3404 /g3505/g1833/g4666/g1876/g3041/g2878/g2869/g1872/g3041/g2878/g2869/uni002C/g1876/g2868/g1872/g2868/g4667/uni0020/g2032/g4666/g1876/g2868/uni002C/g1872/g2868/g4667/g1856/g1876/g2868 \n                                                   (2) \nwhere /g1833/g4666/g1876/g3041/g2878/g2869/g1872/g3041/g2878/g2869/uni002C/g1876/g2868/g1872/g2868/g4667 is the propagator between times t 0 and t.  The propagator in explicit spatial terms i s \n/g1833/g4666/g1876/g3041/g2878/g2869/g1872/g3041/g2878/g2869/uni002C/g1876/g2868/g1872/g2868/g4667/g3404 /g3505/g1766/g1876/g3041/g2878/g2869/uni007C/g1847/g4666/g1872/g3041/g2878/g2869/g1872/g3041/g4667/uni007C/g1876/g3041/g1767/g1766/g1876/g3041/uni007C/g1847/g4666/g1872/g3041/g1872/g3041/g2879/g2869/g4667/uni007C/g1876/g3041/g2879/g2869/g1767/uni2026/uni002E/g1766/g1876/g2869/uni007C/g1847/g4666/g1872/g1872/g2868/g4667/uni007C/g1876/g2868/g1767/uni0020/uni0020/g1856/g1876/g2869/g1856/g1876/g2870/uni2026/uni002E/g1856/g1876/g3041 \n                        (3) \nwhere /g1766/g1876/g3041/uni007C/g1847/g4666/g1872/g3041/g1872/g3041/g2879/g2869/g4667/uni007C/g1876/g3041/g2879/g2869/g1767/g3404/g3495/g3040\n/g2870/g3095/g3036/g1328/g2940/g3047/g1857/g3284/g3288\n/g3118/g1328/g4672/g3299/g3289/g3127/g3299/g3289/g3127/g3117\n/g3188/g3295/g4673/g3118\n/g2940/g3047/uni002E/g1857/g2879/g3284/g3271 \n/g1328/g2940/g3047/g1568/g3495/g3040\n/g2870/g3095/g3036/g1328/g2940/g3047/g2011/g3041/g2879/g2869 is the propagator between two spatial points. \nSubstituting this into Eq. (3) yields /g2032/g4666/g1876/g3041/g2878/g2869/uni002C/g1872/g3041/g2878/g2869/g4667/g3404/g1516/g4672/g3040\n/g2870/g3095/g3036/g1328/g2940/g3047/g4673/g3289/g3126/g3117\n/g3118/g4670/uni0020/g2011/g3041/uni2026/uni002E/uni002E/g2011/g2868/uni0020/g1856/g1876/g3041/uni2026/g1856/g1876/g2869/g4671/uni0020/g2032/g4666/g1876/g2868/g1872/g2868/g4667/uni0020/g1856/g1876/g2868, which can be \nexpressed in  the simple form of: \n/g2032/g4666/g1876/g3041/g2878/g2869/uni002C/g1872/g3041/g2878/g2869/g4667/g3404 /g3505/g3429/g4672/g1865\n/uni0032/g2024/g1861/g1328/uni0394/g1872/g4673/g3041/g2878/g2869\n/g2870/g1857/g3036/g3020 /g4666/g3047/g4667\n/g1328/uni0020/uni0020/uni0020/g1856/g1876/g3041/uni2026/g1856/g1876/g2869/g1856/g1876/g2868/g3433/uni0020/g2032/g4666/g1876/g2868/g1872/g2868/g4667 \n                                                   (4) \nwhere /g2011/g3041/uni2026/uni002E/uni002E/g2011/g2868/g3404 /g1857/g3284/g3268 /g4666/g3295/g4667\n/g1328/uni0020/uni0020/uni0020and /g1845/g4666/g1872/g4667/g3404/g2919\n/g1328/g1516/g3040\n/g2870/g4672/g3031/g3051 \n/g2914/g2930 /g4673/g2870\n/g3398/g1848/g4666/g1876/g3041/g4667/uni0020/g1856/g1872 /g2904\n/g2868  would be the action of the system. Note that /g4672/g3040\n/g2870/g3095/g3036/g1328/g2940/g3047/g4673/g3289/g3126/g3117\n/g3118  \nhas dimension of /g4672/g2869\n/g3039/g4673/g3041/g2878/g2869\n which cancels that due to the volume element /g1856/g1876/g3041/uni2026/g1856/g1876/g2869/g1856/g1876/g2868.  In a dynamic spinor system 5 \n which evolves with the changing b fields, the infin itesimal propagator corresponding \nto /g3495/g3040\n/g2870/g3095/g3036/g1328/g2940/g3047/g2011/g3041/g3404/g1766/g1876/g3041/g2878/g2869/uni007C/g1847/g4666/g1872/g3041/g2878/g2869/g1872/g3041/g4667/uni007C/g1876/g3041/g1767 is \n/g1829/g2011/g3041/g3404 /g3452/g1878/g3041/g2878/g2869/g3628/g1857/g2879/g3036\n/g1328/g3083/g3047/uni0020/g3091/uni0020/g3029/g3560/uni002E/g3097 /g3557/g3628/g1878/g3041/g3456 /g3404 /g3399/g1766/g1878/g3041/g2878/g2869/uni007C/g1878/g3041/g1767/g1857/g2879/g3036\n/g1328/g3083/g3047/uni0020/g3091/uni0020/g3029/g3404 /g3399/g1857/g2879/g3083/g3047/uni0020/g3053/g3126/g3105/g3295/g3053/uni0020/g1857/g2879/g3036\n/g1328/g3083/g3047/uni0020/g3091/g3029  \n                               (5) \nwhere /g1866 and /g1866/g3397/uni0031 corresponds to interval /g1872 and /g1872/g3397/g2012/g1872 , respectively; /g1854/g3560/g3404 /g1854/g1866 /g3548  and /g1866 /g3548  is the unit vector of the b field. \nIn the above, use has been made of the approximatio n /g1766/g1878/g3041/g2878/g2869/uni007C/g1878/g3041/g1767/g3404 /uni0031/g3398/g1878/g3041/g2878/g2869/g2878/g4666/g1878/g3041/g2878/g2869/g3398/g1878/g3041/g4667/g3406 /uni0031/g3398/g1878/g3041/g2878/g2869/g2878/g2034/g3047/g1878/g3041/g2012/g1872 /g3406\n/g1857/g2879/g3036/g3083/g3047/uni0020/g3053 /g3289/g3126/g3105/g3295/g3053/g3289 .  Propagation from /g1/uni007C/g1878/g3041/g1732 to /g1/uni007C/g1878/g3041/g2878/g2869/g1732 depends on whether /g1/uni007C/g1878/g3041/g1732 is parallel or anti-parallel to the b field.  \nLetting  /g1866 /g3548/uni002E/g2026 /g3556 /g3404 /g2026 /g3041 , one could deduce that /g2026/g3041/uni007C/g1/g1878/g3041/g1732/g3404 /g3399/g1/uni007C/g1/g1878/g3041/g1732/g1 for /uni007C/g1/g1878/g3041/g1732/g1 parallel / anti-parallel to /g1854/uni0020/g1866 /g3548, note that the direction \nof spin state /uni007C/g1/g1878/g3041/g1732/g1 is /g1766/g1878/g3041/uni007C/uni0020/g2026 /g3556/uni0020/uni007C/g1878/g3041/g1767/g3404 /g1866 /g3548 . It can be deduced simply by inspection that the action of the system would be \n/g1845/g4666/g1872/g4667/g3404 /g3398/g2020/g1828/g1846/g3397/g1861/g1328 /g1516/g1878/g3041/g2878/g3021\n/g2868/g4666/g1872/g4667/uni0020/g2034/g3047/g1878/g3041/g4666/g1872/g4667/uni0020/g1856/g1872 .  Neglecting the dynamic phase of –/g2020/g1828/g1846  and expanding the action leads to:      \n/g1845 /g3404 /g3399/g1328\n/uni0032/g3505/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016 /g4667/g2034/g1486\n/g2034/g1872 /g3047\n/g2868/uni0020/g1856/g1872 /g3404 /g3399/g1328\n/uni0032/g3505/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016 /g4667/uni0020/g2034/g1486\n/g2034/g1854/g3560/uni002E/g1856/g1854/g3560/g3041\n/g2868 \n                                                     (6)   \nwhere /g3399 corresponds to the parallel / anti-parallel case, respectively.  Here, /g4666/g2016/uni002C/g1486/g4667 is understood to be /g4666/g2016/g3003/uni002C/g1486/g3003/g4667/uni002E/uni0020  The \nabove is the Berry’s phase which can be associated with a gauge field in B space for the system under consideration.  \nThe term /g1328\n/g2870/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016 /g4667/uni0020/g3105/g1486\n/g3105/g3029/g3560 is a gauge potential defined on the S 2 manifold as a regular expression except at /g2016 /g3404 /g2024  .  \nThus the curvature of this term does not represent a regular quantity defined everywhere on a S 2 manifold which is \nparametrized by /g4666/g1870/uni002C/g2016/uni002C/g1486/g4667; in fact it has a singularity on the –z axis (see Appendix).  This problem was resolved by Wu \nand Yang 29,30  by conceiving that the gauge over the S 2 should be represented by at least two different ex pressions, \neach expression covers any part of the manifold but  avoids its own singularity.  The overlap between t he two \nexpressions is the area where transition from one e xpression to the other must be carried out.  In thi s manner, a \n/g1519/uni0020/g3435/g1487/g3561/g3003/g3400/g1827/g4634/g3003/g3439/uni002E/g1856/g1845/g4634 which relies on two cross sections of the gauge po tential functions will be non-vanishing, indeed it \ncan be proven to be a monopole field. In summary, i t has been shown that the path integral of the spin or dynamic in \nB space under adiabatic approximation has thus gene rated a Dirac potential and consequently a Dirac mo nopole field \nwhich can recently be treated as the more generaliz ed Wu-Yang monopole. But the monopole field obtaine d in the B \nspace is not instructive with respect to interpreti ng its physical effect on the dynamics of spin or c harged particle.  \nTransformation of the gauge potential in B space to  the more relevant K or R space will thus be crucia l for providing 6 \n a more insightful understanding. We will now focus on /g3397/g1328\n/g2870/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016 /g4667/uni0020/g3105/g1486\n/g3105/g3029/g3560 which arises due to the spin assuming only \nthe lower energy eigenstate of the magnetic fields,  and discuss the transformation of this gauge poten tial from one \nspace to another.  The gauge field in N space can b e converted to an arbitrary L space by  \n/g1827/g3091/g3013/g3404/g1328\n/uni0032/g1857/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016 /g4667/g2034/g1486\n/g2034/g1866 /g3556/uni002E/g2034/g1866 /g3556\n/g2034/g1864/g3091/g1568 /g1827/g4634/g3015/uni002E/g2034/g1866 /g3556\n/g2034/g1864/g3091 \n                                                                      (8) \nWe have shown that Eq. (8) is a result of adiabatic  evolution of a spin up particle tracking the b fie ld. Alternatively, \none could use the rotation matrix to perform a loca l transformation of the Hamiltonian, resulting in /uni0031 /uni0032/g1865/uni2044/g3435/g1868/g3091/g3398\n/g1861/g1328/g1847/g2034/g3091/g1847/g2878/g4667/g2870 .  It is then worth noting that /g1827/g4634/g3015/uni002E/g3105/g3041/g3556\n/g3105/g3039/g3339  constitutes the top left diagonal term of /g1847/g2034/g3091/g1847/g2878 where U is the \nunitary matrix which rotates the laboratory axis to  the b field where spin is aligned to. The above de scriptions can \nalso be summarized by /g1827/g4634/g3015/uni002E/g3105/g3041/g3556\n/g3105/g3039/g3339/g3404 /g1856/g1861/g1853/g1859/g3427/g1847/g2034 /g3091/g1847/g2878/g3431 /g3404 /g1827/g3040/g3042/g3041 /g3049/g2034/g3091/g1866/g3049, where it is also common to write /g1827/g3040/g3042/g3041 /g3049/g3404 /g1827/g4666/g2016/g4667/g2034/g1486 /g2034/g1866/g3049/uni2044 , \nwhich is none other than the monopole field in N sp ace.  In fact, it is legitimate to ask if the Dirac  gauge potential \nabove has any physical implication to a particle wh ich experiences its presence because such Dirac pot ential is not \nwell-defined ( /g2016 /g3404 /g2024  ) everywhere.  Its surface integral is vanishing a s a consequence.  This suggests that the Dirac \npotential derived through the path integral approac h is mathematically inadequate to represent the mon opole field.  \nTo make a case for the existence of the monopole fi eld, the Dirac string /uni0020\n/g2012/g4666/g1876/g4667/g2012/g4666/g1877/g4667/g2016/g4666/g3398/g1878/g4667  must be avoided.  It becomes clear that negotiatin g the Dirac string and covering the manifold \ncompletely is required to obtain a curvature field which resembles a physical magnetic field.  In appl ied physics, one \ncan thus view the Wu Yang’s treatment of the monopo le field as having reasonably affirmed that the mon opole field \nhere can be regarded as a physical magnetic field w hich influences the dynamics of charged particle in  the same way \nthat physical magnetic field does.  One can by now reason that adiabatic spinor evolution gives rise t o a monopole \nfield, which might have significant physical implic ations to particles which experience its presence.  Since we know \nthat the Dirac string can be negotiated, we will no t show the Dirac string in the derivation of the mo nopole field \n(Dirac string will be covered in the Appendix). The  curvature of the gauge potential in B space, which  according to \nthe electrodynamics tensor is   \n/uni2126/g3089/g3003/g3404/g2034\n/g2034/g1854/g3091/g1827/g3092/g3003/g2013/g3091/g3092/g3089  \n                                                      (9) 7 \n where κνµ,, are the 3 spatial dimensions which sum over double index by Einstein’s convention. Noting the explicit \nform of /uni0020/g1827/g4634/g3003, and setting /g1328/2e  to 1, one obtains    \n/uni2126/g3089/g3003/g3404 /g3437/g3436/uni0031/g3398/g1854/g3053\n/g1854/g3440/g2034/g2870/g1486\n/g2034/g1854/g3091/g2034/g1854/g3092/g3398/g4678/g2034\n/g2034/g1854/g3091/g1854/g3053\n/g1854/g4679/g2034/g1486\n/g2034/g1854/g3092/g3441/g2013/g3091/g3092/g3089 /uni0020 \n                               (10) \nFor clarity, the z component of the curvature can b e deduced directly from Eq. (9) to be  \n/uni2126/g3053/g3003/g3404 /g3436/uni0031/g3398/g1854/g3053\n/g1854/g3440/g4680/g2034\n/g2034/g1854/g3051/uni002C/g2034\n/g2034/g1854/g3052/g4681/g1486/g3398/g3436/g2034\n/g2034/g1854/g3051/g1854/g3053\n/g1854/g3440/g2034/g1486\n/g2034/g1854/g3052/g3397/g4678/g2034\n/g2034/g1854/g3052/g1854/g3053\n/g1854/g4679/g2034/g1486\n/g2034/g1854/g3051 \n                                           (11) \nThe monopole field has been expressed explicitly as  a function of the Zeeman b field.  In SU(2) system , it is normally \nstraightforward to deduce an effective Zeeman b fie ld from the Hamiltonian.  The Zeeman b field would be related to \nthe energy which forms the generator of time transl ation for the spinor part of the wavefunction.  To derive the \ncurvature in B space explicitly, we note that bbz/ cos =θ , x ybb/ tan =φ ; restoring /g1328/2e , the curvature reveals their \nmonopole signatures:  \n/uni2126/g3051/g3003/g3404/g1328\n/uni0032/g1857/g1854/g3051\n/g1854/g2871/uni002C /uni0020/uni0020/uni0020/uni2126 /g3052/g3003/g3404/g1328\n/uni0032/g1857/g1854/g3052\n/g1854/g2871/uni002C/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/uni2126/g3053/g3003/g3404/g1328\n/uni0032/g1857/g1854/g3053\n/g1854/g2871 \n                                                      (12) \nNote that in the above expressions, the Dirac strin g has been deliberately ignored; more discussion of  the Dirac \nstring will be given in the Appendix.  One could ea sily check that had the above derivation been carri ed out in real \nspace, i.e. replacing b with r, expressions of Eq. ( 12) would have the dimension of the magnetic fields .  However, in \napplication this is only possible for the specific case of spin aligning with r, i.e. spin has the sam e orientation as the r \ncoordinates with respect to some spherical center.  One way of achieving this is by means of technolog y, create \nmagnetic fields or local magnetic moments with orie ntations in B space that overlaps the r coordinates  in R space \nwith respect to spherical centers in the resepectiv e spaces.  We will not discuss this in details here , the focus of this \npaper will be the spin orbital or the graphene syst em where topological magnetic field (curvature) cou ld exist \nnaturally in the K space.  \n 8 \n II.  Topological Magnetic Fields in K (reciprocal) space  in various SU(2) systems   \nIn the above, we have shown the path integral deriv ation of the gauge potential and its monopole in th e B space, as \na result of adiabatic spin alignment with the b fie lds.  But the monopole curvature in b field is not useful for \nelucidating even heuristically the orbital dynamics  of a particle which “sees” these fields.  In natur e, there exist many \nsystems that provide the mimic of Zeeman b fields, and these fields normally depend on the momentum (k ).  It is of \ninterest indeed to find the topological magnetic fi eld in the K space which would be more instructive for deriving the \nequation of motion 15,21 , hence for the elucidation of orbital dynamics of particles which “see” these fields.  The SO \nsystems, which have long been known to exist in ato mic physics (hyperfine interaction), semiconductors , metals, are \nthe most conspicuous systems which provide such k-d ependent Zeeman fields.  SO effect manifests in the  \nbandstructure, lifting degeneracy of valence electr ons at momentum other than zero. Other b(k) systems  include the \ncarbon based systems eg. monolayer and bilayer grap hene, and superconducting systems. The formal descr iption of \nSO as well as graphene pseudo spin effect could be derived using the Dirac’s formalism which suggests that these \neffects mimic the vacuum SO effect in the non-relat ivistic limit 28,29; Dirac matrix equations are shown below \n                /g4684/g4672/g1322\n/g3030/g3398/g1857/g1486\n/g3030/g4673/g3398/g1865/g1855 /g3398/g2026 /g3556/uni002E/g1868 /g3556\n/g2026 /g3556/uni002E/g1868 /g3556 /g3398/g4672/g1322\n/g3030/g3398/g1857/g1486\n/g3030/g4673/g3398/g1865/g1855 /g4685/uni002E/g3436/g2032\n/g2031/g3440 /g3404 /uni0030                                   (13)  \n              \n/g4678/g4666/g1322/g3398/g1857/g2038 /g4667/g2870/g3398/g1855/g2870/g1868/g2870/g3398/g1865/g2870/g1855/g2872/g3398/g1861/g1857/g1328/g1855/uni0020/g2026 /g3557/uni002E/g1831 /g3560\n/g3398/g1861/g1857/g1328/g1855/uni0020/g2026 /g3557/uni002E/g1831 /g3560 /g4666/g1322/g3398/g1857/g2038 /g4667/g2870/g3398/g1855/g2870/g1868/g2870/g3398/g1865/g2870/g1855/g2872/g4679/uni002E/g3436/g2032\n/g2031/g3440 /g3404 /uni0030 \n             (14) \nUsing the relation /g3435/g2026 /g3556/uni002E/g1831/g3560/g3439/g4666/g2026 /g3556/uni002E/g1868 /g3556/g4667/g3404 /g1861/g2026/uni002E/g4666/g1831/g3560/g3400/g1868/uni0020/uni0303/uni0020/g4667/g3397/g1831 /g3560/uni002E/g1868 /g3556 , and solving for /g2032 , one obtains \n/g4678/g4666/g1322/g3398/g1857/g2038 /g4667/g2870/g3398/g1855/g2870/g1868/g2870/g3398/g1865/g2870/g1855/g2872/g3398/g1861/g1857/g1328/g1855/g2870/g3435/g1831/g3560/uni002E/g1868 /g3556/g3439\n/g1322/g3398/g1857/g2038/g3397/g1865/g1855/g2870/g3397/g1857/g1328/g1855/g2870/g2026/uni002E/g3435/g1831/g3560/g3400/g1868 /g3556/g3439\n/g1322/g3398/g1857/g2038/g3397/g1865/g1855/g2870/g4679/g2032 /g3404 /uni0030  \n                         (15) \nIn the non-relativistic limit, i.e. assuming a smal l /g2035 /g3404 /g1322/g3398/g1865/g1855/g2870 and /g1857/g2038 , the above reduces to the following standard \nHamiltonian for SO coupling  \n/g4678/g1868/g2870\n/uni0032/g1865/g3397/g1857/g2038/g3397/g1328/g1857\n/uni0034/g1865/g2870/g1855/g2870/g2026/uni002E/g4666/g1868/g3400/g1831/g4667/g3397/g1861/g1328/g1857\n/uni0034/g1865/g2870/g1855/g2870/g1831/uni002E/g1868/g4679/g2032 /g3404 /g2035/g2032  \n                              (16) 9 \n In application to particles in graphene like system s which mimic Dirac fermions due to material bandst ructure, it \nwould be instructive to replace the coupling mass t erm of /g1865/g1855/g2870 for particles in vacuum with a coupling term /uni2206 which \narises due to material bandstructure but plays the same role as the mass term, as far as the Dirac mat rix is \nconcerned.  The coupling term /uni2206 gives rise to the energy dispersion where the effe ctive mass of particles in the \nmaterials can be derived; in other words, particle effective mass is a function of /uni2206 but not vice versa.  For monolayer \ngraphene, /uni2206 vanishes and it can be derived from the energy dis persion relation that particles behave like massles s \nDirac fermions.  In semiconductor physics, SO effec ts have been known to exist both in the conduction and the \nvalence band; its effect on the bandstructure has b een well studied.  In the technology relevant spint ronics, the more \nspecialized type of Rashba and Dresselhaus SO coupl ing have been considered for generating spin curren t in \nsemiconductor spintronics.  SO effects which manife stly appear in the energy dispersion relation often  results in a \ncoupled form of spin-dependent transmission of elec trons across potential barriers. Equation (16) shows  that the SO \nsystem provides a b(k) relation via /g2026/uni002E/g4666/g1868/g3400/g1831/g4667 which is crucial for the derivation of the curvatu re in K space.  The \nother systems which give the required b(k) relation  is the graphene system.  The b(k) in these systems  originate \nfrom the bandstructure which exhibits a linear ener gy dispersion around the two inequivalent, hexagona l corners \n(valley) of the first Brilliouin zone, also known a s the Fermi points. The prevalence of b(k) systems in nature motivate \nus to present a systematic approach to study the ga uge potential and its topological curvature in the K space.  In SO, \ngraphene or superconducting systems, the parameter of interest lie in the K space.  One recalls that i n the system \nwhere the spinor evolves and aligns to the b fields , the gauge fields in B and K spaces, are respectiv ely, /g1827/g4634/g3003/g3404\n/g1328\n/g2870/g3032/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016/g3003/g4667/g3105/g1486/g3251\n/g3105/g3029/g3560 and /g1827/g4634/g3012/g3404/g1328\n/g2870/g3032/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016/g3003/g4667/g3105/g1486/g3251\n/g3105/g3038/g3560; the two expressions are related by /g1827/g3091/g3012/g1568 /g1827/g4634/g3003/uni002E/g3105/g3029/g3560\n/g3105/g3038/g3339.   To find the Dirac \ngauge potential in K space, one merely needs b as a  function of k.  Space conversion for the gauge pot ential is \nstraightforward, but that for the monopole field wi ll be slightly more complex.  But we will show belo w that in fact to \ntransform the monopole field in B space to a topolo gical curvature in K space, it also suffices to hav e the Zeeman b \nas a function of k.  Looking at the z component onl y ,    \n/uni0020/uni2126/g3053/g3012/g3404/g2034/g1827/g3052/g3012\n/g2034/g1863/g3051/g3398/g2034/g1827/g3051/g3012\n/g2034/g1863/g3052 \n                                                                    (17) \nUsing /g1827/g3091/g3012/g1568 /g1827/g4634/g3003/uni002E/g3105/g3029/g3560\n/g3105/g3038/g3339, it is straightforward to obtain 10 \n /uni2126/g3053/g3012/g3404/g2034\n/g2034/g1863/g3051/g4678/g1827/g4634/g3003/uni002E/g2034/g1854/g3560\n/g2034/g1863/g3052/g4679/g3398/g2034\n/g2034/g1863/g3052/g4678/g1827/g4634/g3003/uni002E/g2034/g1854/g3560\n/g2034/g1863/g3051/g4679 /g3404 /g4678/g2034/g1854/g3560\n/g2034/g1863/g3051/uni002E/g2034/g3029/g3560/g4679/g4678/g1827/g4634/g3003/uni002E/g2034/g1854/g3560\n/g2034/g1863/g3052/g4679/g3398/g4678/g2034/g1854/g3560\n/g2034/g1863/g3052/uni002E/g2034/g3029/g3560/g4679/g4678/g1827/g4634/g3003/uni002E/g2034/g1854/g3560\n/g2034/g1863/g3051/g4679 \n                                     (18) \nwhich in tensor form is  \n/uni2126/g3053/g3012/g3404 /g4678/g2034/g1854/g3092\n/g2034/g1863/g3051/uni0020/g2034/g1827/g3091/g3003\n/g2034/g1854/g3100/uni0020/g2034/g1854/g3091\n/g2034/g1863/g3052/g4679/g3398/g4678/g2034/g1854/g3100\n/g2034/g1863/g3052/uni0020/g2034/g1827/g3091/g3003\n/g2034/g1854/g3100/uni0020/g2034/g1854/g3091\n/g2034/g1863/g3051/g4679/g3397/g1827/g3091/g3003/g4678/g2034/g1854/g3100\n/g2034/g1863/g3051/g2034/g2870/g1854/g3091\n/g2034/g1854/g3100/g2034/g1863/g3052/g3398/g2034/g1854/g3100\n/g2034/g1863/g3052/g2034/g2870/g1854/g3091\n/g2034/g1854/g3100/g2034/g1863/g3051/g4679 \n              (19) \nThe last term written in /g1827/g3091/g3003/g3436/g3105/g3118/g3029/g3339\n/g3105/g3038/g3299/g3105/g3038/g3300/g3398/g3105/g3118/g3029/g3339\n/g3105/g3038/g3300/g3105/g3038/g3299/g3440 clearly vanishes as /g3427/g2034/g3051/uni002C/g2034/g3052/g3431 /g3404 /uni0030 in general. \n/uni2126/g3053/g3012/g3404 /g4678/uni0020/g2034/g1827/g3091/g3003\n/g2034/g1854/g3100/g3398/g2034/g1827/g3100/g3003\n/g2034/g1854/g3091/g4679/g2034/g1854/g3100\n/g2034/g1863/g3051/uni0020/uni0020/g2034/g1854/g3091\n/g2034/g1863/g3052/g3404/g2034/g1827/g3036/g3003\n/g2034/g1854/g3037/g3435/g2012/g3036/g3091 /g2012/g3037/g3100 /g3398/g2012/g3036/g3100 /g2012/g3037/g3091 /g3439/g4678/g2034/g1854/g3100\n/g2034/g1863/g3051/uni0020/uni0020/g2034/g1854/g3091\n/g2034/g1863/g3052/g4679 \n                     (20) \nUsing the identity /g3435/g2012/g3036/g3091 /g2012/g3037/g3100 /g3398/g2012/g3036/g3100 /g2012/g3037/g3091 /g3439 /g3404 /g2013/g3036/g3037/g3038/uni0020/g2013/g3091/g3100/g3038  where /g2013/g3036/g3037/g3038/uni0020 is the fully anti-symmetric tensor, /g2012/g3036/g3091  is the kronecker delta, \none gets the the z-component of the curvature in K space \n/uni2126/g3053/g3012/g3404/g2034/g1827/g3036/g3003\n/g2034/g1854/g3037/g2013/g3036/g3037/g3038/uni0020/g4678/g2034/g1854/g3100\n/g2034/g1863/g3051/uni0020/uni0020/g2034/g1854/g3091\n/g2034/g1863/g3052/g4679/g2013/g3091/g3100/g3038  \n                                                   (21) \nLikewise, the other components of this curvature fi eld can be derived, and expressed in the more famil iar vector \ncalculus form:  \n/uni2126/g3051/g3012/g3404/g1854/g3560\n/g1854/g2871/uni002E/g4678/g2034/g1854/g3560\n/g2034/g1863/g3053/g3400/g2034/g1854/g3560\n/g2034/g1863/g3052/g4679 \n/uni2126/g3052/g3012/g3404/g1854/g3560\n/g1854/g2871/uni002E/g4678/g2034/g1854/g3560\n/g2034/g1863/g3051/g3400/g2034/g1854/g3560\n/g2034/g1863/g3053/g4679 \n/uni2126/g3053/g3012/g3404/g1854/g3560\n/g1854/g2871/uni002E/g4678/g2034/g1854/g3560\n/g2034/g1863/g3052/g3400/g2034/g1854/g3560\n/g2034/g1863/g3051/g4679 \n                                               (22)  11 \n where  /g3105/g3002/g3284/g3251\n/g3105/g3029/g3285/g2013/g3036/g3037/g3038/uni0020/g3404/g3029/g3560\n/g3029/g3119 . It has thus been shown that the topological curv ature in K space can be developed by \nsubstituting the Zeeman b field into Eq. (22). It i s of interest to remind readers that had /g1854/g3560/g3404 /g1855/g1863/g3560  or /g1854/g3560/g3404 /g1855/g1863/g3560, no \nconversion is required, and the monopoles will, res pectively be /uni2126/g3091/g3012/g3404/g1328\n/g3030/g2870/g3032/g3038/g3560\n/g3038/g3119  or /uni2126/g3091/g3019/g3404/g1328\n/g3030/g2870/g3032/g3045/uni0303\n/g3045/g3119  when one follows the \nEqs.(9)-(12) in their respective spaces.  When the s pin of a particle inscribes a path on /g1845/g3003/g2870 , the gauge potential it \n“sees” at every point on the path is /g1827/g3003/g3404/g1328\n/g2870/g3032/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016/g3003/g4667/g1856/g1486/g3003 .  However, when a particle inscribes a path on /g1845/g3012/g2870, the \ngauge it “sees” is /g1827/g3012/g3405/g1328\n/g2870/g3032/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016/g3012/g4667/g1856/g1486/g3012 unless the relation /g1854/g3560/g3404 /g1855/g1863/g3560 is obeyed. This is because /g1827 /g3404/g1328\n/g2870/g3032/g4666/uni0031/g3398\n/uni0063/uni006F/uni0073/g2016/g3003/g4667/g1856/g1486/g3003 is obtained as a result of spin assuming the eigen state of /g1854/g3560/g4666/g1872/g4667 in an adiabatic manner, the spin thus \nshares the same polar coordinates with the b field.   To obtain the curvature in K space, one needs to convert /g1827/g3003 to \nand then perform /g1487/g2895/g3400/g1827/g4634/g3012. Since /g1827/g3012/g3405/g1328\n/g2870/g3032/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016/g3012/g4667/g1856/g1486/g3012, it is worth cautioning that while /g1487/g2886/g3400/g1827/g4634/g3003 is a monopole \nfield, /uni0020/g1487/g2895/g3400/g1827/g4634/g3012 might not be one. It is important to note that cur vature in K space cannot be obtained by substitutin g \nthe effective b fields into Eqs. (12); doing this is  merely re-expressing /g1487/g2886/g3400/g1827/g4634/g3003 in B space as a function of k., but not \nperforming the function of Eqs. (22). Equations (22)  are most useful in SO or graphene systems, which c omprise k-\ndependent Zeeman b field in the Lorentz frame of a charge particle, where k is the electron wavevector .   \nBelow, we provide the topological curvature in K sp ace for the cubic Dresselhaus, the linear Dresselha us, the \nPerel-modified Dresselhaus, the linear Rashba and D resselhaus in quantum well systems, as well as the monolayer \nWeyl system of massless Dirac fermions in graphene.   We present below the various SO effects and their  \ncorresponding effective b fields.  Effective Zeeman b fields can be derived by considering the first of  /g1834 /g3404 /g2020/g3003/g1854 /g3404\n/g3032/g3034/g1328\n/g2872/g3040/g1854/uni0020where /g2020/g3003 is the Bohr magneton that has the SI unit of Joule /Tesla; note that /g2020/g3003 is also equivalent to /g1835/g1827  where \n/g1835 is a circulating current and /g1827 is the area enclosed by the circulating current,  \nTable I.  Reference of various spin orbital systems , their Hamiltonian, relativistic effective fields,  and corresponding K-space \nmonopole curvature.   \nSpin Orbit Coupling \nTypes (Material \nSystem) Hamiltonian and Effective b fields /g2743/g3561 is the curvature field in K-space; \n \n \nSemiconductor systems with linear SO coupling (GaAs , GaSb, InAs, InSb, GaInAs, GaInSb, ) \n \nLinear Dresselhaus \n   \n/g1834/g3404/uni03B7/uni0044/g3435/uni03C3/uni0078/uni006B/uni0078/g3398/uni03C3/uni0079/uni006B/uni0079/g3439 \n \n/g1854/g3560/g3404/g2015/g3005\n/g2020/g3003/g3437/g1863/g3051\n/g3398/g1863/g3052\n/uni0030/g3441 \n  \n/uni2126/g3561/g3404/g3399/g1328/g2024\n/g1857/g3437/uni0030\n/uni0030\n/g2012/g2870/g4666/g1863/g4667/g3441 12 \n  \nLinear Rashba \n \n \n \n \n \n  \n/g1834/g3404/uni03B7/uni0052/g3435/uni03C3/uni0078/uni006B/uni0079/g3398/uni03C3/uni0079/uni006B/uni0078/g3439 \n \n/g1854/g3560/g3404/g2015/g3005\n/g2020/g3003/g3437/g1863/g3052\n/g3398/g1863/g3051\n/uni0030/g3441 \n \n \n  \n/uni2126/g3561/g3404/g3399/g1328/g2024\n/g1857/g3437/uni0030\n/uni0030\n/g2012/g2870/g4666/g1863/g4667/g3441 \nGraphene systems  \n \nMonolayer graphene \n(Massless Weyl)  \n/g1834/g3404/g1827/g3435/uni03C3/uni0078/uni006B/uni0078/g3397/uni03C3/uni0079/uni006B/uni0079/g3439 \n \n/g1854/g3560/g3404/g1827\n/g2020/g3003/g3437/g1863/g3051\n/g1863/g3052\n/uni0030/g3441 \n \n  \n/uni2126/g3561/g3404 /g3399/g1328/g2024\n/g1857/g3437/uni0030\n/uni0030\n/g2012/g2870/g4666/g1863/g4667/g3441 \n \nBilayer graphene \n  \n/g1834 /g3404/g1328/g2870\n/uni0032/g1865/g4684/uni03C3/uni0078/uni006B/g3397/uni0032/g3397/uni006B/g3398/uni0032\n/uni0032\n/g3398/uni03C3/uni0079/uni006B/g3398/uni0032/g3398/uni006B/g3397/uni0032\n/uni0032/uni0069/g4679 \n/g1854/g3560/g3404/uni0031\n/uni0032/g2020/g3003/g1865/g4684/g1863/g3051/g2870/g3398/g1863/g3052/g2870\n/uni0032/g1863/g3051/g1863/g3052\n/uni0030/g4685 \n \n  \n \n/uni2126/g3561/g3404 /g3399/g1328/uni0032/g2024\n/g1857/g3437/uni0030\n/uni0030\n/g2012/g2870/g4666/g1863/g4667/g3441 \n \n \nSemiconductor systems with cubic SO coupling  \n \nDresselhaus Cubic \n(Bulk III-V)  \n/g1834/g3404/uni03B7/uni0044/uni0043 /g4672/uni03C3/uni0078/uni006B/uni0078/g4674/uni006B/uni0079/uni0032/g3398/uni006B/uni007A/uni0032/g4675/g4673/g3397/uni0063/uni002E/uni0070/uni002E \n \n/g1854/g3560/g3404/g2015/g3005/g3004 \n/g2020/g3003/g3438/g1863/g3051/g3427/g1863/g3052/g2870/g3398/g1863/g3053/g2870/g3431\n/uni0020/g1863/g3052/g4670/g1863/g3053/g2870/g3398/g1863/g3051/g2870/g4671\n/g1863/g3053/g3427/g1863/g3051/g2870/g3398/g1863/g3052/g2870/g3431/g3442 \n  \n \n/uni2126/g3561/g3404/g3399/g1328\n/uni0032/g1857/g1854/g2871/g4672/g3435/g1863/g3051/g2870/g3398/g1863/g3052/g2870/g3439/g3397/g1855/g1868/g4673/g4684/g1863/g3051\n/uni0020/g1863/g3052\n/g1863/g3053/g4685 \n \n \nDresselhaus (high \nkinetic, Perel) \n  \n/g1834/g3404/uni03B7/uni0044/uni006B /g4672/g3398/uni03C3/uni0078/uni006B/uni0078/uni006B/uni007A/uni0032/g3397/uni03C3/uni0079/uni006B/uni0079/uni006B/uni007A/uni0032\n/g3397/uni03C3/uni007A/uni006B/uni007A/g4674/uni006B/uni0078/uni0032\n/g3398/uni006B/uni0079/uni0032/g4675/g4673 \n \n/g1854/g3560/g3404/g2015/g3005/g3038 \n/g2020/g3003/g4684/g3398/g1863/g3051/g1863/g3053/g2870\n/uni0020/g1863/g3052/g1863/g3053/g2870\n/g1863/g3053/g3427/g1863/g3051/g2870/g3398/g1863/g3052/g2870/g3431/g4685 \n  \n \n/uni2126/g3561/g3404/g3399/g1328\n/uni0032/g1857/g1854/g2871/g3435/g1863/g3052/g2870/g3398/g1863/g3051/g2870/g3439/g4684/g1863/g3053/g2872/g1863/g3051\n/uni0020/g1863/g3053/g2872/g1863/g3052\n/g1863/g3053/g2873/g4685 13 \n  \nHeuristically, the topological magnetic field ( /uni2126) can be regarded as the source providing Lorentz f orces that \naffect electron motion.  The fact that the force ex perienced by the spin up /down particle is opposite  in directions \narising due to /g3399/g3435/g1487/g3400/g1827/g4634/g3439 provides the physical picture of spin transverse s eparation. However, more rigorous \nquantification of such separation, or the Hall cond uctivity would have to come from the Berry’s phase and the \nChern-Simon term which can be interpreted as a summ ation of the Lorentz forces over all electron momen ta. The \nBerry’s phase in K space which has important releva nce to electron transport can be derived from /uni0020/g1486 /g3404/g3032\n/g1328/g1516/g1827/g4634/uni002E/g1856/g1863/g3560/g3404\n/g3032\n/g1328/g1516/g3435/g1487/g3400/g1827/g4634/g3439/uni002E/g1856/g2870/g1863/g3560, where it would be worth noting that /g3435/g1487/g3400/g1827/g4634/g3439 can be viewed as the topological magnetic field th at \nthe electron “feels”.  It would not be the focus of  this paper to conduct extended analysis of the Ber ry’s phase of all \nsystems. But in view of modern trend of quantifying  system conductivity using the Berry’s phase, we me rely provide \nits derivations for common systems only, i.e. linea r Rashba, linear Dresselhaus, monolayer and bilayer  graphene. To \nderive the Dirac potential for the above systems in  the intended space of K, one needs to recall that /g1827/g3091/g3038/g3404\n/g1328\n/g2870/g3032/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016 /g4667/g3105/g1486\n/g3105/g3038/g3339 but noting that /g1486 and /g2016 are angles defined by the B field.  To derive /g1827/g3091/g3038 for the systems above, one \nwould thus need the effective magnetic fields of ea ch system.  Keeping in mind that in the B space, /uni0063/uni006F/uni0073/g2016 /g3404 /g1854 /g3053/g1854/uni2044, \n/uni0074/uni0061/uni006E/g1486 /g3404 /g1854 /g3052/g1854/g3051/uni2044 ,  one derives /g1827/g3091/g3038 . To obtain the Berry’s phase, let us examine the gauge of /g1827/g4634/g3404 /g3399/g1328\n/g2870/g3032/g3038/g3118/g3437/g1868/g1863 /g3052\n/g1869/g1863 /g3051\n/uni0030/g3441 which \nis the general expression for the above-mentioned s ystems at /g2016 /g3404 /g2024 /uni0032/uni2044, i.e. in the plane of the R as well as K space \ntwo-dimensional (2D) system. The Berry’s phase can be derived with /g1519/g1827/g4634/g4672/g2016 /g3404/g3095\n/g2870/g4673/uni002E/g1856/g1863/g3560\n/g3004 where C represents a closed \npath in the plane surrounding the monopole at /g3435/g1863/g3051/uni0020/uni002C/g1863/g3052/g3439 /g3404 /uni0030. Alternatively, one could perform by Stoke’s theor em \n/g1519/uni0020/g3427/g1487/g3400/g1827/g4634/g4666/g2016/uni002C/g1486/g4667/g3431/uni002E/g1856/g2870/g1863/g3560\n/g3020/g3015   where SN represents the entire curved surface of the North hemisphere of the /g1845/g2870 manifold.  \nWe can equivalently evaluate  /g1519/uni0020/g4674/g1487/g3400/g1827/g4634/g4672/g2016 /g3404/g3095\n/g2870/g4673/g4675/uni002E/g1856/g2870/g1863/g3560\n/g3020/g3007   on SF which represents the flat circular surface bound by C.  \nThe results are summarized in Table II for differen t systems corresponding to different values of /g1868/uni002C/g1869 .   \nTable II.  Reference of various spin orbital system s, their gauge, and corresponding Berry’s phase.   \nSpin Orbit Coupling \nTypes (Material \nSystem) Gauge /g1486 is the Berry’s phase  \n/g1486/g3404/g2187\n/g1328/g3505/g2157/g3561/uni002E/g2186/g2193/g3561/g3404/g3515/uni0020/g4674/g2744/g3400/g2157/g3561/g4672/g2242/g3404/g2250\n/g2779/g4673/g4675/uni002E/g2186/g2779/g2193/g3561\n/g2175/g2162  14 \n  \nLinear Dresselhaus \n   \n/g1827/g4634/g3404/g3399/g1328\n/uni0032/g1857/g1863/g2870/g3437/g1863/g3052\n/g3398/g1863/g3051\n/uni0030/g3441 \n  \n \n/g1866/g2024  \n \nLinear Rashba \n \n \n  \n/g1827/g4634/g3404/g3399/g1328\n/uni0032/g1857/g1863/g2870/g3437/g3398/g1863/g3052\n/g1863/g3051\n/uni0030/g3441 \n  \n \n/g1866/g2024  \n \nMassless  Weyl  \n(monolayer \ngraphene)  \n/g1827/g4634/g3404/g3399/g1328\n/uni0032/g1857/g1863/g2870/g3437/g3398/g1863/g3052\n/g1863/g3051\n/uni0030/g3441 \n  \n/g1866/g2024  \n \n \n \nMassive Dirac \n(bilayer graphene) \n  \n/g1827/g4634/g3404/g3399/g1328\n/uni0032/g1857/g1863/g2870/g3437/g3398/uni0032/g1863/g3052\n/uni0032/g1863/g3051\n/uni0030/g3441  \n \n/uni0032/g1866/g2024  \n \n \nWe have introduced a general approach which allows one to derive the monopole field of a specific syst em in the B \nspace using the Dirac gauge potential which is defi ned in the same B space.  We then focused on the va rious \nspintronic and graphene systems in which such monop ole fields are relevant.  In the SO systems relevan t to \nsemiconductors as well as graphene systems where th e Zeeman b fields are k-dependent, we showed how th e \ntopological curvature (magnetic field) in the more useful K space can be derived.  The physical signif icance of the \ncurvature fields depends on the space in which the curvature is taken. The usefulness of the approach in this paper \nis that it provides a unified underlying picture fo r the curvature fields any arbitrary spaces (e.g. i n K and R spaces) \nunder a common origin, i.e. the Dirac gauge potenti al and its monopole field in B space. One merely re quires an \neffective Zeeman b field of the form /g2026/uni002E/g1828/uni0020in the Hamiltonian of a specific system for a quick  derivation of the \ntopological curvature in any space outside of the B  space.   \n \nThe surface integral of the curvature yields a non- vanishing quantized value, which is invariant under  deformation \nof the surface of integration; it is hence a topolo gical object.  In the context of Dirac monopole, th is is associated \nwith the quantization of the electric charge.  In t he SO or graphene systems, this quantity is associa ted with \nquantized magnetic flux or the Berry’s phase.  It t herefore becomes clear that the existence of K-spac e topological \ncurvature (magnetic field) can be related heuristic ally to particle trajectory, and in the case of SO system, spin-15 \n dependent separation of charges would be resulted f rom the spin-dependent curvature fields in K space.  Our \nsummary in Table I provides a unifying picture for the physics of particle motion in various systems e g. spin orbital, \nlocal magnetic, graphene and superconducting system s under the theory of symmetry, gauge and monopoles . The \napproach and results of our work allow experimental ists to seek out material system which exhibit Zeem an-like \nterms for potential spin Hall, quantum spin Hall, S O-induced spin torque, spin oscillation, and other measurements. \nIn modern treatment, Berry’s phase has been associa ted with quantized conductivity of mesoscopic syste ms.  Table \nII thus provides a unifying picture of the various types of quantum Hall effects that could possibly e xist in different \ncondensed matter systems.  \n \nAPPENDIX (Topological Properties of the Monopole Field s) \nTo determine the vector potential for a regular mon opole field is not a trivial task.  In fact a singl e vector potential \nfunction which is regular everywhere on the S 2 manifold probably does not exist.  We would like t o remind readers \nthat a /g3435/g1487/g3561/g3003/g3400/g1827/g4634/g3003/g3439 based on one gauge expression (either North or Sou th pole) will necessarily yield a vanishing \nsurface integral over the entire /g1845/g2870 manifold, due to the Dirac string. Whereas in mode rn understanding where \n/g3435/g1487/g3561/g3003/g3400/g1827/g4634/g3003/g3439 is based on at least two gauge expressions, the cu rvature is regular everywhere even though the \nindividual gauge expression is not, and the surface  integral of such regular curvature will be a non-v anishing \nquantity as the string can be avoided.  Here we wil l show using the South Pole gauge that the Dirac ma gnetic field \nhas a string. The vector form of the Dirac potentia l is:  \n/g1827/g4634/g2285/g3404/g1328/g3435/g1854/g3560/uni0020/g3400/g1863/g3560/g3439\n/uni0032/g1857/g1854/uni0020/g3435/g1854/g3398/g1854 /g3560/uni002E/g1863/g3560/g3439/g3404/uni0020/uni0020/uni0020/g1328/uni0020/g1854/g3052/g1861 /g3398/g1854/g3051/uni0020/g1862\n/uni0032/g1857/g1854/uni0020/g4666/g1854/g3398/g1854/g3053/g4667 \n                                                           (23) \nAt first sight, the curvature of the above potentia l may appear to be /g3435/g1487/g3561/g3003/g3400/g1827/g4634/g2285/g3439 /g3404/g1328/g3029/g3560\n/g2870/g3032/g3029/g3119.  However, the expression is \nincomplete because it does not fully capture the fa ct.  The Dirac potential is not a regular function;  it is singular \nalong /g2016 /g3404 /uni0030  but regular along /g2016 /g3404 /g2024 . A proper approach is to first regularize the Dira c potential by inserting a /g2035/g2870 to \nR so that  /g2016 /g3404 /uni0030  can be negotiated, and derive the regularized magn etic field.  The regularized potential is  \n/g1827/g4634/g2285/g3404/g1328/g3435/g1854/g3560/uni0020/g3400/g1863/g3560/g3439\n/uni0032/g1857/uni0020/g1844/uni0020/g3435/g1844/g3398/g1854 /g3560/uni002E/g1863/g3560/g3439 \n                                                               (24) 16 \n where /g1844/g2870/g3404 /g1854/g3051/g2870/g3397/g1854/g3052/g2870/g3397/g1854/g3053/g2870/g3397/g2035/g2870.  The regularized magnetic field can be shown to b e: \n/uni2126/g3561/g4666/g1854/uni002C/g2035/g4667/g3404/g1328\n/uni0032/g1857/g1854/g3560\n/g1844/g2871/g3398/g1328\n/uni0032/g1857/g4678/g2035/g2870\n/g1844/g2870/g4666/g1844 /g3398/g1854/g3053/g4667/g2870/g3397/g2035/g2870\n/g1844/g2871/g4666/g1844 /g3398/g1854/g3053/g4667/g4679/g1863/g3560 \n                                         (25) \nLifting regularization which simply means presentin g a modified expression which resides in both the r egular and \nthe singular regions, one obtains: \n /uni0020\n/uni0020/uni0020/uni0020/uni0020/uni006C/uni0069/uni006D \n/g3106/g1372/g2868/uni2126/g3561/g4666/g1854/uni002C/g2035/g4667 /g3404/g1328\n/uni0032/g1857/g1854/g3560\n/g1854/g2871/g3398/g1328\n/uni0032/g1857/uni0034/g2024/g2016 /g4666/g1854/g3053/g4667/g2012/g4666/g1854/g3051/g4667/g2012/g4666/g1854/g3052/g4667 \n                                                         (26) \nEquation (26) consists of two parts, the regular mag netic monopole and the singular Dirac string. The c urvature of \nEq. (26) vanishes as /g1519/uni2126/g3561/uni0020/uni002E/g1856/g1845/g4634/g3404/g1519/uni2126/g3561/g3045/uni0020/uni002E/g1856/g1845/g4634/g3397/g1519/uni2126/g3561/g3046/g3047/g3045/g3036/g3041/g3034 /uni0020/uni002E/g1856/g1845/g4634/g3404 /uni0034/g2024/g1859/g3398/uni0034/g2024/g1859 /g3404 /uni0030/uni0020 , noting here that the monopole has been \nseparated into the regular and the string parts.  T his clearly shows that the South pole Dirac potenti al used above \ncannot be a correct representation of the vector po tential for a magnetic monopole.  It is thus necess ary to find a \ncorrect representation and this task was completed with the modern theory of fiber bundle by Wu and Ya ng.  In this \ntreatment, it is important to understand that there  should exist at least two gauge expressions on the  /g1845/g2870 manifold in \norder to provide a non-vanishing /g1519/uni0020/g3435/g1487/g3561/g3003/g3400/g1827/g4634/g3003/g3439/uni002E/g1856/g1845/g4634  over the manifold. In fact by Stokes’ theorem, on e sees that the \nsurface integral of the monopole field is the line integral of the gauge field.  To illustrate this mo re clearly, one \nresorts to the differential form which is related t o the vector quantity as follows: \n/g1827 /g3404/uni0020/g1328\n/uni0032/g1857/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016 /g4667/g3436/g2034/g1486\n/g2034/g1854/g3560/g3440/uni002E/g1856/g1854/g3560/g3404/uni0020/uni0020/uni0020/g1328\n/uni0032/g1857/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016 /g4667/g1856/g1486 \n                                            (27) \n/g1856/g1827 /g3404 /g3435/g1487/g3561/g3400/g1827/g4634/g3439/uni002E/uni0064/uni0053/g3560                                                                          (28) \nIn Eq. (27), /g1827 is expressed as a 1-form.  For comparison in K spa ce, /g1827 /g3404/uni0020/g1328\n/g2870/g3032/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016/g3003/g4667/g4672/g3105/g1486/g3251\n/g3105/g3029/g3560/g4673/uni002E/g4672/g3031/g3029/g3560/uni0020/uni0020\n/g3105/g3038/g3560/g4673/uni002E/g1856/g1863/g3560 is \nconsistent with /g1827/g3012/g3405/g1328\n/g2870/g3032/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016/g3012/g4667/g1856/g1486/g3012.  Performing an exterior differentiation of A foll owing standard definitions in \ndifferential geometry will lead naturally to Eq. (2 8) which is the dot areal product of the curvature;  b is the \nparameter of an arbitrary space eg. the B space and  dS is in that space too.  According to Wu Yang, in  fact Eq. (27) 17 \n is not a unique, nor is it a complete form of the g auge potential. The gauge should be expressed as on e of Eq. (29) \ndepending on the chart which covers the  /g1845/g3029/g2870 surface on which A is defined, b is the radius.  O ne example set of two \ncharts that overlap but completely cover the /g1845/g3029/g2870 manifold is: \n/g1827/g2285/g3404 /g3398/g1328\n/uni0032/g1857/g4666/uni0031/g3397/uni0063/uni006F/uni0073/g2016 /g4667/g1856/g1486/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/g1827/g2280/g3404/g1328\n/uni0032/g1857/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016 /g4667/g1856/g1486 \n                                   (29)        \n/g1856/g1827/g2280 is defined everywhere on /g1845/g2870 except the –z axis, while /g1856/g1827/g2285 is defined everywhere except the +z axis.  Equatio n \n(29) is the differential form of the Dirac gauge po tential in the spherical polar coordinates.  For co mparison in \nCartesian coordinates, the differential form of the  Dirac gauge is: \n/g1827/g2285/g3404/g1328\n/uni0032/g1857/g4678/g3398/g1854/g3052\n/g1854/g4666/g1854 /g3398/g1854/g3053/g4667/uni0020/uni002C/g1854/g3051\n/g1854/g4666/g1854 /g3398/g1854/g3053/g4667/uni002C/uni0030/uni0020/g4679/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/g1827/g2280/g3404/g1328\n/uni0032/g1857/g4678/g3398/g1854/g3052\n/g1854/g4666/g1854 /g3397/g1854/g3053/g4667/uni0020/uni002C/g1854/g3051\n/g1854/g4666/g1854 /g3397/g1854/g3053/g4667/uni002C/uni0030/uni0020/g4679 \n                           (30) \nwhere b is the required coordinates in Cartesian fo rm.  Since /g1827/g2280 is related to /g1827/g2285 by /g1827/g2280/g3404 /g1827/g2285/g3397/uni0064/uni03B8, it is obvious that \n/g1856/g1827 /g2285/g3404 /g1856/g1827/g2280 which shows /g1856/g1827  is unique. In summary, /g1827 is not globally defined throughout /g1845/g3029/g2870, otherwise /g1516/g1856/g1827 /g3405 /uni0030 ; in \nother words, if there exists a vector /g1827/g4634 such that its curvature has no singularity, and /g1827/g4634 is unique, then /g1516/g1856/g1827 /g3405 /uni0030  .  \nNow with the Dirac gauge potential defined on the N orth and South charts, one can then perform the sur face \nintegral of these gauge potentials as /g1519/uni2126/g3561/uni0020/uni002E/g1856/g1845/g4634/g3404/g1519/g3435/g1487/g3400/uni0020/g1827/g4634/g2280/g3439/uni002E/g1856/g1845/g4634/g3397/g3015/g1519/g3435/g1487/g3400/g1827/g4634/g2285/g3439/uni002E/g3020/g1856/g1845/g4634/g3404 /uni0032/g2024/g1859/g3397/uni0032/g2024/g1859 /g3404 /uni0034/g2024/g1859 , thus avoiding \nthe string of each gauge potential.  Although /uni0020/g1827/g2280 and /g1827/g2285 are by no means unique on their own, they uniquely  \ndetermine a 1-form /g2033 on the bundle space P which is a /g1845/g3019/g2871, where R is the radius of the 3-sphere.  Here we p rovide \nthe mathematical origin, the 1-form /g2033 on /g1337/g2872 is given by: \n/g2033 /g3404 /g1861/g4666/g3398/g1877/g2869/g1856/g1876/g2869/g3397/g1876/g2869/g1856/g1877/g2869/g3398/g1877/g2870/g1856/g1876/g2870/g3397/g1876/g2870/g1856/g1877/g2870/g4667 \n                            (31) \nwhich is well-defined on the /g1845/g3019/g2871.  The /g1845/g3019/g2871 is itself defined by /g1845/g3019/g2871/g3404/g4668/g4666/uni0078/g2869/uni002C/uni0079/g2869/uni002C/uni0078/g2870/uni002C/uni0079/g2870/g4667/g1488 /g1337/g2872/uni003A/uni0020/uni0078/g2869/g2870/g3397/uni0079/g2869/g2870/g3397/uni0078/g2870/g2870/g3397/uni0079/g2870/g2870/g3404 /uni0031/g4669.  One \ncan reparametrize /g1845/g3019/g2871 such that /g1845/g3019/g2871/g3404 /g4676/g4666/g1878/g2869/uni002C/g1878/g2870/g4667/g3404 /g4672/uni0063/uni006F/uni0073 /g3087\n/g2870/g1857/g3036/g3084/g3117/uni002C/uni0073/uni0069/uni006E /g3087\n/g2870/g1857/g3036/g3084/g3118/g4673/uni003A/uni0030 /g3409/g3087\n/g2870/g3409/g3095\n/g2870/uni002C/g2013/g2869/uni002C/g2013/g2870/g1488 /g1337/g2872/g4677, while obeying  /uni0078/g2869/g2870/g3397\n/uni0079/g2869/g2870/g3397/uni0078/g2870/g2870/g3397/uni0079/g2870/g2870/g34041. Every point of /g4666/g1878/g2869/uni002C/g1878/g2870/g4667/g3404 /g4672/uni0063/uni006F/uni0073 /g3087\n/g2870/g1857/g3036/g3084/g3117/uni002C/uni0073/uni0069/uni006E /g3087\n/g2870/g1857/g3036/g3084/g3118/g4673 /g1488 /g1845/g3019/g2871 can then be mapped via /g2282/g4666/g2016/uni002C/g2013/g2869/uni002C/g2013/g2870/g4667/g3404\n/g4666/uni0073/uni0069/uni006E/g2016/uni0063/uni006F/uni0073 /g4666/g2013/g2869/g3398/g2013/g2870/g4667/uni002C/uni0073/uni0069/uni006E/g2016/uni0073/uni0069/uni006E /g4666/g2013/g2869/g3398/g2013/g2870/g4667/uni002C/uni0063/uni006F/uni0073/g2016/g4667 to a point on /g1845/g3029/g2870  where  /g2282/uni003A/g1845/g2871/g1372 /g1845/g2870 which is in fact a form of Hopf 18 \n mapping to project the /g2033 on /g1845/g3019/g2871 to /g1845/g3029/g2870. In other words, the Hopf mapping  /g2015/uni003A/uni0020/g1845/g3019/g2871/g3639 /g1845/g3029/g2870 yields a vector potential \nregular everywhere on /g1845/g3019/g2871. It is worth noting that any point of /g4666/g1878/g2869/uni002C/g1878/g2870/g4667/g1488 /g1845/g3019/g2871 satisfies: \n/g4678/uni007C/g1878/g2869/uni007C/uni002C/g1878/g2870/uni007C/g1878/g2869/uni007C\n/g1878/g2869/g4679 /g3404 /g3436/uni0063/uni006F/uni0073 /g2016\n/uni0032/uni002C/uni0073/uni0069/uni006E /g2016\n/uni0032/g1857/g2879/g3036/g3109 /g3440/uni0020/uni003B/uni0020/uni0020/g4678/g1878/g2869/uni007C/g1878/g2870/uni007C\n/g1878/g2870/uni002C/uni007C/g1878/g2870/uni007C/uni002C/g4679 /g3404 /g3436/uni0063/uni006F/uni0073 /g2016\n/uni0032/g1857/g3036/g3109 /uni002C/uni0073/uni0069/uni006E /g2016\n/uni0032/g3440/uni0020/uni002C \n \n                     (32) \nThe former corresponds to /g4666/g1878/g2869/uni002C/g1878/g2870/g4667/g3400/uni007C/g3053/g3117/uni007C\n/g3053/g3117 , the latter /g4666/g1878/g2869/uni002C/g1878/g2870/g4667/g3400/uni007C/g3053/g3118/uni007C\n/g3053/g3118 .  The same point of /g4666/g1878/g2869/uni002C/g1878/g2870/g4667/g1488 /g1845/g3019/g2871 can be re-\nexpressed in Eq. (32) in two different forms known a s the cross section, each corresponding to one part  of the /g1845/g3029/g2870 ; \n/g4672/uni0063/uni006F/uni0073 /g3087\n/g2870/uni002C/uni0073/uni0069/uni006E /g3087\n/g2870/g1857/g2879/g3036/g3109 /g4673/uni0020 corresponds to /g1847/g3015/g3404 /g1845/g3029/g2870/g3398/g4666/uni0030/uni002C/uni0030/uni002C/g3398/uni0031/g4667 , /g4672/uni0063/uni006F/uni0073 /g3087\n/g2870/g1857/g3036/g3109 /uni002C/uni0073/uni0069/uni006E /g3087\n/g2870/g4673 corresponds to /g1847/g3020/g3404 /g1845/g3029/g2870/g3398/g4666/uni0030/uni002C/uni0030/uni002C/g3397/uni0031/g4667 . \nEquations (32) can also be written in formal mathema tics: \n/g1871/g3015/uni0020/g4666/g1870 /g1488 /g1847/g3015/uni0020/g1867/g1866/uni0020/g1845/g2870/g4667/g3404 /g3436/uni0063/uni006F/uni0073 /g2016\n/uni0032/uni002C/uni0073/uni0069/uni006E /g2016\n/uni0032/g1857/g2879/g3036/g3109 /g3440/uni003B/uni0020/uni0020/g1871/g3020/uni0020/g4666/g1870/uni0020 /g1488 /g1847/g3020/uni0020/g1867/g1866/uni0020/g1845/g2870/g4667/g3404/uni0020/g3436/uni0063/uni006F/uni0073 /g2016\n/uni0032/g1857/g3036/g3109 /uni002C/uni0073/uni0069/uni006E /g2016\n/uni0032/g3440/uni002E \n                   (33) \nWe have established the two cross sections of a par ticular point of /g4666/g1878/g2869/uni002C/g1878/g2870/g4667/g1488 /g1845/g3019/g2871, each corresponding to a chart on \nthe  /g1845/g3029/g2870.  We will now work on the inclusion map /g4666/g1350/uni00B0/g4667 of these points on /g1845/g3019/g2871.  We have  \n/g1350/uni00B0/g3436/uni0063/uni006F/uni0073 /g2016\n/uni0032/g1857/g3036/g3084/g3117/uni002C/uni0073/uni0069/uni006E /g2016\n/uni0032/g1857/g3036/g3084/g3118/g3440 /g3404 /g3436/uni0063/uni006F/uni0073 /g2016\n/uni0032/uni0063/uni006F/uni0073/g2013/g2869/uni002C/uni0020/uni0020/uni0073/uni0069/uni006E /g2016\n/uni0032/uni0063/uni006F/uni0073/g2013/g2870/uni002C/uni0020/uni0063/uni006F/uni0073 /g2016\n/uni0032/uni0073/uni0069/uni006E/g2013/g2869/uni002C/uni0020/uni0073/uni0069/uni006E /g2016\n/uni0032/uni0073/uni0069/uni006E/g2013/g2870/uni0020/g3440 \n                         (34) \n/g1350/uni00B0/g1871/g3015/uni0020/g4666/g1870 /g1488 /g1847/g3015/uni0020/g1867/g1866/uni0020/g1845/g2870/g4667/g3404 /g1350/uni00B0/g3436/uni0063/uni006F/uni0073 /g2016\n/uni0032/uni002C/uni0073/uni0069/uni006E /g2016\n/uni0032/g1857/g2879/g3036/g3109 /g3440 /g3404 /g3436/uni0063/uni006F/uni0073 /g2016\n/uni0032/uni002C/uni0030/uni002C/uni0020/uni0073/uni0069/uni006E /g2016\n/uni0032/uni0063/uni006F/uni0073/g1486/uni002C/g3398/uni0073/uni0069/uni006E /g2016\n/uni0032/uni0073/uni0069/uni006E/g1486/uni0020/g3440  \n                 (35) \n/g1350/uni00B0/g1871/g3020/g4666/g1870 /g1488 /g1847/g3020/uni0020/g1867/g1866/uni0020/g1845/g2870/g4667/g3404 /g1350/uni00B0/g3436/uni0063/uni006F/uni0073 /g2016\n/uni0032/g1857/g3036/g3109 /uni002C/uni0073/uni0069/uni006E /g2016\n/uni0032/g3440 /g3404 /g3436/uni0063/uni006F/uni0073 /g2016\n/uni0032/uni0063/uni006F/uni0073/g1486/uni002C/uni0063/uni006F/uni0073 /g2016\n/uni0032/uni0073/uni0069/uni006E/g1486/uni002C/uni0073/uni0069/uni006E /g2016\n/uni0032/uni002C/uni0030/uni0020/g3440 \n                           (36) \nThe above simply means that the /g2033 defined in terms of /g4666/g1876/g2869/uni002C/g1876/g2870/uni002C/g1877/g2869/uni002C/g1877/g2870/uni0020/g4667 can be re-expressed in terms of /g4666/g2016/uni002C/g1486/g4667 on a \nre-parametrized /g1845/g3019/g2871 . In summary, the gauge corresponding to the North  and South charts can be derived as follows: 19 \n /g1827/g2285/g3404/g1328\n/uni0032/g1857/g4666/g1350/uni00B0/g1871/g3015/uni0020/uni0020/g4667/g1499/g2033 /g3404/g1328\n/uni0032/g1857/g4678/g3398/uni0063/uni006F/uni0073 /uni03B8\n/uni0032/uni0073/uni0069/uni006E/g1486/uni002E/g1856/g3436/uni0063/uni006F/uni0073 /uni03B8\n/uni0032/uni0063/uni006F/uni0073/g1486/g3440/g3397/uni0063/uni006F/uni0073 /uni03B8\n/uni0032/uni0063/uni006F/uni0073/g1486/uni002E/g1856/g3436/uni0063/uni006F/uni0073 /uni03B8\n/uni0032/uni0073/uni0069/uni006E/g1486/g3440/g3398/uni0030/uni002E/g1856/g3436/uni0073/uni0069/uni006E /uni03B8\n/uni0032/g3440/g3397/uni0073/uni0069/uni006E /g2016\n/uni0032/uni002E/g1856/g4666/uni0030/g4667/g4679\n/g3404 /g3398/g1328\n/uni0032/g1857/g4666/uni0031/g3397/uni0063/uni006F/uni0073/g2016 /g4667/g1856/g1486 \n                                                    (37) \n/g1827/g2280/g3404/g1328\n/uni0032/g1857/g4666/g1350/uni00B0/g1871/g3015/uni0020/uni0020/g4667/g1499/g2033 /g3404/g1328\n/uni0032/g1857/g4678/g3398/uni0030/uni002E/g1856/g3436/uni0063/uni006F/uni0073 /uni03B8\n/uni0032/g3440/g3397/uni0063/uni006F/uni0073 /uni03B8\n/uni0032/uni002E/uni0064/g4666/uni0030/g4667/g3397/uni0073/uni0069/uni006E /uni03B8\n/uni0032/uni0073/uni0069/uni006E/g1486/uni002E/g1856/g3436/uni0073/uni0069/uni006E /g2016\n/uni0032/uni0063/uni006F/uni0073/g1486/g3440/g3397/uni0073/uni0069/uni006E /g2016\n/uni0032/uni0063/uni006F/uni0073/g1486/uni002E/g1856/g3436/g3398/uni0073/uni0069/uni006E /g2016\n/uni0032/uni0073/uni0069/uni006E/g1486/g3440/g4679\n/g3404/g1328\n/uni0032/g1857/g4666/uni0031/g3398/uni0063/uni006F/uni0073/g2016 /g4667/g1856/g1486 \n (38) \nNoting the above, the gauge potential in the overla p regions are: /g1827/g2280/g3404 /g1827/g2285/g3397/g1856/g4672/uni0032/g1328\n/g2870/g3032/g1486/g4673.  The transition functions are: \n/g1859/g3020/g3015 /g3404/g1878/g2870/uni007C/g1878/g2870/uni007C/uni2044\n/g1878/g2869/uni007C/g1878/g2869/uni007C/uni2044/g3404 /g1857/g2879/g3036/g1486/uni0020/uni0020/uni003B/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/uni0020/g1859 /g3015/g3020 /g3404/g1878/g2869/uni007C/g1878/g2869/uni007C/uni2044\n/g1878/g2870/uni007C/g1878/g2870/uni007C/uni2044/g3404 /g1857/g3036/g1486/uni0020/uni002E \n   (39) \nThe topological magnetic charge is  /uni2126 /g3404/g1516/g1856/g1827/g2280/g3404/g1516/g1856/g1827/g2285/g3397/g1856/g1856/g4672/uni0032/g1328\n/g2870/g3032/g1486/g4673 /g3404/g1516/g1856/g1827/g2285 . It is thus apparent that one could \navoid the singular part of /g1827/g2280 by switching to /g1827/g2285 in the overlapping region, thus avoiding the conce ptual difficulty of \nthe Dirac string.  \n \nReferences \n[1]  H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. A be, T. Dietl, Y. Ohno, and K. 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"    },    {        "title": "2307.00471v1.Unveiling_Stable_One_dimensional_Magnetic_Solitons_in_Magnetic_Bilayers.pdf",        "content": "Unveiling Stable One-dimensional Magnetic Solitons in Magnetic Bilayers\nXin-Wei Jin,1, 2Zhan-Ying Yang,1, 2,∗Zhimin Liao,3Guangyin Jing,1,†and Wen-Li Yang2, 4\n1School of Physics, Northwest University, Xi’an 710127, China\n2Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China\n3School of Physics, Peking University, Beijing, 100871,China\n4Insititute of Physics, Northwest University, Xi’an 710127, China\n(Dated: July 4, 2023)\nWe propose a novel model which efficiently describes the magnetization dynamics in a magnetic\nbilayer system. By applying a particular gauge transformation to the Landau-Lifshitz-Gilbert (LLG)\nequation, we successfully convert the model into an exactly integrable framework. Thus the obtained\nanalytical solutions allows us to predict a 1D magnetic soliton pair existed by tunning the thickness of\nthe spacing layer between the two ferrimagnetic layers. The decoupling-unlocking-locking transition\nof soliton motion is determined at various interaction intensitiy. Our results have implications for\nthe manipulation of magnetic solitons and the design of magnetic soliton-based logic devices.\nIntroduction.— The intricate interplay of multiple\ninteractions in magnetic materials generates a large class\nof localized spin textures —magnetic solitons [1–10].\nThese solitons exhibit distinct and varied configurations\nin different dimensions and hold great promise as\ncandidates for the next generation of magnetic storage\ndevices [11, 12]. Instead of static magnetic interactions,\ndynamic magnetic interactions [13–15] have been recently\npredict and observed by the current-induced torque\nor non-equilibrium spin pumping [16–22]. Within the\ndynamic coupling magnetic interaction, two magnets\ncan be coherently and tunable coupled at the macro\ndistance, presenting a novel avenue for the coherent\ntransfer of magnon excitation between distinct magnetic\nsystems [22, 23]. Furthermore, these developments\nraises also an intriguing question of the existence and\nregulation of attractive magnetic solitons in magnetic\nbilayer structures [24–26].\nExtensive efforts have been dedicated to the quest\nfor stable magnetic solitons in theory, experiments, and\nmicromagnetic simulations [27–37]. The dynamics of\nmagnetic solitons are described by the Landau-Lifshitz-\nGilbert (LLG) equation [21, 38]. However, for decades,\ndue to the intricate nature of this highly nonlinear\ncoupled equations with multiple interactions, finding\nanalytical solutions are extremely challenging and is a\nlong-standing problem [39, 40]. The lack of comprehen-\nsive analytical solutions hinders progress, necessitating\ntime-consuming and labor-intensive experiments and\nsimulations, without the guidance of a solid theoretical\nframework. The dynamic coupling magnetic interaction\nnot only unveils a host of fresh physical phenomena but\nalso amplifies the complexity of solving the coupled LLG\nequation from a theoretical standpoint.\nIn this letter, we establish an exchange-\ncoupled magnetic bilayer structure,\nferromagnetic/normal/ferromagnetic (F/N/F), as a\nmodel system. From the coupled LLG equations\ngoverning the magnetization dynamics in the\nferromagnetic bilayers, a theoretical model at smallamplitude approximation is developed. A gauge\ntransformation is proposed allowing us to convert the\nproblem into an integrable model, which is applicable\nwhen the intermediate layer thickness is appropriately\nchosen. Thereafter, the exact solution of the governing\nequation is achineved, and the analytical magnetic\nsoliton solutions are subsequently obtained. By adjusting\nthe strength of dynamic magnetic coupling, we find that\nthe magnetic soliton pairs in the ferromagnetic bilayer\nundergo a decoupled-unlocking-locking transition. We\nalso examine the influence of Gilbert damping in\nmaterials on the design of practical devices. These\nresults illustrate practical ways to control the one-\ndimentional magnetic solitons, in which three motion\nstates are successfully released: anti-parallel moving,\nsplitting oscillation, and the locking soliton pair.\nModeling.— We consider a magnetic bilayers system\nas illustrated in Fig. 1, which consists of two coupled\nferromagnetic (FM) films and a nonmagnetic interlayer\nwith thicknesse of s. The FM layers are assumed to be\nparallel to each other with equal thicknesses d1=d2=d.\nThe dynamics of the unit magnetization vector miin the\nparallel coupled ferromagnetic layers can be described by\nthe Landau-Lifshitz-Gilbert equation\n∂mi\n∂t=−γimi×Hi\neff+αi\u0012\nmi×∂mi\n∂t\u0013\n−γiJ\nsMs,imi×mj.\n(1)\nwhere γiis the gyromagnetic ratio, αi>0 denotes\nGilbert damping parameter of each FM layer, Ms,iis\nthe saturation magnetization, and Jrepresents coupling\nstrength between miandmjwith i, j= 1,2. Moreover,\nthe effective field of the two FM layers can be obtained\nfrom the free energy density of the system as Hi\neff=\n−1\nµ0δE\nδmi. We assume the total energy incorporates the\ncontributions from the Zeeman energy due to an applied\nmagnetic field H0= (0,0, h), the exchange interaction\nparametrized by an exchange constant Ai, and the\nperpendicular magnetic anisotropy energy. Thus, it takes\nthe form Hi\neff=H0+(2Ai/Ms,i)∇2mi+(2Ki/Ms,i)(mi·arXiv:2307.00471v1  [cond-mat.mes-hall]  2 Jul 20232\nFigure 1: Sketch of the\nferromagnetic/normal/ferromagnetic thin film bilayer\nsystem. The magnetic soliton excitations propagate\nalong x-axis. As a reference, the top (bottom) FM layer\nis labeled i= 1(i= 2). Their corresponding thicknesses\nare represented by d1andd2, respectively. Parameter s\ndenotes the thickness of the nonmagnetic interlayer.\nn)n, where n= (0,0,1) is the unit vector directed along\nthe anisotropy axis. For simplicity, we transform the\ncoupled LLG equation (1) to the dimensionless form\n∂mi\n∂τ=−mi×∂2\n∂ζ2mi−κmi×(mi·n)n−J′mi×mj,\nby rescaling the space and time into ζ=λ−1\nex·x,τ=\nγµ0Ms·t. Here λex=q\n2Ai/(µ0M2\ns,i) is the exchange\nlength, κ= 2Ki/(µ0M2\ns,i) and J′=J/(µ0sM2\ns,i) denote\nthe dimensionless easy-plane anisotropy constant and\ndimensionless coupling strength, respectively. Table I\nsummarizes the realistic physical constants and parame-\nters used for the structure under our consideration.\nTake into account the fact that the magnitude of\nthe magnetization m2\ni= 1 at temperature well below\nthe Curie temperature, we reasonably introduce a\nstereographic transformation Φ j=mx\nj+imy\nj,\u0000\nmz\nj\u00012=\n1−|Φj|2. Furthermore, let us consider small deviations of\nmagnetization mifrom the equilibrium direction (along\nthe anisotropy axis), which corresponds to\u0000\nmx\nj\u00012+\u0000\nmy\nj\u00012≪\u0000\nmz\nj\u00012(or|Φj|2≪1) and therefore mz\nj≈\n1− |Φj|2/2. As a result, the dynamics of the spinor\nΦ= (Φ 1,Φ2)Tcan be expressed as\ni∂\n∂τΦ=∂2\n∂ζ2Φ+ (J′σ1−∆)Φ+S⊙(ΦΦ†)Φ,(2)\nwhere we have defined S=κ(σ3)2/2 +J′σ1/2 and\n∆ =J′+h+2κ, with σ1,2,3of the Pauli matrices. Symbol\n⊙represents the Hadamard product for matrices. A\nnoteworthy remark extracted here is that by maintaining\nTable I: The physical constants and parameters used.\nPhysical constants/parameters Symbol Value Unit\nGyromagnetic ratio γ 1.76×1011 rad\ns·T\nSaturation magnetization Ms 5.8×105 A\nm\nExchange stiffness A 1.3×10−11 J\nm\nMagnetic anisotropy K 5×105 J\nm3\nMagnetic permeability in vacuum µ0 4π×10−7 H\nm\nDamping parameter α 0.01∼0.05a suitable separation between two ferromagnetic layers\n(s=J/2K)), it becomes possible to introduce a gauge\ntransformation Φ 1,2=1√\n2\u0000\nΨ1ei(h+κ)τ±Ψ2ei(h+3κ)τ\u0001\n,\nmaking the dynamic model (1) entirely integrable. Then,\nthe new spinor Ψ= (Ψ 1,Ψ2)Tis determined by the\nManakov equation with arbitrary constant coefficients:\niΨτ=Ψζζ+κ(ΨΨ†)Ψ. (3)\nA diversity of solutions of this equation can be\nconstructed using the methods of exactly integrable\nsystems. One can also easily obtain the formulations\nof three components of magnetization by the inverse\ntransformation from Eq. (2).\nMagnetic soliton solutions.— Non-degenerate soliton\nsolutions of (3) can be constructed with the help of\nthe Hirota bilinear formalism [41, 42], and the first-\nand second- order non-degenerate soliton solutions are\npresented in Supplementary Materials . The final funda-\nmental nondegenerate soliton solutions are characterized\nby four arbitrary complex parameters, describing the\nvelocity and the amplitude of the magnetic soliton in\nboth FM layers, as well as the nonlinear interaction of\nmagnetic solitons between two FM layers.\nFrom the non-degenerate soliton solution of Eq. (3),\nthe formulations of non-degenerate magnetic soliton are\nconstructed. This derived solutions represent several\ncategories of magnetic solitons in this magnetic bilayer\nsystem. Through analyzing these solutions, it becomes\napparent that the magnetic bilayer system possesses\ndiverse spin textures, manifested as dynamical magnetic\nsolitons. As far as we know, experimental observation\nof these magnetic soliton pairs resulting from interlayer\ndynamic interactions are currently lacking. With this\ntheoretical prediction, in the following, we try to discuss\nthe possible generation mechanisms and the practical\napplications by these magnetic soliton pairs in magnetic\nbilayer structures.\nLinear stability analysis.— It has been confirmed that,\nfrom the analytical solution above, there are magnetic\nsolitons allowed in this system, then another important\naspect to be considered is their stability characters. For\npractical applications of magnetic solitons as memory\nunits or drivien objects in spintronics, it is crucial\nto maintain stability of solitons in the presence of\ninterference. The stability property is usually analyzed\nby way of linear stability analysis [43–45]. For this\npurpose, we consider the solitary wave solutions of\nthe form Ψ=Ψ′exp(ibτ), with bbeing propagation\nconstant, then Eq. (5) becomes\n−bΨ′\nτ=Ψ′\nζζ+κ(Ψ′Ψ′†)Ψ′. (4)\nTo analyze the linear stability of the solitary wave,\nwe perturb the relevant wave function as Ψ i=\b\nΨ′\n0i+ [vi(ζ) +wi(ζ)]eλτ+ [v∗\ni(ζ)−w∗\ni(ζ)]eλ∗τ\t\neibτ,3\nFigure 2: Propagations of stable and unstable non-degenerate magnetic solitons. (a) Left panel: Stability regions in\nthe parameter space (Im( k1),Re(l1)). Center panel: mzprofiles of symmetric flat-bottom-double-hump magnetic\nsoliton at t= 0 and t= 30. Right panel: eigenvalue spectrum. Bottom panel: stable propagations of magnetic\nsoliton in two FM layers. (b) Left panel: Stability regions in the parameter space (Re( k1),Re(l1)). Center panel: mz\nprofiles of asymmetric double-hump-double-hump magnetic soliton at t= 0 and t= 30. Right panel: eigenvalue\nspectrum. Bottom panel: unstable propagations of magnetic soliton in two FM layers.\nhere Ψ′\n0ibeing the general complex-valued unperturbed\nwave function calculated from Eq. (3), viand\nwi(i= 1 ,2) are small perturbations for a given\neigenvalue λ. Inserting this perturbed solution in Eq.\n(3) and linearizing thereafter, we obtain the following\nlinear-stability eigenvalue problem:\niL·W=λ·W. (5)\nwhere matrix W= (v1, w1, v2, w2)Tdenotes the normal-\nmode perturbations. The matrix Lcontains the magnetic\nsoliton solution Ψ′\n0irepresenting the linear stability\noperator. The matrix elements and calculation details\nof matrix Lare presented in Supplementary materials .\nIn general, two separate regions can be defined based\non the linear-stability spectrum. The non-degenerate\nsoliton wave is linearly unstable when the spectrum\ncontains eigenvalues with positive real parts, which\ngives an exponential growth rate of perturbations.\nWhile the soliton is regarded as stable if the spectrum\ncontains purely imaginary discrete eigenvalues [43]. The\nwhole spectrum of the linear-stability operator Lare\nnumerically solved by the Fourier collocation method.\nTo verify the predictions of the linear stability analysis\nobtained from the numerical solution of the spectral\nproblem (4), we proceed to numerically simulate the\nnonlinear propagation of the magnetic solitons. The\nevolutions of stable non-degenerate magnetic solitons and\nunstable non-degenerate magnetic solitons are illustrated\nin Figs. 2. The initial conditions for both simulations\nare taken in the form of a soliton solution perturbed\nby a 10% random noise. The upper panels of Fig.\n2(a) depict the stability regions in the parameter space(Im(k1),Re(l1)) of the magnetic soliton and provide an\nexemplary illustration of a stable soliton solution. The\ncenter panel plots the shape of mzcomponent in two\nferromagnetic layers at t= 0 and t= 30. The whole\nstability spectrum of this non-degenerate soliton is shown\nin the upper right corner panel. It can be seen that this\nflat-bottom-double-hump magnetic solitons propagate\nstably and the flat bottom structure in the first FM\nlayer is maintained, which complies with the results of\nthe linear stability analysis. On the other hand, Fig.\n2(b) shows the unstable propagation of the asymmetric\nsingle-double-hump soliton. Stronger instabilities cause\nthe splitting and diffusion of the solitons at relatively\nshort times.\nCoupling and Gilbert-damping.— The successful sta-\nbilization of non-degenerate solitons enlightens us to\ndesign bilayer ferromagnetic spin-electronic devices based\non stable magnetic solitons. Here, we numerically\ninvestigate the propagation behavior of stable magnetic\nsolitons in FM bilayers with various coupling strengths\n(which corresponds to thickness of the nonmagnetic\nspacer).\nOur first step is to construct a stable magnetic soliton\nin each layer, with opposing velocities. When the two\nferromagnetic layers are far apart from each other, their\ninteraction becomes very weak, and the two layers are\ndecoupling ( J′= 0). The two solitons propagate in\nopposite directions respectively, as depicted in Fig. 3(a)\nand 3(b). An increase of the coupling strength leads\nto soliton separation in both FM layers (as shown in\nFig. 3(c) and 3(d)). The interlayer interaction causes\nsolitons to oscillate and propagate towards both ends at4\nFigure 3: The decoupling, unlocking and locking regions\nof magnetic soliton motion. (a)(b) Propagations of mz\n1\nandmz\n2with dimensionless coupling strength J′= 0.\n(c)(d) Propagations of mz\n1andmz\n2with dimensionless\ncoupling strength J′= 10. (e)(f) Propagations of mz\n1\nandmz\n2with dimensionless coupling strength J′= 15.\n(g) Phase diagram for the tristate transition by\nadjusting the interlayer coupling strength.\na constant velocity. This observation can be explained\nas follows. As the thickness of the intermediate layer\nreduces, the long-range dynamic interaction between the\ntwo FM layers, induced by adiabatic spin-pump, starts to\ncome into play. The dynamic magnetization, which arises\nfrom the moving magnetic solitons in the ferromagnetic\nlayer, causes the formation of non-equilibrium spin\nflow between the two layers. This ultimately triggers\nthe bidirectional oscillation transmission of magnetic\nsolitons. We highlight that as the two ferromagnetic\nlayers continue to approach, the interlayer dynamic\ninteraction will exceed a certain threshold, which\nbecomes sufficient to rapidly synchronize the motion of\nmagnetic solitons and balance the spin current. Two\nsolitons thereby get trapped in a stationary position (See\nFig. 3(e) and 3(f)). This dynamic region of soliton\nimmobilization is henceforth referred to as the locking\nregion. These simulation results in the wider range of\nJ′are summarized in Fig. 3(g), which clearly shows the\ndecoupling-to-unlocking-to-locking transition. The black\nand red lines in the figure represent the minimum values\nof soliton signals received by the signal receiving devices\nplaced at both ends of the first layer FM under different\ncoupling strengths.\nThe different behaviors of magnetic solitons in FM\nbilayers under varying coupling strengths inspire usto design a logic signal generator. By adjusting the\nspacing between the two ferromagnetic layers, which\nis highly controllable in practice, it is possible to\nachieve different outputs of logical signals (Decoupling\nstate corresponds to “10” and “01”, unlocking state\ncorresponds to “11”, and locking state corresponds to\n“00”). This suggests a new posibility towards utilizing\nspintronic devices for logic operations. In practical\napplications, the signal attenuation caused by Gilbert\ndamping in ferromagnetic materials must be considered.\nThrough numerical simulation, we find that the damping\neffect has a significant impact on the magnetic solitons in\nthe unlocking state. Fig. 4(a) shows the propagation of\nmagnetic solitons in the unlocking state in the upper FM\nlayer with Gilbert-damping constant α1= 0.05, where\nLmaxrepresents the maximum distance at which the\nsignal attenuates to an unrecognizable state (assuming\nthat the mzcomponent is greater than 0.8). The\ndependence of Lmaxon the damping constant αfor the\nFM layer is shown in Fig. 4(b). It can be observed that\nopting for materials featuring low damping coefficients\ncan significantly increase the separation between signal\nreceivers.\nDiscussion and Conclusion.— To sum up, we have\nderived a model at a small amplitude approximation to\ndescribe the nonlinear dynamics of magnetization in a bi-\nlayer ferromagnetic system. When the intermediate layer\ntakes a characteristic thickness (i.e., 2 nm), s=J/2Kfor\nthe system here, and the dynamic interaction coupling\nFigure 4: The effect of Gilbert-damping on the motion\nof magnetic soliton in unlocking phase. (a)\nPropagations of mz\n1with dimensionless coupling\nstrength J′= 10, damping constant α= 0.05,Lmax\nrepresents the maximum distance at which the signal\nattenuates to an unrecognizable state. (b) Dependence\nof the maximum distance Lmaxfor damping constant α.\n(c) Sketch of maximum separation distance between\nidentifiable magnetic soliton signals.5\nparameter and magnetic anisotropy are taken as 2 mJ /m2\nand 5 ×105J/m3, it is possible to introduce a gauge\ntransformation to transform the equation into a fully\nintegrable constant coefficient Manakov system. The\nfirst-order and second-order non-degenerate magnetic\nsoliton solutions are obtained, as well as their respective\nstability regions. The numerical simulation results of\nmagnetic soliton transmission are well consistent with the\npredictions given by linear stability. These theoretical\nand numerical results confirm the existence of stable one-\ndimensional magnetic soliton pairs in magnetic bilayer\nsystem. To generate such magnetic solitons in a F/N/F\nbilayer system, the magnetization texture based on the\nabove magnetic soliton solution must be manufactured\ninto the two ferromagnetic layers. This excited solitions\ncan be achieved for example by a local magnetic filed or\nspin-polarized electric currents.\nOn the other hand, the intensity of the interlayer\nlong-range dynamic interaction, induced by adiabatic\nspin-pump, can be tailored by manipulating the spacing\nbetween the two FM layers. Through the manipulation\nof the intermediate layer’s thickness, we unveiled three\ndistinct transport states of magnetic solitons: soliton\ndecoupling, unlocking, and locking. With a gradual\nincrement in dynamic interactions, we demonstrated the\nprogression of magnetic soliton motion from decoupling\nto unlocking, and ultimately to locking. It is note that\nthe dynamic exchange coupling strength Jis related\nto the thickness of the spacing layer. We postulate\nan inverse square root relationship between the two\nparameters [22], i.e. J∝1/√s. Through calculations\nbased on the parameters we have considered, it is\ndetermined that when the thickness of the intermediate\nlayer is less than 0.45 nm, magnetic solitons initiate a\ntransition towards the locking state. Note that, the\nthickness of this transition is related to the selection of\nferromagnetic layer and insulating spacer layer materials.\nFor the same material, various material properties\nsuch as the saturation magnetization Ms,iand the\ninterfacial dynamic coupling of the synthetic layers can\nbe controlled within the reach of leading-edge material\nfabrication and deposition techniques [25].\nFinally, we examine the impact of Gilbert damping\nin different ferromagnetic materials on this transitional\nprocess. Our findings reveal that damping predominantly\nresults in the attenuation of magnetic solitons in the\nunlocking state. Furthermore, we have established a\ncorrelation between the damping coefficient and the\nmaximum separation distance between distinguishable\nmagnetic soliton signals. These findings present new\npossibilities for developing spintronic devices for logic\ncomputing based on magnetic solitons, and have ignited\nextensive research on these systems to refine their design\naccording to specific application requirements.\nThe authors thank Prof. H. M. Yu and Prof. C. P. Liufor their helpful discussions. This work was supported\nby the National Natural Science Foundation of China\n(Nos. 12275213, 12174306, 12247103), and Natural\nScience Basic Research Program of Shaanxi (2023-JC-\nJQ-02, 2021JCW-19).\n∗zyyang@nwu.edu.cn\n†jing@nwu.edu.cn\n[1] M. Ahlberg, S. Chung, S. Jiang, A. Frisk, M. Khademi,\nR. Khymyn, A. A. Awad, Q. T. Le, H. Mazraati,\nM. Mohseni, et al. , Nature Communications 13, 2462\n(2022).\n[2] C. Moutafis, S. Komineas, and J. Bland, Physical Review\nB79, 224429 (2009).\n[3] H. Wang, R. Yuan, Y. Zhou, Y. Zhang, J. Chen, S. Liu,\nH. Jia, D. Yu, J.-P. Ansermet, C. Song, et al. , Physical\nReview Letters 130, 096701 (2023).\n[4] C. Liu, J. 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Zeng, Communications in Nonlinear\nScience and Numerical Simulation 102, 105911 (2021)."    },    {        "title": "1111.5115v2.On_magnetic_leaf_wise_intersections.pdf",        "content": "arXiv:1111.5115v2  [math.SG]  13 Feb 2014ON MAGNETIC LEAF-WISE INTERSECTIONS\nYOUNGJIN BAE\nAbstract. In this paper we study leaf-wise intersections in a setting w here the Hamiltonian\nperturbation has magnetic effects. In particular, we establ ish their existence under certain\ntopological assumptions on the ambient manifold.\n1.Introduction\nLet (N,g) be a closed connected orientable Riemannian manifold and τ:T∗N→Nbe\nits cotangent bundle with the canonical symplectic form ωstd=dp∧dq. Here (q,p) are the\ncanonical coordinates on T∗N. Let Σ be a hypersurface in an exact symplectic manifold\n(T∗N,dλ,λ=pdq) such that (Σ ,α:=λ|Σ) is a contact manifold.\nAs a main example, such a hypersurface can be obtained from a s ufficiently high energy\nlevel of a mechanical Hamiltonian\nG=1\n2|p|2+U(q),\ni.e. a closed energy hypersurface G−1(k)⊂T∗N, k > maxq∈NU(q) is a contact manifold\nwith a contact form λ|G−1(k). Here|p|denotes the dual norm of the Riemannian metric gon\nNandU:N→Ris a smooth potential. This Hamiltonian system describes th e motion of\na particle on Nsubject to the conservative force −∇U.\nThe contact hypersurface Σ is foliated by the leaves of the ch aracteristic line bundle which\nis spanned by the Reeb vector field Rofα. LetφΣ\nt: Σ→Σ be the flow of R. Forx∈Σ we\ndenote byLxthe leaf through xwhich can be parametrized as Lx={φΣ\nt(x) :t∈R}. IfLx\nis closed, we call Lxa closed Reeb orbit andxa periodic point .\nIn order to introduce leaf-wise intersections, we need the f ollowing definitions.\nDefinition 1.1. Amagnetic perturbation σ∈C∞(R/Z,Ω2(N)) is a time-dependent closed\n2-form onNwhich satisfies the following conditions:\n•/tildewideσ(t) =dθ(t), for some θ∈C∞(R/Z,Ω1(/tildewideN)) for allt∈R/Z;\n•σ(t) = 0,θ(t) = 0 for all t∈[0,1\n2];\n•θ(t)∈Ω1(/tildewideN) is bounded for all t∈[1\n2,1].\nHere/tildewideσ(t) is the lift of σ(t) to the universal cover /tildewideNand such a 2-form σ(t) which has a\nbounded primitive on the universal cover is called /tildewided-bounded .\nLetMbe the set of such magnetic perturbations and Pbe the set of primitives of magnetic\nperturbations on the universal cover. We consider an R/Z-parametrized symplectic form\nωσ:R/Z→Ω2(T∗N) as follows\nωσ:=ωstd+τ∗σ. (1.1)\nKey words and phrases. Leaf-wise intersection point, Floer homology, Rabinowitz Floer homology, Isoperi-\nmetric inequality, Symplectically hyperbolic manifold.\n12 YOUNGJIN BAE\nDefinition 1.2. We define\nH:={H∈C∞\nc(R/Z×T∗N) :H(t,·) = 0,∀t∈[0,1\n2]}.\nThroughoutthisarticle, wefixan H∈ Handthediffeomorphism ϕwhichisthetime-1-map of\nthe Hamiltonian vector field Xσ\nHdefined byιXσ\nHωσ=−dH. Whenσ≡0, the diffeomorphism\nϕbecomes a (non-magnetic) Hamiltonian diffeomorphism of ( T∗N,ωstd).\nDefinition 1.3. A pointx∈Σ is called a magnetic leaf-wise intersection point , ifϕ(x)∈Lx.\nIn other words, there exists η∈Rsuch that\nφΣ\nη(ϕ(x)) =x. (1.2)\nNote that a leaf-wise intersection point is the σ= 0 case of a magnetic leaf-wise intersection\npoint. The leaf-wise intersection problem asks whether a gi ven diffeomorphism has a leaf-wise\nintersection point in a given hypersurface Σ. If there exist leaf-wise intersections one can ask\nfurther a lower bound on the number of leaf-wise intersectio ns. This problem was introduced\nby Moser in [28], and studied further in [10, 14, 19, 17, 13, 18 , 31, 5, 6, 7, 8, 20, 21, 22, 27].\nSee [4] for the brief history of these problems. In this artic le, we investigate the approaches\nin [3, 24] and generalize their results.\nWe call a hypersurface Σ ⊂T∗Nnon-degenerate if closed Reeb orbits on Σ form a discrete\nset. A generic Σ is non-degenerate, see [11, Theorem B.1]. If Σ is non-degenerate, then\nperiodic leaf-wise intersection points can be excluded by c hoosing a generic Hamiltonian\nfunction, see [3, Theorem 3.3]. In this setting, a leaf-wise intersection point xhas the unique\nreal number ηsuch that (1.2) is satisfied. By these reason, we only conside rnon-periodic\n(magnetic) leaf-wise intersection points.\nIn order to state the main result, we need the following notio n. LetLNbe the free loop\nspace of (N,g). The energy functional E:LN→Ris given by\nE(q) :=/integraldisplay1\n01\n2|˙q|2dt.\nFor given 0 <T <∞, denote by\nLN(T) :=/braceleftbigg\nq∈ LN:E(q)≤1\n2T2/bracerightbigg\n.\nLet Σ be a non-degenerate fiberwise starshaped hypersurface andϕbe a generic diffeomor-\nphism. Given T >0 let us define\nn(T) =nΣ,ϕ(T) := #{x∈T∗N:φΣ\nη(ϕ(x)) =x,for someη∈(0,T)}.\nTheorem 1.4. LetNbe a closed connected oriented manifold of dimension n≥2. LetΣbe\na non-degenerate fiberwise starshaped hypersurface in T∗N. Letgbe a bumpy Riemannian\nmetric onNwithS∗\ngNcontained in the interior of the compact region bounded by Σ. Assume\nthatϕis generic. Then there exists a constant c=c(N,g,Σ,ϕ)>0such that the following\nholds: For all sufficiently large T >0,\nn(T)≥1\nc·rank{ι: H∗/parenleftbig\nLN(c(T−1))/parenrightbig\n→H∗(LN)}. (1.3)\nUnder a certain topological assumption on N, the right hand side of (1.3) grows exponen-\ntially withT. We denote by /tildewideπ1(N) the set of conjugacy classes of π1(N). Then the connectedON MAGNETIC LEAF-WISE INTERSECTIONS 3\ncomponents of LNcorresponds to the elements of /tildewideπ1(N), hence the exponential growth rate\nof/tildewideπ1(N) implies that\nliminf\nT→∞rank{ι: H0(LN(T))→H0(LN)} (1.4)\nhas also exponential growth with respect to T, see [26]. Then the following corollary comes\nfrom exponential growth of (1.4) and Theorem 1.4.\nCorollary 1.5. LetNbe a closed connected oriented manifold of dimension n≥2. Let\nΣbe a non-degenerate fiberwise starshaped hypersurface in T∗N. Suppose that /tildewideπ1(N)has\nexponential growth. If ϕis generic then n(T)grows exponentially with T.\nThe main example of such Nis any surface of genus greater than one. In these cases,\nthe magnetic field σcan be chosen by the volume form of that surface. Other candid ates\nforNare thesymplectically hyperbolic manifolds which will be discussed in Definition 3.1,\nProposition 3.2.\nIn proving Theorem 1.4, we heavily need a recent result by Mar carini, Merry and Paternain\n[24]. With the same assumption as in Theorem 1.4 and a generic non-magnetic Hamiltonian,\nthey showed the exponential growth rate of leaf-wise inters ections. In this paper, we extend\ntheir result to the magnetic case by using a method developed in [9].\nMore precisely we will use a variational approach associate d to magnetic leaf-wise inter-\nsections. The main tools are Rabinowitz Floer homology and i ts variations. In order to show\nthe main result, invariance of Rabinowitz Floer homology is crucial and we take advantage of\ncontinuation map between two different Rabinowitz Floer chai n complexes.\nAcknowledgement: I am grateful to my advisor Urs Frauenfelder for fruitful dis cussions\nand detailed comments. I also thank the anonymous referee. T he author is supported by the\nBasic research fund 2010-0007669 funded by the Korean gover nment.\n2.A perturbation of the Rabinowitz action functional\nLet us begin with a defining Hamiltonian ¯Fof a contact hypersurface Σ ⊂T∗Nwhich is\nconstant outside of a compact set containing Σ.\nDefinition 2.1. Given a fiberwise starshaped hypersurface Σ ⊂T∗N,\n¯D(Σ) :={¯F∈C∞(T∗N) :¯F−1(0) = Σ, X¯F|Σ=R, X¯Fis compactly supported }.\nWiththedefiningHamiltonian ¯F∈¯D(Σ), theRabinowitzactionfunctional A¯F:L×R→R\nis defined by\nA¯F(u,η) :=/integraldisplay1\n0u∗λ−η/integraldisplay1\n0¯F(u(t))dt.\nHereL=LT∗N:=C∞(R/Z,T∗N). The critical points of A¯Fsatisfy\nd\ndtu(t) =ηX¯F(u(t))\n¯F(u(t))dt= 0./bracerightbigg\n(2.1)\nSince the restriction of the Hamiltonian vector field X¯Fto Σ is the Reeb vector field, the\nequations (2.1) are equivalent to\nd\ndtu(t) =ηR(u(t))\nu(t)∈Σ,/bracerightbigg\ni.e.uis a periodic orbit of the Reeb vector field on Σ with period η.4 YOUNGJIN BAE\nIfA¯Fis Morse-Bott, then FH ∗(A¯F) is well-defined, see [11]. By the work of Abbondandolo-\nSchwarz [2] and Cieliebak-Frauenfelder-Oancea [12], we th en have the following non-vanishing\nresult when ∗ /ne}ationslash= 0,1.\nFH∗(A¯F) =/braceleftbigg\nH∗(LN),if∗>1,\nH−∗+1(LN),if∗<0.(2.2)\nHere FH ∗(A¯F) is the Floer homology for A¯FandLNis the free loop space of N.\nNow we introduce a time-dependent defining Hamiltonian in order to consider an action\nfunctional whose critical points give rise to leaf-wise int ersection points.\nDefinition 2.2. Given a fiberwise starshaped hypersurface Σ ⊂T∗N,\nD(Σ) :={F∈C∞(R/Z×T∗N) :F(t,x) =ρ(t)¯F(x),¯F∈¯D(Σ)}.\nHereρ:R/Z→R≥0satisfies\n/integraldisplay1\n0ρ(t)dt= 1 and supp( ρ)⊂(0,1\n2). (2.3)\nRemark that the Hamiltonian vector field XFsatisfies\nXF(t,x) =ρ(t)X¯F(x). (2.4)\nA leaf-wise intersection point x∈Σ with respect to ϕ0∈Hamc(T∗N) can be interpreted\nas a critical point of a perturbed Rabinowitz action functio nalAF\nH:L×R→Rdefined by\nAF\nH(u,η) =/integraldisplay1\n0u∗λ−η/integraldisplay1\n0F(t,u)dt−/integraldisplay1\n0H(t,u)dt.\nHere the additional Hamiltonian H:T∗N→Rgeneratesϕ0. Then a critical point ( u,η) of\nAF\nHis a solution of\nd\ndtu(t) =ηXF(t,u(t))+XH(t,u(t))/integraltext1\n0F(t,u(t))dt= 0./bracerightbigg\n(2.5)\nWe observed in [4] that if ( u,η) is a critical point of AF\nHthenu(1\n2)∈Σ is a leaf-wise inter-\nsection point. For a generic Hamiltonian Hfor which AF\nHis Morse, Albers-Frauenfelder [3]\nconstructed an isomorphism\nFH(AF\nH)∼=FH(A¯F). (2.6)\nNow we construct an action functional whose critical points give rise to magnetic leaf-wise\nintersection points with respect to ϕ=φ1\nXσ\nH. Throughout this article, we fix a primitive θ∈ P\nas in Definition 1.1. We then define an additional term\nBθ:L →R\nu/mapsto→/integraldisplay1\n0/tildewideτ∗θt(/tildewideu(t))[d\ndt/tildewideu(t)]dt.(2.7)\nHere/tildewideτ:T∗/tildewideN→/tildewideN,/tildewideu:R/Z→T∗/tildewideNis a lifting of u. We also fix a fundamental region\nN⊂/tildewideNand assume that /tildewideu(0)∈N.\nA new action functional Aθ:L×R→Ris defined by\nAθ(u,η) =AF\nH,θ(u,η) :=AF\nH(u,η)+Bθ(u)\n=/integraldisplay1\n0u∗λ−η/integraldisplay1\n0F(t,u(t))dt−/integraldisplay1\n0H(t,u(t))dt+/integraldisplay1\n0/tildewideτ∗θt(/tildewideu(t))[d\ndt/tildewideu(t)]dt.ON MAGNETIC LEAF-WISE INTERSECTIONS 5\nA critical point ( u,η)∈ L×RofAθsatisfies\nd\ndtu(t) =ηXF(t,u(t))+Xσ\nH(t,u(t))/integraltext1\n0F(t,u(t))dt= 0./bracerightbigg\n(2.8)\nFor convenience,\nCrit(Aθ) :={w= (u,η)∈ L×R: (u,η) satisfies (2.8) }.\nIn the following proposition we interpret the critical poin t as a magnetic leaf-wise intersection\npoint as in Definition 1.3.\nProposition 2.3. Let(u,η)∈Crit(Aθ). Thenx=u(1\n2)satisfiesϕ(x)∈Lx. Thusxis a\nmagnetic leaf-wise intersection point.\nProof.Fort∈[0,1\n2] we compute, using H(t,·) = 0 for all t≤1\n2,\nd\ndt¯F(u(t)) =d¯F(u(t))·d\ndtu(t)\n=d¯F(u(t))·[η XF(t,u)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=ρ(t)X¯F(u)+Xσ\nH(t,u)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=0] = 0,\nsinced¯F(X¯F) = 0. Hence ¯F(u(t)) =cfor some constant cwhent≤1\n2. Thus,\n0 =/integraldisplay1\n0F(t,u)dt=/integraldisplay1\n0ρ(t)¯F(u(t))dt=c.\nTherefore ¯F(u(t)) =c= 0 and since ¯F−1(0) = Σ, we have u(t)∈Σ fort∈[0,1\n2]. In\nparticular, u(1\n2),u(0) =u(1)∈Σ.\nFort∈[1\n2,1] we have F(t,·) = 0. Thus the loop usolves the equationd\ndtu(t) =Xσ\nH(t,u)\non [1\n2,1], and therefore, u(1) =ϕ(u(1\n2)). We conclude that ϕ(u(1\n2))∈Σ. Fort∈[0,1\n2],\nd\ndtu(t) =ηXF(t,u)+Xσ\nH(t,u) =ηXF(t,u) =ηR, sinceXF|Σ=R. Thismeansthat ϕ(u(1\n2)) =\nu(1) =u(0)∈Lu(1\n2). Thusu(1\n2) is a magnetic leaf-wise intersection point. /square\n2.1.Floer homology for Aθ.In this subsection, we show that FH( Aθ) is well-defined. We\nassume that the readers are familiar with the construction i n Floer theory which can be found\nin [29]. Throughout this subsection, we follow the strategy in [11] with minor modifications.\nDefinition 2.4. Let Σ be a non-degenerate hypersurfacein T∗Nwith a defining Hamiltonian\nF. A diffeomorphism ϕ=φ1\nXσ\nHor a pair (H,θ)∈ H × P is called regularwith respect to\nF∈ D(Σ) if\n(1)Aθ=AF\nH,θis Morse;\n(2)ϕhas no periodic leaf-wise intersection points.\nFor a given non-degenerate closed hypersurface Σ, ϕis regular for generic ( H,θ)∈ H × P .\nWe discuss the regular property further in Appendix A and B. T hroughout this article, we\nassume that ϕis regular.\nRemark 2.5. In order to define gradient flow lines, we need an R/Z-parametrized almost\ncomplex structure J(t) which is compatible with the R/Z-parametrized symplectic form ωσ.\nThis means that\ngt(·,·) :=ωσ(·,J(t)·)6 YOUNGJIN BAE\ndefines a R/Z-parametrized inner product on T∗N. We denote the set of such almost complex\nstructures as Jσ.\nGivenJ(t)∈ Jσ, we denote by ∇JAθthe gradient of Aθwith respect to the inner product\ngJ/parenleftbig\n(ˆu1,ˆη1),(ˆu2,ˆη2)/parenrightbig\n:=/integraldisplay1\n0gt(ˆu1,ˆu2)dt+ ˆη1ˆη2, (2.9)\nwhere (ˆui,ˆηi)∈T(u,η)(L×R) fori= 1,2. One can check that\n∇JAθ(u,η) =/parenleftbigg−J(t,u)/parenleftbigd\ndtu−Xσ\nH(t,u)−ηXF(t,u)/parenrightbig\n−/integraltext1\n0F(t,u)dt/parenrightbigg\n.\nDefinition 2.6. A positive gradient flow line of Aθwith respect to J(t)∈ Jσis a map\nw= (u,η)∈C∞(R,L×R) solving the ODE\nd\ndsw(s)−∇JAθ(w(s)) = 0.\nAccordingtoFloer’s interpretation, thismeans that uandηaresmoothmaps u:R×(R/Z)→\nT∗Nandη:R→Rsatisfying\n∂su+J(t,u)/parenleftbig\n∂tu−Xσ\nH(t,u)−ηXF(t,u)/parenrightbig\n= 0\nd\ndsη+/integraltext1\n0F(t,u)dt= 0./bracerightbigg\n(2.10)\nDefinition 2.7. The energy of a map w∈C∞(R,L×R) is defined as\nE(w) :=/integraldisplay∞\n−∞/ba∇dbld\ndsw(s)/ba∇dbl2\nJds,\nwhere/ba∇dbl·/ba∇dblJ=/radicalbig\ngJ(·,·).\nNote that the energy of a given gradient flow line w∈C∞(R,L×R) with limit conditions\nlims→±∞w(s) =w±∈Crit(Aθ) become\nE(w) =/integraldisplay∞\n−∞/ba∇dbld\ndsw(s)/ba∇dbl2\nJds=/integraldisplay∞\n−∞gJ(∇JAθ(w(s)),d\ndsw(s))ds\n=/integraldisplay∞\n−∞d\ndsAθ(w(s))ds=Aθ(w+)−Aθ(w−).\nRemark 2.8. The most delicate issue in constructing FH( Aθ) is the compactness of the\nmoduli space. Themodulispace of gradient flow lines w(s) = (u,η) ofAθwith the asymptotic\nconditions lim s→±∞w(s) =w±∈Crit(Aθ) can be compactified up to breaking of gradient\nflow lines, see Theorem 2.14. There are three analytic difficul ties we have to overcome:\n(1) a uniform L∞bound onu∈ L;\n(2) a uniform L∞bound onη∈R;\n(3) a uniform L∞bound on the derivatives of u∈ L.\nProperties (1) and (3) are non-trivial but standard problem s in Floer theory and property\n(2) is also treated in [11, 4]. However our gradient flow equat ion (2.10) is slightly generalized\nwith the data σ. Most of the remaining of this subsection is devoted to show p roperty (2) in\nAθcase.\nDefinition 2.9. Define a map c:H×P → [0,∞) by\nc(H,θ) := sup\n(t,u)∈R/Z×L/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0(/tildewideλ+/tildewideτ∗θt)(/tildewideu(t))[/tildewideXσ\nH(t,/tildewideu)]−H(t,u(t))dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle.ON MAGNETIC LEAF-WISE INTERSECTIONS 7\nNote that/tildewideλ+/tildewideτ∗θis a primitive of /tildewideωσon the universal cover T∗/tildewideN. We remind that His\ncompactly supported, and hence c(H,θ) is finite.\nLemma 2.10. Thereexists ǫ>0,/tildewidec>0suchthatif ( u,η)∈ L×Rsatisfies/ba∇dbl∇JAθ(u,η)/ba∇dblJ≤ǫ\nthen\n|η| ≤/tildewidec(|Aθ(u,η)|+1). (2.11)\nHere/ba∇dbl·/ba∇dblJ=/radicalbig\ngJ(·,·).\nProof.The proof consists of 3 steps.\nStep 1:There exist δ >0and a constant cδ<∞such that if u∈ Lsatisfies (t,u(t))∈\nUδ=F−1(−δ,δ)for allt∈[0,1\n2], then\n|η| ≤cδ(|Aθ(u,η)|+/ba∇dbl∇JAθ(u,η)/ba∇dblJ+1).\nThere exists δ>0 such that\nλ(XF(p))≥1\n2+δ,∀p∈Uδ\nWe compute\n|Aθ(u,η)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0u∗λ−/integraldisplay1\n0H(t,u(t))dt−η/integraldisplay1\n0F(t,u(t))dt+Bθ(u(t))/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0/tildewideu∗(/tildewideλ+/tildewideτ∗θt)−/integraldisplay1\n0H(t,u(t))dt−η/integraldisplay1\n0F(t,u(t))dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0(/tildewideλ+/tildewideτ∗θt)(/tildewideu(t))[d\ndt/tildewideu−η/tildewideXF(t,/tildewideu)−/tildewideXσ\nH(t,/tildewideu)]dt\n+η/integraldisplay1\n0λ(u(t))[XF(t,u)]/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≥1\n2+δ−F(t,u(t))/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤δdt\n+/integraldisplay1\n0(/tildewideλ+/tildewideτ∗θt)(/tildewideu(t))[/tildewideXσ\nH(t,/tildewideu)]−H(t,u(t))dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≥1\n2|η|−c′\nθ,δ/ba∇dbld\ndtu−Xσ\nH(t,u)−ηXF(t,u)/ba∇dbl1−c(H,θ)\n≥1\n2|η|−c′\nθ,δ/ba∇dbld\ndtu−Xσ\nH(t,u)−ηXF(t,u)/ba∇dbl2−c(H,θ)\n≥1\n2|η|−c′\nθ,δ/ba∇dbl∇JAθ(u,η)/ba∇dblJ−c(H,θ)\nHere/tildewider[−] means the lifting of [ −] to the universal cover and c′\nθ,δ:=/ba∇dbl(/tildewideλ+/tildewideτ∗θ)|/tildewideUδ/ba∇dbl∞. Set\ncδ= max{2c′\nθ,δ,2c(H,θ),2}then this inequality proves Step 1. Note that the finiteness o fc′\nθ,δ\nis guaranteed by the simple estimate as follows\nc′\nθ,δ=/ba∇dbl/tildewideλ+τ∗θ|/tildewideUδ/ba∇dbl∞\n≤/ba∇dbl/tildewideλ|/tildewideUδ/ba∇dbl∞+/ba∇dbl/tildewideτ∗θ|/tildewideUδ/ba∇dbl∞\n=/ba∇dblλ|Uδ/ba∇dbl∞+/ba∇dblθ|/tildewideτ(/tildewideUδ)/ba∇dbl∞\n≤/ba∇dblλ|Uδ/ba∇dbl∞+/ba∇dblθ/ba∇dbl∞\n<∞.8 YOUNGJIN BAE\nStep 2:There exists ǫ=ǫ(δ)with the following property. If there exists t∈[0,1\n2]with\nF(t,u(t))≥δthen/ba∇dbl∇JAθ(u,η)/ba∇dblJ≥ǫ.\nIf in addition F(t,u(t))≥δ\n2holds for all t∈[0,1\n2] then\n/ba∇dbl∇JAθ(s)(u,η)/ba∇dblJ≥/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0F(t,u(t))dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥δ\n2/integraldisplay1\n0ρ(t)dt=δ\n2.\nOtherwise there exists t′∈[0,1\n2] withF(t,u(t′))≤δ\n2. Thus we may assume without loss of\ngenerality that 0 ≤a < b≤1\n2andδ\n2≤ |F(t,u(t))| ≤δfor allt∈[a,b], and|F(b,u(b))−\nF(a,u(a))|=δ\n2. Then we estimate\n/ba∇dbl∇JAθ(u,η)/ba∇dblJ≥/ba∇dbld\ndtu−Xσ\nH(t,u)−ηXF(t,u)/ba∇dbl2\n≥/parenleftbigg/integraldisplayb\na/ba∇dbld\ndtu−Xσ\nH(t,u)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=0−ηXF(t,u)/ba∇dbl2dt/parenrightbigg1\n2\n≥/integraldisplayb\na/ba∇dbld\ndtu−ηXF(t,u)/ba∇dbldt\n≥1\n/ba∇dbl∇F/ba∇dbl∞/integraldisplayb\na/ba∇dbl∇F(t,u(t))/ba∇dbl·/ba∇dbld\ndtu−ηXF(t,u)/ba∇dbldt\n≥1\n/ba∇dbl∇F/ba∇dbl∞/integraldisplayb\na/vextendsingle/vextendsingle/vextendsingle/vextendsinglegt/parenleftbigg\n∇F(t,u(t)),d\ndtu−ηXF(t,u)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingledt\n=1\n/ba∇dbl∇F/ba∇dbl∞/integraldisplayb\na/vextendsingle/vextendsingle/vextendsingle/vextendsinglegt/parenleftbigg\n∇F(t,u(t)),d\ndtu/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingledt\n=1\n/ba∇dbl∇F/ba∇dbl∞/integraldisplayb\na/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndtF(t,u(t))/vextendsingle/vextendsingle/vextendsingle/vextendsingledt\n≥1\n/ba∇dbl∇F/ba∇dbl∞/integraldisplayb\nad\ndtF(t,u(t))dt\n=δ\n2/ba∇dbl∇F/ba∇dbl∞.\nSince/ba∇dbl∇F/ba∇dbl∞is bounded from above, we set ǫ(δ) := min{δ\n2,δ\n2/ba∇dbl∇F/ba∇dbl∞}. This proves Step 2.\nStep 3:We prove the lemma.\nChooseδas in Step 1, ǫ=ǫ(δ) as in Step 2 and\n/tildewidec=cδ(ǫ+1).\nAssume that /ba∇dbl∇JAθ(u,η)/ba∇dblJ≤ǫthen\n|η| ≤cδ(|Aθ(u,η)|+/ba∇dbl∇JAθ(u,η)/ba∇dblJ+1)≤/tildewidec(|Aθ(u,η)|+1).\nThis proves the lemma. /square\nProposition 2.11. Letw±∈Crit(Aθ)andw= (u,η)be a gradient flow line of Aθwith\nlim\ns→±∞w(s) =w±.\nThen there exists a constant κ=κ(w−,w+)satisfying /ba∇dblη/ba∇dbl∞≤κ.ON MAGNETIC LEAF-WISE INTERSECTIONS 9\nProof.Letǫbe as in Lemma 2.10. For l∈R, letνw(l)≥0 be defined by\nνw(l) := inf{ν≥0 :/ba∇dbl∇JAθ[w(l+ν)]/ba∇dblJ<ǫ}.\nThenνw(l) is uniformly bounded as follows,\nAθ(w+)−Aθ(w−) =/integraldisplay∞\n−∞/ba∇dbld\ndsw(s)/ba∇dbl2ds\n=/integraldisplay∞\n−∞/ba∇dbl∇JAθ(w(s))/ba∇dbl2\nJds\n≥/integraldisplayl+νw(l)\nl/ba∇dbl∇JAθ(w(s))/ba∇dbl2\nJ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≥ǫ2ds\n≥ǫ2νw(l).\nNow, we set\n/ba∇dblF/ba∇dbl∞= max\n(t,x)∈R/Z×T∗N|F(t,x)|, K= max{|Aθ(w+)|,|Aθ(w−)|}.\nSinceF∈ D(Σ), see Definition 2.2, /ba∇dblF/ba∇dbl∞is finite. By definition of νw(l), we get /ba∇dbl∇JAθ[w(l+\nνw(l))]/ba∇dblJ=ǫ. Then we can use Proposition 2.10 to obtain the following est imate\n|η(l+νw(l))| ≤/tildewidec(|Aθ[w(l+νw(l))]|+1)\n≤/tildewidec(K+1).\nBy using the above estimate we get\n|η(l)| ≤ |η(l+νw(l))|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayl+νw(l)\nl˙η(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ |η(l+νw(l))|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayl+νw(l)\nl/integraldisplay1\n0F(t,u(t))dt ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/tildewidec(K+1)+/ba∇dblF/ba∇dbl∞νw(l)\n≤/tildewidec(K+1)+/ba∇dblF/ba∇dbl∞(Aθ(w+)−Aθ(w−))\nǫ2.\nThe right hand side is independent of the gradient flow line wandl∈R. Let\nκ=/tildewidec(K+1)+/ba∇dblF/ba∇dbl∞(Aθ(w+)−Aθ(w−))\nǫ2,\nthen this proves the proposition. /square\nProposition 2.12. Ifσ0∈Ω2(N)is/tildewided-bounded then (ωstd+τ∗σ0)|π2(T∗N)= 0.\nProof.First choose a map f:S2→T∗N, then it suffices to show that/integraltext\nf(S2)ωstd+τ∗σ0= 0.\n/integraldisplay\nf(S2)ωstd+τ∗σ0=/integraldisplay\nf(S2)ωstd+/integraldisplay\nτ◦f(S2)σ0=/integraldisplay\nf(S2)dλ+/integraldisplay\n/tildewideτ◦f(S2)/tildewideσ0\n=/integraldisplay\n∂f(S2)λ+/integraldisplay\n/tildewideτ◦f(S2)dθ0=/integraldisplay\n∂/tildewideτ◦f(S2)θ0\n= 0.\nHere/tildewideτ◦f:S2→/tildewideNis a lifting of τ◦f:S2→N. /square10 YOUNGJIN BAE\nTheorem 2.13. Letwν= (uν,ην)be a sequence of gradient flow lines for which there exists\na<bsuch that\na≤ Aθ/parenleftbig\nwν(s)/parenrightbig\n≤b,∀s∈R.\nThen the sequence {wν}has a subsequence that converges in C∞\nloc(R,L×R).\nProof.As we mentioned in Remark 2.8, we establish compactness by pr oving the following\nuniform bounds:\n(1) a uniform L∞bound onuν;\n(2) a uniform L∞bound onην;\n(3) a uniform L∞bound on the derivatives of uν.\nProperty (1) follows from the convexity at infinity of ( T∗N,ωσ). Proposition 2.11 implies that\nηνis uniformly bounded which guarantees property (2). If the d erivatives would explode we\nthen obtain a non-constant holomorphic sphere as limit whic h is impossible by Proposition\n2.12. Hence we conclude property (3). With (1)-(3), the boot strapping argument gives C∞\nloc-\nconvergence, see [25, Appendix B.4]. /square\nTheorem 2.14. LetΣ⊂T∗Nbe a closed hypersurface with a defining Hamiltonian F. For\na generic choice of (H,θ), the functional Aθis Morse and FH(Aθ)is well-defined.\nProof.Now we are ready to define the Floer homology of Aθ. By choosing a generic pair\n(H,θ)∈ H × P , we may assume that Aθis Morse. Take a critical point w= (u,η) ofAθ.\nSinceAθis Morse,u:R/Z→T∗Nis a non-degenerate orbit. Thus we can associate to ua\nwell-defined integer the Conley-Zehnder index µCZ(u). See [30] and [1] for the definition and\ndetails of the Conley-Zehnder index.\nLet us define µ(w) :=µCZ(u) and denote by\nCrit(a,b)(Aθ) :={(u,η)∈Crit(Aθ) :Aθ(u,η)∈(a,b)};\nCrit(a,b)\nk(Aθ) :={w∈Crit(a,b)(Aθ) :µ(w) =k}.\nNow we define\nFC(a,b)\nk(Aθ) := Crit(a,b)\nk(Aθ)⊗Z2.\nFor a generic almost complex structure J(t)∈ Jσand givenw±∈Crit(a,b)(Aθ), we denote by\n/hatwiderM(w−,w+) :={w(s) :wsatisfies (2.10),lim\ns→±∞w(s) =w±};\nM(w−,w+) :=/hatwiderM(w−,w+)/R.\nThe above R-action is given by translating the s-coordinate. Suppose further that the almost\ncomplex structure J(t) is generic, so that M(w−,w+) is a smooth manifold of dimension\ndimM(w−,w+) =µ(w−)−µ(w+)−1.\nFor the compactification of the moduli space /hatwiderM(w−,w+), we remind the Floer-Gromov\nconvergence as follows. A sequence {(uν,ην)}ν∈Nin/hatwiderM(w−,w+) is said to Floer-Gromov\nconverges to a broken flow line {(uj,ηj)}m\nj=1with\n(uj,ηj)∈/hatwiderM(wj−1,wj), j∈ {1,...,m},\nwherewi∈Crit(Aθ) andw0=w−,wm=w+, if there exist sν\nj∈Rsuch that reparametrized\nsequence (uν,ην)(s+sν\nj) converges to ( uj,ηj)(s) for allj= 1,...,min theC∞\nloc-topology. By\nTheorem 2.13, we are now able to consider a compactification o f/hatwiderM(w−,w+) with respect toON MAGNETIC LEAF-WISE INTERSECTIONS 11\nthe Floer-Gromov convergence. We mention two following sta tements, due to Floer [16, 15],\nwhich is crucial in constructing various type of Floer homol ogies, also containing Aθ-version.\n(1) Ifµ(w−)−µ(w+) = 1, then the moduli space /hatwiderM(w−,w+) is a 1-dimensional compact\nsmoothmanifoldwithrespecttotheFloer-Gromovconvergen ceandhence M(w−,w+)\nis finite.\n(2) LetM(w−,w+) be the compactification of M(w−,w+) with respect to the Floer-\nGromov convergence. If µ(w−)−µ(w+) = 2, then M(w−,w+) is a compact 1-\ndimensional manifold whose boundary is given by\n∂M(w−,w+) =/uniondisplay\nwM(w−,w)×M(w,w+),\nwherew∈Crit(Aθ) runs over µ(w−)−µ(w) = 1.\nBy virtue of (1), the boundary operator ∂k: FC(a,b)\nk(Aθ)→FC(a,b)\nk−1(Aθ) is defined by\n∂kw−:=/summationdisplay\nµ(w+)=k−1#2M(w−,w+)·w+,\nwhere # 2meansZ2-counting. This boundary operator satisfies ∂k−1◦∂k= 0 inZ2-coefficient\nwhich is guaranteed by (2). Then the resulting filtered Floer homology group is defined by\nFH(a,b)\n∗(Aθ) := H ∗(FC(a,b)\n•(Aθ),∂∗).\nBy taking direct and inverse limit, we obtain\nFH∗(Aθ) := lim\n−→alim\n←−\nbFH(−a,b)\n∗(Aθ), a,b → ∞.\n/square\n2.2.Continuation map between FH(AF\nH)andFH(Aθ).In this section, we construct a\ncontinuation homomorphism\n/tildewideΨσ: FH(AF\nH)→FH(Aθ)\nby counting gradient flow lines of the s-dependent perturbed Rabinowitz action functional.\nHere\nσ(s) :=γ(s)σ, θ(s) :=γ(s)θ (2.12)\nares-dependent magnetic perturbation where γ:R→[0,1] is a cut-off function\nγ(s) =/braceleftbigg\n0 fors≤0\n1 fors≥1(2.13)\nand 0≤˙γ(s)≤2 for alls∈R. Then the corresponding action functional is\nAθ(s)(u,η) =AF\nH(u,η)+γ(s)Bθ(u)\nwhereAF\nH(u,η) =/integraltext1\n0u∗λ−η/integraltext1\n0F(t,u(t))dt−/integraltext1\n0H(t,u(t))dt.\nNow we consider the ( s,t)-dependent almost complex structure J(s,t) onT∗Nsuch that\nJ(s,t)∈ Jσ(s)for alls∈[0,1] andJ(s,t) is independent of sfors≤ −1 ands≥1. This\nalmost complex structure induces the ( s,t)-dependent inner product on T∗N\ngs,t(·,·) :=ωσ(s)(·,J(s,t)·),12 YOUNGJIN BAE\nand the following s-dependent inner product on L×R\ngs/parenleftbig\n(ˆu1,ˆη1),(ˆu2,ˆη2)/parenrightbig\n:=/integraldisplay1\n0gs,t(ˆu1,ˆu2)dt+ ˆη1ˆη2, (2.14)\nwhere (ˆui,ˆηi)∈T(u,η)(L×R) fori= 1,2. With the above metric, we obtain\n∇sAθ(s)(u,η) =/parenleftBigg\n−J(s,t,u)/parenleftbig\n∂tu−ηXF(t,u)−Xσ(s)\nH(t,u)/parenrightbig\n−/integraltext1\n0F(t,u)dt./parenrightBigg\nThen the gradient flow line w= (u,η)∈C∞(R×(R/Z),T∗N)×C∞(R,R) satisfies\n∂su+J(s,t,u)/parenleftbig\n∂tu−ηXF(t,u)−Xσ(s)\nH(t,u)/parenrightbig\n= 0\nd\ndsη+/integraltext1\n0F(t,u)dt= 0,/bracerightBigg\n(2.15)\nwith energy\nE(w) :=/integraldisplay∞\n−∞/ba∇dbld\ndsw(s)/ba∇dbl2\nsds,\nwhere/ba∇dbl·/ba∇dbls:=/radicalbig\ngs(·,·).\nIn order to construct the continuation homomorphism /tildewideΨσ, it suffices to show that the\nLagrange multiplier ηand the energy of the time-dependent gradient flow line are un iformly\nbounded. For this purpose, we need the following preliminar y statements.\nLemma 2.15. There exists ǫ>0 and/tildewidec>0 such that if ( u,η)∈C∞(R/Z,T∗N)×Rsatisfies\n/ba∇dbl∇sAθ(s)(u,η)/ba∇dbls<ǫthen\n|η| ≤/tildewidec(|Aθ(s)(u,η)|+1). (2.16)\nHere/ba∇dbl·/ba∇dbls:=/radicalbig\ngs(·,·).\nProof.The proof is basically the same as in Lemma 2.10 by considerin gσ(s) instead of σ.\nHere we omit the proof. /square\nDefinition 2.16. For a magnetic perturbation σ∈Mand its primitive θ∈ P, see Definition\n1.1, theisoperimetric constant C=C(θ) is defined by\nC:=/ba∇dblθ/ba∇dbl∞= max\nt∈R/Z/ba∇dblθt/ba∇dbl∞.\nIn order to state the next proposition, we introduce the nota tions as follows:\ndσ=dH,σ:= sup\ns∈R/ba∇dblXσ(s)\nH/ba∇dbl∞;\ndF:=/ba∇dblXF/ba∇dbl∞.\nProposition 2.17. Letǫ,/tildewidecbe the same as in Lemma 2.15. Let w−∈Crit(AF\nH),w+∈\nCrit(Aθ)andw= (u,η)be a gradient flow line of Aθ(s)withlims→±∞w=w±. If the\nisoperimetric constant C=C(θ)satisfies\nC≤1\n4;\n/parenleftbigg\n8/tildewidecdFC+2/tildewidecdF+4dF\nǫ2/ba∇dblF/ba∇dbl∞/parenrightbigg\nC≤1\n2,(2.17)\nthen there exists a constant κ=κ(w−,w+)such that\n/ba∇dblη/ba∇dbl∞≤κ.ON MAGNETIC LEAF-WISE INTERSECTIONS 13\nProof.We prove the proposition in 3 steps.\nStep 1:Let us first bound the energy of win terms of /ba∇dblη/ba∇dbl∞.\nE(w) =/integraldisplay∞\n−∞/ba∇dbld\ndsw(s)/ba∇dbl2\nsds\n=/integraldisplay∞\n−∞/an}b∇acketle{td\ndsw(s),∇sAθ(s)(w(s))/an}b∇acket∇i}htsds\n=/integraldisplay∞\n−∞d\ndsAθ(s)(w(s))ds−/integraldisplay∞\n−∞˙Aθ(s)(w(s))ds\n=Aθ(1)(w+)−Aθ(0)(w−)−/integraldisplay∞\n−∞˙Aθ(s)(w(s))ds.(2.18)\nWe estimate the last term in (2.18) by using the isoperimetri c constantC\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\n−∞˙Aθ(s)(w(s))ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplay∞\n−∞/vextendsingle/vextendsingle/vextendsingle˙Aθ(s)(w(s))/vextendsingle/vextendsingle/vextendsingleds\n=/integraldisplay∞\n−∞˙γ(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR/Z/tildewideu∗θtdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleds\n≤/integraldisplay∞\n−∞˙γ(s)C/integraldisplay\nR/Z|∂tu|s,tdt ds.(2.19)\nHere|·|s,t:=/radicalbig\ngs,t(·,·). From the gradient flow equation (2.15), we get\n∂tu=J(s,t,u)∂su+ηXF(t,u)+Xσ(s)\nH(t,u). (2.20)\nBy inserting (2.20) into the last term in (2.19), we then obta in\n/integraldisplay∞\n−∞/vextendsingle/vextendsingle/vextendsingle˙Aθ(s)(w(s))/vextendsingle/vextendsingle/vextendsingleds≤/integraldisplay∞\n−∞˙γ(s)C/integraldisplay\nR/Z|∂tu|s,tdt ds\n=/integraldisplay∞\n−∞˙γ(s)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤2C/integraldisplay\nR/Z|J(s,t)∂su+ηXF(t,u)+Xσ(s)\nH(t,u)|s,tdt ds\n≤2C/integraldisplay1\n0/integraldisplay\nR/Z/parenleftBig\n|∂su|s,t+|η||XF(t,u)|s,t+|Xσ(s)\nH(t,u)|s,t/parenrightBig\ndt ds\n≤2C/integraldisplay1\n0/integraldisplay\nR/Z/parenleftBig\n|∂su|2\ns,t+1+|η|/ba∇dblXF/ba∇dbl∞+/ba∇dblXσ(s)\nH/ba∇dbl∞/parenrightBig\ndt ds\n=2CE(u)+2C+2dσC+2/ba∇dblη/ba∇dbl∞dFC\n≤2CE(w)+2C+2dσC+2/ba∇dblη/ba∇dbl∞dFC.(2.21)\nNow by combining the above estimates (2.18) and (2.21), we de duce\nE(w) =Aθ(1)(w+)−Aθ(0)(w−)−/integraldisplay∞\n−∞˙Aθ(s)(w(s))ds\n≤Aθ(1)(w+)−Aθ(0)(w−)+2CE(w)+2C+2dσC+2/ba∇dblη/ba∇dbl∞dFC.14 YOUNGJIN BAE\nBy the assumption on the isoperimetric constant, C≤1\n4and we have\nE(w)≤2Aθ(1)(w+)−2Aθ(0)(w−)+4C+4dσC+4/ba∇dblη/ba∇dbl∞dFC\n=2∆+4C+4dσC+4/ba∇dblη/ba∇dbl∞dFC,(2.22)\nwhere ∆ := Aθ(1)(w+)−Aθ(0)(w−). This finishes Step 1.\nStep 2: Letǫbe as in Lemma 2.10. For l∈Rletνw(l)≥0 be defined by\nνw(l) := inf/braceleftbig\nν≥0 :/ba∇dbl∇sAθ(l+ν)/parenleftbig\nw(l+ν)/parenrightbig\n/ba∇dbls<ǫ/bracerightbig\n.\nIn this step we bound νw(l) in terms of /ba∇dblη/ba∇dbl∞for alll∈Ras follows\nE(w) =/integraldisplay∞\n−∞/ba∇dbld\ndsw(s)/ba∇dbl2\nsds\n=/integraldisplay∞\n−∞/ba∇dbl∇sAθ(s)/ba∇dbl2\nsds\n≥/integraldisplayl+νw(l)\nl/ba∇dbl∇sAθ(s)/ba∇dbl2\ns/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≥ǫ2ds\n≥ǫ2νw(l).(2.23)\nStep 1 and the above estimate finish Step 2.\nStep 3:We prove the proposition.\nFirst set\nK= max{−Aθ(0)(w−),Aθ(1)(w+)}\nBy the definition of νw(l), we get /ba∇dbl∇sAθ[l+νw(l)][w(l+νw(l))]/ba∇dbls<ǫ, which enables us to use\nLemma 2.15. We obtain the following estimate by using (2.16) , (2.21) and (2.22)\n|η(l+νw(l))| ≤/tildewidec/parenleftbig/vextendsingle/vextendsingleAθ[l+νw(l)][w(l+νw(l))]/vextendsingle/vextendsingle+1/parenrightbig\n≤/tildewidec/parenleftbigg\nK+/integraldisplay∞\n−∞/vextendsingle/vextendsingle/vextendsingle˙Aθ(s)/vextendsingle/vextendsingle/vextendsingleds+1/parenrightbigg\n≤/tildewidec(K+2CE(w)+2C+2dσC+2/ba∇dblη/ba∇dbl∞dFC+1)\n≤/tildewidec/bracketleftbig\nK+2C(2∆+4C+4dσC+4/ba∇dblη/ba∇dbl∞dFC)\n+2C+2dσC+2/ba∇dblη/ba∇dbl∞dFC+1/bracketrightbig\n.(2.24)\nBy Step 2 and (2.22), we get the following inequalities\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayl+νw(l)\nl˙η(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayl+νw(l)\nl/integraldisplay1\n0F(t,u(t))dt ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/ba∇dblF/ba∇dbl∞νw(l)\n≤/ba∇dblF/ba∇dbl∞E(w)\nǫ2\n≤/ba∇dblF/ba∇dbl∞\nǫ2(2∆+4C+4dσC+4/ba∇dblη/ba∇dbl∞dFC).(2.25)ON MAGNETIC LEAF-WISE INTERSECTIONS 15\nCombining the above two estimates (2.24) and (2.25), we conc lude the following\n|η(l)| ≤|η(l+νw(l))|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayl+νw(l)\nl˙η(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/tildewidec/bracketleftbig\nK+2C(2∆+4C+4dσC+4/ba∇dblη/ba∇dbl∞dFC)+2C+2dσC+2/ba∇dblη/ba∇dbl∞dFC+1/bracketrightbig\n+/ba∇dblF/ba∇dbl∞\nǫ2(2∆+4C+4dσC+4/ba∇dblη/ba∇dbl∞dFC)\n=/parenleftbigg\n8/tildewidecdFC+2/tildewidecdF+4dF\nǫ2/ba∇dblF/ba∇dbl∞/parenrightbigg\nC\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=:κ1/ba∇dblη/ba∇dbl∞\n+/tildewidec/bracketleftbig\nK+2C(2∆+4C+4dσC)+2C+2dσC+1/bracketrightbig\n+/ba∇dblF/ba∇dbl∞\nǫ2(2∆+4C+4dσC)\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=:1\n2κ.\nNote that the above estimate is valid for all l∈Randκ1≤1\n2by the condition (2.17). Thus\nwe have\n/ba∇dblη/ba∇dbl∞≤1\n2/ba∇dblη/ba∇dbl∞+1\n2κ,\nand hance we conclude\n/ba∇dblη/ba∇dbl∞≤κ. (2.26)\n/square\nLemma 2.18. Letǫ,/tildewidecin (2.27) be the same as in Lemma 2.15. Let w−∈Crit(AF\nH),\nw+∈Crit(Aθ) andw= (u,η) be a gradient flow line of Aθ(s)with lims→±∞w=w±and its\naction value a=Aθ(0)(w−),b=Aθ(1)(w+). If the isoperimetric constant C=C(θ) satisfies\nthe following conditions:\n/tildewidecdFC≤1\n32;\n/parenleftbigg\n2/tildewidecdFC+dF/ba∇dblF/ba∇dbl∞\nǫ/parenrightbigg\nC≤1\n128;\n/parenleftbigg\n1+dσ+2/tildewidecdF+4/tildewidecdFC(1+dσ+4C+4dσC)\n+8dF/ba∇dblF/ba∇dbl∞C\nǫ2(1+dσ)/parenrightbigg\nC≤1\n72,(2.27)\nthen the following assertions hold:\n(1) Ifa≥1\n9, thenb≥a\n2;\n(2) Ifb≤ −1\n9, thena≤b\n2.\nProof.By Proposition 2.17, the Lagrange multiplier ηof the gradient flow line is uniformly\nbounded as follows\n/ba∇dblη/ba∇dbl∞≤2/tildewidec/parenleftbig\nK+2C(2∆+4C+4dσC)+2C+2dσC+1/parenrightbig\n+2/ba∇dblF/ba∇dbl∞\nǫ2(2∆+4C+4dσC).\n(2.28)16 YOUNGJIN BAE\nRecall that K= max{−a,b}and ∆ =b−a. From the fact that E(w)≥0 and (2.22), we\nobtain the following inequality\nb≥a−2C−2dσC−2/ba∇dblη/ba∇dbl∞dFC. (2.29)\nBy combining (2.28) with (2.29), we obtain\nb≥a−2C−2dσC−2/ba∇dblη/ba∇dbl∞dFC\n≥a−2C−2dσC−4/tildewidec/parenleftbig\nK+2C(2∆+4C+4dσC)+2C+2dσC+1/parenrightbig\ndFC\n+4/ba∇dblF/ba∇dbl∞\nǫ2(2∆+4C+4dσC)dFC\n=a−4/tildewidecdFCK−8/parenleftbigg\n2/tildewidecdFC+dF/ba∇dblF/ba∇dbl∞\nǫ/parenrightbigg\nC∆\n−2/parenleftbigg8dF/ba∇dblF/ba∇dbl∞C\nǫ2(1+dσ)+1+dσ+2/tildewidecdF+4/tildewidecdFC(1+dσ+4C+4dσC)/parenrightbigg\nC\n≥a−1\n8K−1\n16(b−a)−1\n36.(2.30)\nHere the last inequality we use the condition (2.27). To prov e the assertion (1), we first\nconsider the case\n|b| ≤a, a ≥1\n9.\nIn this assumption, we induce the following estimate from (2 .30)\nb≥a−1\n8a−1\n8a−1\n36=3\n4a−1\n36≥a\n2.\nNow we want to exclude the case\n−b≥a≥1\n9.\nBut in this case (2.30) implies the following contradiction :\nb≥1\n9+1\n72−1\n16(b−a)−1\n36≥ −1\n16(b−a)>0.\nThis proves the first assumption. To prove the assertion (2), we set\nb′=−a, a′=−b.\nThen (2.30) also holds for b′anda′. Thus we get the following assertion from (1)\n−b≥1\n9=⇒ −a≥ −b\n2\nwhich is equivalent to the assertion (2). This proves the lem ma. /square\nTheorem 2.19. LetΣ⊂T∗Nbe a closed hypersurface with a defining Hamiltonian F. For\na generic choice of (H,θ), the functionals AF\nH,Aθare Morse and\nFH(AF\nH)∼=FH(Aθ).\nProof.For a given magnetic perturbation σ∈Mand its primitive θ∈ P, we first consider\nthe sequence {θi}N\ni=1⊂ Pwhich satisfies the following properties:\n(1)θi:=diθ, where 0 = d0<d1<···<dN= 1;\n(2)Aθi:L×R→Ris Morse for all i= 0,1,...,N;ON MAGNETIC LEAF-WISE INTERSECTIONS 17\n(3)Ci:= (di+1−di)/ba∇dblθ/ba∇dbl∞satisfies the assumption of Proposition 2.17 and Lemma 2.18\nfor alli= 0,1,...,N−1.1\nNow we are ready to use Proposition 2.17 and Lemma 2.18 induct ively. Let us consider\nfollowing homotopies\nθi(s) :=θi+γ(s)(θi+1−θi);\nσi(s) :=σi+γ(s)(σi+1−σi).\nHere/tildewideσi=dθiand see (2.13) for γ(s), and a gradient flow line w= (u,η)∈C∞(R×\n(R/Z),T∗N)×C∞(R,R) of the time dependent action functional Aθi(s)which satisfies\n∂su+J(s,t,u)/parenleftbig\n∂tu−Xσi(s)\nH(t,u)−ηXF(t,u)/parenrightbig\n= 0\nd\ndsη+/integraltext1\n0F(t,u)dt= 0,/bracerightBigg\n(2.31)\nand the limit condition\nlim\ns→−∞w(s) =w−∈Crit(Aθi),lim\ns→∞w(s) =w+∈Crit(Aθi+1).(2.32)\nThe solutions of (2.31), (2.32) form a moduli space\nNi+1\ni(w−,w+) ={w= (u,η) : (u,η) satisfies (2 .31),(2.32)}\nwhich, for a generic homotopy σi(s), is a smooth manifold of dimension µ(w−)−µ(w+).\nForthecompactnessof Ni+1\ni(w−,w+)weneedtoconsidertheanalytic issuesaboutuniform\nbounds on u,η, the derivatives of uand the energy of w. The arguments in Theorem 2.13\nare also valid for the case of the uniform bounds on uand its derivatives. Proposition 2.17\nguarantees the uniform bound on η. Finally (2.22) with Proposition 2.17 implies the uniform\nenergy bound of time-dependent gradient flow lines as follow s,\nE(w)≤2Aθi+1(1)(w+)−2Aθi(0)(w−)+4C+4dσi+1C+4/ba∇dblη/ba∇dbl∞dFC\n≤2Aθi+1(1)(w+)−2Aθi(0)(w−)+4C+4dσi+1C+4κ(w−,w+)dFC.\nWe then consider the compactified moduli space Ni+1\ni(w−,w+) ofNi+1\ni(w−,w+) with respect\nto Floer-Gromov convergence as in Theorem 2.14. When µ(w−)−µ(w+) = 1 we have\n∂Ni+1\ni(w−,w+) =/uniondisplay\nxMi(w−,x)×Ni+1\ni(x,w+)∪/uniondisplay\nyNi+1\ni(w−,y)×Mi+1(y,w+),(2.33)\nwherex,yrun over Crit µ(w+)(Aθi), Critµ(w−)(Aθi+1) respectively.\nNow, by virtue of Lemma 2.18, we define a map for a≤ −1\n9andb≥1\n9\nΨ(a,b)\ni,k: FC(a\n2,b)\nk(Aθi)→FC(a,b\n2)\nk(Aθi+1)\ngiven by\nΨ(a,b)\ni,k(w−) =/summationdisplay\nµ(w+)=µ(w−)#2Ni+1\ni(w−,w+)w+.\nHere # 2means the Z2-counting.\nNow (2.33) gives us Ψ(a,b)\ni,k−1◦∂k=∂k◦Ψ(a,b)\ni,kinZ2coefficient which implies that Ψ(a,b)\ni,kis a\nchain map. Hence we have the following homomorphisms on homo logies as follows\n/tildewideΨ(a,b)\ni: FH(a\n2,b)(Aθi)→FH(a,b\n2)(Aθi+1).\n1SincetheMorse propertyof Aθiis generic, we canguarantee property(2). Notethatwe make Ciarbitrarily\nsmall by choosing small di+1−di.18 YOUNGJIN BAE\nBy taking the inverse and direct limit\nFH∗(Aθi) = lim\nb→∞lim\na→−∞FH(a,b)\n∗(Aθi),\nwe deduce\n/tildewideΨi: FH(Aθi)→FH(Aθi+1).\nWe define\n/tildewideΨσ: FH(AF\nH)→FH(Aθ).\nby/tildewideΨσ=/tildewideΨN−1◦···◦/tildewideΨ1◦/tildewideΨ0. In a similar way, we construct\n/tildewideΨσ: FH(Aθ)→FH(AF\nH),\nby following the homotopies in opposite direction. By a homo topy-of-homotopies argument,\nwe conclude/tildewideΨσ◦/tildewideΨσ= idFH(AF\nH)and/tildewideΨσ◦/tildewideΨσ= idFH(Aθ). Therefore/tildewideΨσis an isomorphism\nwith inverse/tildewideΨσ. /square\nBy these results, if dimH ∗(LN) =∞then we have infinitely many critical points of Aθ.\nThis implies that there are infinitely many magnetic leaf-wi se intersections or a periodicone\nwhich means that the leaf on which it lies forms a closed Reeb o rbit. We exclude the latter\ncase generically, as follows.\nWe recall a hypersurface Σ ⊂T∗Nnon-degenerate if closed Reeb orbits on Σ form a dis-\ncrete set. A generic Σ is non-degenerate, see [11, Theorem B. 1]. If Σ is non-degenerate,\nthen periodic leaf-wise intersection points can be exclude d by choosing a generic Hamiltonian\nfunction, see [3, Theorem 3.3]. With the above generic Hamil tonian, Albers-Frauenfelder con-\nclude that there are infinitely many leaf-wise intersection points on Σ, under the topological\nassumption dimH ∗(LN) =∞. By these reason, we only consider non-periodic (magnetic)\nleaf-wise intersection points. Thus we conclude the follow ing existence result for magnetic\nleaf-wise intersections.\nCorollary 2.20. LetNbe a closed connected orientable manifold of dimension n≥2. Let\nΣbe a non-degenerate hypersurface in T∗N. Suppose that dimH∗(LN) =∞. Ifϕis generic\nthen there exist infinitely many magnetic leaf-wise interse ction points.\nProof of Corollary 2.20. In Theorem 2.19, we have the continuation isomorphism as fol lows\n/tildewideΨσ: FH∗(AF\nH)→FH∗(Aθ). (2.34)\nSince we assume that dimH ∗(LN) =∞, (2.2), (2.6) imply that dimFH ∗(Aθ) =∞and\nconsequently the Morse function Aθhas infinitely many critical points. Now Proposition 2.3\nimplies that there exist infinitely many magnetic leaf-wise intersections or a periodic leaf-wise\nintersection. But, byTheoremB.1, thelatter casecanbeexc ludedforageneric( H,θ)∈ H×P\nwhich generates ϕ. Hence there exist infinitely many magnetic leaf-wise inter sections. /square\n3.On the growth rate of magnetic leaf-wise intersections\nIn [24], Macarini-Merry-Paternain prove the exponential g rowth rate of leaf-wise intersec-\ntions with respect to the period when /tildewideπ1(N) grows exponentially. Recall that /tildewideπ1(N) is the\nset of conjugacy classes of π1(N).ON MAGNETIC LEAF-WISE INTERSECTIONS 19\n3.1.Symplectically hyperbolic manifolds. In this section, we investigate the examples\nand the candidates for the above topological assumption.\nDefinition 3.1. Let (N,ωN) be a closed symplectic manifold of dimension 2 n. If the sym-\nplectic form ωNis/tildewided-bounded, then ( N,ωN) is called symplectically hyperbolic .\nProposition 3.2 (K¸ edra [23]) .Let(N,ωN)be a symplectically hyperbolic manifold then\nπ1(N)grows exponentially.\nAs mentioned above, we are interested in the growth rate of /tildewideπ1(N). It is known that /tildewideπ1(N)\nhas exponential growth rate when Nis a 2-dimensional symplectically hyperbolic manifold.\nBut we don’t know the growth rate of /tildewideπ1(N) for any higher dimensional symplectically hy-\nperbolic manifold.\n3.2.Perturbed F-Rabinowitz action functional. Inordertoshowtheexponentialgrowth\nrate of leaf-wise intersection points, Macarini-Merry-Pa ternain used the F-Rabinowitz action\nfunctional Af:L×R→Ras follows\nAf(u,η) =AF,f\nH(u,η) :=/integraldisplay1\n0u∗λ−f(η)/integraldisplay1\n0F(t,u)dt−/integraldisplay1\n0H(t,u)dt.\nThe above new ingredient f∈C∞(R,R) needs to satisfy the following properties:\n(1)fis a smooth strictly positive, strictly increasing functio n.\n(2) limη→−∞f(η) = 0 andf′satisfies 0<f′(η)≤1 for allη∈R.\nThe additional data f(η) is crucial to the construction of continuation maps betwee n a con-\ncentric family of fiberwise starshaped hypersurfaces, see [ 24, Section 4.2]. We denote by F\nthe set of such f∈C∞(R,R) satisfying the above conditions.\nIf we additionally consider the magnetic perturbation, the n the action functional becomes\nAf\nθ(u,η) =AF,f\nH,θ:=Af(u,η)+Bθ(u).\nOne can check that a critical point of Af\nθsatisfies\nd\ndtu=f(η)XF(t,u)+Xσ\nH(t,u)\nf′(η)/integraltext1\n0F(t,u)dt= 0./bracerightbigg\n(3.1)\nSincef′(η)>0 for allη∈R, it is equivalent to\nd\ndtu=f(η)XF(t,u)+Xσ\nH(t,u)/integraltext1\n0F(t,u)dt= 0./bracerightbigg\n(3.2)\nGiven−∞ ≤a≤b≤ ∞, we adopt the following notations:\nCrit(Af\nθ) :={w= (u,η)∈ L×R: (u,η) satisfies (3.2) };\nCrit(a,b)(Af\nθ) :={(u,η)∈Crit(Af\nθ) :Af\nθ(u,η)∈(a,b)}.\nA (magnetic) leaf-wise intersection point is called positiveornegative ifηin (1.2) is positive or\nnegative respectively. Since f∈ Fis a positive function, we only consider positive(magnetic)\nleaf-wise intersection points. It would be convenient if f(η) =ηon the action window\n(a,b)⊂R+we work with.\nDefinition 3.3. Givena>0,\nF(a) :={f∈ F:f(η) =η,∀η∈[a,∞)}.20 YOUNGJIN BAE\nFor notational convenience, let us denote by\nLW(a,b) = LW Σ,ϕ(a,b) :={x∈T∗N:φΣ\nη(ϕ(x)) =x,for someη∈(a,b)}\nand recall that\nc(H,θ) = sup\n(t,u)∈R/Z×L/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0(/tildewideλ+/tildewideτ∗θt)(/tildewideu(t))[/tildewideXσ\nH(t,/tildewideu)]−H(t,u(t))dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nProposition 3.4. Givena>0, choosef∈ F(a). Then there is a map\nev : Crit(a+c,b−c)(Af\nθ)→LW(a,b)\nev(u,η) =u(1\n2).\nAppendix B guarantees that there is no periodic magnetic lea f-wise intersection point for\ngenericϕ. In this case, evis injective and we then obtain the following estimate\n#LW(a,b)≥#Crit(a+c,b−c)(Af\nθ).\nHerec=c(H,θ).\nProof.Let (u,η) be a critical point of Af\nθ, by the argument in Proposition 2.3, then u(1\n2) is\na magnetic leaf-wise intersection point and its action valu e becomes\nAf\nθ(u,η) =/integraldisplay1\n0(/tildewideλ+/tildewideτ∗θt)/parenleftbig\nf(η)/tildewideXF(t,/tildewideu)+/tildewideXσ\nH(t,/tildewideu)/parenrightbig\n−/integraldisplay1\n0H(t,u)dt\n=f(η)/integraldisplay1\n0(/tildewideλ+/tildewideτ∗θt)[/tildewideXF(t,/tildewideu)]dt\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:♦+/integraldisplay1\n0(/tildewideλ+/tildewideτ∗θt)[/tildewideXσ\nH(t,/tildewideu)]dt−/integraldisplay1\n0H(t,u)dt\n=f(η)+/integraldisplay1\n0(/tildewideλ+/tildewideτ∗θt)/parenleftbig/tildewideXσ\nH(t,/tildewideu)/parenrightbig\n−/integraldisplay1\n0H(t,u)dt.(3.3)\nThe third equality in the above equation (3.3) is deduced fro m the following. Notice here\nthatXF(t,u) =ρ(t)X¯F(u) andρ(t) vanishes on t∈[1\n2,1] whileθt= 0 fort∈[0,1\n2].\n♦=/integraldisplay1\n0(/tildewideλ+/tildewideτ∗θt)[ρ(t)/tildewideX¯F(/tildewideu)]dt=/integraldisplay1\n0ρ(t)/tildewideλ(/tildewideX¯F(/tildewideu))dt\n=/integraldisplay1\n0ρ(t)λ(X¯F(u))dt=/integraldisplay1\n0ρ(t)λ(R(u))dt\n=/integraldisplay1\n0ρ(t)dt= 1.\nThus we obtain\n|Af\nθ(u,η)−f(η)| ≤c(H,θ). (3.4)\nSuppose Af\nθ(u,η)∈(a+c(H,θ),b−c(H,θ)) then\na<f(η)<b.\nSincef∈ F(a), we conclude that\na<η<b.\n/squareON MAGNETIC LEAF-WISE INTERSECTIONS 21\nFor a given almost complex structure J∈ Jω, let∇JAf\nθbe the gradient of Af\nθwith respect\nto the metric gJ(·,·) in (2.9). One can check that\n∇JAf\nθ(u,η) =/parenleftbigg−J(t,u)/parenleftbigd\ndtu−f(η)XF(t,u)−Xσ\nH(t,u)/parenrightbig\n−f′(η)/integraltext1\n0F(t,u)dt/parenrightbigg\n.\nDefinition 3.5. A positive gradient flow line ofAf\nθwith respect to an R/Z-parametrized\nalmost complex structure J(t)∈ Jωσis a mapw:R→ L×Rwhich solves\nd\ndsw−∇JAf\nθ= 0.\nThe above map is interpreted as w= (u,η) whereu:R×\nR/Z→T∗N×R,η:R→Rsuch that\n∂su+J(t,u)/parenleftbigd\ndtu−Xσ\nH(t,u)−f(η)XF(t,u)/parenrightbig\n= 0\nd\ndsη+f′(η)/integraltext1\n0F(t,u)dt= 0./bracerightbigg\n(3.5)\n3.3.Floer homology for Af\nθ.Let us first assume that the perturbed F-Rabinowitz action\nfunctional Af\nθ:L×R→Ris Morse in the sense of Corollary A.5. In order to define the Fl oer\nhomology for Af\nθ, we need to show that the Lagrange multiplier ηis uniformly bounded. We\nfollow the same strategy as in the Aθ-case with minor modifications.\nLemma 3.6. There exist ǫ,c′>0 such that if ( u,η)∈ L×Rsatisfies/ba∇dbl∇JAf\nθ(u,η)/ba∇dblJ≤ǫf′(η)\nthen\n2\n3/parenleftBig\nAf\nθ(u,η)−c′/ba∇dbl∇JAf\nθ(u,η)/ba∇dblJ−c/parenrightBig\n≤f(η)≤2/parenleftBig\nAf\nθ(u,η)+c′/ba∇dbl∇JAf\nθ(u,η)/ba∇dblJ+c/parenrightBig\n.(3.6)\nHerec=c(H,θ) as in Definition 2.9.\nProof.The proof consists of 2 steps.\nStep 1:There exist constants δ,c′>0such that if u∈ Lsatisfies\nu(t)∈Uδ:=F−1(−δ,δ),∀t∈[0,1\n2]\nthen (3.6) holds.\nThere exist δ>0 such that\n1\n2+δ≤λ(XF(p))≤3\n2−δ,∀p∈Uδ.22 YOUNGJIN BAE\nNow we compute\nAf\nθ(u,η) =/integraldisplay1\n0u∗λ−/integraldisplay1\n0H(t,u(t))dt−f(η)/integraldisplay1\n0F(t,u(t))dt+Bθ(u(t))\n=/integraldisplay1\n0/tildewideu∗(/tildewideλ+/tildewideτ∗θt)−/integraldisplay1\n0H(t,u(t))dt−f(η)/integraldisplay1\n0F(t,u(t))dt\n=/integraldisplay1\n0(/tildewideλ+/tildewideτ∗θt)(/tildewideu(t))[d\ndt/tildewideu−f(η)/tildewideXF(t,/tildewideu)−/tildewideXσ\nH(t,/tildewideu)]dt\n+f(η)/integraldisplay1\n0λ(u(t))[XF(t,u)]/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≥1\n2+δ−F(t,u(t))/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤δdt\n+/integraldisplay1\n0(/tildewideλ+/tildewideτ∗θt)(/tildewideu(t))[/tildewideXσ\nH(t,/tildewideu)]−H(t,u(t))dt\n≥/parenleftbigg1\n2+δ−δ/parenrightbigg\nf(η)−c′/ba∇dbld\ndtu−Xσ(s)\nH(t,u)−f(η)XF(t,u)/ba∇dbl1−c(H,θ)\n≥1\n2|f(η)|−c′/ba∇dbld\ndtu−Xσ(s)\nH(t,u)−f(η)XF(t,u)/ba∇dbl2−c(H,θ)\n≥1\n2|f(η)|−c′/ba∇dbl∇JAθ(u,f(η))/ba∇dblJ−c(H,θ),\nwherec′=c′(θ,δ) :=/ba∇dbl(/tildewideλ+/tildewideτ∗θ)|/tildewideUδ/ba∇dbl∞. In a similar way, we get the following estimate\nAf\nθ(u,η) =/integraldisplay1\n0u∗λ−/integraldisplay1\n0H(t,u(t))dt−f(η)/integraldisplay1\n0F(t,u(t))dt+Bθ(u(t))\n=/integraldisplay1\n0/tildewideu∗(/tildewideλ+/tildewideτ∗θt)−/integraldisplay1\n0H(t,u(t))dt−f(η)/integraldisplay1\n0F(t,u(t))dt\n=/integraldisplay1\n0(/tildewideλ+/tildewideτ∗θt)(/tildewideu(t))[d\ndt/tildewideu−f(η)/tildewideXF(t,/tildewideu)−/tildewideXσ\nH(t,/tildewideu)]dt\n+f(η)/integraldisplay1\n0λ(u(t))[XF(t,u)]/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤3\n2−δ−F(t,u(t))/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≥−δdt\n+/integraldisplay1\n0(/tildewideλ+/tildewideτ∗θt)(/tildewideu(t))[/tildewideXσ\nH(t,/tildewideu)]−H(t,u(t))dt\n≤/parenleftbigg3\n2−δ+δ/parenrightbigg\nf(η)+c′/ba∇dbld\ndtu−Xσ(s)\nH(t,u)−f(η)XF(t,u)/ba∇dbl1+c(H,θ)\n≤3\n2|f(η)|+c′/ba∇dbld\ndtu−Xσ(s)\nH(t,u)−f(η)XF(t,u)/ba∇dbl2+c(H,θ)\n≤3\n2|f(η)|+c′/ba∇dbl∇JAθ(u,f(η))/ba∇dblJ+c(H,θ).\nThe above two estimates prove Step 1.\nStep 2:For anyδ>0there existǫ>0such that if (u,η)∈ L×R\n/ba∇dbl∇JAf\nθ(u,η)/ba∇dblJ≤ǫf′(η)ON MAGNETIC LEAF-WISE INTERSECTIONS 23\nthenu(t)∈Uδfor allt∈[0,1\n2].\nBy a similar argument as in Lemma 2.10 Step 2, if F(u(t))≥δ\n2for allt∈[0,1\n2] then\n/ba∇dbl∇JAf\nθ(u,η)/ba∇dblJ≥/vextendsingle/vextendsingle/vextendsingle/vextendsinglef′(η)/integraldisplay1\n0F(t,u(t))dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥f′(η)δ\n2.\nNow, if there exist t1,t2in [0,1\n2] withF(u(t1))≤δ\n2andF(u(t2))≥δthen\n/ba∇dbl∇JAf\nθ(u,η)/ba∇dblJ≥δ\n2/ba∇dbl∇F/ba∇dbl∞.\nIf we set\nǫ=ǫ(δ,F) := min/braceleftbiggδ\n2,δ\n2/ba∇dbl∇F/ba∇dbl∞/bracerightbigg\nand use the fact that f′(η)≤1 for allη∈Rthen this proves Step 2.\nBy combining Step 1 and Step 2, we immediately prove the lemma . /square\nWe need further preliminaries. Now we consider a certain cla ss off∈ F(a) with the\nfollowing condition.\nDefinition 3.7. Givena,r>0,\nF(a,r) :={f∈ F(a) :∃A>0 such that Af′(−A)>r};\n/tildewideF(a) :=/intersectiondisplay\nr>0F(a,r).(3.7)\nRemark 3.8. Givena >0, the set/intersectiontext\nr>0F(a,r) is non-empty and path-connected. An\nexplicit construction of f∈/intersectiontext\nr>0F(a,r) exists. There also exists a homotopy between two\ndifferentf0,f1∈ F(a,r). All these things are explained in [24, Remark 3.24, Lemma 3 .25].\nProposition 3.9. FixF∈ D(Σ)and an action window (a,b)such that 0<a<b< ∞. Let\nc′,ǫ>0be the constants from Lemma 3.6. Choose f∈ F(a\n6,b−a\nmin{ǫ,a/4c′})and a generic pair\n(H,θ)such thatc(H,θ)≤a\n2. Letw±∈Crit(a,b)(Af\nθ)andw= (u,η)be a gradient flow line of\nAf\nθwithlims→±∞w(s) =w±. Then there exists a constant κ=κ(a,b)satisfying /ba∇dblη/ba∇dbl∞≤κ.\nProof.For convenience, set\nǫ1:= min/braceleftBig\nǫ,a\n4c′/bracerightBig\n.\nFirst define a function νw:R→[0,∞) for a given gradient flow line w= (u,η) by\nνw(l) := inf{ν≥0 :/ba∇dbl∇JAf\nθ(w(l+ν))/ba∇dblJ≤ǫ1f′(η(l+ν))}.\nSince lim s→∞f′(η(s)) = 1 and lim s→∞/ba∇dbl∇JAf\nθ((u,η)(s))/ba∇dblJ= 0,νwis well-defined. We get\nthe following estimate\nb−a≥lim\ns→∞Af\nθ(w(s))−lim\ns→−∞Af\nθ(w(s))\n=/integraldisplay∞\n−∞/ba∇dbl∇JAf\nθ(w(s))/ba∇dbl2\nJds\n≥/integraldisplayl+νw(l)\nlǫ2\n1f′(η(s))2ds\n≥νw(l)ǫ2\n1iw(l)2,(3.8)24 YOUNGJIN BAE\nwhereiw(l) := infl≤s≤l+νw(s)f′(η(s)). Hence we obtain\nνw(l)≤b−a\nǫ2\n1iw(l)2.\nNow observe that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayl+νw(l)\nl˙η(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplayl+νw(l)\nl|˙η(s)|ds\n≤/parenleftBigg\nνw(l)/integraldisplayl+νw(l)\nl|˙η(s)|2ds/parenrightBigg1/2\n≤/parenleftBigg\nνw(l)/integraldisplayl+νw(l)\nl/ba∇dbl∇JAf\nθ(w(s))/ba∇dbl2\nJds/parenrightBigg1/2\n≤(νw(l)E(w))1/2\n≤b−a\nǫ1iw(l).(3.9)\nBy Lemma 3.6, we get the following estimate for any l∈R\nf[η(l+νw(l))]≥2\n3/parenleftbigg\nAf\nθ[w(l+νw(l))]−c′/ba∇dbl∇JAf\nθ(u,η)/ba∇dblJ−c(H,θ)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤a\n2/parenrightbigg\n≥2\n3(a−c′ǫ1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤a\n4f′[η(l+νw(l))]/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤1−a\n2)\n≥a\n6.\nSincef∈ F(a\n6), we get\nη(l+νw(l))≥a\n6,\nand hence\nη(l)≥η(l+νw(l))−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayl+νw(l)\nl˙η(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≥a\n6−b−a\nǫ1iw(l)\n>−b−a\nǫ1iw(l).\nThis implies\nf′(η(l))η(l)≥iw(l)η(l)≥ −b−a\nǫ1.\nNow suppose that there exists l0∈Rsuch thatη(l0)<−Athen there must be l1∈Rwith\nη(l1) =−A. This induces the following contradiction by the choice of f∈ F(a\n6,b−a\nǫ1) with\n(3.7),\n−b−a\nǫ1>−f′(−A)A=f′(η(l1))η(l1)>−b−a\nǫ1.\nSo, we conclude that η(l)>−Afor alll∈R.ON MAGNETIC LEAF-WISE INTERSECTIONS 25\nNow consider the upper bound. Start with a new function /tildewideνw:R→[0,∞) by\n/tildewideνw(l) := inf{ν≥0 :/ba∇dbl∇JAf\nθ(w(l+ν))/ba∇dblJ≤ǫ1f′(−A)}.\nBy a similar argument as in (3.8) and (3.9), we see that\n/tildewideνw(l)≤b−a\nǫ2\n1f′(−A)2\nand\n|η(l)−η(l+/tildewideνw(l))|<b−a\nǫ1f′(−A)<A (3.10)\nwhere the last inequality comes from (3.7) again. By Lemma 3. 6, we get\nf[η(l+/tildewideνw(l))]≤2/parenleftBig\nAf\nθ[w(l+/tildewideνw(l))]+c′/ba∇dbl∇JAf\nθ[w(l+/tildewideνw(l))]/ba∇dblJ+c(H,θ)/parenrightBig\n≤2(b+c′ǫ1/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤a\n4f′(−A)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤1+a\n2)\n<2a+2b.\nThis implies that η(l+/tildewideνw(l))<2a+2band by (3.10)\nη(l)<2a+2b+A.\nThus we conclude that\n/ba∇dblη/ba∇dbl∞<κ:= 2a+2b+A.\n/square\nFor simplicity, let us denote by\nA(Af\nθ) :={Af\nθ(w) :w∈Crit(Af\nθ)}.\nTheorem 3.10. FixF∈ D(Σ)andf∈/tildewideF(1\n6), see Definition 3.7. Choose a generic pair\n(H,θ). Ifmax{1,2c(H,θ)}<a<b≤ ∞anda,b /∈A(Af\nθ), thenFH(a,b)(Af\nθ)is well-defined.\nProof.The construction of FH(a,b)(Af\nθ) is the same as in the Aθ-case. For w= (u,η)∈\nCrit(a,b)(Af\nθ), we define the index µ(w) :=µCZ(u). Let us denote by\nCrit(a,b)\nk(Af\nθ) :={w∈Crit(a,b)(Af\nm) :µ(w) =k};\nFC(a,b)\nk(Af\nθ) := Crit(a,b)\nk(Af\nθ)⊗Z2.\nFor a generic almost complex structure J(t)∈ Jσand givenw±∈Crit(a,b)(Af\nθ), we define\n/hatwiderM(w−,w+) :={w(s) :wsatisfies (3.5),lim\ns→±∞w(s) =w±};\nM(w−,w+) :=/hatwiderM(w−,w+)/R.\nThe above R-action is given by translating the s-coordinate. Suppose further that the almost\ncomplex structure J(t) is generic, so that M(w−,w+) is a smooth manifold of dimension\ndimM(w−,w+) =µ(w−)−µ(w+)−1.26 YOUNGJIN BAE\nThe boundary operator ∂k: FC(a,b)\nk(Af\nθ)→FC(a,b)\nk−1(Af\nθ) is defined by\n∂kw−:=/summationdisplay\nµ(w+)=k−1#2M(w−,w+)w+,\nwhere # 2meansZ2-counting. By virtue of Proposition 3.9 with Theorem 2.13, t he boundary\noperator satisfies ∂k−1◦∂k= 0. Then the resulting filtered Floer homology group is\nFH(a,b)\n∗(Af\nθ) = H∗(FC(a,b)\n•(Af\nθ),∂∗).\n/square\n3.4.Continuation map between FH(Af)andFH(Af\nθ).In this section we construct a\ncontinuation homomorphism between FC( Af) and FC( Af\nθ) which induces a map on homolo-\ngies on a suitable action window. The construction is given b y counting gradient flow lines\nof thes-dependent action functional\nAf\nθ(s)(u,η) :=Af(u,η)+γ(s)Bθ(u).\nHereAf(u,η) =/integraltext1\n0u∗λ−f(η)/integraltext1\n0F(t,u(t))dt−/integraltext1\n0H(t,u(t))dtandσ(s) is defined in (2.12).\nWith the same metric as in (2.14), the gradient flow line w= (u,η)∈C∞(R×(R/Z),T∗N)×\nC∞(R,R) satisfies\n∂su+J(s,t,u)/parenleftbigd\ndtu−f(η)XF(t,u)−Xσ(s)\nH(t,u)/parenrightbig\n= 0\nd\ndsη+f′(η)/integraltext1\n0F(t,u)dt= 0./bracerightBigg\n(3.11)\nIn order to construct a continuation map, we need to check tha t the energy/integraltext∞\n−∞/ba∇dbld\ndsw/ba∇dbl2\nsds\nand the Lagrange multiplier ηof gradient flow lines ware uniformly bounded. As in the Aθ\ncase, we start with the fundamental lemma.\nLemma 3.11. There exist ǫ,c′>0 such that if ( u,η)∈ L×Rsatisfies\n/ba∇dbl∇sAf\nθ(s)(u,η)/ba∇dbls≤ǫf′(η)\nthen\n2\n3/parenleftBig\nAf\nθ(s)(u,η)−c′/ba∇dbl∇sAf\nθ(s)(u,η)/ba∇dbls−c/parenrightBig\n≤f(η)≤2/parenleftBig\nAf\nθ(s)(u,η)+c′/ba∇dbl∇sAf\nθ(s)(u,η)/ba∇dbls+c/parenrightBig\n.\nHere\nc=c(H,θ) := sup\ns∈Rsup\n(t,u)∈R/Z×L/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0(/tildewideλ+γ(s)/tildewideτ∗θt)(/tildewideu(t))[/tildewideXσ(s)\nH(t,/tildewideu)]−H(t,u(t))dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nProof.The proof is similar as in Lemma 3.6 with σ(s) instead of σ. So we omit the proof.\nWith a simple computation, one checks that\nǫ=ǫ(δ,F) := min/braceleftbiggδ\n2,δ\n2/ba∇dbl∇F/ba∇dbl∞/bracerightbigg\nand\nc′=c′(θ,δ) := sup\ns∈R/ba∇dbl(/tildewideλ+γ(s)/tildewideτ∗θt)|/tildewideUδ/ba∇dbl∞.\nHereδis chosen to satisfy\n1\n2+δ≤λ(XF(p))≤3\n2−δ,∀p∈Uδ.\n/squareON MAGNETIC LEAF-WISE INTERSECTIONS 27\nBefore state the next proposition, we summarize the notatio ns as follows:\nC=/ba∇dblθ/ba∇dbl∞;\ndσ=dH,σ= sup\ns∈R/ba∇dblXσ(s)\nH/ba∇dbl∞;\ndF=/ba∇dblXF/ba∇dbl∞;\n∆ =Af\nθ(1)(w+)−Af\nθ(0)(w−).(3.12)\nProposition 3.12. FixF∈ D(Σ)and an action window (a,2a)witha≥2. Letc,c′,ǫ>0\nbe the constants from Lemma 3.11. Choose f∈ F(a\n6,2a+1\nmin{ǫ,a/8c′})and a generic pair (H,θ)\nsuch thatc=c(H,θ)≤a\n2. Letwbe a gradient flow line of Af\nθ(s)with the following asymptotic\nconditions\nlim\ns→−∞w(s) =w−∈Crit(a,2a)(Af\nθ(0)),lim\ns→∞w(s) =w+∈Crit(a,2a)(Af\nθ(1)).\nIf the isoperimetric constant C=C(θ)satisfies the following conditions:\nC≤1\n4;\n/parenleftbigg\n16dFC+4dF+4/ba∇dblF/ba∇dbl∞dF\nǫ2/parenrightbigg\nC≤1\n2;\n8(4dFC2+dFC)(2+4C+/ba∇dblF/ba∇dbl∞\nǫ2)+4C≤1\n16;\n8C2+8dσC2+2C+2dσC\n+8(4dFC2+dFC)/ba∇dblF/ba∇dbl∞\nǫ2(2C+2dσC)\n+8(4dFC2+dFC)/parenleftbigg\n8C2+8C2dσ+2C+2dσC+c′ǫ+c+/parenrightbigg\n≤1\n8;\n4C+4dσC+8/ba∇dblF/ba∇dbl∞\nǫ2(2∆+4C+4dσC)C\n+8(4a+8C∆+16C2+16C2dσ+4C+4dσC+2c′ǫ+2c)C≤1,(3.13)\nthen theL∞-norm ofηis uniformly bounded in terms of a constant which only depend s on\nw−,w+.\nProof.The proof consists of 4 steps.\nStep 1:The energy is bounded by /ba∇dblf(η)/ba∇dbl∞.\nBy a similar argument as in Proposition 2.17 Step 1, we obtain/integraldisplay∞\n−∞/vextendsingle/vextendsingle/vextendsingle˙Af\nθ(s)(w(s))/vextendsingle/vextendsingle/vextendsingleds≤2CE(w)+2C+2dσC+2/ba∇dblf(η)/ba∇dbl∞dFC (3.14)\nand\nE(w)≤2∆+4C+4dσC+4/ba∇dblf(η)/ba∇dbl∞dFC, (3.15)\nunder the assumption that the isoperimetric constant C <1\n4.\n. (3.16)\nStep 2:η(s)is uniformly bounded from above .28 YOUNGJIN BAE\nInthisstep, withoutlossofgenerality, weworkontheregio nthatη(s)≥a\n6. Sincef∈ F(a\n6),\nf(η(s)) =η(s) andf′(η(s)) = 1. Then Lemma 3.11 implies the following:\nThere exist ǫ,c,c′>0 such that if ( u,η)∈ L×Rsatisfies\n/ba∇dbl∇sAf\nθ(s)(u,η)/ba∇dbls≤ǫ=ǫf′(η)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=1\nthen\nf(η)≤2/parenleftBig\nAf\nθ(s)(u,η)+c′/ba∇dbl∇sAf\nθ(s)(u,η)/ba∇dbls+c/parenrightBig\n, (3.17)\nfor alls∈Rsatisfyingη(s)≥a\n6. Hereǫ,c,c′>0 are same as in Lemma 3.11.\nNow define\nνw(l) := inf{ν≥0 :/ba∇dbl∇sAf\nθ(l+ν)(w(l+ν))/ba∇dbls<ǫ},\nforl∈Rsuch thatη(l)≥a\n6. Then by a similar argument as in (2.23), we obtain the follow ing\nestimate\nνw(l)≤E(w)\nǫ2. (3.18)\nBy the gradient flow equation (3.11) and (3.18), we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayl+νw(l)\nld\ndsf(η(s))ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayl+νw(l)\nlf′(η(s))/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤1d\ndsη(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayl+νw(l)\nld\ndsη(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayl+νw(l)\nlf′(η(s))/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤1/integraldisplay1\n0F(t,u)dt\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤/ba∇dblF/ba∇dbl∞ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ /ba∇dblF/ba∇dbl∞νw(l)\n≤ /ba∇dblF/ba∇dbl∞E(w)\nǫ2.(3.19)\nLet us note that the following inequality holds for all s∈R\na−/integraldisplay∞\n−∞/vextendsingle/vextendsingle/vextendsingle˙Af\nθ(s)(w(s))/vextendsingle/vextendsingle/vextendsingleds≤ Af\nθ(s)(w(s))≤2a+/integraldisplay∞\n−∞/vextendsingle/vextendsingle/vextendsingle˙Af\nθ(s)(w(s))/vextendsingle/vextendsingle/vextendsingleds. (3.20)\nBy the definition of νw(l) and the above estimates (3.17), (3.20) and (3.14) we get\nf[η(l+νw(l))]≤2/parenleftbigg\nAf\nθ[l+νw(l)][w(l+νw(l))]+c′/ba∇dbl∇sAf\nθ[l+νw(l)][w(l+νw(l))]/ba∇dbls/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n≤ǫ+c/parenrightbigg\n≤2/parenleftbigg\n2a+/integraldisplay∞\n−∞|˙Af\nθ(s)(w(s))|ds+c′ǫ+c/parenrightbigg\n≤2/parenleftbig\n2a+2CE(w)+2C+2dσC+2/ba∇dblf(η)/ba∇dbl∞dFC+c′ǫ+c/parenrightbig\n.(3.21)ON MAGNETIC LEAF-WISE INTERSECTIONS 29\nNow combine (3.19) and (3.21), we then obtain\nf(η(l))≤f[η(l+νw(l))]+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayl+νw(l)\nld\ndsf(η(s))ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤2/parenleftbig\n2a+2CE(w)+2C+2dσC+2/ba∇dblf(η)/ba∇dbl∞dFC+c′ǫ+c/parenrightbig\n+/ba∇dblF/ba∇dbl∞E(w)\nǫ2\n≤/parenleftbigg\n16CdF+4dF+4/ba∇dblF/ba∇dbl∞dF\nǫ2/parenrightbigg\nC/ba∇dblf(η)/ba∇dbl∞\n+4a+8C∆+16C2+16C2dσ+4C+4dσC+2c′ǫ+2c\n+/ba∇dblF/ba∇dbl∞\nǫ2(2∆+4C+4dσC),(3.22)\nwhere for the last inequality we use (3.15). Note that the las t line of the above estimate (3.22)\ndoes not depend on the choice of a gradient flow line wandl∈R. By the 2nd assumption in\n(3.13) on the isoperimetric constant Cwe have\n/parenleftbigg\n16dFC+4dF+4/ba∇dblF/ba∇dbl∞dF\nǫ2/parenrightbigg\nC≤1\n2\nand we conclude\n/ba∇dblf(η)/ba∇dbl∞≤2(4a+8C∆+16C2+16C2dσ+4C+4dσC+2c′ǫ+2c)\n+2/ba∇dblF/ba∇dbl∞\nǫ2(2∆+4C+4dσC) =:κ.(3.23)\nSincef(η(s)) =η(s) fors≥a\n6, this implies that κis an uniform upper bound of η(s).\nStep 3:/integraltext∞\n−∞|˙Af\nθ(s)(w(s))|ds≤a\n8.\nThe above estimates (3.14), (3.15), (3.23) and ∆ <aimply that\n/integraldisplay∞\n−∞|˙Af\nθ(s)|ds≤2CE(w)+2C+2dσC+2/ba∇dblf(η)/ba∇dbl∞dFC\n≤(8dFC2+2dFC)/ba∇dblf(η)/ba∇dbl∞+4aC+8C2+8dσC2+2C+2dσC\n≤8(4dFC2+dFC)(2a+4aC+8C2+8C2dσ+2C+2dσC+c′ǫ+c)\n+8(4dFC2+dFC)/ba∇dblF/ba∇dbl∞\nǫ2(a+2C+2dσC)\n+4aC+8C2+8dσC2+2C+2dσC\n=/parenleftbigg\n8(4dFC2+dFC)(2+4C+/ba∇dblF/ba∇dbl∞\nǫ2)+4C/parenrightbigg\na\n+8(4dFC2+dFC)/parenleftbigg\n8C2+8C2dσ+2C+2dσC+c′ǫ+c/parenrightbigg\n+8(4dFC2+dFC)/ba∇dblF/ba∇dbl∞\nǫ2(2C+2dσC)\n+8C2+8dσC2+2C+2dσC.(3.24)30 YOUNGJIN BAE\nRecall the 3rd, 4th condition in (3.13)\n8(4dFC2+dFC)(2+4C+/ba∇dblF/ba∇dbl∞\nǫ2)+4C≤1\n16;\n8C2+8dσC2+2C+2dσC\n+8(4dFC2+dFC)/ba∇dblF/ba∇dbl∞\nǫ2(2C+2dσC)\n+8(4dFC2+dFC)/parenleftbigg\n8C2+8C2dσ+2C+2dσC+c′ǫ+c+/parenrightbigg\n≤1\n8.\nThen (3.24) is simplified as follows\n/integraldisplay∞\n−∞|˙Af\nθ(s)|ds≤a\n16+1\n8≤a\n8,\nwhere for the last inequality we use a≥2. This proves Step 3.\nStep 4:η(s)is uniformly bounded.\nFirst set\nǫ:= min/braceleftBig\nǫ,a\n8c′/bracerightBig\n,\nand define a function νw:R→R≥0by\nνw(l) := inf{ν≥0 :/ba∇dbl∇sAf\nθ(l+ν)(w(l+ν))/ba∇dbls<ǫf′(η(l+ν))}.\nNow set\niw(l) := inf\nl≤s≤l+νw(l)f′(η(s)).\nBy similar arguments as in (3.8) and (3.9), we obtain the foll owing estimates\nνw(l)≤E(w)\nǫ2iw(l)2\nand\n|η(l)−η(l+νw(l))| ≤E(w)\nǫiw(l).\nLemma 3.11 implies that for any l∈R\nf[η(l+νw(l))]≥2\n3/parenleftBig\nAf\nθ[l+νw(l)][w(l+νw(l))]−c′/ba∇dbl∇sAf\nθ[l+νw(l)][w(l+νw(l))]/ba∇dbls−c/parenrightBig\n≥2\n3/parenleftbigg\nAf\nθ[l+νw(l)][w(l+νw(l))]−c′ǫ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤a\n8f′[η(l+νw(l))]/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≤1−a\n2/parenrightbigg\n≥2\n3/parenleftbigg\nAf\nθ[l+νw(l)][w(l+νw(l))]−5\n8a/parenrightbigg\n.(3.25)\nHere the 2nd inequality in (3.25) comes from the definition of νwand the assumption c≤a\n2,\nthe 3rd inequality use the definition of ǫandf′≤1. The action estimate (3.20) and Step 3\ngive us the following estimate\nAf\nθ(s)(w(s))≥a−/integraldisplay∞\n−∞/vextendsingle/vextendsingle/vextendsingle˙Af\nθ(s)(w(s))/vextendsingle/vextendsingle/vextendsingleds≥7\n8a. (3.26)ON MAGNETIC LEAF-WISE INTERSECTIONS 31\nLet us combine (3.25), (3.26) to obtain\nf[η(l+νw(l))]≥2\n3/parenleftbigg\nAf\nθ[l+νw(l)][w(l+νw(l))]−5\n8a/parenrightbigg\n≥a\n6.\nSincef∈ F(a\n6),\nη(l+νw(l))≥a\n6>0,\nand hence\nη(l)≥a\n6−E(w)\nǫiw(l)≥ −E(w)\nǫiw(l).\nAs a consequence,\n−f′(η(l))η(l)≤ −iw(l)η(l)≤E(w)\nǫ≤1\nǫ(2∆+4C+4dσC+4κdFC),\nwhere the last inequality comes from (3.15) and (3.23). By th e last condition in (3.13) for\nthe isoperimetric constant Cwe estimate\n(4+4dσ+4κdF)C≤1, (3.27)\nthen\n−f′(η(l))η(l)≤2a+1\nǫ,\nhere we use again ∆ <a. Sincef∈ F(a\n6,2a+1\nǫ), there exists A>0 such that\nAf′(−A)>2a+1\nǫ.\nNow suppose that there exists l0∈Rsuch thatη(l0)<−Athen by continuity there exists\nl1∈Rsuch thatη(l1) =−Awhich leads to a contradiction via condition (3.7)\n2a+1\nǫ<f′(−A)A=−f′(η(l1))η(l1)<2a+1\nǫ.\nThus we conclude that η(l)>−Afor alll∈R, and hence\n/ba∇dblη(l)/ba∇dbl∞≤κ:= max{κ,A}.\n/square\nLemma 3.13. FixF∈ D(Σ) and an action window ( a,2a) such that a≥2. Letc,c′,ǫ>0\nbe the constants from Lemma 3.11. Choose f∈ F(a\n6,2a+1\nmin{ǫ,a/8c′}) and a generic pair ( H,θ)\nsuch thatc=c(H,θ)≤a\n2. Letwbea gradient flow lineof Af\nθ(s)with the following asymptotic\nconditions:\nlim\ns→−∞w(s) =w−∈Crit(a,2a)(Af\nθ(0)),lim\ns→∞w(s) =w+∈Crit(a,2a)(Af\nθ(1)).\nIf the isoperimetric constant C=C(θ) satisfies the following condition\n8dF/parenleftbigg/ba∇dblF/ba∇dbl∞\nǫ+4C+1/parenrightbigg\nC≤1\n9;\n2/parenleftbigg\n1+dσ+8dF/ba∇dblF/ba∇dbl∞\nǫ2C+8dFdσ/ba∇dblF/ba∇dbl∞\nǫ2C\n+32dFC2+32dFdσC2+8dFC+8dFdσC+4c′ǫdF+4cdF/parenrightbigg\nC≤1\n9;(3.28)32 YOUNGJIN BAE\nthen\nAf\nθ(1)(w+)≥9\n10Af\nθ(0)(w−)−1\n10.\nProof.For notational simplicity, let us denote by\np=Af\nθ(0)(w−), q=Af\nθ(1)(w+).\nBy Step 2 in Proposition 3.12, f(η) is uniformly bounded as follows,\n/ba∇dblf(η)/ba∇dbl∞≤2(2q+8C(q−p)+16C2+16C2dσ+4C+4dσC+2c′ǫ+2c)\n+2/ba∇dblF/ba∇dbl∞\nǫ2(2(q−p)+4C+4dσC).\nSinceE(w)≥0, we obtain the following inequality from (3.15)\nq≥p−2C−2dσC−2/ba∇dblf(η)/ba∇dbl∞dFC.\nNow we estimate\nq≥p−2C−2dσC−2/ba∇dblf(η)/ba∇dbl∞dFC\n≥p−2C−2dσC−4dFC/parenleftbigg/ba∇dblF/ba∇dbl∞\nǫ2(2(q−p)+4C+4dσC)\n+2q+8C(q−p)+16C2+16C2dσ+4C+4dσC+2c′ǫ+2c/parenrightbigg\n=p+8dF/parenleftbigg/ba∇dblF/ba∇dbl∞\nǫ2+4C/parenrightbigg\nCp−8dF/parenleftbigg/ba∇dblF/ba∇dbl∞\nǫ+4C+1/parenrightbigg\nCq−2/parenleftbigg\n1+dσ+8dF/ba∇dblF/ba∇dbl∞\nǫ2C\n+8dFdσ/ba∇dblF/ba∇dbl∞\nǫ2C+32dFC2+32dFdσC2+8dFC+8dFdσC+4c′ǫdF+4cdF/parenrightbigg\nC\n≥p+8dF/parenleftbigg/ba∇dblF/ba∇dbl∞\nǫ2+4C/parenrightbigg\nC\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n≥0p−1\n9q−1\n9\n≥p−1\n9q−1\n9.\nHere the 4th inequality we use the assumption (3.28) on C. This proves the assertion. /square\nFor convenience, let us abbreviate\nh[p] :=9\n10p−1\n10. (3.29)\nLemma 3.14. FixF∈ D(Σ) and an action window ( a,2a) such that a≥2. Letc,c′,ǫ>0\nbe the constants from Lemma 3.11. Choose f∈ F(a\n6,2a+1\nmin{ǫ,a/8c′}) and a generic pair ( H,θ)\nsuch thatc=c(H,θ)≤a\n2. If the isoperimetric constant C=C(θ) satisfies the conditions in\nProposition 3.12 and Lemma 3.13 then there exists a commutat ive diagram:\nFH(h−2[a],2a)(Af\nθ(0))i(h−2[a],h2[2a])/d47/d47\n/tildewideΦσ/d41/d41❙❙❙❙❙❙❙❙❙❙❙❙❙❙FH(a,h2[2a])(Af\nθ(0))\nFH(h−1[a],h[2a])(Af\nθ(1))/tildewideΦσ/d53/d53❦❦❦❦❦❦❦❦❦❦❦❦❦❦ON MAGNETIC LEAF-WISE INTERSECTIONS 33\nProof.Let us first construct /tildewideΦσ. Letwbe the gradient flow line of Af\nθ(s)satisfying the limit\nconditions:\nlim\ns→−∞w(s) =w−∈Crit(h−2[a],2a)\nk(Af\nθ(0)),lim\ns→∞w(s) =w+∈Crit(h−1[a],h[2a])\nk(Af\nθ(1)).\nLetM(w−,w+) be the moduli space of such gradient flow lines. In order to co mpactify the\nmoduli space M(w−,w+), by similar arguments in Theorem 2.19, it suffices to bound th e\nenergyE(w) =/integraltext∞\n−∞/ba∇dbld\ndsw(s)/ba∇dbl2\nsdsand the Lagrange multiplier ηin terms of w−,w+. By the\nassumption on the isoperimetric constant C, we can use the argument of Proposition 3.12\nand Lemma 3.13. Especially (3.15), (3.23) give us the follow ing uniform energy bound\nE(w)≤2Af\nθ(1)(w+)−2Af\nθ(0)(w−)+4C+4dσC+4/ba∇dblf(η)/ba∇dbl∞dFC\n≤2Af\nθ(1)(w+)−2Af\nθ(0)(w−)+4C+4dσC+4κ(w−,w+)dFC\nand Proposition 3.12 enables us to conclude that the Lagrang e multiplier ηis also uniformly\nbounded.\nIfµ(w−) =µ(w+), thenM(w−,w+) is discrete for a generic almost complex structure\nJ(s,t)∈ Jσ(s). By virtue of Lemma 3.13, we now define a map\nΦσ\n∗: FC(h−2[a],2a)\n∗ (Af\nθ(0))→FC(h−1[a],h[2a])\n∗ (Af\nθ(1))\ngiven by\nΦσ\n∗(w−) =/summationdisplay\nµ(w+)=µ(w−)#2M(w−,w+)w+,\nwhere # 2meansZ2counting. Since the continuation map Φσcommutes with the boundary\noperators, this induces the following homomorphism on homo logies as follows\n/tildewideΦσ: FH(h−2[a],2a)(Af\nθ(0))→FH(h−1[a],h[2a])(Af\nθ(1)).\nNow we consider the inverse homotopy of Af\nθ(s). By modifying the above construction, we\nobtain\n/tildewideΦσ: FH(h−1[a],h[2a])(Af\nθ(1))→FH(a,h2[2a])(Af\nθ(0)).\nBy a homotopy-of homotopies argument, we conclude that /tildewideΦσ◦/tildewideΦσis the identity map on\nFH(h−2[a],h2[2a])(Af\nθ(0)). This proves the lemma. /square\nLemma 3.15. FixF∈ D(Σ) andf∈/tildewideF(1\n6), see Definition 3.7. Choose a generic pair ( H,θ)\nsuch that the isoperimetric constant C=C(θ) satisfies the conditions in Proposition 3.12\nand Lemma 3.13. If max {2,2c(H,θ)}<a<T < ∞then\ndimFH(h−1[a],h[T])(Af\nθ)≥1\n4dimFH(h−2[a],h2[T])(Af) (3.30)\nholds for generic a,T, see (3.29) for h[p].\nProof.By the commutative diagram in Lemma 3.14, we obtain the follo wing dimension\nestimate\ndimFH(h−1[a],h[2a])(Af\nθ)≥rank/parenleftbig\ni(h−2[a],h2[2a])/parenrightbig\n≥dimFH(h−2[a],h2[2a])(Af).(3.31)\nActually if we choose b∈Rsuch thata<b<2athen\ndimFH(h−1[a],h[b])(Af\nθ)≥dimFH(h−2[a],h2[b])(Af)34 YOUNGJIN BAE\nholds under the generic condition h[b]/∈A(Af\nθ) andh2[b]/∈A(Af).\nNow we construct a sequence {ai}∞\ni=1such that the following holds:\n•a1≥max{2,2c(H,θ)};\n•ai+1=h2[2ai];\n•h−1[ai]/∈A(Af\nθ),∀i∈N;\n•h−2[ai],h2[2ai]/∈A(Af),∀i∈N.\nNote thata1determines the sequence and obviously {ai}is strictly increasing. The 3rd and\n4th conditions are guaranteed for a generic a1. Letabe the set of sequences satisfying the\nabove conditions.\nIn order to compare dimFH(h−1[a],h[T])(Af\nθ) and dimFH(h−2[a],h2[T])(Af), we use (3.31) in-\nductively. Choose {ai} ∈athen the following holds:\ndimFH(h−1(a1),h(2ak))(Af\nθ) =k/summationdisplay\ni=1dimFH(h−1[ai],h[2ai])(Af\nθ)\n≥k/summationdisplay\ni=1dimFH(h−2[ai],h2[2ai])(Af).(3.32)\nBut there exist missing action intervals for Afin the last term of (3.32). To cover the missing\nintervals, we first observe that if a≥2 then the length of the action intervals for Af\nθandAf\nh[2a]−h−1[a], h2[2a]−h−2[a]\nare positive and increasing functions with respect to a. By a simple computation, one can\ncheck that its ratio satisfies\nh[2a]−h−1[a]\nh2[2a]−h−2[a]≤4\nfor alla≥2. This implies that there exist 4 sequences {a1\ni},{a2\ni},{a3\ni},{a4\ni} ∈asuch that\na1\n1<a2\n1<a3\n1<a4\n1<a1\n2and\n/parenleftbig\nh−2[a1\nk],h2[2a1\nk]/parenrightbig\n∪k−1/uniondisplay\ni=14/uniondisplay\nj=1/parenleftbig\nh−2[aj\ni],h2[2aj\ni]/parenrightbig\ncovers (h−2[a1\n1],h2[2a1\nk])⊂R+for anyk∈N.\nNow we obtain the following estimate\n4dimFH(h−1[a1\n1],h[2a1\nk])(Af\nθ)≥k−1/summationdisplay\ni=14/summationdisplay\nj=1dimFH(h−1[aj\ni],h[2aj\ni])(Af\nθ)+dimFH(h−1[a1\nk],h[2a1\nk])(Af\nθ)\n≥k−1/summationdisplay\ni=14/summationdisplay\nj=1dimFH(h−2[aj\ni],h2[2aj\ni])(Af)+dimFH(h−2[a1\nk],h2[2a1\nk])(Af)\n≥dimFH(h−2[a1\n1],h2[2a1\nk])(Af).\nThis proves the lemma.\n/squareON MAGNETIC LEAF-WISE INTERSECTIONS 35\nProposition 3.16. FixF∈ D(Σ)andf∈/tildewideF(1\n6), see Definition 3.7. Let c,c′,ǫ >0be the\nconstants from Lemma 3.11. Choose a generic pair (H,θ). Ifmax{h−1[2],h−1[2c(H,θ)]}<\na<T < ∞then there exists\nn=n(N,g,F,c′,ǫ,H,θ)∈N\nsuch that\ndimFH(a,T)(Af\nθ)≥1\n4ndimFH(h−n[a],hn[T])(Af)\nholds for generic a,T, see (3.29) for h[p].\nProof.In order to use Lemma 3.15, we first introduce a sequence of pri mitive of magnetic\nperturbation {θi}n\ni=0⊂ Pwhich satisfies the following properties:\n•θi=diθ, where 0 = d0<d1<···<dn= 1;\n• Af\nθi:L×R→Ris Morse for all i= 0,1,...,n;\n•Ci= (di+1−di)/ba∇dblθ/ba∇dbl∞satisfies the assumption of Proposition 3.12 and Lemma 3.13\nfor alli= 0,1,...,n−1.\nNote that the subdivision number nforθdoes not depend on the action window. Now choose\na,Tsuch that the following conditions hold:\n•max{h−1[2],h−1[2c(H,θ)]}<a<T < ∞;\n•h−n+i[a],hn−i[T]/∈A(Af\nθi)∀i= 0,1,...,n.\nBy the above second condition, FH(h−n+i[a],hn−i[T])(Af\nθi) are well-defined for 0 ≤i≤n. Now\nwe are ready to apply Lemma 3.15. If we use (3.30) inductively then we conclude that\ndimFH(a,T)(Af\nθ) = dimFH(a,T)(Af\nθn)\n≥1\n4dimFH(h−1[a],h[T])(Af\nθn−1)\n≥ ···\n≥1\n4ndimFH(h−n[a],hn[T])(Af\nθ0)\n=1\n4ndimFH(h−n[a],hn[T])(Af\nθ).\nThis proves the lemma. /square\nRemark 3.17. The argument in Proposition 3.16 holds for any F∈ D(Σ) and any generic\n(H,θ). Notec′,ǫdepend onFand aδ−neighborhood of F−1(0). For a given diffeomorphism\nϕ, consider all defining data ( H,θ) forϕsuch thatϕ=φ1\nXσ\nH. Now we consider\nn′:= inf\n(H,θ)inf\n(F,δ)n(N,g,F,c′,ǫ,H,θ)\nthenn′depends only on ( N,g,Σ,ϕ). By abuse of notation, we write n=n′.\nProof of Theorem 1.4. We first fix a defining Hamiltonian Ffor Σ and a defining data ( H,θ)\nforϕ. Choosef∈/tildewideF(1\n6), see Definition 3.7. By the generic assumption, ϕhas no periodicleaf-\nwise intersection point and Af\nθis Morse for the action window/parenleftbig1\n6+c(H,θ),∞/bracketrightbig\n, see Corollary\nA.5.36 YOUNGJIN BAE\nIf we choose a generic action value a,Tsuch that max {h−1[2],h−1[2c(H,θ)]}<a<T < ∞\nthen Proposition 3.4 and Proposition 3.16 imply that\nn(T)≥#Crit(1\n6+c(H,θ),T−c(H,θ))(Af\nθ)\n≥dimFH(a,T−c(H,θ))(Af\nθ)\n≥1\n4ndimFH(h−n[a],hn[T−c(H,θ)])(Af).(3.33)\nHeren=n(N,g,Σ,ϕ)∈Nis the constant from Proposition 3.16 with Remark 3.17.\nNow we recall that LNis the free loop space of ( N,g). The energy functional E:LN→R\nis given by\nE(q) :=/integraldisplay1\n01\n2|˙q|2dt.\nFor given 0 <T <∞, denote by\nLN(T) :=/braceleftbigg\nq∈ LN:E(q)≤1\n2T2/bracerightbigg\n.\nBy the result of Macarini-Merry-Paternain [24, Proof of The orem A, Remark 1.4], there exists\na constantc′=c′(N,g,Σ,ϕ)>0 such that\ndimFH(h−n[a],hn[T−c(H,θ)])(Af)≥rank{ι: H∗/parenleftbig\nLN(c′(T−1))/parenrightbig\n→H∗(LN)}.\nIfc:= max{4n,c′}>0 then finally we obtain\nn(T)≥1\n4nrank{ι: H∗/parenleftbig\nLN(c′(T−1))/parenrightbig\n→H∗(LN)}\n≥1\nc·rank{ι: H∗/parenleftbig\nLN(c(T−1))/parenrightbig\n→H∗(LN)}.\nThis proves the theorem. /square\nAppendix A.The perturbed Rabinowitz action functional is\ngenerically Morse.\nIn this section we study the Morse property of the perturbed R abinowitz action functional.\nNote firstthat theaction functional Aθ=AF\nH,θis determinedby thefollowing data F∈ D(Σ),\nH∈ Handθ∈ P. Weclaimthat AθisMorseforgeneric( H,θ)∈ H×P. Thegenericproperty\nforH∈ His well-studied in [4, Appendix A]. So we additionally consi der the Morse property\nofAθwith respect to θ∈ P. First recall that\nP={θ∈C∞(R/Z,Ω1(/tildewideN)) :θt= 0,∀t∈[0,1\n2] andθtis bounded,∀t∈[1\n2,1]}.\nTheorem A.1. For a generic pair (H,θ)∈ H×P, the perturbed Rabinowitz action functional\nAθis Morse.\nA.1.Preparations. In order to prove the genericity of the Morse property, we fol low the\nstandard method. Let us consider a certain linear operator a nd show its surjectivity then\nTheorem A.1 deduced from Sard-Smale’s theorem. In this proo f we follow the strategy of [4,\nAppendix A].ON MAGNETIC LEAF-WISE INTERSECTIONS 37\nFirst, let us recall the definition of the perturbed Rabinowi tz action functional\nAθ:L×R→R\nAθ(u,η) =/integraldisplay1\n0u∗λ−η/integraldisplay1\n0F(t,u(t))dt−/integraldisplay1\n0H(t,u(t))dt+/integraldisplay1\n0τ∗θt(/tildewideu(t))[d\ndt/tildewideu(t)]dt.\nHere, inthissection, L ≡W1,2(R/Z,T∗N)isthecompleted loopspaceof T∗N. Fornotational\nconvenience we adopt the functionals F:L →RandAη0\nθ:L →Rdefined by\nF(u) :=/integraldisplay1\n0F(t,u)dt,Aη0\nθ(u) :=Aθ(u,η0)\nfor a fixedη0∈R. We note that Aθ(u,η) =Aη0\nθ(u)+(η0−η)F(u), and we obtain\ndAθ(u,η)[ˆu,ˆη] =dAη0\nθ(u)[ˆu]−ˆηF(u)+(η0−η)dF(u)[ˆu]\nwhere ˆu∈Γ1,2(u∗T(T∗N)), the space of W1,2vector fields along uandη∈R. For a critical\npointw0= (u0,η0)∈Crit(Aθ) the Hessian at w0equals\nHAθ(w0)[(ˆu1,ˆη1),(ˆu2,ˆη2)] =HAη0\nθ(u0)[ˆu1,ˆu2]−ˆη1dF(u0)[ˆu2]−ˆη2dF(u0)[ˆu1].\nFor a function ( η0F+H) : [0,1]×T∗N→Rand anR/Z-parametrized symplectic form\nωσ, we consider a Hamiltonian type diffeomorphism ψwhich is a time-1-map of Xσ\nη0F+H. We\nthen define\nLψ:={v∈W1,2([0,1],T∗N) :v(0) =ψ(v(1))}, (A.1)\nthe twisted loop space, and introduce the diffeomorphism Ψ : Lψ→ L\nΨ(v)(t) =ψt(v(t))\nwhereψtis a time-t-map ofXσ\nη0F+H. For a fixed critical point w0= (u0,η0) ofAθwe use\nthis diffeomorphism to pull back Aθ\nAθ:=Aθ◦(Ψ×idR) :Lψ×R→R.\nWe setv0:= Ψ−1◦u0, thusv0= const. Then we simplify the Hessian as follows\nHAθ(v0,η0)[(ˆv1,ˆη1),(ˆv2,ˆη2)] =/integraldisplay1\n0ω(d\ndtˆv1,ˆv2)dt−ˆη1dF(v0)[ˆv2]−ˆη2dF(v0)[ˆv1],\nwhereF:=F ◦Ψ.\nRecall from Definition 2.2 that F(t,x) =ρ(t)¯F(x). Sinceρ(t) = 0fort∈[1\n2,1],ψtpreserves\nthe level of ¯Ffort∈[1\n2,1] andH(t,x),σtvanish fort∈[0,1\n2], we compute\nF(v) =/integraldisplay1\n0F(t,ψt(v))dt=/integraldisplay1\n2\n0F(t,ψt(v))dt\n=/integraldisplay1\n2\n0F(t,v)dt=/integraldisplay1\n0F(t,v)dt.\nThus, the Hessian of Aθbecomes\nHAθ(v0,η0)[(ˆv1,ˆη1),(ˆv2,ˆη2)]\n=/integraldisplay1\n0ω(d\ndtˆv1,ˆv2)dt−ˆη1/integraldisplay1\n0dF(t,v0)[ˆv2]dt−ˆη2/integraldisplay1\n0dF(t,v0)[ˆv1]dt(A.2)38 YOUNGJIN BAE\nA.2.The linearized operator. We denote by\nHk:={H∈Ck(R/Z×T∗N) :H(t,·) = 0,∀t∈[0,1\n2]};\nPk:={θ∈Ck(R/Z,Ω1(/tildewideN)) :θt= 0,∀t∈[0,1\n2] andθtis bounded,∀t∈[1\n2,1]}.\nForv∈ Lψ, see (A.1), we define the bundle Eψ→ Lψby\n(Eψ)v:=L2([0,1],v∗T(T∗N)).\nDefinition A.2. Let (u0,η0) be a critical point of Aθand (v0,η0) the corresponding critical\npoint ofAθ, that is the constant loop v0defined by the equation u0= Ψ(v0). Then we define\nthe linear operator\nL(v0,η0,H,θ):T(v0,η0,H,θ)(Lψ×R×Hk×Pk)→(Eψ)∨×R\nwhere (Eψ)∨is the vertical subspace of the bundle Eψ. Then we obtain\n/an}b∇acketle{tL(v0,η0,H,θ)[ˆv1,ˆη1,ˆH,ˆθ],(ˆv2,ˆη2)/an}b∇acket∇i}ht\n:=HAθ(v0,η0)[(ˆv1,ˆη1),(ˆv2,ˆη2)]+/integraldisplay1\n0(Ψ∗dˆH)(t,v0)[ˆv2(t)]dt\n+/integraldisplay1\n0(Ψ∗τ∗dˆθt)(d\ndtˆv1(t),ˆv2(t))dt.\nProposition A.3. The operator L(v0,η0,H,θ)is surjective. Indeed, L(v0,η0,H,θ)is surjective\nwhen restricted to the space\nV:={(ˆv,ˆη,ˆH,ˆθ)∈T(v0,η0,H,θ)(Lψ×R×Hk×Pk) : ˆv(1\n2) = 0}.\nProof.TheL2-Hessian is a self-adjoint operator. Thus, the operator L(v0,η0,H,θ)has closed\nimage. Therefore, it suffices to prove that the annihilator of the image of L(v0,η0,H,θ)is zero.\nLet (ˆv2,ˆη2) be in the annihilator of the image of L(v0,η0,H,θ), that means\n/an}b∇acketle{tL(v0,η0,H,θ)[ˆv1,ˆη1,ˆH,ˆθ],(ˆv2,ˆη2)/an}b∇acket∇i}ht= 0\nfor all (ˆv1,ˆη1,ˆH,ˆθ)∈T(v0,η0,H,θ)(Lψ×R×Hk×Pk). This is equivalent to the following three\nequations:\nHAθ(v0,η0)[(ˆv1,ˆη1),(ˆv2,ˆη2)] = 0,∀(ˆv1,ˆη1)∈(Tv0Lψ)×R; (A.3)\n/integraldisplay1\n0dˆHt(ψt(v0))[dψt(v0)[ˆv2]] = 0,∀ˆH∈ Hk; (A.4)\n/integraldisplay1\n0dˆθt(ψt(v0))/bracketleftbig\nτ∗dψt(v0)[d\ndtˆv1(t)],τ∗dψt(v0)[ˆv2(t)]/bracketrightbig\ndt= 0,∀(ˆv1,ˆθ)∈Tv0Lψ×Pk.(A.5)\nNote that the Hessian HAθis a self-adjoint operator, equations (A.2) and (A.3) with e lliptic\nregularity implies that ˆ v2∈Ck+1([0,1],Tv0T∗N) and (ˆv2,ˆη2) satisfies the equation\nd\ndtˆv2−ˆη2XF(t,v0) = 0 (A.6)\nand the linearized boundary condition\nˆv2(0) =dψ(v0)[ˆv2(1)]. (A.7)ON MAGNETIC LEAF-WISE INTERSECTIONS 39\nEquation (A.4) implies that\nˆv2(t) = 0,∀t∈[1\n2,1]. (A.8)\nRecall from (2.4) that XF(t,x) =ρ(t)X¯F(x) then (A.6) becomes\nd\ndtˆv2−ˆη2ρ(t)X¯F(v0) = 0.\nThis is a linear ODE in the vector space Tv0T∗Nas follows\nˆv2(t) = ˆv2(0)+ ˆη2/parenleftbigg/integraldisplayt\n0ρ(τ)dτ/parenrightbigg\nX¯F(v0). (A.9)\nRecall from (2.3) that/integraltextt\n0ρ(τ)dτ= 1 for allt∈[1\n2,1]. Substitute this into (A.9) and combine\nwith equation (A.8), we then obtain\n0 = ˆv2(t) = ˆv2(0)+ ˆη2X¯F(v0) (A.10)\nfort≥1\n2. By using equations (A.7) and (A.8) at t= 1, we deduce ˆ v2(0) = 0. Now, put this\ninto (A.10) we have\nˆη2X¯F(v0) = 0\nSince (v0,η0) is deduced from a critical point ( u0,η0) ofAθ, we have ¯F(v0) =¯F(u(0)) =k,\nand we already assume that kis a regular value of ¯F. In particular,\nˆη2= 0 (A.11)\nEquation (A.10), (A.11) and ˆ v2(0) = 0 imply\nˆv2(t) = 0,∀t∈[0,1].\nTherefore, theannihilatoroftheimageof L(v0,η0,H,θ)vanishesandthus L(v0,η0,H,θ)issurjective.\nMoreover, if we restrict the domain of L(v0,η0,H,θ)toVthen the only change occurs in (A.6)\natt=1\n2. By continuity, however, equation (A.6) is still valid for a llt∈[0,1]. /square\nRemark A.4. In the proof of Proposition A.3, we do not use equation (A.5). This means\nthat for a fixed θ∈ Pkthere exists H∈ Hksuch thatL(v0,η0,H,θ)is surjective.\nProof of Theorem A.1. We first define the Banach space bundle E → Lby\nEu=L2(R/Z,u∗T(T∗N))\nforu∈ L. Now consider the section S:L×R×Hk×Pk→ E∨×Rgiven by the differential\nof the Rabinowitz action functional Aθ\nS(u,η,H,θ ) :=dAθ(u,η). (A.12)\nHere (H,θ)∈ Hk× Pkis the additional variables for the perturbation of Aθ. Its vertical\ndifferential DS:T(u0,η0,H,θ)(L×R×Hk×Pk)→TS(u0,η0,H,θ)(E∨×R) at (u0,η0,H,θ)∈S−1(0)\nis\n/an}b∇acketle{tDS(u0,η0,H,θ)[(ˆu1,ˆη1,ˆH,ˆθ)],(ˆu2,ˆη2)/an}b∇acket∇i}ht\n=HAθ(u0,η0)[(ˆu1,ˆη1),(ˆu2,ˆη2)]+/integraldisplay1\n0dˆH(t,u0)[ˆu2(t)]dt+/integraldisplay1\n0dˆθt(τ∗d\ndtˆu1(t),τ∗ˆu2(t))dt\n(A.13)40 YOUNGJIN BAE\nSince (Ψ ×idR×idHk×idPk)∗DS=L(v0,η0,H,θ), the operator DSis surjective. Thus, by the\nimplicit function theorem the moduli space\nM:=S−1(0)\nis a smooth Banach manifold. We consider the projection ΠHk×Pk:M → Hk×Pk. Then the\nAθis Morse if and only if ( H,θ) is a regular value of ΠHk×Pk. By the Sard-Smale theorem\nthis forms a generic set for klarge enough. Moreover, the Morse condition is Ck-open. Thus\nfor function in an open and dense subset of Hk× Pk, the Rabinowitz action functional is\nMorse. Taking the intersection of all kconcludes the proof of Theorem A.1. /square\nNow we discuss the Morse property of Af\nθ. Since we are interested in critical points of\nAf\nθwith positive action value, it suffices to check the Morse prop erty for the positive critical\npoints.\nCorollary A.5. Givena >0and choose f∈ F(a)(see, Definition 3.3). For a generic\n(H,θ)∈ H × P the perturbed F-Rabinowitz action functional Af\nθ=AF,f\nH,θis Morse on the\naction window (a+c(H,θ),∞].\nProof.Letw0= (u0,η0) be a critical point of Af\nθwithAf\nθ(u0,η0)> a−c(H,θ) then by\nthe argument in Proposition 3.4 we obtain f(η0)>a. Sincef∈ F(a), see Definition 3.3, we\nconcludef′(η0) = 1. Hence the argument in the proof of Theorem A.1 definitely holds. This\nproves the corollary. /square\nAppendix B.No periodic magnetic leaf-wise intersection points\nIn this section, we study the second regularity property of ϕ, see Definition 2.4. The claim\nis thatϕhas no periodic leaf-wise intersection points for generic H∈ Handθ∈ P. In\n[3] Albers-Frauenfelder already studied the above propert y with respect to H∈ H. As in\nAppendix A, we work with ( H,θ)∈ H×P and modify the strategy of [3].\nRecall that the hypersurface Σ ⊂T∗Nis called non-degenerate if closed Reeb orbits on Σ\nform a discrete set. A generic Σ is non-degenerate, see [11, T heorem B.1]. If the critical points\nofAθdoes not meet any closed Reeb orbit then there are no periodic leaf-wise intersection\npoints. Thus it suffices to prove the following theorem.\nTheorem B.1. LetΣ⊂T∗Nbe a non-degenerate starshaped hypersurface and Rbe a set of\nclosed Reeb orbit on Σwhich form a discrete set. If dimN≥2then the set\n{(H,θ)∈ H×P :Aθis Morse and im(x)∩im(y) =∅,∀x∈Crit(Aθ), y∈ R} (B.1)\nis generic in H×P, see Definition 1.1 and 1.2.\nProof.We first define the evaluation map ev : M →Σ\nev(u0,η0,H,θ) =u0(1\n2)\nwhereMis the same as in the proof of Theorem A.1. Proposition A.3 wit h Lemma B.2 below\nguarantee that the evaluation map\nev(H,θ):= ev(·,·,H,θ) : Crit(Aθ)→ΣON MAGNETIC LEAF-WISE INTERSECTIONS 41\nis a submersion for a generic choice of ( H,θ). LetRnbe the set of Reeb orbit with period\nless thannwhich is a 1-dimensional set in Σ, then ev−1\n(H,θ)(Rn) does not intersect Crit( Aθ)\nsince dimΣ ≥3. Therefore, the set\n{(H,θ)∈ H×P :Aθis Morse and im( x)∩im(y) =∅,∀x∈Crit(Aθ), y∈ Rn}(B.2)\nis generic in H × Pfor alln∈N. Now, the set (B.1) is a countable intersection of the set\n(B.2), for all n∈N. This proves the Theorem B.1 /square\nThe following lemma is contained in [3].\nLemma B.2. LetE → Sbe a Banach bundle and s:S → Ea smooth section. Moreover,\nlet Φ :S → Cbe a smooth map into the Banach manifold C. 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Geom.\n8, no. 4 (2010), 1-24.\nYoungjin Bae, Department of Mathematics and Research Insti tute of Mathematics, Seoul\nNational University\nE-mail address :jini0919@snu.ac.kr"    },    {        "title": "0712.1087v1.Short_time_critical_dynamics_at_perfect_and_non_perfect_surface.pdf",        "content": "arXiv:0712.1087v1  [physics.comp-ph]  7 Dec 2007Short-time critical dynamics at perfect and non-perfect su rface\nShizeng Lin1,2and Bo Zheng1\n1Zhejiang University, Zhejiang Institute of Modern Physics,\nHangzhou 310027, P.R. China\n2Computational Materials Science Center,\nNational Institute for Materials Science,\nSengen 1-2-1, Tsukuba 305-0047, Japan\nWe report Monte Carlo simulations of critical dynamics far f rom equilibrium on a perfect and\nnon-perfect surface in the 3 dIsing model. For an ordered initial state, the dynamic relax ation of\nthe surface magnetization, the line magnetization of the de fect line, and the corresponding suscep-\ntibilities and appropriate cumulant is carefully examined at the ordinary, special and surface phase\ntransitions. The universal dynamic scaling behavior inclu ding a dynamic crossover scaling form\nis identified. The exponent β1of the surface magnetization and β2of the line magnetization are\nextracted. The impact of the defect line on the surface unive rsality classes is investigated.\nPACS numbers: 64.60.Ht, 68.35.Rh, 05.20.-y\nI. INTRODUCTION\nThe breakdown of space and time translation invari-\nanceleadstogeometricandtemporalsurfaceeffects. The\nformer is very common in a system whose spatial correla-\ntion length is comparable to its dimensions. Such effects\nbecome even more important when nano-scale materials\nare concerned. In a recent experiment, for example, an\nanomalous temperature profile of the phase transitions\nwas observed in the presence of a ferromagnetic surface\n[1]. The latter occurs in a nonequilibrium system, which\nis prepared by suddenly quenching the system to its crit-\nical temperature from any given initial condition.\nThe breakdown of space translation invariance modi-\nfies the critical behaviors near geometric surface and new\ncritical exponents must be introduced. There may ex-\nist several universality classes in one bulk system in the\npresence of free geometric surface. The critical behavior\nof geometric surface has been extensively studied, and\nthe equilibrium phase diagram has been well established\nin the past decades [2, 3, 4, 5]. However, most previous\nstudiesconcentratedonthestaticbehaviors[6, 7, 8, 9,10]\nand the dynamics in the long-time regime[11, 12, 13], i.e.\nsystem only with geometric surface. The critical dynam-\nics of surface in the macroscopic short-time regime, i.e.,\nwhen the system is still far from equilibrium, is much less\ntouched [5].\nOn the other hand, for a system quenched to its crit-\nical temperature, because there is no characteristic time\nscale, the temporal surface has long-lasting effect. This\neffect has very important consequences. One is that in\nnonequilibrium dynamic relaxation of magnetization, if\nin the initial state, there is small, nonvanishing magne-\ntizationm0≪1, the magnetization grows as m0tθwith\nθbeing a new nonequilibrium dynamic exponent[14]. In\nsuch short-time critical dynamics, there exist two com-\npeting nonequilibrium dynamic processes. One is the do-\nmain growth with scaling dimension xiand the other is\nthe critical thermal fluctuation with scaling dimensionx=β/ν. Because the spatial correlation length ξgrows\nast1/z, we can relate θtoxiandxbyθ=xi−x\nz. Gener-\nallyxiis larger than xand the net effect is the domain\ngrowth in the nonequilibrium relaxation process. This\nshort-time critical dynamics of bulk has been established\nin the past decade, and successfully applied to different\nphysical systems [14, 15, 16, 17, 18, 19]. Based on the\nshort-time dynamic scaling, new techniques for the mea-\nsurements of both dynamic and static critical exponents\nas well as the critical temperature have been developed\n[20, 21, 22]. Recent progresses can be found partially in\nRefs. [16, 23, 24, 25, 26].\nObviously, the physical phenomena are more compli-\ncated, when both temporal and geometric surfaces are\nconsidered. The interplay between both surfaces em-\nbraces many interesting physics and is worth for care-\nful studies[5, 27]. Recently it is reported that in non-\nequilibrium states, the surface cluster dissolution may\ntake place instead of the domain growth [28, 29]. In\nthese studies, the dynamic relaxation starting from a\nhigh-temperature state is concerned.\nThe impact of defect on geometric surface is also of\ngreat concern. The presence of imperfection may alter\nthesurfaceuniversityclassesandeventhephasediagram.\nThe former is easily signaled from the non-equilibrium\ndynamics as in the case of bulk[24, 25, 30].\nIn this paper, we study the short-time critical dynam-\nics ona perfect and non-perfect surface with Monte Carlo\nsimulations. We generalize the universal dynamic scaling\nbehaviortothe dynamicrelaxationatgeometricsurfaces,\nstarting from the orderedstate. At the ordinary, spe-\ncial and surface phase transitions, the dynamic scaling\nbehavior of the surface magnetization, susceptibility and\nappropriatecumulant are identified. The static exponent\nβ1of the surface magnetization and β2of the line magne-\ntization of the defect line are extracted from the dynamic\nbehavior in the macroscopic short-time regime. The ro-\nbustness of surface university class against extended de-\nfect is investigated by means of non-equilibrium dynam-2\nics. The surface transition and special transition can also\nbe detected from the short-time dynamics.\nThe remaining part of this paper is organized as fol-\nlows. In Sec. II, the definition of the model and the\nshort-time dynamic scaling analysis are presented. In\nSec. III and IV, the dynamic relaxation on a perfect and\nnon-perfect surface is studied. In Sec. V, the results are\nsummarized.\nII. MODEL AND DYNAMIC SCALING\nANALYSIS\nA. Model\nThe Hamiltonian of the 3 dIsing model with Glauber\ndynamics and line defect on free surface in the absence of\nexternal magnetic field can be written as the sum of bulk\ninteractions, surface interactions and line interactions,\nH=−Jb/summationtextbulk\n<xyz>σxyzσx′y′z′−Js/summationtextsurface\n<xy>σxyzσx′y′z′\n−Jl/summationtextdefect\n<y>σxyzσx′y′z′,\n(1)\nwhere spin σcan take values ±1 and< xyz > indicates\nthe summation over all nearest neighbors. The first sum\nruns over all links including at least one site that does\nnot belong to the surface, whereas the second sum runs\nover all surface links excluding the links that both sites\nare inside the defect line. The last summation extends\nover all links which belong to the defect line. Jb,Jsand\nJlare the coupling constants for the bulk, surface and\ndefect line respectively. For ferromagnetic materials, Jb\nandJsare positive. It is generally believed that the dy-\nnamic universality class of Glauber dynamics is insensi-\ntive to the detailed algorithm used as long as the updat-\ning algorithm is local. Here we use Metropolis spin-flip\nalgorithm. Without explicitly specified, the dynamic ex-\nponent refers to the Ising model with Glauber dynamics\nin the following discussions.\nFor a perfect surface, i.e., Jl=Js, it is well known\nthat there exists a special threshold rsp≡Js/Jbin equi-\nlibrium. For Js/Jb< rsp, the surface undergoes a phase\ntransition at the bulk transition temperature Tb, due to\nthe divergent correlation length in the bulk. This phase\ntransition is called the ordinary transition , and the crit-\nical behavior is independent of Js/Jb. See Fig. 1. This\nis astronguniversality. For Js/Jb> rsp, the surface first\nbecomes ferromagnetic at a surface transition tempera-\ntureTs> Tb, while the bulk remains to be paramagnetic.\nIf the temperature is further reduced, the bulk becomes\nalso ferromagnetic at Tb. The former phase transition is\ncalledthe surface transition and the latter is called the\nextraordinary transition . It is generally believed that the\nsurface transition belongs to the universality class of the\n2dIsing model [2, 3]. Around rspoccurs the crossover\nbehavior. At exactly Js/Jb=rsp, the lines of surface\ntransition, ordinary transition and extraordinary tran-sition meet at this multicritical point with new surface\nexponents. The surface and bulk become critical simul-\ntaneously at this point and this phase transition is called\nthe special transition . The best estimate of rspfor the 3d\nIsing in equilibrium is 1 .5004(20) [31].\nFor a non-perfect surface, we introduce a defect line\nwith coupling strength Jlonto the surface. Generally\nspeaking, the impact of imperfection on a surface is two\nfold. Take a surface with random bond disorder as an\nexample. The randomnessmay reducethe surfacetransi-\ntion temperature, and alterthe globalphasediagramofa\nsemi-infinite system. For example, the special transition\npoint of the Ising model with a amorphous surface is lo-\ncated at rsp= 1.70(1) [32], noticeably larger than that of\nthe Ising model with a perfect surface rsp= 1.5004. An-\nother effect of the randomness is that it may change the\nuniversality class of the surface. The relevance or irrele-\nvanceofrandomimperfections onthe pure surfacecanbe\nassessed by the Harris-type criterion [33]. The extended-\nHarris criterion states that for a surface with random\nbond disorder, the disorder is relevant for α11>0 but\nirrelevant for α11<0. Based on this criterion, the ran-\ndom surface coupling of the Ising model is irrelevant at\nthe ordinary transition since α11<0. In this case, it\nwas rigorously proved βdis\n1=βord\n1by Diehl based on\nthe Griffiths-Kelly-Sherman inequality, where βdis\n1is the\ncritical exponent at the ordinary transition on a random\nbond surface[34]. The situation is less clear at the spe-\ncial transition, for α11is very close to 0. Recent simu-\nlations suggest that α11<0 and hence the disorder is\nirrelevant[10]. The irrelevance at the special transition\nhas also been reported in Ref. [32]. At the surface tran-\nsition, the surface is equivalent to the 2 dIsing model.\nThe disorder only leads to logarithm correction (see [30]\nand reference therein). In the case of defect line, the\ndefect doesn’t shift the transition temperatures of the\nsurface transition, and therefore, the special transition\npointrspat which the surface transition line and ordi-\nrsp=1.5rspTcT\nBF\nSFBP\nSPBF\nSF\nordinarysurface\nextraordinary\nspecial\nFIG. 1: Schematic phase diagram for the semi-infinite Ising\nmodel with bulk coupling Jband surface coupling ratio Js/Jb.\nTbis the bulk transition temperature and the ferromagnetic\nbulk is denoted by FB while the paramagnetic is denoted by\nPB. The surface phases are labeled FS for a ferromagnet and\nPS for a paramagnet.3\nnary transition line meet remains to be unchanged [35].\nWe only consider the robustness of the ordinary, special\nand surface transition in the presence of line defect.\nB. Dynamic scaling analysis\nTABLE I: The bulk critical temperature and critical expo-\nnents of the 3 dIsing model.\nTc ν3d z3d\n4.5115248(6) [36] 0.6298(5) [37] 2.042(6) [38]\nFor a dynamic system, which is initially in a high-\ntemperature state, suddenly quenched to the critical\ntemperature, and then released to the dynamic evolu-\ntion of model A, one expects that there exist univer-\nsal scaling behaviors already in the macroscopic short-\ntime regime [14]. This has been shown both theoreti-\ncally and numerically in a variety of statistical systems\n[14, 15, 16, 17, 18, 19, 23, 25], anditexplainsalsothespin\nglass dynamics. Furthermore, the short-time dynamic\nscaling behavior has been extended to the dynamic re-\nlaxation with an ordered initial state or arbitrary initial\nstate, based on numerical simulations [16, 23, 25, 38, 39].\nRecent renormalization group calculations also support\nthe short-time dynamic scaling form for the ordered ini-\ntial state [26].\nOn the other hand, Ritschel and Czerner have gener-\nalized the short-time critical dynamics in a homogenous\nsystemto that in an inhomogeneousone, i.e., the systems\nwith a free surface, and derived the scaling behavior of\nthe magnetization close to the surface for the dynamic\nrelaxation with a high-temperature initial state [27]. Re-\ncent development can be found in Refs. [28, 29]. In this\npaper, we alternatively focus on the dynamic relaxation\nwith the ordered initial state, and with a non-perfect\nsurface. As pointed out in the literatures [16, 23], the\nfluctuation is less severe in this case. It helps to obtain a\nmore accurate estimate of the critical exponents at sur-\nface. From theoretical point of view, it is also interesting\nto study the dynamic relaxationwith the ordered or even\narbitrary initial state.\nSimilar to the scaling analysis in bulk [14, 16, 26, 38],\nwe phenomenologically assume that, for dynamic relax-\nationwithorderedinitialstate,thesurfacemagnetization\ndecays by a power law,\n< m1(t)>∼t−β1/νszs, (2)\naftera microscopictime scale tmic. Here<·>represents\nthe statistical average, β1is the static exponent of the\nsurface magnetization, νsis the static exponent of the\nspatial correlation length, and zsis the dynamic expo-\nnent. Thisassumptioncanbeunderstoodbynotingthat,\nfor nonequilibrium preparation with ordered initial state\nm0= 1, the dynamic relaxation is governed by criticalthermal fluctuation with scaling dimension x1=β1/νs.\nFor theordinary andspecialtransitions where the crit-\nicality of surface originates from the divergence of the\ncorrelation length in bulk, there are no genuine new sur-\nface dynamic exponent zsand static exponent νs.νsand\nzsare just the same as those in the bulk, i.e. νs=ν3d\nandzs=z3d, whileβ1is neither that of the 2 dIsing\nmodel nor that of the 3 dIsing model [11]. For the sur-\nfacetransition where the critical fluctuation of surface\nis of the universality class of the 2 dIsing model, it is\ngenerally believed that all static and dynamic exponents\nare the same as those of the 2 dIsing model [2, 3]. i.e.\nβ1=β2d= 1/8,νs=ν2d= 1 and zs=z2d≈2.16(2)\n[16].\nAnother important observable is the second moment\nof the surface magnetization, or the so-called time-\ndependent surface susceptibility, defined as\nχ11=L2[< m2\n1>−< m1>2]. (3)\nSimple finite-size scaling analysis [16] reveals that\nχ11(t)∼tγ11/νszs. (4)\nHere the exponent γ11/νsis related to β1/νsbyγ11/νs=\nd−1−2β1/νs, withd= 3 being the spatial dimension of\nbulk. This is nothing but the scaling law in equilibrium\nbetween the exponent of the surface susceptibility and\nthe exponent of the surface magnetization. One can also\nunderstand the scaling behavior in Eq. (4) in an intuitive\nway. In equilibrium, χ11behaves as χ11∼Lγ11/νswith\nLbeing the lattice size. In the dynamic evolution, χ11(t)\nshould be related to the non-equilibrium spatial correla-\ntion length ξ(t) withχ11(t)∼ξ(t)γ11/νs, since the finite\nsize effect is negligible. Then the growth law ξ(t)∼t1/zs\nof the non-equilibrium spatial correlation length imme-\ndiately leads to Eq. (4).\nAlternatively, one can also construct the appropriate\ntime-dependent cumulant U(t) =< m2\n1> / < m 1>2−1.\nObviously, U(t)∼t(γ11+2β1)/νszs. From the scaling law\nγ11/νs=d−1−2β1/νs, onederives( γ11+2β1)/νs=d−1.\nThescalingbehaviorof U(t)then reducestothestandard\nform [16, 38],\nU(t)∼t(d−1)/zs, (5)\nwithd−1 being the spatial dimension of the surface.\nIn other words, from Eqs. (2) and (4), or from Eqs. (2)\nand(5), weobtainindependentmeasurementsoftwocrit-\nical exponents, e.g., β1/νsandzs. Alternatively, if we\ntakeνsandzsas input, we have two independent esti-\nmates of the static exponent β1of the surface magneti-\nzation. This may testify the consistency of our dynamic\nscaling analysis.\nAll foregoing equations involve the bulk exponents νs\nandzs. Therefor an accurate estimate of the surface crit-\nical exponents β1needs precise values of ν3dandz3d, as\nwell asz2d. Since the 3 dbulk Ising model has been ex-\ntensively studied with various methods, many accurate4\nTABLE II: Summary of the surface critical exponents at the or dinary and special transition in the 3 dIsing model, as obtained\nby different techniques. MF: mean-field, MC: Monte Carlo simu lations, FT: field-theoretical methods, CI: conformal inva riance.\nThe data marked with ∗are calculated by using scaling law 2 β1+γ11= (d−1)νs.\nMF [2]MC [8] MC [9] MC [40] MC [10] MC [41] MC+CI [36] FT [42] this work\nβord\n110.78(2)0.807(4)0.80(1)0.796(1) − 0.798(5) 0.7960.795(6)\nβsp\n11/20.18(2)0.238(2) −0.229(1) 0.237(5) − 0.2630.220(3)\nγsp\n111/20.96(9)0.788(1) −0.802(3)∗0.785(11)∗− 0.7340.823(4)\nresults of the critical exponents and transition tempera-\nture are available. We concentrate our attention to the\nsurface exponents and take the bulk exponents as input.\nThe results of the bulk exponents of the 3 dIsing model\nare summarized in Table I. The criteria to choose those\nvalues are their relative accuracy, as well as the methods\nused to extract these exponents.\nC. Simulations\nIn this paper, with Monte Carlo simulations we study\nthe dynamic relaxationof the 3 dIsing model on a perfect\nand non-perfect surface at the transition temperature,\nquenched from a completely ordered initial state. The\nstandard Metropolis algorithm is adopted in the simula-\ntions. In order to investigate the surface critical behav-\nior, we apply the periodic boundary condition in the xy\nplane and open boundary condition in the z direction to\ntheL×L×Lcubic lattice.\nThe main results are obtained with the lattice size L=\n128 and L= 80, and additional simulations with other\nlattice sizes are also performed to study the finite-size\neffect. For a perfect surface, the surface magnetization is\ndefined as\nm1=1\n2L2L/summationdisplay\nxy(σxy1+σxyL), (6)\nand its critical exponent is denoted by β1. For a non-\nperfect surface, the defect line is placed at surface posi-\ntionx=L/2 and the line magnetization is defined as\nm2=1\n2LL/summationdisplay\nxy(σL\n2y1+σL\n2yL), (7)\nand its critical exponent is denoted by β2. The spin σxyz\ndenotes the spin sitting at site ( x,y,z). We measure the\nsurfaceandlinemagnetizationduringthenonequilibrium\nrelaxation. We averagefrom 5000to 20000runs with dif-\nferent random numbers to achieve a good statistics. Er-\nror bars are estimated by dividing the total samples into\ntwo subgroups, and by measuring the exponents at dif-\nferent time intervals. Most of the simulations are carried\nout on the Dawning 4000A supercomputer. The total\nCPU time is about 3 node-year.III. SHORT-TIME DYNAMICS ON A PERFECT\nSURFACE\nIn this section we study the nonequilibrium critical dy-\nnamics on a perfect surface, i.e. Jl=Js. To investi-\ngate the critical behavior on the surface, it is important\nto know the special transition point rsp. For a perfect\nsurface of the 3 dIsing model with ferromagnetic interac-\ntions, there exist rather accurate estimates of rspin equi-\nlibrium, e.g., rsp= 1.5004(20)in Ref. [31]. We adopt this\nvalue as the special transition point. As illustrated later,\nthespecialtransitionpoint rspcanalsobeestimatedfrom\nthe scaling plot of a dynamic crossover scaling relation.\nFor the ordinary phase transition, the dynamic relax-\nation of the surface magnetization with different Js/Jb\nare shown in Fig. 2. The curves of Js/Jb= 1.0 with\nL= 40 and L= 80 overlap up to t≥300MCS(Monte\nCarlo sweep per site). It confirms that the finite-size\neffect is negligibly small for L= 80 up to at least\nt= 1000MCS, since the correlating time of a finite sys-\ntem increases by tL∼Lz. In Fig. 2, a power-law be-\nhavior is observed for all Js/Jb. The microscopic time\nscaletmic, after which the short-time universal scaling\nbehavior emerges, in other words, after which the cor-\nrection to scaling is negligible, gradually increases as the\nsurface coupling is being enhanced. For Js/Jb= 0.2,\ntmic∼10MCS, while for Js/Jb= 1.2,tmic∼100MCS.\n10 100 10000.010.11\ntm1(t)Js/Jb=1.5004\nPerfect surfaceJs/Jb=1.2\n1.0\n0.8\n0.5\n0.2Ordinary transitionSpecial transition\nFIG. 2: Dynamic relaxation of the surface magnetization is\ndisplayed with solid lines on a double-log scale, at the ordi -\nnary transition with various Js/Jb, and at the special tran-\nsitionJs/Jb=rsp= 1.5004. The temperature is set to the\nbulk critical temperature Tc, and the lattice size is L= 80.\nOpen circle are the data for Js/Jb= 1.0 andL= 40. Well\naway from the special transition, the slope of the curves is\nindependent of Js/Jb.5\n50 500 100010-310-310-210-2\nU11\nχ11χ11(t) U11(t)\nJs/Jb=1.5004\ntSpecial transitionPerfect surface\nFIG. 3: Dynamic relaxation of the surface susceptibility an d\ncumulant at the special transition is plotted on a double-lo g\nscale. The lattice size is L= 128 and T=Tc.\nA direct observation in Fig. 2 is that the curves for\nJs/Jb< rspare parallel to each other. By fitting these\ncurves to Eq. (2), we obtain βord\n1= 0.790(7), 0 .792(6),\n0.795(6), 0 .786(6) and 0 .755(12) for Js/Jb= 0.2, 0.5,\n0.8, 1.0 and 1.2 respectively. The values of βord\n1at\nJs/Jb= 0.2,Js/Jb= 0.5 andJs/Jb= 0.8 are well con-\nsistent with each other within error. It indicates that\nthe ordinary transition is universal over a wide range in\ntheJs/Jbspace. Deviation occurs for Js/Jb>1.0 and\nmanifests itself as the effect of the crossover to the spe-\ncial transition. This is in agreement with the observa-\ntion in Ref. [8]. From our analysis, βord\n1= 0.795(6) is a\ngood estimate for the ordinarytransition. In Table II, we\nhave compile all the existing results which were obtained\nwith simulations and analytical calculations in equilib-\nrium, and our measurements from the non-equilibrium\ndynamic relaxation. A reasonable agreement in βord\n1can\nbe observed. Part of the statistical error in our dynamic\nmeasurements is from the input of the bulk exponents νs\nandzs.\nWithm1(t) at hand, one may proceed to investigate\nthe time-dependent susceptibility, χ11(t). In the case of\nthe ordinary transition of the 3 dIsing model, however,\nγ11is negative. Therefore the χ11is suppressed during\nthe time relaxation according to Eq. (4) and fluctuating\naround 0 if nonequilirium preparation is an ordered ini-\ntial state χ11(0) = 0. The power-law behavior in Eq. (4)\ncould not be observed. Nevertheless, at the special tran-\nsition, where γ11is positive, the situation is different.\nThe power-law behavior of the surface susceptibility and\ncumulant shows up.\nIn Fig. 2 and 3, the surface magnetization, surface\nsusceptibility and appropriate cumulant are displayed at\nthe special transition Js/Jb=rsp. A power-law behav-\nior is observed for all three observables. From the slope\nof the curve of the surface magnetization, we measure\nβsp\n1/νszs= 0.171(2), and then obtain βsp\n1= 0.220(3)\nwithνsandzsin Table I as input. From the curve of the\nsurface susceptibility, we measure γsp\n11/νszs= 0.640(3),\nand then calculate γsp\n11= 0.823(4). From the scaling law\nγ11/νs=d−1−2β1/νs, one derives βsp\n1= 0.218(2),\nwhich is in good agreement with βsp\n1= 0.220(3) esti-\nmated from the surface magnetization. The scaling be--1.2 -0.8 -0.4 0\ntφ/νszs(Js/Jb-rsp)0.51\nJs/Jb=1.30\nJs/Jb=1.35\nJs/Jb=1.37\nJs/Jb=1.40\nJs/Jb=1.43\nJs/Jb=1.45\nJs/Jb=1.47\nJs/Jb=1.49m1(t)tβ/νszsPerfect surface\nFIG. 4: The scaling plot of m1(t) according to Eq. (9) around\nthespecial transition Js/Jb=rsp= 1.5004. The timewindow\nin this plot is within [10 ,1000]. The lattice size is L= 80 and\nT=Tc.\nhaviors in Eqs. (2) and (4) indeed hold.\nThe remarkable feature of the cumulant on the surface\nis that its scaling behavior in Eq. (5) does not involve\nthe exponent β1of the surface magnetization. From the\ncurve in Fig. 3, we obtain ( d−1)/zs= 0.996(11), then\ncalculatethe bulkdynamiccriticalexponent zs= 2.01(2).\nThis value of zsis very close to z3d= 2.04(1) measured\nin numerical simulations in the bulk in Table I, and it\nconfirms that the dynamic exponent on the surface is\nthe same as that in the bulk.\nIn order to describe the dynamic behavior of the sur-\nface magnetization around rsp, we need to introduce a\ncrossoverscalingrelation. Tounderstandthescalingrela-\ntioninnon-equilibriumstates,wefirstrecallthecrossover\nscaling relation in equilibrium. In equilibrium, m1(τ)\nnear the special transition is described by a crossover\nscaling relation\nm1(τ)τ−βsp\n1=Meq(τ−φ(Js/Jb−rsp)),(8)\nwhereτ= 1−T/Tcis the reduced temperature, and φis\nthe crossover exponent. From the crossover scaling rela-\ntion ofm1(τ) , one can determine the special transition\npointrspaswellas βsp\n1andφ[8]. Nevertheless, uptonow\nit has not been studied whether there also exists a cor-\nresponding crossover scaling relation in non-equilibrium\nstates. Here we will verify that such a dynamic crossover\nscaling form indeed exists. For simplicity, we consider\nthe case when Js/Jbapproaches the special transition\nfromr−\nspand the system is at the bulk critical tempera-\nture. Now the non-equilibrium spatial correlation length\nξ(t)∼t1/ztakes the place of the equilibrium spatial cor-\nrelation length τ−ν. By substituting t−1/νszsforτinto\nEq. (8), we obtain\nm1(t)tβsp\n1/νszs=Mneq(tφ/νszs(Js/Jb−rsp)).(9)\nWe have performed non-equilibrium simulations at\nJs/Jb= 1.30, 1.35, 1.37, 1.40, 1.43, 1.45, 1.47, 1.49, and\nmade a scaling plot according to Eq. (9). This is demon-\nstrated in Fig. 4. All curves of different Js/Jbcollapse\ninto a single master curve, and it indicates that Eq. (9)6\n200 1000t0.550.75\n0.65m1(t)\nT = 4.950\n4.9604.955Surface transition\nat perfect surface\nJs/Jb = 2.0\nFIG. 5: Determination of the surface transition temperatur e\nTsforJs/Jb= 2.0. The dashed line is a power-law fit to the\ncurve of T= 4.955. The lattice size is L= 80.\ndoesdescribe the crossover behavior during the dynamic\nrelaxation. ThescalingplotinFig.4yieldstheexponents\nφ= 0.52 andβsp\n1= 0.220, as well as the special transi-\ntion point rsp= 1.50. The crossover exponent φis very\nclose to the mean-field value 0 .5 [8], and βsp\n1andrspare\nin agreement with the existing results from simulations\nin equilibrium in Table II and in Ref. [31]. Although the\nprecision of rspand critical exponents obtained here are\nnot very high, it is still theoretically interesting. The dy-\nnamiccrossoverscalingforminEq.(9) shouldbe general,\nand hold in various statistical systems.\nTo carry out the simulation at the surface transition,\nwe fixJs/Jbat 2.0, well above rsp. At the surface tran-\nsition, where the critical fluctuation is essentially two\ndimensional, νsandzsin Eq. (2) become ν2dandz2d.\nAround the transition temperature, the surface mag-\nnetization obeys a dynamic scaling form < m1(t)>∼\nt−β1/νszsF(t1/νszsτ) [16]. To determine the surface tran-\nsition temperature Ts, one may search for a best-fitting\npower-law curve to the surface magnetization. Then the\ncorresponding temperature is identified as the transition\ntemperature Ts. We perform the simulations with three\ntemperatures around the transition temperature Ts, and\nmeasure the surface magnetization. The results are dis-\nplayed in Fig. 5. Interpolating the surface magnetization\nto other temperatures around these three temperatures,\n200 1000 2000t10-310-310-210-210-110-1\nχ11(t) U11(t)U11\nχ11Surface transition\nat perfect surface\nJs/Jb = 2.0    T = 4.955\nFIG. 6: Dynamic relaxation of the surface susceptibility an d\ncumulant at the surface transition Js/Jb= 2.0 plotted on\ndouble-log scale. The lattice size is L= 80 and Ts= 4.955.\nThe dashed lines are power-law fits to the curves.one finds the best power-lawbehaviorofthe surfacemag-\nnetization at Ts= 4.955. The corresponding slope of the\ncurve gives β1/zs= 0.0570(10) at Ts= 4.955, and it is\nin agreement with the value in the 2 dIsing model [16].\nTherefore we take Ts= 4.955 as the surface transition\ntemperature, which is consistent with Ts= 4.9575(75)\nobtained with Monte Carlo simulations in equilibrium\n[43].\nThetime-dependentcumulant Uandsusceptibility χ11\nat the surface transition are measured, and displayed\nin Fig. 6. The slope of cumulant is 0 .916(15), in a\ngood agreement with 2 /z2d= 0.926(9) of the 2 dIsing\nmodel [16]. Consistence is also observed for the suscep-\ntibility where the slope is 0 .824(10), in comparison with\nγ2d/z2d= 0.810(8) in the 2 dIsing model. We thus con-\nfirm that the surface transition belongs to the universal-\nity class of the 2 dIsing model. Meanwhile, Ts= 4.955 is\na good estimate of the surface transition temperature.\nIV. SHORT-TIME DYNAMICS ON A\nNON-PERFECT SURFACE\nIn this section we investigate the nonequilibrium criti-\ncal dynamics on a non-perfect surface, i.e. Jl/negationslash=Js. The\nstaticanddynamicpropertiesofanon-perfectsurfaceare\nimportantand interesting, becauserealsurfacesareoften\nrough, due to the impurity or limitation of experimental\nconditions [5]. Furthermore, the advance in nano-science\nallows experimentalists to create other structures on top\nof films artificially. We study the line defect on a surface,\nand the procedure can be generalized to other extended\ndefects.\nWe first consider the dynamic behavior of m2at the\nordinary transition. For convenience, we fix Js/Jb= 1.0.\nThe profiles of m2(t) withJl= 0.5Js,Jl= 1.0Jsand\nJl= 1.5Jsare depicted in Fig. 7. All lines look paral-\nlel to each other, and it indicates that they may belong\nto a same universality class. By fitting these curves to\nthe power law in Eq. (2), we estimate βord\n2= 0.792(18),\n0.786(6) and 0 .797(33) with Jl= 1.5Js,Jl= 1.0Jsand\n10 100 10000.010.11\ntm2(t)\nJl/Js = 1.5Ordinary transition\n1.0\n0.5at non-perfect surface\nJs/Jb = 1.0\nFIG. 7: Dynamic relaxation of the line magnetization at the\nordinary transition on a non-perfect surface with various Jl\nis plotted on a double-log scale. The slope of the curves is\nindependent of Jl.7\n10 100 10000.21\n10 100 1000 100000.31\ntm2(t)\nJl/Js = 0.41.6\n1.2\n1.0\n0.8Jl/Js=1.6 m2(t)\ntSpecial transition\nat non-perfect surface\nJs/Jb = rsp = 1.5004\nFIG. 8: Dynamic relaxation of the line magnetization at the\nspecial transition Js/Jb=rsp= 1.5004 on a non-perfect sur-\nface with various Jlis plotted on a double-log scale. The open\ncircles are a fit to Eq. (10) with a correction to scaling. The\ninset displays the line magnetization at Jl= 1.6Jsbut with\na longer simulation time. The lattice size is L= 128 and\nT=Tc. The slope of the curves is dependent on Jleven after\ntaking the correction to scaling into account.\nJl= 0.5Jsrespectively. These values are consistent with\neach other and with βord\n1on the perfect surface reported\nin the previous section. It confirms that the defect in\nthe ordinary transition is irrelevant, in term of the renor-\nmalization group argument. This conclusion echoes that\nin Ref. [32], where the impact of random bonds on\nthe surface is investigated in equilibrium . According to\nthe generalized Harris criterion [33], defects with random\nbonds or diluted bonds on a surface are irrelevant. The\nshort-time dynamic approach shows its merits in identi-\nfying the universal behavior of the surface magnetization\n[32, 40, 43, 44]. Here we note that the line magneti-\nzation is one-dimensional, and therefore somewhat more\nfluctuating than the surface magnetization.\nNow we turn to the special transition. We perform\nsimulations with various Jlat the special transition, and\nthe line magnetization is presented in Fig. 8. From the\nslopesofthecurves,onemeasurestheexponent βsp\n2/νszs,\nand then calculates βsp\n2= 0.260(4), 0 .230(3), 0 .219(5),\n0.204(6) and 0 .162(3) for Jl= 0.4Js,Jl= 0.8Js,Jl=\n1.0Js,Jl= 1.2JsandJl= 1.6Jsrespectively, with νsand\n100 10000.60.9\ntm2(t)\nJl/Js = 0.51.5\n1.02.0\nSurface transition\nat non-perfect surfaceJs/Jb = 2.0    T = 4.955\nFIG. 9: Dynamic relaxation of the line magnetization with\nvariousJlat the surface transition is plotted on a double-log\nscale. The lattice size is L= 80 and T=Ts. The slope of the\ncurves is dependent on Jl, and given in Table III.100 1000t10-310-210-1\nJl/Js = 0.5\n1.0\n1.5\n2.0χ22(t)\nSurface transition\nat non-perfect surfaceJs/Jb = 2.0    T = 4.955\nFIG. 10: Dynamic relaxation of the line susceptibility with\nvariousJlat the surface transition plotted on a double-log\nscale. The dashed lines are power-law fits. The slope of the\ncurves is dependent on Jl, and given in Table III.\nzstaken as input from Table I. Obviously βsp\n2changes\ncontinuously with Jl.\nSince there exists certain deviation from a power law\nin shorter times for the curves with a larger ratio Jl/Js\nin Fig. 8, one may wonder whether the small variation\ninβsp\n2may stem from the correction to scaling induced\nby the defect line. Therefore, a careful analysis of the\ncorrection to scaling is necessary in this case. Assuming\na power-law correction to scaling, m2(t) should evolve\naccording to\nm2(t) =at−βsp\n2/νszs(1−bt−c). (10)\nAs shown in Fig. 8, such an ansatz fits the numeri-\ncal data very well, and yields βsp\n2= 0.258(1), 0 .235(7),\n0.228(3),0 .214(6)and0 .171(3)for Jl= 0.4Js,Jl= 0.8Js,\nJl= 1.0Js,Jl= 1.2JsandJl= 1.6Jsrespectively. For\nJl= 1.6Js, we extend our simulations up to a maximum\ntimet= 10000MCS to gain more confidence on our re-\nsults. Still βsp\n2variescontinuouslywith Jl, andthestrong\nuniversality is violated. This is different from the case on\na random surface, where the generalized Harris criterion\nstates that the enhancement of the short-range random-\nness on the surface is irrelevant at the surface transition\nin the 3dIsing model [33]. Our result is, however, not\nsurprising, for the defect line is not a short-range ran-\ndomness [5], but an extended one. As the short-range\nrandom surface is close to being relevant [5], it is not\nsurprising that the defect line modifies the surface uni-\nversality class. This can also be understood. The reduc-\ntion of the coupling in the defect line is somewhat like\nturning the local surface from the special transition to\nthe ordinary one, and therefore gives rise to a large value\nof the critical exponent βsp\n2.\nTo investigate the impact of the line defect at the sur-\nface transition, we fix Js/Jb= 2.0. We measure the\ntime evolution of the line magnetization at its transition\ntemperature Ts= 4.955 with Jl= 0.5Js,Jl= 1.0Js,\nJl= 1.5JsandJl= 2.0Js. In Fig. 9, one observes\nthat after a microscopictime tmic∼100MCS,the power-\nlaw behavior emerges. However, the exponent β2isJl-\ndependent, and the strong universality is violated. This8\nTABLE III: Comparison between the numerical simulations of surface transition with a non-perfect surface and the theor y of\nthe two-dimensional Ising model with a defect line. νs=ν2d= 1 and zs=z2d= 2.16(2) have been taken as input [16].\nline magnetization susceptibility cumulant\nexponent β2/zs (1−2β2)/zs 1/zs\nSimulation Theory Simulation Theory Simulation Theory\nJl= 0.5Js0.0923(36) 0 .0936(9) 0.282(4) 0 .276(3) 0.462(2) 0 .463(4)\nJl= 1.0Js0.0570(10) 0 .0579(5) 0.356(5) 0 .347(3) 0.475(2) 0 .463(4)\nJl= 1.5Js0.0301(24) 0 .0307(3) 0.405(4) 0 .402(4) 0.468(2) 0 .463(4)\nJl= 2.0Js0.0149(12) 0 .0145(1) 0.428(9) 0 .434(4) 0.459(9) 0 .463(4)\nis similar to the case in Ref. [43, 44], where a non-\nuniversal behavior of the edge and corner magnetization\nhas been found at the surface transition.\nSince the surface transition is essentially two-\ndimensional, one may relate this non-perfect surface to\nthe 2dIsing model with a defect line without the pres-\nence of bulk . The violation of the strong universality of\nthe 2dIsing model with a line or a ladder defect is rig-\norously proved by Bariev [45]. For the line defect, exact\ncalculations show that\nβ2=2\nπ2arctan2(κl), (11)\nwith\nκl= exp(−2(Jl−J)/kBTc). (12)\nThecriticalexponent β2reducesmonotonically,whenthe\ndefect coupling Jlisenhanced. We measurethe exponent\nβ2and compare it with the exact values obtained from\nEqs. (11) and (12). The results are summarized in Table\nIII. One finds a goodagreementbetweensimulations and\nexact results. A similar behavior of the edge magnetiza-\ntion, which can be viewed as a line defect at the surface\ntransition, is also observed in Ref. [43]. Our results sup-\nport that at the surface transition, the critical exponent\nβ2will change in the presence of a small perturbation.\nFinally, the susceptibility χ22(t) and cumulant U22(t)\nof the line magnetization, which are similarly defined as\n100 1000t10-310-210-1100\nU22(t)Jl/Js = 0.5\n1.0\n1.5\n2.0\nSurface transition\nat non-perfect surfaceJs/Jb = 2.0    T = 4.955\nFIG. 11: Dynamic relaxation of the cumulant with various Jl\nat the surface transition plotted on a double-log scale. The\ndashed lines are power-law fits. The slope of the curves is\nindependent of Jl, and given in Table III.those of the surface magnetization, are also measured.\nThe results are plotted in Fig. 10 and 11. Simple scal-\ning analysis shows that χ22(t)∼t(d−2−2β2/νs)/zsand\nU22(t)∼t(d−2)/zs. The estimated exponents are also\ncompiled in Table III, and a good consistency with the\ntheory can be spotted.\nV. CONCLUSION\nWithMonteCarlosimulations,wehavestudiedthedy-\nnamic relaxation on a perfect and non-perfect surface in\nthe 3dIsing model, starting from an ordered initial state.\nOn the perfect surface, the dynamic behavior of the sur-\nface magnetization, susceptibility and appropriatecumu-\nlant is carefully analyzed at the ordinary, special and\nsurface transition. The universal dynamic scaling behav-\nior is revealed, and the static exponent β1of the surface\nmagnetization, the static exponent γ11of the surface sus-\nceptibility and the dynamic exponent zsare estimated.\nAll the results for β1are compiled in Table II. Since the\nexponents νsandzscan be identified as those at bulk,\nit is convenient to study different phase transitions from\nthe non-equilibrium dynamic relaxation. Especially, the\ndynamic crossover scaling form in Eq. (9) is interesting.\nBecause of the existence of new scaling variable Js/Jb,\nthe nonequilibrium relaxation of magnetization at the\ncritical temperature may not obey a power law, which\nis quite different from the general systems investigated\nso far where a power law behavior was always expected.\nThis unusual nonequilibrium behavior is a consequence\nof the presence of geometric surface.\nOn the non-perfect surface, i.e., with a defect line in\nthe surface, the universality class of the ordinary tran-\nsition remains the same as that at the perfect surface.\nOn the other hand, for the special and surface transi-\ntions, the critical exponent β2of the line magnetization\nvaries with the coupling Jlstrength of the defect line.\nThe susceptibility and appropriate cumulant of the line\nmagnetization also exhibit the dynamic scaling behavior\nand yield the static exponent γ22and the dynamic expo-\nnentzs. The short-time dynamic approach is efficient in\nunderstanding the surface critical phenomena.\nAcknowledgements: This work was supported in\npart by NNSF (China) under Grant No. 10325520. The9\nauthors would like to thank M. Pleimling for helpful dis-\ncussions. One of the authors (SZL) would like to thank\nL. Y. Wang for critical reading this manuscript. Thecomputations are partially carried out in Shanghai Su-\npercomputer Center.\n[1] M.A. Torija, A.P. Li, X.C. Guan, E.W. Plummer, and J.\nShen, Phys. Rev. Lett. 95, 257203 (2005).\n[2] K. Binder, in Phase Transition and Critical Phenomena\n(Academic Press, London, 1987), vol. 8, p.1, and refer-\nences therein.\n[3] H.W. Diehl, in Phase Transition and Critical Phenom-\nena(Academic Press, London, 1987), vol. 10, p.76, and\nreferences therein.\n[4] H.W. Diehl, Int. J. Mod. Phys. B11, 3503 (1997).\n[5] M. Pleimling, J. Phys. A37, R79 (2004).\n[6] K. Binder and P.C. Hohenberg, Phys. Rev. B6, 3461\n(1972).\n[7] K. Binder and D.P. 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Khuntia1, 5,†\n1Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India\n2Department of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea\n3Center for Artificial Low Dimensional Electronic Systems,\nInstitute for Basic Science, Pohang 37673, Republic of Korea\n4Dresden High Magnetic Field Laboratory (HLD-EMFL),\nHelmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany\n5Quantum Centre of Excellence for Diamond and Emergent Materials,\nIndian Institute of Technology Madras, Chennai 600036, India.\n(Dated: January 25, 2024)\nCompeting magnetic interactions, frustration-driven quantum fluctuations, and spin-correlation\noffer an ideal route for the experimental realization of emergent quantum phenomena and exotic\nquasi-particle excitations in three-dimensional frustrated magnets. In this context, trillium lat-\ntice, wherein magnetic ions decorate a three-dimensional chiral network of corner-shared equilateral\ntriangular motifs, provides a viable ground. Herein, we present the crystal structure, magnetic\nsusceptibility, specific heat, electron spin-resonance (ESR), muon spin-relaxation ( µSR) results on\nthe polycrystalline samples of K 2CrTi(PO 4)3wherein the Cr3+ions form a perfect trillium lat-\ntice without any detectable anti-site disorder. The Curie-Weiss fit of the magnetic susceptibility\ndata above 100 K yields a Curie-Weiss temperature θCW=−23 K, which indicates the presence\nof dominant antiferromagnetic interactions between S= 3/2 moments of Cr3+ions. The specific\nheat measurements reveal the occurrence of two consecutive phase transitions, at temperatures TL\n= 4.3 K and TH= 8 K, corresponding to two different magnetic phases. Additionally, it unveils\nthe existence of short-range spin correlations above the ordering temperature TH. The power-law\nbehavior of ESR linewidth suggests the persistence of short-range spin correlations over a relatively\nwide critical region ( T−TH)/TH>0.25 in agreement with the specific heat results. The µSR results\nprovide concrete evidence of two different phases corresponding to two transitions, coupled with the\ncritical slowing down of spin fluctuations above TLand persistent spin dynamics below TL, consis-\ntent with the thermodynamic results. Moreover, the µSR results reveal the coexistence of static and\ndynamic local magnetic fields below TL, signifying the presence of complex magnetic phases owing\nto the entwining of spin correlations and competing magnetic interactions in this three-dimensional\nfrustrated magnet.\nI. INTRODUCTION\nFrustrated quantum materials, where competing inter-\nactions between localized spin moments and frustration-\ninduced strong quantum fluctuations prevent classical\nN´ eel order, are highlighted as promising contenders for\nthe discovery of emergent physical phenomena such as\nquantum spin liquid (QSL), and spin ice with exotic ex-\ncitations that goes beyond conventional symmetry break-\ning paradigms in condensed matter [1–4].\nA QSL state is a highly entangled state, in which\nfrustration-induced strong quantum fluctuations defy\nlong-range magnetic order even at absolute zero temper-\natures, despite strong exchange coupling between mag-\nnetic moments in the host spin-lattice [5–9]. Instead,\ntwo spins in the corresponding lattice form resonating\nspin-singlet pairs or long-range entangled states with ex-\notic fractionalized excitations such as spinons coupled\nto emergent gauge fields [10]. Experimental realization\n∗choisky99@skku.edu\n†pkhuntia@iitm.ac.inof fractional quantum numbers, their identification and\ninteraction are of profound importance in elucidating\nthe underlying mechanism of some of the groundbreak-\ning phenomena such as high-temperature superconduc-\ntivity, fractional Hall effect, topological spin-textures,\nand monopoles in quantum condensed matter [11–18]\nthat could have far-reaching ramifications in both fun-\ndamental physics and innovative quantum technologies\n[16]. The realization of QSL state is well established in\none-dimensional magnets, however, its identification in\nhigher dimensional spin system remains a long-standing\nchallenge in modern condensed matter physics.\nIn the quest of this long-sought goal, significant effort has\nbeen devoted to the exploration of two-dimensional (2D)\nfrustrated lattices [4], which has led to the identification\nof a few QSL candidates in 2D systems [19, 20]. Mean-\nwhile, three-dimensional (3D) frustrated spin-lattices\nsuch as hyperkagome and pyrochlore lattice are found\nto host QSL as well. Despite the fact that quantum\nfluctuations are less pronounced in 3D lattices, certain\nfrustrated 3D lattices, including transition metal-based\nhyperkagome lattices PbCuTe 2O6[21, 22] and Na 4Ir3O8\n[23, 24] along with pyrochlore lattice NaCaNi 2F7[25–27]\nas well as rare-earth-based pyrochlore lattice Ce 2Zr2O7arXiv:2401.13445v1  [cond-mat.str-el]  24 Jan 20242\n[28], shows signatures of QSL state driven by frustration-\ninduced strong quantum fluctuations [29]. However, the\nexperimental realization of QSL in 3D spin lattice re-\nmains scarce, as most of the candidate materials show\nspin-freezing or magnetic ordering owing to perturbing\nterms in the spin Hamiltonian, lattice imperfections, and\nexchange anisotropy. In this respect, it is pertinent to\ninvestigate promising frustrated magnets on 3D spin lat-\ntices wherein competition between emergent degrees of\nfreedom offers an alternate route to stabilize exotic quan-\ntum and topological states [4, 30–33]\nTo date, hexagonal-based layered frustrated magnets\nhave been studied rigorously for their 2D QSL and certain\ncubic lattice systems for their 3D counterparts [34]. Re-\ncently, there is growing interest in non-centrosymmetric\nfrustrated quantum materials which crystallize in the cu-\nbic space group P213 [35–37]. The lack of inversion sym-\nmetry in such non-centrosymmetric materials gives rise\nto interesting physical phenomena owing to anisotropic\nDzyaloshinskii-Moriya (DM) interactions [35, 38–42]. For\nexample, non-centrosymmetric cubic quantum materials\nare known to host a wide range of quantum phenomena\nincluding magnetic skyrmions in insulator Cu 2OSe 2O3\n[43], and in itinerant magnets MnSi [44], and FeGe [45],\nand topological Hall effect in MnGe [46]. Apart from\nquantum phenomena induced by DM interactions, the\nspin frustration induced quantum phenomena in such 3D\nnon-centrosymmetric chiral magnets have garnered sig-\nnificant attention in the quest to achieve distinct quan-\ntum phenomena. It is observed that in such quantum\nmaterials that crystallizes in the cubic structure ( P213)\nthe magnetic ions form a trillium lattice i.e., a 3D chi-\nral network of corner-shared equilateral triangular mo-\ntifs which provides the origin of spin frustration and may\nhost a myriads of frustration-induced physical phenom-\nena such as QSL. It is a well known that a geometrically\nfrustrated lattice has the potential to harbor frustration-\ninduced new quantum phenomena, as observed in inter-\nmetallic compounds. Notable examples include the pres-\nence of a dynamic state above the transition temperature\nin EuPt(Si/Ge) [47], a spin-ice-like state in CeIrSi [48],\nand the occurrence of pressure-induced quantum phase\ntransitions in MnSi [49].\nNevertheless, oxide-based materials having a single tril-\nlium lattice structure are extremely rare. Recently, the\ntrillium lattice, composed of corner-sharing equilateral\ntriangles with six nearest neighbors, has emerged as\nan alternative route to further explore the influence of\nquantum fluctuations in the 3D QSL state and to re-\nalize unique quantum phenomena [37, 50]. Remark-\nably, a recent study on the langbeinite family member\nK2Ni2(SO 4)3, which crystallizes in a cubic ( P213) crys-\ntal structure, reveals that Ni2+(S= 1) ions form a cou-\npled trillium lattice [51]. This material demonstrates\nseveral magnetic sublattices with competing magnetic\ninteractions and exhibits a 3D QSL state induced by\nan applied magnetic field [51]. More recently, contin-\nuum spin excitations driven by strong quantum fluctu-ations are revealed by inelastic neutron scattering mea-\nsurement on single crystals of K 2Ni2(SO 4)3[52]. Fur-\nthermore, despite possessing a high spin moment, the\nsignature of a spin-liquid state has been observed in the\ntrillium compounds KSrFe 2(PO 4)3(Fe3+;S= 5/2) [53]\nand Na[Mn(HCOO) 3] (Mn2+;S= 5/2) [54], that invokes\nfurther investigation of the ground state and associated\nquasi-particle excitations in analogous trillium lattice an-\ntiferromagnets. In addition, the spin correlations, inter-\nplay between competing interactions and the topology\nof the electronic band structure in non-centrosymmetric\nmagnets stabilizing in 3D spin lattice make them an ex-\nciting class of materials with potential technological ap-\nplications [55, 56].\nHerein, we focus on a new phosphate langbeinite\nK2CrTi(PO 4)3(henceforth KCTPO), wherein Cr3+ions\nwithS= 3/2 form a trillium lattice without any de-\ntectable anti-site disorder between constituent atoms.\nMagnetization data reveals the presence of dominant an-\ntiferromagnetic interactions between S= 3/2 moments\nof Cr3+ions. Specific heat measurements demonstrate\ntwo magnetic transitions, namely at TL= 4.3 K and TH\n= 8 K, as well as the development of short-range spin\ncorrelations above TH, which is supported by critical-like\nbehavior of the ESR line width. Furthermore, µSR data\nuncovers an intriguing evolution of static and dynamic\nspin correlations, notably below TL, and a critical slow-\ning down of spin dynamics above TL. Our experimental\nfindings reveal a complex magnetic landscape with coex-\nisting states, successive phase transitions, and dynamic\nbehavior of Cr3+moments decorating a frustrated 3D\ntrillium spin-lattice over a wide temperature regime.\nII. EXPERIMENTAL DETAILS\nPolycrystalline samples of KCTPO were prepared by a\nconventional solid-state reaction method. The appropri-\nate stoichiometric amounts of K 2CO3(Alfa Aesar, 99.997\n%), Cr 2O3(Alfa Aesar, 99.97 %), TiO 2(Alfa Aesar,\n99.995 %) and (NH 4)2HPO 4(Alfa Aesar, 98 %) were\nmixed. Prior to use, the reagent K 2CO3was preheated\nin air at 100◦C to eliminate moisture. The stoichiomet-\nric mixture was pelletized, and the pellet was sintered at\n300◦C for 4 hrs. This sintering procedure was repeated at\nseveral intermediate temperatures before annealing the\nsample at 800◦C for 48 hrs to achieve a single-phase\ncomposition. Powder x-ray diffraction (XRD) data were\ncollected using a smartLAB Rigaku x-ray diffractometer\nwith Cu K αradiation (λ= 1.54 ˚A) at room tempera-\nture.\nMagnetization measurements were performed using the\nVSM option of Physical Properties Measurement System\n(PPMS, Quantum Design) in the temperature range of\n2 K≤T≤300 K and in magnetic fields up to 7 T.\nSpecific heat measurements were performed using PPMS\nby thermal relaxation method in the temperature range\nof 1.9 K≤T≤250 K and in magnetic fields up to 73\n10203040506070\nIntensity (arb. u)2\n/s113/s32(degrees) Iobs. \nIcal. \nBragg peaks       I\nobs.-Ical.\nK+C\nr3+T\ni4+P\n5+O\n2-(a)( b)(c)O\n1O\n4\nFIG. 1. (a) Rietveld refinement pattern of the room temperature powder x-ray diffraction data of K 2CrTi(PO 4)3. The orange\ncircle, black line, olive vertical bars, and blue line show the experimentally observed points, the result of Rietveld fitting,\nexpected Bragg reflection positions, and the difference between observed and calculated intensities, respectively. (b) Schematic\ndepicting the one unit cell of K 2CrTi(PO 4)3. The Cr3+ions form distorted CrO 6octahedra (blue), and the P5+ions form\nPO4tetrahedra (pink). These are connected through a shared oxygen ion, resulting in the formation of a Cr-O-P-O-Cr super-\nexchange bridge with nearest CrO 6octahedra. (c) The nearest-neighbor Cr3+ions (solid blue line; 6.10 ˚A) arrange themselves\nin a trillium lattice, featuring six nearest neighbors. Trillium lattice is composed of three motifs of equilateral triangle of Cr3+\nions with possible nearest-neighbor superexchange routes. The second (8.32 ˚A), third (9.79 ˚A), and fourth (10.06 ˚A) nearest\nneighbors are shown by the dashed orange, red, and olive lines, respectively. Other atoms are omitted to ensure a clearer view\nof the trillium lattice and potential nearest-neighbor exchange paths.\n.\nT. High-field magnetization measurement was conducted\nat Dresden High Magnetic Field Laboratory, sweeping a\nmagnetic field up to 30 T at 4 K using a nondestructive\npulsed magnet. The obtained data were scaled to the\nisothermal magnetization taken with PPMS at 4 K.\nX-band (ν= 9.5 GHz) electron spin resonance (ESR)\nmeasurements were performed using a Bruker EMXplus-\n9.5/12/P/L spectrometer with a continuous He flow cryo-\nstat in the temperature range of 4 K ≤T≤300 K. The\nµ+SR experiments in zero field and in weak transverse\nmagnetic field (32 G) were performed at the surface muon\nbeamline M20 at TRIUMF in the temperature range\n1.93 K≤T≤80 K. Powder samples (approximately 0.5\ng) were placed into a thin envelope composed of Mylar\ntape coated with aluminum ( ∼50µm thick), which was\nmounted on a Cu fork sample stick. A standard4He flow\ncryostat was employed to achieve the base temperature\nof 1.93 K for TRIUMF/M20. The obtained µSR data\nwas analyzed using the musrfit software package [57].\nIII. RESULTS\nA. Rietveld refinement and crystal structure\nIn order to ensure phase purity and determine atomic\nparameters of the polycrystalline samples of KCTPO, the\nRietveld refinement of powder XRD data was performed\nusing GSAS software [58]. For the Rietveld refinement,\nthe initial atomic parameters were obtained from ref-\nerences [59–61]. The Rietveld refinement reveals that\nKCTPO crystallizes in a cubic space group ( P213) and\nthe resulting refinement pattern is shown in Fig. 1 (a).No secondary phase was found, confirming the success-\nful synthesis of KCTPO in its single phase. The ob-\nserved sharp and well-defined XRD peaks imply that\nhigh-quality polycrystalline samples have been used in\nthis study. The obtained lattice parameters as well as\ngoodness factors from the Rietveld refinement of powder\nXRD are tabulated in table I. Our analysis shows the\nabsence of any detectable anti-site disorders between the\natoms that constitute KCTPO.\nFigure 1 (b) depicts the one unit cell of KCTPO, where\nCr3+ions form distorted CrO 6octahedra with nearest-\nneighbor O2−that are separated from adjacent CrO 6oc-\ntahedra by PO 4tetrahedra. The bonding between the\nCrO 6octahedra and TiO 4tetrahedra through a com-\nmon oxygen ion is expected to construct a 3D Cr-O-P-O-\nCr superexchange pathway between two nearest-neighbor\nCr3+ions (see table II). Most interestingly, the nearest-\nneighbor Cr3+ions (6.10 ˚A) form a trillium lattice, i.e.,\na 3D network of corner-shared equilateral triangular lat-\ntice with coordination number six (Fig. 1 (c)), potentially\nserving as a platform for hosting a frustrated spin-lattice\nin 3D.\nFrom a structural point of view, KCTPO belongs to the\nlangbeinite family K 2M2(SO 4)3(M= Mg, Co, Ni,..),\nwhich crystallizes in the cubic crystal structure (space\nstructureP213). Two crystallographic 4 asites occupy\ndivalent magnetic ions that form a double trillium lat-\ntice in the sulfate langbeinites without any anti-site dis-\norder [51]. Among various phosphate langbeinites, the\nrecently reported phosphate langbeinite KSrFe 2(PO 4)3\nshows a similar configuration, featuring a double trillium\nlattice with two magnetic sites of Fe3+ions and K/Sr\nanti-site disorder [53]. Unlike the aforementioned dou-4\nTABLE I. Rietveld refinement of x-ray diffraction data at 300\nK yielded structural parameters of K 2CrTi(PO 4)3. (Space\ngroup:P213,a=b=c= 9.796 ˚A,α=β=γ= 90◦andχ2\n= 2.86, R wp= 5.29 %, R p= 3.56 %, and R exp= 1.84%)\nAtom Wyckoff position x y z Occ.\nCr 4 a 0.666 0.666 0.666 1\nK1 4a 0.960 0.960 0.960 1\nK2 4a 0.186 0.186 0.186 1\nTi 4 a 0.391 0.391 0.391 1\nP 12 b 0.474 0.703 0.368 1\nO1 12b 0.609 0.860 0.737 1\nO2 12b 0.463 0.314 0.232 1\nO3 12b 0.506 0.573 0.313 1\nO4 12b 0.585 0.721 0.457 1\nTABLE II. Bond lengths and angles between atoms that\nresult in distinct antiferromagnetic interactions between Cr3+\nspins.\nBond length ( ˚A) Bond angle (◦)\nCr-O 1= 2.101 ∠Cr-O 1-P = 153.510\nCr-O 4= 2.262 ∠O1-P-O 4= 120.635\nP-O 4= 1.405 ∠P-O 4-Cr = 143.804\nP-O 1= 1.273 ∠Cr-Cr-Cr = 60\nCr-P = 3.494 ∠Cr-P-Cr = 128.355\nO1-O4= 2.32 ∠Cr-O 1-Cr = 154.692\nble trillium lattice-based langbeinites, KCTPO, a phos-\nphate langbeinite, accommodates a single magnetic site\n(4a) hosting Cr3+ions, alongside another 4 asite oc-\ncupied by non-magnetic Ti4+ions. This characteristic\npositions KCTPO as a considerably simpler 3D lattice,\nsuited for the experimental realization of theoretically\nproposed physical phenomena intrinsic to the trillium lat-\ntice [36, 62].\nB. Magnetic susceptibility\nFigure 2 (a) depicts the temperature dependence of\nmagnetic susceptibility χ(T) in several magnetic fields\nup to 7 T. Upon lowering temperature, a sharp increase\nofχ(T) was observed below 10 K, tending to saturate\nin a magnetic field µ0H= 1 T. This behavior indicates\nthe presence of long-range magnetic order in KCTPO\nconsistent with the specific heat result presented in this\nwork. However, this behavior is suppressed significantly\nin magnetic fields µ0H≥3 T. A minor bump around\n50 K that appeared in all magnetic fields is attributed\nto residual oxygen trapped in the polycrystalline sample\nthat was wrapped with teflon during measurement [63].\nThis signal is not an intrinsic characteristic of the inves-\ntigated compound in this study.\nIn order to determine the dominant magnetic interaction\nbetween the S= 3/2 moments of Cr3+ions, the linear\nregion (100 K≤T≤300 K ) of the 1/ χ(T) data ina magnetic field µ0H= 1 T was fitted by the Curie-\nWeiss (CW) law, i.e., χ=χ0+C/(T−θCW). Here,\nχ0is the sum of temperature-independent core diamag-\nnetic susceptibility ( χcore) and Van Vleck paramagnetic\nsusceptibility ( χVV),Cis the Curie constant used to es-\ntimate the effective magnetic moment ( µeff=√\n8C µ B)\nandθCWis the Curie-Weiss temperature associated with\nthe exchange interaction between magnetic moments of\nCr3+ions. The corresponding CW fit (red line in the in-\nset of Fig. 2 (a)) yields χ0=−7.42×10−4cm3/mol,C\n= 1.88 cm3K/mol, and θCW=−23±0.15 K. The esti-\nmated effective magnetic moment µeff= 3.87µB, is close\nto the value g/radicalbig\nS(S+ 1) = 3.87 µBexpected for S= 3/2\nmoments of free Cr3+ions. The negative θCWindicates\nthe presence of dominant antiferromagnetic interactions\nbetween the S= 3/2 spin of Cr3+ions. As depicted in the\ninset of Fig. 2 (a), the Curie-Weiss fit begins to diverge\nbelow 80 K, signaling the onset of antiferromagnetic spin\ncorrelations upon lowering the temperature [64]. Inter-\nestingly, the material shows anomalies at low tempera-\ntures associated with magnetic phase transitions.\nTo further understand the effect of magnetic fields on the\nlong-range magnetic ordered state in KCTPO, the zero-\nfield-cooled (ZFC) and field-cooled (FC) χmeasurements\nwere performed in weak magnetic fields µ0H≥0.005 T.\nFigure 2 (b) reveals the presence of a clear bifurcation be-\ntween ZFC and FC χbelow the temperature ( TL) = 4.3\nK in a magnetic field of µ0H= 0.005 T, which implies the\npresence of a ferromagnetic component below TL. Nev-\nertheless, as the magnetic field strength increases, the\ndeviation between ZFC and FC magnetic susceptibility\ngradually diminishes, almost converging at a magnetic\nfieldµ0H= 0.05 T.\nIn order to shed more insights concerning the nature\nof magnetically ordered phases at different temperature\nregimes, isotherm magnetization measurements were per-\nformed at several temperatures as shown in the inset\nof Fig. 2 (c). Several features can be observed in the\nisotherm magnetization curves such as a weak S-shaped\ncurvature in low-fields at 2 K, indicating the existence of\na ferromagnetic component consistent with the bifurca-\ntion between ZFC and FC susceptibility below TL[65–\n67]. Notably, the magnetization at 2 K does not reach\nsaturation in a high magnetic field (9 T); instead, the lin-\near behavior of the magnetization curves implies the pres-\nence of a canted antiferromagnetic state below TL. A sim-\nilar scenario, attributed to the canted antiferromagnetic\nphase, has also been observed in several frustrated mag-\nnets [68, 69]. As the temperature increases, the S-shaped\ncurvature is gradually suppressed due to the quenching of\nthe ferromagnetic component, finally disappearing. The\nresulting magnetization curve becomes linear at 10 K, in-\ndicating the crossover to an additional antiferromagnetic\nphase. These observations imply that KCTPO exhibits\na canted antiferromagnetic state below TL, above which\nit hosts an additional antiferromagnetic state corrobo-\nrated by the specific heat results described below. To\nfind the saturation magnetic moment in high magnetic5\n11 01 000.00.10.20.30.40\n1 002003000501001502000\n24680122\n3 4 5 1100\n1 02 00123χ (cm3/mol)T\n (K) 1 T  3 T  \n5 T 7 T1 /c (mol/cm3)T\n (K) 1 T \nCW fitM\n (/s109B/f.u)/s109\n0H (T) 2 K   5 K \n10 K 15 K \n25 K(a)( b)( c)χ (cm3/mol)T\n (K) ZFC 50 Oe        FC            \n \nZFC  100 Oe      Fc  \n \nZFC 500 Oe       FCM\n (/s109B/f.u)/s109\n0H (T) 4 K\nFIG. 2. (a) Temperature dependence of magnetic susceptibility, χ(T), of K 2CrTi(PO 4)3in several magnetic fields. The inset\ndepicts the temperature dependence of inverse magnetic susceptibility at µ0H= 1 T. The red line represents the Curie-Weiss\nfits to the high-temperature inverse susceptibility data. (b) Temperature dependence of zero-field cooled (ZFC) and field-cooled\n(FC) magnetic susceptibility in the temperature range 2 K ≤T≤5 K in magnetic fields 0.005 T ≤µ0H≤0.05 T. The dotted\nvertical line is at temperature TL= 4.3 K, below which ZFC and FC bifurcation begins in a magnetic field of µ0H= 0.005 T.\n(c) Magnetization as a function of an external magnetic field up to 30 T at 4 K. The inset shows the isotherm magnetization\nup to 9 T at several temperatures.\n.\nfields, magnetization measurements were performed at 4\nK using a pulsed field magnet up to 30 T.\nFigure 2 (c) represents the high-field magnetization data\nat 4 K, calibrated using the VSM-SQUID data at 4 K.\nIt is worth to note that a weak ferromagnetic compo-\nnent persists in low fields µ0H≤0.012 T and beyond\nthis magnetic-field, the magnetization continues to in-\ncrease linearly until µ0H≤14 T and reaches a satura-\ntion magnetic moment of 3.04 µBin 24 T. We recall that\nthe saturation field is comparable to the CW tempera-\nture discussed above. The obtained saturation magnetic\nmoment of 3.04 µBis close to the expected theoretical\nvalue of 3.0 µBper Cr3+ions (S= 3/2). The absence\nof a fractional “1/3” magnetization plateau indicates ei-\nther the classical behavior of Cr3+spins or the presence\nof perturbation terms such as magnetic anisotropy and\nlonger-range interactions [70, 71].\nC. Specific heat\nSpecific heat is an excellent probe to track magnetic\norder and associated low-lying excitations in frustrated\nmagnets. In order to discern the evidence of magnetic\nlong-range order, temperature-dependent specific heat\nmeasurements were performed in several magnetic fields.\nFigure 3 (a) shows the temperature dependence of spe-\ncific heat in the whole measured temperature range in\nzero-magnetic field. As shown in Fig. 3 (b), below T=\n10 K, the specific heat exhibits two anomalies: one at\nthe temperature TL= 4.3 K and another at the temper-\natureTH= 8 K in zero-magnetic field, suggesting the\noccurrence of successive magnetic phase transitions in\nKCTPO. It is worth to note that the low-temperature\nanomaly occurring at TLin specific heat coincides withthe temperature below which the splitting of ZFC and FC\nχ(Fig. 2 (b)) as well as the appearance of a ferromag-\nnetic component in magnetization are observed (Fig. 2\n(c)). This coincidence suggests that the anomaly at TL\nis most likely related to the transition temperature of the\ncanted antiferromagnetic phase in KCTPO. On the other\nhand, the anomaly at TH= 8 K is attributed to the tran-\nsition temperature of additional antiferromagnetic phase\nof KCTPO. We recall that the two successive magnetic\nphase transitions are also observed in the two coupled\ntrillium lattice K 2Ni2(SO 4)3[51].\nFigure 3 (b) displays the specific heat in several mag-\nnetic fields in the low-temperature regime. The salient\nobservation is that the low-temperature anomaly at TL\nshifts towards higher temperatures upon the application\nof a magnetic field, eventually vanishing at a magnetic\nfield ofµ0H= 3 T. The increase in TLwith increasing\nmagnetic field can be ascribed to either strong quantum\nfluctuations or the presence of competing anisotropic in-\nteractions. While the applied magnetic field increases,\nthe anomaly at THwidens, shifts to slightly lower temper-\natures, and a clear λ-like anomaly is observed in µ0H= 7\nT, suggesting the presence of a typical antiferromagnetic\nsecond-order phase transition at THin KCTPO. More-\nover, in the temperature range 8 K ≤T≤20 K, the\nspecific heat exhibits field dependency, indicating that\nspecific heat is of magnetic origin in this temperature\nrange.\nIn order to estimate the magnetic entropy and under-\nstand the low-temperature magnetic properties relevant\nto this trillium lattice antiferromagnet, one must sub-\ntract the lattice specific heat ( Clatt(T)) from the total\nspecific heat ( Cp(T)). To extract the lattice contribu-\ntions, we used a combination of one Debye and three6\n05 010015020025001002003000\n4 8 1 21 62 002460\n48121620242805101\n1 00246Cp (J/mol.K)T\n (K) 0 T \nLattice(a)0\n48121620240510Cp (J/mol.K)T\n (K)C\nP (J/mol.K)T\n (K) 0 T   1 T \n3 T   5 T \n7 T    LatticeT\nHTL(b)(\nc)Δ\nS (J/mol.K)T\n (K) 0 T \n1 T \n3 T \n5 T \n7 T(d)R ln(2S+1)0  T1 T3\n T5 T7\n T \n∼T1.28 \n∼T1.05 \n∼T3Cmag. (J/mol.K)T\n (K)THT LT *\nFIG. 3. (a)Temperature dependence of specific heat ( Cp) of K 2CrTi(PO 4)3in zero-magnetic field. The solid red line represents\nthe lattice contributions obtained by combining one Debye and three Einstein functions as described in text. The inset shows\na closer view of the low-temperature specific heat. (b) Temperature dependence of Cp(T) in several magnetic fields in the\ntemperature range 2 K ≤T≤20 K. The dashed vertical lines indicate two anomalies: one at temperature TL= 4.30 K and\nanother at temperature TH= 8 K in zero-field. (c) Temperature dependence of Cmag(T) in several magnetic fields in semi-log\nscale. Below TL= 4.3 K, the solid lines tentatively represent ∼Tnpower-law behavior of magnetic specific heat, where nvaries\nfrom 1.3 to 1 in an applied magnetic field of 7 T. The dashed orange line represents the Cmag(T)∼T3behavior typical for\nconventional antiferromagnets. (d) Temperature dependence of entropy change in several magnetic fields with the horizontal\npink line indicating the expected entropy of Rln(4) forS= 3/2 spin of Cr3+ions.\n.\nEinstein terms, i.e.,\nClatt(T) =CD[9kB/parenleftbiggT\nθD/parenrightbigg3/integraldisplayθD/T\n0x4ex\n(ex−1)2dx]\n+3/summationdisplay\ni=1CEi[3R/parenleftbiggθEi\nT/parenrightbigg2exp(θEi\nT)\n(exp(θEi\nT)−1)2], (1)\nwhereθDis the Debye temperature, θEisare the Einstein\ntemperatures of the three modes, RandkBare the molar\ngas constant and Boltzmann constant, respectively. The\ncorresponding lattice fit, yielding θD= 223 K,θE1= 376\nK,θE2= 615 K, and θE3= 1344 K, is represented by the\nsolid red line on top of the experimental data point inFig. 3 (a). In order to minimize the fitting parameters,\nthe coefficients were set at a fixed ratio of CD:CE1:\nCE2:CE3= 1 : 1 : 1.5 : 6, closely matching with the\nratio of the number of heavy atoms (K, Cr, Ti, P) to\nlight atoms (O) in KCTPO.\nAfter subtraction of the lattice contributions, the ob-\ntained temperature dependence of magnetic specific heat\nCmag(T) is shown in Fig. 3 (c). As the temperature\ndecreases in a zero-magnetic field, Cmagincreases and\nexhibits a broad maximum around temperature T∗= 14\nK, followed by a weak kink at TH. The broad maximum is\nattributed to the presence of short-range spin correlations\n[51]. The presence of a moderate frustration parameter f,\nroughly estimated through the ratio f=|θCW|/TH≈3,7\n2003 004 005 0001 002 003 001.9621.9641.9661.9681.9701.9720\n.11 1 030323436 ΔHw(T/TL-1) \n∼T −0.035 \n∼T −0.097 \nΔHw(T/TH-1) \n∼ T −0.032 \n∼T −0.072(\nT -TL)/TLΔHw (mT)(T-TH)/TH(\nc)0.11 1 0  \n \n300 K \nFit(a) \n 100 K \n 25 K \n 10 K \n 8 K \n 7 K \n 6 K \nDerivative of the ESR absorption spectra (arb. units)/s109\n0H (mT) 4 Kg\n-factorT\n (K)(b)\nFIG. 4. (a) Derivative of the ESR absorption spectra of K 2CrTi(PO 4)3at selected temperatures. The solid red lines indicate\nthe fitting to the Lorentzian line shape above 8 K. (b) Temperature dependence of g-factor in semi-log scale. (c) Semi-log plot\nof the ESR linewidth ∆ Hwas a function of the reduced temperature expressed as Trl= (T−TL)/TLon the bottom x-axis and\nTrh= (T−TH)/THon the upper x-axis. The dashed lines are power-law fits.\n.\ncould account for both the short-range order and the sup-\npression of magnetic order. The anomaly at THis rela-\ntively weak compared to the anomaly at TLsimilar to\nthe observed anomalies in K 2Ni2(So4)3[51]. Neverthe-\nless, upon increasing the magnetic field, the canted an-\ntiferromagnetic phase diminishes, leading to an increase\nin the anomaly observed at THinµ0H= 7 T. Most no-\ntably, as shown in Fig. 3 (c), the magnetic specific heat\nfollowsCmag∼Tn(n= 1.28) power-law in zero field\nfor temperatures below TL, which significantly deviates\nfrom theCmag(T)∼T3behavior typical of conventionalantiferromagnets. In the presence of a magnetic field up\nto 7 T, the average value of nwas found to be 1.10.\nIt is worth noting that in future low temperature spe-\ncific heat measurements are required to confirm whether\nthe power-law behavior persists even in the sub-Kelvin\ntemperature range. In contrast to KCTPO, the trillium\nlattice structure with two interpenetrating sublattices of\nK2Ni2(SO 4)3, well below the transition temperature, the\nmagnetic specific heat follows Cmag∼T2behavior [51].\nTo quantify the entropy release associated with phase\ntransitions and spin dynamics, it is crucial to account the8\n02 4 6 8 -0.2-0.10.00.10.21\n1 00.11101001\n1 00.111\n100.11Asymmetryt\n (/s109s) 1.95 K   2.88 K   4.05 K \n4.5         6 K        16 K(a)( b)( c)λL (/s109s-1)T\n (K) wTFλTF (/s109s-1)T\n (K) wTFf\nT\n (K)\nFIG. 5. (a) Time dependence of the muon asymmetry in K 2CrTi(PO 4)3in a weak transverse field ( BTF= 32 G) at a several\ntemperatures. The solid lines are fits obtained using Eq. 2. (b) Temperature dependence of muon spin relaxation rate in\ntransverse applied field on a double logarithmic scale. The inset shows the fraction of the oscillating component as a function of\ntemperature. (c) Temperature dependence of muon spin relation rate due to internal magnetic field in the investigated sample.\nThe dashed vertical lines indicate the position of characteristic transition temperatures at TL= 4.3 K and TH= 8 K.\n.\nmagnetic-specific heat below 2 K. In the absence of no\nmore phase transition below 2 K, we extend the corre-\nsponding power-law to zero temperature. Next, the en-\ntropy was obtained by integrating Cmag/Twith respect\nto temperature, as shown in Fig. 3 (d). The obtained sat-\nuration entropy is found to be 11.16 J/mol ·K at 30 K that\nis close the theoretically expected value 11.52 J/mol ·K\nforS= 3/2 moments. The missing ∼3.12 % entropy\nis ascribed to either the existence of short-range mag-\nnetic correlations well above the transition temperature\nor an overestimation of the lattice contribution given the\nlimitations of the model employed here [72, 73]. Addi-\ntionally, around the transition temperature THof 8 K,\napproximately 76 % of the saturation entropy is released,\nsuggesting that roughly 20 % originates from short-range\nspin correlations that begin forming as high as at ∼3TH.\nHowever, the lack of a few percentage of entropy because\nof short-range order persisting above the transition tem-\nperature is typical for frustrated magnets [64, 72]. This\nobservation is corroborated by the ESR results presented\nin the following subsection.\nD. Electron spin resonance\nTo provide microscopic insights into the temperature\nevolution of spin correlations, we conducted X-band ESR\nmeasurements down to 4 K [74]. The obtained ESR spec-\ntra at several temperatures are shown in Fig. 4 (a), which\nfits well with a derivative of a Lorentzian curve above 8\nK. This indicates that the ESR signal is exchange nar-\nrowed. At temperatures below T= 10 K, the spectra\ncannot be fitted by a single Lorentzian curve, indicat-\ning the development of antiferromagnetic resonance mode\nand supporting the conclusions drawn from thermody-\nnamic results. The temperature dependence of estimated\ng-factors from the fit is shown in Fig. 4 (b). Above T=\n100 K, the obtained gvalue remains relatively constantat approximately g= 1.972, slightly lower than the stan-\ndardg= 2 for a free Cr3+ion. This value is typical\nfor a less-than-half-filled Cr3+ion with a negligible spin-\norbit interaction in an octahedral ligand coordination, as\nobserved in similar compounds [73]. The g-factor starts\nto decrease with temperature below 100 K, which corre-\nsponds to the deviation from the CW fit (see the inset of\nFig. 2 (a)). This is attributed to the persistent magnetic\ncorrelations up to several |θCW|, often observed in frus-\ntrated magnets [64].\nThe ESR linewidth is directly proportional to the spin\ncorrelations in transition metal-based systems with min-\nimum spin-orbit coupling. Figure 4 (c) shows the esti-\nmated ESR linewidth (∆ Hw) of KCTPO as a function of\nreduced temperatures Trl= (T−TL)/TLin the bottom x-\naxis andTrh= (T−TH)/THin the upper x-axis. Above\nthe transition temperature TH, the observed linewidth\nexhibits a power-law behavior of Trl/Trh∼T−pin two\ndistinct temperature ranges. For temperatures spanning\nfrom 10 K to 24 K, the power exponent ptakes values\nof 0.097 and 0.072, while in the range of 30 K to 100\nK, the values of pare 0.035 and 0.032 for TrlandTrh,\nrespectively [64, 73, 75, 76]. The critical-like broadening\nin the temperature range 10 K ≤T≤24 K, alongside\nthe weak hump at T∗inCmagand the small percentage\nof entropy released above 8 K, supports the notion of the\ncritical-like spin fluctuations. Conversely, the broadening\nobserved in the temperature range of 30 K ≤T≤100 K is\nascribed to cooperative paramagnetic behavior between\ntheS= 3/2 spin of Cr3+ions. Additionally, a deviation\nfrom the Curie-Weiss fit has also been noted below 100\nK (refer to Fig. 2 (a)). The presence of two stage broad-\nening is also observed in other high-spin based frustrated\nmagnets alluding to the two-step thermal evolution of\nmagnetic correlations [64, 75, 76].9\nE. Muon spin relaxation\nTo capture the microscopic details of spin dynamics\nand the internal magnetic field distribution, µSR mea-\nsurements were performed on polycrystalline samples of\nKCTPO in zero-field (ZF) and weak transverse field\n(wTF). When a weak transverse field (here BTF= 32\nG) is applied perpendicular to the initial direction of the\nmuon spin polarization, the implanted muon spin pre-\ncesses around the applied transverse field in the para-\nmagnetic state of the sample, leading to an oscillatory\nsignal with a frequency of γµBTF/2π, whereγµ= 2π\n×135.5 MHz/T represents the muon gyromagnetic ratio\n[77]. In addition, when a static magnetic field is present\nin the sample, the muon spin polarization exhibits a non-\noscillatory signal, as observed in the material under in-\nvestigation [72, 78, 79]. Figure 5 (a) shows the evolution\nof the TF spectra at selected temperatures that were fit-\nted according to\nAwTF(t) =A0fcos(γµBTFt+ϕ)e−λTFt+ (1−f)A0e−λLt,\n(2)\nwhich combines an exponentially decaying oscillatory\ncomponent, corresponding to muon spins experiencing\nzero-static field, and an exponentially decaying non-\noscillatory component that accounts for the component\nof the static local field parallel to the initial muon polar-\nization. In Eq. 2, A0represents the initial asymmetry at\ntime zero,ϕdenotes the relative phase, λTFstands for\nthe muon spin relaxation rate in the applied transverse\nfield, andfquantifies the fraction of the oscillatory com-\nponent while λLrefers to the muon spin relaxation rate\ncaused by the internal magnetic fields. At short time\nscales, we observe a significantly damped signal with de-\ncreasing temperature without any loss of initial asymme-\ntry, which is a characteristic signature of a magnetically\nordered state in KCTPO.\nThe estimated fraction of the oscillatory signal f, associ-\nated to the volume fraction experiencing zero-static field\n(inset of Fig. 5 (b)), remains constant to a value of 0.93\nin the temperature range 4.3 K ≤T≤78 K. This tem-\nperature independent value of f <1 is attributed to the\npresence of weak static local fields above 4.3 K which is\nalso observed in ESR results. However, it gradually de-\ncreases below TL= 4.3 K, indicating a crossover to anther\nmagnetically ordered state. It is noteworthy that fdoes\nnot sharply drop to zero below the transition tempera-\ntureTL, suggesting the existence of dynamic local fields\nin the ordered state. Upon lowering the temperature,\nthe transverse field muon spin relaxation rate gradually\nincreases (see Fig. 5 (b)), as expected in the extreme mo-\ntional narrowing limit of fluctuating magnetic moments.\nA sharp peak in λTFatTLindicates the critical slowing\ndown of spin fluctuations above the transition tempera-\nture as evident from unsaturated magnetic entropy. The\nnotable observation is the occurrence of the critical spin\nrelaxation below TH[47]. Conversely, the muon spin re-\nlaxation rate caused by the static local field (Fig. 5 (c))exhibits a weak kink at TH, attributed to the antifer-\nromagnetic phase transition similar to that observed in\nspecific heat. This is followed by a rapid rise towards TL,\nindicating the enhancement of internal field fluctuations\nthrough spin reconfiguration to the canted antiferromag-\nnetic state. It is interesting to note that, below TL, the\nλTFremains constant down to 1.93 K without display-\ning a sharp fall, persistent spin fluctuations even in the\nmagnetically ordered state, similar to that observed in\ncoupled trillium lattice K 2Ni2(SO 4)3[51].\nTo validate the spin dynamics in the ordered states, ZF-\nµSR measurements were performed in the wide temper-\nature range. Figure 6 (d) shows the ZF- µSR spectra at\ntemperatures above TLwithout any change of the initial\nasymmetry. Furthermore, despite an anomaly in specific\nheat associated to the magnetic phase transition, the ab-\nsence of any oscillation or a strong damped signal even\nin short-time scale, suggests that a dynamical behavior\nof electronic spins associated with the magnetic phase\nsimilar to that observed in trillium lattice antiferromag-\nnet K 2Ni2(SO 4)3. The corresponding µSR data (Fig. 6)\ncan be well fitted to the phenomenological model P(t) =\nA0e−(λZFt)βZF, whereA0is the initial asymmetry, λZF\nis the muon spin relaxation rate in zero-field and βZFis\nthe stretched exponent. The estimated relaxation rate\nexhibits several features as shown in Fig. 6 (b). Upon\nlowering the temperature from 78 K, the λZFremains\nconstant in the temperature range 8 K ≤T≤78 K.\nThe essentially same behavior is observed by λTF(see\nFig. 5 (b)). Below 8 K, the λZFshows a rapid increase,\nascribed to the presence of the canted antiferromagnetic\nphase transition accompanied by a critical slowing down\nof spin fluctuation consistent with our wTF- µSR results.\nThe obtained stretched exponent, measuring the distri-\nbution of electronic moment, is shown Fig. 6 (c). At high-\ntemperatures, βZFis close to one, corresponding to fast\nfluctuation limit. As the temperature decreases, the βZF\nvalue gradually decreases, notably dropping to 0.85 at\nTH∼8 K. Below 8 K, the βZFvalue gradually decreases,\nbut there is a sudden drop at TLto a value of 0.6. It is\nobserved that the ZF- µSR data (for T≥4.41 K) cannot\nbe accurately fitted using a single exponential function\n(βZF= 1) or models with multi-exponential components\n[64]. The obtained βZF<1 is associated with the pres-\nence of a distribution of relaxation times, as expected\nwhen various spins fluctuate in different timescales due\nto competing magnetic interactions in KCTPO. The ob-\nservedβZFof 0.6 (>1/3) value, as seen in Fig. 6 (c),\nimplies that the splitting in ZFC and FC χdata at 4.3\nK is due to canted antiferromagnetic phase rather than\nspin freezing [80].\nFigure 6 (d) displays the time evolution of ZF- µSR spec-\ntra at several temperatures below 4.5 K in the short-time\nscale. As the temperature decreases below 4.5 K, the\npolarization initially exhibits a moderate departure from\nstretched exponential behavior [81]. As it reaches around\n1.9 K, it undergoes an abrupt transition into a strongly\nnon-exponential signal. The shorter time domain clearly10\n101 000.20.40.60\n2 4 6 8 0.00.30.50.81.01\n01 000.60.70.80.912\n3 4 0102030402\n3450.00.51.00\n.00.20.40.60.81.00.00.20.40.60.81.0λ\nZF (µs−1)T\n (K)  ZF(c)Polarizationt\n (µs) 4.41 K   4.55 K \n4.63 K    5.00  K \n7.57 K    50 KZ\nF/s98\nZFT\n (K) ZF(a)( b)2\n3 4 012 \nλf  λsT\n (K)λf (/s109s-1)0\n.000.020.040.06λ\ns (/s109s-1)Δ\n0 (MHz)T\n (K) Δ0RT\n (K) R(d)Polarizationt\n (/s109s) 1.9 K   2.92 K  3.3 K    4.05 \n4.30 K  4.39 (e)( f)\nFIG. 6. (a) Time dependence of zero-field µSR spectra of K 2CrTi(PO 4)3at temperatures higher than the temperature of\ncanted antiferromagnetic phase transition. The solid lines represent the fitted curve by a stretched exponential function. (b),\n(c) Temperature dependence of muon spin relaxation rate and stretched exponent as a function of temperature in zero magnetic\nfield on a double logarithmic scale. The dashed vertical bars indicate the position of characteristic transition temperatures\natTL= 4.30 K and TH= 8 K. (d) Time evolution of zero-field spectra at temperatures below the canted antiferromagnet\nstate. The solid lines are the fits of Eq. 3 to the data. (e) Temperature dependence of fast (left y-axis) and slow (right y-axis)\nmuon spin relaxation rate on a double logarithmic scale. (f) Mean value of the Gaussian-broadened Gaussian distribution as\na function of temperature where the solid lines are the fitted curves with critical scaling behavior of internal field distribution\nwith critical exponent β= 0.37. Inset shows the temperature dependence of Ri.e, the ratio of the width of distributions (W)\nto the distribution mean(∆ 0).\n.\nshows a substantially damped signal, making it challeng-\ning to simulate the muon polarization throughout the\nwhole temperature range using a stretched exponential\nrelaxation function. Furthermore, the absence of any\npeak in the first Fourier transform of the correspond-\ning spectra indicates that the damped signal can not be\nattributable to coherent oscillations typically observed in\nlong-range ordered magnets [79, 82, 83]. Instead, it sug-\ngests the existence of disordered static moments of Cr3+\nions below 4.3 K. After attempting with several models\n[84–86], it turns out that the ZF- µSR data can be mod-\neled with the following polarization function\nPZF=fzPGDG(t; ∆0,W)e−λst+ (1−fz)e−λft,(3)\nwhich takes into account a phenomenological Gaussian-\nbroadened Gaussian (GBG) function with a slow expo-\nnential decay with relaxation rate λsand a fast exponen-\ntial decay with relaxation rate λf[87, 88]. The parame-\nterfzdefines the fraction of Cr3+moments that produce\nstatic local fields, while the remaining part contributes todynamic fields. The following GBG polarization function\nPGBG(t) =1\n3+2\n3(1\n1 +R2∆2\n0t2)3/2(1−∆2\n0t2\n1 +R2∆2\n0t2)\nexp[−∆2\n0t2\n2(1 + R2∆2\n0t2)]\n(4)\nis an extension of the Gaussian Kubo-Toyabe function\nwith a Gaussian distribution width Wand a mean value\n∆0, whereR=W/∆0. This function is commonly\nemployed when the internal field distribution exceeds\nthe Gaussian field distribution. It effectively captures\nthe existence of disordered static magnetic moments\nwith short-range correlations as observed in 3D hyper-\nkagome compound such as Na 4Ir3O8which exhibits\nZFC and FC splitting in dc susceptibility similar to the\ncompound studied here [24]. The extracted temperature\ndependence of fast relaxation rate ( λf; lefty-axis),\nrepresenting a few fraction of Cr3+moments exhibiting\ndynamic behavior, and the slow relaxation rate ( λs;11\nrighty-axis) corresponds to the remaining fraction Cr3+\nmoments displaying static behavior are shown in Fig. 6\n(e). Both λfandλsstart to increase below TLand\nattain a constant value at low temperatures indicating\nthe coexistence of static and dynamic components of\nCr3+moments in KCTPO.\nIn KCTPO, the origin of static moments can be asso-\nciated to the canted antiferomagnetic phase involving\na small fraction of static Cr3+moments. Simulta-\nneously, the rest of the Cr3+moments maintain a\ndynamic state most likely due to the interplay of\ncompeting magnetic interactions. In geometrically\nfrustrated magnets, the coexistence of dynamic and\nstatic electronic moments is a common scenario, as also\nobserved in two coupled trillium lattice K 2Ni2(SO 4)3\nand pyrochlore lattice NaCaNi 2F7[27]. ∆ 0undergoes\nan order-parameter-like decrease as the temperature\nincreases from 1.9 K towards TL. It may be noted that\nthe relative distribution width Rsteadily increases and\nreaches a value of one at TL(inset of Fig. 6 (f)). The\ncritical-like behavior of the local field distribution near\nTLis described by phenomenological expression [89, 90]\n∆0(T) = ∆ 0(T= 0 K)/bracketleftig\n1−T\nTL/bracketrightigβ\nthat yields β=\n0.37 that is typical for 3D Heisenberg magnets (see the\ndashed line in Fig. 6 (f)). Using ∆ 0= 35 MHz at T=\n1.9 K, the calculated internal magnetic field is found\nto be⟨Bloc⟩≈ 400 G. In order to confirm such static\nmagnetic field is of electronic origin, we also performed\nµSR measurements at 3.3 K in longitudinal field (not\nshown here), which decouples the static field owing to\nnuclear origin. It is observed that approximately 2 kG\n>⟨Bloc⟩longitudinal field is required to recover full\npolarization, confirming the presence of both dynamic\nand static moments of Cr3+ions in KCTPO. Similar to\nKCTPO, the observed muon spin asymmetry of triangu-\nlar lattice antiferromagnet NiGa 2S4was characterized by\na stretched exponential function at high temperatures,\nassociated to multichannel relaxation, while the data\nat low temperatures were primarily influenced by static\nlocal fields [81]. The contrasting time evolution of the\nmuon spin polarization function, observed both above\nand below the characteristic temperature of TL, may be\nassociated with distinct magnetic phases in KCTPO,similar to those found in other 3D frustrated magnets\n[91–94].\nIV. CONCLUSION\nIn summary, we have thoroughly investigated the\nstructural and magnetic properties of nearly perfect S\n= 3/2 trillium lattice K 2CrTi(PO 4)3through magneti-\nzation, specific heat, as well as ESR and µSR techniques.\nMagnetic susceptibility results suggest that the presence\nof antiferromagnetic interactions between S= 3/2 mo-\nments of Cr3+ions. Specific heat measurements unveil\ntwo consecutive phase transitions, one at TL= 4.3 K, cor-\nresponding to the canted antiferromagnetic phase, and\nanother atTH= 8 K due to an antiferromagnetic phase.\nAboveTH, the existence of short-range magnetic order\nis supported by the observation of maximum entropy re-\nlease and the manifestation of critical-like behavior in the\nESR line width. Furthermore, µSR provides concrete ev-\nidence of the presence of static and dynamic local mag-\nnetic fields, particularly below TL, as well as a critical\nslowing down of spin dynamics above TLand persistent\nspin dynamics to the lowest temperatures. Further inves-\ntigations on single crystals is essential to gain a detailed\nunderstanding of the low-energy excitations associated\nto the short range order above the transition temper-\nature and the multi-stage spin dynamics in this three-\ndimensional frustrated antiferromagnet.\nACKNOWLEDGMENTS\nP.K. acknowledges the funding by the Science and\nEngineering Research Board, and Department of Sci-\nence and Technology, India through Research Grants.\nThe work at SKKU was supported by the National Re-\nsearch Foundation (NRF) of Korea (Grant no. RS-2023-\n00209121, 2020R1A5A1016518). We further acknowl-\nedge the support of HLD at HZDR, member of the Eu-\nropean Magnetic Field Laboratory (EMFL).\n[1] L. Balents, Nature 464, 199 (2010).\n[2] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133\n(1966).\n[3] P. Khuntia, J. Magn. Magn. Mater. 489, 165435 (2019).\n[4] J. Khatua, B. Sana, A. Zorko, M. Gomilˇ sek, K. Sethu-\npathi, M. R. Rao, M. Baenitz, B. Schmidt, and\nP. 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Mater. 4,\n074405 (2020)."    },    {        "title": "1106.2297v1.Qutrit__entanglement_dynamics_in_the_finite_qutrit_chain_in_the_consistent_magnetic_field.pdf",        "content": "arXiv:1106.2297v1  [quant-ph]  12 Jun 2011Qutrit: entanglement dynamics in the finite qutrit chain in t he consistent magnetic\nfield\nE. A. Ivanchenko∗\nInstitute for Theoretical Physics, National Science Cente r “Institute of Physics and Technology”,\n1, Akademicheskaya str., 61108 Kharkov, Ukraine\n(Dated: May 8, 2022)\nBased on the Liouville-von Neumann equation, we obtain clos ed system of equations for the\ndescription of a qutrit or coupled qutrits in arbitrary time -dependent external magnetic field. The\ndependenceofthedynamicsontheinitial statesandmagneti cfieldmodulationisstudiedanalytically\nand numerically. We compare the relative entanglement meas ure’s dynamics in the bi-qutrit system\nwith permutation particle symmetry. We find the magnetic fiel d modulation which retains the\nentanglement in the system of two coupled qutrits. Analytic al formulas for entanglement measures\nin the chain from 2 to 6 qutrits are presented.\nPACS numbers: 03.67.Bg Entanglement production and manipu lation\n03.67.Mn Entanglement measures, witnesses, and other char acterizations\nI. INTRODUCTION\nMulti-level quantum systems are studied intensively, since they hav e wide applications. Some of the existent ana-\nlytical results for spin 1 [1] are derived in terms of the coherent vec tor [2]. The class of exact solutions for a three-level\nsystem is given in Ref. [3]. The application of coupled multi-level system s in quantum devices is actively studied\n[4]. The study of these systems is topical in view of possible application s for useful work in microscopic systems [5].\nExact solutions for two uncoupled qutrits interacting with vacuum a re obtained in Ref. [6]. For the case of the qutrits\ninteracting with stochastic magnetic field exact solutions are obtain ed in Ref. [7]. Exact solutions for coupled qutrits\nin magnetic field as far as we know were not found.\nThe entanglement in multi-particle coupled systems is an important re source for many problems in quantum infor-\nmation science, but its quantitative value is difficult because of differe nt types of entanglement. Multi-dimensional\nentangled states are interesting both for the study of the found ations of quantum mechanics and for the topicality\nof developing new protocols for quantum communication. For examp le, it was shown that for maximally entangled\nstates of two quantum systems, the qudits break the local realism stronger than the qubits [8], and that the entangled\nqudits are less influenced by the noise than the entangled qubits. Us ing entangled qutrits or qudits instead of qubits\nis more protective from interception. From the practical point of v iew, it is clear that generating and saving the\nentanglement in the controlled manner is the primary problem for the realization of the quantum computers. The\nmaximally entangled states are best suited for the protocols of qua ntum teleportation and quantum cryptography.\nThe entanglement and the symmetry are the basic notions of the qu antum mechanics. We study the dynamics of mul-\ntipartite systems, which are invariant at any subsystem permutat ion. The aim of our work is finding exact solutions\nfor the dynamics of coupled qutrits interacting with alternating mag netic field as well as the comparative analysis of\nthe entanglement measures in the chain of qutrits.\nThe rest of the paper is organized as following. The Hamiltonian of the anisotropic qutrit in arbitrary alternating\nmagnetic field is described in Sec. II. Then the system of equations f or the description of the qutrit dynamics is\nderived in the Bloch vector representation. We introduce the cons istent magnetic field, which describe entire class of\nfield forms. In section III we derive the system of equations for th e description of the dynamics of two coupled qutrits\nin the consistent field and find the analytical solution for the density matrix in the case of anisotropic interaction.\nAnalytical formulas, which describe the entanglement in spin chains f rom 2 to 6 qutrits, are presented in Sec. IV.\nThe results are demonstrated graphically in Sec. V for concrete pa rameters. The brief conclusions are given in Sec.\nVI. The auxiliary analytical results are presented in the Appendices .\n∗Electronic address: yevgeny@kipt.kharkov.ua2\nII. QUTRIT\nA. Qutrit Hamiltonian\nWe take the qutrit Hamiltonian (for the spin-1 particle) in the space o f one qutrit C3in the basis |1>=\n(1,0,0),|0>= (0,1,0),| −1>= (0,0,1), in external magnetic field− →h= (h1,h2,h3) with anisotropy, in the\nform\nˆH=h1S1+h2S2+h3S3+Q(S2\n3−2/3E)+d(S2\n1−S2\n2), (1)\nwhereh1, h2, h3are the Cartesian components of the external magnetic field in the frequency units (we assume\n/planckover2pi1= 1, Bohr magneton µB= 1);S1, S2, S3are the spin-1 matrices (see Appendix A); Estands for the 3 ×3 unity\nmatrix;Q, dare the anisotropy constants. When the constants Q, dare zeros, then the two Hamiltonian eigenvalues\nare symmetrically placed in respect to the zero level.\nB. Liouville-von Neumann equation\nThe qutrit dynamics in the magnetic field we describe in the density mat rix formalism with the Liouville-von\nNeumann equation\ni∂tρ= [ˆH, ρ], ρ(t= 0) =ρ0. (2)\nIt is convenient to rewrite Eq. (2) presenting the density matrix ρin the decomposition with the full set of orthogonal\nHermitian matrices Cα[9] (further the summation over Greek indices will be from 0 to 8 and o ver the Latin ones\nfrom 1 to 8, see Appendix A)\nρ=1√\n6CαRα=\n1\n3+R3√\n6+R6√\n18R1+R7−i(R2+R5)√\n12−iR4+R8√\n6\nR1+R7+i(R2+R5)√\n121\n3−2R6√\n18R1−R7−i(R2−R5)√\n12\niR4+R8√\n6R1−R7+i(R2−R5)√\n121\n3−R3√\n6+R6√\n18\n. (3)\nSince TrCi= 0 for 1 ≤i≤8, then from the condition Tr ρ=R0it follows that R0= 1. And although the results are\nindependent of the basis choice, in this basis the functions Ri= TrρCihave the concrete physical meaning [10]. The\nvaluesR1,R2,R3are the polarization vector Cartesian components; R4is the two-quantum coherence contribution\ninR2;R5is the one-quantum anti-phase coherence contribution in R2;R6is the contribution of the rotation between\nthe phase and anti-phase one-quantum coherence; R7is the one-quantum anti-phase coherence contribution in R1;\nR8is the two-quantum coherence contribution in R1.\nUnder the unitary evolution the length of the generalized Bloch vect or\nb=/radicalig\nR2\ni (4)\nis conserved. The length of the generalized vector (4) for pure st ates equals to√\n2. Sincei∂tρn= [ˆH,ρn] (n=\n1,2,3,...), then under unitary evolution there is countable number of the co nservation laws Tr ρ=c1= 1,Trρ2=\nc2,..., from which only c2, c3are algebraically independent [11]. Additional quadric invariants of mo tion can be\neasily obtained after equating the matrix elements in defining the pur e state. For example, two of these invariants,\nwhich follow from the expression ( ρ2−ρ)13= 0, have the form\nR2\n1−R2\n2+R2\n5−R2\n7−2/radicalbigg\n2\n3(1−√\n2R6)R8= 0, R5R7−R1R2+2√\n3(1√\n2−R6)R4= 0. (5)\nFor numerical calculations, these invariants control also the signs of the values Riand thus the using of the invariants\nis useful when the analytical solutions are difficult to find. According to the Kelly-Hamilton theorem, the density\nmatrixρsatisfies to its characteristic equation\nρ3−ρ2+2−b2\n6ρ−detρE= 0. (6)3\nFrom equation (6) it follows that the density matrix determinant det ρ= (Trρ3−Trρ2)/3+(2−b2)/18 is also the\nmotion invariant. The Liouville-von Neumann equation in terms of the f unctionsRitakes the form of the closed\nsystem of 8 real differential first-order equations. This system o f equations in the compact form can be written as\nfollowing [11, 12]:\n∂tRl=eijlhiRj, (7)\nwhereeijlaretheantisymmetricalstructureconstants, hi= 2(h1,h2,h3,0,0,Q√\n3,0,d)aretheHamiltoniancomponents\n(1) in the basis Cα(see Appendix A).\nC. The consistent field\nConsider the qutrit dynamics in the alternating field of the form\n/vectorh(t) = (ω1cn(ωt|k), ω1sn(ωt|k), ω0dn(ωt|k)), (8)\nwhere cn,sn,dn are the Jacobi elliptic functions [13]. Such field modulation under th e changing of the elliptic\nmoduluskfrom 0 to 1 describes the whole class of field forms from trigonometr ic (cn(ωt|0) = cosωt,sn(ωt|0) =\nsinωt,dn(ωt|0) = 1 ) [14] to the exponentially impulse ones (cn( ωt|1) =1\nchωt,sn(ωt|1) = thωt,dn(ωt|1) =1\nchωt)\n[15]. The elliptic functions cn( ωt|k) and sn(ωt|k) have the real period4K\nω, while the function dn( ωt|k) has the two\ntimes smaller period. Here Kis the full elliptic integral of the first kind [13]. In other words, even t hough the field is\nperiodic with common real period4K\nω, but as we can see, the frequency of the longitudinal field amplitude modulation\nis two times higher than the one of the transverse field. Such field we call consistent.\nLet us make use of the substitution ρ=α−1\n1rα1. Then we obtain the equation for the matrix rin the form\ni∂tr= [α1ˆHα−1\n1−iα1∂t(α−1\n1),r] (9)\nwith the matrix\nα1=\nf0 0\n0 1 0\n0 0f−1\n, (10)\nwheref(ωt|k) = cn(ωt|k)+isn(ωt|k).Since\nα1S1α−1\n1=S1cn(ωt|k)−S2sn(ωt|k), α1S2α−1\n1=S1sn(ωt|k)+S2cn(ωt|k), α1S3α−1\n1=S3, (11)\nthen the equation for the matrix rwithout taking into account the anisotropy can be written as followin g\ni∂tr= [ω1S1+δdn(ωt|k)S3,r], r(t= 0) =ρ0, δ=ω0−ω. (12)\nAtk= 0 equation (12) describes the dynamics of the qutrit in the circular ly polarized field [14, 16, 17]. The exact\nsolutions of this equation are known and at some initial conditions the explicit formulas are given in Ref. [18]. At\nexact resonance, ω=ω0it is straightforward to present Eq. (2) in the deformed field ( k/\\e}atio\\slash= 0) (8) for the given initial\nconditionρ=ρ0:\nρ(t) =α−1\n1e−iω1tS1ρ0eiω1tS1α1. (13)\nExplicit solutions for some specific initial conditions are given in the App endix B, Eqs. (39) – (44). From the\nexplicit exact solutions in the deformed field at resonance δ= 0 one can see that the populations and the transition\nprobabilities do not depend on the field deformation (it is independent of thekmodulus).\nConsider the solution of Eq. (12) far from the resonance in the for m ofδpower expansion\nr(t) =r(0)(t)+r(1)(t)+···. (14)\nThen we put the expansion (14) in Eq. (12) and equate the same deg ree terms. As the result we obtain the system\nof equations for finding r(l)(t):\ni∂tr(0)=ω1[S1,r(0)], (15a)4\ni∂tr(l)=ω1[S1,r(l)]+δdn(ωt|k)[S3,r(l−1)], l= 1,2,.... (15b)\nWe multiply Eq. (15) to the left by the matrix eiω1tS1and to the right by the matrix e−iω1tS1for formation of the\nintegrating multiplier [19]. Now finding the terms r(l)in the series (14) is defined by the previous ones r(l−1)as\nfollowing\nr(l)(t) =−iδ/integraldisplayt\n0dt′eiω1(t′−t)S1dn(ωt′|k)[S3,r(l−1)(t′)]e−iω1(t′−t)S1. (16)\nIII. BI-QUTRIT\nIn the space C3⊗C3the two-qutrit density matrix can be written in the Bloch represent ation\n̺=1\n6RαβCα⊗Cβ, R00= 1, ̺(t= 0) =̺0, (17)\nwhere⊗denotes the direct product. The functions Rm0,R0mcharacterise the individual qutrits and functions Rmn\ncharacterise their correlations. The length of the generalized Bloc h vector/radicalig\nR2\nαβ−1 for pure states equals 2√\n2.\nConsider the Hamiltonian of the system of two qutrits with anisotrop ic and exchange interaction in magnetic field in\nthe following form\nH2= (− →h− →S+Q(S2\n3−2/3E)+d(S2\n1−S2\n2))⊗E+\nE⊗(− →¯h− →S+¯Q(S2\n3−2/3E)+¯d(S2\n1−S2\n2))+JSi⊗Si=1\n2hαβCα⊗Cβ, (18)\nwhere− →hand− →¯hare the magnetic field vectors in frequency units, which operate on the first and the second qudits\nrespectively, and Jis the constant of isotropic exchange interaction.\nThe system of equations for two qutrits takes the real form in ter ms of the functions Rm0,R0m,Rmnas the closed\nsystem of 80 differential equations [12], supplemented by the initial c onditions\n∂tRm0=/radicalbigg\n2\n3epim(hp0Ri0+hplRil), ∂tR0m=/radicalbigg\n2\n3epim(h0pR0i+hlpRli), (19a)\n∂tRmn=epim/bracketleftigg/radicalbigg\n2\n3(hpnRi0+hp0Rin)+grlnhprRil/bracketrightigg\n+epin/bracketleftigg/radicalbigg\n2\n3(hmpR0i+h0pRmi)+grlmhrpRli/bracketrightigg\n,(19b)\nwhere by definition\nTrρCα⊗Cβ=2\n3Rαβ (20)\nandhp0=√\n6(− →h,0,0,Q√\n3,0,d),h0p=√\n6(− →¯h,0,0,¯Q√\n3,0,¯d), h11=h22=h33= 2Jare the Hamiltonian expansion\ncoefficients in the basis Cα⊗Cβ(other coefficients equal to zero). In equations (19) Latin indices m, ntake the values\nfrom 1 to 8. Numerical values for the structure constants epim, grlmare given in Appendix A.\nThe energy of the coupled qutrits in terms of the correlation funct ions has the following form\nE(t) =1\n3(hp0Rp0+h0pR0p+3/summationdisplay\ni=1hiiRii). (21)\nWe study the dynamics of two qutrits in the magnetic field− →h= (ω1cn(ωt|k)), ω1sn(ωt|k), ω0dn(ωt|k),− →¯h=\n(̟1cn(̟t|k), ̟1sn(ωt|k), ̟0dn(ωt|k)) at the anisotropy constants equal to 0. We transform the mat rix density\n̺=α−1\n2r2α2with the matrix α2=α1⊗α1. Then equation for the matrix r2takes the form i∂tr2= [/tildewideH,r2] with the\ntransformed Hamiltonian5\n/tildewideH=\nJ+D(−2ω+̟0+ω0)̟1√\n20ω1√\n20 0 0 0 0\n̟1√\n2D(ω0−ω)̟1√\n2Jω1√\n20 0 0 0\n0̟1√\n2D(ω0−̟0)−J0Jω1√\n20 0 0\nω1√\n2J 0 D(̟0−ω)̟1√\n20ω1√\n20 0\n0ω1√\n2J̟1√\n20̟1√\n2Jω1√\n20\n0 0ω1√\n20̟1√\n2D(ω−̟0) 0 Jω1√\n2\n0 0 0ω1√\n2J 0D(̟0−ω0)−J̟1√\n20\n0 0 0 0ω1√\n2J̟1√\n2D(ω−ω0)̟1√\n2\n0 0 0 0 0ω1√\n20̟1√\n2J+D(2ω−̟0−ω0)\n.\nSinceDdef≡dn(ωt|k)|k=0= 1, then the transformed Hamiltonian /tildewideHdoes not depend on time and the solution for the\ndensity matrix in the circularly polarized field has the form\n̺(t) =α−1\n2e−i/tildewideHt̺0ei/tildewideHtα2|k=0. (22)\nIn the consistent field at resonance ω=̟0=ω0=hat equal̟1=ω1the Hamiltonian eigenvalues equal to\n−2J,−J,J,J−2ω1,−J−ω1,J−ω1,−J+ω1,J+ω1,J+2ω1. This allows to find the exact solution in the closed\nform for any initial condition, since the matrix exponent ei/tildewideHtin this case can be calculated analytically.\nFor larger number of the qutrits with pairwise isotropic interaction, the generalization is evident. In the case of\ninteraction of qudits with different dimensionality, the reduction of t he original system to the system with constant\ncoefficients can be done by choosing, for example, the transforma tion matrix for spin-3/2 and spin-2 in the form\ndiag(f3/2,f1/2, f−1/2, f−3/2)⊗diag(f2,f,1,f−1, f−2). (23)\nHowever, the Hamiltonian eigenvalues cannot be found in the simple an alytic form because of the lowering the system\nsymmetry.\nIV. ENTANGLEMENT IN THE QUTRITS\nA. Entanglement in the bi-qutrit\nFor the initial maximally entangled state, which is symmetrical at the p article permutation,\n|ψ>=1√\n31/summationdisplay\ni=−1|i>⊗|i>, (24)\nin the consistent field at the resonance ω=̟0=ω0=hat equal̟1=ω1,the exact solution for the correlation\nfunctions is given in Appendix C. The correlation functions have the p ropertyRαβ=Rβα, i.e. the symmetry is\nconserved during the evolution, since the initial state and Hamiltonia n are symmetric in respect to the particle\npermutation.\nGiven the exact solution, one can find the negative eigenvalues of th e partly transposed matrix ̺pt= (T⊗E)̺(here\nTdenotes the transposition):\nǫ1=ǫ2=−1\n27√\n69+28cos3 Jt−16cos6Jt, ǫ3=−1\n27(5+4cos3Jt). (25)\nThe absolute value of the sum of these eigenvalues mV W=|ǫ1+ǫ1+ǫ3|defines the entanglement measure (negativity)\nbetween the qutrits [20].\nThe entanglement between the qutrits can be described quantitat ively with the measure [21]\nmSM=/radicalbigg\n1\n8(Rij−Ri0R0j)2. (26)\nThis measure equals to 0 for the separable state and to 1 for the ma ximally entangled state, and it is applicable for\nboth pure and mixed states.\nThat is why for the maximally entangled initial state of two qutrits, th e entanglement in the consistent field is defined\nby the formulae with the found solution for the density matrix\nmSM=1√\n6561√\n4457+2776cos3 Jt−632cos6Jt−56cos9Jt+16cos12Jt. (27)6\nThis measure is numerically equivalent to the measure mV W[20, 22], which is defined by the absolute value of the\nsum of the negative eigenvalues (25) of the partly transposed mat rix.\nAccording to the definition [23] for 2-qutrit pure state, the entan glement measure equals to\nη2=1\n22/summationdisplay\ni=1Si, (28)\nwhereSi=−Trρilog3ρiis the reduced von Neumann entropy, the index inumerates the particles, i.e. the other\nparticle are traced out.\nSince the qutrit reduced matrix eigenvalues equal to λ1=λ2=1\n27(5+4cos3Jt), λ3=1\n27(17−8cos3Jt),then the\nentanglement measure in the bi-qutrit takes the form\nη2=−3/summationdisplay\ni=1λilog3λi. (29)\nNormalized by the unity, the measure I-concurrence, which is easy to calculate, is defined by the formulae [24]\nmI=√\n3\n2/radicalig\n2(1−Trρ2\n1) =1\n9√\n57+32cos3 Jt−8cos6Jt, (30)\nwhereρ1=1√\n6CαRα0is the reduced qutrit matrix.\nThe time-dependence of the measure mSMfor the symmetrical initial state\n|s>=1√\n12/summationdisplay\ni/negationslash=j(|i>⊗|j>+|j >⊗|i>) (31)\ntakes the form\nm|s>\nSM=1√\n209952√\n102679+19136cos3 Jt+29312cos6 Jt−1024cos9Jt+800cos12 Jt; (32)\natt= 0 this measure equals to/radicalbig\n23/32.\nThemeasures mVW, mSM, η2, mI, m|s>\nSMdonotdependontheparametersoftheconsistentfield, signofth eexchange\nconstant at zero anisotropy parameters. It should be noted tha t the Wooters entanglement measure (concurrence) in\nthe system of two qubits with the isotropic interaction in the circular ly polarized field at resonance is also independent\nof the alternating field amplitude [25], but depends on the exchange c onstant magnitude and the initial conditions\nonly.\nAt zero external field the entanglement measure (24) takes the a nalytic form at equal non-zero anisotropy parameters\nQ=d=d=Q\nmSM(Q) =1\n(9J2+8QJ+16Q2)2/radicaltp/radicalvertex/radicalvertex/radicalbt4/summationdisplay\nk=0qkcos/parenleftig\nk/radicalbig\n9J2+8QJ+16Q2t/parenrightig\n, (33)\nwhereq0= 4457J8+11616QJ7+47392Q2J6+85888Q3J5+163072Q4J4+194560Q5J3+221184Q6J2+131072Q7J+\n65536Q8;q1= 8J2(J+2Q)2/parenleftbig\n347J4+518QJ3+1440Q2J2+1504Q3J+1024Q4/parenrightbig\n;\nq2=−8J2(J+2Q)2/parenleftbig\n79J4+76QJ3+320Q2J2+448Q3J+256Q4/parenrightbig\n;\nq3=−8J3(7J−4Q)(J+2Q)3(J+4Q),q4= 16J4(J+2Q)4.\nB. Entanglement in the chain of qutrits\nWe consider the Hamiltonian of the chain of Nqutrits with the pairwise isotropic interaction in the magnetic field− →ωin the following form\nHN=/summationdisplay\n(− →ω− →S⊗N−1/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nE⊗···⊗E+J− →S⊗− →S⊗N−2/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nE⊗···⊗E), (34)7\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s49/s48/s46/s50/s48/s46/s51/s80/s43\n/s73\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s49/s48/s46/s50/s48/s46/s51/s80/s48\n/s73/s73\nFIG. 1: The time-averaged populations for the initial pure s tate|−1>versus the normalized Larmor frequency ω0/ωat the\nparameters k= 0.85 (solid line), k= 0.2 (dashed line), d=Q= 0,ω1= 1/3, ω= 1 (Ishows the upper level |1>population;\nIIshows the middle level |0>population).\nwhere the summation is overdifferent possible positions of− →Sin the direct products. Becausethe maximally entangled\nstate ofNqutrits\n|φ>=1√\n31/summationdisplay\ni=−1|i>⊗N(35)\nand the Hamiltonian (34) have the permutation symmetry, it follows t hat the density matrix of Nqutrits has the\nsymmetric correlation functions. The length of the generalized Bloc h vector for pure states equals√\n3N−1.\nThe entanglement measures for the many-particle multi-level quan tum systems are not studied enough and difficult\nto calculate in the analytic form, that is why we will present only analyt ic formulas for the entropy measure ηN[23],\nwhich is defined by the eigenvalues of the reduced one-particle matr ices for each qutrit. As the result of the mentioned\nsymmetry, the reduced matrices are equal to each other. There fore the entanglement measure for Nqutrits is defined\nby the formulae\nηN=−3/summationdisplay\ni=1rilog3ri. (36)\nThe eigenvalues of the reduced matrices for 3, 4, 5, and 6 qutrits a re presented in the table below\nN\\ri r1=r2 r3\n329−4cos5Jt\n7517+8cos5 Jt\n75\n4905−98cos3Jt−72cos7Jt\n2205395+196cos3 Jt+144cos7 Jt\n2205\n516919−1944cos5 Jt−800cos9 Jt\n425258687+3888cos5 Jt+1600cos9 Jt\n42525\n621977−1694cos3 Jt−1936cos7 Jt−560cos11 Jt\n533619407+3388cos3 Jt+3872cos7 Jt+1120cos11 Jt\n53361.(37)\nThe measures η3, η4, η5, η6do not depend on sign of the exchange constant like the measure η2.\nV. NUMERICAL RESULTS\nIn Fig. 1 we present the populations of the upper and middle levels in th e qutrit averaged over the time interval\nτ→ ∞:P+=1\nτ/integraltextτ\n0dt/parenleftig\n1\n3+1√\n6R3(t)+1\n3√\n2R6(t)/parenrightig\n,P0=1\nτ/integraltextτ\n0dt(1\n3−1\n3√\n2R6(t)) in dependence on the normalized\nLarmor frequency ω0/ω. The population of the upper level in qutrit coincides in form with the u pper level occupation\nin a two-level system [19], i.e. this demonstrates the magnetic reson ance position stabilization and the presence of\nthe parametric resonances.\nIn Fig. 2 we note the considerable suppression of the qutrit spin osc illationsSy= cn(ωt|k)sinω1tandSz=−cosω1t\nby the environment (fluctuator) in the case of the resonance ω=ω0,− →̟= 0.8\n/s49/s50/s48 /s50/s52/s48/s45/s49/s48/s49/s32/s83\n/s121/s32/s44/s32/s83\n/s122\n/s116/s32/s83\n/s121\n/s83\n/s122\nFIG. 2: Dynamics of the spin vector components Sy, Szfor the initial pure state | −1>(dashed lines) in the circularly\npolarized field with the parameters: k= 0, ω1= 0.02, ω=ω0= 1, d=Q= 0. Solid lines demonstrate the deformation of the\nspin components due to the influence of the second spin (the flu ctuator) with J= 0.1 for the initial pure state |−1>⊗|−1>.\n/s48 /s49/s53 /s51/s48 /s52/s53/s48/s46/s53/s49/s46/s48/s101/s110/s116/s97/s110/s103/s108/s101/s109/s101/s110/s116\n/s116/s49\n/s50/s51\n/s52/s53\nFIG. 3: Disentanglement dynamics of the initially maximall y entangled state in the bi-qutrit: in the zero external field with\nequal anisotropy constants Q=d=d=Q= 0.02507, J=−0.1 (curve 1) and for J= 0.1 (curve 2);in the consistent field\nthe curve 3 (thick line) demonstrates complete coincidence of the measures mV WandmSMatJ=0.1 and zero anisotropy\nconstants; the curve 4 demonstrates the entropy measure η2;I-concurrence is presented by the curve 5 at J=0.1.\nThe bi-qutrit energy (21) in the consistent field at isotropic interac tion in the case of the solution (45) is constant\nand equal to2\n3J.\nAlthough the analytic expressions for the measures in a bi-qutrit mV W, mSMare different, but the numerical values\nare practically identical. Maximal deviation in the rectangle (1 ≥J≥0.01)×(100≥t≥0) equals 0.014, where ×\ndenotes the Cartesian product.\nMeasuresη2andmIqualitatively coincide with the measures mVW, mSM.\nWe have found that the anysotropy of the qutrits disentangles th em, namely the entanglement is decreased down to\n0.0010 (see graphs 1 and 2 in Fig. 3).\nIn the constant longitudinal field− →ω=−− →̟= (0,0,ω0) (the bi-qutrit Hamiltonian eigenvalues are equal to\nJ,J,x1,x2,x3,−p,−p,p,p, wherex1,x2,x3are the roots of the equation x3+2x2J−p2x−2J3= 0, p=/radicalbig\nJ2+ω2\n0)\nthe Hamiltonian containsthe antisymmetricpart, thus it followsthat the density matrix forthe initial symmetricstate\nwillnotbesymmetricbecauseofthebreakingthesymmetryofthep articlepermutations. Theanalyticsolutioniscum-\nbersome. In the constant longitudinal impulse field− →ω=−− →̟= (0,0,2(θ((t−17)(t−60))+θ((40−t)(57−t)(t−60))))\nthe entanglement dynamics is blocked at ω0≫J(Fig. 4 ). This points to the possibility to control the entanglement.\nIn Fig. 5 we present the comparative dynamics of the entropy enta nglement measure for 2 to 6 qutrits. The disen-\ntanglement dynamics of the measures η3,η4,η5,η6is similar to the one in the case of two qutrits, but with smaller\noscillation amplitude, i.e. larger number of the qutrits disentangles les s than two qutrits (0 .889≤η3≤1).\nVI. CONCLUSION\nWe have shown that the time-averaged upper-level occupation pr obability for the qutrit in the consistent field in\ndependence on the normalized Larmor frequency ω0/ωcoincides in form with the upper-level occupation probability\nin the two-level system and the parametric resonances appear (F ig. 1). In the qutrit coupled to another qutrit9\n/s48 /s50/s48 /s52/s48 /s54/s48/s48/s46/s53/s49/s46/s48\n/s109\n/s83/s77\n/s116\nFIG. 4: Disentanglement of the maximally entangled state (2 4) (solid line) in the impulse field ω1=̟1= 0, ω0=−̟0=\n2, J= 0.178. The dashed curve presents the evolution in zero externa l field.\n/s48 /s50/s48 /s52/s48 /s54/s48/s48/s46/s53/s49/s46/s48\n/s116\nFIG. 5: Disentanglement of the maximally entangled state (2 8) in the chain of 2,3,4,5 and 6 qutrits with J= 0.1.\n(fluctuator), the spin oscillations are essentially suppressed.\nThe comparative analysis of the bi-qutrit entanglement measures o n the base of the analytic solution for the density\nmatrix demonstrates that, in spite of the different approaches to the derivation of the formulas for the entanglement,\nall the formulas give quite close results (Fig. 3), and the measures mVWandmSMare practically equal. This is in\naccordance with the general results for the entanglement in the s ystems with the permutational symmetry [22].\nThe analytical formulas for the entanglement measures η3,η4,η5,η6are similar to the disentanglement measure for\ntwo qutrits η2, but with numerically smaller oscillation amplitude, i.e. the larger number of the qutrits disentangles\nless than two qutrits.\nAcknowledgments\nThe author is grateful to A. A. Zippa for fruitful discussions and c onstant invaluable support. Many thanks are\ndue to S. N. Shevchenko for help in editing and useful comments.\nVII. APPENDIX A\nThe matrix representation of the full set of Hermitian orthogonal operators for spin-1 has the form\nC1=S1=1√\n2\n0 1 0\n1 0 1\n0 1 0\n, C2=S2=i√\n2\n0−1 0\n1 0−1\n0 1 0\n,C3=S3=\n1 0 0\n0 0 0\n0 0−1\n, (38a)10\nC4=i\n0 0−1\n0 0 0\n1 0 0\n,C5=i√\n2\n0−1 0\n1 0 1\n0−1 0\n, C6=√\n3(S2\n3−2/3E) =1√\n3\n1 0 0\n0−2 0\n0 0 1\n, (38b)\nC7=1√\n2\n0 1 0\n1 0−1\n0−1 0\n,C8=S2\n1−S2\n2=\n0 0 1\n0 0 0\n1 0 0\n, C0=/radicalbigg\n2\n3E, (38c)\nwhereEis the unity 3 ×3 matrix. These matrices have the property of the trace equal to zero TrCa= 0 and\northogonality Tr CaCb= 2δab, 1≤a,b≤8. The connection between the basis Caand the Gell-Mann basis λais the\nfollowing:\nC1=1√\n2(λ1+λ6), C2=1√\n2(λ2+λ7), C3=1\n2λ3+√\n3\n2λ8,C4=λ5, C5=1√\n2(λ2−λ7), C6=√\n3\n2λ3−1\n2λ8, C7=\n1√\n2(λ1−λ6), C8=λ4.\nNon-zero structure constants eabc(gabc) antisymmetric (symmetric) in respect to the permutation of any p air of\nindices for the commutators [ Ca,Cb] = 2ieabcCc(anticommutators {Ca,Cb}=4\n3Eδab+ 2gabcCc) are respectively\nequal according to the definitions eabc=1\n4iTr[Ca,Cb]Cc:\ne123=e147=e158=−e245=e278=−e357=1\n2, e156=e267=√\n3\n2, e348=−1;gabc=1\n4Tr{Ca,Cb}Cc:g336=g446=\n−g666=g688=1√\n3, g116=g226=g556=g677=−1\n2√\n3, g118=g124=g137=−g228=g235=−g457=g558=−g778=\n1\n2. Hence, the product of the generators is equal to CaCb=2\n3Eδab+(gabc+ieabc)Cc.\nVIII. APPENDIX B\nFor the initial state |−1>at non-zero detuning δ=ω0−ωthe density matrix elements in the circularly polarized\nfield have the form\nρ11=ω4\n1\nΩ4sin4Ωt\n2,ρ12=−√\n2ω3\n1\nΩ4sin3Ωt\n2e−iωt/parenleftbigg\niΩcosΩt\n2+δsinΩt\n2/parenrightbigg\n, (39)\nρ13=−ω2\n1\n2Ω4sin2Ωt\n2e−2iωt/parenleftbig\nω2\n1−2iδΩsinΩt+/parenleftbig\n2δ2+ω2\n1/parenrightbig\ncosΩt/parenrightbig\n, (40)\nρ22=ω2\n1sin2Ωt\n2\nΩ4/parenleftbig\n2δ2+ω2\n1(1+cosΩt)/parenrightbig\n, ρ23=−ω1√\n2Ω4e−iωt/parenleftbig\n2δ2+ω2\n1(1+cosΩt)/parenrightbig/parenleftbigg\nδsin2Ωt\n2+iΩ\n2sinΩt/parenrightbigg\n,(41)\nρ33=1\n4Ω4/parenleftbig\n2δ2+ω2\n1(1+cosΩt)/parenrightbig2,ρik=ρ∗\nki, (42)\nwhere Ω =/radicalbig\nω2\n1+δ2is the Rabi frequency.\nFor the initial doubly stochastic state1√\n3(|−1>+|0>+|1>) and for the states |0>,1\n2|−1>+1√\n2|0>+1\n2|1>|\nat exact resonance δ= 0 in the consistent field, the density matrices are respectively equ al\n\n1\n12(cos2ω1t+3)1\n12f−1/parenleftbig\ni√\n2sin2ω1t+4/parenrightbig1\n12f−2(cos2ω1t+3)\n1\n12f/parenleftbig\n4−i√\n2sin2ω1t/parenrightbig1\n6(3−cos2ω1t)1\n12f−1/parenleftbig\n4−i√\n2sin2ω1t/parenrightbig\n1\n12f2(cos2ω1t+3)1\n12f/parenleftbig\ni√\n2sin2ω1t+4/parenrightbig1\n12(cos2ω1t+3)\n, (43)\n\n1\n2sin2ω1t−if−1sin2ω1t\n2√\n21\n2f−2sin2ω1t\nifsin2ω1t\n2√\n2cos2ω1tif−1sin2ω1t\n2√\n2\n1\n2f2sin2ω1t−ifsin2ω1t\n2√\n21\n2sin2ω1t\n,\n1\n16(5−cos2ω1t)−if−1sin2ω1t\n8√\n21\n8f−2sin2ω1t\nifsin2ω1t\n8√\n21\n8(cos2ω1t+3)if−1sin2ω1t\n8√\n2\n1\n8f2sin2ω1t−ifsin2ω1t\n8√\n21\n16(5−cos2ω1t)\n.(44)\nFor both the initial middle level and the doubly stochastic pure initial s tate, the populations of the upper and bottom\nlevels are equal. [18]. This property is fulfilled for the mixed state as we ll.11\nIX. APPENDIX C\nThe exact solution for the correlation functions of the initial state (24), which is symmetric under the particle\npermutation, in the consistent field at resonance ω=̟0=ω0=hand equal̟1=ω1takes the form\nR0,1=R0,2=R0,3= 0,R0,4=8\n3/radicalbigg\n2\n3cos2ω1tcnusnusin23Jt\n2, R0,5=−4\n3/radicalbigg\n2\n3cnusin23Jt\n2sin2ω1t,\nR0,6=2\n9√\n2(3cos2ω1t−1)sin23Jt\n2, R0,7=4\n3/radicalbigg\n2\n3snusin2ω1tsin23Jt\n2,\nR0,8=4\n3/radicalbigg\n2\n3cos2ω1t/parenleftbig\n1−2sn2u/parenrightbig\nsin23Jt\n2,(45a)\nR1,1=1\n36/parenleftbig\n16+12(cos3 Jt+2)/parenleftbig\ncn2u−sn2u/parenrightbig\ncos2ω1t+2cos3Jt−3cos(3J−2ω1)t\n−12cos2ω1t−3cos(3J+2ω1)t), R1,2=2\n3(cos3Jt+2)cnusnucos2ω1t,\nR1,3=1\n3(cos3Jt+2)snusin2ω1t, R1,4=1\n3snusin3Jtsin2ω1t,\nR1,5=1\n6/parenleftbig\n2/parenleftbig\ncn2u−sn2u/parenrightbig\ncos2ω1t+3cos2ω1t−1/parenrightbig\nsin3Jt,R1,6=1√\n3cnusin3Jtsin2ω1t,\nR1,7=−2\n3cos2ω1tcnusnusin3Jt, R1,8=1√\n3R1,6, (45b)\nR22=1\n18/parenleftbig\n6(cos3Jt+2)/parenleftbig\nsn2u−cn2u/parenrightbig\ncos2ω1t+cos3Jt−3(cos3Jt+2)cos2ω1t+8/parenrightbig\n,\nR23=−1\n3(cos3Jt+2)cnusin2ω1t, R24=1√\n3R16, R25=−R17,R26=√\n3R14,\nR27=1\n6/parenleftbig\n2/parenleftbig\ncn2u−sn2u/parenrightbig\ncos2ω1t−3cos2ω1t+1/parenrightbig\nsin3Jt, R28=−1√\n3R26, (45c)\nR33=1\n18(−2cos3Jt+3cos(3J−2ω1)t+12cos2ω1t+3cos(3J+2ω1)t+2),\nR34=−2\n3cos2ω1t/parenleftbig\ncn2u−sn2u/parenrightbig\nsin3Jt,R35=−R14, R36= 0,\nR37=−1\n3cnusin3Jtsin2ω1t, R38=4\n3cos2ω1tcnusnusin3Jt, (45d)\nR44=1\n72/parenleftbig\n−72/parenleftbig\n1−8cn2usn2u/parenrightbig\ncos4ω1t+8cos3Jt−12(2cos3Jt+1)cos2ω1t+9cos4ω1t+19/parenrightbig\n,\nR45=1\n12snu/parenleftbig\n24/parenleftbig\nsn2u−3cn2u/parenrightbig\nsinω1tcos3ω1t+2(2cos3Jt+1)sin2ω1t−3sin4ω1t/parenrightbig\n,\nR46=−2\n3√\n3cos2ω1t(2cos3Jt−9cos2ω1t+7)cnusnu,\nR47=1\n12cnu/parenleftbig\n−24/parenleftbig\ncn2u−3sn2u/parenrightbig\nsinω1tcos3ω1t−2(2cos3Jt+1)sin2ω1t+3sin4ω1t/parenrightbig\n,\nR48= 4cos4ω1tcnusnu/parenleftbig\ncn2u−sn2u/parenrightbig\n, (45e)\nR55=1\n18/parenleftbig\n6(cos3Jt+6cos2ω1t−4)/parenleftbig\nsn2u−cn2u/parenrightbig\ncos2ω1t−cos3Jt+3(cos3Jt+2)cos2ω1t\n−9cos4ω1t+1), R56=1\n6√\n3cnu/parenleftbigg\n4sin23Jt\n2sin2ω1t−9sin4ω1t/parenrightbigg\n,\nR57=1\n6(2cos3Jt+cos(3J−2ω1)t+4cos2ω1t+6cos4ω1t+cos(3J+2ω1)t−2)cnusnu,\nR58=1\n12cnu/parenleftbig\n−24/parenleftbig\ncn2u−3sn2u/parenrightbig\nsinω1tcos3ω1t+2(2cos3Jt+1)sin2ω1t−3sin4ω1t/parenrightbig\n,(45f)12\nR66=1\n36(−4cos3Jt+6cos(3J−2ω1)t−12cos2ω1t+27cos4ω1t+6cos(3J+2ω1)t+13),\nR67=1\n6√\n3snu(2(cos3Jt−1)sin2ω1t+9sin4ω1t),\nR68=1\n3√\n3cos2ω1t(−2cos3Jt+9cos2ω1t−7)/parenleftbig\ncn2u−sn2u/parenrightbig\n, (45g)\nR77=1\n18/parenleftbig\n(cos3Jt−1)/parenleftbig\n3cn2u−3sn2u−1/parenrightbig\n+9cos4ω1t/parenleftbig\ncn2u−sn2u−1/parenrightbig\n+3(cos3Jt+2)cos2ω1t/parenleftbig\ncn2u−sn2u+1/parenrightbig/parenrightbig\n,\nR78=1\n6snu/parenleftbig\n6/parenleftbig\n3cn2u−sn2u/parenrightbig\ncos2ω1t+2cos3Jt−3cos2ω1t+1/parenrightbig\nsin2ω1t, (45h)\nR88=1\n72/parenleftbig\n72/parenleftbig\n1−8cn2usn2u/parenrightbig\ncos4ω1t+8cos3Jt−12(2cos3Jt+1)cos2ω1t+9cos4ω1t+19/parenrightbig\n,(45i)\nwhereu= (ht|k).It isstraightforwardtofind theanalyticsolutionforlargernumber ofqutritsatthe sameconditions.\n[1] F. 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A 23, 2496 (1981).\n[16] J. B. Miller, B. H. Suits, and A. N. Garroway, Journal of M ag. resonance 151, 228 (2001).\n[17] M. Grifoni and P. Hanggi, Physics Reports 304, 229 (1998).\n[18] M. R. Nath, S. Sen, and G. Gangopadhyay, Pramana-Journa l of Physics 61, 1089 (2003).\n[19] E. A. Ivanchenko, Low Temp. Phys. 31, 577 (2005).\n[20] G. Vidal and R. F. Werner, Phys. Rev. A 65, 32314 (2002).\n[21] J. Schlienz and G. Mahler, Phys. Rev. A 52, 4396 (1995).\n[22] G. Toth and O. G¨ uhne, Phys. Rev. Letters 102, 170503 (2009).\n[23] F. Pan, D. Liu, G. Y. Lu, and J. P. Draayer, Int. J. Theor. P hys.43, 1241 (2004).\n[24] F. Mintert, A. R. R. Carvalho, M. Kus, and A. Buchleitner , Physics Reports 415, 207 (2005).\n[25] S.-X. Zhang, Q.-S. Zhu, and X.-Y. Kuang, Commun. Theor. Phys. (Beijing, China) 50, 883 (2008)."    },    {        "title": "2009.01680v2.Coincidence_inelastic_neutron_scattering_for_detection_of_two_spin_magnetic_correlations.pdf",        "content": "arXiv:2009.01680v2  [cond-mat.str-el]  27 Feb 2021Coincidence inelastic neutron scattering for detection of two-spin magnetic\ncorrelations\nYuehua Su,∗Shengyan Wang, and Chao Zhang\nDepartment of Physics, Yantai University, Yantai 264005, P eople’s Republic of China\nInelastic neutron scattering (INS) is one powerful techniq ue to study the low-energy single-spin\ndynamics of magnetic materials. A variety of quantum magnet s show novel magnetic correlations\nsuch as quantum spin liquids. These novel magnetic correlat ions are beyond the direct detection\nof INS. In this paper we propose a coincidence technique, coi ncidence inelastic neutron scattering\n(cINS), which can detect the two-spin magnetic correlation s of the magnetic materials. In cINS\nthere are two neutron sources and two neutron detectors with an additional coincidence detector.\nTwo neutrons from the two neutron sources are incident on the target magnetic material, and\nthey are scattered by the electron spins of the magnetic mate rial. The two scattered neutrons are\ndetected by the two neutron detectors in coincidence with th e coincidence probability described by\na two-spin Bethe-Salpeter wave function. Since the two-spi n Bethe-Salpeter wave function defines\nthe momentum-resolved dynamical wave function with two spi ns excited, cINS can explicitly detect\nthe two-spin magnetic correlations of the magnetic materia l. Thus, it can be introduced to study\nthe various spin valence bond states of the quantum magnets.\nI. INTRODUCTION\nThe novel magnetic correlations in various quantum\nmagnetshaveattractedmuchattentionin thecondensed-\nmatter field. Quantum spin liquids with strong frustra-\ntion and quantum fluctuations are one special type of\nexample1–4. One experimental technique in the study\nof these novel magnetic correlations is inelastic neutron\nscattering (INS), which can provide the single-spin dy-\nnamical responses of magnetic materials and thus can\nshow the relevant physics of single-spin excitations5–9.\nHowever, as most novel magnetic correlations in the\nquantum magnets are beyond that of the single-spin\nmagnons, the spectrum of INS cannot provide explicit\ninformation on these novel magnetic correlations. It is\nimperative to develop experimental techniques which can\nexplicitly detect these novel magnetic correlations.\nRecently, coincidence angle-resolved photoemission\nspectroscopy (cARPES) was proposed for detection of\ntwo-particle correlations of material electrons10. In this\npaper we will follow the idea of cARPES to propose\nanother coincidence technique, coincidence INS (cINS),\nwhich can explicitly detect the two-spin magnetic cor-\nrelations of magnetic materials. There are two neutron\nsources and two neutron detectors in the experimental\ninstrument of cINS, with an additional coincidence de-\ntector. The twoneutron sourcesemit twoneutronswhich\nareincidentonthetargetmagneticmaterialandarescat-\ntered by the material electron spins. These two scattered\nneutrons are then detected by the two neutron detectors\nin coincidence with the coincidence probability relevant\nto a two-spin Bethe-Salpeter wave function.\nThe two-spin Bethe-Salpeter wave function is defined\nas\nφ(ij)\nαβ(q1t1,q2t2) =/an}b∇acketle{tΨβ|TtS(i)\n⊥(q1,t1)S(j)\n⊥(q2,t2)|Ψα/an}b∇acket∇i}ht,\n(1)\nwhere|Ψα/an}b∇acket∇i}htand|Ψβ/an}b∇acket∇i}htare the eigenstates of the electron\nspins of the target magnetic material, S(i)\n⊥(q,t) is theith component of the spin operator within a perpendic-\nular plane normal to the momentum q, andTtis a time-\nordering operator. This Bethe-Salpeter wave function\ndescribes the time dynamical evolution of the magnetic\nmaterial with two spins excited at times t1andt2in\ntime ordering. The coincidence probability of cINS can\nprovide the Fourier transformation of the time dynami-\ncalBethe-Salpeterwavefunction, withthecenter-of-mass\nfrequency defined by the sum of the two transfer energies\nin the two-neutron scattering and the relative frequency\ndefined by the difference of the two transfer energies.\nTherefore, the coincidence detection of cINS can pro-\nvide the momentum-resolved dynamics of the two-spin\nmagnetic correlations, with the physics of both the cen-\nter of mass and the relative degrees of freedom of two\nexcited spins of the magnetic material. Thus, it can be\nintroduced to study the spin valence bond states of the\nquantum magnets.\nOur paper is organized as follows. In Sec. II the the-\noretical formalism of the coincidence detection of cINS\nwill be provided. In Sec. III the coincidence probabilities\nof cINS for a ferromagnet and an antiferromagnet with\nlong-range magnetic order will be presented. Discussion\nof the experimental detection of cINS will be given in\nSec. IV, where a brief summary will also be provided.\nII. THEORETICAL FORMALISM FOR cINS\nIn this section we will establish the theoretical formal-\nism for the coincidence detection of cINS. First, we will\nreview the principle of the single-spin INS in Sec. IIA.\nWe will then provide the theoretical formalism for cINS\nin Sec. IIB.2\nA. Review of INS\nSuppose the incident neutrons have momentum qiand\nspinβiwith a spin distribution function P1(βi). The in-\ncident neutrons interact with the electron spins of the\ntarget magnetic material via the electron-neutron mag-\nnetic interaction\n/hatwideVs=/summationdisplay\nqiqfg(q)/hatwideσqfqi·S⊥(q), (2)\nwhereg(q)≡gF0(q), withgbeing an interaction con-\nstant andF0(q) being a magnetic form factor, and q=\nqf−qi, with/hatwideq=q\nq. The operator /hatwideσqfqiis defined for\nneutrons,\n/hatwideσqfqi=/summationdisplay\nβiβfd†\nqfβfσβfβidqiβi, (3)\nwheredqβandd†\nqβare the respective neutron annihila-\ntion and creation operators and σis the Pauli matrix.\nThe electron spin operator S(q) is defined by\nS(q) =/summationdisplay\nlSle−iq·Rl,Sl=/summationdisplay\nα1α2c†\nlα2Sα2α1clα1,(4)\nwhereclαandc†\nlαaretheannihilationandcreationopera-\ntors ofthe Wannier electrons at position Rl, respectively,\nandS=σ\n2isthe spin angularmomentum operator. Here\nwe assume that the material electrons which have a dom-\ninant interaction with the incident neutrons are the local\nWannier electrons. It is noted that S⊥(q) is defined as\nS⊥(q) =S(q)−/hatwideq(S(q)·/hatwideq). (5)\nA simple review of the electron-neutron magnetic inter-\naction/hatwideVsis given in Appendix A.\nOne incident neutron with momentum qican be scat-\ntered by the material electrons into the state with mo-\nmentum qf. The relevant scattering probability is de-\nfined as\nΓ(1)(qf,qi) =1\nZ/summationdisplay\nαββiβfe−βEαP1(βi)\n×|/an}b∇acketle{tΦβ|/hatwideS(1)(+∞,−∞)|Φα/an}b∇acket∇i}ht|2,(6)\nwhere the initial state |Φα/an}b∇acket∇i}ht=|Ψα;qiβi/an}b∇acket∇i}htand the final\nstate|Φβ/an}b∇acket∇i}ht=|Ψβ;qfβf/an}b∇acket∇i}htand|Ψα/an}b∇acket∇i}htand|Ψβ/an}b∇acket∇i}htare the elec-\ntron eigenstates whose eigenvalues are EαandEβ, re-\nspectively. /hatwideS(1)(+∞,−∞) is the first-order expansion of\nthe time-evolution Smatrix ofthe perturbation electron-\nneutron magnetic interaction /hatwideVsand is defined as\n/hatwideS(1)(+∞,−∞) =−i\n/planckover2pi1/integraldisplay+∞\n−∞dt/hatwideVI(t)Fθ(t),(7)\nwhere/hatwideVI(t) =eiH0t//planckover2pi1/hatwideVse−iH0t//planckover2pi1, withH0being the sum\noftheHamiltoniansofthematerialelectronsandtheneu-\ntrons.Fθ(t) defines the interaction perturbation time,\nFθ(t) =θ(t+∆t/2)−θ(t−∆t/2),(8)whereθ(t) is the step function.\nIt should be noted that in the above scattering prob-\nability, we have defined implicitly the initial and final\nstates by the density matrices as follows:\n/hatwidePI=1\nZ/summationdisplay\nαβie−βEαP1(βi)|Ψα;qiβi/an}b∇acket∇i}ht/an}b∇acketle{tβiqi;Ψα|,\n/hatwidePF=/summationdisplay\nββf|Ψβ;qfβf/an}b∇acket∇i}ht/an}b∇acketle{tβfqf;Ψβ|. (9)\nInthispaperwewillfocusonthecaseswheretheincident\nneutronsarethethermalneutronsinthe spinmixedstate\ndefined by\n/summationdisplay\nβiP1(βi)|βi/an}b∇acket∇i}ht/an}b∇acketle{tβi|=1\n2(| ↑/an}b∇acket∇i}ht/an}b∇acketle{t↑ |+| ↓/an}b∇acket∇i}ht/an}b∇acketle{t↓ |).(10)\nWe introduce an imaginary-time Green’s function\nG(q,τ) =−/summationtext\nij/an}b∇acketle{tTτSi(q,τ)S†\nj(q,0)/an}b∇acket∇i}ht(δij−/hatwideqi/hatwideqj). Its\ncorresponding spectrum function χ(q,E) is defined as\nχ(q,E) =−2 ImG(q,iνn→E+iδ+), which follows\nχ(q,E) =2π\nZ/summationdisplay\nαβije−βEα/an}b∇acketle{tΨα|S†\ni(q)|Ψβ/an}b∇acket∇i}ht/an}b∇acketle{tΨβ|Sj(q)|Ψα/an}b∇acket∇i}ht\n×(δij−/hatwideqi/hatwideqj)n−1\nB(E)δ(E+Eβ−Eα).(11)\nThe scattering probability can easily be shown to follow\nΓ(1)(qf,qi) =|g(q)|2∆t\n/planckover2pi1χ(q,E(1))nB(E(1)),(12)\nwhere the transfer momentum and energy are defined as\nq=qf−qi,E(1)=E(qf)−E(qi),(13)\nwithE(q) =(/planckover2pi1q)2\n2mn(mnis the neutron mass), and nB(E)\nistheBosedistributionfunction. Intheabovederivation,\nwe have assumed that the time interval ∆ tis large and\nsin2(ax)\nx2→πaδ(x) whena→+∞.\nLet us consider the scattering cross section. We define\nthe incident neutron flux by JI=nIvI, where the den-\nsitynI=1\nVI(VIis the renormalization volume for one\nneutron) and the velocity vI=/planckover2pi1qi\nmn. The scattering cross\nsection per scatter σfollows\nJIσ=1\nNm∆t/summationdisplay\nqfΓ(1)(qf,qi), (14)\nwhereNmis the number of scatter electrons in the in-\ncident neutron beam. The double-differential scattering\ncross section is shown to follow\nd2σ\ndΩdEf=(γRe)2\n2πNmqf\nqi|F0(q)|2χ(q,E(1))nB(E(1)),(15)\nwhereEfis the energy of the scattered neutrons, γ=\n1.91 is a constant for the neutron gyromagnetic ratio,\nandReis the classical electron radius, defined as\nRe=µ0e2\n4πme=e2\n4πε0mec2, (16)3\nwithµ0being the free-space permeability and ε0being\nthe vacuum permittivity. This double-differential cross\nsectionwe haveobtainedis the same asthat from Fermi’s\ngolden rule5–7. Physically, the scattering probability and\nthe scattering cross section of INS come from the con-\ntribution of the first-order perturbation of the electron-\nneutron magnetic interaction.\nB. Theoretical formalism for cINS\nIn this section we will present a coincidence technique,\ncoincidence inelastic neutron scattering, which we call\ncINS.Itisproposedforthedetectionofthetwo-spinmag-\nnetic correlations of the target magnetic material. The\nschematic diagram of cINS is shown in Fig. 1. There are\ntwo neutron sources which emit two neutrons with mo-\nmentaqi1andqi2. These two neutrons are incident on\nthe target magnetic material and interact with the elec-\ntron spins. The two incident neutrons are then scattered\noutside of the material into the states with momenta qf1\nandqf2. Two single-neutron detectors detect the two\nscattered neutrons, and a coincidence detector records\nthe coincidence counting probability when each of the\ntwo single-neutron detectors detects one single neutron\nsimultaneously.\nFIG. 1: (Color online) Schematic diagram of cINS. The two\nred dashed lines represent two incident neutrons, and the tw o\ngreen solid lines represent two scattered neutrons. D 1and\nD2are two single-neutron detectors, and D 12is a coincidence\ndetector which records one counting when D 1and D 2each\ndetect one single neutron simultaneously.\nThe coincidence counting probability of the two scat-\ntered neutrons is described by\nΓ(2)(qf1qf2,qi1qi2) =1\nZ/summationdisplay\nαββiβfe−βEαP2(βi1,βi2)\n×|/an}b∇acketle{tΦβ|/hatwideS(2)(+∞,−∞)|Φα/an}b∇acket∇i}ht|2,(17)\nwhere the initial state |Φα/an}b∇acket∇i}ht=|Ψα;qi1βi1qi2βi2/an}b∇acket∇i}htand the\nfinal state |Φβ/an}b∇acket∇i}ht=|Ψβ;qf1βf1qf2βf2/an}b∇acket∇i}ht.P2(βi1,βi2) de-\nfines the spin distribution function of the incident ther-\nmal neutrons. In the following, we will consider the cases\nwithP2(βi1,βi2) =P1(βi1)P1(βi2)./hatwideS(2)(+∞,−∞) is the\nsecond-order expansion of the time-evolution Smatrixand is defined by\n/hatwideS(2)(+∞,−∞)\n=1\n2!/parenleftbigg\n−i\n/planckover2pi1/parenrightbigg2/integraldisplay/integraldisplay+∞\n−∞dt1dt2Tt[/hatwideVI(t1)/hatwideVI(t2)]Fθ(t1,t2).\n(18)\nHere the time function Fθ(t1,t2) is defined as Fθ(t1,t2) =\nFθ(t1)Fθ(t2). Physically, the coincidence probability of\ncINS is determined by the second-order perturbation of\nthe electron-neutron magnetic interaction.\nFollowing the theoretical treatment for cARPES10, we\nintroduce the two-spin Bethe-Salpeter wave function de-\nfined in Eq. (1). With the two-spin Bethe-Salpeter wave\nfunction, we can show that the coincidence probability of\ncINS can be expressed as\nΓ(2)= Γ(2)\n1+Γ(2)\n2, (19)\nwhere\nΓ(2)\n1=1\nZ/summationdisplay\nαββiβfe−βEαP1(βi1)P1(βi2)\n×1\n/planckover2pi14/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay/integraldisplay+∞\n−∞dt1dt2Mαβ,1(t1,t2)Fθ(t1,t2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n,\nΓ(2)\n2=1\nZ/summationdisplay\nαββiβfe−βEαP1(βi1)P1(βi2)\n×1\n/planckover2pi14/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay/integraldisplay+∞\n−∞dt1dt2Mαβ,2(t1,t2)Fθ(t1,t2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n.\nHere the matrix elements Mαβ,1andMαβ,2are defined\nas\nMαβ,1=g(q1)g(q2)/summationdisplay\nijφ(ij)\nαβ(q1t1,q2t2)\n×σ(i)\nβf1βi1σ(j)\nβf2βi2ei(E(2)\n1t1+E(2)\n2t2)//planckover2pi1,\nMαβ,2=g(q1)g(q2)/summationdisplay\nijφ(ij)\nαβ(q1t1,q2t2)\n×σ(i)\nβf1βi2σ(j)\nβf2βi1ei(E(2)\n1t1+E(2)\n2t2)//planckover2pi1,\nwhere the transfer momenta are defined as\nq1=qf1−qi1,q2=qf2−qi2,\nq1=qf1−qi2,q2=qf2−qi1,(20)\nand the transfer energies are defined as\nE(2)\n1=E(qf1)−E(qi1),E(2)\n2=E(qf2)−E(qi2),\nE(2)\n1=E(qf1)−E(qi2),E(2)\n2=E(qf2)−E(qi1).(21)\nPhysically, there are two different classes of microscopic\nneutron scattering processes involved in the coincidence\nscattering. One is with the state changes of the two neu-\ntronsas|qi1βi1/an}b∇acket∇i}ht → |qf1βf1/an}b∇acket∇i}htand|qi2βi2/an}b∇acket∇i}ht → |qf2βf2/an}b∇acket∇i}ht, and4\nthe other one is with |qi1βi1/an}b∇acket∇i}ht → |qf2βf2/an}b∇acket∇i}htand|qi2βi2/an}b∇acket∇i}ht →\n|qf1βf1/an}b∇acket∇i}ht. The matrix elements Mαβ,1andMαβ,2and the\ncorresponding coincidence probabilities Γ(2)\n1and Γ(2)\n2de-\nscribe these two different classes of microscopic neutron\nscattering processes, respectively. It should be noted\nthat here we have ignored the quantum interference of\nthese two different scattering contributions as they come\nfrom different scattering channels of energy transfer with\nenergy-conservation-like resonance features at different\nenergies.\nWe define the center-of-mass time tc=1\n2(t1+t2)\nand the relative time tr=t1−t2and denote the two-\nspin Bethe-Salpeter wave function φ(ij)\nαβ(q1,q2;tc,tr) =\nφ(ij)\nαβ(q1t1,q2t2). We can introduce the Fourier transfor-\nmations of φ(ij)\nαβ(q1,q2;tc,tr) as follows:\nφ(ij)\nαβ(q1,q2;tc,tr)\n=/integraldisplay/integraldisplay+∞\n−∞dΩdω\n(2π)2φ(ij)\nαβ(q1,q2;Ω,ω)e−iΩtc−iωtr,\nφ(ij)\nαβ(q1,q2;Ω,ω)\n=/integraldisplay/integraldisplay+∞\n−∞dtcdtrφ(ij)\nαβ(q1,q2;tc,tr)eiΩtc+iωtr.\nFor the incident thermal neutrons in the spin mixed state\ndefined by Eq. (10), the coincidence probability is shown\nto follow\nΓ(2)=1\n/planckover2pi141\nZ/summationdisplay\nαβije−βEα\n×[C1/vextendsingle/vextendsingleφ(ij)\nαβ,1(q1,q2)/vextendsingle/vextendsingle2+C2/vextendsingle/vextendsingleφ(ij)\nαβ,2(q1,q2)/vextendsingle/vextendsingle2],(22)\nwhere the two factors are defined as\nC1=|g(q1)g(q2)|2,C2=|g(q1)g(q2)|2,(23)\nand the two wave functions φ(ij)\nαβ,1(q1,q2) and\nφ(ij)\nαβ,2(q1,q2) are defined as\nφ(ij)\nαβ,1(q1,q2)\n=/integraldisplay/integraldisplay+∞\n−∞dΩdω\n(2π)2φ(ij)\nαβ(q1,q2;Ω,ω)Y1(Ω,ω),(24)\nφ(ij)\nαβ,2(q1,q2)\n=/integraldisplay/integraldisplay+∞\n−∞dΩdω\n(2π)2φ(ij)\nαβ(q1,q2;Ω,ω)Y2(Ω,ω).(25)Here the functions Y1(Ω,ω) andY2(Ω,ω) are given by\nY1(Ω,ω) =sin[(E(2)\n1//planckover2pi1−Ω/2−ω)∆t/2]\n(E(2)\n1//planckover2pi1−Ω/2−ω)/2\n×sin[(E(2)\n2//planckover2pi1−Ω/2+ω)∆t/2]\n(E(2)\n2//planckover2pi1−Ω/2+ω)/2,(26)\nY2(Ω,ω) =sin[(E(2)\n1//planckover2pi1−Ω/2−ω)∆t/2]\n(E(2)\n1//planckover2pi1−Ω/2−ω)/2\n×sin[(E(2)\n2//planckover2pi1−Ω/2+ω)∆t/2]\n(E(2)\n2//planckover2pi1−Ω/2+ω)/2.(27)\nIn large, but finite, ∆ t, we can make the approxima-\ntion that/integraltext∆t/2\n−∆t/2dt2/integraltext∆t/2\n−∆t/2dt1→/integraltext∆t/2\n−∆t/2dtc/integraltext∆t/2\n−∆t/2dtr.\nIn this case the functions Y1(Ω,ω) andY2(Ω,ω) can be\napproximated as\nY1(Ω,ω) =sin[(Ω−E(2)\n1//planckover2pi1−E(2)\n2//planckover2pi1)∆t/2]\n(Ω−E(2)\n1//planckover2pi1−E(2)\n2//planckover2pi1)/2\n×sin[(ω−E(2)\n1/2/planckover2pi1+E(2)\n2/2/planckover2pi1)∆t/2]\n(ω−E(2)\n1/2/planckover2pi1+E(2)\n2/2/planckover2pi1)/2,(28)\nY2(Ω,ω) =sin[(Ω−E(2)\n1//planckover2pi1−E(2)\n2//planckover2pi1)∆t/2]\n(Ω−E(2)\n1//planckover2pi1−E(2)\n2//planckover2pi1)/2\n×sin[(ω−E(2)\n1/2/planckover2pi1+E(2)\n2/2/planckover2pi1)∆t/2]\n(ω−E(2)\n1/2/planckover2pi1+E(2)\n2/2/planckover2pi1)/2.(29)\nIn the limit with ∆ t→+∞, it can be shown that\nΓ(2)=1\n/planckover2pi141\nZ/summationdisplay\nαβije−βEα[C1/vextendsingle/vextendsingleφ(ij)\nαβ(q1,q2;Ω1,ω1)/vextendsingle/vextendsingle2+C2/vextendsingle/vextendsingleφ(ij)\nαβ(q1,q2;Ω2,ω2)/vextendsingle/vextendsingle2], (30)5\nwhere the transfer frequencies are defined as\nΩ1=1\n/planckover2pi1(E(2)\n1+E(2)\n2),ω1=1\n2/planckover2pi1(E(2)\n1−E(2)\n2),Ω2=1\n/planckover2pi1(E(2)\n1+E(2)\n2),ω2=1\n2/planckover2pi1(E(2)\n1−E(2)\n2). (31)\nThe coincidence probability Γ(2)in Eq. (30) shows that cINS can explicitly detect the frequency Bet he-Salpeter wave\nfunction, which describes the dynamical magnetic physics of the ta rget material with two-spin excitations involved.\nThis can be seen more clearly from the following spectrum expression of the frequency Bethe-Salpeter wave function:\nφ(ij)\nαβ(q1,q2;Ω,ω) = 2πδ[Ω+(Eβ−Eα)//planckover2pi1]φ(ij)\nαβ(q1,q2;ω), (32)\nwhereφ(ij)\nαβ(q1,q2;ω) follows\nφ(ij)\nαβ(q1,q2;ω) =/summationdisplay\nγ/bracketleftBigg\ni/an}b∇acketle{tΨβ|S(i)\n⊥(q1)|Ψγ/an}b∇acket∇i}ht/an}b∇acketle{tΨγ|S(j)\n⊥(q2)|Ψα/an}b∇acket∇i}ht\nω+iδ++(Eα+Eβ−2Eγ)/2/planckover2pi1−i/an}b∇acketle{tΨβ|S(j)\n⊥(q2)|Ψγ/an}b∇acket∇i}ht/an}b∇acketle{tΨγ|S(i)\n⊥(q1)|Ψα/an}b∇acket∇i}ht\nω−iδ+−(Eα+Eβ−2Eγ)/2/planckover2pi1/bracketrightBigg\n.(33)\nObviously, the frequency Bethe-Salpeter wave func-\ntion involves the following dynamical magnetic physics\nof two spins of the target magnetic material: (1)\nthe center-of-mass dynamics of the two spins de-\nscribed byδ[Ω+(Eβ−Eα)//planckover2pi1], which shows the trans-\nfer energy conservation with the center-of-mass de-\ngrees of freedom involved, and (2) the two-spin\nrelative dynamics φ(ij)\nαβ(q1,q2;ω), which has reso-\nnance structures peaked at ∓(Eα+Eβ−2Eγ)/2/planckover2pi1\nwith weights /an}b∇acketle{tΨβ|S(i)\n⊥(q1)|Ψγ/an}b∇acket∇i}ht/an}b∇acketle{tΨγ|S(j)\n⊥(q2)|Ψα/an}b∇acket∇i}htand\n/an}b∇acketle{tΨβ|S(j)\n⊥(q2)|Ψγ/an}b∇acket∇i}ht/an}b∇acketle{tΨγ|S(i)\n⊥(q1)|Ψα/an}b∇acket∇i}ht, respectively. There-\nfore, cINS can provide the momentum-resolved dynami-\ncal two-spin magnetic correlations of the target magnetic\nmaterial.\nIII. COINCIDENCE PROBABILITIES OF THE\nFERROMAGNET AND ANTIFERROMAGNET\nIn this section we will study the coincidence probabil-\nities of cINS for a ferromagnet and an antiferromagnet\nwhich have long-range magnetic order with well-defined\nmagnon excitations.\nProvided that (1) the two incident neutrons are in-\ndependent following a spin distribution function as\nP2(βi1,βi2) =P1(βi1)P1(βi2) and (2) the single-spin\nmagnetic excitations of the target material have well-\ndefined momenta and are decoupled from each other, the\ncoincidence probability of cINS has a simple product be-\nhavior, which can be expressed mathematically as\nΓ(2)= Γ(1)(qf1,qi1)·Γ(1)(qf2,qi2)\n+ Γ(1)(qf1,qi2)·Γ(1)(qf2,qi1).(34)\nThis is a general result which can be exactly proven from\nthe definitions of the scattering probability of INS and\nthe coincidence probability of cINS, Eq. (6) and (17).\nWe will consider localized spin magnetic systems with\na cubic crystal lattice, the Hamiltonians of which aredefined by\nH=J\n2/summationdisplay\nlδSl·Sl+δ, (35)\nwhereδ=±aex,±aey,±aez. The localized spins are\nin a low-temperature ordering state with the magnetic\nmoments ordered along the ezaxis.\nA. Ferromagnet\nLet us consider a ferromagnet with J <0. We in-\ntroduce the Holstein-Primakoff transformation, S+\nl=/radicalBig\n2S−a†\nlalal,S−\nl=a†\nl/radicalBig\n2S−a†\nlal,Sz\nl=S−a†\nlal, where\nalanda†\nlare the bosonic magnon operators. In linear\nspin-wave theory, the spin Hamiltonian can be approxi-\nmated as\nHFM=/summationdisplay\nkεka†\nkak, (36)\nwhereεk=|J|zS(1−γk), withγk=1\nz/summationtext\nδeik·δand\ncoordination number z= 6. Hereak=1√\nN/summationtext\nlale−ik·Rl.\nLet us first study the scattering probability of the\nsingle-spin INS. Suppose the incident thermal neutrons\nare in the spin mixed state defined by Eq. (10). It can\nbe shown from Eq. (12) that the scattering probability\nΓ(1)follows\nΓ(1)\nFM(qf,qi)\n=|g(q)|2∆t\n/planckover2pi1nB(E(1))/bracketleftBig\nχxx(q,E(1))(1−/hatwideq2\nx)\n+χyy(q,E(1))(1−/hatwideq2\ny)+χzz(q,E(1))(1−/hatwideq2\nz)/bracketrightBig\n,\n(37)6\nwhere the spin spectrum functions χii(q,E) are given by\nχxx(q,E) =χyy(q,E) =πNS[δ(E−εq)−δ(E+ε−q)],\nχzz(q,E) = 2π/summationdisplay\nk[nB(εk)−nB(εk+q)]δ(E+εk−εk+q).\n(38)\nHere the transfer momentum and energy, qandE(1),\nare defined as in Eq. (13). While the transverse spin\nflips lead to single-magnon peak structures in the scat-\ntering probability, the longitudinal spin fluctuations con-\ntribute magnon density fluctuations. Besides these in-\nelastic scattering contributions, there is one additional\nelastic scattering contribution from the magnon conden-\nsation, which gives\nΓ(1)\nFM,c=2π|g(q)|2∆t\n/planckover2pi1(NmFM)2δ(E(1))δq,0(1−/hatwideq2\nz),\n(39)\nwheremFM=1\nN/summationtext\nl/an}b∇acketle{tSz\nl/an}b∇acket∇i}htis the ordered spin magnetic\nmoment per site. It is noted that in experiment, Nis the\nnumber of local Wannier electron spins in the incident\nneutronbeam. Whenconsideringonlythesingle-magnon\ncontributions without that of the magnon density fluctu-\nations, the inelastic scattering probability of INS for the\nordered ferromagnet follows\nΓ(1)\nFM(qf,qi) =πNS|g(q)|2∆t\n/planckover2pi1nB(E(1))(1+/hatwideq2\nz)\n×[δ(E(1)−εq)−δ(E(1)+ε−q)].(40)\nNow let us study the coincidence probability of cINS\nfor the ordered ferromagnet. Suppose the two incident\nthermal neutrons with momenta qi1andqi2are scat-\ntered into the states with momenta qf1andqf2and the\nincident neutrons are in spin states with P2(βi1,βi2) =\nP1(βi1)P1(βi2) andP1(βi) defined in Eq. (10). Since\nthe magnons are well-defined single-spin excitations with\nthe momentum being a good quantum number, the coin-\ncidence probability of cINS for the ordered ferromagnet\nwith only contributions from the single-magnon excita-\ntions has a product behavior described by Eq. (34), i.e.,\nΓ(2)\nFM= Γ(1)\nFM(qf1,qi1)·Γ(1)\nFM(qf2,qi2)\n+ Γ(1)\nFM(qf1,qi2)·Γ(1)\nFM(qf2,qi1),(41)\nwhere the four Γ(1)\nFM(qf,qi)’s are the scattering prob-\nabilities of the single-magnon relevant INS defined inEq. (40). The magnon density fluctuations are not well-\ndefined excitations, and their contribution would break\ndown this simple product behavior.\nB. Antiferromagnet\nNow let us consider an antiferromagnet in a cubic\ncrystal lattice with long-range magnetic order. It has\na spin lattice Hamiltonian defined by Eq. (35) with\nJ >0. We introduce the spin rotation transformation\nasSx\nl=eiQ·RlSx\nl,Sy\nl=Sy\nl,Sz\nl=eiQ·RlSz\nl, where\nQ= (π/a,π/a,π/a ) is the characteristic antiferromag-\nnetic momentum. Introducing the Holstein-Primakoff\ntransformation for the new spin operators, the spin\nHamiltonian can be approximated in a linear spin-wave\ntheory as\nHAF=/summationdisplay\nk′\nψ†\nk/parenleftbigg\nA Bk\nBkA/parenrightbigg\nψk,ψk=/parenleftbiggak\na†\n−k/parenrightbigg\n,(42)\nwhereA=JzS,Bk=−JzSγk, andψkis a bosonic\nNambu spinor operator. Here the sum over kinvolves\neach pair ( k,−k) once. With the canonical transforma-\ntion\n/parenleftbiggak\na†\n−k/parenrightbigg\n=/parenleftbigg\nukvk\nvkuk/parenrightbigg/parenleftbiggβk\nβ†\n−k/parenrightbigg\n,(43)\nthe Hamiltonian can be diagonalized into the form\nHAF=/summationdisplay\nk′Ek(β†\nkβk+β†\n−kβ−k),(44)\nwhereEk=/radicalbig\nA2−B2\nk. Hereu2\nk=A+Ek\n2Ek,v2\nk=A−Ek\n2Ek,\nandukvk=−Bk\n2Ek.\nIt can easily be shown that the neutron scatter-\ning probability of INS for the ordered antiferromagnet\nΓ(1)\nAF(qf,qi) follows an expression similar to Eq. (37)\nfor Γ(1)\nFM(qf,qi), with the corresponding spin spectrum\nfunctionsχii(q,E) given by\nχxx(q,E) =χyy(q,E)\n=πNSA+Bq\nEq[δ(E−Eq)−δ(E+Eq)] (45)\nand\nχzz(q,E) = 2π/summationdisplay\nk/braceleftBig\n[C(1)\nkqδ(E+εk−εk+q+Q)−C(4)\nkqδ(E−εk+εk+q+Q)][nB(εk)−nB(εk+q+Q)]\n+ [C(2)\nkqδ(E−εk−εk+q+Q)−C(3)\nkqδ(E+εk+εk+q+Q)][1+nB(εk)+nB(εk+q+Q)]/bracerightBig\n.(46)\nHereC(1)\nkq=u2\nk+q+Qu2\nk+uk+q+Qvk+q+Qukvk,C(2)\nkq=u2\nk+q+Qv2\nk+uk+q+Qvk+q+Qukvk,C(3)\nkq=v2\nk+q+Qu2\nk+7\nuk+q+Qvk+q+Qukvk, andC(4)\nkq=v2\nk+q+Qv2\nk+uk+q+Qvk+q+Qukvk. Similar to the ordered ferromagnet, there is\nalso one additional elastic scattering contribution due to the magno n condensation,\nΓ(1)\nAF,c=2π|g(q)|2∆t\n/planckover2pi1(NmAF)2δ(E(1))δq,Q(1−/hatwideq2\nz), (47)\nwheremAF=1\nN/summationtext\nleiQ·Rl/an}b∇acketle{tSz\nl/an}b∇acket∇i}htis the ordered antiferromagnetic moment per site. Here the trans fer momentum and\nenergy,qandE(1), are also defined in Eq. (13). In the approximation with only the single -magnon contributions,\nthe inelastic scattering probability of INS for the ordered antiferr omagnet follows\nΓ(1)\nAF(qf,qi) =πNS|g(q)|2∆t\n/planckover2pi1A+Bq\nEqnB(E(1))(1+/hatwideq2\nz)[δ(E(1)−Eq)−δ(E(1)+Eq)]. (48)\nNow let us consider cINS with the thermal neutrons\nwhich have initial incident momenta qi1andqi2and fi-\nnal scattered momenta qf1andqf2. The incident neu-\ntrons are independent, with the spin state defined by Eq.\n(10). In linear spin-wave theory defined by the approxi-\nmateHamiltonian(42), theNambuspinoroperatorswith\ndifferent momenta are decoupled. This means that the\nsingle-magnon excitations in the ordered antiferromag-\nnet are decoupled. Therefore, in the linear spin-wave\ntheory with only contributions from the single-magnon\nexcitations, the conditions for the product behavior of\nthe coincidence probability in Eq. (34) are also satisfied\nin the ordered antiferromagnet. In this approximation\nthe coincidence probability of cINS for the ordered an-\ntiferromagnet follows a similar product behavior defined\nas\nΓ(2)\nAF= Γ(1)\nAF(qf1,qi1)·Γ(1)\nAF(qf2,qi2)\n+ Γ(1)\nAF(qf1,qi2)·Γ(1)\nAF(qf2,qi1),(49)\nwhere the four Γ(1)\nAF(qf,qi)’s are the scattering proba-\nbilities of the single-magnon relevant INS defined in Eq.\n(48).\nIV. DISCUSSION AND SUMMARY\nIn this paper we have proposed a coincidence tech-\nnique, cINS, which has two neutron sources and two neu-\ntron detectors, with an additional coincidence detector.\nThe two neutron sources emit two neutrons which are\nscattered by the electron spins of the magnetic material\nand are then detected by the two neutron detectors. The\ncoincidence detector records the coincidence probability\nofthe two scatteredneutrons, which givesinformation on\na two-spin Bethe-Salpeter wave function. This two-spin\nBethe-Salpeter wave function defines the momentum-\nresolved dynamical wave function of the magnetic ma-\nterial with two spins excited. Thus, cINS can explicitly\ndetect the two-spin magnetic correlations of the mag-\nnetic material. The coincidence probabilities of cINS for\na ferromagnet and an antiferromagnet with long-range\nmagnetic order have been calculated and show a prod-\nuct behavior contributed by the single-magnon relevantINSs. This trivial product behavior for the ordered ferro-\nmagnet and antiferromagnet is consistent with the mag-\nnetic properties dominated by the nearly free magnon\nexcitations, which have no intrinsic two-spin magnetic\ncorrelations.\nOn the experimental instrument of cINS, we remark\nthat the two incident neutrons can come from one neu-\ntron source. In this case the initial momenta of the two\nincident neutrons follow qi2=qi1+δq, withδq→0.\nThese two incident neutrons can be regarded equiva-\nlently to be emitted from two different neutron sources\nbut with nearly the same momenta. Thus, the theoret-\nical formalism for cINS with one neutron source can be\nsimilarly established following the one we established in\nSec. IIB for cINS with two neutron sources. There are\ntwo main challenges in the experimental realization of\ncINS. One is to develop a two-neutron coincidence detec-\ntor, and the other one is accurate control of the coinci-\ndence detection. The two-photon coincidence measure-\nment in modern quantum optics11and the coincidence\ndetection of the photoelectron and the Auger electron in\ndouble-photoemission spectroscopy12may provide a use-\nful guideline.\nThe cINS we have proposed is one potential technique\nto study novel magnetic correlations which are far be-\nyond the physics of the single-spin magnons. For exam-\nple, the long-sought quantum spin liquids1–4from strong\nfrustrationand quantum fluctuations show novelphysics,\nsuch as various spin valence bond states13–16and novel\nquantum criticality17. Experimental study of the spin\nvalence bond states by cINS would provide new insights\ninto quantum spin liquids. The various quantum mag-\nnetic materials with spin dimers, such as TlCuCl 318,\nSrCu2(BO3)219, and BaCuSi 2O620, could be the first fo-\ncus in a cINS experiment. Quantum spin liquid materials\nin triangular, honeycomb, kagome, and hyperkagome lat-\ntices (e.g., the materials reviewed in Ref. [4,21]) are also\ninteresting target materials for a cINS experiment.\nIn summary, we have proposed a coincidence tech-\nnique, cINS, which can explicitly detect the two-spin\nmagnetic correlations of magnetic materials. It can be\nintroduced to study the dynamical physics of the spin\nvalence bond states of quantum magnets.8\nACKNOWLEDGMENTS\nWe thank H. Shao and D. Z. Cao for invaluable discus-\nsions. This work was supported by the National Natural\nScience Foundation of China (Grants No. 11774299 and\nNo. 11874318) and the Natural Science Foundation of\nShandong Province (Grants No. ZR2017MA033 and No.\nZR2018MA043).\nAppendix A: Electron-neutron magnetic interaction\nLet us review the electron-neutron magnetic\ninteraction5–7. We define the neutron spin mag-\nnetic moment as µn=−γµNσ, whereγ= 1.91 is a\nconstant for the neutron gyromagnetic ratio, µN=e/planckover2pi1\n2mp\nis the nuclear magneton, with mpbeing the proton mass,\nandσis the Pauli matrix. We define the electron spin\nmagnetic moment as µs=−gsµBSand the electron\norbital magnetic moment as µl=−glµBL, where the\ngfactors are set as gs= 2 andgl= 1 andµB=e/planckover2pi1\n2meis the Bohr magneton. The spin angular momentum\noperator Shas eigenvalues ±1\n2, and the orbital angular\nmomentum operator is defined as L=1\n/planckover2pi1re×pe. Suppose\nthere is an electron at position rewhich can produce a\nmagnetic field at position rnas\nB=µ0\n4π∇×/bracketleftbigg\n(µs+µl)×R\nR3/bracketrightbigg\n,(A1)\nwhereµ0is the free-space permeability and R=rn−re.\nTheelectron-neutronmagneticinteractioncanbedefined\nbyV=−µn·B, which follows\nV=−µ0\n4πγµNµBσ·∇×/bracketleftbigg\n(gsS+glL)×R\nR3/bracketrightbigg\n.(A2)\nHere we have introduced the orbital angular momentum\nLto describe the orbital motions of the electrons7. It\nis more convenient in the study of the orbital motions\nof electrons in compounds with transition metal and/or\nrare earth atoms.\nLet us present the second quantization of the electron-\nneutron magnetic interaction. Introduce the single-\nneutronstates {|qβ/an}b∇acket∇i}ht}, whereqisthe neutronmomentum\nandβdefines the neutron spin, and the single-electron\nstates{|λ/an}b∇acket∇i}ht}, whereλinvolves the momentum, orbital,\nand spin degrees of freedom, etc. Let us introduce the\nfollowing identities:\n1 =1\nV1/integraldisplay\ndre|re/an}b∇acket∇i}ht/an}b∇acketle{tre|\nfor the electrons, and\n1 =1\nV2/integraldisplay\ndrn|rn/an}b∇acket∇i}ht/an}b∇acketle{trn|\nfor the neutrons. Here V1andV2are the renormalization\nvolumes for the single-electron and single-neutron states,respectively. The electron-neutron magnetic interaction\nin second quantization can be expressed as\n/hatwideV=/hatwideVs+/hatwideVl, (A3)\nwhere\n/hatwideVs=4πAs\nV2/summationdisplay\nqiqf/hatwideσqfqi·[/hatwideq×(Ds(q)×/hatwideq)],(A4)\n/hatwideVl=4πAl\nV2/summationdisplay\nqiqf/hatwideσqfqi·[/hatwideq×(Dl(q)×/hatwideq)].(A5)\nHere the momentum q=qf−qi, and/hatwideq=q\nq. It is noted\nthat/hatwideq×(D(q)×/hatwideq) can be reexpressed as D⊥(q):\nD⊥(q) =D(q)−/hatwideq(D(q)·/hatwideq). (A6)\nIn the electron-neutron magnetic interaction /hatwideV, the con-\nstantsAsandAlare defined as\nAs=−µ0\n4πγgsµNµB,Al=−µ0\n4πγglµNµB,(A7)\nand the operator /hatwideσqfqiis defined as\n/hatwideσqfqi=/summationdisplay\nβiβfd†\nqfβfσβfβidqiβi, (A8)\nwheredqβandd†\nqβare the annihilation and creation op-\nerators for the neutrons. The operators DsandDlin/hatwideV\nare defined as\nDs(q) =/summationdisplay\nλ1λ2c†\nλ2M(s)\nλ2λ1(q)cλ1,(A9)\nDl(q) =/summationdisplay\nλ1λ2c†\nλ2M(l)\nλ2λ1(q)cλ1,(A10)\nwherecλandc†\nλare the annihilation and creation oper-\nators for the electrons and\nM(s)\nλ2λ1(q) =1\nV1/integraldisplay\ndre[ψ∗\nλ2(re)Sψλ1(re)]e−iq·re,\nM(l)\nλ2λ1(q) =1\nV1/integraldisplay\ndre[ψ∗\nλ2(re)Lψλ1(re)]e−iq·re.\nHereψλ(re) is the single-electron wave function.\nLet us focus on the spin degrees of freedom of the elec-\ntrons and ignore the orbital ones. We consider the elec-\ntrons to be in the local Wannier states {|lα/an}b∇acket∇i}ht}with posi-\ntionRland spinα.Ds(q) can be approximately defined\nas\nDs(q) =F0(q)S(q), (A11)\nwhere the spin operator S(q) is defined as\nS(q) =/summationdisplay\nlSle−iq·Rl,Sl=/summationdisplay\nα1α2c†\nlα2Sα2α1clα1,(A12)9\nand the magnetic form factor F0(q) is given by\nF0(q) =1\nV1/integraldisplay\ndaψ∗\nl(a)ψl(a)e−iq·a,a=re−Rl.(A13)\nHere we have made an approximation to consider only\nthe on-site intraorbital integrals and ignore all the other\ncontributions. For the itinerant electrons in the Bloch\nstates{|kα/an}b∇acket∇i}ht}, the operator Ds(q) can be given by\nDs(q) =/summationdisplay\nk1k2Fk2k1(q)Sk2k1, (A14)\nwhere the spin operator is defined by\nSk2k1=/summationdisplay\nα1α2c†\nk2α2Sα2α1ck1α1,(A15)\nand the form factor Fk2k1(q) is given by\nFk2k1(q) =1\nV1/integraldisplay\ndreψ∗\nk2(re)ψk1(re)e−iq·re.(A16)\nHereψk(re) is the Bloch-state wave function. In the\napproximation with ψk(re) =eik·re,Ds(q) can be sim-\nplified as\nDs(q) =/summationdisplay\nkSk,k+q. (A17)\nIn summary, the electron-neutronmagneticinteraction\nwith only the spin degreesof freedom of the electrons can\nbe given as follows. For the local Wannier electrons,\n/hatwideVs=/summationdisplay\nqiqfg(q)/hatwideσqfqi·S⊥(q), (A18)\nwhereg(q)≡gF0(q), withg=4πAs\nV2, andS⊥(q) is the\nprojection of S(q) in the perpendicular plane normal to\nthe momentum qand is defined similarly to D⊥(q) in\nEq. (A6). For the itinerant Bloch electrons,\n/hatwideVs=/summationdisplay\nqiqfk1k2gk2k1(q)/hatwideσqfqi·Sk2k1,⊥,(A19)\nwheregk2k1(q)≡gFk2k1(q) andSk2k1,⊥is defined simi-\nlarly toD⊥(q) in Eq. (A6). It should be noted that the\nform factors F0(q) andFk2k1(q) have strong qdepen-\ndence.\nOne remark is that in the above electron-neutronmag-\nnetic interaction, the contributions from the spin and or-\nbital magnetic moments are independently derived. In\nthis case, the spin-orbit coupling is weak like for the elec-\ntrons of the transition metal atoms. In the case with\nstrong spin-orbit coupling such as that of the electrons\nof rare earth atoms, the total angular momentum Jis\nconserved. In this case we can introduce the total mag-\nnetic moment µJ=−g(JLS)µBJ, with the Land´ e gfac-\ntorg(JLS) defined following glL+gsS=g(JLS)J. A\nsimilar derivation can give us an electron-neutron mag-\nnetic interaction in this case. Another remark is that the\nDebye-Waller factor5,6from the crystal lattice effects is\nignored in our discussion on the neutron scattering prob-\nability of the inelastic neutron scattering.Appendix B: Calculations for scattering probability\nof INS\nLet us introduce the imaginary-time Green’s functions\nGij(q,τ) =−/an}b∇acketle{tTτSi(q,τ)S†\nj(q,0)/an}b∇acket∇i}htwithi,j=x,y,z.\nThe corresponding spectrum functions are defined as\nχij(q,E) =−2 ImGij(q,iνn→E+iδ+). Then we have\nG(q,τ) =/summationdisplay\nijGij(q,τ)(δij−/hatwideqi/hatwideqj),(B1)\nand\nχ(q,E) =/summationdisplay\nijχij(q,E)(δij−/hatwideqi/hatwideqj).(B2)\nFirst, let us consider the ferromagnet in a cubic crys-\ntal lattice with a long-range magnetic order. We intro-\nduce the imaginary-time Green’s function for the ferro-\nmagnetic magnons, Ga(q,τ) =−/an}b∇acketle{tTτak(τ)a†\nk(0)/an}b∇acket∇i}ht. Its fre-\nquency Fourier transformation is given by\nGa(k,iνn) =1\niνn−εk, (B3)\nwhere the magnon energy dispersion εkis defined in Eq.\n(36). It can be shown that in the linear spin-wave ap-\nproximation,\nGxx(q,iνn) =Gyy(q,iνn)\n=NS\n2[Ga(q,iνn)+Ga(−q,−iνn)] (B4)\nand\nGzz(q,iνn) =−1\nβ/summationdisplay\nk,iν1Ga(k+q,iν1+iνn)Ga(k,iν1).\n(B5)\nThe other Green’s functions follow\nGij(q,iνn) = 0,fori/ne}ationslash=j. (B6)\nFrom these results, we can obtain the spectrum functions\nχij(q,E) in Eq. (38) for the ordered ferromagnet.\nNow let us consider the antiferromagnet in a cubic\ncrystal lattice with a long-range magnetic order. We in-\ntroduce the imaginary-time Green’s function of a Nambu\nspinor operator,\nGψ(k,τ) =−/an}b∇acketle{tTτψk(τ)ψ†\nk(0)/an}b∇acket∇i}ht, (B7)\nwhereψkis defined in Eq. (42). It can be shown that\nthe frequency Green’s function follows\nGψ(k,iνn) =iνnτ3+A−Bkτ1\n(iνn)2−E2\nk,(B8)\nwhereAandBkare defined in Eq. (42) and the magnon\nenergyEkis given in Eq. (44). Here τi(i= 1,2,3) are\nthe Pauli matrices.10\nIt can be shown that\nGxx(q,iνn) =NS\n2Tr[Gψ(q+Q,iνn)+Gψ(q+Q,iνn)τ1],\nGyy(q,iνn) =NS\n2Tr[Gψ(q,iνn)−Gψ(q,iνn)τ1],(B9)and\nGzz(q,iνn) =−1\nβ/summationdisplay\nk,iν1[G(11)\nψ(k+q+Q,iν1+iνn)G(11)\nψ(k,iν1)+G(21)\nψ(k+q+Q,iν1+iνn)G(12)\nψ(k,iν1)].(B10)\nThe other Green’s functions Gij(q,iνn) = 0 for the cases\nwithi/ne}ationslash=j. With these results, we can obtain the spec-trum functions χij(q,E) in Eqs. 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Harrison, C. D.\nBatista, N. Kawashima, Y. Kazuma, G. A.\nJorge, R. Stern, I. Heinmaa, S. A. Zvyagin,\net al., Phys. Rev. Lett. 93, 087203 (2004), URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.93.087 203.\n21J. R. Chamorro, T. T. Tran, and T. M.\nMcQueen, arXiv:2006.10882 (2020), URL\nhttps://ui.adsabs.harvard.edu/abs/2020arXiv20061088 2C."    },    {        "title": "0904.3533v1.Dynamic_equations_for_three_different_qudits_in_a_magnetic_field.pdf",        "content": "arXiv:0904.3533v1  [quant-ph]  22 Apr 2009Dynamic equations for three different qudits in a magnetic fie ld\nE. A. Ivanchenko∗\nNational Science Center “Kharkov Physicotechnical\nInstitute”, Institute for Theoretical Physics,\nAkademicheskaya Str. 1, Kharkov 61108, Ukraine\n(Dated: October 29, 2018)\nAbstract\nA closed system of equations for the local Bloch vectors and s pin correlation functions of three\nmagnetic qudits, which are in an arbitrary, time-dependent , external magnetic field, is obtained\nusingdecomplexification oftheLiouville-von Neumannequa tion. Thealgorithmofthederivationof\nthe dynamic equations is presented. In the basis convenient for the important physical applications\nstructure constants of algebra su(2S+1) are calculated.\nPACS numbers: 03.67.-a, 03.67.Ac, 03.67.Mn\n∗yevgeny@kipt.kharkov.ua\n1I. INTRODUCTION\nAt present it is not known which experiment will lead to the reliable, pro totypical quan-\ntum computing device. Quantum systems with two states, called qub its, are taken to be the\nbasic unit for quantum information processing. The quantum syste ms realised on coupled\nqubits are widely investigated for the purpose of creation of a quan tum computer. In the\nsame time there is a possibility to solve this problem, using the systems consisting from cou-\npled qudits (multilevel systems). Qudits possess a number of prope rties which differ from\nproperties qubits and may have advantages for quantum informat ion processing. For exam-\nple, two qutrits can be entangled more strongly than two qubits [1]. The main purpose of\nthis study is the derivation of dynamic equations for three qudit sys tem, which is placed in a\nmagnetic field, for entanglement study [2] and other applications. T he article is organized as\nfollows. Section II contains the model Hamiltonian including the three -particle interactions.\nIn section III the Liouville-von Neumann equation for the matrix den sity of three qudits in\nthe variable magnetic field is obtained in the Bloch’s representation in t erms of local vectors\nand spin correlation functions taking into account three-particle in teractions. We describe\nconservation laws which control numerical calculations effectively. The Appendix describes\nthe Hermitian basis and analytic formulas for structure constants .\nII. MODEL HAMILTONIAN\nThe Hamiltonian of three coupled different qudits (particles with spin 1 /2,1,3/2,2, ...)\nin the external ac magnetic field h= (h1,h2,h3), where h1,h2,h3are the Cartesian com-\nponents of the external magnetic field, we will present in a form of d ecomposition on a\ncomplete set orthogonal Hermitian matrices Cα, operating in the Hilbert space of three\nquditsHS1⊗HS2⊗HS3with (2S1+1)(2S2+1)(2S3+1) dimension\nˆH=1\n2h̺στCS1\n̺⊗CS2\nσ⊗CS3\nτ, (1)\nwhereS1,S2,S3are qudit spins, ⊗- denotes a direct product. CS1̺,CS2σ,CS3τare the Hermi-\ntian matrices operating in spaces HS1,HS2,HS3accordingly (see Appendix). The functions\nh̺στcontain one- , two- and three-qudit interactions [3], [4], [5].\n2III. DECOMPLEXIFICATIN OF THE LIOUVILLE-VON NEUMANN EQUA-\nTION\nThe Liouville-von Neumann equation for the density matrix ρ, describing the dynamics\nof a 3-qudit system, has the form\ni∂tρ= [ˆH,ρ], ρ(0) =ρ0, ρ+=ρ,Trρ= 1. (2)\nIt is convenient to rewrite the equation Eq. (2) having presented t he density matrix ρ\nas well as the Hamiltonian ˆH, in the form of decomposition on a complete set orthogonal\nHermitian matrices Cα\nρ=1\nc1(2S1+1)c2(2S2+1)c3(2S3+1)RαβγCS1\nα⊗CS2\nβ⊗CS3\nγ, (3)\nwherec1,2,3=/radicalBig\nS1,2,3(S1,2,3+1)\n3. We define the three coherence Bloch vectors Rm00,R0n0,R00p\n, which are widely used in the theory of magnetic resonance charact erize the local properties\nof the individual qudits,\nc1c2c3Rm00= TrρCS1\nm⊗CS2\n0⊗CS3\n0, (4a)\nc1c2c3R0n0= TrρCS1\n0⊗CS2\nn⊗CS3\n0, (4b)\nc1c2c3R00p= TrρCS1\n0⊗CS2\n0⊗CS3\np. (4c)\nwhile the tensors Rmn0,Rm0p,R0np,Rmnp\nc1c2c3Rmn0= TrρCS1\nm⊗CS2\nn⊗CS3\n0, (5a)\nc1c2c3Rm0p= TrρCS1\nm⊗CS2\n0⊗CS3\np, (5b)\nc1c2c3R0np= TrρCS1\n0⊗CS2\nn⊗CS3\np, (5c)\nc1c2c3Rmnp= TrρCS1\nm⊗CS2\nn⊗CS3\np, (5d)\ndescribe the spin correlations. In the formulas Eq. (4), Eq. (5) CSi\n0≡CSi\n0,z=/radicalBig\nSi(Si+1)\n3ESi,\nESiis the unit matrix in dimension (2 Si+ 1) and each of the Latin indices designates a\nset of matrices CSi\nkSi,qSi;x,CSi\nkSi,qSi;y,CSi\nkSi;z, where 1 ≤kSi≤2Si, and 1 ≤qSi≤kSiin\nCS\nkSi,qSi;x,CS\nkSi,qSi;yand 1≤kSi≤2Si, inCS\nk,zin steps of 1, i= 1,2,3.\nLet’sformulatethelinear algorithmof adecomplexification oftheLiou ville-von Neumann\n3equation.\n1. Insert the Hamiltonian Eq. (1) and the density matrix Eq. (3), de composed on Hermitian\nbasis into the Liouville-von Neumann equation Eq. (2).\n2. Multiply the Liouville-von Neumann equation by all elements of the ba sisCS1η⊗CS2\nθ⊗CS3\nϑ\nin turn.\n3. Execute operation of a trace taking for each equation.\n4. Apply the formula Tr( CS1α⊗CS2\nβ⊗CS3γ)(CS1ǫ⊗CS2ε⊗CS3\nζ)(CS1η⊗CS2\nθ⊗CS3\nϑ) =\nTrCS1αCS1ǫCS1ηTrCS2\nβCS2εCS2\nθTrCS3γCS3\nζCS3\nϑ.\n5. Express a trace from the three matrices through structure c onstants according to formu-\nlas Eqs. (A4,A5).\n6. As a result the use of the structure constants symmetry the r eal terms in each equation\nare cancelled out, and the purely imaginary terms are duplicated.\n7. The imaginary unit iis cancelled.\nThis algorithm is easy to apply for the system of more than 3 qudits.\nIn terms of the functions Rαβγthe Liouville-von Neumann equation becomes real and com-\nprises a closed system of first-order differential equations for th e local Bloch vectors and\nspin correlation functions\n∂tRm00=c2c3eS1\njim(hj00Ri00+hjk0Rik0+hj0kRi0k+hjkrRikr), (6a)\n∂tR0n0=c1c3eS2\nikn(h0i0R0k0+hji0Rjk0+h0ijR0kj+hjirRjkr), (6b)\n∂tR00p=c1c2eS3\nijp(h00iR00j+hk0iRk0j+h0kiR0kj+hlkiRlkj), (6c)\n∂tRmn0=c2c3eS1\njimhj00Rin0+c1c3eS2\njknh0j0Rmk0+c2c3eS1\njimhjn0Ri00+c1c3eS2\nqknhmq0R0k0+\nc3(eS1\njimgS2\nkqn+gS1\njimeS2\nkqn)hjq0Rik0+c2c3eS1\njimhj0qRinq+c1c3eS2\njknh0jqRmkq+\n+c2c3eS1\njimhjnqRi0q+c1c3eS2\niknhmiqR0kq+c3(eS1\njimgS2\nlkn+gS1\njimeS2\nlkn)hjlqRikq,\n(6d)\n∂tRm0p=c2c3eS1\njimhj00Ri0p+c1c2eS3\nkqph00kRm0q+c2c3eS1\njimhj0pRi00+c1c2eS3\nkqphm0kR00q+\nc2(eS1\njimgS3\nkqp+gS1\njimeS3\nkqp)hj0kRi0q+c2c3eS1\njimhjk0Rikp+c1c2eS3\nlqph0klRmkq+\nc1c2eS3\nlqphmklR0kq+c2c3eS1\njimhjkpRik0+c2(eS1\njimgS3\nrqp+gS1\njimeS3\nrqp)hjkrRikq,(6e)\n4∂tR0np=c1c3eS2\niknh0i0R0kp+c1c2eS3\niqph00iR0nq+c1c3eS2\niknh0ipR0k0+c1c2eS3\nkqph0nkR00q+\nc1(eS2\nikngS3\njqp+gS2\nikneS3\njqp)h0ijR0kq+c1c3eS2\nlknhil0Rikp+c1c2eS3\nlqphi0lRinq+\nc1c3eS2\nqknhiqpRik0+c1c2eS3\nlqphinlRi0q+c1(eS2\nlkngS3\nrqp+gS2\nlkneS3\nrqp)hirlRikq,(6f)\n∂tRmnp=c2c3eS1\njimhjnpRi00+c1c3eS2\nqknhmqpR0k0+c1c2eS3\nkqphmnkR00q+c2c3eS1\njimhj0pRin0+\nc1c3eS2\njknh0jpRmk0+c3(eS1\njimgS2\nqkn+gS1\njimeS2\nqkn)hjqpRik0+c2c3eS1\njimhjn0Ri0p+\nc1c2eS3\nkqph0nkRm0q+c2(eS1\njimgS3\nlqp+gS1\njimeS3\nlqp)hjnlRi0q+c1c3eS2\niknhmi0R0kp+\nc1c2eS3\niqphm0iR0nq+c1(eS2\nikngS3\nlqp+gS2\nikneS3\nlqp)hmilR0kq+c2c3eS1\njimhj00Rinp+\nc1c3eS2\njknh0j0Rmkp+c3(eS1\njimgS2\nlkn+gS1\njimeS2\nlkn)hjl0Rikp+c1c2eS3\njqph00jRmnq+\nc2(eS1\njimgS3\nlqp+gS1\njimeS3\nlqp)hj0lRinq+c1(eS2\njkngS3\nlqp+gS2\njkneS3\nlqp)h0jlRmkq+\n(eS1\njimgS2\nlkngS3\nrqp+gS1\njimeS2\nlkngS3\nrqp+gS1\njimgS2\nlkneS3\nrqp−eS1\njimeS2\nlkneS3\nrqp)hjlrRikq.\n(6g)\nAsi∂tρn= [ˆH,ρn] (n= 1,2,3,...) at unitary evolution there is the numerable number of\nconservation laws Tr ρ=C1= 1,Trρ2=C2,..., whereCnare the constants of motion,\nfrom which only the first (2 S1+1)(2S2+1)(2S3+1) are algebraically independent [6]. From\nthe conservation of purity, for which ( ρ2)ikdef≡(ρ)ik, the polynomial (square-law) invariants\nare obtained. The square polynomials also control the signs Rαβγ. In the external dc field\nthe energy of system is the constant:\nE= TrˆHρ. (7)\nUnitary evolution preserves the length of the generalized Bloch vec torbS1S2S3\nbS1S2S3=/radicalBig\nR2\nm00+R2\n0n0+R2\n00p+R2\nmn0+R2\nm0p+R2\n0np+R2mnp. (8)\nThe qudit with (2 S1+1) states in the environment of two other qudits is described by th e\nreduced matrix ρS1\nρS1=1\nc1(2S1+1)(R000CS1\n0+Rm00CS1\nm), (9)\nin which R000≡1, and functions Rm00are determinated by the system solution Eq. (6), as\nthe equations for the reduced matrices are not closed.\nFor two different coupled qudits we have ˆH=1\n2hαβCS1α⊗CS2\nβ,\nρ=1\nc1(2S1+1)c2(2S2+1)RγδCS1\nγ⊗CS2\nδ, R00= 1. (10)\n5The dynamic equation Eq. (2) for two different qudits takes on the r eal form in terms of the\nfunctions Rm0,R0m,Rmnas a closed system of differential equations Eqs. (11,12,13) for the\nset of initial conditions:\n∂tRm0=c2eS1\npim(hp0Ri0+hplRil), (11)\n∂tR0m=c1eS2\npim(h0pR0i+hlpRli), (12)\n∂tRmn=eS1\npim/bracketleftbig\nc2(hpnRi0+hp0Rin)+gS2\nrlnhprRil/bracketrightbig\n+\neS2\npin/bracketleftbig\nc1(hmpR0i+h0pRmi)+gS1\nrlmhrpRli/bracketrightbig\n, (13)\nwhere by definition\nTrρCS1\nα⊗CS2\nβ=c1c2Rαβ. (14)\nThe functions Rm0,R0mdescribe the individual qudits and the functions Rmndefine their\ncorrelations. The Liouville-von Neumann equation for one qudit take s on the real form in\nterms of the functions Rjas a closed system of differential equations [7]:\n∂tRl=eS1\nijlhiRj. (15)\nThe set of equations for 3 qubits has been obtained in [8].\nThe set of equations Eq. (6) with the initial conditions has wide applica tions, since\nthe magnetic field enters in the form of arbitrary functions. It allow s to make numerical\ncalculations for continuous (a paramagnetic resonance in a continu ous mode), as well as\nfor pulse modes (a nuclear magnetic resonance). By means of this s ystem it is possible to\ninvestigate the entanglement dynamics of qudits in a magnetic field as the entanglement\nmeasures are expressed in terms of the reduced density matrices or of populations. Another\nimportant application of the system Eq. (6) is quantum approach to the Carnot cycle [9],\n[10], [11], [12], [13], when a working body is a finite spin chain.\nIV. CONCLUSION\nThe simple algorithm of the derivation of equation system for coupled qudits, which\nare in an arbitrary, time-dependent external magnetic field has be en presented. It is not\n6necessary for the basis to be Hermitian since the results of calculat ions are independent of\nthe choice of base, but there is the main advantage with the Hermitia n basis. It is that\nthe Liouville-von Neuman equation not involve any complex numbers an d can be solved\nusing real algebra. This is not true for non-Hermitian bases. Real a lgebra makes numerical\ncalculations faster and simplifies the interpretation of the equation system Eqs. (6).\nThis basis forms a natural basis for calculations on coupled spin syst ems [14] because all\nthe single-spin operators are part of the complete basis when the u nit operator is part of\nthe single-spin basis.\nAcknowledgments\nThe author is grateful to Zippa A. A. for constant invaluable suppo rt.\nAPPENDIX A\nLet{CS\n1,CS\n2,...,CS\nn}be a base of su(2S+1) algebra, where S= 1/2,1,3/2,...is the spin\nquantum number, n= (2S+1)2−1. We have according to [15]\nCS\niCS\nj=c\ndEδij+zS\nijkCS\nk,TrCS\ni= 0,TrCS\niCS\nj=cδij, (A1)\nzS\nijk=gS\nijk+ieS\nijk, (A2)\nhence\n−i[CS\ni,CS\nj] = 2eS\nijkCS\nk,{CS\ni,CS\nj}=c\ndEδij+2gS\nijkCS\nk, (A3)\neS\nijk=1\n2icTr[CS\ni,CS\nj]CS\nk, (A4)\ngS\nijk=1\n2cTr{CS\ni,CS\nj}CS\nk, (A5)\nwhered= 2S+1,Eis the unit matrix in dimension d×d,cis a constant, Tr is a symbol for\ntrace. Itiseasytoseethatthestructureconstants eS\nijkandgS\nijkarecompletely antisymmetric\nand symmetric in the displacement of any pair of indices.\n71. Hermitian basis\nThe structure constants of su(2S+1) algebra have important physical applications. In\norder to calculate the structure constants we have to choose th e basis. The basis is based on\nlinearcombinationsofirreducibletensoroperators. Thematrixrep resentationsofirreducible\ntensor operators TS\nk,q[16] can be calculated using the Wigner 3 jmsymbols:\nTS\nk,q=/radicalbig\n(2S+1)(2k+1)S/summationdisplay\nm,m′=−S(−1)S−m/parenleftbigS k S\n−m q m′/parenrightbig\n|S,m >< S,m′|,(A6)\nwhere 0≤k≤2S, and−k≤q≤kin steps of 1. The normalization is such that TS\n0,0=E.\nIt is known that the Cartesian product operators Sx,Sy, andSzfor spinS=1\n2are Hermitian\nand can be calculated from irreducible tensor operators [17]\nS1\n2x=1\n2√\n2(T1\n2\n1,−1−T1\n2\n1,1) =1\n2\n0 1\n1 0\n, (A7a)\nS1\n2y=i\n2√\n2(T1\n2\n1,−1+T1\n2\n1,1) =1\n2\n0−i\ni0\n, (A7b)\nS1\n2z=1\n2T1\n2\n1,0=1\n2\n1 0\n0−1\n. (A7c)\nAllard and H¨ ard [14] have formed linear combinations of the irreduc ible tensor operators\nnot only for single-quantum coherences, but also for all coherenc es according to\nCS\nk,qx=/radicalbigg\nS(S+1)\n6(TS\nk,−q+(−1)qTS\nk,q), q/ne}ationslash= 0, (A8a)\nCS\nk,qy=i/radicalbigg\nS(S+1)\n6(TS\nk,−q−(−1)qTS\nk,q), q/ne}ationslash= 0, (A8b)\nCS\nk,z=/radicalbigg\nS(S+1)\n3TS\nk,0, q= 0,k≥1, (A8c)\nCS\n0,z=/radicalbigg\nS(S+1)\n3E, (A9)\nwhere 1 ≤k≤2S, and 1≤q≤kinCS\nk,qx,CS\nk,qyand 1≤k≤2S, inCS\nk,zin steps of 1.\nThe matrices Eqs. (A8) are traceless and their number is equal to ( 2S+1)2−1. Using the\nwell-known relations for the irreducible tensor operators\n(TS\nk,q)+= (−1)qTk,−q, (A10)\n8we can see that matrices Eqs. (A8) are Hermitian. Using the formula from [16]\nTrTS\nk,qTS\nk′,q′= (−1)q(2S+1)δk,k′δq,−q′ (A11)\nit is easy to show that the basis is normalized so that Sx=CS\n1,x,Sy=CS\n1,y,Sz=CS\n1,z,\nirrespective of the spin quantum number S, i.e.\n(Cr,Cs) = TrCrCs=δr,sS(S+1)(2S+1)\n3. (A12)\nThe set Eqs. (A8,A9) is complete. The matrices Ck,zare diagonal\n[CS\nk,z,CS\nk′,z] = 0. (A13)\nThere also exist the other useful bases [18], [19]. From the physical point of view, for\nimportant physical applications the basis [14] is preferred.\n2. Analytic formulas for structure constants\nThere are 27 combinations in threes including the repetitions: XX′X′′,XX′Y′′,\nXX′Z′′..., where X=CS\nk,qx,X′=CS\nk′,q′x,Y′′=CS\nk′′,q′′y,Z′′=CS\nk′′,zand so on. The\nuse of the symmetrical properties of the Wigner 3 jmsymbols and the formula 2.4(23)[16]\nTrTS\nk,qTS\nk′,q′TS\nk′′,q′′= (−1)2S+k+k′+k′′(2S+1)3\n2\n[(2k+1)(2k′+1)(2k′′+1)]1\n2{k k′k′′\nS S S}/parenleftBig\nk k′k′′\nq q′q′′/parenrightBig\n, (A14)\nwhere{k k′k′′\nS S S}is the 6jsymbol, allows us after substitution of Eqs. (A8) in Eq. (4),Eq. (5)\n, to calculate all structure constants of su(2S+1) algebra. Let us introduce the function\nF(k,k′,k′′,S) =(−1)2S\n√\n3/radicalbig\nS(S+1)(2S+1)(2k+1)(2k′+1)(2k′′+1){k k′k′′\nS S S}.(A15)\nAll antisymmetric structure constants are zero for K=k+k′+k′′even and nonvanishing\nantisymmetric structure constants in terms of 3 jmand 6jsymbols have the explicit form\nare presented by formulas Eqs. (A16a,A16b,A16c) for Kodd:\neS\nXX′Y′′=−F√\n2/bracketleftBig\n(−1)q/parenleftBig\nk k′k′′\nq−q′−q′′/parenrightBig\n+(−1)q′/parenleftBig\nk k′k′′\n−q q′−q′′/parenrightBig\n+(−1)q′′/parenleftBig\nk k′k′′\nq q′−q′′/parenrightBig/bracketrightBig\n,(A16a)\neS\nYY′Y′′=F√\n2/bracketleftBig\n(−1)q/parenleftBig\nk k′k′′\n−q q′q′′/parenrightBig\n+(−1)q′/parenleftBig\nk k′k′′\nq−q′q′′/parenrightBig\n+(−1)q′′/parenleftBig\nk k′k′′\nq q′−q′′/parenrightBig/bracketrightBig\n, (A16b)\n9eS\nXY′Z′′=−F(−1)q/parenleftBig\nk k′k′′\nq−q′0/parenrightBig\n. (A16c)\nAll symmetric structure constants are zero for K=k+k′+k′′odd and nonvanishing\nsymmetric structure constants in terms of 3 jmand 6jsymbols have the explicit form are\npresented by formulas Eqs. (A17a,A17b,A17c) for Keven:\ngS\nXX′X′′=F√\n2/bracketleftBig\n(−1)q/parenleftBig\nk k′k′′\nq−q′−q′′/parenrightBig\n+(−1)q′/parenleftBig\nk k′k′′\n−q q′−q′′/parenrightBig\n+(−1)q′′/parenleftBig\nk k′k′′\nq q′−q′′/parenrightBig/bracketrightBig\n,(A17a)\ngS\nXY′Y′′=F√\n2/bracketleftBig\n−(−1)q/parenleftBig\nk k′k′′\nq−q′−q′′/parenrightBig\n+(−1)q′/parenleftBig\nk k′k′′\n−q q′−q′′/parenrightBig\n+(−1)q′′/parenleftBig\nk k′k′′\n−q−q′q′′/parenrightBig/bracketrightBig\n,(A17b)\ngS\nXX′Z′′=gS\nYY′Z′′=F(−1)q/parenleftBig\nk k′k′′\nq−q′0/parenrightBig\n, gS\nZZ′Z′′=F(−1)q/parenleftBig\nk k′k′′\n0 0 0/parenrightBig\n. (A17c)\nWe have in X,Y1≤k,k′,k′′≤2S, 1≤q≤k,1≤q′≤k′,1≤q′′≤k′′and inZ\n1≤k,k′,k′′≤2Sin steps of 1.\nThe straightforward calculation confirms that the structure con stantseS\nijkandgS\nijkare com-\npletely antisymmetric and symmetric in the displacement of any pair of operators. In other\nwords it is eS\nXX′Y′′=−eS\nXY′′X′,gS\nXX′Z′′=gS\nYY′Z′′and so on.\n10[1] Jing-Ling Chen, Dagomir Kaszlikowski, L. C. Kwek, Marek Zukowski, and C. H. Oh,\narXiv:quant-ph/0103099v1 2001.\n[2] Ming Li, Shao-Ming Fei and Zhi-Xi Wang, arXiv:0809.1022 v1 [quant-ph] 2008; Xin-Gang\nYang, Zhi-Xi Wang, Xiao-Hong Wang and Shao-Ming Fei, arXiv: 0809.1556v1 [quant-ph] 2008;\nMichael J. Bremner, Dave Bacon, and Michael A. Nielsen, arXi v:quant-ph/0405115v1 2004;\nDafa Li, Xiangrong, Hongtao Huang, Xinxin Li, arXiv:quant- ph/0604147 v1 2006 .\n[3] J. K. Pachos, quant-ph/0505225 v1 2005.\n[4] J. K. and P. L. Knight, Phys. Rev. Lett. 91, 107902 (2003).\n[5] J.Zhang, X. Peng,and D. Suter, quant-ph/0512229 v1 2005 .\n[6] V. Tapia, arXiv:math-ph/0702001v1 2007.\n[7] F. T. Hioe and J. H. Eberly, Phys. Rev. Letters, 47, 838, (1981).\n[8] E. A. Ivanchenko, Low Temp. Physics, 33(4), 336, (2007); quant-ph/0610176.\n[9] H. Scovill and E. O. Schulz-Dubois, Phys. Rev. Lett. 2, 262 (1959); J. E. Geusic, E. O.\nSchulz-Dubois and H. Scovill, Phys. Rev. 156, 343 (1967).\n[10] S. Carnot, Refl´ ections sur la Puissance Motrice du Feu et sur les Machin es Propres ` a\nD´ evelopper Cette Puissance (Bachier, Paris, 1824).\n[11] Tova Feldmann and Ronnie Kosloff, Phys. Rev. E 70, 046110 (2004).\n[12] Tova Feldmann and Ronnie Kosloff, Phys. Rev. E 68, 016101 (2003).\n[13] Yair Rezek, Ronnie Kosloff, arXiv:quant-ph/0601006v2, 2006.\n[14] P. Allard and T. H¨ ard, J. Mag. Resonance, 153, 15, (2001).\n[15] G. Kimura and A. Kossakowski, Open Systems & Informatio n Dynamics. 12, 207 (2005);\nquant-ph/0408014.\n[16] D. A. Varshalovich A. N. Moskalev, and V. K. Khersonskii ,Quantum Theory of Angular\nMomentum (Leningrad: ”Nauka” edition) 1975.\n[17] R.R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in\nOne and Two Dimensions , Oxford Univ. Press, Oxford, 1987.\n[18] R. A. Bertmann and P. Krammer, arXiv:0706.1743 (2007).\n[19] Maurice R. Kibler, arXiv:0810.4418v1 [quant-ph] 2008 .\n11"    },    {        "title": "0909.1328v1.On_the_dynamic_efficiency_of_internal_shocks_in_magnetized_relativistic_outflows.pdf",        "content": "arXiv:0909.1328v1  [astro-ph.HE]  7 Sep 2009Mon. Not.R. Astron. Soc. 000, 000–000 (0000) Printed 17 November 2021 (MN L ATEXstyle file v2.2)\nOnthedynamice fficiencyofinternalshocksin magnetized\nrelativisticoutflows\nP. Mimica1⋆and M. A. Aloy1\n1Departamento deAstronom´ ıa yAstrof´ ısica, Universidad d e Valencia, 46100, Burjassot, Spain\n17 November 2021\nABSTRACT\nWe study the dynamic e fficiency of conversion of kinetic-to-thermal /magneticenergy of in-\nternal shocks in relativistic magnetized outflows. We model internal shocks as being caused\nby collisions of shells of plasma with the same energy flux and a non-zero relative velocity.\nThecontactsurface,wheretheinteractionbetweenthe shel lstakesplace,can breakupeither\ninto two oppositelymovingshocks(in the frame where the con tact surface is at rest), or into\na reverse shock and a forward rarefaction. We find that for mod erately magnetized shocks\n(magnetizationσ≃0.1), the dynamic efficiency in a single two-shell interaction can be as\nlargeas40%.Thus,thedynamice fficiencyofmoderatelymagnetizedshocksislargerthanin\nthe corresponding unmagnetized two-shell interaction. If the slower shell propagates with a\nsufficientlylargevelocity,the e fficiencyisonlyweaklydependentonitsLorentzfactor.Con-\nsequently, the dynamic e fficiency of shell interactions in the magnetized flow of blazar s and\ngamma-raybursts is e ffectively the same. These results are quantitativelyrather independent\non the equation of state of the plasma. The radiative e fficiency of the process is expected to\nbe a fraction fr<1 of the estimated dynamic one, the exact value of frdepending on the\nparticularities of the emission processes which radiate aw ay the thermal or magnetic energy\noftheshockedstates.\nKey words: Hydrodynamics – (magnetohydrodynamics) MHD – Shock waves – gamma-\nrays:bursts– galaxies:BL Lacertaeobjects:general\n1 INTRODUCTION\nInternal shocks (Rees & Meszaros 1994) are invoked to explai n\nthe variability of blazars (see e.g., Spada et al. 2001; Mimi ca et al.\n2005) and the light curves of the prompt phase of gamma-ray\nbursts (GRBs) (Sari&Piran 1995, 1997; Daigne & Mochkovitch\n1998). A possible problem in this model is the question wheth er\nthis mechanism is e fficient enough to explain the relation between\nthe observed energies both in the prompt GRB phase and in the\nafterglow (see e.g., Kobayashi et al. 1997 (KPS97), Belobor odov\n2000, Kobayashi & Sari 2000, Fan& Piran 2006). To asses the ef -\nficiency of the internal shock model, most of the previous wor ks\nfocusonthecomparisonbetweentheobservedlightcurves an dthe\nmodel predictions employing a simple inelastic collision o f two-\npoint masses (KPS97; Lazzatiet al. 1999; Nakar & Piran 2002;\nTanihata et al. 2003; Zhang &M´ esz´ aros 2004). Less attenti on has\nbeenpaidtothehydrodynamice ffectsduringtheshellcollision(but\nsee Kobayashi & Sari 2000; Kinoet al. 2004; Mimica etal. 2004 ,\n2005; Boˇ snjaket al.2009).\nThe ejecta in GRBs and blazars may be rather magnetized,\nparticularly if they are originated as a Poynting-flux-domi nated\nflow (e.g., Usov 1992) Forming shocks in highly magnetized me -\n⋆E-mail: Petar.Mimica@uv.esdia is challenging (Rees & Gunn 1974; Kennel & Coroniti 1984) .\nTherefore, to account for the observed phenomenology it is n ec-\nessary to address how e fficient the process of internal collisions in\narbitrarily magnetized flows is. This question has been part ly con-\nsidered by a few recent works (e.g., Fanetal. 2004; Mimica et al.\n2007).\nThe base to study the e fficiency of internal collisions is the\ndeterminationofthedynamice fficiencyofasinglebinarycollision,\ni.e., the efficiency of converting the kinetic energy of the colliding\nfluid into thermal and /or magnetic energy. Note that the radiative\nefficiency(i.e.,theefficiencyofconvertingthekineticenergyofthe\nflowintoradiation)isexpectedtobesomewhat smaller.Acco rding\nto, e.g., Panaitescuetal. (1999) and Kumar (1999), it can be as\nlow asfr∼0.1. As we shall show in this paper, binary collisions\nin relativistic, magnetized flows can be an e fficient enough way to\ndissipateamajorfractionofthebulkkineticenergyofarel ativistic\nejecta. Therefore, it will depend on the e fficiency of the particular\nradiationmechanism, that produces the observed emission ( i.e.,on\nthe factor fr), that the model of internal shocks be e fficient enough\ntoexplaintheobservations(particularly,thedistributi onofenergies\nbetweenthe prompt GRBphase and the afterglowphase).\nWe model internal shocks as shells of plasma with the same\nenergy flux and a non-zero relative velocity. The contact sur face,\nwhere the interaction between the shells takes place, can br eak up\nc/circleco√yrt0000 RAS2P.Mimica andM. A. Aloy\neither into two oppositely moving shocks (in the frame where the\ncontactsurfaceisatrest),orintoareverseshockandaforw ardrar-\nefaction. The determination of whether one or the other poss ibil-\nityoccurs is computed by estimating the invariant relative velocity\nbetween the fastest and the slowest shell, i.e., by solving t he Rie-\nmann problem posed by the piecewise uniform states given by t he\nphysicalquantitiesonthetwointeractingshells(Section 2).InSec-\ntion3wedefinepreciselythenotionofdynamice fficiency,bothfor\nshocks and rarefactions. We perform a parametric study of th e bi-\nnaryshellcollisiondynamice fficiencyinSection4.Thediscussion\nand conclusions are listed in Section 5. Finally, we have ext ended\nour analysis on the dynamic e fficiency of internal shocks in mag-\nnetized, relativistic plasma to consider more realistic eq uations of\nstate inthe Appendix.\n2 RELATIVISTICMAGNETOHYDRODYNAMIC\nRIEMANNPROBLEM\nWemodeltheinteractionbetweenpartsoftheoutflowwithvar ying\nproperties by considering Riemann problems, i.e. relativi stic mag-\nnetohydrodynamic initial-value problems with two constan t states\nseparated by a discontinuity in planar symmetry. We note, th at we\ncould use a more sophisticated approach consisting on perfo rming\nnumerical relativisticmagnetohydrodynamic (RMHD)simul ations\nof the interaction of parts of the outflow with di fferent velocities.\nHowever,suchanapproachdemandshugecomputationalresou rces\n(evenperformingonedimensionalsimulationsusingthesam ecode\nas in Mimica et al. 2009), and we are interested in sampling ve ry\nfinelyalargeparameterspacewithourmodels.Apartfromthi snu-\nmerical reason, itisinorder topoint out that,bythe intern al shock\nphase,thelateralexpansionoftheflowisverysmall,sincet heflow\nis probably cold and ultrarelativistic. Thus, a descriptio n of the in-\nteractions assuming planar symmetry su ffices to compute the dy-\nnamic efficiency of such interactions (rather than a more complex\nsphericallysymmetric approach).\nIn the following we use subscripts LandRto denote prop-\nerties of the (faster) left and (slower) right state, respec tively. To\navoid repeated writing of a factor 4 πand the speed of light c, we\nnormalize the rest-mass density ρtoρR, the energy density to ρRc2\nand themagnetic fieldstrengthto c/radicalbig\n4πρR.\n2.1 Initialstates of theRiemannproblem\nFor the initial thermal pressure of both states we assume tha t it\nis small fraction of the density, pL=χρLandpR=χ. We as-\nsume magnetic fields perpendicular to the direction of the flo w\npropagation. The remaining parameters determining the RMH D\nRiemann problem are: the density contrast ρL, the Lorentz fac-\ntor of the right state ΓR, the relative Lorentz factor di fference\n∆g:=(ΓL−ΓR)/ΓR,andthe magnetizations of leftandright states,\nσL:=B2\nL/(Γ2\nR(1+∆g)2ρL) andσR:=B2\nR/Γ2\nR, whereBLandBR\nare the lab frame magnetic field strengths of left and right st ates,\nrespectively.Furthermore,wedefinethetotal(thermal +magnetic)\npressure\np∗:=p+B2\n2Γ2=p+σρ\n2, (1)\nthe totalspecific enthalpy\nh∗:=1+ǫ+p/ρ+σ, (2)\nande∗:=ρ(1+ǫ)+σρ\n2. (3)\nwhereǫdenotes the specific internal energy and is dependant on\nthe equation of stateused (see Section3.1).\nThegeneralsolutionofaRMHDRiemannproblemwasfound\nby Giacomazzo & Rezzolla (2006), and recently used in RMHD\nnumerical codes bye.g.,van der Holst etal.(2008).However , here\nwe deal with a degenerate RMHD configuration, which solution\nwas first found by Romeroet al. (2005). The typical structure of\nthe flow after the break up of the initial discontinuity consi sts of\ntwoinitialstates,andtwointermediatestatesseparatedb yacontact\ndiscontinuity(CD).Thetotalpressureandvelocityarethe sameon\nbothsides ofthe CD.The quantity σ/ρisuniform everywhere, ex-\ncept across the CD, where it can have a jump. We denote the tota l\npressure of intermediate states p∗\nS, and rest-mass density left and\nrightoftheCDasρS,LandρS,R.Inthecontextofinternalshocks, if\nthe flow is ultrarelativistic in the direction of propagatio n, the ve-\nlocitycomponents perpendicular totheflowpropagationsho uld be\nnegligiblysmall and, hence, theyare set uptozeroinour mod el1.\n2.2 Conditionsfor theexistence of atwo-shocksolution\nOne ofthe key steps insolving a Riemannproblem is todetermi ne\nunder whichconditions internal shocks can form.Statesahe ad and\nbehind the shock front are related by the Lichnerowicz adiab at\n(Romeroet al.2005)\n(h∗\nb)2−(h∗\na)2−/parenleftBiggh∗\nb\nρb−h∗\na\nρa/parenrightBigg\n(p∗\nb−p∗\na)=0. (4)\nFollowing Rezzolla &Zanotti (2001), we study the relative v eloc-\nity between the states ahead (a) and behind (b) the shock fron t (all\nvelocities are measured inthe rest frame of the shock, and al l ther-\nmodynamic properties are measured inthe fluidrest frame),\nvab:=va−vb\n1−vavb=/radicalBigg\n(p∗\nb−p∗\na)(e∗\nb−e∗\na)\n(e∗\na+p∗\nb)(e∗\nb+p∗\na). (5)\nIn our case states ahead of the shock are the initial (L, R)\nstates, while states behind the shock are the intermediate s tates.\nSincevabis Lorentz-invariant, we can measure the velocity ahead\nof the left-propagating ( reverse) shock (RS) in the frame in which\nthe CDis atrest,\nvl=/radicalBigg\n(p∗\nS−p∗\nL)(e∗\nS,L(p∗\nS)−e∗\nL)\n(e∗\nL+p∗\nS)(e∗\nS,L(p∗\nS)+p∗\nL). (6)\nLikewise, the velocity ahead of the right-going ( forward) shock\n(FS)measured inthe CD frameis\nvr=−/radicalBigg\n(p∗\nS−p∗\nR)(e∗\nS,R(p∗\nS)−e∗\nR)\n(e∗\nR+p∗\nS)(e∗\nS,R(p∗\nS)+p∗\nR), (7)\nwheree∗\nS,Lande∗\nS,Rare the energy densities of the states to the left\nandtotherightoftheCD,respectively.Therest-massdensi tiesρS,R\nandρL,Rcanbe obtained from (4)and (2).\nSince both FS and RS only exist if p∗\nS>p∗\nRandp∗\nS>p∗\nL,\nrespectively, with decreasing p∗\nSeither the FS will disappear first\n(forp∗\nS=p∗\nR>p∗\nL,givingvr=0)ortheRSwilldisappear first(for\np∗\nS=p∗\nL>p∗\nR, givingvl=0). Using equations (6) and (7) and the\n1If such velocities were significant, appreciable changes in the Rie-\nmann structure may result as pointed out in Aloy &Rezzolla (2 006) or\nAloy & Mimica (2008).\nc/circleco√yrt0000 RAS,MNRAS 000, 000–000Efficiency ofinternalshocks 3\ninvariance of the relative velocity between the left and rig ht states,\nvlr:=(vl−vr)/(1−vlvr), we can determine the minimum relative\nvelocityfor whicha two-shock solution ispossible\n(vlr)2S=/radicalBigg\n(p∗\nL−p∗\nR)(e∗\nS,R(p∗\nL)−e∗\nR)\n(e∗\nS,R(pL∗)+p∗\nR)(e∗\nR+p∗\nL)ifp∗\nL=p∗\nS>p∗\nR\n/radicalBigg\n(p∗\nR−p∗\nL)(e∗\nS,L(p∗\nR)−e∗\nL)\n(e∗\nS,L(p∗\nR)+p∗\nL)(e∗\nL+p∗\nR)ifp∗\nL<p∗\nR=p∗\nS(8)\nGenerally, the quantity ( vlr)2Scan be only determined numer-\nically. If ( vlr)<(vlr)2S, a single shock and a rarefaction emerge\nfrom theinitialdiscontinuity. Itiseven possible that ins teadoftwo\nshocks tworarefactions form (see,Rezzolla & Zanotti 2001) .\n3 ENERGYDISSIPATIONEFFICIENCYOFINTERNAL\nSHOCKS\nInternal shocks in relativistic outflows are invoked as mode rately\nefficient means of conversion of the kinetic energy of the flow int o\nradiation.Inthissectionwepresentour modelforinhomoge neous,\nultrarelativisticoutflowsandprovide anoperativedefinit ionforthe\nefficiency of conversion of the initial energy of the outflow into\nthermal and magnetic energy produced by internal shocks. We as-\nsume that a fraction of this thermal and magnetic energy will be\nradiatedaway.\n3.1 Outflowmodel\nTostudyinternalshocksweidealizeinteractionsofpartso ftheout-\nflowmovingwithdi fferentvelocitiesascollisionsofhomogeneous\nshells.Inourmodelthefaster(left)shellcatchesupwitht heslower\n(right) one yielding, in some cases, a pair of shocks propaga ting\nin opposite directions (as seen from the CD frame). In order t o\ncover a wide range of possible flow Lorentz factors and shell m ag-\nnetizations, we assume that initially, the flux of energy in t he lab\nframe is uniform and the same in both shells2. The energy flux for\nashellwithrest-massdensity ρ,ratioofthermalpressuretodensity\nχ, magnetizationσand Lorentz factor Γis (e.g., Leismannet al.\n2005).\nFτ:=ρ/bracketleftBig\nΓ2(1+ǫ+χ+σ)−Γ/bracketrightBig√\n1−Γ−2. (9)\nUsing the notation introduced in Section 2.1 and assuming th e\nequality of Fτin both shells we find that the density contrast ρL\nbetween leftandright shells is\nρL=(1+∆g)−2/bracketleftBig\n1+ǫ+χ+σR−Γ−1\nR/bracketrightBig/radicalBig\n1−Γ−2\nR\n/bracketleftBig\n1+ǫ+χ+σL−Γ−1\nR(1+∆g)−1/bracketrightBig/radicalBig\n1−Γ−2\nR(1+∆g)−2.(10)\nConsideringσL,σR,ΓRand∆gas parameters, we can use (10)\nto compute the rest of the variables needed to set-up the Riem ann\nproblem. We thencompute the break up of the initialdisconti nuity\nbetween both shells using the exact Riemann solver develope d by\nRomeroet al.(2005).\nIn the following we use a polytropic equation of state withan\nadiabatic index ˆγ=4/3:\n2In Mimica etal. (2009) we use a similar model to compare after glow\nejecta shells with di fferent levels of magnetization, with a slight di fference\nthat in that study, instead of having the same flux of energy, a ll the ejecta\nshells wereassumed to contain the sametotal energy.ǫ:=p\n(ˆγ−1)ρ(11)\nAs we show in the Appendix, the Riemann solver of Romeroet al.\n(2005)has beensuitablymodifiedtoinclude a morerealistic equa-\ntion of state. However, we find no qualitative di fferences between\nthe results using a polytropic EoS (witheither ˆ γ=4/3 or ˆγ=5/3)\nand ˆγ-variable EoS. Furthermore, the quantitative di fferences are\nverysmall interms of dynamic e fficiency.\n3.2 Efficiencyof energydissipationbyashock\nTo model the dynamic e fficiency of energy dissipation we follow\nthe approach described in Mimica et al. (2007), suitably mod ified\ntoaccount for the fact that inthe present work, there can occ ur sit-\nuationswhereeithertheFSortheRSdonotexist(seeSection 2.2).\nByusingtheexactsolverwedeterminetheexistenceofshock sand\n(in case one or two shocks exist) obtain the hydrodynamic sta te\nof the shocked fluid. We use subscripts S,LandS,Rto denote\nshocked portions of left and right shells, respectively. In the fol-\nlowingwe treatthe e fficiency of eachshock separately.\n3.2.1 Reverse shock\nTo compute the efficiency we need to compare the energy content\nof the initial(unshocked) faster shell withthat of the shoc ked shell\nat the moment when RS has crossed the initial shell. Assuming an\ninitial shell width ∆x, we define total initial kinetic, thermal and\nmagnetic energy (see also equations (A.1) - (A.3) of Mimica e t al.\n2007)\nEK(Γ,ρ,∆x) := Γ(Γ−1)ρ∆x\nET(Γ,ρ,p,∆x) :=[(ρε+p)Γ2−p]∆x\nEM(Γ,ρ,σ,∆x) :=/parenleftBigg\nΓ2−1\n2/parenrightBigg\nρσ∆x. (12)\nWhentheRScrosses the whole initialshell,the lengthof the com-\npressed shell (i.e., the fluid between the RS and the CD) is ζL∆x,\nwhere\nζL:=vCD−vS,L\nvL−vS,L<1\nandvCDandvS,Lare velocities (in the lab frame) of the contact\ndiscontinuity and the RS, both obtained from the solver. Wit hout\nloss of generality, we can normalize the initial shell width so that\n∆x=1. Thenwe define the dynamic thermalefficiency\nεT,L:=ET(ΓS,L,ρS,L,pS,L,ζL)−ET(ΓR(1+∆g),ρL,χρL,1)\nE0,(13)\nandthe dynamic magnetic efficiency\nεM,L:=EM(ΓS,L,ρS,L,σS,L,ζL)−EM(ΓR(1+∆g),ρL,σL,1)\nE0,(14)\nwhereE0is the totalinitialenergy of bothshells\nE0:=EK(ΓR(1+∆g),ρL,1)+ET(ΓR(1+∆g),ρL,χρL,1)\n+EM(ΓR(1+∆g),ρL,σL,1)+EK(ΓR,1,1)\n+ET(ΓR,1,χ,1)+EM(ΓR,1,σR,1).(15)\nEquations (13) and (14) express the fraction of the initial e n-\nergy that the RS has converted into thermal and magnetic ener gy,\nrespectively.\nc/circleco√yrt0000 RAS, MNRAS 000, 000–0004P.Mimica andM. A. Aloy\n3.2.2 Forward shock\nIn complete analogy, we define the thermal and magnetic e fficien-\ncies for the forwardshock,\nεT,R:=ET(ΓS,R,ρS,R,pS,R,ζR)−ET(ΓR,1,χ,1)\nE0, (16)\nεM,R:=EM(ΓS,R,ρS,R,σS,R,ζR)−EM(ΓR,1,σR,1)\nE0, (17)\nwhere\nζR:=vS,R−vCD\nvS,R−vR<1.\nvS,Ris the velocity of the FS in the lab frame. Here we set εT,R=\nεM,R=0if the forwardshock isabsent.\nCombining equations (13), (14), (16) and (17) we define the\ndynamic thermal andmagnetic e fficiency of internalshocks\nεT:=εT,L+εT,R (18)\nεM:=εM,L+εM,R. (19)\nWe point outthat these definitions ofe fficiency generalize the\nones typically used when cold, unmagnetized shell collisio ns are\nconsidered. In that case, initially one only has bulk kineti c energy\nintheshells(i.e., E0=EK(ΓL,ρL,1)+EK(ΓR,1,1)).Incaseofcolli-\nsionsofarbitrarilymagnetizedshellswitharbitrarilyin itialthermal\ncontent,E0canbesubstantiallylargerthantheinitialkineticenergy\ninthe shells.\n3.3 Efficiencyof energy dissipationbyararefaction\nIn a rarefaction there is a net conversion of magnetic and /or ther-\nmal energy into kinetic energy, thus the net dynamic e fficiency\nproduced by a rarefaction, defined as in e.g., Eq. (14), shoul d be\nnegative. The consequence of this is that, whenwe obtain a sh ock-\ncontact-rarefactionorrarefaction-contact-shockstruc tureasasolu-\ntiontotheRiemannproblem, itmayhappen thatthetotal(lef tplus\nright)thermalormagnetice fficiency(Eqs.(18)-(19))wasnegative.\nHowever, this situation does not correctly model the fact th at, in\ncases where a shock exists only in one of the shells, it is stil l able\nto radiate away part of the thermal or magnetic energy behind it,\neven though there is a rarefaction propagating through the o ther\nshell.Wealsopoint outthatrarefactions happen inour case ,where\nwe model initially cold flows, as a result of a net conversion o f\nmagnetic intokinetic energy. Thiskinetic energy canbe fur ther re-\ncycled by the flow, and dissipated in the course of ulterior bi nary\ncollisions. Therefore, directly summing the (positive) dy namic ef-\nficiency of conversion of kinetic-to-thermal /magnetic energy in a\nshock to the (negative) dynamic e fficiency of conversion in a rar-\nefactionisinadequate.Thetotaldynamice fficiencyinacasewhere\nonly one shock forms will be determined only by the e fficiency in\nthe shocked shell. Thus, we set εT,L=εM,L=0 (εT,R=εM,R=0)\nif the reverse (forward) shock is absent. We note that this co ntrasts\nwith previous works on internal collisions of unmagnetized shells\n(e.g., Kinoet al. 2004), and may yield higher values of the ne t dy-\nnamic efficiency.\n4 PARAMETRICSTUDYOFTHE DYNAMIC\nDISSIPATIONEFFICIENCY\nNext we study the dynamic dissipation e fficiency in the process\nof collision of cold, magnetized shells.The shells are assu med toFigure1. Contours:totaldynamice fficiencyεT+εM(eqs.(18)and(19))in\ntheblazar regime(ΓR=10,∆g=1)fordifferent combinations of( σL,σR).\nContours indicate the e fficiency in percent and their levels are 1, 2, 3, 4, 5,\n6, 7, 8, 9, 10, 11, 12 and 13. In the region of the parameter spac e above\nthe dashed line there is no forward shock, while the reverse s hock is al-\nways present for the considered parametrization. Filled co ntours: magnetic\nefficiencyεMin percent.\nbe cold because in the standard fireball model (e.g., Piran 20 05)\n, almost all the internal energy of the ejecta has been conver ted\nto kinetic energy beforeinternal shocks start to show up. Thus,\na regime where the parameter χis large does not properly model\nanefficientlyacceleratedejectabynon-magnetic processes.Ont he\notherhand,iftheejectawereacceleratedbymagneticfields (likein\nPoynting-dominated flow models; e.g., Usov 1992), then the fl ow\nis cold all the way from the beginning to the internal shock ph ase,\nandthenχshould alsobe small insuch a case.\nFor all the models in this paper, in order to reduce the dimen-\nsions ofthe parameter space, wefix χ=10−4uniform everywhere,\nto model initially cold shells and, unless otherwise specifi ed, set\n∆g=1 as a reference value. Wechoose χsufficiently small sothat\nit does not influence the solution of the Riemann problem. In t he\nfirst two subsections we consider blazar and GRB regimes. The n,\nwe study the flow structure for three representative Riemann prob-\nlems,andendthesectionwithadiscussionoftheimpactofva rying\n∆gonthe efficiency.\n4.1 Blazar regime\nIntheblazarregimeweset ΓR=10asatypicalvalueoftheLorentz\nfactor of the outflowing material.We continuously vary σLandσR\nandshow contours of totale fficiency (εT+εM) inFig.1.\nThe maximum efficiency is attained for moderately magne-\ntized slower shells ( σR≈0.2) and highly magnetized left shells\n(σL≈1). The broad region to the right of the e fficiency maximum\nisindependent ofσLbecauseinacollisionwithsuchahighlymag-\nnetized fast shell almost all the energy is dissipated by the FS. In\nthe region above the dashed line of Fig. 1 the FS is absent and,\nthus, since only the RS dissipates the initial energy, the e fficiency\nc/circleco√yrt0000 RAS,MNRAS 000, 000–000Efficiency ofinternalshocks 5\nFigure 2. Sameas Fig. 1,but in the GRB regime ( ΓR=100,∆g=1).\nslightly drops. However, the transition between the regime where\nthe twoshocks operate or onlythe RSexists issmooth. Therea son\nbeing that the efficiency below the separatrix of the two regimes\nand close toitisdominated by the contributionof RS.\nAs expected, when either σLorσRapproach low values, the\ndynamic efficiency ceases to depend on them. This can be seen in\nthe center ofthe leftside of Fig.1where, for σL≪1,the dynamic\nefficiencyonlydepends on σR.Theconverse istrueinthecenterof\nthe lower side of the figure, where σR≪1.\n4.2 GRB regime\nThe results in the GRB regime ( ΓR=100,∆g=1) are shown\nin Fig. 2. The general shape of the contours is similar to Fig. 1,\nwhich is expected since both in the blazar and in the GRB regim e\nthe flow is utrarelativistic and most of the quantities which de-\npend on the relativevelocity between the faster and the slower\nshell depend only weakly on ΓR,∆gbeing the crucial parame-\nter (Daigne &Mochkovitch 1998). For example, the dashed cur ve\nwhich delimits regions with and without a forward shock does not\ndepend onΓRbutonly on∆g.\nThemaximumofthedynamice fficiencyintheGRBregimeis\nlocalizedatroughlythesamespotasintheblazarregime.Ho wever,\ncomparedtotheformercasetheregionofmaximume fficiency(i.e.,\nwhereεT+εM/greaterorsimilar0.13) issmaller.\n4.3 Flow structure\nIn this subsection we study the flow structure for three repre sen-\ntative models in the GRB regime. Their location in the param-\neter space is marked by letters A,BandCin Fig. 2. Model A\ncorresponds to a prototype of interaction between non-magn etized\nshells (σL=σR=10−6). Model Bis picked up to illustrate the\nflow structure at the maximum e fficiency (σL=0.8, σR=0.2).\nFinally, model Ccorresponds to the case when the FS is absent\n(σL=1,σR=102).Weshow therestmassdensityprofileofthese10-210-1100101ρA\nB\nC CD\nFigure 3. Rest-mass density profile of models A,BandC(see legends)\nwhose parameters are given in Sect. 4.3. Profiles have been sh ifted so that\nthe CD for all models coincides. In models AandBthe FS and the RS are\nclearly visible, whilein themodel Cararefaction waveisvisible asasmall\n“step” to theright of the CD.\nmodels in Fig. 3. Model A(thick full line on Fig. 3) shows two\nstrong shocks which dissipate kinetic into thermal energy. In con-\ntrast, model B(dashed line) has much weaker shocks due to non-\nnegligible magnetization in both shells. Finally, model C(thin full\nline) does not have forward shock due to very high magnetizat ion\ninthe slower shell.\nAll three models have a substantial dynamic e fficiency, but\nthere is a qualitative di fference among them. In model Ainternal\nshocksdissipatekinetictothermalenergyonly(thermale fficiency).\nIn model Bshocks mainly compress the magnetic field (magnetic\nefficiency) and dissipate only a minor fraction of the initial ki netic\nandmagneticenergytothermalenergy.Finally,inthemodel Conly\nthe reverse shock is active, compressing the faster shell ma gnetic\nfield.\n4.4 Dependenceon ∆gandon∆s\nThe choice of a relatively small value of ∆gin the previous sec-\ntion is motivated by the results of numerical simulations of rel-\nativistic outflows (e.g Aloyet al. 2000, 2005; Mizuta et al. 2 006;\nZhanget al. 2003, 2004; Morsony et al. 2007; Lazzati et al. 20 09;\nMizuta & Aloy 2009) where ∆g<2 between adjacent parts of the\nflow that may catch up (but see, e.g., Kino&Takahara 2008, who\nfind∆g∼1−19appropriatetomodelMrk421).Thisadjacentflow\nregions can be assimilatedtopairs of shells whose binary co llision\nwe areconsidering here.\nHowever, ithas beenconfirmedbyseveralindependent works\n(KPS97; Beloborodov 2000; Kobayashi & Sari 2000; Kinoet al.\n2004, etc.) that, in order to achieve a high e fficiency (more than\na fewpercent) ininternal collisions of unmagnetized shell s,the ra-\ntio between the maximum ( Γmax) and the minimum ( Γmin) Lorentz\nfactor of the distributionof initialshellsshould be Γmax/Γmin>10.\nInview ofthese results,we have alsomade anextensive anal-\nysisofthedependence ofthedynamice fficiencyonthevariationof\n∆g. Since we are also interested in evaluating the influence of t he\nmagnetic fields onthe results, we define a new variable\n∆s=1+σL\n1+σR, (20)\nc/circleco√yrt0000 RAS, MNRAS 000, 000–0006P.Mimica andM. A. Aloy\nFigure4. Thegray scale indicates thevalue ofthemaximumtotal dynam ic\nefficiency (inpercent) asafunction ofthe parameter pair ( ∆g,∆s).Theval-\nues of the rest of the parameters are fixed to ΓR=100, andχ=10−4.\nContours: magnetization of the slowest shell: σR=0.1,0.5,1, 5and 10.\nand plot (Fig. 4) the value of the maximum e fficiency reached for\nevery combination ( ∆g,∆s) and fixed values of the rest of the pa-\nrameters toΓR=100, andχ=10−4. To be more precise, for\nfixed∆gand∆swe need to look for the maximum of the ef-\nficiency of all models whose σLandσRsatisfy Eq. (20). It is\nevident from Fig. 4 that the maximum total dynamic e fficiency\ngrows (non-monotonically) with increasing ∆g, in agreement with\nthe above mentioned works (where unmagnetized collisions h ave\nbeen considered). Indeed, a large value ∆g/greaterorsimilar10 yields dynamic\nefficiency values∼40% if both shells are moderately magnetized\n(σR∼σL/lessorsimilar0.1).Nevertheless,theamountofincreaseofe fficiency\nwith∆gdepends strongly on ∆s. For|∆s|/greaterorsimilar1, corresponding to\ncases where the slower shell is highly magnetized ( σR/greaterorsimilar4), the\nmaximum dynamic e fficiency is almost independent of ∆g; while\nfor|∆s|/lessorsimilar1, the maximum dynamic e fficiency displays a strong,\nnon-monotonic dependence on ∆g.\nIt is remarkable that values 5 /lessorsimilar∆s/lessorsimilar100 yield dynamic ef-\nficiencies in excess of ∼20%, regardless of the relative Lorentz\nfactor betweenthetwoshells.Inthis regionofthe paramete r space\nthemaximaldynamice fficiencyhappenswhenbothshellsaremag-\nnetized (σR>10,σL>50), and the total dynamic magnetic e ffi-\nciency dominates the total dynamic e fficiency.\n5 DISCUSSION\nWe have focused in this paper on the estimation of the dynamic\nefficiency of conversion of kinetic-to-thermal /magnetic energy in\ncollisions (internal shocks) of magnetized shells in relat ivistic out-\nflows. A fundamental di fference between the internal collisions in\nmagnetized and unmagnetized outflows is the fact that in the f or-\nmer case not only shocks but alsorarefactions canform. Thus , one\nwouldnaturallyexpectareduceddynamic e fficiencyinthemagne-\ntized case. However, we have shown that such dynamic e fficiencymay reach values∼10%−40%, in a wide range of the parame-\nter space typical for relativistic outflows of astrophysica l interest\n(blazars and GRBs). Thus, the dynamic e fficiency of moderately\nmagnetized shell interactions is larger than in the corresp onding\nunmagnetizedcase.Thisisbecause whentheshellsaremoder ately\nmagnetized, most of the initial shell kinetic energy is conv erted to\nmagnetic energy, rather thantothermal energy.\nThe difference in efficiency between flows with moderate\nLorentz factors (ΓR=10) and ultrarelativistic ones ( ΓR=100)\nis very small, because in the ultrarelativistic kinematic l imit (i.e.,\nΓ≫1), once the energy fluxand the magnetizations of both shells\narefixed,thekeyparametergoverningthedynamice fficiencyis∆g\nrather thanΓR. From numerical simulations one expects that any\nefficiently accelerated outflow will not display huge variation s in\nthe velocity between adjacent regions of flow. Therefore, va lues of\n∆g≃1 seem reasonable and ∆g=1 has been taken as a typi-\ncal value for both blazars and GRB jets. A fixed value of ∆g=1\nbrings maximum e fficiency when the magnetizations of the collid-\ningshells are (σL,σR)≃(1,0.2). Largerdynamic e fficiency values\n∼40% are reached by magnetized internal shocks if ∆g/greaterorsimilar10 and\n|∆s|/lessorsimilar1, corresponding to cases where the magnetization of both\nshells ismoderate ( σR≃σL/lessorsimilar0.1).\nConsistent with our previous work (Mimica etal. 2007), in\nthe limit of low magnetization of both shells, the kinetic en ergy\nis mostly converted into thermal energy, where the increase d mag-\nnetic energy in the shocked plasma is only a minor contributi on\nto the total dynamic e fficiency, i.e.,εT≪εM. Here we find that\nas the magnetization of the shells grows, the roles of εTandεM\nare exchanged, so that εT< εM(at the maximum dynamic e ffi-\nciencyεT≃0.1εM). If the magnetization of both shells is large,\nthe dynamic efficiency decreases again because producing shocks\nin highly magnetized media is very di fficult. All these conclusions\nare independent on the EoS used to model the plasma, i.e., the y\nare both qualitatively and quantitatively basically the sa me inde-\npendent on whether a polytropic EoS with a fixed adiabatic ind ex\nis taken (either ˆγ=4/3 or ˆγ=5/3) or a more general, analytic\napproximation to the exact relativistic EoS (the TMEoS; see Ap-\npendix) isconsidered.\nThe comparison of our resultswithprevious analytic or semi -\nanalytic works (e.g.,KPS97; Beloborodov 2000; Kobayashi & Sari\n2001; Kobayashi etal. 2002) is not straightforward. Genera lly,\nthese works compute the e fficiency of the collision of shells with-\noutcomputingtheir(magneto-)hydrodynamicevolutionand ,onthe\nother hand, these works include not only a single collision, but the\nmultiple interactions of a number of dense shells. The botto m line\nin these previous works is that internal collisions of unmag netized\nshells can be extremely e fficient; the efficiency exceeding 40%, or\neven∼100% (Beloborodov 2000) if the spread of the Lorentz fac-\ntor(i.e.,theratiobetweentheLorentzfactorofthefaster ,Γmax,and\nof the slowerΓminshell in the sample) is large ( Γmax/Γmin=103;\ne.g., KPS97, Kobayashi et al. 2002). For a more moderate spre ad\nof the Lorentz factor Γmax/Γmin=10, the efficiency is∼20%. We\nnotethatthesehighe fficienciesarereachedbecausealargenumber\nof binary collisions is included in the model (not only a sing le one\nas in our case). Thus, the kinetic energy which is not dissipa ted in\nthefirstgenerationofcollisions(betweentheinitiallyse tupshells),\ncan be further converted into internal energy as subsequent gener-\nations of collisions take place. In contrast, we find that mod erate\nmagnetizations ofbothshells( σ/lessorsimilar0.1)and∆g/greaterorsimilar10(whichwould\nroughly correspond to Γmax/Γmin=9) are enough for a single bi-\nnarycollisiontoreach atotal dynamic e fficiencyof∼40%.\nWepointoutthattheenergyradiatedinthecollisionofmagn e-\nc/circleco√yrt0000 RAS,MNRAS 000, 000–000Efficiency ofinternalshocks 7\ntizedshells isonlya fraction, fr≃0.1(e.g.,Panaitescuet al.1999;\nKumar 1999) to fr≃1 (e.g., Beloborodov 2000) of the energy dy-\nnamicallyconverted intothermal or magnetic energy. Thus, the ra-\ndiativeefficiencyofthe process, measuredasthefractionoftheto-\ntalinitialenergyconvertedintoradiation,willbe1 /frtimessmaller\nthanthecomputeddynamice fficiency.Evenconsideringthisfactor,\nsinglebinarycollisionsbetweenmoderatelymagnetizedsh ellsmay\nyield efficiencies∼0.4fr, which can obviously rise if many binary\ncollisions take place inthe flow reprocessing the remaining kinetic\nenergy of the first generation of interacting shells (in the s ame sta-\ntisticalwayasdiscussedbyKobayashi & Sari2000).Therefo re,on\nthe light of our results, binary collisions in relativistic magnetized\nflowsareefficientenough,fromthedynamicalpointofview,tobea\nvalid mechanism to dissipate the bulk kinetic energy of rela tivistic\nejecta.Hence,themainrestrictionontheradiativee fficiencycomes\nfrom the radiationmechanism settingthe limitingfactor fr.\nWe stress that we are not assuming any particular radi-\nation mechanism in this study (thus, determining a value for\nfr). Therefore, we compute the dynamic e fficiency including not\nonly the increased thermal energy in the flow, but also the ex-\ntra magnetic energy resulting from the magnetic field compre s-\nsion. This is justified because, although standard shock acc elera-\ntion mechanisms are ine fficient in very magnetized shocks (e.g.,\nSironi& Spitkovsky 2009), other mechanisms might extract t he\nenergy from the whole volume of a very magnetized fluid (e.g.,\nThompson 1994;Giannios &Spruit 2007).\nThe estimated dynamic e fficiency in the binary collision of\nmagnetized shells will be completed in a future work by accou nt-\ningforthenumericalMHDevolutionofsuchbuildingblockso fthe\ninternal shock models. A step further would be to compute the ra-\ndiativeefficiencyusingthemethoddevisedinMimica et al.(2009).\nACKNOWLEDGMENTS\nMAA is a Ram´ on y Cajal Fellow of the Spanish Ministry of Ed-\nucation and Science. We acknowledge the support from the Spa n-\nish Ministry of Education and Science through grants AYA200 7-\n67626-C03-01 and CSD2007-00050. We thank Jos´ e-Mar´ ıa Mar t´ ı\nand Jos´ e-Mar´ ıa Ib´ a˜ nez for their support and critical di scussions.\nThe authors thankfully acknowledge the computer resources , tech-\nnical expertise and assistance provided by the Barcelona Su per-\ncomputing Center - CentroNacional de Supercomputaci´ on.\nAPPENDIXA: EQUATIONOFSTATE\nIn this Appendix we discuss the e ffects of using a more re-\nalistic equation of state on our results. We choose the TM\nanalytic approximation to the Synge equation of state (EoS)\n(de Berredo-Peixoto etal. 2005; Mignone et al. 2005). In the TM\nEoS the specific enthalpy can be written as (using the notatio n of\nSection2.1)\nh∗\nTM:=5\n2p∗\nρ−σ\n4+9\n4/parenleftBiggp∗\nρ−σ\n2/parenrightBigg2\n+11/2\n. (A1)\nand thespecific internal energy\nǫTM:=3\n2p\nρ+9\n4/parenleftBiggp\nρ/parenrightBigg2\n+11/2\n−1. (A2)Figure A1. Sameas Fig. 2, butfor the TMequation of state.\nFrom (A1) it can be seen that the limit σ=0 the effective adia-\nbatic index of this EoS lies between 4 /3 and 5/3. We modified the\nRomeroet al.(2005) solver toinclude the TMEoS.\nOn Fig. A1 we show the dynamical e fficiency in the GRB\nregimeusingthe TMEoS.ComparisonofFig.2andFig.A1shows\nthat,overall, the dynamical e fficiencyis higher when usingthe TM\nEoS, but the qualitative features remain the same in both cas es.\nAlso, as expected, in the highly magnetized regime the di fferences\nare minor, since inboth cases (polytropic or TMEoS)the effective\nadiabatic index approaches 2 in such a regime. Figure A2 show s\nthe influence of the EoS on the existence of the FS. The only dif -\nference between models with di fferent EoS appears in the region\nwherethefastershellisweaklymagnetized.Thereaslightl yhigher\n(lower) magnetization of the slower shell is needed to suppr ess the\nFSwhenusing TMthanwhenusingapolytropicEoSwith ˆ γ=4/3\n(ˆγ=5/3). The overlap of all three curves in Fig. A2 in the limit\nof high magnetization of both shells, shows again the result that\nthe choice of EoS plays no role in such a regime, as expected. W e\nnote that the separatrix between the regions of existence an d non-\nexistence of the FS corresponding to the case ˆ γ=5/3 lies closer\ntothat of the TMEoS than that corresponding to the case ˆ γ=4/3.\nThis is a natural consequence of the fact that the initial she lls are\nboth cold, and thus, the e ffective adiabatic index is closer to 5 /3\nthanto4/3, at lowmagnetizations.\nWe note that in the case of the external shocks, where rela-\ntivistic fluid encounters a cold external medium the choice o f the\nrealistic EoS can influence results much more dramatically t han in\nthe case of the internal shocks (see e.g.,Meliani et al.2008 ).\nREFERENCES\nAloyM. A.,Janka H.-T.,M¨ uller E.,2005, A&A, 436, 273\nAloyM. A.,Mimica P.,2008, ApJ, 681, 84\nAloy M. A., M¨ uller E., Ib´ a˜ nez J. M., Mart´ ı J., MacFadyen A .,\n2000, ApJL, 531, L119\nAloyM. A.,Rezzolla L.,2006, ApJ, 640, L115\nc/circleco√yrt0000 RAS, MNRAS 000, 000–0008P.Mimica andM. A. 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Moler1, 2\n1Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory,\n2575 Sand Hill Road, Menlo Park, CA 94025, USA\n2Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA\n3Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA\n4Department of Chemistry, Princeton University, Princeton, New Jersey 08540, USA\n(Dated: March 28, 2019)\nHolmium titanate (Ho 2Ti2O7) is a rare earth pyrochlore and a canonical example of a classical\nspin ice material. Despite the success of magnetic monopole models, a full understanding of the en-\nergetics and relaxation rates in this material has remained elusive, while recent studies have shown\nthat defects play a central role in the magnetic dynamics. We used a scanning superconducting\nquantum interference device (SQUID) microscope to study the spatial and temporal magnetic \ruc-\ntuations in three regions with di\u000berent defect densities from a Ho 2Ti2O7single crystal as a function\nof temperature. We found that the magnetic \rux noise power spectra are not determined by sim-\nple thermally-activated behavior and observed evidence of magnetic screening that is qualitatively\nconsistent with Debye-like screening due to a dilute gas of low-mobility magnetic monopoles. This\nwork establishes magnetic \rux spectroscopy as a powerful tool for studying materials with complex\nmagnetic dynamics, including frustrated correlated spin systems.\nKeywords: Condensed matter physics\nI. INTRODUCTION\nClassical spin ices such as holmium titanate\n(Ho2Ti2O7) have generated intense interest, both the-\noretical and experimental, in the last two decades.1{11\nThe pyrochlore lattice of corner-sharing tetrahedra hosts\na magnetic holmium atom at each shared vertex, and\nthe local crystal \feld environment causes each holmium\nmoment to point directly into one or the other of the\nadjacent tetrahedra. The term \\spin ice\" comes from\nthe analogy between the ground state manifold, which\nhas two spins pointing in and two spins pointing out of\neach tetrahedron (the ice rule ), and the freezing of water\nice, in which the two electron lone pairs on the oxygen\natom line up with the hydrogen atoms of adjacent water\nmolecules.1\nA single spin \rip from the ground state manifold re-\nsults in an excitation that is dipole-like. A subsequent\n\rip of a nearest-neighbor spin can restore the ice rule in\none of the tetrahedra while violating it in the adjacent\ntetrahedron, extending the dipole to next nearest tetra-\nhedra. Continuing this process, one end of the dipole\ncan be taken away entirely, leaving a single monopole-\nlike excitation behind.8Much of the recent theoretical\nwork has focused on these emergent magnetic monopole\nmodels.3,5,6,8,9,12{14Nevertheless, it has subsequently be-\ncome clear that defects, such as oxygen vacancies and\nstu\u000bed spins (additional spins from Ho atoms occupying\nTi sites), must be accounted for in understanding the full\nmagnetic dynamics of spin ice.15{17Progress in under-\nstanding these dynamics has been limited by a relative\nlack of suitable tools for microscopic magnetic studies.\nIn this paper, we demonstrate the utility of magnetic\n\rux noise spectroscopy as such a tool for studying frus-\ntrated correlated spin systems. We used a scanning su-perconducting quantum interference device (SQUID) mi-\ncroscope with a gradiometric SQUID magnetometer that\nhas been described previously.18With a spatial resolu-\ntion of 4:6\u0016m and a magnetic \rux noise \roor of or-\nder 1\u0016\b0=p\nHz, we measured the temperature depen-\ndence of spatial magnetic correlations and of the mag-\nnetic \rux noise spectrum in three regions with di\u000berent\ndefect densities from a Ho 2Ti2O7single crystal. We ob-\nserved qualitative deviations from simple thermally acti-\nvated behavior, which would predict a Lorentzian noise\nspectrum with a characteristic time that follows an Ar-\nrhenius law in temperature. Furthermore, we found ev-\nidence of screening at low frequencies and high tem-\nperatures, which we compare to a model for Debye-like\nscreening from the theoretically predicted gas of mag-\nnetic monopoles.19,20\nII. EXPERIMENTAL SETUP\nWe measured two samples from a single crystal grown\nvia the \roating-zone method which has previously been\ncharacterized elsewhere;17sample A is from near the cen-\nter of the growth boule while Sample B is from near\nthe edge, where the density of defects was somewhat\nhigher. Sample A was transparent pink, while Sample\nB was cloudy but translucent pink, with a dark, opaque\nregion at one corner. We measured both regions of Sam-\nple B, which we will subsequently refer to as Samples B1\nand B2, respectively. We fractured sample A to obtain\na smooth but not \rat surface with roughly [111] orienta-\ntion. For Sample B, we prepared a polished [111] surface\nwith<1\u0016m grit polishing \flm and isopropanol.\nFor two-dimensional image data, we de\fned the scan\nsurface by determining the height at which the SQUID\nwas in contact with the sample at a series of locationsarXiv:1903.11465v1  [cond-mat.str-el]  27 Mar 20192\n0.43 K 18 hours later\n145 µm\n1 hour later\n-1\n/uni0394Φ (mΦ0)1\n0\nFIG. 1. Magnetometry scans of Sample A at 430 mK taken over 18 hours. Overlay in \frst panel is scale drawing of SQUID\npickup loop (orange) and \feld coil (blue). The pickup loop size sets the spatial resolution of the images; resolution-limited\nfeatures in the scans are qualitatively similar from one scan to the next, but are di\u000berent in the details. This demonstrates\nthat magnetic dynamics persist, even over long timescales and at the lowest measured temperatures, and that the magnetic\ntexture fails to order or otherwise converge.\nand \ftting a two-dimensional, second order polynomial to\nthe surface topography. We rastered the SQUID parallel\nto this surface at a nominal height of 1 \u0016m. For one\ndimensional scans, we acquired a single row of the image\nrepeatedly to discern the time evolution.\nWe obtained magnetic \rux noise power spectra by\nplacing the SQUID in contact with the sample and col-\nlecting each spectrum with an SR760 FFT Spectrum An-\nalyzer in three overlapping segments (3.8 mHz{1.52 Hz,\n488 mHz{195 Hz, and 125 Hz{49.9 kHz), using a Han-\nning window function and 128 exponentially weighted av-\nerages.\nIII. RESULTS\nWe present magnetometry scans of Sample A taken at\nour base temperature of 430 mK in Fig. 1. At this tem-\nperature, features which are limited by the spatial resolu-\ntion of the SQUID magnetometer (4.6 \u0016m) dominate each\nscan, suggesting that the dynamics are slow compared to\nthe scan speed. Repeated scans appear as di\u000berent pan-\nels, at times indicated at the top of each panel, showing\nlong timescale \ructuations despite qualitative similarity\nfrom scan to scan. As expected for a truly frustrated spin\nsystem, the sample fails to show convergence or ordering\nof the magnetic texture even over the hold time of our\ncryostat, in excess of two days, at 430 mK.\nIn magnetometry scans conducted as a function of tem-\nperature, from base temperature to 810 mK, we observe\nthat magnetic dynamics quicken as the temperature is in-\ncreased. In Fig. 2, we show both two-dimensional scans\n[Fig. 2(a)], each acquired over several minutes, and one-\ndimensional scans as a function of time [Fig. 2(b)], each\nacquired over 200 minutes with each row taking approxi-\nmately 12 seconds, taken at various temperatures. As in\nFig. 1, scans at the lowest temperatures (\frst panels in\neach part of the \fgure) show resolution-limited features,implying that temporal dynamics occur on timescales\nlong compared to the sampling rate. The \frst panel\nin Fig. 2(a) shows this explicitly, as the features persist\nover many rows, suggesting a correlation time of order\nhours. As the temperature is increased, the features in\nthe scans in Fig. 2(a) vary on shorter length scales, indi-\ncating that there are faster temporal variations coming\ninto play. This is made manifest by comparing adjacent\nrows in the various panels of Fig. 2(b), where the corre-\nlation time falls to order seconds by 610 mK. Due to the\nlimited scan speed of the SQUID, these measurements\ncannot resolve magnetic dynamics at temperatures above\n1 K, as it is di\u000ecult to unambiguously distinguish spatial\nand temporal variations. To measure at higher tempera-\ntures, we instead \fx the position of the SQUID in contact\nwith the sample surface and measure the magnetic \rux\nas a function of time only. By taking the Fourier trans-\nform of time series data, we obtain a magnetic \rux noise\npower spectrum.\nThe key results of this paper are contained in plots of\nthe natural logarithm of the magnetic \rux noise power\nspectra, in units of \b2\n0=Hz, as a function of the standard\nlogarithm of frequency, log( f), and the inverse tempera-\nture, 1=T. Were the sample an ensemble of identical but\nnon-interacting, thermally-excited Ising spins, we would\nobserve a Lorentzian noise spectrum, S\b=c\u001c=(1+!2\u001c2),\nwithcconstant,!the measurement frequency, and \u001ca\ntemperature-dependent characteristic time. The thermal\nexcitation over the Ising barrier would yield an Arrhenius\nlaw for that temperature dependence, \u001c=\u001c0eEa=kBT,\nwhere\u001c0is a microscopic attempt time, Eais an activa-\ntion energy, kBis the Boltzmann constant, and Tis the\ntemperature. In this illustrative example, the contours\nof constant noise power would be vertical at high tem-\nperatures and linear at low temperatures. For Lorentzian\nnoise spectra, the noise power monotonically increases for\ndecreasing frequency, down to a characteristic frequency\nat which it plateaus. The line formed by the maxima of3\n0.43 K\n0.81 K 0.79 K 0.77 K 0.75 K 0.73 K0.71 K 0.69 K 0.67 K 0.65 K 0.63 K0.61 K 0.59 K 0.57 K 0.55 K 0.53 K0.51 K 0.49 K 0.47 K 0.45 Ka.\n-1\n/uni0394Φ\n(mΦ0)1\n0\nb.\n0.47 K 0.50 K 0.52 K 0.55 K 0.58 K 0.61 K40 µm\n40 µm\n200 min\nFIG. 2. Magnetometry scans of Sample A as a function of\ntemperature. (a) Two-dimensional scans, as in Fig. 1, from\nbase temperature to 810 mK. As the temperature increases,\nthe observed \ructuations become sub-resolution, suggesting\nthat there are temporal \ructuations that are fast compared\nto the scan speed. (b) One-dimensional scans vs. time from\n470 mK to 610 mK. Each series is 200 minutes long, and the\nvertical correlations of pixels from row to row characterize the\ncorrelation time, which is of order hours at 470 mK but falls\nto seconds by 610 mK.\n0.5 1 1.5 2\n1/T [1/K]-2-101234log10(f [Hz])\n-18-14-10-6\nlog10(Flux PSD [Φ02/Hz])\nTmax[21]\nTmax[22]\nTmax[23]\nfmax[23]\nTmaxFIG. 3. Magnetic \rux noise power spectra as a function of\ntemperature for Sample A. Overlaid red line is a linear \ft to\nmaxima for each row, while markers indicate ac susceptibility\ndata from previous works for comparison.\nhorizontal line cuts gives the Arrhenius law: the slope\ngivesEawhile the intercept is related to \u001c0.\nIn Fig. 3, we show the magnetic \rux noise spectra for\nSample A from 430 mK to 4 K. The data are manifestly\nnon-Arrhenius: the contours of constant noise power at\nlow temperatures \rare out, with the noise power from\n1{100 Hz at 430 mK higher than would be expected for\na Lorentzian following an Arrhenius law. For tempera-\ntures above 650 mK, the \rux noise power as a function of\ndecreasing frequency reaches a maximum and then falls\no\u000b sharply at lower frequencies, below 1 Hz at 1 K. This\nfeature suggests that, regardless of whether the observed\nmagnetic \rux noise spectra result from the dynamics of\nmagnetic monopoles or some other microscopic origin,\nthere is a source of magnetic screening within the sam-\nple. The overlaid red line in Fig. 3 is a best \ft line (Ar-\nrhenius law) to the maxima of each row. Comparing the\nextracted Arrhenius law to previously reported bulk ac\nsusceptibility data,21{23we \fnd that it is in close agree-\nment, suggesting that we are measuring the same mag-\nnetic dynamics as have previously been reported.14,21{31\nTo understand the impact on the magnetic dynamics of\ndefects, such as those introduced by additional magnetic\nholmium atoms on titanium sites (stu\u000bed spins),16,17,32,33\nwe measured two regions from an additional sample\n(Sample B) taken from nearer to the edge of the growth\nboule. The additional \rux noise spectra are shown in\nFig. 4, together with those from Sample A. The predom-\ninant Arrhenius-like feature smears out considerably and\nbecomes somewhat less steep as the defect density is in-\ncreased, implying a broadening distribution of activation\nenergies that are lower on average. This is consistent with\nexpectations for increased defect densities, as magnetic\ndisorder broadens and reduces the barrier for individual\nspins in the sample to \rip.\nThe qualitative deviations from Arrhenius behavior\nseen in Sample A can be seen more clearly in Sample B.4\n0.5 1 1.5 2\n1/T [1/K]-2-101234log10(f [Hz])\n-18-14-10-6\nlog10(Flux PSD [Φ02/Hz])\n0.5 1 1.5 2\n1/T [1/K]-2-101234log10(f [Hz])\n-18-14-10-6\nlog10(Flux PSD [Φ02/Hz])a.\nb.\nFIG. 4. Comparison of magnetic \rux noise power spectra\nfrom two locations on second sample (a. B1; b. B2), demon-\nstrating di\u000berent defect densities. The central, Arrhenius-like\nfeature seen in Sample A is also seen in both data sets; the\nfeature becomes broader, more di\u000buse, and more shallow as\nthe defect density is increased, consistent with the barrier to\nspin \rips shortening and becoming broadened with increased\nmagnetic disorder. Qualitative deviations from Arrhenius be-\nhavior, seen as excess noise at center left and bottom right\nof each panel, and screening at bottom left of each panel, ap-\npear in all data sets and are quantitatively similar in their\nfrequency and temperature dependence. These comparisons\nsuggest that the central, Arrhenius-like feature is a\u000bected by\nmagnetic defects, while the deviations from Arrhenius behav-\nior are universal.\nThe \raring out of the contours of constant noise power\nat low temperatures have corresponding features at high\ntemperatures, seen above 1.3 K from 0.1{100 Hz. The\ncorrespondence of the excess noise features at the highest\nand lowest temperatures suggests that there is an addi-\ntional source of magnetic dynamics in these samples at\nlower frequencies than has been accessible in previous ac\nsusceptibility measurements. Furthermore, we see that\nthe screening behavior in Sample A is also seen in Sam-\nple B, and that it is quantitatively comparable across all\nthree samples, independent of the defect density.IV. DISCUSSION\nHaving shown the \rux noise spectra from our samples,\nwe now turn to the question of what we expect from a\ndilute gas of monopoles. The basic form of our model is\na Lorentzian noise term (this form for the noise due to\nmagnetic monopoles was previously robustly justi\fed by\nRyzhkin20), modi\fed by a Debye-like screening term:\nS\b=C\u0012\u001cMon\n1 +!2\u001c2\nMon\u0013!2=!2\nc\n1 +!2=!2c(1)\nwhereCis an overall scaling constant, !is the angular\nfrequency,!cis a characteristic cuto\u000b frequency for De-\nbye screening as described below, and \u001cMon is the char-\nacteristic time associated with spin relaxation, \u001cMon =\n\u001cMon; 0=x(T), where\u001cMon; 0is the microscopic monopole\nhopping time and x(T) is the temperature-dependent\nmonopole concentration. This relaxation time is respon-\nsible for monopole hopping by way of the \ructuation-\ndissipation theorem.19\nThe Debye-H uckel model, as applied to the case of\nmagnetic monopoles in spin ice by Castelnovo, Moess-\nner, and Sondhi,8implies that the magnetic \felds due\nto a source magnetic charge will be screened by a cloud\nof magnetic monopoles equal and opposite in charge to\nthe source. This screening occurs over a length scale, the\nDebye length lD, which is given by:\nlD=s\nkBTV0\n\u00160Q2x(2)\nwithTthe temperature, V0the volume of the dia-\nmond lattice site, Qthe monopole charge, and xthe\nmonopole concentration. The Debye-H uckel concentra-\ntion for monopoles, x(T), can be calculated iteratively\nas described in Ref.8.\nThe Debye length is of order 50 nm for the lowest tem-\nperatures at which we performed \rux spectroscopy, and\nmonotonically decreases as the temperature rises, such\nthat it is always far smaller than the spatial resolution\nfor the SQUID. This suggests that in the presence of\nmonopoles there would be no observable magnetic \ructu-\nations whatsoever; however, because the monopoles have\na \fnite mobility, only slowly varying magnetic \felds are\nscreened.\nThe mobility, \u0016, appears in the characteristic Debye\nfrequency by way of the di\u000busivity, D, and the Einstein\nrelation:\n!c=D\nl2\nD=\u0016kBT\nl2\nD: (3)\nThe mobility itself has been calculated from Monte Carlo\nsimulations:8\n\u0016=4\n27a2\nd\nkBT\u001c(4)5\n0.5 1 1.5 2\n1/T [1/K]-2-101234log10(f [Hz])\n-18-14-10-6\nlog10(Flux PSD [Φ02/Hz])\n10/one.superior s10⁰ s10/hyphen.superior/one.superior s10/hyphen.superior/two.superior s10/hyphen.superior/three.superior s\nFIG. 5. Temperature dependence of the Debye screening criti-\ncal frequency, !c, for di\u000berent values of the monopole hopping\ntime. The contours indicate !c(T) for each hopping time, as\nlabeled.\nwhereadis the lattice constant and \u001cis the Monte Carlo\nstep time, which we identify as the same magnetic relax-\nation time as is used in the noise calculation above.\nSince the Debye-H uckel concentration can be com-\nputed directly, the noise and screening due to magnetic\nmonopoles can be modeled with only two free param-\neters, the microscopic monopole hopping time and an\noverall scale factor that takes into account geometric fac-\ntors that couple the \ructuating population of monopoles\nand the SQUID magnetometer.\nIn Fig. 5, we overlay contours that give the characteris-\ntic frequency as a function of temperature, !c(T), for the\nDebye screening for various values of the monopole hop-\nping time, as indicated. Noting that the characteristic\nfrequency is where the screening is of order unity, while\nthe blue region in the bottom left corner of Fig. 5 is where\nthe noise is already reduced by orders of magnitude, we\nidentify the monopole hopping time as 1{10 ms, in agree-\nment with some previous measurements.6,23,26,34,35We\nalso note the qualitative agreement between the data and\nplotted contours for the temperature dependence of the\nscreening.\nTaking the monopole hopping time \u001c0= 3 ms, we plot\nthe full monopole model including the noise and screen-\ning terms in Fig. 6, in arbitrary units. We see that the\nmonopole dynamics account not only for the screening at\nhigh temperatures and low frequencies, but can also qual-\nitatively account for the non-Arrhenius source of noise\nas well. A full modeling of the measured noise spec-\ntrum would also require a model for the Arrhenius-like\nnoise behavior. The defect series that we have measured\nhere suggests that this noise feature is due to defects,\nmost likely stu\u000bed spins. Previous studies have shown\nthat even nominally stochiometric Ho 2Ti2O7grown by\nthe \roating zone method contains approximately 3% Ho\nstu\u000eng on the Ti site, or roughly 0.06 stu\u000bed Ho spins\nper tetrahedron.17Given that this exceeds the calcu-\n0.5 1 1.5 2\n1/T [1/K]-2-101234log10(f [Hz])FIG. 6. A model of the expected magnetic dynamics for\nmonopoles in Ho 2Ti2O7. A band of noise, plotted in arbi-\ntrary units, is present above the critical frequency, as the\nmonopole gas is too dilute and immobile to completely screen\nitself. The monopole hopping time used here is 3 ms and is\nthe only free parameter; the resulting model is qualitatively\nconsistent with all deviations from Arrhenius behavior in our\ndata.\nlated monopole density at all but the highest temper-\natures measured (at which the calculated monopole den-\nsity reaches nearly 0.15 monopoles/tetrahedron), it is un-\nsurprising that defect dynamics would produce magnetic\n\rux noise of a similar magnitude to monopoles.\nOne possible route towards modeling these dynamics\nwould be to calculate the distribution of activation ener-\ngies for stu\u000eng defects in a transverse \feld Ising model.\nA holmium atom on a titanium site has 6 nearest neigh-\nbor holmium spins that form a closed hexagon in the py-\nrochlore lattice. In their normal Ising orientations, these\nspins each provide an in-plane \feld for the defect spin.\nIf the ice state manifold is taken into account in deter-\nmining the frequency with which di\u000berent orientations\nof these nearest neighbor spins will occur, a distribu-\ntion of activation energies could be calculated. However,\nthis would not yield the distribution of attempt times\nwhich is also necessary for a full accounting of the mag-\nnetic dynamics of the defects. Other defects, such as\nnon-magnetic substitutions on the holmium site or oxy-\ngen vacancies,16could also give rise to magnetic dynam-\nics that would not be accounted for in this model, and\n\u001c0could also be temperature dependent for other rea-\nsons not considered here, such as non-trivial spin-phonon\ncoupling.8\nV. CONCLUSION\nWe have demonstrated the utility of scanning SQUID\nmagnetic \rux spectroscopy by measuring the magnetic\n\rux noise power spectra as a function of temperature in\nthree locations on two samples of the classical spin ice6\nHo2Ti2O7. In these measurements, we observe a dom-\ninant Arrhenius-like feature that matches the behavior\nobserved in previous bulk ac susceptibility measurements\non similar samples. We identify this feature as the result\nof the magnetic dynamics of defects in the sample, which\nwe speculate are stu\u000bed spins.\nWe further identify three qualitative deviations from\nArrhenius behavior in all three datasets, namely excess\nnoise below 10 Hz at the lowest temperatures and be-\nlow 100 Hz at the highest temperatures and screening of\nthe noise at high temperatures and low frequencies. We\n\fnd that all three of these behaviors are consistent with\nthe expected dynamics of a dilute, low-mobility gas of\nmagnetic monopoles.\nOur measurements represent a new technique that is\ncomplementary to existing magnetic probes used in the\nstudy of frustrated magnetic systems. We demonstrate\nthe importance of quantitative modeling for the mag-\nnetic dynamics of defects in these systems and the utility\nof scanning SQUID magnetic \rux spectroscopy in disen-tangling the overlapping magnetic signals of such defects\nand the essential physics of the system under study, with\npotential further applications in the study of other, re-\nlated magnetic systems such as spin liquids.\nACKNOWLEDGMENTS\nWe thank C. Castelnovo, B. Gaulin, M. Gingras, G.\nLuke, R. Moessner, H. Noad, J. Rau, K. Ross, and S.L.\nSondhi for helpful discussions. This work was primar-\nily supported by the Department of Energy, O\u000ece of\nScience, Basic Energy Sciences, Materials Sciences and\nEngineering Division, under Contract No. DE-AC02-\n76SF00515; Ilya Sochnikov was partially supported by\nthe Gordon and Betty Moore Foundation through grant\nGBMF3429. The crystal growth was supported by the\nUS Department of Energy, Division of Basic Energy Sci-\nences, grant de-sc0019331.\n\u0003These two authors contributed equally\n1B. C. den Hertog and M. J. P. Gingras, Phys. Rev. Lett.\n84, 3430 (2000).\n2S. T. Bramwell, M. J. Harris, B. C. den Hertog, M. J. P.\nGingras, J. S. Gardner, D. F. McMorrow, A. R. Wildes,\nA. L. Cornelius, J. D. M. Champion, R. G. Melko, and\nT. Fennell, Phys. Rev. Lett. 87, 047205 (2001).\n3T. Fennell, P. P. Deen, A. R. Wildes, K. Schmalzl, D. Prab-\nhakaran, A. T. Boothroyd, R. J. Aldus, D. F. McMorrow,\nand S. T. Bramwell, Science 326, 415 (2009).\n4D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke,\nC. 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Avishai2,3,4, Alexander Shnirman3,5,6\n1Department of Chemistry, Department of Physics,\nDepartment of Electro-Optics, and the Ilse Katz Center for N ano-Science,\nBen-Gurion University, Beer-Sheva 84105, Israel\n2New York University and the NYU-ECNU Institute of Physics at NYU Shanghai,\n3663 Zhongshan Road North, Shanghai, 200062, China\n3Department of Physics, and the Ilse Katz Center for Nano-Sci ence,\nBen-Gurion University, Beer-Sheva 84105, Israel\n4Yukawa Institute for Theoretical Physics, Kyoto, Japan\n5Institut f¨ ur Theorie der Kondensierten Materie,\nKarlsruhe Institute of Technology, D-76128 Karlsruhe, Ger many\n6Institute of Nanotechnology, Karlsruhe Institute of Techn ology, D-76344 Eggenstein-Leopoldshafen, Germany\nAn analysis of a single-domain magnetic needle (MN) in the pr esence of an external magnetic\nfieldBis carried out with the aim of achieving a high precision magn etometer. We determine the\nuncertainty ∆ Bof such a device due to Gilbert dissipation and the associate d internal magnetic\nfield fluctuations that give rise to diffusion of the MN axis dir ectionnand the needle orbital angular\nmomentum. The levitation of the MN in a magnetic trap and its s tability are also analyzed.\nA rigid single-domain magnet with large total spin,\ne.g.,S≃1012/planckover2pi1, can be used as a magnetic needle magne-\ntometer (MNM). Recently Kimball, Sushkov and Budker\n[1] predicted that the sensitivity of a precessing MNM\ncan surpass that of present state-of-the-art magnetome-\nters by orders of magnitude. This prediction motivates\nour present study of MNM dynamics in the presence of\nan external magnetic field B. Such analysis requires in-\nclusion of dissipation of spin components perpendicular\nto the easy magnetization axis (Gilbert damping). It is\ndue to interactions of the spin with internal degrees of\nfreedom such as lattice vibrations (phonons), spin waves\n(magnons), thermal electric currents, etc. [2, 3]. Once\nthere is dissipation, fluctuations are also present [6], and\nresult in a source of uncertainty that can affect the ac-\ncuracy of the magnetometer. Here we determine the un-\ncertainty in the measurement of the magnetic field by a\nMNM. We also analyze a related problem concerning the\ndynamics of the needle’s levitation in an inhomogeneous\nmagnetic field, e.g., a Ioffe-Pritchard trap [8].\nThe Hamiltonian for a MN, treated asa symmetric top\nwith body-fixed moments of inertia IX=IY≡ I ∝negationslash=IZ,\nsubject to a uniform magnetic field Bis,\nH=1\n2IˆL2+(1\n2IZ−1\n2I)ˆL2\nZ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nHR−(ω0//planckover2pi1)(ˆS·ˆn)2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nHA−ˆµ·B/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nHB,\n(1)\nwhere a hat denotes quantum operator. In the rotational\nHamiltonain HR,ˆLis the orbital angular momentum op-\nerator and ˆLZ=ˆL·ˆZis its component along the body-\nfixed symmetry axis. ˆSis the needle spin angular mo-\nmentum operator, and ˆnis the operator for nthat is the\nunit vector in the direction of the easy magnetization\naxis. The frequency appearing in the anisotropy Hamil-\ntonianHA[4] isω0= 2γ2KS/V, whereKis the strengthofthe anisotropy, Vis the needle volume, and γ=gµB//planckover2pi1\nis the gyromagnetic ratio, in which µBis the Bohr mag-\nnetron, and gis theg-factor (taken to be a scalar for\nsimplicity). In the expression for the Zeeman Hamilto-\nnianHB,ˆµ=gµBˆSis the magnetic moment operator.\nThe Heisenberg equations of motion are\n˙ˆS=−gµBB׈S+2ω0\n/planckover2pi1(ˆS׈n)(ˆS·ˆn),(2)\n˙ˆL=-2ω0\n/planckover2pi1(ˆS׈n)(S·ˆn), (3)\n˙ˆJ=−gµBB׈S, (4)\n˙ˆn=I−1\n/planckover2pi1[ˆL׈n+i/planckover2pi1ˆn], (5)\nwhereˆJ=ˆL+ˆSis the total angularmomentum operator\nandIis the moment of inertia tensor.\nThe dynamics of a MN can be treated semiclassically\nbecause Sis very large. A mean–field approximation\n[9–11] is obtained by taking quantum expectation values\nof the operator equations and assuming that for a given\noperator ˆA, the inequality/radicalBig\n∝angbracketleftˆA2∝angbracketright−∝angbracketleftˆA∝angbracketright2≪ |∝angbracketleftˆA∝angbracketright|holds,\n(an assumption warranted for large S). Hence, the ex-\npectation values of a product of operators on the RHS\nof Eqs. (2)-(5) can be replaced by a product of expecta-\ntion values. The semiclassical equations are equivalent\nto those obtained in a classical Lagrangian formulation.\nDissipation is accounted for by adding the Gilbert term\n[2, 4]−αS×(˙S//planckover2pi1−Ω×S//planckover2pi1) to the RHS of the expecta-\ntion value of Eq. (2) and subtracting it from the RHS of\nEq. (3). Here αis the dimensionless friction parameter,\nand the term Ω×Stransforms from body fixed to space\nfixed frames. Note that Gilbert damping is due to inter-\nnalforces, hence Jis not affected and Eq. (4) remains\nintact.2\nIt is useful to recast the semiclassical dynamical equa-\ntions of motion in reduced units by defining dimension-\nless vectors: the unit spin m≡S/S, the orbital angu-\nlar momentum ℓ≡L/S, the total angular momentum,\nj=m+ℓand the unit vector in the direction of the\nmagnetic field b=B/B:\n˙m=ωBm×b+ω0(m×n)(m·n)−αm×(˙m−Ω×m),(6)\n˙ℓ=−ω0(m×n)(m·n)+αm×(˙m−Ω×m),(7)\n˙n=Ω×n, (8)\n˙j=ωBm×b, (9)\nwhere the angular velocity vector Ωis given by\nΩ= (ω3−ω1)(ℓ·n)n+ω1ℓ\n= (ω3−ω1)[(j−m)·n]n+ω1(j−m).(10)\nHereωB=γ|B|is the Larmor frequency, ω1=S/IX,\nandω3=S/IZ. Similar equations were obtained in\nRef. [5], albeit assuming that the deviations of n(t) and\nm(t) frombare small. We show below that the dynam-\nics can be more complicated than simply precession of\nthe needle about the magnetic field, particularly at high\nmagnetic fields where nutation can be significant.\nFor the numerical solutions presented below we are\nguided by Ref. 1, which uses parameters for bulk cobalt,\nand take ω1= 100 s−1,ω3= 7000 s−1, anisotropy fre-\nquencyω0= 108s−1, Gilbert constant α= 0.01, tem-\nperature T= 300 K, and N=S//planckover2pi1= 1012. First, we elu-\ncidate the effects of Gilbert dissipation, and consider the\nshorttimebehaviorin aweakmagneticfield, ωB= 1s−1.\nThe initial spin direction is intentionally chosen notto be\nalong the easy magnetic axis; n(0) = (1 /2,1/√\n2,1/2),\nm(0) = (1 /√\n2,1/√\n2,0),ℓ(0) = (0 ,0,0). Figure 1(a)\nshows the fast spin dissipation as it aligns with the easy\naxis of the needle, i.e., m(t)→n(t) after a short time,\nand Fig. 1(b) shows relaxation of the oscillations in ℓ(t),\nwhileℓx(t) andℓy(t) approach finite values. Figure 1(c)\nshowsthe innerproduct m·n, which clearlytendstounity\nonthe timescaleofthe figure. Increasing αleadsto faster\ndissipation of m(t), but the short-time saturation values\nof bothm(t) andℓ(t) are almost independent of α.\nWe consider now the long time dynamics (still in\nthe weak field regime) and take the initial value of the\nspin to coincide with the easy magnetization axis, e.g.,\nm(0) =n(0) = (1 /√\n2,1/√\n2,0), with all other param-\neters unchanged. The spin versus time is plotted in\nFig. 2(a). The unit vectors m(t) andn(t) are almost\nidentical, andsincetheir z-componentisnearlyzero,they\nmove together in the x-yplane. In this weak field case,\nthe nutation is small, and the fast small-oscillations due\nto nutation are barely visible. The orbital angular mo-\nmentum dynamics is plotted in Fig. 2(b) [note the differ-\nenttimescalein (a)and(b)] andshowsthat ℓ(t) oscillates\nwith a frequency equal to that of the fast tiny-oscillation\u0001\u0001\u0001\u0002\u0001\u0003\u0001\n\u0001\u0002\u0001\u0003\u0001\u0002\u0004\u0001\u0001\u0002\u0004\u0003\u0001\u0002\u0005\u0001\u0001\u0002\u0005\u0003\u0001\u0002\u0006\u0001\u0004\u0001\u0001\u0002\n-\u0001\u0002\u0003\u0001\u0002\u0003\u0004\u0002\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003\u0007\u0001}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0003\u0004\u0002\u0001\u0004\u0002\u0003\u0005\u0002\u0001\u0005\u0002\u0003\u0006\u0002\u0001\u0002\n-\u0001\u0002\u0001\u0005\u0001-\u0001\u0002\u0001\u0004\u0003-\u0001\u0002\u0001\u0004\u0001-\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0004\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003}\n\u0001\u0002\u0001\u0003\u0001\u0002\u0004\u0001\u0001\u0002\u0004\u0003\u0001\u0002\u0005\u0001\u0001\u0002\u0005\u0003\u0001\u0002\u0006\u0001\u0004\u0001\u0001\u0001\u0001\u0002\u0007\u0001\u0001\u0002\u0007\u0003\u0004\u0002\u0001\u0001{\u0002\u0002\u0003}\nFIG. 1: (color online) (a) The normalized spin vector mver-\nsus time for the low-field case at short times (5 orders of\nmagnitude shorter than in Fig. 2) when the initial spin is\nnot along the fast axis. (b) The reduced orbital angular mo-\nmentum vector ℓ(t). (c) The inner product m(t)·n(t) (the\nprojection of the spin on the fast magnetic axis of the needle .\nofm(t) [the oscillation amplitude is 0 .02|m(t)|]. Fig-\nure 2(c) shows a parametric plot of m(t) versus time.\nThe nutation is clearly very small; the dynamics of m(t)\nconsists almost entirely of precession at frequency ωB.\nFigure 3 shows the dynamics at high magnetic field\n(ωB= 105s−1) with all the other parametersunchanged.\nFigure 3(a) shows mversus time, and now the nutation\nis clearly significant. For the high magnetic field case,\nm(t) is also almost numerically equal to n(t).ℓ(t) is\nplotted in Fig. 3(b). Its amplitude is very large, ℓ(t)≈\n40m(t). However, its oscillation frequency is comparable\nwith that of m(t). In contrast with the results in Fig. 2,\nhere, in addition to precession of the needle, significant\nnutation is present, as shown clearly in the parametric\nplot of the needle spin vector m(t) in Fig. 3(c).\nWe now determine the uncertainty of the MNM due to\ninternal magnetic field fluctuations related to the Gilbert\ndamping. A stochastic force ξ(t), whose strength is de-\ntermined by the fluctuation–dissipation theorem [6], is3\n\u0001\u0001\u0001\u0002\u0001\u0003\u0001\n\u0001\u0002\u0003\u0004\u0005\u0006\u0002\n-\u0005\u0007\u0006-\u0006\u0007\b\u0006\u0007\b\u0005\u0007\u0006{\u0001\u0001\t\u0001\u0002\t\u0001\u0003\t\u0001}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0003\u0004\u0002\u0001\u0004\u0002\u0003\u0005\u0002\u0001\u0005\u0002\u0003\u0006\u0002\u0001\u0002\n-\u0001\u0002\u0001\u0005\u0001-\u0001\u0002\u0001\u0004\u0003-\u0001\u0002\u0001\u0004\u0001-\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0004\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003}\nFIG. 2: (color online) Dynamics for thelow-field case ( ωB= 1\ns−1), over relatively long timescales relative to those in Fig. 1.\n(a)mversus time in units of seconds (note that nis indistin-\nguishable from mon the scale of the figure). (b) ℓ(t) (note\nthat it stays small compared to S). (c) Parametric plot of the\nneedle spin vector m(t) showing that nutation is almost im-\nperceptible for small fields [contrast this with the large fie ld\nresult in Fig. 3(c)]; only precession is important.\nadded to Eq. (6), in direct analogy with the treatment of\nBrownianmotionwherebothdissipationandastochastic\nforce are included [12]:\n˙m=m×(ωBb+ξ)+ω0(m×n)(m·n)\n−αm×(˙m−Ω×m). (11)\nξ(t) is internal to the needle and therefore it does not\naffect the total angular momentum jdirectly, i.e., ξ(t)\ndoes not appear in Eq. (9) [since the term −m×ξis also\nadded to the RHS of (7)]. However, as shown below, ξ(t)\naffectsℓas well as m, causing them to wobble stochas-\ntically. This, in turn, makes jstochastic as well via the\nZeeman torque [see Eq. (9)].\nThe fluctuation-dissipation theorem [6] implies\n∝angbracketleftξαξβ∝angbracketrightω≡/integraldisplay\ndt∝angbracketleftξα(t)ξβ(0)∝angbracketrighteiωt\n=δαβαωcoth(/planckover2pi1ω/2kBT)\nN≈δαβ2αkBT\n/planckover2pi1N,(12)\u0001\u0001\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0004\u0001\u0002\u0001\u0005\u0001\u0002\u0001\u0006\u0001\u0002\u0007\u0001\u0002\n-\u0007\u0002\u0001-\u0001\u0002\b\u0001\u0002\b\u0007\u0002\u0001{\u0001\u0001\t\u0001\u0002\t\u0001\u0003}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0004\u0001\u0002\u0001\u0005\u0001\u0002\u0001\u0006\u0001\u0002\u0007\u0001\u0002\n-\u0004\u0001-\u0003\u0001\u0003\u0001\u0004\u0001{\u0001\u0001\n\b\u0001\u0002\n\u0000\u0001\u0003}\nFIG. 3: (color online) High-field case ( ωB= 105s−1). (a)\nm(t) [which is almost numerically equal to n(t)]. (b)ℓ(t)\n(note the ordinate axis scale is [ −40,40]). (c) Parametric plot\nof the needle spin vector m(t) showing that strong nutation\noccurs for large fields in addition to precession.\nwhereN=S//planckover2pi1, and the last approximation is ob-\ntained under the assumption that /planckover2pi1ω≪kBT. Note that\nEq. (11) should be solved together with Eqs. (8) and (9).\nThe presence of the anisotropy term in Eq. (11) makes\nnumerical solution difficult for large ω0. Hence, we con-\nsider a perturbative expansion in powers of λ≡ω1/ω0:\nm(t) =n0(t)+λδm(t)+...,n(t) =n0(t)+λδn(t)+...,\nj(t) =j0(t) +λδj(t) +.... Sinceω0is the largest fre-\nquency in the problem, the inequalities αω0≫ωB,ω1,ω3\nhold. Moreover, the Gilbert constant αis large enough\nto effectively pin m(t) ton(t) [hencej(t) =ℓ(t)+m(t)≈\nℓ(t)+n(t)]. Therefore, anadiabaticapproximationtothe\nset of dynamical stochastic equations can be obtained.\nThe zero order term in λreads:\n˙j0=ωBn0×b,˙n0=ω1j0×n0,(13)4\nwhereΩwas approximated by Ω0= (ω3−ω1)(j0·n0−\n1)n0+ω1(j0−n0) in Eqs. (8) and (10) in obtaining (13)\n[7]. The solution to Eqs. (13) [for times beyond which\nGilbert dissipation is significant so m(t)≈n(t)] is very\nclose to that obtained from Eqs. (6)-(8).\nExpanding Eq. (11) in powers of λand keeping only\nthe first order terms (the zeroth order term on the LHS\nvanishes since m0=n0), we get: ω1(δm−δn)×n0=\n˙n0−ωBn0×b+αn0×(˙n0−Ω0×n0)−n0×ξ. Taking\nEq. (13) into account and introducing the notation δη≡\nδm−δn, we obtain\nδη×n0=j0×n0−(ωB/ω1)n0×b−(1/ω1)n0×ξ,(14)\nand from Eqs. (8) and (9) we find\nd\ndtδj=ωB(δn+δη)×b, (15)\nd\ndtδn=ω1(j0−n0)×δn+ω1(δj−δn−δη)×n0\n=ω1j0×δn+ω1(δj−δη)×n0. (16)\nTo first order in λ,δn⊥n0(sincenmust be a unit\nvector), and δm⊥n0, henceδη⊥n0. Therefore, δη×\nb= [j0−(j0·n0)n0]×b+(ωB/ω1)[b−(b·n0)n0]×b+\nω−1\n1[ξ−(ξ·n0)n0]×bon the RHS of Eq. (15) and\nd\ndtδj=ωBδn×b+ωB[j0−(j0·n0)n0]×b\n−ω2\nB\nω1(b·n0)n0×b+ωB\nω1[ξ−(ξ·n0)n0]×b.(17)\nEquations (13), (16) and (17) form a closed system of\nstochastic differential equations [upon using Eq. (14) to\nsubstitute for δη×n0on the RHS of Eq. (16)]. With\nthe largest frequency ω0eliminated, a stable numerical\nsolution is obtained. Moreover, for small magnetic field\n(whereωBis the smallest frequency in the system), an\nanalytic solution of these equations is achievable. To ob-\ntain an analytic solution to Eqs. (13), let us transform\nto the frame rotating around Bwith frequency ωBto\nget equations of the formd\ndτv=d\ndtv+ωBb×v(which\ndefinesτ):\nd\ndτn0=−ω1n0×/parenleftbigg\nn0−j0+ωB\nω1b/parenrightbigg\n,(18)\nd\ndτj0=ωBb×/parenleftbigg\nn0−j0+ωB\nω1b/parenrightbigg\n.(19)\nIf the initial condition is n0(0)−j0(0)+(ωB/ω1)b= 0,\nthen, in the rotating frame j0(τ) andn0(τ) are constant\nvectors. Note that this initial condition is only slightly\ndifferent from the “ordinary” initial condition n0(0) =\nj0(0)since( ωB/ω1)≪1forsmallmagneticfields. Hence,\nin the rotating frame,\nd\ndτδn=ω1n0×(δn−δj+δη),(20)d\ndτδj=−ωBb×(δn−δj+δη).(21)\nWith the special initial conditionbeing satisfied, Eq. (14)\nbecomes δη×n0=−(1/ω1)n0×ξ, and Eqs. (20)-(21)\nbecome a set of first order differential equations with\ntime-independent coefficients. Their solution for initial\nconditions, δn(t= 0) = 0, δj(t= 0) = 0 is,\n/parenleftbiggδn(t)\nδj(t)/parenrightbigg\n=t/integraldisplay\n0dt1exp[C(t−t1)]C/parenleftbiggδη(t1)\n0/parenrightbigg\n,(22)\nwhere the constant matrix C=/parenleftbiggA−A\n−B B/parenrightbigg\nhas di-\nmension 6 ×6 and the 3 ×3 matrices AandBare given by\nAij=−ω1ǫijknk\n0,Bij=−ωBǫijkbk. Without loss of gen-\neralitywecanchoose n0=ˆzandb=ωB(cosθˆz+sinθˆx),\nwhereθis the angle between the easy magnetization\naxis and the magnetic field. In this basis, ∝angbracketleftδηxδηx∝angbracketrightω=\n∝angbracketleftδηyδηy∝angbracketrightω≈ω−2\n0∝angbracketleftξxξx∝angbracketrightω=ω−2\n0∝angbracketleftξyξy∝angbracketrightω=Sa(ω), and\n∝angbracketleftδηzδηz∝angbracketrightω= 0. Here ∝angbracketleftxx∝angbracketrightω≡/integraltext\ndteiωt∝angbracketleftx(t)x(0)∝angbracketrightand [see\nEq. (12)] Sa(ω) =αωcoth(/planckover2pi1ω/2kBT)\nω2\n0N≈2αkBT\nN/planckover2pi1ω2\n0.\nWe are particularly interested in the quantities\n∝angbracketleftδn2\ny(t)∝angbracketright ≡ ∝angbracketleftδny(t)δny(t)∝angbracketrightand∝angbracketleftδj2\ny(t)∝angbracketright ≡ ∝angbracketleftδjy(t)δjy(t)∝angbracketright\nbecause, in the basis chosen above, the y-axis is the di-\nrection of precession of n0aroundb. Using Eq. (22) we\nobtain∝angbracketleftδn2\ny(t)∝angbracketright ≈tω2\n1Sa(ω∼ω1). Assuming the pre-\ncession of nis measured, [or equivalently, the precession\nofm, since they differ only for short timescales of or-\nder (αω0)−1], the uncertainty in the precession angle is\n∝angbracketleft(∆ϕ)2∝angbracketright ≈tω2\n1Sa(ω∼ω1). We thus arrive at our central\nresult: the precision with which the precession frequency\ncan be measured is, ∆ ωB=√\n/angbracketleft(∆ϕ)2/angbracketright\nt≈ω1\nω0/radicalBig\n2αkBT\n/planckover2pi1N1√\nt.\nEquivalently, the magnetic field precision is,\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBω1\nω0/radicalbigg\n2αkBT\n/planckover2pi1N1√\nt.(23)\nFor the parameters used in this paper we find ∆ B≈\n5×10−18√\nt[s]Tesla (independent of ωB). This result should\nbe compared with the scaling ∆ B∝t−3/2obtained in\nRef. 1. Therein, the initial uncertainty of the spin di-\nrection relative to the needle axis was estimated from\nthe fluctuation-dissipation relation and the deterministic\nprecession resulted in the t−3/2scaling of the precession\nangle uncertainty (in addition this angle was assumed to\nbe small). In contrast, we consider the uncertainty ac-\nquired due to Gilbert dissipation duringthe precession,\nallowing the precession angle to be large. Thus, the stan-\ndard1/√\ntdiffusion scalingis obtained and dominates for\ntimes that are even much longer than those considered\nin Ref. 1.\nIntheSupplementalMaterial[13]wediscussthreerele-\nvant related issues. (a) The time at which diffusion stops\nbecause equipartition is reached (we estimate the time5\nwhen the energy stored in stochastic orbital motion be-\ncomes of order kBT). (b) The uncertainty of the mag-\nnetic field for experiments in which the fast precession of\nnaroundjis averaged out in the measurement, and the\ndiffusion of jdetermines ∆ B. (c) We consider the related\nproblem of the dynamics and stability of a rotating MN\nin an inhomogeneous field (e.g., levitron dynamics in a\nIoffe-Pritchard trap [14, 15]).\nIn conclusion, we show that ∆ Bdue to Gilbert damp-\ning is very small; external noise sources, as discussed in\nRef. [1], will dominate over the Gilbert noise for weak\nmagnetic fields. A closed system of stochastic differen-\ntial equations, (13), (16) and (17), can be used to model\nthe dynamics and estimate ∆ Bfor large magnetic fields.\nA rotating MN in a magnetic trap can experience levi-\ntation, although the motion does not converge to a fixed\npoint or a limit cycle; an adiabatic–invariant stability\nanalysis confirms stability [13].\nThis work was supported in part by grants from the\nDFG through the DIP program (FO703/2-1). Useful\ndiscussions with Professor Dmitry Budker are gratefully\nacknowledged. A. S. was supported by DFG Research\nGrant No. SH 81/3-1.\n[1] D. F. J. Kimball, A. O. Sushkov, and D. Budker, Phys.\nRev. Lett. 116, 190801 (2016).\n[2] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004)\n[3] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjet. 8153\n(1935). In L. D. Landau, Collected Papers. Ed. by D. ter\nHaar, (Gordon and Breach, New York, 1967), p. 101.\n[4] W. F. Brown Jr., Phys. Rev. 130, 1677 (1963).\n[5] H. Keshtgar, et al., Phys. Rev. B 95, 134447 (2017).\n[6] H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).\n[7] We note in passing that Eqs. (13) are equivalent to the\nequations of motion of a symmetric top in a gravita-\ntional field when the top is anchored at a point a=an\non its axis a distance afrom the center of mass. The\nequations of motion are: dL/dt=T, where Land\nT=an×(−mgz) are taken with respect to the fixedpoint, and dn/dt=Ω×n. The angular velocity is given\nbyΩ=I−1\n1[L−(L·n)n] +I−1\n3(L·n)n, where the mo-\nments of inertia ( I1,I1,I3) are calculated relative to the\nfixed point. Introducing a characteristic scale L0so that\nL=L0j(jis not a unit vector and its length is not\nconserved) we obtain Eqs. (13) with ωB=mga/L 0and\nω1=L0/I1. Here, the analog of the magnetic field is the\ngravitational field and the analog of bisz.\n[8] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87,\n3989 (2000), and references therein.\n[9] O. Zobay and B. M. Garraway, Phys. Rev. A 61, 033603\n(2000); J. Liu, L. Fu, B.-Y. Ou, S.-G. Chen, D.-I. Choi,\nB. Wu, and Q. Niu, Phys. Rev. A 66, 023404 (2002).\n[10] Y. B. Band, I. Tikhonenkov, E. Pazyy, M. Fleischhauer,\nand A. Vardi, J. of Modern Optics 54, 697-706 (2007).\n[11] Y. B. Band, Phys. Rev. E 88, 022127 (2013); Y. B. Band\nand Y. Ben-Shimol, Phys. Rev. E 88, 042149 (2013).\n[12] H. P. Breuer and F. Petruccione, The Theory of\nOpen Quantum Systems (Oxford University, Cambridge,\n2002); M. Schlosshauer, Decoherence and the Quantum-\nto-Classical Transition (Springer, Berlin, 2007).\n[13] See Supplemental Material at\nhttp://link.aps.org/supplemental/10.1103/PhysRevLet t.121.160801\nwhich contains a discussion of the three issues enumer-\nated in the text, and which includes Refs. 16-20.\n[14] M. V. Berry, Proc. R. Soc. A 452, 1207 (1996).\n[15] A movie showing the dynamics of a Levitron can be seen\nathttps://www.youtube.com/watch?v=wyTAPW_dMfo .\n[16] Y. B. Band, Y. Avishai, A. Shnirman, “Dynamics of a\nMagnetic Needle Magnetometer: Sensitivity to Landau–\nLifshitz–Gilbert Damping”, Phys. Rev. Lett. (to be pub-\nlished).\n[17] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87,\n3989 (2000), and references therein; D. E. Pritchard,\nPhys. Rev. Lett. 51, 15 (1983).\n[18] C. C.Rusconi, V.P¨ ochhacker, K.Kustura, J.I.Ciracan d\nO. Romero-Isart, Phys. Rev. Lett. 119, 167202 (2017);\nC. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac\nand O. Romero-Isart, Phys. Rev B 96, 134419 (2017); C.\nC. Rusconi and O. Romero-Isart, Phys. Rev B 93, 054427\n(2016).\n[19] S. Earnshaw, Trans. Camb. Phil. Soc. 7, 97-112 (1842).\n[20] D. R. Merkin, Introduction to the Theory of Stability ,\n(Springer–Verlag, New York, 1997); F. Verhulst, Non-\nlinear Differential Equations and Dynamical Systems ,\n(Springer–Verlag, Berlin, 1990).arXiv:1803.10064v2  [physics.gen-ph]  19 Oct 2018Supplemental Material for “Dynamics of a Magnetic Needle Ma gnetometer:\nSensitivity to Landau–Lifshitz–Gilbert Damping”\nY. B. Band1,2, Y. Avishai2,3,4, Alexander Shnirman3,5,6\n1Department of Chemistry, Department of Physics,\nDepartment of Electro-Optics, and the Ilse Katz Center for N ano-Science,\nBen-Gurion University, Beer-Sheva 84105, Israel\n2New York University and the NYU-ECNU Institute of Physics at NYU Shanghai,\n3663 Zhongshan Road North, Shanghai, 200062, China\n3Department of Physics, and the Ilse Katz Center for Nano-Sci ence,\nBen-Gurion University, Beer-Sheva 84105, Israel\n4Yukawa Institute for Theoretical Physics, Kyoto, Japan\n5Institut f¨ ur Theorie der Kondensierten Materie,\nKarlsruhe Institute of Technology, D-76128 Karlsruhe, Ger many\n6Institute of Nanotechnology, Karlsruhe Institute of Techn ology, D-76344 Eggenstein-Leopoldshafen, Germany\nIn this supplemental material we expand the discussion of the main t ext [1] and address the following three issues.\n(a) The time τeat which the diffusion of the magnetic needle axis direction nand the magnetic needle orbital\nangular momentum ℓstops because equipartition is reached, i.e., we estimate the time req uired for the energy stored\nin stochastic orbital motion to become of order kBT. (b) The uncertainty ∆ Bof the magnetic field for experiments\nin which the fast precession of naroundjis averaged out in the measurement process and the uncertainty ∆ Bis\ndetermined by the diffusion of j. (c) The dynamics of a magnetic needle in an inhomogeneous field, e.g., levitron\ndynamics of a rotating magnetic needle in a Ioffe-Pritchard trap [2], s ee Refs. [3–5].\n(a):τecan be estimated by noting that the diffusion determined in [1] stops once equipartition is reached. The\nenergy ∆ Estored in stochastic orbital motion is given by\n∆E∼/planckover2pi1ω1N/angbracketleftδℓ2/angbracketright, (1)\nwhere where N=S//planckover2pi1(note that δj−δn=δℓ). By requiring ∆ E∼kBTwe can estimate that the diffusion given\nby Eqs. (20-21) of [1] stops when τe∼ω2\n0/(αω3\n1) (this result can also be obtained by expanding Eq. (11) further in\npowers of λ≡ω1/ω0). For the parameters used in [1] this is an extremely long time ( τe∼1012s∼5 years). Hence,\nwe conclude that the diffusion of Eqs. (20-21) and the error estima tes given for ∆ Bin Ref. [1] are relevant for all\nreasonable times.\n(b): In [1] we calculate ∆ Bassuming the experimental measurement follows the temporal dyn amics of nandj.\nAn alternative assumption is that the precession of naroundjis averaged out by the measurement process and one\nmeasures the diffusion of j. For the latter we obtain the leading term\n/angbracketleftδj2\ny(t)/angbracketright ≈tω2\nBcos2θSa(ω∼ω1), (2)\nwhereSa(ω) is given in Eq. (23) of [1]. At θ=π/2 the leading contribution obtained in Eq. (2) vanishes and the\nremaining sub-leading term is\n/angbracketleftδn2\ny(t)/angbracketright ≈t2ω4\nB\nω2\n1Sa(ω∼ω1), (3)\nhence for θ/negationslash=π/2 we obtain\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBωB\nω0cosθ/radicalbigg\n2αkBT\n/planckover2pi1N1√\nt, (4)\nwhereas at θ=π/2,\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBω2\nB\nω0ω1/radicalbigg\n4αkBT\n/planckover2pi1N1√\nt. (5)\nTakingωB= 1s−1we obtain ∆ B≈cosθ×5×10−23√\nt[s]Tesla for θ/negationslash=π/2, and ∆ B≈7×10−25√\nt[s]Tesla for θ=π/2.2\n(c): A rotating magnet can be levitated in an inhomogeneous magnet ic field [3–5]. This is possible despite Earn-\nshaw’s theorem [6] from which one can conclude that levitation of a non-rotating ferromagnetin a static magnetic field\nis not possible. Two important factors regarding magnetic levitation are the forces on the magnet and its stability\n(ensuring that it does not spontaneously slide or flip into a configura tion without lift). The dynamics of a magnetic\nneedle in an inhomogeneous magnetic field can be modelled using Eqs. (6 ), (7) and (8) of [1] augmented by the\nequations of motion for the center of mass (CM) degrees of freed om of the needle,\n˙p=∇(µ·B(r)), (6)\n˙r=p/m , (7)\nwhererandpare the needle CM position and momentum vectors. Our numerical re sults show levitation of the\nmagnetic needle when the initial rotational angular momentum vecto r of the needle is sufficiently large and points\nin the direction of magnetic field at the center of the trap. We shall s ee that the dynamical variables do not evolve\nto a fixed point or a simple cyclic orbit. Moreover, a linear stability analy sis yields a 15 ×15 Jacobian matrix with\neigenvalues having a positive real part, so the system is unstable. However, a stability analysis of the system using\nthe adiabatic invariant |µ||B|[3] does yield a stable fixed point (contrary to the full numerical re sults which show a\nmore complicated levitation dynamics).\nFigure 1 shows the dynamics of the system over time in the trap. We u se the same magnetic needle parameters\nused in Fig. 2 of [1] and a Ioffe-Prichard magnetic field [2]\nB(r) =ex/parenleftbigg\nB′x−B′′\n2xz/parenrightbigg\n+ey/parenleftbigg\nB′y−B′′\n2zy/parenrightbigg\n+ez/parenleftbigg\nB0+B′′\n2(z2−x2+y2\n2)/parenrightbigg\n, (8)\nwith field bias B0, gradient B′, and curvature B′′parameters chosen so that the Zeeman energy and its variation ov er\nthe trajectory of the needle in the trap are substantial (as is clea r from the results shown in the figure). We start\nthe dynamics with initial conditions: r(0) = (0,0,0),p(0) = (0,0,0),m(0) = (0,0.0011/2,−(1−0.001)1/2) (almost\nalong the −zdirection), n(0) =m(0),ℓ(0) = (0 ,0,0.001) [this is large orbital angular momentum since ℓis the\norbital angular momentum divided by S]. Figure 1(a) shows the needle CM position r(t) versus time. Fast and slow\noscillations are seen in the xandymotion, whereas z(t) remains very close to zero. Figure 1(b) shows oscillations of\nthe CM momentum p(t) with time. px(t) andpy(t) oscillate with time, and pz(t) remains zero. Figure 1(c) plots the\nspinm(t) versus time. Initially, m(0) points almost in the −zdirection, and the tip of the needle n(t) =m(t) carries\nout nearly circular motion in the nx-nyplane. Figure 1(d) plots the orbital angular momentum ℓ(t). The components\nℓx(t) andℓy(t) undergo a complicated oscillatory motion in the ℓx(t)-ℓy(t) plane but ℓz(t)≈ℓz(0). Figure 1(e) is a\nparametric plot of m(t); the motion consists of almost concentric rings that are slightly dis placed one from the other.\nThe full dynamics show levitation but they do not converge to a fixed point or a limit cycle.\nQuite generally, for a system of dynamical equations, ˙ yi(t) =fi(y1,...,y n),i= 1,...n, a linear stability analysis\nrequires calculating the eigenvalues of the Jacobian matrix evaluate d at the equilibrium point y∗wheref(y∗) =0,\nJij=/parenleftBig\n∂fi\n∂yj/parenrightBig\ny∗[7]. The system is unstable against fluctuations if any of the eigenvalu es ofJijhave a positive real\npart. Equations (6), (7) and (8) of [1] together with Eqs. (6) and (7) above have a Jacobian matrix with eigenvalues\nwhose real part are positive, so the linear stability test fails. Howev er, if the Zeeman force −∇HZin Eq. (6) is\nreplaced by the gradient of the adiabatic invariant, µ·∇|B(r)|, none of the eigenvalues of the Jacobian matrix have\na positive real part and the system is linearly stable, i.e., the stability a nalysis using the adiabatic-invariant predicts\nstability. Note that substituting the adiabatic invariant for the Zee man energy in the full equations of motion yields\nr(t) andp(t) vectors that are constant with time and n(t),m(t) andℓ(t) are similar to the results obtained with\nthe full equations of motion (but the parametric plot of m(t) is a perfectly circular orbit). Thus, adiabatic–invariant\nstability analysis of a rotating magnetic needle in a magnetic trap confi rms stability of its levitation as obtained in\nthe numerical solution of the dynamical equations.3\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0001\u0001\u0004-\u0001\u0005\u0001\u0001\u0003\u0001\u0005\u0001\u0001\u0001\u0001\u0005\u0001\u0001\u0003\u0001\u0005\u0001\u0001\u0004\n\u0001{\u0001\u0001\u0002\u0001\u0003}\u0002\u0003\u0004\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0003\u0001-\u0001\u0005\u0001\u0002\u0001\u0005\u0001\u0001\u0001\u0005\u0001\u0002\u0001\u0005\u0003\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0003\u0005\u0001-\u0001\u0005\u0006-\u0001\u0005\u0007-\u0001\u0005\b-\u0001\u0005\u0004\u0001\u0005\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0001\u0001\u0001\u0006-\u0001\u0005\u0001\u0001\u0001\u0004\u0001\u0005\u0001\u0001\u0001\u0001\u0001\u0005\u0001\u0001\u0001\u0004\u0001\u0005\u0001\u0001\u0001\u0006\u0001\u0005\u0001\u0001\u0001\u0007\u0001\u0005\u0001\u0001\u0001\b\u0001\u0005\u0001\u0001\u0003\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\nFIG. 1: (color online) Dynamics of a needle in a Ioffe-Pritcha rd magnetic field. (a) rversus time, (b) pversus time, (c) m\nversus time (note that n(t) is indistinguishable from m(t) on the scale of the figure). (d) ℓversus time (note that |ℓ(t)|is small\ncompared to Sbut rotational angular momentum L(t) =Sℓ(t) is large since S= 1012). (e) Parametric plot of the needle spin\nvectorm(t) (nutation is very small for this case of small magnetic field ).4\n[1] Y. B. Band, Y. Avishai, A. Shnirman, “Dynamics of a Magnet ic Needle Magnetometer: Sensitivity to Landau–Lifshitz–\nGilbert Damping”, Phys. Rev. Lett. (to be published).\n[2] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87, 3989 (2000), and references therein; D. E. Pritchard,\nPhys. Rev. Lett. 51, 15 (1983).\n[3] M. V. Berry, Proc. R. Soc. A 452, 1207 (1996).\n[4] A movie a a Levitron can be seen at https://www.youtube.com/watch?v=wyTAPW_dMfo .\n[5] C. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac and O. Romero-Isart, Phys. Rev. Lett. 119, 167202 (2017); C. C.\nRusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac and O. Romer o-Isart, Phys. Rev B 96, 134419 (2017); C. C. Rusconi and\nO. Romero-Isart, Phys. Rev B 93, 054427 (2016).\n[6] S. Earnshaw, Trans. Camb. Phil. Soc. 7, 97-112 (1842).\n[7] D. R. Merkin, Introduction to the Theory of Stability , (Springer–Verlag, New York, 1997); F. Verhulst, Nonlinear Differential\nEquations and Dynamical Systems , (Springer–Verlag, Berlin, 1990)."    },    {        "title": "1806.04782v3.Dynamical_and_current_induced_Dzyaloshinskii_Moriya_interaction__Role_for_damping__gyromagnetism__and_current_induced_torques_in_noncollinear_magnets.pdf",        "content": "arXiv:1806.04782v3  [cond-mat.other]  9 Dec 2020Dynamical and current-induced Dzyaloshinskii-Moriya int eraction: Role for damping,\ngyromagnetism, and current-induced torques in noncolline ar magnets\nFrank Freimuth1,2,∗Stefan Bl¨ ugel1, and Yuriy Mokrousov1,2\n1Peter Gr¨ unberg Institut and Institute for Advanced Simula tion,\nForschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, German y and\n2Institute of Physics, Johannes Gutenberg University Mainz , 55099 Mainz, Germany\nBoth applied electric currents and magnetization dynamics modify the Dzyaloshinskii-Moriya in-\nteraction (DMI), which we call current-induced DMI (CIDMI) and dynamical DMI (DDMI), respec-\ntively. We report a theory of CIDMI and DDMI. The inverse of CI DMI consists in charge pumping\nbyatime-dependentgradient ofmagnetization ∂2M(r,t)/∂r∂t, while theinverseofDDMIdescribes\nthe torque generated by ∂2M(r,t)/∂r∂t. In noncollinear magnets CIDMI and DDMI depend on\nthe local magnetization direction. The resulting spatial g radients correspond to torques that need\nto be included into the theories of Gilbert damping, gyromag netism, and current-induced torques\n(CITs) in order to satisfy the Onsager reciprocity relation s. CIDMI is related to the modification of\norbital magnetism induced by magnetization dynamics, whic h we call dynamical orbital magnetism\n(DOM), and spatial gradients of DOM contribute to charge pum ping. We present applications of\nthis formalism to the CITs and to the torque-torque correlat ion in textured Rashba ferromagnets.\nI. INTRODUCTION\nSince the Dzyaloshinskii-Moriya interaction (DMI)\ncontrols the magnetic texture of domain walls and\nskyrmions, methods to tune this chiral interaction by\nexternal means have exciting prospects. Application of\ngatevoltage[1–3]orlaserpulses[4]arepromisingwaysto\nmodify DMI. Additionally, theory predicts that in mag-\nnetic trilayer structures the DMI in the top magnetic\nlayer can be controlled by the magnetization direction\nin the bottom magnetic layer [5]. Moreover, methods to\ngeneratespin currentsmaybe usedto induceDMI, which\nis predicted by the relations between the two [6, 7]. Re-\ncent experiments show that also electric currents modify\nDMI in metallic magnets, which leads to large changes in\nthe domain-wallvelocity [8, 9]. However,a rigoroustheo-\nretical formalism for the investigation of current-induced\nDMI (CIDMI) in metallic magnets has been lacking so\nfar, and the development of such a formalism is one goal\nof this paper.\nRecently, a Berry phase theory of DMI [6, 10, 11]\nhas been developed, which formally resembles the mod-\nern theory of orbital magnetization [12–14]. Orbital\nmagnetism is modified by the application of an electric\nfield, which is known as the orbital magnetoelectric re-\nsponse [15]. In the case of insulators it is straightfor-\nward to derive the expressions for the magnetoelectric\nresponse directly. However, in metals it is much easier\nto derive expressions instead for the inverse of the mag-\nnetoelectric response, i.e., for the generation of electric\ncurrents by time-dependent magnetic fields [16]. The in-\nverse current-induced DMI (ICIDMI) consists in charge\npumping by time-dependent gradients of magnetization.\nDue to the analogies between orbital magnetism and the\nBerryphasetheoryofDMIonemayexpectthatinmetals\nit is convenient to obtain expressions for ICIDMI, which\ncanthen be used todescribe the CIDMI byexploitingthereciprocity between CIDMI and ICIDMI. We will show\nin this paper that this is indeed the case.\nIn noncentrosymmetric ferromagnets spin-orbit inter-\naction (SOI) generates torques on the magnetization –\nthe so-called spin-orbit torques (SOTs) – when an elec-\ntric current is applied [17]. The Berry phase theory of\nDMI [6, 10, 11] establishes a relation to SOTs. The for-\nmal analogies between orbital magnetism and DMI have\nbeen shown to be a very useful guiding principle in the\ndevelopment of the theory of SOTs driven by heat cur-\nrents [18]. In particular, it is fruitful to consider the DMI\ncoefficients as a spiralization, which is formally analo-\ngous to magnetization. In the theory of thermoelectric\neffects in magnetic systems the curl of magnetization de-\nscribes a bound current, which cannot be measured in\ntransport experiments and needs to be subtracted from\nthe Kubo linear response in order to obtain the mea-\nsurable current [19–21]. Similarly, in the theory of the\nthermal spin-orbit torque spatial gradients of the DMI\nspiralization, which result from the temperature gradi-\nent together with the temperature dependence of DMI,\nneed to be subtracted in order to obtain the measurable\ntorqueandto satisfyaMott-likerelation[10, 18]. In non-\ncollinear magnets the question arises whether gradients\nof the spiralization that are due to the magnetic texture\ncorrespond to torques like those from thermal gradients.\nWe will show that indeed the spatial gradients of CIDMI\nneed to be included into the theory of current-induced\ntorques (CITs) in noncollinear magnets in order to sat-\nisfy the Onsager reciprocity relations [22].\nWhen the system is driven out of equilibrium by mag-\nnetization dynamics rather than electric current one may\nexpect DMI to be modified as well. The inverse effect of\nthis dynamical DMI (DDMI) consists in the generation\nof torques by time-dependent magnetization gradients.\nIn noncollinear magnets the DDMI spiralization varies\nin space. We will show that the resulting gradient cor-2\nresponds to a torque that needs to be considered in the\ntheory of Gilbert damping and gyromagnetism in non-\ncollinear magnets.\nThis paper is structured as follows. In section IIA\nwe give an overview of CIT in noncollinear magnets and\nintroduce the notation. In section IIB we describe the\nformalism used to calculate the response of electric cur-\nrent to time-dependent magnetization gradients. In sec-\ntion IIC we show that current-induced DMI (CIDMI)\nand electric current driven by time-dependent magneti-\nzation gradients are reciprocal effects. This allows us\nto obtain an expression for CIDMI based on the formal-\nism of section IIB. In section IID we discuss that time-\ndependent magnetization gradients generate additionally\ntorques on the magnetization and show that the inverse\neffect consists in the modification of DMI by magnetiza-\ntion dynamics, which we calldynamical DMI (DDMI). In\nsection IIE we demonstrate that magnetization dynam-\nics induces orbital magnetism, which we call dynamical\norbitalmagnetism (DOM) and showthat DOM is related\nto CIDMI. In section IIF we explain how the spatial gra-\ndients of CIDMI and DOM contribute to the direct and\nto the inverse CIT, respectively. In section IIG we dis-\ncuss how the spatial gradients of DDMI contribute to the\ntorque-torque correlation. In section IIH we complete\nthe formalism used to calculate the CIT in noncollinear\nmagnets by adding the chiral contribution of the torque-\nvelocity correlation. In section III we finalize the theory\nof the inverse CIT by adding the chiral contribution of\nthe velocity-torque correlation. In section IIJ we fin-\nish the computational formalism of gyromagnetism and\ndamping by adding the chiral contribution of the torque-\ntorque correlation and the response of the torque to the\ntime-dependent magnetization gradients. In section III\nwe discuss the symmetry properties of the response to\ntime-dependent magnetizationgradients. In section IVA\nwe present the results for the chiral contributions to the\ndirectand the inverseCITin the Rashbamodel andshow\nthat both the perturbation by the time-dependent mag-\nnetization gradient and the spatial gradients of CIDMI\nand DOM need to be included to ensure that they are\nreciprocal. In section IVB we present the results for the\nchiralcontributiontothe torque-torquecorrelationinthe\nRashba model and show that both the perturbation by\nthe time-dependent magnetization gradient and the spa-\ntialgradientsofDDMI need tobe included toensurethat\nit satisfies the Onsager symmetry relations. This paper\nends with a summary in section V.II. FORMALISM\nA. Direct and inverse current-induced torques in\nnoncollinear magnets\nEven in collinearmagnets the application of an electric\nfieldEgenerates a torque TCIT1on the magnetization\nwhen inversion symmetry is broken [17, 23]:\nTCIT1\ni=/summationdisplay\njtij(ˆM)Ej, (1)\nwheretij(ˆM) is the torkance tensor, which depends on\nthe magnetization direction ˆM. This torque is called\nspin-orbit torque (SOT), but we denote it here CIT1,\nbecause it is one contribution to the current-induced\ntorques (CITs) in noncollinear magnets. Inversely, mag-\nnetization dynamics pumps a charge current JICIT1ac-\ncording to [24]\nJICIT1\ni=/summationdisplay\njtji(−ˆM)ˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n,(2)\nwhereˆejis a unit vector that points into the j-th spa-\ntial direction. Generally, JICIT1can be explained by\nthe inverse spin-orbit torque [24] or the magnonic charge\npumping [25]. We denote it here by ICIT1, because it\nis one contribution to the inverse CIT in noncollinear\nmagnets. In the special case of magnetic bilayers one im-\nportantmechanism responsiblefor JICIT1arisesfrom the\ncombination of spin pumping and the inverse spin Hall\neffect [26, 27].\nIn noncollinear magnets there is a second contribution\nto the CIT, which is proportional to the spatial deriva-\ntives of magnetization [28]:\nTCIT2\ni=/summationdisplay\njklχCIT2\nijklEjˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n.(3)\nThe description of noncollinearity by the derivatives\n∂ˆM/∂rlisonlyapplicablewhenthe magnetizationdirec-\ntion changes slowly in space like in magnetic skyrmions\nwith large radius and in wide magnetic domain walls. In\norder to treat noncollinear magnets such as Mn 3Sn [29],\nwhere the magnetization direction varies strongly on the\nscale of one unit cell, Eq. (3) needs to be modified, which\nis beyond the scope of the present paper. The adia-\nbatic and the non-adiabatic [30] spin transfer torques\nare two important contributions to χCIT2\nijkl, but the in-\nterplay between broken inversion symmetry, SOI, and\nnoncollinearity can lead to a large number of additional\nmechanisms [22, 31]. Similarly, the current pumped\nby magnetization dynamics contains a contribution that\nis proportional to the spatial derivatives of magnetiza-3\ntion [22, 32, 33]:\nJICIT2\ni=/summationdisplay\njklχICIT2\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n.\n(4)\nTCIT2\niandJICIT2\nican be considered as chiral contribu-\ntionsto the CIT and to the ICIT, respectively, because\nthey distinguish between left- and right-handed spin spi-\nrals. Due to the reciprocity between direct and inverse\nCIT [22, 24] the coefficients χICIT2\nijklandχCIT2\njiklare related\naccording to\nχICIT2\nijkl(ˆM) =χCIT2\njikl(−ˆM). (5)\nB. Response of electric current to time-dependent\nmagnetization gradients\nIn order to compute JICIT2based on the Kubo lin-\near response formalism it is necessary to split it into\ntwo contributions, JICIT2aandJICIT2b. While JICIT2a\nis obtained as linear response to the perturbation by\natime-dependent magnetization gradient in a collinear\nferromagnet, JICIT2bis obtained as linear response to\nthe perturbation by magnetization dynamics in a non-\ncollinear ferromagnet. Therefore, as will become clear\nbelow,JICIT2acan be expressed by a correlation func-\ntion of two operators, because it describes the response\nof the current to a time-dependent magnetization gradi-\nent: A time-dependent magnetization gradient is a single\nperturbation, which is described by a single perturbing\noperator. In contrast, JICIT2binvolves the correlation\nof three operators, because it describes the response of\nthe current to magnetization dynamics in the presence of\nperturbation by noncollinearity. These are twoperturba-\ntions: One perturbation by the magnetization dynamics,\nandasecondperturbationtodescribethenoncollinearity.\nIn the Kubo formalism the expressions for the response\nonethe onehand toatime-dependent magnetizationgra-\ndient, which is described by a single perturbing operator,\nand the response on the other hand to a time-dependent\nmagnetization in the presence of a magnetization gradi-\nent, which is described by two perturbing operators, are\ndifferent. Therefore, we split JICIT2into these two con-\ntributions, which we call JICIT2aandJICIT2b. In the\nremainder of this section we discuss the calculation of\nthe contribution JICIT2a. The contribution JICIT2bis\ndiscussed in section III below.\nJICIT2ais determined by the second derivative of mag-\nnetization with respect to time and space variables and\ncan be written as\nJICIT2a\ni=/summationdisplay\njkχICIT2a\nijk∂2ˆMj\n∂rk∂t. (6)\nAnonzerosecondderivative∂2ˆMj\n∂rk∂tis what we referto asa\ntime-dependent magnetization gradient . Wewillshowbe-low that in special cases∂2ˆMj\n∂rk∂tcan be expressed in terms\nof the products∂ˆMl\n∂rk∂ˆMl\n∂t, which will allow us to rewrite\nJICIT2a\niin the form of Eq. (4) in the cases relevant for\nthe chiral ICIT. However, as will become clear below,\nEq. (6) is the most general expression for the response\nto time-dependent magnetization gradients, and it can-\nnot generally be rewritten in the form of Eq. (4): This\nis only possible when it describes a contribution to the\nchiral ICIT.\nJICIT2aoccurs in two different situations, which need\nto be distinguished. In one case the magnetization gra-\ndient varies in time like sin( ωt) everywhere in space. An\nexample is\nˆM(r,t) =\nηsin(q·r)sin(ωt)\n0\n1\n, (7)\nwhereηis the amplitude and the derivatives at t= 0 and\nr= 0 are\n∂ˆM(r,t)\n∂ri/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=∂ˆM(r,t)\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0= 0 (8)\nand\n∂2ˆM(r,t)\n∂ri∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\nηqiω\n0\n0\n. (9)\nIn the other case the magnetic texture varies like a\npropagating wave, i.e., proportional to sin( q·r−ωt). An\nexample is given by\nˆM(r,t) =\nηsin(q·r−ωt)\n0\n1−η2\n2sin2(q·r−ωt)\n,(10)\nwhere the derivatives at t= 0 and r= 0 are\n∂ˆM(r,t)\n∂ri/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\nηqi\n0\n0\n, (11)\n∂ˆM(r,t)\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\n−ηω\n0\n0\n (12)\nand\n∂2ˆM(r,t)\n∂ri∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\n0\n0\nη2qiω\n. (13)\nIn the latter example, Eq. (10), the second derivative,\nEq. (13), is along the magnetization ˆM(r= 0,t= 0),\nwhile in the former example, Eq. (7), the second deriva-\ntive, Eq. (9), is perpendicular to the magnetization when\nr= 0 andt= 0.4\nWe assume that the Hamiltonian is given by\nH(r,t) =−/planckover2pi12\n2me∆+V(r)+µBˆM(r,t)·σΩxc(r)+\n+1\n2ec2µBσ·[∇V(r)×v],\n(14)\nwhere the first term describes the kinetic energy, the sec-\nond term is a scalar potential, Ωxc(r) in the third term is\nthe exchange field, and the last term describes the spin-\norbit interaction. Around t= 0 and r= 0 we can de-\ncompose the Hamiltonian as H(r,t) =H0+δH(r,t),\nwhereH0is obtained from H(r,t) by replacing ˆM(r,t)\nbyˆM(r= 0,t= 0) and\nδH(r,t) =∂H0\n∂ˆMxηsin(q·r)sin(ωt)\n=µBΩxc(r)σxηsin(q·r)sin(ωt)(15)\nin the case of the first example, Eq. (7). In the case of\nthe second example, Eq. (10),\nδH(r,t)≃∂H\n∂ˆMxηsin(q·r−ωt)\n+∂H\n∂ˆMzη2sin(q·r)sin(ωt),(16)\nwhereforsmall randtonlythesecondterm ontheright-\nhand side contributes to∂2H(r,t)\n∂rk∂t. We consider here only\nthe time-dependence of the exchange field direction and\nignore the time-dependence of the exchange field mag-\nnitude Ωxc(r) that is induced by the time-dependence\nof the exchange field direction. While the variation of\nthe exchange field magnitude drives currents and torques\nas well, as shown in Ref. [34], the variation of the ex-\nchange field magnitude is a small response and therefore\nthese secondary responses are suppressed in magnitude\nwhen compared to the direct primary responses of the\ncurrent and torque to the variation in the exchange field\ndirection. We will use the perturbations Eq. (15) and\nEq. (16) in order to compute the response of current and\ntorque within the Kubo response formalism. An alterna-\ntive approach for the calculation of the response to time-\ndependent fields is variational linear-response, which has\nbeen applied to the spin susceptibility by Savrasov [35].\nThe perturbation by the time-dependent gradient can\nbe written as\nδH=∂H\n∂ˆM·∂2ˆM\n∂ri∂tsin(qiri)\nqisin(ωt)\nω,(17)\nwhich turns into Eq. (15) when Eq. (9) is inserted. When\nEq. (13) is inserted it turns into the second term in\nEq. (16).\nIn Appendix A we derive the linear response to pertur-\nbations of the type of Eq. (17) and show that the corre-\nsponding coefficient χICIT2a\nijkin Eq. (6) can be expressedas\nχICIT2a\nijk=ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nviRvkRROjR+viRRvkROjR+\n−viRROjRvkR−viRvkROjAA\n+viROjAvkAA+viROjAAvkA\n−viRvkRROjA−viRRvkROjA\n+viRROjAvkA+viAvkAOjAA\n−viAOjAvkAA−viAOjAAvkA/bracketrightBig\n,(18)\nwhereR=GR\nk(E) andA=GA\nk(E) are shorthands for the\nretarded and advanced Green’s functions, respectively,\nandOj=∂H/∂ˆMj.e >0 is the positive elementary\ncharge.\nIn the case of the perturbation of the type Eq. (7) the\nsecond derivative∂2ˆM\n∂ri∂tis perpendicular to M. In this\ncase it is convenient to rewrite Eq. (6) as\nJICIT2a\ni=/summationdisplay\njkχICIDMI\nijkˆej·/bracketleftBigg\nˆM×∂2ˆM\n∂rk∂t/bracketrightBigg\n,(19)\nwhere the coefficients χICIDMI\nijkare given by\nχICIDMI\nijk=ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nviRvkRRTjR+viRRvkRTjR+\n−viRRTjRvkR−viRvkRTjAA\n+viRTjAvkAA+viRTjAAvkA\n−viRvkRRTjA−viRRvkRTjA\n+viRRTjAvkA+viAvkATjAA\n−viATjAvkAA−viATjAAvkA/bracketrightBig\n,(20)\nand\nT=ˆM×∂H\n∂ˆM(21)\nis the torque operator. In Sec. IIC we will explain that\nχICIDMI\nijkdescribes the inverse of current-induced DMI\n(ICIDMI).\nIn the case of the perturbation of the type of Eq. (10)\nthe second derivative∂2ˆMj\n∂rk∂tmay be rewritten as product\nof the first derivatives∂ˆMl\n∂tand∂ˆMl\n∂rk. This may be seen5\nas follows:\n∂H\n∂ˆM·∂2ˆM\n∂ri∂t=∂2H\n∂t∂ri=\n=∂\n∂t/bracketleftBigg/parenleftbigg\nˆM×∂H\n∂ˆM/parenrightbigg\n·/parenleftBigg\nˆM×∂ˆM\n∂ri/parenrightBigg/bracketrightBigg\n=\n=/bracketleftBigg/parenleftBigg\n∂ˆM\n∂t×∂H\n∂ˆM/parenrightBigg\n·/parenleftBigg\nˆM×∂ˆM\n∂ri/parenrightBigg/bracketrightBigg\n=\n=/bracketleftBigg/parenleftBigg/parenleftBigg\nˆM×∂ˆM\n∂t/parenrightBigg\n׈M/parenrightBigg\n×∂H\n∂ˆM/bracketrightBigg\n·/bracketleftBigg\nˆM×∂ˆM\n∂ri/bracketrightBigg\n=\n=−/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n·/bracketleftBigg\nˆM×∂ˆM\n∂ri/bracketrightBigg/bracketleftbigg\nˆM·∂H\n∂ˆM/bracketrightbigg\n=\n=−∂ˆM\n∂t·∂ˆM\n∂ri/bracketleftbigg\nˆM·∂H\n∂ˆM/bracketrightbigg\n.\n(22)\nThis expression is indeed satisfied by Eq. (11), Eq. (12)\nand Eq. (13):\n∂ˆM\n∂ri·∂ˆM\n∂t=−∂2ˆM\n∂ri∂t·ˆM (23)\natr= 0,t= 0. Consequently, Eq. (6) can be rewritten\nas\nJICIT2a\ni=/summationdisplay\njkχICIT2a\nijk∂2ˆMj\n∂rk∂t=\n=−/summationdisplay\njklχICIT2a\nijk∂ˆMl\n∂rk∂ˆMl\n∂t[1−δjl]\n=/summationdisplay\njklχICIT2a\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n,\n(24)\nwhere\nχICIT2a\nijkl=−/summationdisplay\nmχICIT2a\niml[1−δjm]δjk.(25)\nThus, Eq. (24) and Eq. (25) can be used to express\nJICIT2a\niin the form of Eq. (4).\nC. Direct and inverse CIDMI\nEq. (20) describes the response of the electric current\nto time-dependent magnetization gradients of the type\nEq. (15). The reciprocal process consists in the current-\ninduced modification of DMI. This can be shown by ex-\npressing the DMI coefficients as [10]\nDij=1\nV/summationdisplay\nnf(Ekn)/integraldisplay\nd3r(ψkn(r))∗Dijψkn(r)\n=1\nV/summationdisplay\nnf(Ekn)/integraldisplay\nd3r(ψkn(r))∗Ti(r)rjψkn(r),\n(26)where we defined the DMI-operator Dij=Tirj. Using\nthe Kubo formalism the current-induced modification of\nDMI may be written as\nDCIDMI\nij=/summationdisplay\nkχCIDMI\nkijEk (27)\nwith\nχCIDMI\nkij=1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(28)\nwhere\n∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) =−i∞/integraldisplay\n0dteiωt∝an}bracketle{t[Dij(t),vk(0)]−∝an}bracketri}ht(29)\nis the Fourier transform of a retarded function and Vis\nthe volume of the unit cell.\nSince the position operator rin the DMI operator\nDij=Tirjis not compatible with Bloch periodic bound-\nary conditions, we do not use Eq. (28) for numerical\ncalculations of CIDMI. However, it is convenient to use\nEq. (28) in order to demonstrate the reciprocity between\ndirect and inverse CIDMI.\nInverseCIDMI (ICIDMI) describes the electric current\nthat responds to the perturbation by a time-dependent\nmagnetization gradient according to\nJICIDMI\nk=/summationdisplay\nijχICIDMI\nkijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\n.(30)\nThe perturbation by a time-dependent magnetization\ngradient may be written as\nδH=−/summationdisplay\njm·∂2ˆM\n∂t∂rjrjΩxc(r)sin(ωt)\nω=\n=/summationdisplay\njT·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\nrjsin(ωt)\nω\n=/summationdisplay\nijDijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\nsin(ωt)\nω.(31)\nConsequently, the coefficient χICIDMI\nkijis given by\nχICIDMI\nkij=1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tvk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n.(32)\nUsing\n∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω,ˆM) =−∝an}bracketle{t∝an}bracketle{tvk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω,−ˆM) (33)\nwe find that CIDMI and ICIDMI are related through the\nequations\nχCIDMI\nkij(ˆM) =−χICIDMI\nkij(−ˆM). (34)\nIn order to calculate CIDMI we use Eq. (20) for ICIDMI\nand then use Eq. (34) to obtain CIDMI.6\nThe perturbation Eq. (16) describes a different kind\nof time-dependent magnetization gradient, for which the\nreciprocaleffect consists in the modification of the expec-\ntation value ∝an}bracketle{tσ·ˆMrj∝an}bracketri}ht. However, while the modification\nof∝an}bracketle{tTirj∝an}bracketri}htby an applied current can be measured [8, 9]\nfrom the change of the DMI constant Dij, the quantity\n∝an}bracketle{tσ·ˆMrj∝an}bracketri}hthas not been considered so far in ferromagnets.\nIn noncollinear magnets the quantity ∝an}bracketle{tσrj∝an}bracketri}htcan be used\ntodefinespintoroidization[36]. Therefore,whiletheper-\nturbation of the type of Eq. (15) is related to CIDMI and\nICIDMI, which are both accessible experimentally [8, 9],\nin the case of the perturbation of the type of Eq. (16)\nwe expect that only the effect of driving current by the\ntime-dependent magnetization gradient is easily accessi-\nble experimentally, while its inverse effect is difficult to\nmeasure.\nD. Direct and inverse dynamical DMI\nNot only applied electric currents modify DMI, but\nalso magnetization dynamics, which we call dynamical\nDMI (DDMI). DDMI can be expressed as\nDDDMI\nij=/summationdisplay\nkχDDMI\nkijˆek·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n.(35)\nIn Sec. IIG we will show that the spatial gradient of\nDDMI contributes to damping and gyromagnetism in\nnoncollinear magnets. The perturbation used to describe\nmagnetization dynamics is given by [24]\nδH=sin(ωt)\nω/parenleftBigg\nˆM×∂ˆM\n∂t/parenrightBigg\n·T.(36)\nConsequently, the coefficients χDDMI\nkijmay be written as\nχDDMI\nkij=−1\nVlim\nω→0/bracketleftbigg1\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tDij;Tk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightbigg\n.(37)\nSince the position operator in Dijis not compatible\nwith Bloch periodic boundary conditions, we do not use\nEq. (37) for numerical calculations of DDMI, but instead\nwe obtain it from its inverse effect, which consists in the\ngeneration of torques on the magnetization due to time-\ndependent magnetization gradients. These torques can\nbe written as\nTIDDMI\nk=/summationdisplay\nijχIDDMI\nkijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\n,(38)\nwhere the coefficients χIDDMI\nkijare\nχIDDMI\nkij=1\nVlim\nω→0/bracketleftbigg1\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tTk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightbigg\n,(39)becausethe perturbationby the time-dependent gradient\ncan be expressed in terms of Dijaccording to Eq. (31)\nand because the torque on the magnetizationis described\nby−T[23]. Consequently,DDMIandIDDMIarerelated\nby\nχDDMI\nkij(ˆM) =−χIDDMI\nkij(−ˆM). (40)\nFor numerical calculations of IDDMI we use\nχIDDMI\nijk=i\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvkRRTjR+TiRRvkRTjR+\n−TiRRTjRvkR−TiRvkRTjAA\n+TiRTjAvkAA+TiRTjAAvkA\n−TiRvkRRTjA−TiRRvkRTjA\n+TiRRTjAvkA+TiAvkATjAA\n−TiATjAvkAA−TiATjAAvkA/bracketrightBig\n,(41)\nwhichisderivedinAppendix A. InordertoobtainDDMI\nwecalculateIDDMIfromEq.(41)andusethereciprocity\nrelation Eq. (40).\nEq.(38)is validfortime-dependent magnetizationgra-\ndients that lead to perturbations of the type of Eq. (15).\nPerturbations of the second type, Eq. (16), will induce\ntorques on the magnetization as well. However, the in-\nverse effect is difficult to measure in that case, because it\ncorresponds to the modification of the expectation value\n∝an}bracketle{tσ·ˆMrj∝an}bracketri}htby magnetization dynamics. Therefore, while\nin the case of Eq. (15) both direct and inverse response\nare expected to be measurable and correspond to ID-\nDMI and DDMI, respectively, we expect that in the case\nof Eq. (16) only the direct effect, i.e., the response of the\ntorque to the perturbation, is easy to observe.\nE. Dynamical orbital magnetism (DOM)\nMagnetization dynamics does not only induce DMI,\nbut also orbital magnetism, which we call dynamical or-\nbital magnetism (DOM). It can be written as\nMDOM\nij=/summationdisplay\nkχDOM\nkijˆek·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n,(42)\nwhere we introduced the notation\nMDOM\nij=e\nV∝an}bracketle{tvirj∝an}bracketri}htDOM, (43)\nwhich defines a generalized orbital magnetization, such\nthat\nMDOM\ni=1\n2/summationdisplay\njkǫijkMDOM\njk (44)7\ncorresponds to the usual definition of orbital magnetiza-\ntion. The coefficients χDOM\nkijare given by\nχDOM\nkij=−1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tvirj;Tk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(45)\nbecause the perturbation by magnetization dynamics is\ndescribed by Eq. (36). We will discuss in Sec. IIF that\nthe spatial gradient of DOM contributes to the inverse\nCIT. Additionally, we will show below that DOM and\nCIDMI are related to each other.\nIn order to obtain an expression for DOM it is conve-\nnient to consider the inverse effect, i.e., the generation of\natorquebythe applicationofa time-dependent magnetic\nfieldB(t) that actsonly onthe orbitaldegreesoffreedom\nof the electrons and not on their spins. This torque can\nbe written as\nTIDOM\nk=1\n2/summationdisplay\nijlχIDOM\nkijǫijl∂Bl\n∂t, (46)\nwhere\nχIDOM\nkij=−1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tTk;virj∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(47)\nbecause the perturbation by the time-dependent mag-\nnetic field is given by\nδH=−e\n2/summationdisplay\nijkǫijkvirj∂Bk\n∂tsin(ωt)\nω.(48)\nTherefore, thecoefficientsofDOMandIDOM arerelated\nby\nχDOM\nkij(ˆM) =−χIDOM\nkij(−ˆM). (49)\nIn Appendix A we show that the coefficient χIDOM\nijkcan\nbe expressed as\nχIDOM\nijk=−ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvkRRvjR+TiRRvkRvjR+\n−TiRRvjRvkR−TiRvkRvjAA\n+TiRvjAvkAA+TiRvjAAvkA\n−TiRvkRRvjA−TiRRvkRvjA\n+TiRRvjAvkA+TiAvkAvjAA\n−TiAvjAvkAA−TiAvjAAvkA/bracketrightBig\n.(50)\nEq. (50) and Eq. (20) differ only in the positions of\nthe two velocity operators and the torque operator be-\ntween the Green functions. As a consequence, IDOM\nare ICIDMI are related. In Table I and Table II we list\nthe relations between IDOM and ICIDMI for the Rashba\nmodel Eq. (83). We will explain in Sec. III that IDOM\nandICIDMI arezeroin the Rashbamodel when themag-\nnetization is along the zdirection. Therefore, we discussin Table I the case where the magnetization lies in the xz\nplane, and in Table II we discuss the case where the mag-\nnetization lies in the yzplane. According to Table I and\nTable II the relation between IDOM and ICIDMI is of\nthe formχIDOM\nijk=±χICIDMI\njik. This is expected, because\nthe indexiinχIDOM\nijkis connected to the torque operator,\nwhile the index jinχICIDMI\nijkis connected to the torque\noperator.\nTABLEI:Relations betweentheinverseofthemagnetization -\ndynamics induced orbital magnetism (IDOM) and inverse\ncurrent-inducedDMI (ICIDMI)in the 2d Rashbamodel when\nˆMlies in the zxplane. The components of χIDOM\nijk(Eq. (50))\nandχICIDMI\nijk(Eq. (20)) are denoted by the three indices ( ijk).\nICIDMI IDOM\n(211) (121)\n(121) (211)\n-(221) (221)\n(112) (112)\n-(212) (122)\n-(122) (212)\n(222) (222)\n(231) (321)\n(132) (312)\n-(232) (322)\nTABLE II: Relations between IDOM and ICIDMI in the 2d\nRashba model when ˆMlies in the yzplane.\nICIDMI IDOM\n(111) (111)\n-(211) (121)\n-(121) (211)\n(221) (221)\n-(112) (112)\n(212) (122)\n(122) (212)\n-(131) (311)\n(231) (321)\n(132) (312)\nF. Contributions from CIDMI and DOM to direct\nand inverse CIT\nIn electronic transport theory the continuity equation\ndetermines the current only up to a curl field [37]. The\ncurl of magnetization corresponds to a bound current\nthat cannot be measured in electron transport experi-\nments such that\nJ=JKubo−∇×M (51)\nhastobeusedtoextractthetransportcurrent Jfromthe\ncurrentJKuboobtained from the Kubo linear response.8\nThe subtraction of ∇×Mhas been shown to be impor-\ntant when calculating the thermoelectric response [37]\nand the anomalous Nernst effect [20]. Similarly, in the\ntheory of the thermal spin-orbit torque [10, 18] the gra-\ndients of the DMI spiralization have to be subtracted in\norder to obtain the measurable torque:\nTi=TKubo\ni−/summationdisplay\nj∂\n∂rjDij, (52)\nwhere the spatial derivative of the spiralization arises\nfrom its temperature dependence and the temperature\ngradient.\nSince CIDMI and DOM depend on the magnetization\ndirection, they vary spatially in noncollinear magnets.\nSimilar to Eq. (52) the spatial derivatives of the current-\ninduced spiralization need to be included into the theory\nof CIT. Additionally, the gradients of DOM correspond\ntocurrentsthatneedtobeconsideredinthetheoryofthe\ninverse CIT, similar to Eq. (51). In section IV we explic-\nitly show that Onsager reciprocity is violated if spatial\ngradients of DOM and CIDMI are not subtracted from\nthe Kubo response expressions. By trial-and-error we\nfind that the following subtractions are necessary to ob-\ntain response currents and torques that satisfy this fun-\ndamental symmetry:\nJICIT\ni=JKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂MDOM\nij\n∂ˆM(53)\nand\nTCIT\ni=TKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂DCIDMI\nij\n∂ˆM,(54)\nwhereJICIT\niis the current driven by magnetization dy-\nnamics, and TCIT\niis the current-induced torque.\nInterestingly, we find that also the diagonal elements\nMDOM\niiare nonzero. This shows that the generalized def-\ninition Eq. (43) is necessary, because the diagonal ele-\nmentsMDOM\niido not contribute in the usual definition\nofMiaccording to Eq. (44). These differences in the\nsymmetry properties between equilibrium and nonequi-\nlibrium orbital magnetism can be traced back to sym-\nmetry breaking by the perturbations. Also in the case\nof the spiralization tensor Dijthe nonequilibrium cor-\nrectionδDijhas different symmetry properties than the\nequilibrium part (see Sec. III).\nThe contribution of DOM to χICIT2\nijklcan be written as\nχICIT2c\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χDOM\njil\n∂ˆM/bracketrightBigg\n(55)\nand the contribution of CIDMI to χCIT2\nijklis given by\nχCIT2b\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χCIDMI\njil\n∂ˆM/bracketrightBigg\n.(56)G. Contributions from DDMI to gyromagnetism\nand damping\nThe response to magnetization dynamics that is de-\nscribed by the torque-torque correlation function con-\nsists of torques that are related to damping and gyro-\nmagnetism [24]. The chiral contribution to these torques\ncan be written as\nTTT2\ni=/summationdisplay\njklχTT2\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n,\n(57)\nwhere the coefficients χTT2\nijklsatisfy the Onsager relations\nχTT2\nijkl(ˆM) =χTT2\njikl(−ˆM). (58)\nSinceDDMIdependsonthemagnetizationdirection,it\nvaries spatially in noncollinear magnets and the resulting\ngradients of DDMI contribute to the damping and to the\ngyromagnetic ratio:\nTTT\ni=TKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂DDDMI\nij\n∂ˆM.(59)\nThe resulting contribution of the spatial derivatives of\nDDMI to the coefficient χTT2\nijklis\nχTT2c\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χDDMI\njil(ˆM)\n∂ˆM/bracketrightBigg\n.(60)\nH. Current-induced torque (CIT) in noncollinear\nmagnets\nThe chiral contribution to CIT consists of the spatial\ngradient of CIDMI, χCIT2b\nijklin Eq. (56), and the Kubo\nlinear response of the torque to the applied electric field\nin a noncollinear magnet, χCIT2a\nijkl:\nχCIT2\nijkl=χCIT2a\nijkl+χCIT2b\nijkl. (61)\nIn orderto determine χCIT2a\nijkl, we assume that the magne-\ntization direction ˆM(r) oscillates spatially as described\nby\nˆM(r) =\nηsin(q·r)\n0\n1\n1/radicalBig\n1+η2sin2(q·r),(62)\nwherewewilltakethelimit q→0attheendofthecalcu-\nlation. Since the spatial derivative of the magnetization\ndirection is\n∂ˆM(r)\n∂ri=\nηqicos(q·r)\n0\n0\n+O(η3),(63)9\nthe chiralcontributiontothe CIToscillatesspatiallypro-\nportional to cos( q·r). In order to extract this spatially\noscillating contribution we multiply with cos( q·r) and\nintegrate over the unit cell. The resulting expression for\nχCIT2a\nijklis\nχCIT2a\nijkl=−2e\nVηlim\nq→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tTi(r);vj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n,\n(64)\nwhereVis the volume of the unit cell, and\nthe retarded torque-velocity correlation function\n∝an}bracketle{t∝an}bracketle{tTi(r);vj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) needs to be evaluated in the\npresence of the perturbation\nδH=Tkηsin(q·r) (65)\ndue to the noncollinearity (the index kin Eq. (65) needs\nto match the index kinχCIT2a\nijkl).\nIn Appendix B we show that χCIT2a\nijklcan be written as\nχCIT2a\nijkl=−2e\n/planckover2pi1Im/bracketleftBig\nW(surf)\nijkl+W(sea)\nijkl/bracketrightBig\n,(66)\nwhere\nW(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nTiGR\nk(E)vlGR\nk(E)vjGA\nk(E)TkGA\nk(E)\n+TiGR\nk(E)vjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−TiGR\nk(E)vjGA\nk(E)TkGA\nk(E)vlGA\nk(E)\n+/planckover2pi1\nmeδjlTiGR\nk(E)GA\nk(E)TkGA\nk(E)/bracketrightBigg(67)\nis a Fermi surface term ( f′(E) =df(E)/dE) and\nW(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)/bracketleftBigg\n−Tr[TiRvlRRvjRTkR]−Tr[TiRvlRTkRRvjR]\n−Tr[TiRRvlRvjRTkR]−Tr[TiRRvjRvlRTkR]\n+Tr[TiRRvjRTkRvlR]+Tr[TiRRTkRvjRvlR]\n+Tr[TiRRTkRvlRvjR]−Tr[TiRRvlRTkRvjR]\n−Tr[TiRvlRRTkRvjR]+Tr[TiRTkRRvjRvlR]\n+Tr[TiRTkRRvlRvjR]+Tr[TiRTkRvlRRvjR]\n−/planckover2pi1\nmeδjlTr[TiRRRTkR]−/planckover2pi1\nmeδjlTr[TiAAATkA]\n−/planckover2pi1\nmeδjlTr[TiAATkAA]/bracketrightBigg(68)\nis a Fermi sea term.I. Inverse CIT in noncollinear magnets\nThe chiral contribution JICIT2(see Eq. (4)) to the\ncharge pumping is described by the coefficients\nχICIT2\nijkl=χICIT2a\nijkl+χICIT2b\nijkl+χICIT2c\nijkl,(69)\nwhereχICIT2a\nijkldescribes the response to the time-\ndependentmagnetizationgradient(seeEq.(18),Eq.(25),\nand Eq. (24)) and χICIT2c\nijklresults from the spatial gra-\ndient of DOM (see Eq. (55)). χICIT2b\nijkldescribes the re-\nsponseto the perturbation bymagnetizationdynamics in\na noncollinear magnet. In order to derive an expression\nforχICIT2b\nijklwe assume that the magnetization oscillates\nspatially as described by Eq. (62). Since the correspond-\ning response oscillates spatially proportional to cos( q·r),\nwe multiply by cos( q·r) and integrate over the unit cell\nin order to extract χICIT2b\nijklfrom the retarded velocity-\ntorque correlation function ∝an}bracketle{t∝an}bracketle{tvi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω), which\nis evaluated in the presence of the perturbation Eq. (65).\nWe obtain\nχICIT2b\nijkl=2e\nVηlim\nq→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tvi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n,\n(70)\nwhich can be written as (see Appendix B)\nχICIT2b\nijkl=2e\n/planckover2pi1Im/bracketleftBig\nV(surf)\nijkl+V(sea)\nijkl/bracketrightBig\n,(71)\nwhere\nV(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBig\nviGR\nk(E)vlGR\nk(E)TjGA\nk(E)TkGA\nk(E)\n+viGR\nk(E)TjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−viGR\nk(E)TjGA\nk(E)TkGA\nk(E)vlGA\nk(E)/bracketrightBig(72)\nis the Fermi surface term and\nV(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\n−Tr[viRvlRRTjRTkR]−Tr[viRvlRTkRRTjR]\n−Tr[viRRvlRTjRTkR]−Tr[viRRTjRvlRTkR]\n+Tr[viRRTjRTkRvlR]+Tr[viRRTkRTjRvlR]\n+Tr[viRRTkRvlRTjR]−Tr[viRRvlRTkRTjR]\n−Tr[viRvlRRTkRTjR]+Tr[viRTkRRTjRvlR]\n+Tr[viRTkRRvlRTjR]+Tr[viRTkRvlRRTjR]/bracketrightBig(73)\nis the Fermi sea term.\nIn Eq. (70) we use the Kubo formula to describe the\nresponse to magnetization dynamics combined with per-\nturbation theory to include the effect of noncollinearity.10\nThereby, the time-dependent perturbation and the per-\nturbation by the magnetization gradient are separated\nand perturbations of the form of Eq. (15) or Eq. (16)\nare not automatically included. For example the flat cy-\ncloidal spin spiral\nˆM(x,t) =\nsin(qx−ωt)\n0\ncos(qx−ωt)\n (74)\nmoving inxdirection with speed ω/qand the helical spin\nspiral\nˆM(y,t) =\nsin(qy−ωt)\n0\ncos(qy−ωt)\n (75)\nmovinginydirectionwith speed ω/qbehavelikeEq.(10)\nwhentandraresmall. Thus, these movingdomainwalls\ncorrespond to the perturbation of the type of Eq. (10)\nand the resulting contribution JICIT2afrom the time-\ndependent magnetization gradient is not described by\nEq. (70) and needs to be added, which we do by adding\nχICIT2a\nijklin Eq. (69).\nJ. Damping and gyromagnetism in noncollinear\nmagnets\nThe chiral contribution Eq. (57) to the torque-torque\ncorrelation function is expressed in terms of the coeffi-\ncient\nχTT\nijkl=χTT2a\nijkl+χTT2b\nijkl+χTT2c\nijkl, (76)\nwhereχTT2c\nijklresults from the spatial gradient of DDMI\n(see Eq. (60)), χTT2a\nijkldescribes the response to a time-\ndependent magnetization gradient in a collinear magnet,\nandχTT2b\nijkldescribes the response to magnetization dy-\nnamics in a noncollinear magnet.\nIn order to derive an expression for χTT2b\nijklwe as-\nsume that the magnetization oscillates spatially accord-\ning to Eq. (62). We multiply the retarded torque-torque\ncorrelation function ∝an}bracketle{t∝an}bracketle{tTi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) with cos(qlrl)\nand integrate over the unit cell in order to extract the\npart of the response that varies spatially proportional to\ncos(qlrl). We obtain:\nχTT2b\nijkl=2\nVηlim\nql→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tTi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n.\n(77)\nIn Appendix B we discuss how to evaluate Eq. (77) in\nfirst order perturbation theory with respect to the per-\nturbation Eq.(65) and showthat χTT2b\nijklcan be expressedas\nχTT2b\nijkl=2\n/planckover2pi1Im/bracketleftBig\nX(surf)\nijkl+X(sea)\nijkl/bracketrightBig\n,(78)\nwhere\nX(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nTiGR\nk(E)vlGR\nk(E)TjGA\nk(E)TkGA\nk(E)\n+TiGR\nk(E)TjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−TiGR\nk(E)TjGA\nk(E)TkGA\nk(E)vlGA\nk(E)/bracketrightBigg(79)\nis a Fermi surface term and\nX(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBigg\n−(TiRvlRRTjRTkR)−(TiRvlRTkRRTjR)\n−(TiRRvlRTjRTkR)−(TiRRTjRvlRTkR)\n+(TiRRTjRTkRvlR)+(TiRRTkRTjRvlR)\n+(TiRRTkRvlRTjR)−(TiRRvlRTkRTjR)\n−(TiRvlRRTkRTjR)+(TiRTkRRTjRvlR)\n+(TiRTkRRvlRTjR)+(TiRTkRvlRRTjR)/bracketrightBigg\n(80)\nis a Fermi sea term.\nThe contribution χTT2a\nijklfrom the time-dependent gra-\ndients is given by\nχTT2a\nijkl=−/summationdisplay\nmχTT2a\niml[1−δjm]δjk,(81)\nwhere\nχTT2a\niml=i\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvlRROmR+TiRRvlROmR+\n−TiRROmRvlR−TiRvlROmAA\n+TiROmAvlAA+TiROmAAvlA\n−TiRvlRROmA−TiRRvlROmA\n+TiRROmAvlA+TiAvlAOmAA\n−TiAOmAvlAA−TiAOmAAvlA/bracketrightBig\n,(82)\nwithOm=∂H/∂ˆMm(see Appendix A).\nIII. SYMMETRY PROPERTIES\nIn this section we discuss the symmetry properties of\nCIDMI, DDMI and DOM in the case of the magnetic\nRashba model\nHk(r) =/planckover2pi12\n2mek2+α(k׈ez)·σ+∆V\n2σ·ˆM(r).(83)11\nAdditionally, we discuss the symmetry properties of the\ncurrents and torques induced by time-dependent magne-\ntization gradients of the form of Eq. (10).\nWe consider mirror reflection Mxzat thexzplane,\nmirror reflection Myzat theyzplane, and c2 rotation\naround the zaxis. When ∆ V= 0 these operations leave\nEq. (83) invariant, but when ∆ V∝ne}ationslash= 0 they modify the\nmagnetization direction ˆMin Eq. (83), as shown in Ta-\nble III. At the same time, these operations affect the\ntorqueTandthecurrent Jdrivenbythe time-dependent\nmagnetization gradients (see Table III). In Table IV and\nTable V we show how ˆM×∂ˆM/∂rkis affected by the\nsymmetry operations.\nAflat cycloidalspin spiralwith spinsrotatingin the xz\nplane is mapped by a c2 rotation around the zaxis onto\nthe same spin spiral. Similarly, a flat helical spin spiral\nwith spins rotating in the yzplane is mapped by a c2 ro-\ntationaroundthe zaxisontothesamespinspiral. There-\nfore, when ˆMpoints inzdirection, a c2 rotation around\nthezaxis does not change ˆM×∂ˆM/∂ri, but it flips the\nin-plane current Jand the in-plane components of the\ntorque,TxandTy. Consequently, ˆM×∂2ˆM/∂ri∂tdoes\nnot induce currents or torques, i.e., ICIDMI, CIDMI, ID-\nDMI and DDMI are zero, when ˆMpoints inzdirection.\nHowever, they become nonzero when the magnetization\nhas an in-plane component (see Fig. 1).\nSimilarly, IDOM vanishes when the magnetization\npoints inzdirection: In that case Eq. (83) is invariant\nunder the c2 rotation. A time-dependent magnetic field\nalongzdirection is invariant under the c2 rotation as\nwell. However, TxandTychange sign under the c2 rota-\ntion. Consequently, symmetryforbidsIDOM inthiscase.\nHowever, when the magnetization has an in-plane com-\nponent, IDOM and DOM become nonzero (see Fig. 2).\nThat time-dependent magnetization gradients of the\ntype of Eq. (7) do not induce in-plane currents and\ntorqueswhen ˆMpoints inzdirectioncan alsobe seendi-\nrectly from Eq. (7): The c2 rotation transforms q→ −q\nandMx→ −Mx. Since sin( q·r) is odd in r, Eq. (7) is in-\nvariantunder c2rotation, whilethe in-planecurrentsand\ntorques induced by time-dependent magnetization gradi-\nents change sign under c2 rotation. In contrast, Eq. (10)\nis not invariant under c2 rotation, because sin( q·r−ωt)\nis not odd in rfort>0. Consequently, time-dependent\nmagnetization gradients of the type of Eq. (10) induce\ncurrents and torques also when ˆMpoints locally into\nthezdirection. These currents and torques, which are\ndescribed by Eq. (24) and Eq. (82), respectively, need to\nbe added to the chiral ICIT and the chiral torque-torque\ncorrelation. While CIDMI, DDMI, and DOM are zero\nwhen the magnetization points in zdirection, their gra-\ndients are not (see Fig. 1 and Fig. 2). Therefore, the gra-\ndients of CIDMI, DOM, and DDMI contribute to CIT, to\nICIT and to the torque-torque correlation, respectively,\neven when ˆMpoints locally into the zdirection.TABLE III: Effect of mirror reflection Mxzat thexzplane,\nmirror reflection Myzat theyzplane, and c2 rotation around\nthezaxis. The magnetization Mand the torque Ttransform\nlike axial vectors, while the current Jtransforms like a polar\nvector.\nMxMyMzJxJyTxTyTz\nMxz−MxMy−MzJx−Jy−TxTy−Tz\nMyzMx−My−Mz−JxJyTx−Ty−Tz\nc2-Mx-MyMz-Jx−Jy−Tx−TyTz\nTABLE IV: Effect of symmetry operations on the magneti-\nzation gradients. Magnetization gradients are described b y\nthree indices ( ijk). The first index denotes the magnetiza-\ntion direction at r= 0. The third index denotes the di-\nrection along which the magnetization changes. The second\nindex denotes the direction of ∂ˆM/∂rkδrk. The direction of\nˆM×∂ˆM/∂rkis specified by the number below the indices\n(ijk).\n(1,2,1) (1,3,1) (2,1,1) (2,3,1) (3,1,1) (3,2,1)\n3-2 -3 1 2-1\nMxz(-1,2,1)(-1,-3,1) (2,-1,1) (2,-3,1) (-3,-1,1) (-3,2,1)\n-3 -2 3 -1 2 1\nMyz(1,2,1) (1,3,1)(-2,-1,1) (-2,3,1) (-3,-1,1) (-3,2,1)\n3-2 -3 -1 2 1\nc2(-1,2,1)(-1,-3,1) (-2,1,1) (-2,-3,1) (3,1,1) (3,2,1)\n-3 -2 3 1 2-1\n.\nTABLE V: Continuation of Table IV\n(1,2,2) (1,3,2) (2,1,2) (2,3,2) (3,1,2) (3,2,2)\n3 -2 -3 1 2-1\nMxz(-1,-2,2) (-1,3,2) (2,1,2) (2,3,2)(-3,1,2)(-3,-2,2)\n3 2-3 1-2 -1\nMyz(1,-2,2) (1,-3,2) (-2,1,2)(-2,-3,2) (-3,1,2)(-3,-2,2)\n-3 2 3 1-2 -1\nc2(-1,2,2) (-1,-3,2) (-2,1,2)(-2,-3,2) (3,1,2) (3,2,2)\n-3 -2 3 1 2-1\nA. Symmetry properties of ICIDMI and IDDMI\nInthefollowingwediscusshowTableIII,TableIV,and\nTable V can be used to analyze the symmetry of ICIDMI\nandIDDMI.AccordingtoEq.(19)thecoefficient χICIDMI\nijk\ndescribes the response of the current JICIT2a\nito the time-\ndependent magnetization gradient ˆej·[ˆM×∂2ˆM\n∂rk∂t]. Since\nˆM×∂2ˆM\n∂rk∂t=∂\n∂t[ˆM×∂ˆM\n∂rk] fortime-dependent magnetiza-\ntion gradients of the type Eq. (7) the symmetry proper-\nties ofχICIDMI\nijkfollow from the transformation behaviour\nofˆM×∂ˆM\n∂rkandJunder symmetry operations.\nWe consider the case with magnetization in xdirec-\ntion. The component χICIDMI\n132describes the current in x\ndirection induced by the time-dependence of a cycloidal\nmagnetizationgradientin ydirection(withspinsrotating12\nFIG. 1: ICIDMI in a noncollinear magnet. (a) Arrows illus-\ntrate the magnetization direction. (b) Arrows illustrate t he\ncurrentJyinduced by a time-dependent magnetization gra-\ndient, which is described by χICIDMI\n221. When ˆMpoints in z\ndirection, χICIDMI\n221andJyare zero. The sign of χICIDMI\n221and\nofJychanges with the sign of Mx.\nFIG. 2: DOM in a noncollinear magnet. (a) Arrows illustrate\nthe magnetization direction. (b) Arrows illustrate the orb ital\nmagnetization induced by magnetization dynamics (DOM).\nWhenˆMpoints in zdirection, DOM is zero. The sign of\nDOM changes with the sign of Mx.\nin thexyplane).Myzflips both ˆM×∂ˆM\n∂yandJx, but\nit preserves ˆM.Mzxpreserves ˆM×∂ˆM\n∂yandJx, but it\nflipsˆM. A c2 rotation around the zaxis flips ˆM×∂ˆM\n∂y,\nˆMandJx. Consequently, χICIDMI\n132(ˆM) is allowed by\nsymmetry and it is even in ˆM. The component χICIDMI\n122\ndescribes the current in xdirection induced by the time-\ndependence of a helical magnetization gradient in ydi-\nrection (with spins rotating in the xzplane).Myzflips\nˆM×∂ˆM\n∂yandJx, but it preserves ˆM.MzxflipsˆM×∂ˆM\n∂y\nandˆM, but it preserves Jx. A c2 rotation around the z\naxis flipsJxandˆM, but it preserves ˆM×∂ˆM\n∂y. Conse-\nquently,χICIDMI\n122is allowed by symmetry and it is odd in\nˆM. The component χICIDMI\n221describes the current in y\ndirection induced by the time-dependence of a cycloidal\nmagnetization gradient in xdirection (with spins rotat-\ning in thexzplane).Mzxpreserves ˆM×∂ˆM\n∂x, but it flipsJyandˆM.Myzpreserves ˆM,Jy, andˆM×∂ˆM\n∂x. The\nc2 rotation around the zaxis preserves ˆM×∂ˆM\n∂x, but\nit flipsˆMandJy. Consequently, χICIDMI\n221is allowed by\nsymmetry and it is odd in ˆM. The component χICIDMI\n231\ndescribes the current in ydirection induced by the time-\ndependence of a cycloidal magnetization gradient in xdi-\nrection (with spins rotating in the xyplane).Mzxflips\nˆM×∂ˆM\n∂x,ˆM, andJy.Myzpreserves ˆM×∂ˆM\n∂x,ˆMand\nJy. The c2 rotation around the zaxis flips ˆM×∂ˆM\n∂x,Jy,\nandˆM. Consequently, χICIDMI\n231is allowed by symmetry\nand it is even in ˆM.\nThese properties are summarized in Table VI. Due to\nthe relations between CIDMI and DOM (see Table I and\nTable II), they can be used for DOM as well. When the\nmagnetization lies at a general angle in the xzplane or in\ntheyzplaneseveraladditionalcomponentsofCIDMIand\nDOMarenonzero(seeTableIandTableII,respectively).\nTABLE VI: Allowed components of χICIDMI\nijkwhenˆMpoints\ninxdirection. + components are even in ˆM, while - compo-\nnents are odd in ˆM.\n132 122 221 231\n+ - - +\nSimilarly, one can analyze the symmetry of DDMI. Ta-\nble VII lists the components of DDMI, χDDMI\nijk, which are\nallowed by symmetry when ˆMpoints inxdirection.\nTABLEVII:Allowedcomponentsof χDDMI\nijkwhenˆMpointsin\nxdirection. +componentsareevenin ˆM, while -components\nare odd in ˆM.\n222 232 322 332\n- + + -\nB. Response to time-dependent magnetization\ngradients of the second type (Eq. (10))\nAccording to Eq. (13) the time-dependent magneti-\nzation gradient is along the magnetization. Therefore,\nin contrast to the discussion in section IIIA we can-\nnot use ˆM×∂2ˆM\n∂rk∂tin the symmetry analysis. Eq. (24)\nand Eq. (25) show that χICIT2a\nijjldescribes the response of\nJICIT2a\nitoˆej·/bracketleftBig\nˆM×∂ˆM\n∂t/bracketrightBig\nˆej·/bracketleftBig\nˆM×∂ˆM\n∂rl/bracketrightBig\nwhileχICIT2a\nijkl=\n0 forj∝ne}ationslash=k. According to Eq. (23) the symmetry prop-\nerties of/bracketleftBig\nˆM×∂ˆM\n∂t/bracketrightBig\n·/bracketleftBig\nˆM×∂ˆM\n∂rl/bracketrightBig\nagree to the symmetry\nproperties of ˆM·∂2ˆM\n∂rl∂t. Therefore, in order to under-\nstand the symmetry properties of χICIT2a\nijjlwe consider\nthe transformation of JandˆM·∂2ˆM\n∂rl∂tunder symmetry\noperations.\nWe consider the case where ˆMpoints inzdirection.\nχICIT2a\n1jj1describes the current driven in xdirection, when13\nthe magnetization varies in xdirection. MxzflipsˆM,\nbut preserves JxandˆM·∂2ˆM/(∂x∂t).MyzflipsˆM,Jx,\nandˆM·∂2ˆM/(∂x∂t). c2 rotation flips ˆM·∂2ˆM/(∂x∂t)\nandJx, but preserves ˆM. Consequently, χICIT2a\n1jj1is al-\nlowed by symmetry and it is even in ˆM.\nχICIT2a\n2jj1describes the current flowing in ydirection,\nwhen magnetization varies in xdirection. MxzflipsˆM\nandJy, but preserves ˆM·∂2ˆM/(∂x∂t).MyzflipsˆM,\nandˆM·∂2ˆM/(∂x∂t), but preserves Jy. c2 rotation\nflipsˆM·∂2ˆM/(∂x∂t) andJy, but preserves ˆM. Conse-\nquently,χICIT2a\n2jj1is allowed by symmetry and it is odd in\nˆM.\nSimilarly, one can show that χICIT2a\n1jj2is odd in ˆMand\nthatχICIT2a\n2jj2is even in ˆM.\nAnalogously, one can investigate the symmetry prop-\nerties ofχTT2a\nijjl. We find that χTT2a\n1jj1andχTT2a\n2jj2are odd\ninˆM, whileχTT2a\n2jj1andχTT2a\n1jj2are even in ˆM.\nIV. RESULTS\nIn the following sections we discuss the results for the\ndirect and inverse chiral CIT and for the chiral torque-\ntorque correlation in the two-dimensional (2d) Rashba\nmodel Eq. (83), and in the one-dimensional (1d) Rashba\nmodel [38]\nHkx(x) =/planckover2pi12\n2mek2\nx−αkxσy+∆V\n2σ·ˆM(x).(84)\nAdditionally, we discuss the contributions of the time-\ndependent magnetization gradients, and of DDMI, DOM\nand CIDMI to these effects.\nWhile vertex corrections to the chiral CIT and to\nthe chiral torque-torque correlation are important in the\nRashba model [38], the purpose of this work is to show\nthe importance ofthe contributionsfrom time-dependent\nmagnetization gradients, DDMI, DOM and CIDMI. We\ntherefore consider only the intrinsic contributions here,\ni.e., we set\nGR\nk(E) =/planckover2pi1[E −Hk+iΓ]−1, (85)\nwhere Γ is a constant broadening, and we leave the study\nof vertex corrections for future work.\nThe results shown in the following sections are ob-\ntained for the model parameters ∆ V= 1eV,α=2eV˚A,\nand Γ = 0 .1Ry = 1.361eV, when the magnetization\npoints inzdirection, i.e., ˆM=ˆez. The unit of χCIT2\nijkl\nis charge times length in the 1d case and charge in the\n2d case. Therefore, in the 1d case we discuss the chiral\ntorkance in units of ea0, wherea0is Bohr’s radius. In the\n2d case we discuss the chiral torkance in units of e. The\nunit ofχTT2\nijklis angular momentum in the 1d case and\nangular momentum per length in the 2d case. Therefore,\nwe discussχTT2\nijklin units of /planckover2pi1in the 1d case, and in units\nof/planckover2pi1/a0in the 2d case.-2 -1 0 1 2\nFermi energy [eV]-0.02-0.0100.010.020.030.040.05χijklCIT2 [ea0]2121\n1121\n2121 (gauge-field)\n1121 (gauge-field)\nFIG. 3: Chiral CIT in the 1d Rashba model for cycloidal gra-\ndients vs. Fermi energy. General perturbation theory (soli d\nlines) agrees to the gauge-field approach (dashed lines).\nA. Direct and inverse chiral CIT\nIn Fig. 3 we show the chiral CIT as a function of the\nFermi energyfor cycloidalmagnetization gradients in the\n1d Rashba model. The components χCIT2\n2121andχCIT2\n1121are\nlabelled by 2121 and 1121, respectively. The component\n2121ofCITdescribesthe non-adiabatictorque, while the\ncomponent 1121 describes the adiabatic STT (modified\nby SOI). In the one-dimensional Rashba model, the con-\ntributionsχCIT2b\n2121andχCIT2b\n1121(Eq. (56)) from the CIDMI\nare zero when ˆM=ˆez(not shown in the figure). For cy-\ncloidal spin spirals, it is possible to solve the 1d Rashba\nmodel by a gauge-field approach [38], which allows us to\ntest the perturbation theory, Eq. (66). For comparison\nwe show in Fig. 3 the results obtained from the gauge-\nfield approach, which agree to the perturbation theory,\nEq. (66). This demonstrates the validity of Eq. (66).\nIn Fig. 4 we show the chiral ICIT in the 1d Rashba\nmodel. The components χICIT2\n1221andχICIT2\n1121are labelled\nby 1221and 1121, respectively. The contribution χICIT2a\n1221\nfrom the time-dependent gradient is of the same order of\nmagnitude as the total χICIT2\n1221. Comparison of Fig. 3 and\nFig. 4 shows that CIT and ICIT satisfy the reciprocity\nrelationsEq. (5), that χCIT2\n1121is odd in ˆM, and thatχCIT2\n2121\nis even in ˆM, i.e.,χCIT2\n2121=χICIT2\n1221andχCIT2\n1121=−χICIT2\n1121.\nThe contribution χICIT2a\n1221from the time-dependent gradi-\nents is crucial to satisfy the reciprocity relations between\nχCIT2\n2121andχICIT2\n1221.\nIn Fig. 5 and Fig. 6 we show the CIT and the ICIT, re-\nspectively, for helical gradients in the 1d Rashba model.\nThe components χCIT2\n2111andχCIT2\n1111are labelled 2111 and\n1111, respectively, in Fig. 5, while χICIT2\n1211andχICIT2\n1111\nare labelled 1211 and 1111, respectively, in Fig. 6. The\ncontributions χCIT2b\n2111andχCIT2b\n1111from CIDMI are of the14\n-2 -1 0 1 2\nFermi energy [eV]-0.0200.020.04χijklICIT2 [ea0]1221 \n1121\nχ1221ICIT2a\nFIG. 4: Chiral ICIT in the 1d Rashba model for cycloidal\ngradients vs. Fermi energy. Dashed line: Contribution from\nthe time-dependent gradient.\nsame order of magnitude as the total χCIT2\n2111andχCIT2\n1111.\nSimilarly, the contributions χICIT2c\n1211andχICIT2c\n1111from\nDOM are of the same order of magnitude as the to-\ntalχICIT2\n1211andχICIT2\n1111. Additionally, the contribution\nχICIT2a\n1111from the time-dependent gradient is substantial.\nComparisonofFig.5andFig.6showsthatCITandICIT\nsatisfy the reciprocity relation Eq. (5), that χCIT2\n2111is odd\ninˆM, and thatχCIT2\n1111is even in ˆM, i.e.,χCIT2\n1111=χICIT2\n1111\nandχCIT2\n2111=−χICIT2\n1211. These reciprocity relations be-\ntween CIT and ICIT are only satisfied when CIDMI,\nDOM, and the response to time-dependent magnetiza-\ntion gradients are included. Additionally, the compar-\nison between Fig. 5 and Fig. 6 shows that the contri-\nbutions of CIDMI to CIT ( χCIT2b\n1111andχCIT2b\n2111) are re-\nlated to the contributions of DOM to ICIT ( χICIT2c\n1111and\nχICIT2c\n1211). These relations between DOM and ICIT are\nexpected from Table I.\nIn Fig. 7 and Fig. 8 we show the CIT and the ICIT,\nrespectively, for cycloidal gradients in the 2d Rashba\nmodel. In this case there are contributions from CIDMI\nand DOM in contrast to the 1d case with cycloidal gra-\ndients (Fig. 3). Comparison between Fig. 7 and Fig. 8\nshows that χCIT2\n1121andχCIT2\n2221are odd in ˆM, thatχCIT2\n1221\nandχCIT2\n2121are even in ˆM, and that CIT and ICIT sat-\nisfy the reciprocity relation Eq. (5) when the gradients\nof CIDMI and DOM are included, i.e., χCIT2\n1121=−χICIT2\n1121,\nχCIT2\n2221=−χICIT2\n2221,χCIT2\n1221=χICIT2\n2121, andχCIT2\n2121=χICIT2\n1221.\nχCIT2\n1121describesthe adiabatic STT with SOI, while χCIT2\n2121\ndescribes the non-adiabatic STT. Experimentally, it has\nbeen found that CITs occur also when the electric field\nis applied parallel to domain-walls (i.e., perpendicular to\ntheq-vector of spin spirals) [39]. In our calculations, the\ncomponents χCIT2\n2221andχCIT2\n1221describe such a case, where\nthe applied electric field points in ydirection, while the-2 -1 0 1 2\nFermi energy [eV]-0.04-0.0200.020.040.06χijklCIT2 [ea0]1111\n2111\nχ1111CIT2b\nχ2111CIT2b\nFIG. 5: Chiral CIT for helical gradients in the 1d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.0200.020.040.06χijklICIT2 [ea0]1111\n1211\nχ1111ICIT2a\nχ1111ICIT2c\nχ1211ICIT2c\nFIG. 6: Chiral ICIT for helical gradients in the 1d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted line: Contribution from the time-\ndependent magnetization gradient.\nmagnetization direction varies with the xcoordinate.\nIn Fig. 9 and Fig. 10 we show the chiral CIT and\nICIT, respectively, for helical gradients in the 2d Rashba\nmodel. The component χCIT2\n2111describes the adiabatic\nSTT with SOI and the component χCIT2\n1111describes the\nnon-adiabatic STT. The components χCIT2\n2211andχCIT2\n1211\ndescribe the case when the applied electric field points\ninydirection, i.e., perpendicular to the direction along\nwhich the magnetization direction varies. Comparison\nbetween Fig. 9 and Fig. 10 shows that χCIT2\n1111andχCIT2\n2211\nare even in ˆM, thatχCIT2\n1211andχCIT2\n2111are odd in ˆMand\nthat CIT andICIT satisfythe reciprocityrelationEq.(5)\nwhenthegradientsofCIDMIandDOMareincluded, i.e.,15\n-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklCIT2 [e]1121\n2221\n1221\n2121\nχ2221CIT2b\nχ1221CIT2b\nχ2121CIT2b\nFIG. 7: Chiral CIT for cycloidal gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklICIT2 [e]1121\n1221\n2121\n2221\nχ2221ICIT2a\nχ1221ICIT2a\nχ2121ICIT2c\nχ1221ICIT2c\nχ2221ICIT2c\nFIG. 8: Chiral ICIT for cycloidal gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted lines: Contributions from the time-\ndependent gradients.\nχCIT2\n1111=χICIT2\n1111,χCIT2\n2211=χICIT2\n2211,χCIT2\n1211=−χICIT2\n2111, and\nχCIT2\n2111=−χICIT2\n1211.\nB. Chiral torque-torque correlation\nIn Fig. 11 we show the chiral contribution to the\ntorque-torque correlation in the 1d Rashba model for\ncycloidal gradients. We compare the perturbation the-\nory Eq. (78) plus Eq. (82) to the gauge-field approach\nfrom Ref. [38]. This comparison shows that perturba-\ntion theory provides the correct answer only when the\ncontribution χTT2a\nijkl(Eq. (82)) from the time-dependent-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklCIT2 [e]2211\n1111\n1211\n2111\nχ2111CIT2b\nχ1211CIT2b\nχ2211CIT2b\nχ1111CIT2b\nFIG. 9: Chiral CIT for helical gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.004-0.00200.0020.0040.006χijklICIT2 [e]1111\n1211\n2111\n2211\nχ1111ICIT2a\nχ2221ICIT2a\nχ1111ICIT2c\nχ2111ICIT2c\nχ1211ICIT2c\nχ2211ICIT2c\nFIG. 10: Chiral ICIT for helical gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted lines: Contributions from the time-\ndependent gradient.\ngradients is taken into account. The contributions χTT2a\n1221\nandχTT2a\n2221fromthe time-dependent gradientsarecompa-\nrable in magnitude to the total values. In the 1d Rashba\nmodel the DDMI-contribution in Eq. (60) is zero for cy-\ncloidal gradients (not shown in the figure). The compo-\nnentsχTT2\n2121andχTT2\n1221describe the chiral gyromagnetism\nwhile the components χTT2\n1121andχTT2\n2221describe the chi-\nral damping [38, 40, 41]. The components χTT2\n2121and\nχTT2\n1221are odd in ˆMand they satisfy the Onsagerrelation\nEq. (58), i.e., χTT2\n2121=−χTT2\n1221.\nIn Fig. 12 we show the chiral contributions to the\ntorque-torque correlation in the 1d Rashba model for\nhelical gradients. In contrast to the cycloidal gradients16\n-2 -1 0 1 2\nFermi energy [eV]-0.00500.0050.01χijklTT2 [h_]2121\n1221\n2221\n1121\nχ1221TT2a\nχ2221TT2a\n2121 (gf)\n1221 (gf)\n1121 (gf)\n2221 (gf)\nFIG. 11: Chiral contribution to the torque-torque correla-\ntion for cycloidal gradients in the 1d Rashba model vs. Fermi\nenergy. Perturbation theory (solid lines) agrees to the gau ge-\nfield (gf) approach (dotted lines). Dashed lines: Contribut ion\nfrom the time-dependent gradient.\n(Fig. 11) there are contributions from the spatial gra-\ndients of DDMI (Eq. (60)) in this case. The Onsager\nrelation Eq. (58) for the components χTT2\n2111andχTT2\n1211is\nsatisfied only when these contributions from DDMI are\ntaken into account, which are of the same order of mag-\nnitude as the total values. The components χTT2\n2111and\nχTT2\n1211are even in ˆMand describe chiral damping, while\nthe components χTT2\n1111andχTT2\n2211are odd in ˆMand de-\nscribe chiral gyromagnetism. As a consequence of the\nOnsager relation Eq. (58) we obtain χTT2\n1111=χTT2\n2211= 0\nfor the total components: Eq. (58) shows that diagonal\ncomponents of the torque-torque correlation function are\nzero unless they are even in ˆM. However, χTT2a\n1111,χTT2c\n1111,\nandχTT2b\n1111=−χTT2a\n1111−χTT2c\n1111are individually nonzero.\nInterestingly, the off-diagonal components of the torque-\ntorquecorrelationdescribechiraldampingforhelicalgra-\ndients, while for cycloidal gradients the off-diagonal ele-\nments describe chiral gyromagnetism and the diagonal\nelements describe chiral damping.\nIn Fig. 13 we show the chiral contributions to the\ntorque-torque correlation in the 2d Rashba model for cy-\ncloidal gradients. In contrast to the 1d Rashba model\nwith cycloidal gradients (Fig. 11) the contributions from\nDDMIχTT2c\nijkl(Eq.(60))arenonzerointhiscase. Without\nthesecontributionsfromDDMI theOnsagerrelation(58)\nχTT2\n2121=−χTT2\n1221is violated. The DDMI contribution is\nof the same order of magnitude as the total values. The\ncomponents χTT2\n2121andχTT2\n1221are odd in ˆMand describe\nchiral gyromagnetism, while the components χTT2\n1121and\nχTT2\n2221are even in ˆMand describe chiral damping.\nIn Fig. 14 we show the chiral contributions to the\ntorque-torque correlation in the 2d Rashba model for he-\nlical gradients. The components χTT2\n1211andχTT2\n2111are even-2 -1 0 1 2\nFermi energy [eV]-0.00500.0050.01χijklTT2 [h_]1111\n2111\n1211\n2211\nχ1111TT2c\nχ2111TT2c\nχ1211TT2c\nχ2211TT2c\nχ1111TT2a\nχ2111TT2a\nFIG. 12: Chiral contribution to the torque-torque correla-\ntion for helical gradients in the 1d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\n-2 -1 0 1 2\nFermi energy [eV]-0.000500.00050.001χijklTT2 [h_/a0]1121\n2121\n1221\n2221\nχ1221TT2a\nχ2221TT2a\nχ2121TT2c\nχ1221TT2c\nFIG. 13: Chiral contribution to the torque-torque correla-\ntion for cycloidal gradients in the 2d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\ninˆMand describe chiral damping, while the compo-\nnentsχTT2\n1111andχTT2\n2211are odd in ˆMand describe chiral\ngyromagnetism. The Onsager relation Eq. (58) requires\nχTT2\n1111=χTT2\n2211= 0 andχTT2\n2111=χTT2\n1211. Without the\ncontributions from DDMI these Onsager relations are vi-\nolated.17\n-2 -1 0 1 2\nFermi energy [eV]-0.000500.00050.001χijklTT2 [h_ /a0]1111\n2111\n1211\n2211\nχ1111TT2a\nχ2111TT2a\nχ1111TT2c\nχ1211TT2c\nχ2211TT2c\nFIG. 14: Chiral contribution to the torque-torque correla-\ntion for helical gradients in the 2d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\nV. SUMMARY\nFinding ways to tune the Dzyaloshinskii-Moriya inter-\naction (DMI) by external means, such as an applied elec-\ntriccurrent,holdsmuchpromiseforapplicationsinwhich\nDMI determines the magnetic texture of domain walls or\nskyrmions. In order to derive an expression for current-\ninduced Dzyaloshinskii-Moriya interaction (CIDMI) we\nfirst identify its inverse effect: When magnetic textures\nvary as a function of time, electric currents are driven by\nvarious mechanisms, which can be distinguished accord-\ningtotheirdifferentdependenceonthetime-derivativeof\nmagnetization, ∂ˆM(r,t)/∂t, and on the spatial deriva-\ntive∂ˆM(r,t)/∂r: One group of effects is proportional\nto∂ˆM(r,t)/∂t, a second group of effects is propor-\ntional to the product ∂ˆM(r,t)/∂t ∂ˆM(r,t)/∂r, and\na third group is proportional to the second derivative\n∂2ˆM(r,t)/∂r∂t. We show that the response of the elec-\ntric current to the time-dependent magnetization gradi-\nent∂2ˆM(r,t)/∂r∂tcontais the inverse of CIDMI. We\nestablish the reciprocity relation between inverse and di-\nrectCIDMI and therebyobtainan expressionforCIDMI.\nWe find that CIDMI is related to the modification of\norbital magnetism induced by magnetization dynamics,\nwhich we call dynamical orbital magnetism (DOM). We\nshow that torques are generated by time-dependent gra-\ndients of magnetization as well. The inverse effect con-\nsists in the modification of DMI by magnetization dy-\nnamics, which we call dynamical DMI (DDMI).\nAdditionally, we develop a formalism to calculate the\nchiral contributions to the direct and inverse current-\ninduced torques (CITs) and to the torque-torque correla-tion in noncollinear magnets. We show that the response\nto time-dependent magnetization gradients contributes\nsubstantially to these effects and that the Onsager reci-\nprocityrelationsareviolated when it is not takeninto ac-\ncount. InnoncollinearmagnetsCIDMI,DDMIandDOM\ndepend on the local magnetization direction. We show\nthat the resulting spatial gradients of CIDMI, DDMI\nand DOM have to be subtracted from the CIT, from\nthe torque-torque correlation, and from the inverse CIT,\nrespectively.\nWe apply our formalism to study CITs and the torque-\ntorque correlation in textured Rashba ferromagnets. We\nfind that the contribution of CIDMI to the chiral CIT is\noftheorderofmagnitudeofthe totaleffect. Similarly, we\nfind that the contribution of DDMI to the chiral torque-\ntorque correlation is of the order of magnitude of the\ntotal effect.\nAcknowledgments\nWeacknowledgefinancialsupportfromLeibnizCollab-\norative Excellence project OptiSPIN −Optical Control\nofNanoscaleSpin Textures. Weacknowledgefundingun-\nder SPP 2137 “Skyrmionics” of the DFG. We gratefully\nacknowledge financial support from the European Re-\nsearch Council (ERC) under the European Union’s Hori-\nzon 2020 research and innovation program (Grant No.\n856538, project ”3D MAGiC”). The work was also sup-\nported by the Deutsche Forschungsgemeinschaft (DFG,\nGerman Research Foundation) −TRR 173 −268565370\n(project A11). We gratefully acknowledge the J¨ ulich\nSupercomputing Centre and RWTH Aachen University\nfor providing computational resources under project No.\njiff40.\nAppendix A: Response to time-dependent gradients\nIn this appendix we derive Eq. (18), Eq. (20), Eq. (41),\nand Eq. (82), which describe the response to time-\ndependent magnetization gradients, and Eq. (50), which\ndescribesthe responsetotime-dependentmagneticfields.\nWe consider perturbations of the form\nδH(r,t) =Bb1\nqωsin(q·r)sin(ωt).(A1)\nWhenweset B=∂H\n∂ˆMkandb=∂2ˆMk\n∂ri∂t, Eq.(A1)turnsinto\nEq. (17), while when we set B=−eviandb=1\n2ǫijk∂Bk\n∂t\nwe obtain Eq. (48). We need to derive an expression for\nthe response δA(r,t) of an observable Ato this pertur-\nbation, which varies in time like cos( ωt) and in space like\ncos(q·r), because∂2ˆM(r,t)\n∂ri∂t∝cos(q·r)cos(ωt). There-\nfore, weusethe Kubolinearresponseformalismtoobtain18\nthe coefficient χin\nδA(r,t) =χcos(q·r)cos(ωt), (A2)\nwhich is given by\nχ=i\n/planckover2pi1qωV/bracketleftBig\n∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n−∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(−/planckover2pi1ω)/bracketrightBig\n,(A3)\nwhere∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) is the retarded\nfunction at frequency ωandVis the volume of the unit\ncell.\nThe operator Bsin(q·r) can be written as\nBsin(q·r) =1\n2i/summationdisplay\nknm/bracketleftBig\nB(1)\nknmc†\nk+nck−m−B(2)\nknmc†\nk−nck+m/bracketrightBig\n,\n(A4)\nwherek+=k+q/2,k−=k−q/2,c†\nk+nis the cre-\nation operator of an electron in state |uk+n∝an}bracketri}ht,ck−mis the\nannihilation operator of an electron in state |uk−m∝an}bracketri}ht,\nB(1)\nknm=1\n2∝an}bracketle{tuk+n|[Bk++Bk−]|uk−m∝an}bracketri}ht(A5)\nand\nB(2)\nknm=1\n2∝an}bracketle{tuk−n|[Bk++Bk−]|uk+m∝an}bracketri}ht.(A6)\nSimilarly,\nAcos(q·r) =1\n2/summationdisplay\nknm/bracketleftBig\nA(1)\nknmc†\nk+nck−m+A(2)\nknmc†\nk−nck+m/bracketrightBig\n,\n(A7)\nwhere\nA(1)\nknm=1\n2∝an}bracketle{tuk+n|/bracketleftbig\nAk++Ak−/bracketrightbig\n|uk−m∝an}bracketri}ht(A8)\nand\nA(2)\nknm=1\n2∝an}bracketle{tuk−n|/bracketleftbig\nAk++Ak−/bracketrightbig\n|uk+m∝an}bracketri}ht.(A9)\nIt is convenient to obtain the retarded response func-\ntion in Eq. (A3) from the correspondingMatsubarafunc-\ntion in imaginary time τ\n1\nV∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(τ) =\n=1\n4i/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/bracketleftBig\nA(1)\nknmB(2)\nkn′m′Z(1)\nknmn′m′(τ)\n−A(2)\nknmB(1)\nkn′m′Z(2)\nknmn′m′(τ)/bracketrightBig\n,\n(A10)\nwhered= 1,2 or 3 is the dimension,\nZ(1)\nknmn′m′(τ) =∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(0)ck+m′(0)∝an}bracketri}ht\n=−GM\nm′n(k+,−τ)GM\nmn′(k−,τ),\n(A11)Z(2)\nknmn′m′(τ) =∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(0)ck−m′(0)∝an}bracketri}ht\n=−GM\nm′n(k−,−τ)GM\nmn′(k+,τ),\n(A12)\nand\nGM\nmn′(k+,τ) =−∝an}bracketle{tTτck+m(τ)c†\nk+n′(0)∝an}bracketri}ht(A13)\nis the single-particle Matsubara function. The Fourier\ntransform of Eq. (A10) is given by\n1\nV∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=i\n4/planckover2pi1β/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\np/bracketleftBig\nA(1)\nknmB(2)\nkn′m′GM\nm′n(k+,iEp)GM\nmn′(k−,iEp+iEN)\n−A(2)\nknmB(1)\nkn′m′GM\nm′n(k−,iEp)GM\nmn′(k+,iEp+iEN)/bracketrightBig\n,\n(A14)\nwhereEN= 2πN/βandEp= (2p+ 1)π/βare bosonic\nandfermionicMatsubaraenergypoints, respectively, and\nβ= 1/(kBT) is the inverse temperature.\nIn order to carry out the Matsubara summation over\nEpwe make use of\n1\nβ/summationdisplay\npGM\nmn′(iEp+iEN)GM\nm′n(iEp) =\n=i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iEN)GM\nm′n(E′+iδ)\n+i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iδ)GM\nm′n(E′−iEN)\n−i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iEN)GM\nm′n(E′−iδ)\n−i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′−iδ)GM\nm′n(E′−iEN),(A15)\nwhereδis a positive infinitesimal. The retarded function\n∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(ω) is obtained from the Mat-\nsubara function ∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(iEN) by the\nanalytic continuation iEN→/planckover2pi1ωto real frequencies. The\nright-hand side of Eq. (A15) has the following analytic\ncontinuation to real frequencies:\ni\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′+/planckover2pi1ω)GR\nm′n(E′)\n+i\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′)GA\nm′n(E′−/planckover2pi1ω)\n−i\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′+/planckover2pi1ω)GA\nm′n(E′)\n−i\n2π/integraldisplay\ndE′f(E′)GA\nmn′(E′)GA\nm′n(E′−/planckover2pi1ω).(A16)\nTherefore, we obtain\nχ=−i\n8π/planckover2pi12qω/integraldisplayddk\n(2π)d[Zk(q,ω)−Zk(−q,ω)\n−Zk(q,−ω)+Zk(−q,−ω)],(A17)19\nwhere\nZk(q,ω) =\n=/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′+/planckover2pi1ω)BkGR\nk+(E′)/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′)BkGA\nk+(E′−/planckover2pi1ω)/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′+/planckover2pi1ω)BkGA\nk+(E′)/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGA\nk−(E′)BkGA\nk+(E′−/planckover2pi1ω)/bracketrightBig\n.(A18)\nWe consider the limit lim q→0limω→0χ. In this limit\nEq. (A17) may be rewritten as\nχ=−i\n2π/planckover2pi12/integraldisplayddk\n(2π)d∂2Zk(q,ω)\n∂q∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq=ω=0.(A19)\nThe frequency derivative of Zk(q,ω) is given by\n1\n/planckover2pi1∂Zk\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0=/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAk∂GR\nk−(E′)\n∂E′BkGR\nk+(E′)/bracketrightBigg\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAkGR\nk−(E′)Bk∂GA\nk+(E′)\n∂E′/bracketrightBigg\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAk∂GR\nk−(E′)\n∂E′BkGA\nk+(E′)/bracketrightBigg\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAkGA\nk−(E′)Bk∂GA\nk+(E′)\n∂E′/bracketrightBigg\n.\n(A20)\nUsing∂GR(E)/∂E=−GR(E)GR(E)//planckover2pi1we obtain\n∂Zk\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0=−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−GR\nk−BkGR\nk+/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−BkGA\nk+GA\nk+/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−GR\nk−BkGA\nk+/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGA\nk−BkGA\nk+GA\nk+/bracketrightBig\n.\n(A21)\nMaking use of\nlim\nq→0∂GR\nk+\n∂q=1\n2GR\nkv·q\nqGR\nk (A22)we finally obtain\nχ=−i\n2π/planckover2pi12/integraldisplayddk\n(2π)dlim\nq→0lim\nω→0∂2Z(q,ω)\n∂q∂ω=\n=−i\n4π/planckover2pi12q\nq·/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nAkRvRRBkR+AkRRvRBkR\n−AkRRBkRvR−AkRvRBkAA\n+AkRBkAvAA+AkRBkAAvA\n−AkRvRRBkA−AkRRvRBkA\n+AkRRBkAvA\n+AkAvABkAA−AkABkAvAA\n−AkABkAAvA/bracketrightBig\n,(A23)\nwhere we use the abbreviations R=GR\nk(E) andA=\nGA\nk(E). When we substitute B=∂H\n∂ˆMj,A=−evi, and\nq=qkˆek, we obtain Eq. (18). When we substitute B=\nTj,A=−evi, andq=qkˆek, we obtain Eq. (20). When\nwe substitute A=−Ti,B=Tj, andq=qkˆek, we obtain\nEq. (41). When we substitute B=−evj,A=−Ti,\nandq=qkˆek, we obtain Eq. (50). When we substitute\nB=∂H\n∂ˆMj,A=−Ti, andq=qkˆek, we obtain Eq. (82).\nAppendix B: Perturbation theory for the chiral\ncontributions to CIT and to the torque-torque\ncorrelation\nIn this appendix we derive expressionsfor the retarded\nfunction\n∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) (B1)\nwithin first-orderperturbation theory with respect to the\nperturbation\nδH=Bηsin(q·r), (B2)\nwhich may arise e.g. from the spatial oscillation of the\nmagnetization direction. As usual, it is convenient to ob-\ntain the retarded response function from the correspond-\ning Matsubara function\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ) =−∝an}bracketle{tTτcos(q·r)A(τ)C(0)∝an}bracketri}ht.\n(B3)\nThe starting point for the perturbative expansion is\nthe equation\n−∝an}bracketle{tTτcos(q·r)A(τ1)C(0)∝an}bracketri}ht=\n=−Tr/bracketleftbig\ne−βHTτcos(q·r)A(τ1)C(0)/bracketrightbig\nTr[e−βH]=\n=−Tr/braceleftbig\ne−βH0Tτ[Ucos(q·r)A(τ1)C(0)]/bracerightbig\nTr[e−βH0U],(B4)20\nwhereH0is the unperturbed Hamiltonian and we con-\nsider the first order in the perturbation δH:\nU(1)=−1\n/planckover2pi1/integraldisplay/planckover2pi1β\n0dτ1Tτ{eτ1H0//planckover2pi1δHe−τ1H0//planckover2pi1}.(B5)\nThe essentialdifference between Eq. (A3) and Eq. (B4) is\nthat in Eq. (A3) the operator Benters together with the\nfactor sin( q·r)sin(ωt) (see Eq. (A1)), while in Eq. (B4)\nonly the factor sin( q·r) is connected to Bin Eq. (B2),\nwhile the factor sin( ωt) is coupled to the additional op-\neratorC.\nWe use Eq. (A4) and Eq. (A7) in order to express\nAcos(q·r) andBsin(q·r) in terms of annihilation and\ncreation operators. In terms of the correlators\nZ(3)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(τ1)ck−m′(τ1)c†\nk−n′′ck−m′′∝an}bracketri}ht\n(B6)\nand\nZ(4)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(τ1)ck−m′(τ1)c†\nk+n′′ck+m′′∝an}bracketri}ht\n(B7)\nand\nZ(5)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(τ1)ck+m′(τ1)c†\nk+n′′ck+m′′∝an}bracketri}ht\n(B8)\nand\nZ(6)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(τ1)ck+m′(τ1)c†\nk−n′′ck−m′′∝an}bracketri}ht\n(B9)\nEq. (B4) can be written as\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ1) =\n=ηV\n4i/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay/planckover2pi1β\n0dτ/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′/bracketleftBigg\n−B(2)\nknmA(1)\nkn′m′Ck−n′′m′′Z(3)\nknmn′m′n′′m′′(τ,τ1)\n−B(2)\nknmA(1)\nkn′m′Ck+n′′m′′Z(4)\nknmn′m′n′′m′′(τ,τ1)\n+B(1)\nknmA(2)\nkn′m′Ck+n′′m′′Z(5)\nknmn′m′n′′m′′(τ,τ1)\n+B(1)\nknmA(2)\nkn′m′Ck−n′′m′′Z(6)\nknmn′m′n′′m′′(τ,τ1)/bracketrightBigg(B10)\nwithin first-order perturbation theory, where we de-\nfinedCk−n′′m′′=∝an}bracketle{tuk−n′′|C|uk−m′′∝an}bracketri}htandCk+n′′m′′=\n∝an}bracketle{tuk+n′′|C|uk+m′′∝an}bracketri}ht.\nNote that Z(5)can be obtained from Z(3)by replac-\ningk−byk+andk+byk−. Similarly, Z(6)can be\nobtained from Z(4)by replacing k−byk+andk+by\nk−. Therefore, we write down only the equations forZ(3)andZ(4)in the following. Using Wick’s theorem\nwe find\nZ(3)\nknmn′m′n′′m′′(τ,τ1) =\n=−GM\nm′n(k−,τ1−τ)GM\nmn′(k+,τ−τ1)GM\nm′′n′′(k−,0)\n+GM\nmn′(k+,τ−τ1)GM\nm′′n(k−,−τ)GM\nm′n′′(k−,τ1)\n(B11)\nand\nZ(4)\nknmn′m′n′′m′′(τ,τ1) =\n=−GM\nmn′(k+,τ−τ1)GM\nm′n(k−,τ1−τ)GM\nm′′n′′(k+,0)\n+GM\nmn′′(k+,τ)GM\nm′n(k−,τ1−τ)GM\nm′′n′(k+,−τ1).\n(B12)\nThe Fourier transform\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ1)(B13)\nof Eq. (B10) can be written as\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=ηV\n4i/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′/bracketleftBigg\n−B(2)\nknmA(1)\nkn′m′Ck−n′′m′′Z(3a)\nknmn′m′n′′m′′(iEN)\n−B(2)\nknmA(1)\nkn′m′Ck+n′′m′′Z(4a)\nknmn′m′n′′m′′(iEN)\n+B(1)\nknmA(2)\nkn′m′Ck+n′′m′′Z(5a)\nknmn′m′n′′m′′(iEN)\n+B(1)\nknmA(2)\nkn′m′Ck−n′′m′′Z(6a)\nknmn′m′n′′m′′(iEN)/bracketrightBigg(B14)\nin terms of the integrals\nZ(3a)\nknmn′m′n′′m′′(iEN) =/integraldisplay/planckover2pi1β\n0dτ/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1×\n×GM\nmn′(k+,τ−τ1)GM\nm′′n(k−,−τ)GM\nm′n′′(k−,τ1) =\n=1\n/planckover2pi1β/summationdisplay\npGM\nk+mn′(iEp)GM\nk−m′′n(iEp)GM\nk−m′n′′(iEp+iEN)\n(B15)\nand\nZ(4a)\nknmn′m′n′′m′′(iEN) =/integraldisplay/planckover2pi1β\n0dτ/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1×\n×GM\nmn′′(k+,τ)GM\nm′n(k−,τ1−τ)GM\nm′′n′(k+,−τ1) =\n=1\n/planckover2pi1β/summationdisplay\npGM\nk+mn′′(iEp)GM\nk−m′n(iEp)GM\nk+m′′n′(iEp−iEN),\n(B16)\nwhereEN= 2πN/βis a bosonic Matsubara energy point\nand we used\nGM(τ) =1\n/planckover2pi1β∞/summationdisplay\np=−∞e−iEpτ//planckover2pi1GM(iEp),(B17)21\nwhereEp= (2p+1)π/βis a fermionic Matsubara point.\nAgain Z(5a)is obtained from Z(3a)by replacing k−by\nk+andk+byk−andZ(6a)is obtained from Z(4a)in\nthe same way.\nSummation overMatsubarapoints Epin Eq.(B15) and\nin Eq. (B16) and analytic continuation iEN→/planckover2pi1ωyields\n2πi/planckover2pi1Z(3a)\nknmn′m′n′′m′′(/planckover2pi1ω) =\n−/integraldisplay\ndEf(E)GR\nk+mn′(E)GR\nk−m′′n(E)GR\nk−m′n′′(E+/planckover2pi1ω)\n+/integraldisplay\ndEf(E)GA\nk+mn′(E)GA\nk−m′′n(E)GR\nk−m′n′′(E+/planckover2pi1ω)\n−/integraldisplay\ndEf(E)GA\nk+mn′(E−/planckover2pi1ω)GA\nk−m′′n(E−/planckover2pi1ω)GR\nk−m′n′′(E)\n+/integraldisplay\ndEf(E)GA\nk+mn′(E−/planckover2pi1ω)GA\nk−m′′n(E−/planckover2pi1ω)GA\nk−m′n′′(E)\n(B18)\nand\n2πi/planckover2pi1Z(4a)\nknmn′m′n′′m′′(/planckover2pi1ω) =\n−/integraldisplay\ndEf(E)GR\nk+mn′′(E)GR\nk−m′n(E)GA\nk+m′′n′(E−/planckover2pi1ω)\n+/integraldisplay\ndEf(E)GA\nk+mn′′(E)GA\nk−m′n(E)GA\nk+m′′n′(E−/planckover2pi1ω)\n−/integraldisplay\ndEf(E)GR\nk+mn′′(E+/planckover2pi1ω)GR\nk−m′n(E+/planckover2pi1ω)GR\nk+m′′n′(E)\n+/integraldisplay\ndEf(E)GR\nk+mn′′(E+/planckover2pi1ω)GR\nk−m′n(E+/planckover2pi1ω)GA\nk+m′′n′(E).\n(B19)\nIn the next step we take the limit ω→0 (see Eq. (64),\nEq. (70), and Eq. (77)):\n−1\nVlim\nω→0Im∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ω=\n=η\n4/planckover2pi1Im/bracketleftBig\nY(3)+Y(4)−Y(5)−Y(6)/bracketrightBig\n,(B20)where we defined\nY(3)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(2)\nknmA(1)\nkn′m′Ck−n′′m′′×\n×∂Z(3a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(4)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(2)\nknmA(1)\nkn′m′Ck+n′′m′′×\n×∂Z(4a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(5)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(1)\nknmA(2)\nkn′m′Ck+n′′m′′×\n×∂Z(5a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(6)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(1)\nknmA(2)\nkn′m′Ck−n′′m′′×\n×∂Z(6a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\n(B21)\nwhich can be expressed as Y(3)=Y(3a)+Y(3b)and\nY(4)=Y(4a)+Y(4b), where\n2π/planckover2pi1Y(3a)=1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)\n+AkGR\nk−(E)Ck−GA\nk−(E)GA\nk−(E)BkGA\nk+(E)/bracketrightBigg\n=/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)×\n×Tr/bracketleftBig\nAkGR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)/bracketrightBig\n(B22)\nand\n2π/planckover2pi1Y(3b)=−1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGA\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)Ck−GR\nk−(E)BkGR\nk+(E)\n+AkGA\nk−(E)Ck−GA\nk−(E)GA\nk−(E)BkGA\nk+(E)/bracketrightBigg\n.(B23)22\nSimilarly,\n2π/planckover2pi1Y(4a)=1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)GA\nk+(E)\n−AkGR\nk−(E)GR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)\n−AkGR\nk−(E)BkGR\nk+(E)GR\nk+(E)Ck+GA\nk+(E)/bracketrightBigg\n=/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)×\n×Tr/bracketleftBig\nAkGR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)/bracketrightBig\n(B24)\nand\n2π/planckover2pi1Y(4b)=−1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGA\nk−(E)BkGA\nk+(E)Ck+GA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)BkGR\nk+(E)Ck+GR\nk+(E)\n+AkGR\nk−(E)BkGR\nk+(E)GR\nk+(E)Ck+GR\nk+(E)/bracketrightBigg\n.(B25)\nWe call Y(3a)andY(4a)Fermi surface terms and Y(3b)\nandY(4b)Fermi sea terms. Again Y(5)is obtained from\nY(3)by replacing k−byk+andk+byk−andY(6)is\nobtained from Y(4)in the same way.\nFinally, we take the limit q→0:\nΛ =−2\n/planckover2pi1VηIm lim\nq→0lim\nω→0∂\n∂ω∂\n∂qi∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n=1\n2/planckover2pi1lim\nq→0∂\n∂qiIm/bracketleftBig\nY(3)+Y(4)−Y(5)−Y(6)/bracketrightBig\n=1\n2/planckover2pi1Im/bracketleftBig\nX(3)+X(4)−X(5)−X(6)/bracketrightBig\n,\n(B26)\nwhere we defined\nX(j)=∂\n∂qi/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq=0Y(j)(B27)\nforj= 3,4,5,6. Since Y(4)andY(6)are related by\nthe interchange of k−andk+it follows that X(6)=\n−X(4). Similarly, since Y(3)andY(5)arerelated by the\ninterchange of k−andk+it follows that X(5)=−X(3).\nConsequently, we need\nΛ =1\n/planckover2pi1Im/bracketleftBig\nX(3a)+X(3b)+X(4a)+X(4b)/bracketrightBig\n,(B28)\nwhere X(3a)andX(4a)are the Fermi surface terms and\nX(3b)andX(4b)are the Fermi sea terms. The Fermisurface terms are given by\nX(3a)=−1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nAkGR\nk(E)vkGR\nk(E)CkGA\nk(E)BkGA\nk(E)\n+AkGR\nk(E)CkGA\nk(E)vkGA\nk(E)BkGA\nk(E)\n−AkGR\nk(E)CkGA\nk(E)BkGA\nk(E)vkGA\nk(E)\n+AkGR\nk(E)∂Ck\n∂kGA\nk(E)BkGA\nk(E)/bracketrightBigg(B29)\nand\nX(4a)=−/bracketleftBig\nX(3a)/bracketrightBig∗\n. (B30)\nThe Fermi sea terms are given by\nX(3b)=−1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBigg\n−(ARvRRCRBR)+(AACAABAvA)\n−(ARRvRCRBR)−(ARRCRvRBR)\n+(ARRCRBRvR)−(AAvACABAA)\n−(AACAvABAA)+(AACABAvAA)\n+(AACABAAvA)−(AAvACAABA)\n−(AACAvAABA)−(AACAAvABA)\n−(ARR∂C\n∂kRBR)−(AA∂C\n∂kAABA)\n−(AA∂C\n∂kABAA)/bracketrightBigg(B31)\nand\nX(4b)=−/bracketleftBig\nX(3b)/bracketrightBig∗\n. (B32)\nIn Eq. (B31) we use the abbreviations R=GR\nk(E),A=\nGA\nk(E),A=Ak,B=Bk,C=Ck. It is important\nto note that Ck−andCk+depend on qthrough k−=\nk−q/2 andk+=k+q/2 . Theqderivative therefore\ngenerates the additional terms with ∂Ck/∂kin Eq. (B29)\nand Eq. (B31). In contrast, AkandBkdo not depend\nlinearly on q.\nEq. (B28) simplifies due to the relations Eq. (B30) and\nEq. (B32) as follows:\nΛ =2\n/planckover2pi1Im/bracketleftBig\nX(3a)+X(3b)/bracketrightBig\n. (B33)\nIn order to obtain the expression for the chiral con-\ntribution to the torque-torque correlation we choose the\noperators as follows:\nB→ Tk\nA→ −Ti\nC→ Tj\n∂C\n∂k= 0\nv→vl.(B34)23\nThis leads to Eq. (78), Eq. (79) and Eq. (80) of the main\ntext.\nIn order to obtain the expression for the chiral contri-\nbution to the CIT, we set\nB→ Tk\nA→ −Ti\nC→ −evj\n∂C\n∂k→ −e/planckover2pi1\nmδjl\nv→vl.(B35)\nThis leads to Eq. (66), Eq. (67) and Eq. (68).\nIn order to obtain the expression for the chiral contri-\nbution to the ICIT, we set\nB→ Tk\nA→ −evi\nC→ Tj\n∂C\n∂k→0\nv→vl.(B36)\nThis leads to Eq. (71), Eq. (72) and Eq. (73).\n∗Corresp. author: f.freimuth@fz-juelich.de\n[1] K. Nawaoka, S. 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Bauer2, 1\n1Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n2Institute for Materials Research and WPI-AIMR, Tohoku University, Sendai 980-8577, Japan\n(Dated: November 5, 2018)\nWe present a theory for the coherent magnetization dynamics induced by a focused ultrafast\nlaser beam in magnetic films, taking into account nonthermal (inverse Faraday effect) and thermal\n(heating) actuation. The dynamic conversion between spin waves and phonons is induced by the\nmagnetoelastic coupling that allows efficient propagation of angular momentum. The anisotropy of\nthe magnetoelastic coupling renders characteristic angle dependences of the magnetization propa-\ngation that are strikingly different for thermal and nonthermal actuation.\nPACS numbers: 75.80.+q, 75.30.Ds, 75.78.-n, 78.20.Ls\nIntroduction — Since the discovery of laser-induced ul-\ntrafast spin dynamics in Nickel by Beaurepaire et al. [1],\nthe spin manipulation in ferromagnetic system by fem-\ntosecond laser pulses has attracted much attention since\ncombining the intellectual challenge of new physics with\nthe application potential of ultrafast magnetization re-\nversal [2]. Intense light can cause many effects in mag-\nnets, such as the coherent inverse Faraday effect (IFE)\nas well as the excitation of the coupled electron, magnon\nand phonon subsystems on various time scales. The asso-\nciated modulation of the magnetic anisotropy and mag-\nnetization modulus allows coherent control of the mag-\nnetic order [3, 4]. The transient magnetic field generated\nby the IFE allows non-thermal ultrafast magnetization\ncontrol [5, 6] that may be distinguished from heating-\ninduced effects by switching the light polarization. Nev-\nertheless, heating is essential for light-induced magneti-\nzation reversal [7]. Toggle switching of the magnetization\nby heat alone has also been reported [8]. Understanding\nand controlling the relative magnitude of thermal and\nnon-thermal excitation is therefore an important but un-\nsolved issue.\nThe optical ultrafast pump-probe technique as shown\nin Fig. 1 is an established powerful method to study\nmatter. Here we will show that the symmetry of the\nspatiotemporal magnetization distribution excited by a\nfocused laser beam reveals the relative contributions of\nthermal and non-thermal excitations. This phenomenon\noriginates from the magnetoelastic coupling (MEC) [9–\n11], i.e., the coupling between spin waves (magnons) and\nacoustic lattice waves (phonons).\nIn the regions of phase space in which the magnon\nand phonon dispersion come close, the MEC hybridizes\nmagnons and phonons into coherently mixed quasipar-\nticles [“magnon-polarons”(MPs)]. This implies that\nphonons in magnets can be converted into magnetiza-\ntion and become detectable via magneto-optical [12–14]\nor electrical [15, 16] techniques. In this Letter, we present\na study of the spatial magnetization dynamics in mag-\nnetic thin films after focused-laser excitation [17]. We\nconsider here magnetic insulators that are not affected\nFIG. 1: Pump-probe study of the dynamics of magnetic films\nby pulsed lasers: An external magnetic field aligns the equi-\nlibrium magnetization along the xdirection. The pump laser\nhits the sample at the origin in time and space. The tem-\nporal distribution of the out-of-plane ( z) component of the\nmagnetization is detected by the Kerr rotation of the linearly\npolarized probe pulse.\nby conduction or photo-excited free carriers. We find\nthat laser-generated phonons efficiently excite magneti-\nzation when the diameter of the laser spot is comparable\nwith the wavelength of the MPs. The spatial dynamics\nof such phonon-induced magnetization shows a different\nangular symmetry from that of the magnetization gener-\nated directly by laser via the IFE. Dedicated experiments\nsuggested here can therefore help understanding the fun-\ndamental nature of light-matter interaction in magnets.\nFormalism — The basic theory for MPs in special sym-\nmetry directions by Kittel [9] and Akhiezer et al. [10],\nextended by Schl¨ omann [11] to arbitrary propagating di-\nrections, was developed more than half a century ago.\nThe energy density of the minimal model reads [9]\nH=Hex+HZ+Hel+Hmec+Hdip. (1)\nWe adopt a cubic unit cell and consider the thin film\nlimit in which the magnetization is spatially constant\nover the film thickness. This assumption holds for films\nup to 100µm [17], which for wide-gap insulators is still\nless than the penetration depth of the light and al-\nlows us to use a two-dimensional model. With ex-\nternal magnetic field Hand equilibrium magnetization\nvector M0/bardblx(|M0|=M0saturation magnetization)\nHex=A[(∇my)2+ (∇mz)2] andHZ=−µ0HM 0+arXiv:1508.02094v2  [cond-mat.mes-hall]  5 Nov 20152\n(µ0HM 0/2)(m2\ny+m2\nz) represent the (linearized) exchange\nand Zeeman energies, respectively, where myandmzare\nthe transverse magnetization components of m=M/M0.\nHelis the lattice energy with both kinetic and elas-\ntic contributions Hel= (1/2)ρ˙R·˙R+ (1/2)λ(/summationtext\niSii)2+\nµ/summationtext\nijS2\nijwith strain tensor Sij= (∂iRj+∂jRi)/2 and\nRrepresenting the lattice displacement with respect to\nequilibrium, ρthe mass density, and λandµelastic con-\nstants. The MEC in Eq. (1) reads Hmec=/summationtext\ni,j∈{y,z}(b+\naδij)Sijmimj+ 2b/summationtext\ni∈{y,z}Sixmi, whereaandbare\nmagnetoelastic coupling coefficients. By adopting the\nshort-wave length limit of the magnetostatic dipolar in-\nteractionHdip≈(µ0M2\n0/2)m2\nysin2θ,we disregard the\nDamon-Eshbach surface modes [18] and simplify the dis-\npersion of the volume modes, which is allowed for small\nlaser spot sizes with response being dominated by high-\nmomentum wave vectors [19].\nBy introducing the forces and torques Facting on\nthe displacement vector Φ= (my,mz,Rl,Rt,Rz)T, one\ncan write out the linearized equations of motion as\nshown in the Supplemental Material [20]. Here, the\nlattice displacement is rewritten in the form of longi-\ntudinal (Rl), in-plane transverse ( Rt) and out-of-plane\ntransverse ( Rz) modes. Strictly speaking, the damp-\ning of phonons and magnon are not necessarily inde-\npendent, since magnetization is affected by phonon at-\ntenuation via the MEC [27]. We treat Gilbert damp-\ning constant αand phonon relaxation time τpas inde-\npendent parameters since Gilbert damping can also be\ncaused by magnetic disorder, surface roughness or de-\nfects [28]. We define the anisotropic spin wave frequency\nΩ0=γµ0/radicalBig\nH(H+M0sin2θ) and the MEC frequency\nparameter ∆( k) =/radicalbig\nγb2k2/(4M0ρΩ0) withθbeing the\nangle between magnetic field and in-plane wave vector k.\nThe spatiotemporal dynamics of Φ(r,t) reads\nΦi(r,t) =/integraldisplay\ndr/primedt/primeGij(r−r/prime,t−t/prime)Fj(r/prime,t/prime),(2)\nwhereGijare the components of the Green function\nmatrix (propagator) associated with the magnetoelastic\nequations of motion specified in the Supplement [20]. A\nfemtosecond laser pulse generates forces via the inverse\nFaraday effect [2, 17, 29] and heating [14, 30–32] that\nare instantaneous on the scale of the lattice and magne-\ntization dynamics. The relative importance of these two\nmechanisms depends on the material and light and is still\na matter of controversy. Here we find that spot excitation\nof thin magnetic films is an appropriate method to sep-\narate the two, since they lead to conspicuous differences\nin the time and position dependent response.\nWe consider circularly polarized light along zthat by\nthe IFE generates an effective magnetic field along the\nsame direction. For a femtosecond Gaussian pulse with\nspot sizeW, the generated magnetic field has a spatial\ndistribution HIFE(r,t) = ˆzHIFEf(t) exp(−r2/W2) wheretemporal shape f(t)≈τlδ(t) with pulse duration τl; the\namplitudeHIFE=βIinσis proportional to laser intensity\n(Iin) and IFE coefficient ( β), respectively. The latter is\nrelated to the Verdet constant ( V) asβ=Vλ0/(2πc0),\nwhereλ0andc0are wavelength and velocity of the\nlight [33].σ= 1(−1) for left(right)-handed polarization.\nThe torque Fmy(r,t) =γτlµ0HIFEδ(t) exp(−r2/W2).\nOn the other hand, the light pulse generates a sud-\nden increase of the local lattice temperature δT(r,t) =\n[ΓIinτl/(ρCv)]Θ(t) exp(−r2/W2), where Γ is the light-\nabsorption coefficient. By choosing the Heaviside step\nfunction Θ( t) we assume that the lattice locally equili-\nbrates much faster than the response time of the coherent\nmagnetization (a few picoseconds [34]), while the subse-\nqent cooling of the lattice by diffusion is slow. The re-\nsulting in-plane thermoelastic stress FRl=η(3λ+2µ)∂rT\ngenerates longitudinal (pressure) waves [31, 35], where η\nis the thermoelastic expansion coefficient. The local ther-\nmal expansion also generates a “bulge” shear stress [36]\nat a free surface, i.e., an out-of-plane displacement Rz.\nFRz=ζη∇2T, whereζis a parameter proportional to\nthe film thickness and controlled by the substrate and an\neventual cap layer, leads to displacement proportional to\nlocal temperature gradient (see numerical results below).\nIn the Supplemental Material [20], we specify the mate-\nrial parameters for yttrium iron garnet (YIG) adopted in\nour calculations.\nOne-dimensional dynamics — We start with a spin\nwave propagating along the external magnetic field, i.e.,\nθ= 0, which by symmetry couples only with the trans-\nverse phonons. The IFE generates the torque Fmy(x,t) =\nm0δ(t) exp(−x2/W2) withm0=γτlµ0HIFE. This can be\nrealized by a line-shaped excitation spot [17].\nThe calculated magnetization profiles at Ω 0t= 50 and\n75 without and with MEC are plotted in Fig. 2(a) and (b)\nseparately for α= 10−4. Without MEC the magnetiza-\ntion is localized at the exposure spot and broadens only\nvery weakly with time while the MEC strongly enhances\nthe broadening of primary magnetization packet, with a\nwavefront propagating with the sound velocity ct. This\nphenomenon illustrates that the lattice plays an essential\nrole for spin transport in magnetic films.\nFig. 2(c) illustrates the sound-assisted propagation for\nGilbert damping α= 0.1. Instead of the expanding wave\nfront in Fig. 2(b), we now find two packets escaping the\nexcitation region into opposite directions. The packets\nhave a much longer lifetime than the coherently gener-\nated IFE magnetization, hence dominate at long time\nscales. This behavior is recovered by the asymptotic\nexpression obtained when α/greatermuch˜∆(˜kc) at the magnon-\nphonon dispersion crossing wave vector ( ˜kc= 1/˜ct),\nmz(˜x,˜t)/similarequalm0e−α˜t−˜x2sin/parenleftbig˜t/parenrightbig\n+ (2α)−1m0e−˜t/˜τp˜∆2(˜kc)\n×/braceleftBigg\n2α˜c2\nt/bracketleftbig\nΛ1(˜ct˜t−˜x) + Λ 1(˜ct˜t+ ˜x)/bracketrightbig\n, ˜ct/lessmuch1,\n˜c−1\nt√π/bracketleftbig\nΛ2(˜ct˜t−˜x) + Λ 2(˜ct˜t+ ˜x)/bracketrightbig\n,˜ct≥1,(3)3\n-0.3 0 0.3 0.6\n-100 -50  0  50  100mz (10-2m0)\nx/WFull calculation\nα = 0.1 (c)-0.1 0 0.1 0.2mz (m0) Full calculation\nα = 10-4\n(b) 0 0.5 1mz (m0)\nα = 10-4(a)Without MECΩ0t = 50\n75\n 0 0.5 1\n-2  0  2 0 0.5 1\n-2  0  2\nFIG. 2: (Color online) One-dimensional model for the dynam-\nics of the out-of-plane magnetization component mzinduced\nby the inverse Faraday effect: (a) At times Ω 0t= 50 and 75 for\na Gaussian laser intensity spot in the absence of MEC with\nGilbert damping α= 10−4. The dashed envelopes are the\nmodulus (|m−ˆx|). (b) with α= 10−4and (c) with α= 0.1\nare computed for MEC parameter ˜∆(˜kc) = 0.02. Common\nare the exchange parameter ˜D= 0.02, sound velocity ˜ ct= 1,\nand sound attenuation rate ˜ τ−1\np= 10−3. Note the change of\nscale between (b) and (c). The insets provide an expanded\nview of the laser spot.\nwhere Λ 1(ξ) = (1/√π)/integraltext∞\n0d˜k˜k2sin(˜kξ) exp(−˜k2/4) and\nΛ2(ξ) = exp/bracketleftbig\n−(ξ2α2+ 1)/(2˜ct)2/bracketrightbig\nsin(ξ/˜ct) andWand\nΩ0have been by rendered dimensionless as explaind in\nthe Supplemental Material [20]. The (purely magnetic)\nfirst term on the right-hand side represent the exponen-\ntial decay of the initially excited wave packet, while the\nsecond term is a propagating MP mode. The latter de-\ncays with the phonon damping rate, hence may have a\nvery long mean free path for materials with high acous-\ntic quality like YIG (assuming that doping affects the\nmagnetization without increasing sound attenuation).\nWhen the laser spot size is large relative to the MP\nwave length, i.e., ˜ ct/lessmuch1, according to Eq. (3) the\nratio between MP amplitude and IFE strength scales\nas ˜c2\nt˜∆2(˜kc), i.e., increases with sound velocity and de-\ncreases with spot size. In the other limit, ˜ ct/greatermuch1, the am-\nplitude of the long-lived signal is inversely proportional\nto ˜ct, therefore decreases with increasing ˜ ct. We therefore\nestimate this ratio to be maximal e−1/4√π˜∆2(˜kc)/(2α)\nwhen the laser spot size matches the MP wave length.\nThe peak amplitude of MPs in Fig. 2(c) is around\n3×10−3, in good agreement with e−1/4√π˜∆2(˜kc)/(2α)≈\n-1 0 1\n-100 -50  0  50  100-1 0 1mz (m1)\nRz (unit)\nx/WFIG. 3: (Color online) Out-of-plane magnetization dynamics\nmzwith Gilbert damping α= 10−4(dotted curve) and 0 .1\n(solid curve) induced by the spot heating by a laser pulse.\nThe blue dashed curve shows the displacement profile ( Rz).\nOther parameters are those in Fig. 2.\n2.7×10−3.\nThermal actuation is caused by the shear force\ngenerated by the laser heating profile FRz(x,t) =\n(ζηc2\ntΓIinτl/Cv)Θ(t)∂2\nxexp(−x2/W2), since the pressure\nwave is decoupled from the spin wave at θ= 0. The\nasymptotic expression for α/greatermuch˜∆(˜kc) becomes\nmz(x,t) =m1\n×\n\n˜ct{Λ3(˜x)−(1/2)e−˜t/˜τp\n×[Λ3(˜ct˜t+ ˜x)−Λ3(˜ct˜t−˜x)]}, ˜ct/lessmuch1,\n˜ct(1−e−αtcost)Λ3(˜x) + ˜c−1\nt(√π/4)\n×e−˜t/˜τp/bracketleftbig\nΛ4(˜ct˜t+ ˜x)−Λ4(˜ct˜t−˜x)/bracketrightbig,˜ct≥1,\n(4)\nwhere Λ 3(ξ) =ξexp(−ξ2) and Λ 4(ξ) = exp[−(ξ2α2+\n1)/(2˜ct)2] cos(ξ/˜ct). For YIG, the parameter m1=\nγbζηΓIinτl/(2M0ρctCv)∼103ζδTm−1K−1. Compared\nto Eq. (3), the heat-induced magnetization has ( i) odd\nparity in real space, i.e., mz(−x,t) =−mz(x,t), (ii) a\nlong-lived localized signal near the excitation spot, and\n(iii) maximum amplitude of propagation at ˜ ct/similarequal1, cf.\nFig. 3. We also plot the amplitude of the thermally gen-\nerated phonon wave front that is trailed by the magneti-\nzation.\nTwo Dimensions — In the following, magnetization is\noriented along ˆ xby an external in-plane magnetic field\nµ0H= 50 mT corresponding to γµ0H/(2π)/similarequal1.4 GHz.\nThe spot size W= 1µm and the dimensionless veloc-\nities are ˜ct≈0.43 and ˜cl≈0.82. Fig. 4 summarizes\nour main results for the IFE and heat induced dynam-\nics in terms of the out-of-plane magnetization compo-\nnentmz. We plot a snapshot at t= 5 ns in the x-y\n(film) plane from the calculation with low ( α= 10−4)\nand enhanced ( α= 0.1) magnetic damping in (a) and\n(b), respectively. Fig. 4(a, left) displays IFE actuated\noutgoing rays that broaden with distance from the exci-\ntation spot. This feature is insensitive to MEC strength\nand can be understood by the angular dependent group\nvelocities of magnetostatic spin wave dispersion around4\n(a)-40 -20  0  20  40\nx (µm)-40-20 0 20 40 y (µm)-0.1  0  0.1\nv\ngtvp\n-20  0  20  40\nx (µm)-0.08 -0.04  0  0.04  0.08\n-20  0  20  40\nx (µm)-0.4 -0.2  0  0.2  0.4\n(b)(×10-3)\n-40 -20  0  20  40\nx (µm)-40-20 0 20 40 y (µm)-0.3  0  0.3\n-20  0  20  40\nx (µm)-0.02 -0.01  0  0.01  0.02\n-20  0  20  40\nx (µm)-0.2 -0.1  0  0.1  0.2\nFIG. 4: (Color online) Two dimensional profile of out-of-plane\nmagnetization, mz, att= 5 ns due to (left) IFE field Fmy,\n(middle) pressure stress FRland (right) shear stress FRz. We\nnormalize the result by m0for (left),m2=γbηΓIinτl(3λ+\n2µ)/(2M0ρ2Cvc2\ntΩ0) for (middle) and m1for (right). The\nGilbert damping coefficient α= 10−4and 0.1 for (a) and (b),\nrespectively. The dashed (olive) curve in (a, left) illustrates\nthe angular-dependent spin wave group velocity.\nthe average modulus of the wave vectors k0. As discussed\nin the Supplemental Material [20], the group velocity\nvg/similarequalˆθ[γµ0M0/(2k0)] sin 2θ/radicalBig\nH/(H+M0sin2θ) gener-\nates an expansion of the initial wave packet as shown by\nthe dashed (olive) curve, while the the star-like interfer-\nence fringes are governed by the phase velocity. At larger\nmagnetic damping, cf. Fig. 4(b, left), the star-like fea-\ntures in the ˆ xdirection are suppressed in favor of MP\npropagation with transverse sound velocity ct. Dotted\nfeature around cltare caused by interference of the lon-\ngitudinal MP and the damped residue of the initial mag-\nnetization wave packet with θ/similarequal0, which has relative\nlonger lifetime. Note the mirror symmetry with respect\nto theyaxis,mz(x,y) =mz(−x,y).\nThe quadrupolar features in Figs. 4(middle) with\nnodes along the xandyaxes and sin 2 θsymmetry are\ninduced by the pressure FRlcaused by a heat pulse. The\nradii of the circular wave fronts correspond to the longitu-\ndinal sound velocity. Figs. 4(right) illustrate that a shear\nstress induces MPs that spread with transverse sound ve-\nlocityct,which are thereby clearly distinguishable from\nthe pressure induced signals: FRzgenerates dipolar sym-\nmetric features with nodes at the yaxis, which follows\nfrom the cos θsymmetry of the MEC coupling. Clearly,\nboth heat-induced signals are antisymmetric with respect\nto reflection at the yaxis,mz(x,y) =−mz(−x,y), which\nallows discrimination from the IFE response. Moreover,\nwe identify a non-propagating signal in the vicinity of\nthe excitation spot [see the center of Figs. 4(b, middle)\nand (b, right)], a “smoking gun” for thermally exciteddynamics.\nIt is not easy to predict the absolute and relative mag-\nnitude of the two mechanisms for a given light intensity\nfrom first principles due to uncertainties in the strongly\nnon-equilibrium processes after an intense fs light pulse.\nMicroscopic theories address the ultra-fast physics of an-\ngular momentum and energy transfer from the light to\nthe magnetic order [37, 38] and the lattice [34] and should\nideally be employed to fix the initial conditions for our\ncalculations. But also the long-time response depends\non several temperature and frequency dependent mate-\nrials parameters that govern the IFE, light absorption\netc. Satoh et al. [17] find a Faraday rotation of the\nprobe pulse of the order of milli-rad for 110 µm thick\nbismuth-doped iron garnet, which corresponds to a light-\ninduced torque of m0∼0.004 for a Verdet constant of\n104rad m−1T−1[20]. With thermal expansion coefficient\nη∼10−5/K [39], the thermal torque m2∼4×10−3δT/K,\nwhich can be larger than m0for pulsed laser-induced\nheating [40]. Similar values may be anticipated for m1\nwhen effective thicknesses ζ∼µm. We should also\nnot forget that the fast light-induced demagnetization [1]\nshould affect the response directly under the excitation\nspot, but its diffusion should be slower than the ballistic\nresponse computed here.\nConclusion and Discussion — We modeled the spa-\ntiotemporal laser-induced magnetization dynamics in\nmagnetic thin films, concluding that magnetoelastic cou-\npling is essential for spin angular momentum transport\nbecause the phonon group velocity is much larger than\nthat of the magnons. An experimental study of the sym-\nmetry of time-resolved magnetization wave fronts radi-\nating from the excitation spot allows discriminating dif-\nferent laser excitation mechanisms, thereby helping to\nanswer the long-standing question on the physical ori-\ngin of ultra-fast magnetization dynamics, i.e. whether\nit is caused by coherent light-induced magnetic fields or\nsudden heating of the lattice. Moreover, we clarified the\noptimal size of the excitation laser spot to be the MP\nwavelength; for YIG at an applied field of 50 mT it is\n∼1µm.\nThe essential role of the MEC coupling might have\nlarger ramifications. For example, a number of recent ex-\nperiments on the spin Seebeck effect on YIG came to the\nconclusion that the thermal spin pumping is not caused\nby terhahertz magnons at energies around kBT, but by\nspin waves in a low energy band close to the gap [41–43].\nSpin information was found to propagate in YIG diffusely\nover large distances [44, 45]. 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Kehlberger, J. Cramer, G. Jakob, and M.\nKl¨ aui, arXiv:1506.06037.\n[44] L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef,\nand B. J. van Wees, Nature Phys. (in press).\n[45] B. L. Giles, Z. Yang, J. Jamison, and R. C. Myers,\narXiv:1504.02808.\n[46] N. Ogawa, W. Koshibae, A. J. Beekman, N. Nagaosa, M.\nKubota, M. Kawasaki, and Y. Tokura, PNAS 112, 8977\n(2015).Supplementary material to “Laser-induced spatiotemporal dynamics of magnetic\nfilms”\nKa Shen1and Gerrit E. W. Bauer2, 1\n1Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n2Institute for Materials Research and WPI-AIMR, Tohoku University, Sendai 980-8577, Japan\nThis supplement includes the linearized equations of motion in Fourier space with choice of parameters, as well\nas analytical expression for the IFE-induced dynamical response to line and point sources and estimation on the\nmagnitude of IFE.\n1. LINEARIZED EQUATIONS OF MOTION IN FOURIER SPACE\nAfter introducing the forces and torques Facting on the displacement vector Φ= (my,mz,Rl,Rt,Rz)T, the\nlinearized equations of motion in Fourier space read [1]\n(ω2−2iωτ−1\np−c2\ntk2)ρRt=ibkcos 2θmy−FRt, (S1)\n(ω2−2iωτ−1\np−c2\ntk2)ρRz=ibkcosθmz−FRz, (S2)\n(ω2−2iωτ−1\np−c2\nlk2)ρRl=ibksin 2θmy−FRl, (S3)\niωmy+ (γHk+iαω)mz=i∆1cosθRz+Fmy, (S4)\niωmz−(γH/prime\nk+iαω)my=−i∆1(cos 2θRt+ sin 2θRl) +Fmz, (S5)\nwhere ∆ 1=γbk/M 0,H/prime\nk=Hk+µ0M0sin2θwithHk=µ0H+γ−1(2A/M 0)k2,andθis the angle between magnetic\nfield and in-plane wave vector k.αandτpare the Gilbert damping constant and phonon relaxation time [2].\nWe adopt the material parameters of yttrium iron garnet (YIG). Specifically, the dipolar field and gyromagnetic\nratio areµ0M0= 175 mT and γ/2π= 2.8×1010Hz/T, respectively [3]. The MEC constant b= 6.96×105J/m3[4],\nthe exchange parameter 2 A/M 0= 9.2×10−6m2/s [5], and the mass density ρ= 5170 kg/m3[6]. The transverse and\nlongitudinal sound velocities are ct= 3.8×103m/s andcl= 7.2×103m/s [4]. The phonon relaxation rate in YIG\nis of the order of 1 µs−1[7]. In high quality films, the Gilbert damping coefficient is typically around 10−4[3]. For\ncomparison, we also show results for α= 0.1, which is close to the upper limit for Bi-doped YIG [8].\n2. ANALYTICAL EXPRESSION FOR IFE-INDUCED ONE-DIMENSIONAL DYNAMICS\nThe time evolution of the out-of-plane magnetization Eq. (2) of the main text can be simplified to\nmz(˜x,˜t) =m0√π/summationdisplay\n±/integraldisplay∞\n0d˜ke−˜k2/4Im/parenleftBigg\n˜ω∓−˜Ω0\n˜ω∓−˜ω±ei˜ω±˜t/parenrightBigg\ncos˜k˜x,\nwhereω±=/bracketleftbigg\n˜Ω0+˜Ωp±/radicalBig\n(˜Ω0−˜Ωp)2+ 8˜∆2/bracketrightbigg\n/2. Here ˜Ω0= (1 +iα)(1 + ˜D˜k2),˜Ωp= ˜ct˜k+i˜τ−1\np,˜k=kW, ˜x=x/W ,\n˜τp= Ω 0τp,˜∆ = ∆/Ω0˜ct=ct/(WΩ0),˜t= Ω 0t, and ˜D= 2A/(M0Ω0W2) are all dimensionless.\n3. ANALYSIS OF STAR-LIKE FEATURE DUE TO IFE FROM POINT SOURCES\nThe IFE-amplitude induced by an axially symmetric light pulse as plotted in Fig. 4(a,left) displays suprisingly\ncomplex interference patterns. These features can be explained in terms of the angular-dependent group velocity at\nconstantk0=/angbracketleft|k|/angbracketright[yellow circle in the (volume) spin wave dispersion shown in Fig. S1(a)]. The thin (black) arrows\nindicate the group velocities vg≡ ∇ kωk|k0,θ/similarequalˆθ[γµ0M0/(2k0)] sin(2θ)/radicalBig\nH/(H+M0sin2θ), where we used the\nsimplified dispersion ωk/similarequalγµ0/radicalBig\nH(H+M0sin2θ) that is accurate for not too small k0/greaterorsimilar1/W.In this approximationarXiv:1508.02094v2  [cond-mat.mes-hall]  5 Nov 20152\n(a)(GHz)\n-2  0  2\nkx (1/µm)-2 0 2ky (1/µm) 1  2  3\nvavbvc\nkakckb\n(b)-40 -20  0  20  40\nx (µm)-40-20 0 20 40 y (µm)-0.1  0  0.1\nka\nkckb-0.1  0  0.1\nFIG. S1: (Color online) (a) Approximate (bulk) spin wave spectrum ωk=γµ0/radicalbig\nH(H+M0sin2θ) ink-space. The thin (black)\narrows show angular dependence of the group velocities vg=∇kωkat the average modulus k0=/angbracketleft|k|/angbracketright. Those for three typical\nwave vectors ka,b,c represented by thick (purple) arrows are labeled by va,b,c. (b) Replotting of Fig.4(a,left), IFE-induced\nmagnetization profile in real space at 5 ns, with three thin (black) [thick (purple)] arrows representing tva,b,c[ka,b,c].\nvg·k= 0,i.e., the group velocity is purely transverse. The thick (purple) arrows Fig. S1(a) represent three typical\nwave vectors ka,b,c, whose corresponding group velocities are labeled by va,b,crespectively. In Fig. S1(b), we replot\nthe IFE-induced magnetization profile in real space with dashed (olive) curve representing t|vg|and the vectors ka,b,c\nandtva,b,cas thick (purple) and thin (black) arrows for t= 5 ns. We can understand this pattern in terms of the time\ndependence of a linear superposition of wave packets arranged around the origin: Each wave packet centered around a\ncertain k0is modulated in the direction ˆk0with wavelength 1 /k0. The whole package moves rigidly in the transverse\ndirection with group velocity vgthat is finite by the dipolar interaction. The initial wave packet at the origin thereby\nblows up like|tvg|. The interference pattern initially normal to k0is in real space shifted and combines with the\npatterns generated by other wave packets on the ring to form complex features that in the snapshot plotted in the\nfigures look like a star. Moreover, the lifetime of spin waves is determined by 1 /(αωk), leading to τka> τkb> τkc.\nThis explains that even for large dampings a radial pattern is still visible near the y-axis while that along the x-axis\nhas disappeared [see Fig. 4(b,left)].\nAnalytically, the out-of-plane magnetization at r= (rcosϕ,rsinϕ) with respect to the center of initial spot can be\nexpressed as\nmz(r,ϕ,t )∝/integraldisplay2π\n0dθe−αtγµ 0√\nH(H+M0sin2θ)/integraldisplay∞\n0d˜k˜ke−˜k2/4cos/parenleftbigg\n˜k˜rcos(θ−ϕ)−tγµ0/radicalBig\nH(H+M0sin2θ)/parenrightbigg\n.(S6)\nFar from the excitation spot, i.e., ˜ r/greatermuch1, the ˜k-integral is dominated by the spin waves satisfying cos( θ−ϕ)∼0 or\nθ∼ϕ±π/2. Taylor-expanding in the phase factor of the cosine leads to\nmz(r,ϕ,t )∼e−αtγµ 0√\nH(H+M0cos2ϕ)/integraldisplay∞\n0d˜k˜ke−˜k2/4/integraldisplayδθ\n−δθdθ/primecos/parenleftBig\n˜k(˜r−t|vg|)θ/prime−tγµ0/radicalbig\nH(H+M0cos2ϕ)/parenrightBig\n∼2 cos/parenleftBig\ntγµ0/radicalbig\nH(H+M0cos2ϕ)/parenrightBig\ne−αtγµ 0√\nH(H+M0cos2ϕ)/integraldisplay∞\n0d˜ksin/parenleftBig\n˜k(˜r−t|vg|)δθ/parenrightBig\n˜r−t|vg|e−˜k2/4(S7)\nwhereδθis a small cut-off angle and |vg|= (γµ0M0/˜k)|sin 2ϕ|/radicalbig\nH/(H+M0cos2ϕ). Eq. (S7) catches most features\nobserved in Figs. 4(left) as summarized above: ( i) cos/parenleftBig\ntγµ0/radicalbig\nH(H+M0cos2ϕ)/parenrightBig\nis an angle-dependent oscillations,\ni.e., reproduces the star-like feature. ( ii) exp[−αtγµ 0/radicalbig\nH(H+M0cos2ϕ)] explains the angular dependent attenu-\nation, revealing that rays with ϕ/similarequalπ/2 decay slower than those for ϕ/similarequal0. At large damping a radial pattern is\nvisible near the ybut not in the x-direction [see Fig. 4(b,left)]. ( iii) The radial integral is a function of ˜ r−t|vg|only,\ntherefore predicts that the wave packet expands with the group velocity |vg|.3\n4. MAGNITUDE OF IFE\nHere we estimate the magnitude of the IFE based on the experiments by Satoh et al. [9], who excited spin\nwaves in a 110 µm thick Gd 4/3Yb2/3BiFe 5O12film by laser light with pump fluence of 200 mJ(cm)−2and pulse\nwidth of 120 fs at wavelength 1400 nm, and measured Faraday rotations of a linearily polarized probe pulse (at wave\nlength 792 nm) of the order of ϑF∼1 mrad. Satoh et al. do not measure the Verdet constant. In Bi-YIG (with\nunspecified doping concentration) the Verdet constants are V∼5800, 1050, and 380 rad m−1T−1for wavelengths\n543nm, 633nm, and 780nm, respectively [10]. The Verdet constant in (BiYbTb) 3Fe5O12at 780 nm was found to be\nV∼8.5×104rad m−1T−1[10]. Here we adopt V= 104rad m−1T−1for both pump and probe fields.\nThe IFE and FE may be formulated in terms of a phenomenological free energy [11, 12]\nF=βpump(µ0Mz)Iin, (S8)\nwhereβpump is a magneto-optical susceptibility at the pump laser frequency and Iinis the intensity of (right-hand)\ncircularly polarized light. The effective field HIFE=−(1/µ0)∂MzF=−βpumpIin.In the same theoretical framework,\nthe Faraday rotation induced in the external field-free limit by B=µ0Mzreads\nϑF=2πc0\nλ0βprobe(µ0Mz)L=Vprobe(µ0Mz)L, (S9)\nwherec0,λ0, andLare the velocity of light in vacuum, the wavelength of the probe pulse, and the film thickness,\nrespectively. This relation allow us to express βin terms of V.\nWe can now estimate (assuming Vpump≈Vprobe)|µ0HIFE|=µ0Vλ0Iin/(2πc0)∼0.2 T for the experimental\npump intensity. HIFEwhen applied as a pulse with width τlgenerates a magnetization My=γτlµ0HIFEM0and\nm0=My/M0∼0.004. Disregarding damping, the precession in the external field Hˆxpreserves this amplitude such\nthatMz≈My.This leads to ϑF≈V(γτlµ2\n0HIFEM0)L∼0.5 mrad, in reasonable agreement with the observations [9].\n[1] E. Schl¨ omann, J. Appl. Phys. 31, 1647 (1960).\n[2] C. Kittel, Phys. Rev. 110, 836 (1958).\n[3] S. A. Manuilov, S. I. Khartsev, and A. M. Grishin, J. Appl. Phys. 106, 123917 (2009).\n[4] A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves (CRC Press, Boca Raton, 1996).\n[5] A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Appl. Phys. 43, 264002 (2010).\n[6] M. A. Gilleo and S. Geller, Phys. Rev. 110, 73 (1958).\n[7] M. F. Lewis and E. Patterson, J. Appl. Phys. 39, 1932 (1968).\n[8] G. G. Siu, C. M. Lee, and Y. Liu, Phys. Rev. B 64, 094421 (2001).\n[9] T. Satoh, Y. Terui, R. Moriya, B. A. Ivanov, K. Ando, E. Saitoh, T. Shimura, and K. Kuroda, Nat. Photonics 6, 662\n(2012).\n[10] J. G. Bai, G.-Q. Lu, and T. Lin, Sensors and Actuators A: Physical 109, 9 (2003).\n[11] P. S. Pershan, J. P. van der Ziel, and L. D. Malmstrom, Phys. Rev. 143, 574 (1966).\n[12] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010)."    },    {        "title": "0707.2344v1.Magnetodipolar_interlayer_interaction_effect_on_the_magnetization_dynamics_of_a_trilayer_square_element_with_the_Landau_domain_structure.pdf",        "content": "1Magnetodipolar interlayer interaction effect on the magnetization dynamics\nof a trilayer square element with the Landau domain structure\nD.V. Berkov, N.L. Gorn\nInnovent Technology Development, Prüssingstr. 27b, D-07745 Jena, Germany\nABSTRACT\nWe present a detailed numerical simulation study of the effects caused by the magnetodipolar\ninteraction between ferromagnetic (FM) layers of a trilayer magnetic nanoelement on its mag-\nnetization dynamics. As an example we use a Co/Cu/Ni80Fe20 element with a square lateral\nshape where the magnetization of FM layers forms a closed Landau-like domain pattern. First\nwe show that when the thickness of the non-magnetic (NM) spacer is in the technology\nrelevant region h ~ 10 nm, magnetodipolar interaction between 90o Neel domain walls in FM\nlayers qualitatively changes the equilibrium magnetization state of these layers. In the main of\nthe paper we compare the magnetization dynamics induced by a sub-nsec field pulse in a\nsingle-layer Ni80Fe20 (Py) element and in the Co/Cu/Py trilayer element. Here we show that\n(i) due to the spontaneous symmetry breaking of the Landau state in the FM/NM/FM trilayer\nits domains and domain walls oscillate with different frequencies and have different spatial\noscillation patterns; (ii) magnetization oscillations of the trilayer domains are strongly supp-\nressed due to different oscillation frequencies of domains in Co and Py; (iii) magnetization\ndynamics qualitatively depends on the relative rotation sense of magnetization states in Co\nand Py layers and on the magnetocrystalline anisotropy kind of Co crystallites. Finally we\ndiscuss the relation of our findings with experimental observations of magnetization dynamics\nin magnetic trilayers, performed using the element-specific time-resolved X-ray microscopy.\nKeywords: thin magnetic films; magnetic nanoelements; magnetization dynamics; numerical\nsimulations\nPACS numbers: 75.40.Mg; 75.40.Gb; 75.60.Ch; 75.75.+a;2I. INTRODUCTION\nStudies of nanoelements patterned out of magnetic multilayers constitute now a rapidly gro-\nwing research area due to their already existing and very promising future applications. Using\nmodern patterning technologies it is possible to produce arrays of nanoelements with lateral\nsizes of  ~ 102 - 103 nm and nearly arbitrary shapes. Such elements and their arrays can be\nemployed in magnetic random access memory (MRAM) cells, miniaturized magnetoresistan-\nce sensors (in read/write hard disk heads), advanced high-density storage media, spintronic\ndevices [1] etc.\nSmall lateral sizes of these single- and multilayered structures lead to qualitatively new featu-\nres of their magnetization dynamics, with the quantization of their spin wave eigenmodes\nbeing the most famous example (see, e.g., [2, 3, 4, 5]). Thorough understanding of this novel\nfeatures is crucially important both for the progress of the fundamental research in this area\nand for the development of reliable high-technology products based on such systems.\nIn the last decade extensive experimental and theoretical effort was dedicated to the studies of\nmagnetization dynamics of single-layer nanoelements. Among them the nanodisks possessing\nclosed magnetization configuration with the central vortex (for continuous disks) or without it\n(for rings with the hole in the middle) represent the simplest non-trivial example due to their\ncircular form and hence - axially symmetric magnetization configuration. Magnetization\ndynamics of these nanodisks has been extensively studied using advanced experimental\ntechniques, analytical theories and numerical simulations [6] and is satisfactory understood.\nThe next complicated case is a square or rectangular single-layer nanoelement with either\nsaturated magnetization state or closed Landau domain structure. In the state close to satura-\ntion the main non-trivial effect is due to the strong demagnetizing field near the element edges\nperpendicular to the field and magnetization direction; corresponding dynamics could also be\nunderstood quantitatively combining experimental and theoretical methods (see, e.g., [2, 4]\nand references therein). The closed Landau magnetization pattern is much more demanding at\nleast from the theoretical point of view, because magnetization dynamics of this structure\nexhibits both highly localized (oscillation of the central core and domain walls) and extended\n(oscillation of domain areas) modes. But many important features of this dynamics could be\nalso understood very recently using such advanced experimental methods as time-resolved\nKerr microscopy and space resolved quasielastic Brilloin light scattering techniques,\nsupported by detailed numerical simulations [7, 8, 9, 10, 11].\nHowever, single-layer elements are not very interesting from the point of view of potential\napplications, because almost any technical device based on magnetic nanolayers employs - for\nvarious, but fundamental reasons - mainly multilayer structures. An additional layer (or seve-\nral such layers) is required, e.g., as a reference layer with 'fixed' magnetization to detect via\nsome MR effect the resistance change when the 'free' layer changes its magnetization direc-\ntion, or as an electron spin polarizer in spintronic devices, etc. For this reason the magnetiza-\ntion dynamics of multilayered structures is of the major interest.\nIn such structures the interlayer interaction effects play often a very important role. Even if\nwe leave aside a strong exchange (RKKY) coupling present in structures with very thin non-\nmagnetic spacers consisting of some specific materials like Ru, we are still left with the un-\navoidable magnetodipolar coupling between the layers. This coupling is identically zero only\nfor a multilayer structure with infinitely extended and homogeneously magnetized layers - the\nsituation which virtually never is encountered in practice. For this reason understanding of the\nmagnetodipolar interlayer interaction influence is absolutely necessary for further progress.3This interaction is especially strong in situations, where the layer magnetization is - at least in\nsome regions - perpendicular to the free layer surface, thus inducing very large \"surface mag-\nnetic charges\" and consequently - high stray fields. Typical example of such systems are mul-\ntilayers with a perpendicular magnetic anisotropy and nanoelements where the magnetization\nlies in the layer plane, but is nearly saturated, so that large stray fields emerge near the side\nedges of a multilayer stack. The influence of the interlayer interaction in such systems has\nbeen extensively investigated in the past for the quasistatic magnetization structures (see, e.g.,\nthe review [12] and references therein) and very recently - by studies of the magnetization\ndynamics [13].\nTo avoid the strong interaction caused by the magnetization directed normally to the free sur-\nface, a commonly used idea is to employ multilayer nanoelements with closed magnetization\nstructures. In nanodisks such a structure is represented by an in-plane rotating magnetization,\ncontaining a central vortex as the only element producing strong demagnetizing field. Magne-\ntodipolar interaction between the cores of such vortices in such multilayered circular nanodots\nhas been investigated recently in [13].\nFor the next commonly used nanoelement shape - magnetic rectangle - the closed magnetiza-\ntion structure is achieved by the famous Landau pattern with four homogeneously magnetized\ndomains and four 90o domain walls (in case of a square element). For sufficiently thin films\ncommonly used in technologically relevant systems these walls are the so called Neel walls\n[14] with the magnetization lying almost in the element plane. Such a magnetization\nconfiguration contains only volume 'magnetic charges' (no free poles on the element surface,\nand hence - no surface charges), which are usually weaker than surface charges. For this\nreason the interlayer interaction mediated by the domain walls is multilayer elements with the\nclosed magnetization structure is expected to be weaker than in multilayer stacks with\nsaturated  magnetization.\nHowever, recently several research groups [15, 16, 17, 18] have demonstrated that the stray\nfield caused by the magnetization of vortex and/or Neel walls in one layer of a multilayered\nsystem can still strongly affect the magnetic state of other layers. Most of these studies use\nsimplified DW models which enable a semianalytical treatment of the problem (see, e.g., [15,\n16]), but rigorous micromagnetic simulations have confirmed that the stray field of a vortex\nwall with the Neel cap [17] or of a purely Neel wall [18] can strongly affect the magnetic state\nof other layers even if the interlayer separation is as large as ~ 100 nm [18].\nIn this study we consider multilayer square elements with lateral sizes ~ 1 mkm and thicknes-\nses of magnetic layers and non-magnetic spacers ~ 10 nm (i.e., geometry typical for numerous\napplications). We shall demonstrate that in such systems the magnetodipolar interlayer inter-\naction due to the Neel domain walls of the closed Landau structure is strong enough to change\nqualitatively both the quasistatic magnetization structure and magnetization dynamics of a\nsystem. The paper is organized as follows. After the brief description of our simulation me-\nthodology (Sec. II.A) we show reference results for a square single-layer element which will\nserve for comparison with multilayered systems. Afterwards we analyze in detail the effect of\nthe interlayer interaction on the quasistatic magnetization structure and magnetization dyna-\nmics for a square FM/NM/FM trilayer (Sec. II.C), considering both the influence of the initial\nmagnetization state - compare Sec. II.C and D, and the effect of the random polycrystalline\ngrain structure - compare Sec. II.D and E. In Sec. III we compare our results with (unfor-\ntunately very few) available experimental and numerical studies of similar systems, and\ndiscuss the possibility of an experimental verification of our simulation predictions.4II. NUMERICAL SIMULATION RESULTS\nA. Numerical simulations setup\nIn this study we have simulated the trilayer element Co/Cu/Py with the following geometry\nshown in Fig. 1 (when not stated otherwise): lateral sizes 1 x 1 mkm2, Co and Py layer thick-\nnesses hCo = hPy = 25 nm, Cu interlayer thickness (non-magnet spacer thickness between Co\nand Py layers) hsp = 10 nm. Both magnetic layers were discretized into Nx x Nz x Ny = 200 x\n200 x 4 rectangular prismatic cells. We have checked that the discretization into at least 4 in-\nplane sublayers was necessary to reproduce correctly the 3D magnetization structure of 90o\ndomain walls (DW) present in the equilibrium Landau magnetization state of square\nnanoelements.\nThe following magnetic parameters have been used: for Py - saturation magnetization\nPy\nS 860 M = G, exchange stiffness constant APy = 1\u000110-6 erg/cm and cubic magnetocrystalline\ngrain anisotropy Py 3\ncub510 K =× erg/cm3; for Co - Co\nS 1400 M = G, ACo = 3\u000110-6 erg/cm and\nmagnetocrystalline grain anisotropies Co 5\ncub610 K =× erg/cm3 for the cubic fcc modification of\nCo and Co 6\nun410 K =× erg/cm3 for its uniaxial hcp modification (see [19] for discussion of all\nvalues for Co). The average grain (crystallite) size 10 D\u0001\u0002= nm (in all directions) with 3D\nrandomly oriented anisotropy axes of various crystallites was used for both magnetic\nmaterials. There was no correlation between the crystallites in Py and Co layers.\nSimulations of both the equilibrium magnetization structure and magnetization dynamics\nwere performed using our commercially available MicroMagus package (see [20] for\nimplementation details). For simulations of the magnetization dynamics the package employs\nthe optimized Bulirsch-Stoer method with the adaptive step-size control to integrate the\nLandau-Lifshitz-Gilbert equation for the magnetization motion with the standard linear\nGilbert damping (damping constant was set to l = 0.01 throughout our simulations). Due to\nthe small amplitude of magnetization oscillations studied here we believe that this simplest\ndamping form adequately describes the energy dissipation in our system. We also did not take\ninto studies additional damping caused by the spin pumping effect in magnetic multilayers\n(see, e.g., [21]), because this is beyond the scope of this paper.\nWe have studied magnetization dynamics of our system in a pulsed magnetic field applied\nperpendicularly to the element plane. To obtain magnetization excitation eigenmodes of an\nequilibrium magnetization state we have applied a short field pulse in the out-of-plane\ndirection with the maximal field value Hmax = 100 Oe and the trapezoidal time dependence\nwith rise and fall times tr = tf = 100 ps and the plateau duration tpl = 300 ps. To obtain the\neigenmode spectrum, we have set the dissipation constant to zero (l = 0) and recorded\nmagnetization trajectories of each cell during the pulse and for Dt = 10 ns after the pulse was\nover. Spatial profiles of the eigenmodes (spatial maps of the oscillation power distribution in\nthe element plane) were then obtained in the meanwhile standard way [2, 22] using the\nFourier analysis of these magnetization trajectories after the pulse decay. Because the applied\nfield pulse was spatially homogeneous, we could observe only eigenmodes which symmetry\nwas not lower than the symmetry of the equilibrium magnetization state of the studied system.\nIn principle, the analysis of the eigenvalues and eigenvectors of the energy Hessian matrix\n[23] allows to obtain all eigenmodes, but our method can be used for much larger systems5because it does not require the explicit search of eigenvalues for large matrices with sizes Z x\nZ proportional to total number of discretization cells Z  ~ Nx x Nz x Ny. For the qualitative\nanalysis of the influence of various physical factors on the magnetization dynamics aimed in\nthis paper our method provides enough information.\nTo study the transient magnetization dynamics which could be compared to real experiments\nwe have applied the same field pulse as described above and recorded magnetization time\ndependencies for each discretization cell during the pulse and for Dt = 3 ns after the pulse.\nHere the dissipation constant was set to l = 0.01 - the value commonly reported in literature\nfor thin Py films; the same constant was used for Co layer. We have checked that increasing\nthis value up to l = 0.05 led, as expected, to faster overall oscillation decay, but did not\nproduce any qualitative changes in the magnetization dynamics.\nB. Single layer Py element as the reference system\nKeeping in mind that we are going to study interaction effects in a trilayer system, we first\npresent reference results for the single Py nanoelement with the same parameters as the Py\nlayer of the complete trilayer (the square element 1 x 1 mkm2, with the thickness hPy = 25 nm,\nPy\nS 860 M = G, APy = 1\u000110-6 erg/cm, Py 3\ncub510 K =× erg/cm3). Fig. 2a shows the equilibrium\nmagnetization structure of such a nanoelement obtained starting from the initial state\nconsisting of four homogeneously magnetized domains in corresponding triangles (as shown,\ne.g., in Fig. 4a), whereby magnetic moments of four central cells were oriented perpendicular\nto the layer plane (along the y-axis). As expected, the very small random grain anisotropy of\nPermalloy has virtually no influence on the magnetization state, so that the equilibrium\nmagnetization forms a nearly perfect closed Landau magnetization pattern with four 90o Neel\ndomain walls between the domains and the central vortex showing upwards.\nSpectrum of magnetization excitations for this Landau pattern is shown in Fig. 2b together\nwith spatial maps of the oscillation power of the out-of-plane magnetization component for\neach significant spectral peak. We remind (see Sec. IIA) that the field pulse used to draw the\nmagnetization out of its equilibrium state was spatially homogeneous, and hence only modes\nwith the corresponding symmetry could be excited. Excitation modes of the Landau domain\npattern have been recently studied in detail in [10, 11], so here we will only briefly mention\nseveral issues important for further comparison with the trilayer system.\nThe lowest peak in the excitation spectrum in Fig. 2b corresponds, in a qualitative agreement\nwith the results from [10], to the domain wall oscillations, whereby due to the spatial\nsymmetry of the exciting field pulse we observe only the oscillation mode where all domain\nwalls oscillate in-phase. We are not aware of any analytical theory which would allow to cal-\nculate the frequency of a 90o domain wall oscillations and thus could be compared to our\nsimulations. From the qualitative point of view, DW oscillations are exchange-dominated and\ntheir frequency is the lowest one among other exchange-dominated magnetization excitations,\nbecause the equilibrium magnetization configuration inside a DW is inhomogeneous and thus\nits stiffness with respect to small deviations from the equilibrium is smaller than for a colline-\nar magnetization state.\nPeaks with higher frequencies correspond to the oscillations within four triangular domains of\nthe Landau structure; again, only symmetric in-phase oscillations have been observed. Accor-\nding to the analysis performed in [10, 11] domain excitations can be classified into the\nfollowing types. First, there exist modes which power distribution has nodes (between the\npeaks) in the radial direction, i.e., from the square center to its edges. Corresponding wave6vector is perpendicular to the magnetization direction in the domains (^kM). Such modes\nare similar to Damon-Eshbach modes in extended thin films and are called radial (wave vec-\ntor in the radial direction) [10] or transverse (because ^kM) [11]. Second, there exist modes\nwith power distribution nodes along the contour around the square center. In this case regions\nwith high power form elongated bands from the center to the edges of the square. For these\nmodes the wave vector of their spatial power distribution is roughly parallel to the local\nmagnetization direction in each domain (kM\u0001); they behaviour is similar to the backward\nvolume modes in extended thin films. For obvious reasons this second type is called azi-\nmuthal [10] or longitudinal [11] modes.\nAs it can be seen from Fig. 2b, our field pulse excites mainly an azimuthal mode with the\nfrequency f \u0002 3.2 GHz and several modes which can be classified as mixed radial-azimuthal\nmodes, because their spatial power distribution has nodes along both the radial direction and\nthe contours around the square center. These our results can be compared to simulations from\n[11], where the Py element with the same lateral sizes 1000 x 1000 nm2, but with the smaller\nthickness h = 16 nm was studied. Qualitatively our power maps are very similar to those\nshown in [11], but there are some important discrepancies. First of all, our overall excitation\nspectrum is very different from that presented in [11] (compare our Fig. 2b with Fig. 1d from\n[11]), although several peak positions are very close. Our power maps for specific modes also\nhave some qualitative similarities to several maps presented in [11], but detailed comparison\ndoes not make much sense due to the different total power spectra as mentioned above. All\nthese difference may arise because the simulated nanoelement in [11] was not discretized in\nthe layer plane,  but we believe that the major reasons are (i) the much shorter excitation pulse\n(td = 2.5 ps pulse length) used in [11] compared to our (300 ps) and (ii) the presence of the\nfinite damping in simulations from [11]. This problem requires further investigation, but is\nbeyond the scope of this paper.\nTransient magnetization dynamics for the single-layer Py element after the application of the\nsame field pulse as used for the studies of the excitation spectrum is shown in Fig. 3. We\nremind that for these simulations we have used the non-zero damping l = 0.01 typical for Py\nfilms. Fig 3 shows the time dependence of the angle between the average layer magnetization\nand the element plane () () tmt Y^\u0002  (panel (a)), spatial maps of the out-of-plane magnetizati-\non projection during the pulse (b) and after the pulse (c). By displaying the out-of-plane mag-\nnetization projection we have subtracted the equilibrium magnetization eq() m r, so that maps\nin Fig. 3 (and all other figures where the transient magnetization dynamics is shown) repre-\nsent the difference eq(,) () m m tm D^ ^ ^ = -r r. Homogeneous grey background around the mag-\nnetic element shows the reference grey intensity for 0 mD^=.\nFirst of all, we emphasize that even the small damping l = 0.01 used here leads to relatively\nfast oscillation decay (within ~ 3 ns after the pulse). After the initial increase of the perpendi-\ncular magnetization projection due to the field pulse (see the bright contrast across the whole\nsquare in Fig. 3b) is over, the time dependence of the average magnetization is dominated by\nrelatively fast oscillations of the domains, slightly modulated by oscillations of a lower frequ-\nency due to the domain wall motion. Corresponding patterns can be seen in Fig. 3c, where\none can directly recognize that domain walls and domains themselves oscillate with very\ndifferent frequencies. Further, comparison of the time-dependent maps from Fig. 3c with the\nspatial power maps in Fig. 2b shows the qualitative relation between the eigenmodes and the\ntransient dynamics of the Py square in this case: not only the contrast due to the DW oscilla-7tions, but also characteristic wave patterns inside the domains and near the outer regions of\ndomain walls agree qualitatively with the eigenmode power distributions shown in Fig. 2b.\nAnalogous simulations (with qualitatively similar results) have been carried out in [24] in\norder to explain the magnetization dynamics observed there using the time-resolved X-ray\nmicroscopy. We shall return to the analysis of these results by comparing our simulation with\nexperimental data in Sec. III.\nC. Trilayer Co/Cu/Py element: Landau structures with the same rotation sense in both\nmagnetic layers\n1. Deformation of the quasistatic magnetization structure\nIn this section we consider the trilayer element Co(25nm)/Cu(10nm)/Py(25nm), 1 x 1 mkm2\nin-plane size, with magnetic parameters given in Sec. II.B and cubic random anisotropy of Co\ngrains with Co 5\ncub610 K =× erg/cm3. In order to determine the equilibrium magnetization state\nof any system by minimizing its magnetic free energy we have to choose the initial (starting)\nmagnetization state.  As such a state we take in this section for both Co and Py layers the\nclosed in-plane magnetization configuration with sharply formed four triangular domains and\nfour magnetic moments in the middle of each layer pointing in the same out-of-plane direc-\ntion - along the +y-axis (to fix the orientation of the central vortex). An important point is that\nthe rotation sense of the starting magnetization state is the same for both layers. This initial\nstate is shown schematically in Fig. 4a. The situation, when the initial state consists of two\nclosed magnetization configurations with opposite rotation senses in Co and Py layers, is\nconsidered in the next subsection.\nThe corresponding equilibrium state which comes out as the result of the energy minimization\nis shown in Fig. 4c. The most striking feature of this state is the strong deformation of a 'nor-\nmal' Landau pattern (see Fig. 2a). Namely, central vortices in Co and Py layers are signifi-\ncantly displaced in opposite directions and domain walls are bended - also in opposite direc-\ntions for Co and Py. Equilibrium domains in both layers do not have anymore a shape of iso-\nsceles triangles, but rather form 'triangles' with slightly bended sides of different lengths. The\ndegree of the deformation described above depends both on the Cu spacer thickness and the\nlateral size of the squared trilayer structure (results not shown).\nThe reason for this unusual deformation can be understood by analyzing the intermediate\nmagnetization configurations arising during the energy minimization. At the first stage of this\nprocess the 'normal' Landau domain configuration inside each layer is formed. It is well\nknown that the 90o Neel domain walls of this magnetization configuration possess both\nvolume and surface 'magnetic charges' along them. We consider in detail the configuration of\nsurface charges, because the density of these 'charges' is simply proportional to the out-of-\nplane magnetization component () m^r and is thus easier to visualize. The corresponding map\nof the out-of-plane magnetization () m^r for the equilibrium Landau state of a 25 nm thick\nsquare Py nanoelement is shown in Fig. 4b. One can clearly see significant enhancement of\nthe () m^r-magnitude along all four domain walls. Thus two lines of the opposite 'surface\nmagnetic' charges are formed along each wall, building a kind of a 'linear dipole' (shown\nschematically in Fig. 4b with arrows and + and – signs). Now, it is important to realize that\nthe orientation of these dipoles for the given domain wall is the same on both upper and lower\nsurfaces of the nanoelement. For this reason, for the geometry shown in Fig. 1 and initial\nmagnetization states with the same rotation senses (as shown in Fig. 4a) we have on the upper8Co surface and lower Py surface linear dipoles with the same (parallel) orientations along all\nfour walls in each layer. The volume 'charges' formed due to the non-zero magnetization\ndivergence in the nanoelement volume have a qualitatively similar distribution. They also\ncontribute to the effect described below; with the decrease of the film thickness, when the\nNeel wall tends to a perfectly 'in-plane' magnetization structure, the contribution from these\nvolume charges becomes dominating.\nThe linear dipoles described above dipoles obviously repel each other, and due to the small\ninterlayer distance (which is in this case significantly smaller than the wall width) this\nrepulsion is very strong. With other words, corresponding 90o Neel domain walls in Co and\nPy layers 'feel' a strong mutual repulsion, so that they start to move away from each other. As\nthe result, domain walls in Co and Py layers shift in opposite directions, forming the final\nequilibrium structure displayed in Fig. 4c. We note in passing that for the starting state used in\nthese simulations, the central vortices in Co and Py layer have the same orientation and thus\nattract each other. However, due to the small vortex area this attraction can not compensate\nfor the strong repulsion of all domain walls, although the surface density of magnetic charges\nwithin the vortices is much higher than along the walls to due a large values of () m^r within\nthe vortex.\nFor further consideration it also important to note that the equilibrium magnetization state of\nCo is disturbed by its random grain anisotropy more than for the Py layer, for which the influ-\nence of this anisotropy is very small. Corresponding disturbance can be seen on the spatial\nmap of () m^r for Co (Fig. 4c, middle panel of the upper row), but for the moderate cubic ani-\nsotropy of Co Co 5\ncub610 K =× erg/cm3 and the small average crystallite size 10 D\u0001\u0002= nm, this\ndisturbance is still rather weak.\n2. Eigenmodes and transient magnetization dynamics\nStrong deformation of the equilibrium domain structure discussed in the previous subsection\nhas a qualitative impact on magnetization dynamics in the trilayer as compared to a single-\nlayer systems.\nFirst of all, spectrum of eigenmodes of the Py layer from Co/Cu/Py trilayer (shown in Fig. 5,\nupper panel) is qualitatively different from the corresponding single-layer Py square (Fig. 2b).\nIrregular domain structure results in a quasi-continuos (for our resolution) oscillation power\nspectrum, because peaks corresponding to the oscillations of each domain and domain wall\nare located at different positions. In particular, all domain walls oscillate with various frequ-\nencies, as shown in Fig. 5 by the peaks of the 1st group (a, b, c). These peaks can be attributed\nto oscillations of different domain walls as displayed on the power spatial maps in the upper\nrow of these maps. Eigenmode frequencies for oscillations within the domains also differ\nsignificantly for different Py domains, as shown by the peak positions of the 2nd group in the\nspectrum. In addition, the power distribution patterns within each domain become highly\nirregular, as shown by corresponding maps in the second map row in the same figure. For\nhigher frequencies (group 3), the power distribution is even more complicated, although some\ntypical attributes of longitudinal and transverse modes can still be recognized (3rd map row).\nTransient magnetization dynamics of the same trilayer system with the finite damping (it was\nset to l = 0.01 for both Co and Py layers) also strongly differs from the monolayer case. Cor-\nresponding time dependencies for the out-of-plane angles () () tmt Y^\u0002 of the average magne-9tization are shown in Fig. 6a for the Co layer (thin solid line), Py layer (dashed line) and the\ntotal system (thick solid line).\nThe out-of-plane magnetization deviation during the field pulse is smaller for the Co layer\nthan for the Py layer, due to the higher saturation magnetization of Co which leads to larger\ndemagnetizing field caused by the out-of-plane excursion of Co magnetization. Due to the\nsame larger saturation magnetization of Co (and equal Co and Py layer thicknesses), the basic\noscillation frequency is now close to the oscillation frequency of domains for the single-layer\nCo nanosquare: Co layer in the trilayer element determines the overall oscillation frequency,\n'locking' (capturing) the frequency of the Py layer domains also. Because the eigenfrequencies\nof Co and Py domains do not coincide, this phenomenon leads to a much faster decay of the\ndomain oscillations in the Py layer compared to the case of the single-layer Py element (com-\npare the dashed lines in Fig. 6a to Fig. 3a). In fact, shortly after the pulse is over (t > 0.6 ns),\nPy domain oscillations are barely visible both in the average magnetization time-dependence\n(Fig. 6a) and spatial maps of the out-of-plane magnetization component (Fig. 6c). Low-frequ-\nency oscillations of the average magnetization of Py are entirely determined by the oscillati-\nons of bended DWs. Note that oscillations of different walls are out of phase due to the diffe-\nrent eigenfrequencies of the four DWs in the disturbed Landau structure (see Fig. 5), so that\nfor the given time moment different walls (and even different regions of one and the same\nwall) can exhibit opposite magnetization contrasts as displayed on the last images in both map\nrows in Fig. 6b and 6c. We shall return to this important circumstance by comparing our\nsimulations to experimental data in Sec. III.\nD. Trilayer Co/Cu/Py element: Landau structures with opposite rotation senses in Co\nand Py magnetic layers\nEquilibrium magnetization configuration. It is well known that the initial magnetization state\nused to start the energy minimization in micromagnetics can have a decisive influence on the\nequilibrium configuration resulting from this minimization, because any realistic ferromagne-\ntic system possesses many energy minima due to several competing interactions present in\nferromagnets. For this reason we have studied the influence of the starting configuration on\nthe equilibrium magnetization and dynamical properties of our trilayer system, choosing as an\nalternative starting state the same Landau-like domains structure as described at the beginning\nof subsection II.C.1, but with opposite rotation senses for Co and Py layers (see Fig. 7a).\nIn this case Landau patterns are also formed at the initial energy minimization stage in both\nmagnetic layers. However, closed magnetization states in Co and Py layers have now opposite\nrotation senses. For this reason linear magnetic dipoles appearing along each domain wall at\nthe upper Co and lower Py surfaces as described in subsection II.C.1 above, are oriented anti-\nparallel. Hence the domain walls (which form these dipoles) attract each other, so that these\nwalls become wider and do not move across the layers. This naturally leads to a nearly sym-\nmetrical magnetization configurations in both Co and Py. This configuration is qualitatively\nsimilar to a 'normal' Landau pattern in a single square nanoelement, but domain walls are\nmuch broader (compare Fig. 7b to Fig. 2a).\nTable 1. Energies of equilibrium magnetization states shown in Fig. 4c and 7b.10Initial magn. state Total energy\n(nanoerg)Anisotropy\nenergyExchange\nenergyDemag.\nenergy\nThe same rotation\nsenses in Co and Py4.193 2.851 0.715 0.627\nOpposite rotation\nsenses in Co and Py3.602 2.838 0.510 0.254\nIt is instructive to compare the energies of equilibrium magnetization configurations obtained\nfrom the two different starting states as explained above. From Table 1 one can see that the\ntotal energy of the configuration with opposite rotation senses of closed magnetization states\nin Co and Py layers (Fig. 7b) is lower than the energy of the state with the same rotation sen-\nses in both layers (Fig. 4c). The total energy decrease is mainly due to the smaller exchange\nenergy (wider domain walls) and demagnetizing energy (attraction of domain wall dipoles) in\nthe 'opposite' state. However, due to the dominant contribution of the magnetocrystalline\nanisotropy energy (which is nearly equal in both cases) the total energy difference is not very\nlarge, so in experimental realizations both states can be expected.\nMagnetization dynamics: eigenmodes. The almost symmetrical equilibrium magnetization\nstate results in the excitation spectrum with much narrower peaks than for the strongly\ndisturbed asymmetrical state considered in the previous subsection. Corresponding spectrum\nof eigenmodes for the Py layer (from the Co/Cu/Py trilayer) is shown in Fig. 8 together with\nspatial maps of the oscillation power. All domain walls have now nearly the same oscillation\nfrequency (similar to the situation for the single-layer Py element), but due to the increased\nwidth of the domain walls corresponding oscillation regions are also much wider - compare\nthe 1st map on Fig. 8 with the 1st map on Fig. 2b. Broadening of domain walls manifests itself\nalso in the significant decrease of the corresponding oscillation frequency (\u0002 2.2 GHz for the\nPy layer within the trilayer vs \u0002 3.2 GHz for the single-layer Py element).\nSpectral peaks corresponding to the oscillations of domains themselves are also much narro-\nwer than for the highly asymmetrical state discussed above, so that several modes can be well\nresolved (Fig. 8). Although the oscillation frequencies for various domains coincides (within\nour resolution \u0003f ~ 0.1 GHz) and oscillation power patterns for various domains are very\nsimilar (at least for modes 2 and 3 shown in Fig. 8), the absolute values of the spatial power\nsignificantly differs from domain to domain. We attribute this effect to the random anisotropy\nfluctuations of the Co layer. It is well known that due to the small average crystallite size\nthese fluctuations are largely 'averaged out' [25]. However, remaining small fluctuations of\nthe out-of-plane magnetization in the Co layer on a large spatial scale have a significant\ninfluence on the Py layer eigenmodes due to the large saturation magnetization of Co and\nsmall spacer thickness. In particular, these fluctuations may lead to the redistribution of the\noscillation power between the domains as can be seen on the spatial power maps in Fig. 8.\nMagnetization dynamics: transient behaviour. Due to the qualitatively different equilibrium\nmagnetization state for the trilayer with oppositely oriented Landau structures in Co and Py\nlayers its transient magnetization oscillations (after the field pulse) for the finite damping case\nare also very different from both the single-layer square and the trilayer possessing Landau\nstructures with the same rotation senses in both Co and Py layers. Corresponding simulation\nresults are shown in Fig. 9 in the same format as in Fig. 6.11First of all, due to the largely restored symmetry of the equilibrium magnetization configura-\ntion, magnetization oscillations of different domain walls and different domains are now 'in-\nphase'. Due the much 'softer' magnetization configurations of the domains their oscillations\nhave now a much higher amplitude than for the trilayer with 'parallel' Landau structures\n(compare after-pulse oscillations and magnetization maps in Fig. 6 and Fig. 9). For this reason\nthe relative contribution of domain wall oscillations to the time dependence of the average\nmagnetization is almost negligible. It can be seen that domain oscillations are dominated by\nthe propagating spin wave which is excited at the square center (core of the Landau structure).\nIts wavefront has initially a nearly squared form, but when the wave propagates towards the\nelement edges, its front becomes circular. Several nodes appear along this wave front for the\nsufficiently long propagation time (see several last maps in Fig. 9) in accordance with the\nspatial power distribution of the system eigenmodes. However, the propagating time shown in\nFig. 9 is too short to establish the nodal structure with as many nodes as shown in Fig. 6.\nE. Trilayer Co/Cu/Py element: influence of the Co anisotropy type\nIt is well known that thin polycrystalline Co films may possess two kinds of the magnetocrys-\ntalline grain anisotropy, according to the two possible grain types: fcc grains have the cubic\nanisotropy Co 5\ncub610 K =× erg/cm3 (the case which was analyzed above) and hcp grains have\nthe much stronger uniaxial anisotropy Co 6\nun410 K =× erg/cm3 (see [19] and original\nexperiments in [26] for the corresponding discussion). Films with mixed fcc-hcp structure are\nalso possible. For this reasons we have studied the effect of the Co anisotropy type on the\nequilibrium magnetization structure and magnetization dynamics of our trilayer simulating the\nsystem with all parameters as given above, and with the starting Landau states with the same\nrotation sense in both magnetic layers, but with the uniaxial anisotropy of Co grains\nCo 6\nun410 K =× erg/cm3. Grain anisotropy axes were again distributed randomly in 3D.\nCorresponding results are presented in Fig. 10 and 11.\nThe major effect of such a large magnetocrystalline grain anisotropy is the strong disturbance\nof the equilibrium magnetization structure, as it can be seen from Fig. 10. Despite the small\naverage grain size 10 D\u0001\u0002= nm, the anisotropy fluctuations in Co even after their 'averaging-\nout' [25] are sufficiently strong to induce large deviations from the ideal Landau structure and\nto enforce significant randomly varying out-of-plane magnetization component on the lateral\nCo surfaces. The stray field induced on the Py layer by this Co() m^r-component nearly\ndestroys the original Landau magnetization structure of this layer, so that only the overall\nmagnetization rotation sense is preserved. The initially triangular domains of the Landau\nstructure now have a highly irregular form and only small pieces of domain walls (mainly\nnear the Py square corners) can be recognized (Fig. 10b).\nCorrespondingly, the magnetization dynamics of such a trilayer element again differs quali-\ntatively from all cases studied above (Fig. 11). Oscillations of the domain walls are almost\ninvisible. Average magnetization time dependence is entirely dominated by the circular wave\nemitted from the central vortex as shown in Fig. 11b and 11c. Due to the strongly disturbed\ndomain structure and absence of well defined domain walls (at least in the middle of the Py\nsquare) the wave front is roughly circular from the very beginning. However, irregularities of\nthe equilibrium magnetization structure within the domains lead to large modulations of the\noscillation amplitude along the wave front, as it can be recognized already for the initial stage\nof the wave propagation (Fig. 11b).12III. VERIFICATION OF SIMULATION PREDICTIONS AND COMPARISON WITH\nEXPERIMENTAL OBSERVATIONS\nAlthough, as already mentioned in the Introduction, both static magnetization structures and\nmagnetization dynamics in multilayer nanoelements have been extensively studied in the last\nseveral years, we are not aware of any experiments which could be used for direct confirma-\ntion or disprove of our simulation results.\nIn principle, our predictions concerning the equilibrium magnetization structure in square\nmkm-sized multilayer elements  (Fig. 4, 8, 10, 12) can be verified quite easily. Fabrication of\nmkm- and sub-mkm patterned multilayer elements of corresponding sizes is possible using\nseveral experimental techniques. Deformation of the 'normal' Landau structure predicted by us\nis strong enough for both the same and opposite magnetization rotation senses in individual\nlayers of the magnetic element. Hence it should be possible to detect this deformation with the\nstate-of-the-art methods for the observation of magnetization structures in nm-thick layers,\ne.g., using the meanwhile standard high-resolution MFM-facilities. We believe, that in an\narray of square nanoelements composed as described in this paper both types of the equilibri-\num magnetization states (depending on random initial fluctuations of the magnetization) will\nbe formed, so that structures shown both in Fig. 4 and Fig. 7 can be found.\nMagnetization dynamics of thin film systems can be measured nowadays not only with a very\nhigh lateral and temporal resolution, but also element-specific using the synchrotron X-ray\nradiation [24, 27, 28] with the potential resolution of several tens of nanometers. Layer-\nselective measurements are also possible using the Kerr microscopy technique [29], whereby\nthe resolution lies in the sub-mkm region (see, e.g., the recent detailed study of the magnetiza-\ntion dynamics of the Landau state for Py squares with sizes ~ 10 - 40 mkm in [32]). As\nalready mentioned in the Introduction, most papers on this topic are devoted to the\nmagnetization dynamics of mkm-sized single-layer elements. Excitations for trilayered\ncircular Py/Cu/Py nanodots at different external fields were studied in thermodynamical\nequilibrium in [30];  mainly the interlayer interaction effects due to the magnetic poles on the\nedges of nearly saturated layers have been described. There are also a few papers where the\nmagnetization switching of rectangular magnetic trilayers is studied (see, e.g., [31]), where\nthe major effect is also due to the strong stray fields induced near the edges of a nanoelement\nin a magnetically saturated state.\nWe are aware of only two experimental studies which results can be more or less directly rela-\nted to the subject of this paper, namely, the interlayer dipolar interaction dominated by the\nnearly in-plane domain walls of the closed (Landau-like) magnetization configuration. In both\ncases [24, 27] the magnetization dynamics of Co/Cu/Py square trilayers was studied using the\nelement-specific time-dependent synchrotron X-ray microscopy.\nIn the pioneering paper [24] the transient magnetization dynamics of a relatively large 4 x 4\nmkm2 trilayer Co(50nm)/Cu(2nm)/Py(50nm) was studied by the pump-and-probe X-ray\nmagnetic circular dichroism (XMCD) microscopy in the field pulse perpendicular to the\nsample plane. This technique has allowed to investigate the magnetization dynamics with the\ntemporal resolution ~ 50 ps and potential spatial resolution ~ 20 nm (however, the actual reso-\nlution achieved in [24] is much poorer and is difficult to estimate due to a significant shot\nnoise). Stoll et al. [24] did not study the equilibrium magnetization state of their system and\nhave presented spatial maps of the out-of-plane magnetization component of the Py layer only\nfor various time moments during and after the field pulse. Magnetization maps shown in [24]\nare effectively differential images between the excited and equilibrium magnetic states, so13that the contrasts due to the static domain walls and central vortex of the closed magnetization\nstructure are excluded.\nThe main qualitative features of experimental images presented in [24] are the following: (i)\nbright and relatively narrow bands along the diagonals of the squared magnetic element, i.e.,\nwhere the domain walls of the 'normal' equilibrium Landau pattern are located; (ii) at the\nsame time different domain walls exhibit contrast of different brightness and even of different\nsigns, indicating that oscillations phases and/or frequencies for different walls are different;\n(iii) after the decay of the field pulse virtually no contrast within the domains themselves can\nbe seen, so that the average out-of-plane magnetization component after the field pulse is zero\n(Py()0 m^\u0001\u0002=r ) within the experimental resolution.\nThe authors of [24] attributed the narrow contrast bands mentioned above to the domain walls\noscillations. To support their experimental findings, Stoll et al. have performed dynamic mic-\nromagnetic simulations, where they have included the Py layer only and discretized this layer\nonly in the lateral plane. Simulated out-of-plane magnetization images shown in [24]\ndemonstrate, of course, the time-dependent contrast between the oscillations of domain walls\nand domains themselves, but clearly fail to reproduce all other qualitative features of their\nexperimental images listed above.\nWe have also performed simulations of the Py single layer element with the sizes used in\n[24], discretizing it into 400 x 400 x 4 (totally 6.4\u0001105) cells. We note that due to the low\nanisotropy and relatively low saturation magnetization of  Py the size of our discretization\ncells 10 x 10 x 12.5 nm3 was small enough to reproduce main features of the Py dynamics.\nProper simulation of the magnetic trilayer with the same lateral sizes including the 50 nm\nthick Co layer would require to halve the cell size in each dimension, so that the overall cell\nnumber would be prohibitively large for the state-of-the-art micromagnetic simulations. Our\nPy() m^r-images (Fig. 12) qualitatively agree with simulation data from [24], demonstrating\nonce more that when the interaction with the Co layer is neglected, in-phase oscillations of all\nfour DWs of the Landau pattern should be observed. In addition, the strong contrast within\nthe domains after the field pulse is clearly seen in Fig. 12c, manifesting itself also in strong\nafter-pulse oscillations of the average out-of-plane magnetization Py() m t^\u0001\u0002 as shown in Fig.\n12a. The amplitude of these after-pulse oscillations is comparable with the maximal value of\nPym^\u0001\u0002 achieved during the pulse. Taking into account that the domain contrast during the\npulse is clearly seen in the experimental images presented in [24], it is unlikely that\napproximately the same contrast after the pulse would be completely overlooked. All in one,\nexperimental findings from [24] can not be explained satisfactory when the dynamics of Co\nlayer and the interlayer interaction in the trilayer Co/Cu/Py is neglected.\nTaking into account that we could not simulate (at least not with proper resolution) the comp-\nlete system studied in [24], we can compare the results of Stoll et al. with our simulation data\nonly qualitatively. First of all, we note that the straight lines corresponding to the domain wall\noscillations indicate that Co and Py layers in this experiment possess Landau magnetization\nstates with opposite rotation senses, because for the trilayer with the same magnetization\nrotation senses in both magnetic layers domain walls should be strongly bended (see Fig. 4\nand 6 above).\nFrom the remaining possibilities, dynamic magnetization images of the trilayer with 'opposite'\nLandau patterns and fcc Co crystallites (Fig. 9) demonstrate well pronounced straight lines14corresponding to the domain wall oscillations similar to those observed in [24]. However, due\nto the high symmetry of the equilibrium state, all domain walls oscillate in-phase and with the\nsame amplitude, in contrast with strongly out-of-phase DW-oscillations with different\namplitudes seen in Fig. 2 from [24]. For the same trilayer with hcp-Co magnetization\ndynamics images are asymmetric (Fig. 11), but due to the very blurred boundaries between Py\ndomains virtually no contrast is observed along the square diagonals (which would\ncorrespond to DW-oscillations).\nAt this point it should be noted that the thickness of magnetic layers studied in [24] (hCo = hPy\n= 50 nm) is twice as large as in our simulated system (hCo = hPy = 25 nm). In a system with\nsuch thick layers, domain wall structure disturbed due to the interlayer interaction, could be\npartially recovered due to thicker magnetic layers. To check this idea, we have  simulated the\nCo/Cu/Py trilayer with the same parameters and initial magnetization structure, as for the sys-\ntem shown in Fig. 10 and 11, but with the Py thickness hPy = 50 nm; here the Py layer was\ndiscretized in 8 in-plane sublayers, so that the size of the discretization cell was preserved.\nCorresponding simulation results are shown in Fig. 13 (equilibrium state) and 14\n(magnetization dynamics). One can see, that for such increased Py thickness domain walls in\nthe equilibrium magnetization state are, indeed, partially recovered (see Fig. 13), so that their\noscillations are clearly visible in the dynamic patterns (Fig. 14). Due to the remaining asym-\nmetry of the magnetization structure, oscillations of different DWs have different spatial\npatterns, amplitudes and frequencies, in a qualitative agreement with the images displayed in\n[24]. The average out-of-plane magnetization projection exhibits only very weak oscillations\nafter the field pulse, which is also in agreement with [24]. However, we observe significant\nmagnetization contrast near the square center which is due to the wave emitted by the vortex\ncore; this contrast was not found experimentally [24].\nIn the second paper mentioned above [27] the magnetization dynamics of 1 x 1 mkm2\nCo(20nm)/Cu(10nm)/Py(20nm) trilayer was studied in the in-plane pulsed magnetic field, so\nthat mainly the central vortex motion could be seen both in Co and Py layers. The authors\ndisplay also the equilibrium magnetization structures of both magnetic layers (see Fig. 1 in\n[27]), also obtained using XMCD-microscopy. Unfortunately, the resolution of these images\nis still not good enough to make any quantitative statements, so one can only say that both Co\nand Py layers possess closed magnetization structures with the same rotation senses and that\nthese structures are somewhat disturbed (compared to 'ideal' Landau patterns). However, no\nmeaningful quantitative comparison to our results presented in Fig. 4 is possible.\nIV. CONCLUSION\nIn this paper we have studied the effects of magnetodipolar interlayer interaction in trilayer\nelements with lateral sizes in sub-mkm region and magnetic layers and spacer thicknesses of\nseveral nanometers. We have shown that due to such a small interlayer distance even relative-\nly weak stray field induced by the 90o Neel domain walls of the closed magnetization state\n(Landau-like pattern) causes qualitative changes of both the equilibrium magnetization struc-\nture and magnetization dynamics in these systems. We have also demonstrated that the effect\nof such an interaction may be very different, depending not only on the initial magnetization\nstate used to find the equilibrium magnetization pattern of a system, but also on the crystallo-\ngraphic structure of magnetic layers. This random crystal grain structure significantly affects\nthe magnetization dynamics also for very small crystallite size, where the random magneto-\ncrystalline anisotropy of the grains is largely 'averaged-out'. The statement about this anisotro-\npy averaging is often used to justify the neglect of this random anisotropy when simulating15the corresponding magnetization dynamics; our results reveal that in many important cases\nsuch a neglect may be the prohibitive oversimplification of a problem.\nOur simulations clearly demonstrate that for the qualitative and especially quantitative under-\nstanding of magnetization dynamics in multilayers, magnetodipolar interlayer interaction\neffects must be included into consideration, even when the equilibrium magnetization struc-\nture forms a closed flux state and thus its stray field is believed to be relatively weak.\nAlthough we are not aware of any experimental studies which results could be directly com-\npared to our simulation data, our main predictions can be relatively easily verified with avail-\nable experimental techniques, as discussed in detail in Sec. III.. 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Schultz, Small-amplitude magnetization dynamics in\npermalloy elements investigated by time-resolved wide-field Kerr microscopy, Phys. Rev., B71 (2005) 13440518Figure captions:\nFig. 1. Geometry of the simulated system, co-ordinate axes and the pulsed field direction used\nin simulations.\nFig. 2. Magnetic properties of the single-layer squared Permalloy element (1000 x 1000 x 25\nnm3): (a) - equilibrium Landau magnetization structure in zero external field shown as grey\nscale maps of magnetization projections; (b) - spectrum of eigenmodes excited by the small\nout-of-plane pulsed homogeneous field for the state shown in (a). Grey-scale maps below the\nspectrum show the spatial distribution of the oscillation power for corresponding peaks\n(bright areas correspond to a large oscillation power)\nFig. 3. Magnetization dynamics of a single-layer Permalloy element with the same sizes as in\nFig. 2 in the pulsed out-of-plane field: (a) - time dependence of the angle \u0001perp between the\naverage layer magnetization and the element plane (perp~ ()ym Y\u0001\u0002r, see Fig. 1); the trapezoi-\ndal pulse form is shown at the same panel as thin solid line; (b) - grey scale maps of my(r)\n(out-of-plane magnetization projection) during the pulse; (c) grey scale maps of my(r) after\nthe pulse. Oscillations of both domain walls and domains themselves are clearly seen.\nFig. 4. To the formation of a static equilibrium magnetization structure in zero external field\nfor the trilayer Co/Cu/Py element with the lateral sizes 1000 x 1000 nm2, Co and Py thicknes-\nses hCo = hPy = 25 nm and spacer thickness hCu = 10 nm. The Co layer possesses a fcc poly-\ncrystalline structure with the average crystallite size 10 D\u0001\u0002= nm and cubic grain anisotropy\nKcub = 6\u0001105 erg/cm3. (a) - initial magnetization structure used as the starting state by the\ncalculation of the equilibrium structure shown in (c) as grey scale maps of magnetization\nprojections; (b) initial distribution of the surface charges responsible for the repulsion of 90o\ndomain walls initially located along the main diagonals of the square in Co and Py layers.\nFig. 5. Eigenmodes spectrum for a Permalloy layer of the trilayer element with the static\nmagnetization structure shown in Fig. 4c. Due to the symmetry breaking of the underlying\nmagnetization state spectral lines corresponding to the oscillations of different domain walls\nare positioned at different frequencies (spectral group 1). Spectral peaks corresponding to the\ndomain oscillations form two quasi-continuous groups (groups 2 and 3), whereby each line\nwithin a group corresponds to magnetization oscillations within a specific domain as shown\nby grey-scale maps of the oscillation power distributions below.\nFig. 6. Magnetization dynamics of a trilayer Co/Cu/Py element with the static magnetization\nstructure shown in Fig. 4c in the pulsed out-of-plane field: (a) - time dependence of the angle\n\u0001perp between the average magnetization and the element plane for the magnetization of the\ntotal element (thick green line), Co layer (thin blue line) and Py layer (thick dashed line).\nGrey scale maps of the out-of-plane magnetization projection of the Py layer during the pulse\n(b) and after the pulse (c). Due to different eigenfrequencies oscillations of different domain\nwalls are out-of-phase here and oscillation of domains themselves are strongly suppressed\ncompared to the case of a single-layered Py element (Fig. 3c).\nFig. 7. Static equilibrium magnetization structure in zero external field for the same Co/Cu/Py\nelement as shown in Fig. 4, but starting from Landau magnetization states with opposite\nrotation senses in Co and Py layers (a). Resulting equilibrium state is shown at the panel (b)19as grey scale maps of magnetization projections. It can be seen that due to the attraction of\ndomain walls for the starting magnetization states the symmetry of the final equilibrium\nmagnetization structure is nearly preserved.\nFig. 8. Eigenmodes spectrum for a Permalloy layer of the trilayer element with the static\nmagnetization structure shown in Fig. 7b. Due to the largely preserved symmetry of domain\nwalls their oscillations have nearly the same frequency (spectral line 1 and grey-scale map 1).\nOscillation power distribution in domain regions (maps 2-4) is still asymmetric due to magne-\ntodipolar interaction with the Co layer which magnetization has a noticeable and spatially\nvarying out-of-plane component due to a significant magnetocrystalline anisotropy (fcc Co).\nFig. 9. Magnetization dynamics of a trilayer Co/Cu/Py element with the static magnetization\nstructure shown in Fig. 7b in the pulsed out-of-plane field presented in the same way as in\nFig. 6. It can be seen that the magnetization dynamics is largely dominated by the propagation\nof the spin wave excited at the central vortex; its wave front has initially the square shape\nwhich transforms during the propagation into a nearly circular one.\nFig. 10. Static equilibrium magnetization state for Hext = 0 for the same Co/Cu/Py element as\nshown in Fig. 7 (starting from Landau magnetization states with opposite rotation senses in\nCo and Py layers - see (a)) but with the Co layer having a hcp polycrystalline structure with\nthe uniaxial grain anisotropy Kun = 4\u0001106 erg/cm3. Due to such a large random anisotropy\nvalue the symmetry of the final magnetization state is strongly disturbed (b) and boundary\nregions between the domains are very wide.\nFig. 11. Magnetization dynamics of a trilayer Co/Cu/Py element with the static magnetization\nstructure shown in Fig. 10b in the pulsed field presented in the same way as in Fig. 9. Due to\nthe strong disturbance of the static magnetization state the oscillations of domain walls are\nnearly invisible. Although the front of the dominating spin wave remains approximately\ncircular, the wave amplitude shows significant inhomogeneities along this wave front.\nFig. 12. Simulated transient magnetization dynamics for a single-layer Py element with lateral\nsizes 4 x 4 mkm2 and thickness hPy = 50 nm (after the same field pulse and presented in the\nsame way as in Fig. 3). Strong magnetization oscillations within the domain regions after the\nfield pulse can be seen.\nFig. 13. Static equilibrium magnetization state for Hext = 0 for the same Co/Cu/Py element as\nshown in Fig. 10 (starting from Landau states with opposite rotation senses in Co and Py\nlayers - see (a)) but with the thicker Py layer: hPy = 50 nm. Due to the increased Py layer\nthickness domain walls within this layer are partially recovered (see the grey-scale map of\nmy(r) for Py in Fig. 13b).\nFig. 14. Magnetization dynamics of a trilayer Co/Cu/Py element with the static magnetization\nstate from Fig. 13b presented in the same way as in Fig. 11. In contrast to the case shown in\nFig. 11, oscillations of domain walls can be clearly seen. However, these oscillations remain\nstrongly asymmetric, what can be seen especially well on the magnetization maps after the\nfield pulse (c).20xy\nzCoPy\nhCohPy\ndHext\nFig. 1\nf, GHz\n0 2 4 6 8 10 12 14Pav(my(r))\n010x10-620x10-630x10-640x10-6\n(a)\n(b)\nFig. 2\n21t, ns\n0.0 0.5 1.0 1.5 2.0 2.5 3.0Yperp\n0.0000.0050.010\n(a)\n(b)\n(c)\nFig. 322Co Py\n(a)\n(c)\n(b)\nFig. 423\nf, GHz\n0 2 4 6 8 10 12 14<P(my(r))>\n05x10-610x10-615x10-6\n   1\na b c     2\na  b  c\n     3\na  b  c\n1:\n2:\n3:a b c\na b\nbc a\nFig. 524t, ns\n0.0 0.5 1.0 1.5 2.0Yperp\n0.0000.0050.010 Total magn.\nCo layer\nPy layer(a)\n(c)\n(b)\nFig. 625(a)\n(b)Co Py\nFig. 7\nf, GHz\n0 2 4 6 8 10 12<P(my(r))>\n050x10-6100x10-6\n1\n23 4\n1 2 3 4\nlog(P)\nFig. 826\nt, ns\n0.0 0.5 1.0 1.5 2.0Yperp\n-0.0050.0000.0050.0100.015\nTotal element \nCo layer\nPy layer\n(a)\n(c)(b)\nFig. 927\n(a)\nCo Py\n(b)\nFig. 1028t, ns0.0 0.5 1.0 1.5 2.0Yperp\n-0.0050.0000.0050.0100.015\nTotal element \nCo layer\nPy layer\n(a)\n(c)(b)\nFig. 1129t, ns\n0.0 0.5 1.0 1.5 2.0 2.5 3.0Yperp\n-0.0050.0000.0050.0100.015\n(a)\n(c)\n(b)\nFig. 1230(a)\nCo Py\n(b)\nFig. 1331t, ns\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Yperp\n0.0000.0050.010Total element \nCo layer\nPy layer\n(a)\n(c)(b)\nFig. 14"    },    {        "title": "1806.05395v1.Boundary_effects_on_the_magnetic_Hamiltonian_dynamics_in_two_dimensions.pdf",        "content": "BOUNDARY EFFECTS ON THE MAGNETIC HAMILTONIAN\nDYNAMICS IN TWO DIMENSIONS\nTHO.NGUYEN DUC, NICOLAS RAYMOND, SAN V ˜U NGO.C\nAbstract. We study the Hamiltonian dynamics of a charged particle submitted to a pure\nmagnetic field in a two-dimensional domain. We provide conditions on the magnetic field\nin a neighbourhood of the boundary to ensure the confinement of the particle. We also\nprove a formula for the scattering angle in the case of radial magnetic fields.\n1.Introduction\n1.1.Magnetic Hamiltonian dynamics. This article is concerned with the dynamics of a\ncharged particle in a smooth bounded domain \n\u001aR2in the presence of a non homogeneous\nmagnetic field B. The motion of a particle of charge eand massmunder the action of the\nLorentz force can be expressed by Newton’s equation\nmq=e_q\u0002B; (1.1)\nwhereq= (q1;q2;q3)T2R3. To simplify our discussion, we assume that e= 1andm= 1.\nThe vector field B, defined on \n, is assumed to be smooth and to satisfy the Maxwell\nequationr\u0001B= 0. For our target problem in two dimensions, we suppose that Bis\nperpendicular to the plane R2,i.e.,B(q) = (0;0;b(q)). This assumption forces particles\nlying in the R2plane and whose initial velocities are in the plane to stay in this same plane\nfor all time. Since a vector field in R3can be identified with a 2-form, we write the magnetic\nfield asB=b(q)dq1^dq2. Then, if there is a 1-form A=A1dq1+A2dq2such that d A=B,\nwe can write (1.1) in Hamiltonian form. Consider, for all (q;p)2R2\u0002R2,\nH(q;p) =kp\u0000A(q)k2\n2; (1.2)\nwherek:kdenotes the Euclidean norm on R2.\nThe matrix representing the right cross product with Bin the canonical basis is\nMB=JT\nA\u0000JA;\nwhereJAis the Jacobian matrix of A. Hence Newton’s equation (1.1) becomes\nq=MB_q;\nso thatd\ndt( _q+A(q)) =JT\nA_q:\nBy introducing the momentum variable p= _q+A(q), we see thatH(q;p) =1\n2k_qk2is the\nkinetic energy of the system, and (q;p)evolves according to the Hamiltonian flow associated\nwithH: \u001a_q=@pH(q;p)\n_p=\u0000@qH(q;p): (1.3)\nWe shall always assume that q7!b(q)is locally Lipschitz-continuous, ensuring that the\nsystem (1.3) has a unique local maximal solution, thanks to the Cauchy-Lipschitz theorem.\nThen, the vector potential Awill always be chosen to be C1-smooth.\n2010Mathematics Subject Classification. 70H05, 37N05.\nKey words and phrases. Magnetic Hamiltonian, dynamics, confinement, scattering, boundary.\n1arXiv:1806.05395v1  [math.DS]  14 Jun 20182 D. T. NGUYEN, N. RAYMOND, S. V ˜U NGO.C\n1.2.Two questions. From now on, we call bthe magnetic field and it is identified with\nthe2-form\nb(q1;q2)dq1^dq2=d(A1dq1+A2dq2):\nThis article addresses two classical dynamical problems: confinement and scattering.\n- (Confinement) Consider a charged particle in the magnetized region \n. A natural question\nis the following:\n“Will the particle reach the boundary in finite time?”\nWe will provide a precise answer to this question, depending on the behaviour of the\nmagnetic field at the boundary and on the initial conditions. Our results will improve\nrecent results by Martins in [4]. In particular, we will see that, even if the magnetic field\nis infinite at the boundary, some trajectories can escape from \n. This kind of (open)\nproblems is mentioned in [2, Section 1.4].\n- (Scattering)Considerachargedparticleoutsidethemagnetizedregion \n. Beforeitreaches\nthe region \n, the trajectory is a straight line. If it enters the region \n, does the particle\nescape from it in finite time? And, if it does so, what is the deviation angle between the\ningoing and outgoing directions? We will explicitly answer these questions in the case of\nradial magnetic fields and when \nis a disc. In this case, the angular momentum commutes\nwith the Hamiltonian and allows a reduction to a one degree of freedom system.\nFor both problems, we provide numerical illustrations of our results.\nThese questions have intrinsic physical motivations. Their answers allow a better under-\nstanding of the classical dynamics of charged particles in magnetic fields. The description\nof the classical trajectories has also many applications, for instance, at the quantum level.\nThe quantum aspect of the trapped trajectories can be related to the essentially self-adjoint\ncharacter of the magnetic Laplacian (see [2, 5, 6, 8]). It is also a key point to describe\nthe spectrum/resonances of magnetic Laplacians. As far as the authors know, whereas the\ndescription of the magnetic dynamics has allowed to estimate the spectrum of magnetic\nLaplacians (see [7, 3]), no result seems to exist to estimate their resonances near the real\naxis. Investigating the trapped trajectories is a necessary step in this direction.\nIn the regime of large magnetic field and small energy, a special treatment of the con-\nfinement problem can be done and takes advantage of the near-integrable structure of the\nHamiltonian dynamics, either via Birkhoff normal form [7], or KAM theorems [1]. On the\ncontrary, our results here will give more explicit initial conditions and allow regimes where\nthe guiding center motion is not necessarily meaningful.\n1.3.Organization of the article. The article is organized as follows. In Section 2, we\nstate our main results about confinement and scattering. Section 3 is devoted to the proofs.\n2.Statements\n2.1.Confinement problem.\n2.1.1.Tubular coordinates. Inordertostateourresults, itisconvenienttointroducetubular\ncoordinates near the boundary of \n, following the analysis of [4].\nWe assume that the connected components of @\nareC2-smooth closed curves without\nself-intersections. Let Cbe a connected component of @\n. It can be parametrized by its arc\nlength\r:R=LZ!CwhereLis the length ofC.\nThere exists \u000e>0such that\n :\u001a(0;\u000e)\u0002R=LZ!\nC(\u000e)\n(n;s)7!\r(s) +nN(s) =q(2.1)\nis a smooth diffeomorphism. N(s)denotes the inward pointing normal at \r(s)and\n\nC(\u000e) =fq2\n :d(x;C)<\u000eg:BOUNDARY EFFECTS ON THE MAGNETIC HAMILTONIAN DYNAMICS 3\nNote that\nB=b(q)dq1^dq2=b( (n;s))(1\u0000n\u0014(s))ds^dn; (2.2)\nwhere\u0014(s)is the signed curvature of Cat\r(s). In these coordinates, we can write\nA=An(n;s)dn+As(n;s)ds\nwithAn;Asdefined on (0;\u000e)\u0002R=LZsuch that\n@As\n@n\u0000@An\n@s=:B(n;s) =\u0000b( (n;s))(1\u0000n\u0014(s)): (2.3)\nVia the tubular coordinates, we can define the symplectic change of coordinates\n\t :(\n(0;\u000e)\u0002R=LZ\u0002R2!\nC(\u000e)\u0002R2\n(n;s;pn;ps)7!( (n;s);((d )\u00001\n(n;s))T(pn;ps)) = (q;p);(2.4)\nwhere we have explicitly p= (1\u0000n\u0014(s))\u00001ps\r0(s) +pnN(s).\nThe Hamiltonian takes the form (see Lemma A.1):\nH(n;s;pn;ps) =1\n2(pn\u0000An(n;s))2+(ps\u0000As(n;s))2\n2(1\u0000\u0014(s)n)2: (2.5)\n2.1.2.General confinement theorems. We can now state our confinement results. Our first\ntheorem provides a sufficient condition on Bso that no trajectory can escape from \n.\nTheorem 2.1. For every connected component Cof@\n, we assume that\nlim\nn!0\f\f\f\fZ\u000eC\nnZLC\n0B(\u0011;\u0018)d\u0018d\u0011\f\f\f\f= +1; (2.6)\nand that there exists MC\u00150such that, for all (n;s)2(0;\u000eC)\u0002R=LCZ,\n\f\f\f\fB(n;s)\u00001\nLCZLC\n0B(n;\u0018)d\u0018\f\f\f\f\u0014MC: (2.7)\nThen the magnetic Hamiltonian dynamics is complete (i.e. no solution of (1.3), starting in\n\n, reaches@\nin finite time).\nOf course, given a starting point q2\n, only the components Cthat bound the connected\ncomponent of qin\nneed to be taken into account. Actually, there is a more quantitative\nversion of the previous theorem.\nTheorem 2.2. Consider a connected component Cof@\n. Let\nK= sup\ns2R=LZj\u0014(s)j; K0= sup\ns2R=LZj\u00140(s)j:\nWe assume that, for some \u000f2(0;1),\u000esatisfies 0<\u000e\u0014\u000f=K. We assume that there exists\nM\u00150such that, for all (n;s)2(0;\u000e)\u0002R=LZ,\n\f\f\f\fB(n;s)\u00001\nLZL\n0B(n;\u0018)d\u0018\f\f\f\f\u0014M: (2.8)\nConsiderT > 0andq(t) = (n(t);s(t))a trajectory contained in \nC(\u000e)fort2[0;T]with\nenergyH0. Let\nf(n) =\u00001\nLZ\u000e\nnZL\n0B(\u0011;\u0018)d\u0018d\u0011 (2.9)\nand assume that\nlim inf\nn!0jf(n)j>C(T); (2.10)4 D. T. NGUYEN, N. RAYMOND, S. V ˜U NGO.C\nwhere\nC(T) =\f\f\f\f_s(0)[1\u0000\u0014(s(0))n(0)] +Z\u000e\nn(0)ZL\n0B(\u0011;\u0018)d\u0018d\u0011\f\f\f\f\n+p\n2H0(1 +\u000f) +\u0012\nMp\n2H0+2H0K0N\n1\u0000\u000f\u0013\nT:\nLetg1be a continuous and strictly decreasing function such that\nlim\nn!0g(n) = lim inf\nn!0jf(n)j; g\u0014jfjon [0;\u000e]:\nThen,gtakes the value C(T)and, for all t2[0;T),\nn(t)>g\u00001(C(T)): (2.11)\nRemark 2.1.Theorems 2.1 and 2.2 are improvements of [4, Theorems 1&2]. They tell us\nthat a particle in \nnever reaches the boundary of \n. In [4], it is assumed that @sBis\nintegrable:\nsup\ns2CZN\n0j@sB(m;s)jdm< +1; (2.12)\nand the question of removing this assumption was explicitly mentioned as important ( op.\ncit., section 3 ). Our theorems give a partially positive answer to this question, thus allowing\nfor magnetic fields having wilder tangential behaviors.\n- Theorem 2.1 generalizes [4, Theorem 1] by replacing the integrability assumption by (2.7).\nThis allows in particular to consider a magnetic field (on the unit disc) of the form\nB(n;s) =1\nn+ sin\u0012\u001f(s)\nn\u0013\n;\nwhere\u001fis a smooth function supported in (\u0000\u0019;\u0019)such that\u001f0(0)6= 0and\u001f(0) = 0.\nFor this magnetic field, it is easy to check that (2.12) is not satisfied. In fact, the C1\nsmoothness is actually not required; in order to draw Figure 1, we took, for simplicity, a\nsmall perturbation of \u001f(s) = arcsin(sin( s)).\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81\nFigure 1. A trajectory obtained with a magnetic field on the unit disc that\nis strong near the boundary with a non-integrable tangential derivative:\nB(q) =1\n1\u0000p\nq2\n1+q2\n2+ sin \narcsin(q2)\n1\u0000p\nq2\n1+q2\n2!\n+ 5q3\n1\u00007q2:\n1such a function g always exists.BOUNDARY EFFECTS ON THE MAGNETIC HAMILTONIAN DYNAMICS 5\n- Anexplicitlowerboundfortheescapingtimeofamagnetizedregionisgivenin[4,Theorem\n2] in the case when\nB(n;s) =M\nn\u000b+h(n;s); \u000b\u00151: (2.13)\nwhereM6= 0andhis bounded and smooth in \nC(\u000e), and so that (2.12) holds. Theorem\n2.2 implies [4, Theorem 2], and also provides an explicit lower bound for magnetic fields\nthatarenotintheform(2.13), seeFigure2wherethemagneticfieldchangessigninfinitely\nmany times.\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81\nFigure 2. A trajectory obtained with a magnetic field on the unit disc that\nstrongly oscillates near the boundary:\nB(q) =1\n2\u0000sin\u0012\n1\n1\u0000p\nq2\n1+q2\n2\u0013\n(1\u0000p\nq2\n1+q2\n2)2+ 10q1\u00002q2\n1\u000010q2\n2:\n2.1.3.Confinement results in the radial case. When \n =D(0;1)and whenBis radial, the\ndynamics is completely integrable, and hence can be entirely described by a one degree of\nfreedom Hamilonian; concerning the confinement problem, this of course leads to stronger\nresults.\nProposition 2.3. Letq(t) = (q1(t);q2(t))be a solution to (1.3)starting att= 0from inside\nthe unit disc. If the initial data (q(0);_q(0))satisfies either H1orH2below:\nH1:\nlim inf\nr!1\u0000\f\f\f\f1\n2\u0019Z\nkq(0)k\u0014kqk\u0014rB(q)dq\u0000det(q(0);_q(0))\f\f\f\f>k_q(0)k; (2.14)\nH2:\nlim inf\nr!1\u0000\f\f\f\f1\n2\u0019Z\nkq(0)k\u0014kqk\u0014rB(q)dq\u0000det(q(0);_q(0))\f\f\f\f=k_q(0)k; (2.15)\nand\nlim sup\nr!1\u0000\f\f\f1\n2\u0019R\nkq(0)k\u0014kqk\u0014rB(q)dq\u0000det(q(0);_q(0))\f\f\f\u0000k_q(0)k\nr\u00001<0; (2.16)\nthen the solution exists for all t\u00150, and there exists \u00112[0;1)such that\n8t\u00150;kq(t)k<\u0011: (2.17)6 D. T. NGUYEN, N. RAYMOND, S. V ˜U NGO.C\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81\nFigure 3. B(r) =e\u0000r\u00002\nr.\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81\nFigure 4. B(r) = ln2(1\u0000r): the particle is confined or not.\nOne can find situations where none of the hypothesis of Proposition 2.3 hold and the\ntrajectory can be arbitrarily close to the boundary. (see Figure 3: this unusual behavior\ncan be explained by a critical point of the radial Hamiltonian at r= 1, see (2.20)).\nIf the magnetic field is L1-integrable near the boundary of \n, we can prove that there\nexist trajectories escaping from \nin finite time. In particular, even if the magnetic field is\ninfinite at the boundary, the confinement is not ensured.\nProposition 2.4. When\nlim sup\nr!1\u0000\f\f\f\fZ\nD(0;r)B(q)dq\f\f\f\f<+1; (2.18)\nthere exists a trajectory starting in \nand reaching the boundary in finite time.\nOf course, even under assumption (2.18), some trajectory may be confined, depending on\ninitial conditions (see Figure 4 where the simulations are performed with B(r) = ln2(1\u0000r)).\n2.2.Scattering in the radial case. Let us now describe our scattering result in the radial\ncase. We assume that Bj\nadmits a locally Lipschitz extension in a neighbourhood of \n.\nIn polar coordinates, we have\nB=B(r)rdr^d\u0012=d(G(r)d\u0012);\nwhere\nG(r) =Zr\n0\u001cB(\u001c)d\u001c:\nVia the symplectic change of coordinates\nR\u0003\n+\u0002R=2\u0019Z\u0002R2!(Dnf0g)\u0002R2\n(r;\u0012;pr;p\u0012)7!\u0012\nrcos\u0012;rsin\u0012;cos\u0012pr\u0000sin\u0012\nrp\u0012;sin\u0012pr+cos\u0012\nrp\u0012\u0013\n= (q;p);BOUNDARY EFFECTS ON THE MAGNETIC HAMILTONIAN DYNAMICS 7\nthe Hamiltonian becomes\n~H(r;\u0012;pr;p\u0012) =p2\nr\n2+(p\u0012\u0000G(r))2\n2r2; (2.19)\nIn particular, the angular momentum p\u0012is constant along the flow and we consider the\nreduced one dimensional Hamiltonian on T\u0003R\u0003\n+\nH(r;pr) :=p2\nr\n2+V(r); V (r) :=(p\u0012\u0000G(r))2\n2r2; (2.20)\nwhereV2C1(R\u0003\n+). We notice that (see, for example, Lemma A.1)\nvr=pr; v\u0012=r\u00001(p\u0012\u0000G(r));\nwherevrandv\u0012are the classical radial and tangential components of the velocity v.\nWe consider a charged particle with energy H0arriving into the disk with velocity v1. In\nparticular,H0=1\n2kv1k2. If the particle escapes from the disc with velocity v2(see Figure 5),\nwe havekv2k=kv1k, and a natural question is to compute the (scattering) angle between\nthese two vectors. Let !2(\u0000\u0019;\u0019]be the oriented angle between v1andv2.\nTheorem 2.5. Consider a trajectory starting on @\n, with velocity v16= 0and entering \n.\nThis means that either vr<0, orvr= 0andB(1)\nv\u0012<\u00001. We define \rthe angle between the\noutward pointing normal and v1.\nWe also assume\ni.either that the equation V(r) =H0has a solution for r2(0;1)and that the closest\nsolution to 1, denoted by r\u0003, satisfiesV0(r\u0003)<0.\nii.or, only when p\u0012= 0, that the equation V(r) =H0has no solution.\nThen the trajectory escapes from \nin finite time with velocity v2, and we can compute\nthe scattering angle !mod 2\u0019:\ni.either the trajectory does not pass through the origin and\n!=\u000b+\u0019\u00002\r;\nwhere\n\u000b= 2Z1\nr\u0003p\u0012\u0000G(r)\nrp\n2H0r2\u0000(p\u0012\u0000G(r))2dr; (2.21)\nii.or the trajectory passes through the origin (in this case p\u0012= 0) and\n!=\u000b\u00002\r;\nwhere\n\u000b= 2Z1\n0\u0000G(r)\nrp\n2H0r2\u0000G(r)2dr: (2.22)\n3.Proofs\n3.1.Proof of Theorems 2.1 and 2.2. To reach the boundary, the particle has to be close\nto a connected component Cof@\n. Thus, we can assume that, for all t2[0;T),\nq(t)2\nC(\u000e):\nModifying the vector potential corresponds to a symplectic transformation of the form\n(q;p)7!(q;p+dS(q)), forsomesmoothfunction S, andhencedoesnotmodifythetrajectory\nof the particle. Thus, we consider the function\n\u000b(n;s) =s\nLZL\n0B(n;\u0018)d\u0018\u0000Zs\n0B(n;\u0018)d\u0018:8 D. T. NGUYEN, N. RAYMOND, S. V ˜U NGO.C\n~ v1\n~ v2r\u0003\u00121\u00122\r\n\u000b\nFigure 5. The scattering arrows.\nNotice that \u000b(n;\u0001)isL-periodic. Recalling (2.9) and letting A=\u000b(n;s)dn+f(n)ds, we\nhaveB=dA.\nBy (2.5), the corresponding Hamiltonian is\nH(n;s;pn;ps) =(pn\u0000\u000b(n;s))2\n2+(ps\u0000f(n))2\n2(1\u0000\u0014(s)n)2:\nConcerning Hamilton’s equations, we have in particular\n_n=pn\u0000\u000b(n;s); _ps=~B(n;s) _n\u0000(ps\u0000f(n))2\n(1\u0000\u0014(s)n)3\u00140(s)n;\nwhere\n~B(n;s) =1\nLZL\n0B(n;\u0018)d\u0018\u0000B(n;s):\nWe recall that, for all t2[0;T),H(n(t);s(t);pn(t);ps(t)) =H0. We get\nj_nj\u0014p\n2H0\njps\u0000f(n)j\u0014p\n2H0(1 +\u000f)\f\f\f\f(ps\u0000f(n))2\n(1\u0000\u0014(s)n)3\u00140(s)n\f\f\f\f\u00142H0K0\u000e\n1\u0000\u000f;(3.1)\nwhere in the last estimates we have used the notation of Theorem 2.2 and in particular\nj\u0014jn\u0014K\u000e\u0014\u000f. With our assumption (2.8) on ~B(n;s), we find, for all t2[0;T),\njps(t)j\u0014jps(0)j+\u0012\nMp\n2H0+2H0K0\u000e\n1\u0000\u000f\u0013\nT;\nand thus\njf(n(t))j \u0014 jps(t)j+jps(t)\u0000f(n(t))j\u0014C(T); (3.2)BOUNDARY EFFECTS ON THE MAGNETIC HAMILTONIAN DYNAMICS 9\nwith\nC(T) =jps(0)j+p\n2H0(1 +\u000f) +\u0012\nMp\n2H0+2H0K0\u000e\n1\u0000\u000f\u0013\nT:\nIf the trajectory reaches the boundary at t=T, then\nlim\nt!Tn(t) = 0:\nThis, with (3.2) and (2.6), gives a contradiction. This proves Theorem 2.1.\nNow, consider a function gas in Theorem 2.2. We have, for all t2[0;T),\ng(n(t))\u0014jf(n(t))j\u0014C(T):\nFrom (2.10), we have limn!0g(n)> C (T); hencegmust take the value C(T)and the\nconclusion follows.\n3.2.Proof of Proposition 2.3. Let us recall (2.20). The assumptions of Proposition 2.3\ncan be written in terms of V.\n(H1) If\nlim inf\nr!1\u0000V(r)>H 0; (3.3)\nweconsider \u0011= supfx2(0;1) :V(x) =H0g2(0;1). Consideratrajectory (q(t);p(t))\nwithq(0)2D(0;1). We can assume that q(0)6= 0. LetTbe the maximal time of\nexistence in D(0;1). By energy conservation, we have, for all t2[0;T),\nV(r(t))\u0014H0;\nso thatr(t)\u0014\u0011.\nNote that (3.3) means\nlim inf\nr!1\u0000jG(r)\u0000p\u0012j>p\n2H0:\nUsing the usual complex coordinate in the plane R2, we can write _q=\u0010\n_r+i_\u0012r\u0011\nei\u0012\nand thus\ndet(q(t);_q(t)) =r2(t)_\u0012(t) =p\u0012\u0000G(r(t)):\nFinally, we notice that k_q(0)k=p2H0and write\nG(r)\u0000p\u0012=G(r)\u0000G(r(0))\u0000[p\u0012\u0000G(r(0))];\nwhich gives (2.14).\n(H2) If\nlim inf\nr!1\u0000V(r) =H0; (3.4)\nand\nlim sup\nr!1\u0000V(r)\u0000H0\nr\u00001<0;\nthen we must again have\nsupfx2(0;1) :V(x) =H0g<1;\nand we can proceed as above.10 D. T. NGUYEN, N. RAYMOND, S. V ˜U NGO.C\n3.3.Proof of Proposition 2.4. Considerp\u0012= 0. LetjVj1:= supr2(0;1)jV(r)j. By\nassumption,jVj1<+1.\nLetr(0)2(0;1)and choose pr(0)>0such thatp2\nr(0) = 2 (jVj1\u0000V(r(0))) +v2, with\nv>0. Since, for all t2[0;T),\np2\nr(t)\n2+V(r(t)) =p2\nr(0)\n2+V(r(0));\nwe get _r(t) =pr(t)>vso that\nr(t)>vt +r(0):\nThe particle escapes at t=1\u0000r(0)\nv.\n3.4.Proof of Theorem 2.5. We distinguish between the cases p\u0012= 0andp\u00126= 0.\n3.4.1.Case when p\u00126= 0.In this case, limr!0V(r) = +1; hence, due to energy conserva-\ntion, the trajectory does not approach the origin.\ni. Assume that pr(0)<0. We haveV(1)<H 0and we can consider the right most turning\npointr\u00032(0;1). By definition V(r\u0003) =H0, and necessarily V0(r\u0003)\u00140.\nIfV0(r\u0003)<0, it is easy to check that rreachesr\u0003in finite time, say t=t\u0003. This time\nis given by\nt\u0003=Z1\nr\u0003drp\n2(H0\u0000V(r)):\nBy symmetry, the escape time is 2t\u0003. Since _\u0012=p\u0012\u0000G(r)\nr2, we have\n\u0012(t\u0003)\u0000\u0012(0) =Zt\u0003\n0p\u0012\u0000G(r)\nr2dt=Zt\u0003\n0(p\u0012\u0000G(r)) _r\nr2prdt=Zt\u0003\n0\u0000(p\u0012\u0000G(r)) _r\nr2p\n2(H0\u0000V(r))dt;\nso that\n\u0012(t\u0003)\u0000\u0012(0) =Z1\nr\u0003p\u0012\u0000G(r)\nr2p\n2(H0\u0000V(r))dr:\nBy symmetry, we have\n\u0012(2t\u0003)\u0000\u0012(0) = 2Z1\nr\u0003p\u0012\u0000G(r)\nr2p\n2(H0\u0000V(r))dr:\nIfV0(r\u0003) = 0,(r\u0003;0)is a critical point of the Hamiltonian and we get that rreaches\nr\u0003in infinite time (see Figure 6).\nii. Assume that pr(0) = 0. ThenV(1) =H0. By assumption (the trajectory enters\nD(0;1)), we haveV0(1)\u00150,i.e.,(p\u0012\u0000G(1))B(1) + (p\u0012\u0000G(1))2\u00140. IfV0(1) = 0, the\nparticle sits at a fixed point of the Hamiltonian system, and hence r(t)\u00111is constant.\nIfV0(1)>0, it entersD(0;1)and the discussion is the same as previously.\n3.4.2.Case when p\u0012= 0.In this case, since G(0) = 0,V(r) =1\n2r2G(r)2admits a continuous\nextension at r= 0.\ni. Assume that pr(0)<0. We haveV(1)<H 0. The existence of r\u0003such thatV(r\u0003) =H0\nis not ensured. If V(r)<H 0on[0;1], the particle reaches r= 0in finite time t=t\u0003:\nt\u0003=Z1\n0drp\n2(H0\u0000V(r)):\nWe get, by symmetry,\n\u0012(2t\u0003)\u0000\u0012(0) = 2Z1\n0\u0000G(r)\nr2p\n2(H0\u0000V(r))dr+\u0019:BOUNDARY EFFECTS ON THE MAGNETIC HAMILTONIAN DYNAMICS 11\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81\nFigure 6. B(r) =e\u0000r\u00002\nr.\nIf there exists r\u00032(0;1)such thatV(r\u0003) =H0, the trajectory does not reach the origin\nand the discussion is the same as in the case p\u00126= 0.\nii. Assume that pr(0) = 0. The discussion is the same as when p\u00126= 0.\n3.4.3.Scattering angle. We can now end the proof of Theorem 2.5. In terms of complex\nnumbers, we can write\nv1= (vr(0) +iv\u0012(0))ei\u00121; v 2= (\u0000vr(0) +iv\u0012(0))ei\u00122:\nThe scattering angle is\n\u00122\u0000\u00121+ Arg\u0012\u0000vr(0) +iv\u0012(0)\nvr(0) +iv\u0012(0)\u0013\n:\nIf\rdenotes the argument of vr(0) +iv\u0012(0), the scattering angle is thus\n\u00122\u0000\u00121+\u0019\u00002\r:\nAppendix A.Tubular coordinates\nLemma A.1. We write A=A1dq1+A2dq2. With(2.1), we have\nA=Andn+Asds; ~A= (An;As)T= (d )T(A1;A2)T:\nWe have\nH(n;s;pn;ps) =H\u000e\t(n;s;pn;ps) =(pn\u0000An(n;s))2\n2+(ps\u0000As(n;s))2\n2(1\u0000\u0014(s)n)2:(A.1)\nMoreover,vn=pn\u0000An(n;s)andvs= (1\u0000n\u0014(s))\u00001(ps\u0000As)are the normal and tangential\ncomponent of v.\nProof.We write\n2H(q;p) =kp\u0000Ak2=k(d \u00001)T(~p\u0000~A)k2=h(d \u00001)(d \u00001)T(~p\u0000~A);~p\u0000~Ai;\nwith ~p= (pn;ps)T. Note that\n(d \u00001)T= [N(s);(1\u0000n\u0014(s))\r0(s)]: (A.2)\nWe get\n(d \u00001)(d \u00001)T=\u0012\n1 0\n0 (1\u0000n\u0014(s))\u00002\u0013\n:\nConcerning the velocity v, we write\nv=p\u0000A= (d \u00001)T(~p\u0000~A);12 D. T. NGUYEN, N. RAYMOND, S. V ˜U NGO.C\nand we use (A.2). \u0003\nReferences\n[1] C. Castilho. The motion of a charged particle on a Riemannian surface under a non-zero magnetic field.\nJ. Differential Equations , 171(1):110–131, 2001.\n[2] Y. Colin de Verdière and F. Truc. Confining quantum particles with a purely magnetic field. Ann. Inst.\nFourier (Grenoble) , 60(7):2333–2356 (2011), 2010.\n[3] B. Helffer, Y. Kordyukov, N. Raymond, and S. V˜ u Ngo .c. Magnetic wells in dimension three. Anal. PDE ,\n9(7):1575–1608, 2016.\n[4] G.Martins.TheHamiltoniandynamicsofplanarmagneticconfinement. Nonlinearity , 30(12):4523, 2017.\n[5] G. Nenciu and I. Nenciu. On confining potentials and essential self-adjointness for Schrödinger operators\non bounded domains in Rn.Ann. Henri Poincaré , 10(2):377–394, 2009.\n[6] G. Nenciu and I. Nenciu. On essential self-adjointness for magnetic Schrödinger and Pauli operators on\nthe unit disc in R2.Lett. Math. Phys. , 98(2):207–223, 2011.\n[7] N. Raymond and S. V˜ u Ngo .c. Geometry and spectrum in 2D magnetic wells. Ann. Inst. Fourier (Greno-\nble), 65(1):137–169, 2015.\n[8] M. Reed and B. Simon. Methods of Modern Mathematical Physics: Fourier analysis, self-adjointness .\nMethods of Modern Mathematical Physics. Academic Press, 1975."    },    {        "title": "0811.0999v1.Magnetic_exchange_interaction_between_rare_earth_and_Mn_ions_in_multiferroic_hexagonal_manganites.pdf",        "content": "arXiv:0811.0999v1  [cond-mat.mtrl-sci]  6 Nov 2008Magnetic exchange interaction between rare-earth and Mn io ns in\nmultiferroic hexagonal manganites\nD. Talbayev,1,∗A. D. LaForge,2S. A. Trugman,1N.\nHur,3A. J. Taylor,1R. D. Averitt,1,4and D. N. Basov2\n1Center for Integrated Nanotechnologies, MS K771,\nLos Alamos National Laboratory, Los Alamos, NM 87545, USA\n2Department of Physics, University of California San Diego,\n9500 Gilman Drive, La Jolla, CA 92093, USA\n3Department of Physics, Inha University, Incheon 402-751, K orea\n4Department of Physics, Boston University,\n590 Commonwealth Avenue, Boston, MA 02215, USA\n(Dated: November 3, 2018)\nAbstract\nWe report a study of magnetic dynamics in multiferroic hexag onal manganite HoMnO 3by far-\ninfrared spectroscopy. Low-temperature magnetic excitat ion spectrum of HoMnO 3consists of\nmagnetic-dipole transitions of Ho ions within the crystal- field split J= 8 manifold and of the\ntriangular antiferromagnetic resonance of Mn ions. We dete rmine the effective spin Hamiltonian\nfor the Ho ion ground state. The magnetic-field splitting of t he Mn antiferromagnetic resonance\nallows us to measure the magnetic exchange coupling between the rare-earth and Mn ions.\n1Superexchange is one of the main concepts of magnetism in solids. Re cently, superex-\nchangewas foundtomediate magnetoelectric(ME) coupling inmultife rroics -materialswith\ncoexisting magneticandferroelectric orders1,2,3. MEcoupling may findimportanttechnolog-\nicalapplications, forexample, inanelectric-writemagnetic-readme moryelement4. Multifer-\nroic hexagonal manganites display robust room-temperature fer roelectricity and remarkable\nME behavior5that allow technological exploitation6. In these multiferroics, the superex-\nchange interaction between the rare-earth and Mn ions is respons ible for the strong ME\ncoupling that allows the control of ferromagnetism by an electric fie ld2. The details of the\nrare-earth/Mn superexchange coupling have heretofore remain ed unknown. Here, we report\nthe measurement of the rare-earth/Mn superexchange in multife rroic hexagonal HoMnO 3\nvia the detection of an antiferromagnetic resonance by far-infra red (far-IR) spectroscopy.\nThese results demonstrate the ferromagnetic nature of the rar e-earth/Mn exchange that\nenables the electric-field control of magnetism in HoMnO 3.\nHoMnO 3(HMO) crystallizes in a hexagonal lattice, space group P63cm. The crystal\nstructure consists of layers of corner-sharing trigonal MnO 5bipyramids separated by lay-\ners of Ho3+ions2. Ferroelectric polarization along the caxis (TFE= 875 K) results from\nelectrostatic and size effects that lead to the buckling of MnO 5bipyramids and the displace-\nment of the Ho3+ions out of the a-bplane2,7. The magnetic structure is formed by Mn3+\nions in a two-dimensional triangular network in the a-bplane coupled by antiferromagnetic\n(AF) exchange and also by Ho3+(5I8) ions located at Ho(1) and Ho(2) lattice sites with\npoint symmetries C3vandC3, respectively8,9. The AF exchange between Mn3+magnetic\nmoments leads to their 120oordering at TN= 72 K10,11, and easy plane anisotropy confines\nthe Mn spins to the a-bplane. Two in-plane spin reorientation transitions occur at temper-\naturesTSR1≈38 K and TSR2≈5 K associated with changes of magnetic symmetry to P6′\n3cm′\nand toP63cm, respectively8,10,12. The spin reorientation at TSR2≈5 K is accompanied by\nantiferromagnetic ordering of the Ho3+ions located at the Ho(2) lattice sites8. Several man-\nifestations of ME coupling in HMO have been documented, one of the m ost prominent being\nthe presence of the reentrant phase with a strong magnetodielec tric response2,13,14,15,16.\nThe HMO single crystals were grown using an optical floating zone fur nace11. The ma-\nterial’s magnetic phase diagram11at temperatures above 5 K displays the reentrant phase\ndiscovered by Lorenz et al14. Polarized far-IR transmission of a 0.4 mm thick, (110) ori-\nented crystal was measured as a function of temperature and ap plied magnetic field using a\n2/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s51/s48\n/s53/s52\n/s49/s44/s51/s50/s32\n/s32/s84/s114/s97/s110/s115/s109/s105/s115/s115/s105/s111/s110\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s99/s109/s45/s49\n/s41/s104/s32/s124/s124/s32/s99\n/s50\n/s104/s32/s124/s32/s99\nFIG. 1: Power transmission spectra of HoMnO 3with polarized far-IR light incident along the\ncrystal’s [110] direction at T=10 K. Arrows indicate magnet ic absorption lines. The dotted line\nillustrates the simulation of the transmission spectrum us ing Eqs. (1) and (2) and the parameters\nǫ0= 9.07,σ= 0.9cm−1, with magnetic resonances centered at 27 .5, 43.5, and 52 .1cm−1.\nBruker 66 Fourier-transform spectrometer coupled to an 8 Tesla split-coil superconducting\nmagnet17. Static magnetic field Bwas applied both parallel and perpendicular to the caxis.\nFigure1shows far-IRpower transmission spectra oftheHMOcrys tal atlowtemperatures\nand zero magnetic field for two polarizations of the light incident along the [110] direction -\nthe magnetic h-field of the lightwave parallel ( h||c) and perpendicular ( h⊥c) to thecaxis.\nThe spectra in Fig. 1 are normalized to the transmission through an e mpty aperture of the\nsamesizeasthesample. Thefringeswiththeperiodof ≈3cm−1seeninthespectraaredueto\nmultiple reflections of light within the sample. The transmission minima ind icated by arrows\ncorrespond to magnetic resonance frequencies. The broad abso rption in both polarizations\nat frequencies above 70 cm−1is of non-magnetic origin, as it exhibits no magnetic field\ndependence. The dotted line in Fig. 1 shows the simulated transmissio n, assuming that the\n3magnetic and dielectric response of HMO consists of a collection of lor entzian oscillators:\nǫ(ω) =ǫ0+i4πσ/ω+f0\nω2\n0−ω2−iγ0ω, (1)\nµ(ω) = 1+/summationdisplay\njfj\nω2\nj−ω2−iγjω, (2)\nwhereωj,γj, andfjare frequency, relaxation rate, and oscillator strength of magne tic\nresonances. ǫ0andσare constants, and ω0,γ0, andf0are introduced to account for the\nnon-magnetic absorption above 70 cm−1.\nColor-coded transmission maps in Fig. 2 illustrate the magnetic field de pendence of the\nresonance frequencies at T=10 K. To create the maps, we record ed the transmission spectra\nin 1 T steps between 0 and 8 T. The darker color in Fig. 2 indicates lower transmission\nand the white color indicates where the transmission is the highest; t he symbols specify\nthe positions of magnetic resonances extracted from the numeric al simulation of the far-IR\ntransmission. In the remainder of this Letter, we show that the ma gnetic field dependence of\nthe resonance frequencies carries anunambiguous signatureof t he superexchange interaction\nbetween magnetic Ho3+and Mn3+ions that allows us to quantify the interaction.\nWe attribute the absorption lines labeled 1-4 in Figs. 1 and 2 to crysta l field excitations\nof Ho3+ions. The crystal fields of C3vandC3symmetry have qualitatively similar effects\non the Ho ion. According to the work of Elliott and Stevens18, the crystal field potential\nVCFcan be written in terms of spin operators J,Jz, andJ±for the manifold of states of\nconstant J. The diagonal part of VCFdepends only on JandJz. The off-diagonal part of\nVCFhas non-zero matrix elements only between the states mandm′that differ by ∆ m= 6,\nm′=m±6, in the field of C3symmetry and by ∆ m= 3,6,m′=m±3,6, in the field\nofC3vsymmetry. In zero applied magnetic field, the degenerate J= 8 manifold splits\ninto a set of singlet and doublet states of the form |±6,±3,0/angbracketrightand|±8,±5,±2,∓1,∓4,∓7/angbracketright,\nrespectively. We restrict our discussion to a qualitative description of the observed crystal\nfield transitions, as the frequency range of 20 −80cm−1in our far-IR measurements does not\nallow us to characterize all magnetic-dipole active crystal field tran sitions and determine\nthe crystal field parameters. For example, the 12 .5 cm−1transition observed by inelastic\nneutron scattering10is outside of our accessible frequency range.\nThe magnetic-dipole selection rule for the ( h⊥c) polarization allows only transitions with\n∆m=±1. This rule forbids transitions between singlets, but allows transitio ns between\n4/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s56/s48/s57/s48\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s56/s48/s57/s48\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s56/s48/s57/s48\n/s50\n/s49/s53/s84/s61/s49/s48/s32/s75/s44/s32/s66/s32/s124/s124 /s32/s99\n/s32\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s99/s109/s45/s49\n/s41/s48 /s48/s46/s48/s51/s57/s48/s54 /s48/s46/s48/s55/s56/s49/s51 /s48/s46/s49/s49/s55/s50 /s48/s46/s49/s53/s54/s51 /s48/s46/s49/s57/s53/s51 /s48/s46/s50/s51/s52/s52 /s48/s46/s50/s53/s48/s48/s48/s48/s46/s50/s53\n/s50/s50\n/s71/s83\n/s66\n/s49/s53/s50/s84/s61/s49/s48/s32/s75/s44/s32/s66/s32/s124/s32 /s99\n/s32\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s32/s48 /s48/s46/s48/s49/s56/s55/s53 /s48/s46/s48/s51/s55/s53/s48 /s48/s46/s48/s53/s54/s50/s53 /s48/s46/s48/s55/s53/s48/s48 /s48/s46/s48/s57/s51/s55/s53 /s48/s46/s49/s49/s50/s53 /s48/s46/s49/s50/s48/s48/s48 /s48/s46/s49/s50\n/s32\nFIG. 2: Transmission maps for the ( h⊥c) polarization where darker color indicates lower transmis -\nsion. Left panel - static field B||c, right panel - B⊥c. The bars above the panels set the absolute\ntransmission scale. Symbols represent magnetic resonance positions extracted from fitting the\nmeasured transmission using Eqs. (1) and (2). Resonances ar e labeled by the same numbers used\nin Fig. 1. The inset of the left panel is a scheme of the transit ion 2, which happens from the\nground state (GS) doublet. The final state of the transition i s assumed to be a singlet. In low\nmagnetic field, the GS doublet splits and two resonance lines are observed as illustrated by the two\nvertical arrows on the left. In high magnetic field, the spitt ing of the GS becomes large enough\nfor a thermal depopulation of the upper branch. Only one reso nance line is observed, as the single\nvertical arrow on the right illustrates.\nsinglets and doublets and between different doublets. Transitions 1 and 2 are strong in the\n(h⊥c) polarization (Fig. 2) but are very faint in the ( h||c) polarization, which suggests that\nthey are allowed in the former, but forbidden in the latter, polarizat ion. Since we observe\ntransition 2 in the highest applied magnetic fields, it must be the trans ition from the ground\nstate. Thefinalstateofthetransitioncouldbeeither adoubletor asinglet. Withamagnetic\nfieldBapplied alongthe caxis(left panel ofFig. 2anditsinset) thetransition splitslinearly,\na characteristic of the Zeeman splitting of a doublet. The lower bran ch of the Zeeman-split\ntransition disappears when the applied field increases beyond 3 T, wh ich is indicative of a\nthermal depopulation of the upper branch of the Zeeman-split gro und state doublet. The\n5slope of the upper branch of transition 2 allows an estimation of the g-factor in the effective\nspin Hamiltonian Heff=gµBB/tildewideS,/tildewideS= 1/2, that describes the ground state. Assuming that\nthe final state of the transition is a singlet with energy independent of the applied field (at\nsufficiently low fields), we get gz≈8.8. The magnetic field Bperpendicular to the caxis\nproduces no Zeeman splitting of the ground state doublet. Theref ore, neglecting the mixing\nof the ground state with higher energy levels at sufficiently low field B, we setgx=gy= 0.\nThe upper branch of transition 2 is further split by about 5 cm−1in magnetic fields along\nthecaxis, which we attribute to hyperfine interactions.\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56/s52/s48/s52/s50/s52/s52/s52/s54/s52/s56/s53/s48/s53/s50\n/s52/s48/s52/s50/s52/s52/s52/s54/s52/s56/s53/s48/s53/s50\n/s32/s32\n/s32/s109/s101/s97/s115/s117/s114/s101/s100/s32 /s66 /s124/s124 /s99\n/s32/s109/s101/s97/s115/s117/s114/s101/s100/s32 /s66 /s32/s124/s32 /s99\n/s32/s99/s97/s108/s99/s117/s108/s97/s116/s101/s100/s32 /s66 /s124/s124 /s99/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s99/s109/s45/s49\n/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\nFIG. 3: Field dependence of triangular AFMR frequencies at T =10 K. The measured splitting of\nthe resonance by magnetic field is about twice the splitting c alculated using the free energy of Eq.\n(3).\nWenowturntothemagneticresponseofMnionsinHMOandassignthe sharpabsorption\nline at 43 .4cm−1in the (h⊥c) polarization to the antiferromagnetic resonance (AFMR) of\nthetriangularlatticeofMnmagneticmoments (transition5inFigs.1, 2). Thisassignment is\nsupportedbytheobservationofthetriangularAFMRatthesamef requency( ≈43cm−1)and\n6light polarization in hexagonal YMnO 319, a related compound with the same Mn magnetic\nstructure and similar exchange and anisotropy parameters20. The AFMR corresponds to the\ngappedk= 0 magnon detected by inelastic neutron scattering at 47cm−1in HMO10. The\nsimilarity between the k= 0 magnon frequency and the position of the AFMR in our data\nfurther supports the assignment of the resonance. The magnet ic field dependence of the\nAFMR agrees qualitatively with the expected linear splitting induced by the field applied\nalong the caxis (B||c) (left panel of Fig. 2). A slight increase in the AFMR frequency and\nno splitting of the resonance are observed in fields up to 5 T applied pe rpendicular to the\ncaxis (B⊥c). In fields of 5 T and higher, the AFMR line is hard to identify, as it beco mes\na part of the broad absorption band due to the crystal field levels o f Ho ions (right panel\nof Fig. 2). The same qualitative behaviour, i.e., linear splitting with B||cand no splitting\nwithB⊥c, in fields up to 5.4 T was recorded for the AFMR in YMnO 319.\nTo quantify the magnetic-field behaviour of the AFMR, we use the fo llowing free energy:\nF=λ/summationdisplay\ni/negationslash=jMi·Mj+K/summationdisplay\ni(Mz\ni)2−B/summationdisplay\niMz\ni, (3)\nwhereMi,i= 1,2,3, are the three sublattice magnetizations within the a−bplane,λ\ndescribes the antiferromagnetic exchange between the sublattic es, andKrepresents the\neasy-plane anisotropy. The B= 0 excitation spectrum of the free energy (3) consists of a\nzero-frequency mode corresponding to global rotations of the s ublattices about the caxis\nand a doubly-degenerate gapped mode whose degeneracy is lifted b y the application of B||c.\nThe analytical expression for the frequency of the gapped mode in B||cwas given by Palme\net al21:\nω2\n±=ab\n2−b(a−b)\n2(a+b)2B2\n±bB\n2(a+b)2/radicalbig\nB2b(b−2a)+2ab(a+b)2, (4)\nwherea= 2Hd,b= 3Hex, the exchange field Hex=λM0, the anisotropy field Hd=KM0,\nandM0is the sublattice magnetization. Equation (3) is a molecular-field vers ion of the\nHamiltonian used by Vajk et alto describe the spinwave dispersion in HMO10. In this\npicture,Hex= 3SJ= 127 T and Hd=SD= 5.75 T, where JandDare the exchange and\nanisotropy parameters measured by inelastic neutron scattering andSis the total spin of\nMn ions. The value of Hdwas lowered slightly to reproduce the zero-field AFMR frequency\n7observed in our far-IR study. The field dependence of the AFMR ca lculated using Eq.\n(4) and the g-factorg= 2 is shown in Fig. 3. The calculated frequencies largely disagree\nwith the measured ones as the measured splitting is about twice the c alculated splitting. For\ncomparison, thesplitting of the AFMR in YMnO 3is ingoodagreement with our calculation:\nthe measured splitting of 4 .8cm−1at≈5.4 T (g= 1.9±0.1)19agrees with the calculated\nvalue of 4 .9cm−1, as determined from exchange and anisotropy fields measured by in elastic\nneutron scattering20in YMnO 322.\nWhat causes the large discrepancy between the measured and calc ulated splitting of\nthe AFMR line in HMO? Our calculation describes well the AFMR in YMnO 3, a material\nthat is largely similar to HMO, but with one important difference: the Y3+ions are not\nmagnetic. Therefore, we conclude that the magnetic Ho ions in HMO a re responsible for the\ndiscrepancy. The Mn ions in HMO interact with the surrounding Ho ions via an exchange\ncoupling of the form J′\nijJHo\ni·SMn\nj, whereSMnandJHoare the angular momenta of Mn\nand Ho ions. Using the equivalence of matrix elements gJJz=gz/tildewideSz, wheregJ= 5/4 is the\nLande factor, gzand/tildewideSzare the effective g-factor and spin for the Ho ion ground state, we\ncan write the exchange Hamiltonian as18Hex=Jx/tildewideSxSMn\nx+Jy/tildewideSySMn\ny+Jz/tildewideSzSMn\nz,where\nJz= (gz/gJ)J′\nij, etc. We found earlier that gz≈8.8 andgx=gy= 0, which reduces\nthe Hamiltonian to Hex=Jz/tildewideSzSMn\nz. In magnetic field Bapplied along the caxis, we can\nreplace/tildewideSzby its thermal average/angbracketleftBig\n/tildewideSz/angbracketrightBig\n=χB/gzµBand write Hex= (Jzχ)/(gzµB)BSMn\nz,\nwhereχis the magnetic susceptibility of Ho ions. The last expression shows th at the Ho-Mn\n(HM) exchange interaction is equivalent to an effective magnetic field acting on Mn spins.\nTo describe the effect of the HM exchange on the AFMR, we add the t erm\nFHM=JzχN\ngzgµ2\nBB/summationdisplay\niMz\ni=λHMB/summationdisplay\niMz\ni, (5)\nwhereN=6 is the number of Ho ions neighboring a Mn ion, to the free energy of the Eq.\n(3). The total magnetic field acting on Mn ions is then the sum of the e xternal static field\nand the effective field of Ho ions, Btot= (1−λHM)B, and can be larger or smaller than the\nexternalfield, thusenhancingormitigatingitseffectontheAFMR.C omparingthemeasured\nAFMR splitting and the splitting calculated using Eq. (3), we find that λHM=−1, which\ncorresponds to ferromagnetic HM coupling.\nWe can account for the two kinds of Ho lattice sites (Ho(1) and Ho(2 )) by writing Jz=\n(2J1\nz+4J2\nz)/N, whereJ1\nzandJ2\nzare the HM exchange constants for each site. Each Mn\n8ion has two Ho(1) ions and four Ho(2) ions as Ho nearest neighbors. The two Ho lattice sites\nare distinct because of the ferroelectric displacements. Such disp lacements are significantly\nsmaller that interionic distances in HoMnO 3, and the approximation J1\nz≈ J2\nz≈ Jzis\nexpected to be a good one. Otherwise, we must use the above comb ination of J1\nzandJ2\nz\nin the definition of λHMin Eq. (5).\nBefore we conclude, we consider the possible effect of phonon coup ling on the frequencies\nof the studied magnetic excitations. The most prominent signature of such coupling is the\nanti-crossing of the coupled magnetic and lattice modes. Such anti- crossing is expected to\nbe strongest when the frequencies of the two modes are similar, as shown on the right panel\nof Fig. 2, where the strong magnetic absorption line overlaps at high magnetic field with the\nnon-magnetic absorption at 70cm−1. At all magnetic fields, we see no measurable changes to\nthe 70cm−1excitation. Thus, we conclude that we detect no measurable couplin g between\nmagnetic and lattice vibrations.\nThe measured ferromagnetic HM exchange plays a significant role in a ll aspects of the\nphysics of HoMnO 3. It causes the electric-field induced ferromagnetic ordering of Ho mag-\nnetic moments along the caxis2and may also be responsible for the multitude of low-\ntemperature (T <8 K) magnetic phases and the two Mn spin reorientation transitions in\nHMO10. The reentrant phase that is induced in HMO by magnetic field displays a strong\nmagnetodielectric effect and was explained by the formation of magn etic domain walls dur-\ning spin reorientation, which reduces the local magnetic symmetry a nd allows the coupling\nbetween magnetic and ferroelectric polarizations13,14. Theoretical descriptions of such mag-\nnetoelectricity must include domain wall contributions to the free en ergy from lattice distor-\ntions and magnetic superexchange interactions. Such description s are incomplete without\nthe consideration of the magnetic exchange between the Ho and Mn ions. Our measurement\nof the ferromagnetic HM exchange provides an important contribu tion to the modeling of\nthe interplay between magnetism and ferroelectricity in HoMnO 3.\nWe thank Darryl Smith for useful theoretical discussions and Jon athan Gigax for the\nsimulation of the Ho crystal field eigenstates. The work at LANL was supported by the\nLDRD program and the Center for Integrated Nanotechnologies. The work at UCSD was\n9supported by NSF DMR 0705171.\n∗Electronic address: diyar@lanl.gov\n1N. Hur, S. Park, P. A. Sharma, J. S. Ahn, S. Guha, and S.-W. Cheo ng, Nature 429, 392 (2004).\n2T. Lottermoser, T. Lonkai, U. Amann, D. Hohlwein, J. Ihringe r, and M. Fiebig, Nature 430,\n541 (2004).\n3S.-W. Cheong and M. Mostovoy, Nature Mat. 6, 13 (2007).\n4J. Scott, Science 315, 954 (2007).\n5M. Fiebig, T. Lottermoser, D. Frohlich, A. Goltsev, and R. 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Schotte, Solid State Comm. 76, 873 (1990).\n22Satoet al.20find that the chemical triangular lattice of Mn ions is slight ly trimerized with\nexchange strength J1between the nearest neighbors belonging to the same trimer a nd exchange\nstrength J2between the ions belonging to neighboring trimers. The zero wave-vector magnon\nspectrum of the trimerized chemical lattice is calculated u sing Eq. (4) and the substitution\nJ→(J1+2J2)/3.\n11"    },    {        "title": "1801.05552v2.Magnetic_effect_on_dynamical_tide_in_rapidly_rotating_astronomical_objects.pdf",        "content": "Magnetic e\u000bect on dynamical tide in rapidly rotating\nastronomical objects\nXing Wei\nInstitute of Natural Sciences, School of Mathematical Sciences, Shanghai Jiao Tong\nUniversity\nxing.wei@sjtu.edu.cn\nReceived ; acceptedarXiv:1801.05552v2  [astro-ph.SR]  19 Jan 2018{ 2 {\nABSTRACT\nBy numerically solving the equations of rotating magnetohydrodynamics\n(MHD), the magnetic e\u000bect on dynamical tide is studied. It is found that mag-\nnetic \feld has a signi\fcant impact not only on the \row structure, i.e. the internal\nshear layers in rotating \row can be destroyed in the presence of a moderate or\nstronger magnetic \feld (in the sense that the Alfv\u0013 en velocity is at least of the\norder of 0.1 of the surface rotational velocity), but also on the dispersion relation\nof waves excited by tidal force such that the range of tidal resonance is broad-\nened by magnetic \feld. A major result is that the total tidal dissipation scales\nas square of the \feld strength, which can be used to estimate the strength of\ninternal magnetic \feld in the astronomical object of a binary system. Moreover,\nwith a moderate or stronger \feld the ratio of the magnetic dissipation to the\nviscous dissipation is almost inversely proportional to magnetic Prandtl number\n(i.e. the ratio of viscosity to magnetic di\u000busivity), and therefore in the astrophys-\nical situation at small magnetic Prandtl number magnetic dissipation dominates\nover viscous dissipation with a moderate or stronger \feld.\nKeywords: magnetic \felds, rotation, binaries: general\n1. Introduction\nIn a binary system, e.g. binary stars, star and planet, planet and satellite, etc.,\nthe angular momentum transfers between the orbital motion and the rotational motion\nof astronomical objects such that the binary system evolves to the equilibrium state\nof minimum energy, the so-called synchronization and circularization processes. Tidal\ndissipation plays an important role in these processes. Tide consists of two parts,{ 3 {\nequilibrium tide which is a response to the external tidal potential and a slow (compared\nto the dynamical time scale) quasi-hydrostatic displacement at the surface of astronomical\nobjects (Ogilvie 2014), and dynamical tide which is the internal \ruid waves resonantly\nexcited by the external tidal force. Equilibrium tide \u0018obtained by the hydrostatic\nbalance is equal to the sum of the tidal potential \t and the Eulerian perturbation of\nthe self-gravitational potential \b0divided by the local gravitational acceleration g, i.e.\n\u0018=\u0000(\t + \b0)=g(Souchay et al. 2013; Ogilvie 2014). The self-gravitational potential\nperturbation \b0is an order-of-unit multiple of the tidal potential \t and the major\ncomponent of tidal potential is on the ( l= 2;m= 2) mode, where landmare respectively\ndegree and order of spherical harmonics. Therefore, equilibrium tide is on a large\nlength-scale, but the \ruid waves of dynamical tide can have small length scales. Energy\ncan dissipate either through the small-scale turbulent convective eddies on the large-scale\nequilibrium tide (Zahn 1977; Goodman & Oh 1997) or through the small-scale dynamical\ntide (Zahn 1975; Goldreich & Nicholson 1989).\nThere exist various \ruid waves in the \ruid part (e.g. atmosphere, ocean, interior) of\nan astronomical object, and these waves have various length and time scales. When the\nwave frequency is close to the tidal frequency, resonance occurs and the tidal response and\ndissipation can be greatly improved. At the surface of an astronomical object, gravity\ninduces surface gravity waves, namely f mode (equilibrium tide can be regarded as a forced\nf mode). The frequency of f mode is of the order of dynamical frequency and f mode is too\nfast to be resonantly excited by tidal force. In a compressible \ruid, pressure induces sound\nwaves, namely p mode, and again p mode is too fast to be resonantly excited by tidal force.\nIn the radiative zone, buoyancy force in a stably strati\fed layer induces internal gravity\nwaves, namely g mode. The frequency of g mode, i.e. the buoyancy frequency, can be\nclose to tidal frequency and g mode can resonate with external tidal force. Internal gravity\nwaves transport angular momentum and dissipate through radiation. In a rapidly rotating{ 4 {\nastronomical object, Coriolis force induces inertial waves which can also resonate with tidal\nforce. In the convective zone, inertial waves transport angular momentum and dissipate\nthrough small-scale turbulent eddies. Internal gravity wave and inertial wave that resonate\nwith tidal force are both dynamical tide.\nInternal gravity wave has been extensively studied. Zahn started this study in his\nthree seminal papers (1970; 1975; 1977), Goldreich & Nicholson (1989) gave a physical\ninterpretation and applied to early-type stars (1989), Savonije & Papaloizou (1983) studied\nthe non-adiabatic e\u000bect, Goodman & Dickson (1998) applied to solar-type stars, Fuller &\nLai (2012) applied to white dwarfs with a sharp composition jump, etc. Inertial wave has\nalso been extensively studied in (Ogilvie & Lin 2007), (Goodman & Lackner 2009), (Ogilvie\n& Lin 2004), (Ogilvie 2009), (Ogilvie 2013), (Wu 2005a), (Wu 2005b), etc. Nonlinear e\u000bect\non dynamical tide was considered in (Kumar & Goodman 1996), (Ogilvie & Lin 2007),\n(Barker & Ogilvie 2010), (Weinberg et al. 2012), (Favier et al. 2014), (Essick & Weinberg\n2016), (Wei 2016b), etc. The summary and details of the studies about dynamical tide can\nbe found in the recent review paper by Ogilvie (2014).\nHowever, the magnetic e\u000bect on dynamical tide is not very clear. Bu\u000bett (2010)\nestimated the magnetic (or Ohmic) tidal dissipation in the Earth's \ruid core to extrapolate\nthe strength of Earth's magnetic \feld in the \ruid core. He performed numerical calculations\nfor rotating magnetohydrodynamics (MHD) in a spherical shell and assumed that magnetic\ndissipation focuses in the internal shear layers built by the propagation of inertial waves.\nIn his paper how magnetic \feld in\ruences the internal shear layers was not considered,\ni.e. the Lorentz force was neglected in the equation of \ruid motion, and the dynamics was\ncompletely controlled by rotation. In (Wei 2016a) I performed a local WKB analysis in\nrotating MHD about the magnetic e\u000bect on dynamical tide, and found two results. The\n\frst is that magnetic dissipation wins out viscous dissipation even with a weak magnetic{ 5 {\n\feld because the presence of magnetic \feld modi\fes the dispersion relation of waves of\ndynamical tide, i.e. inertial waves with only rotation become magneto-inertial waves with\nboth rotation and magnetic \feld. The second is that the frequency-averaged dissipation\nis constant regardless of rotation and magnetic \feld. But in the local analysis the global\nspherical geometry was not considered. Most recently, in an online published paper (Lin &\nOgilvie 2017) the magnetic e\u000bect on dynamical tide was thoroughly studied in the presence\nof either a uniform vertical or dipolar \feld in a spherical shell with the vortical forcing\nmethod developed by Ogilvie (2005). Lin & Ogilvie (2017) found the similar results to\nWei (2016a), namely magnetic \feld changes the dissipations because wave dispersion is\nin\ruenced by magnetic \feld such that magnetic dissipation can win out viscous dissipation,\nand the frequency-averaged dissipation is independent of dissipation mechanism. Moreover,\nthey found that a su\u000eciently strong magnetic \feld changes \row pattern of internal shear\nlayers, and obtained the scaling law for the competition between rotation and magnetic\n\feld, which is very useful for the estimation in astrophysical situation.\nIn the current paper I will perform numerical calculations for rotating MHD in a\nspherical shell to study the magnetic e\u000bect on dynamical tide, and the impact of magnetic\n\feld on \row structure and waves will be considered. Although the problem and the setup\nin this study are similar to (Lin & Ogilvie 2017), there are still some di\u000berences. The \frst\nis that I use the boundary \row method instead of the vortical forcing method and the\ntime-stepping method instead of a super-large matrix solver (matrix is very large especially\nin the case of strong \feld). The second is that I study the regime of a stronger \feld. The\nthird is that I try to \fnd the scaling laws about the tidal dissipation versus the magnetic\n\feld strength. In Section 2 the model of this numerical study will be built, in Section 3 the\nnumerical results will be shown and discussed, and in Section 4 a brief summary will be\ngiven.{ 6 {\n2. Model\nWe use a spherical shell with the radius ratio Ri=Ro= 1=2 as the model for a spherical\nastronomical object. The shell rotates rapidly at a constant angular velocity \n= \n^z,\nwhere ^zis the unit vector in the rotational axis, and a conducting \ruid resides within the\nshell to mimic the convective zone of a star or giant planet or the \ruid core of a terrestrial\nplanet or satellite. The inner sphere rotates uniformly to mimic the radiative zone of a star\nor giant planet or the solid inner core of a terrestrial planet or satellite. A uniform vertical\nmagnetic \feld B=B^zis imposed throughout the shell and the motion of the conducting\n\ruid in the shell will induce a new \feld b, which is weak compared to B. We numerically\nsolve the equations of rotating MHD in a frame rotating at the angular frequency \n.\nBecause sound waves in a compressible \ruid are too fast to be resonantly excited by tidal\nforce, we consider an incompressible \ruid to \flter out sound waves. This will bring us a big\nnumerical advantage to avoid the very small time-step for capturing sound waves. We study\nthe linear regime and neglect the nonlinear quadratic terms. We solve the dimensionless\nrather than dimensional equations, because dimensionless equations can give us more, e.g.\nscaling laws to extrapolate to the real parameter regime which is not accessible under the\ncurrent computational power.\nThe dimensionless equation of \ruid motion in the rotating frame reads\n@u\n@t=\u0000rp0+Er2u+ 2u\u0002^z+S2(r\u0002b)\u0002^z: (1)\nIn Equation (1), the term on the left-hand-side is the acceleration of \ruid motion, the\nterms on the right-hand-side are successively the Eulerian perturbation of pressure gradient\nincorporating the centrifugal potential perturbation, the viscous force, the Coriolis force\narising from rotation and the Eulerian perturbation of Lorentz force arising from magnetic\n\feld, and the nonlinear quadratic terms u\u0001ruand (r\u0002b)\u0002bare neglected. The\ntidal force will be implemented by the outer boundary condition. In Equation (1), length{ 7 {\nis normalized with the outer radius Ro, time with the inverse of rotational speed \n\u00001,\nvelocity with \n Ro, and magnetic \feld with the strength of the imposed \feld B. The two\ndimensionless numbers are the Ekman number\nE=\u0017\n\nR2\no(2)\nwhich is the ratio of the rotational time-scale \n\u00001to the viscous time-scale R2\no=\u0017(\u0017being\nviscosity) and measures the strength of rotation \n relative to viscosity \u0017, and theSnumber\nS=Bp\u001a\u0016\nRo(3)\nwhich is the ratio of the Alfv\u0013 en speed B=p\u001a\u0016(\u0016being magnetic permeability) to the\nangular speed \n Roand measures the strength of the imposed \feld Brelative to rotation \n.\nTheSnumber is called the Lenhert number (Lehnert 1954).\nOn the inner boundary, the horizontal velocity is taken to be stress-free to minimize\nthe e\u000bect of boundary layer and the radial velocity vanishes, i.e. \ruid cannot penetrate\nacross the inner boundary. On the outer boundary, the horizontal velocity is also taken to\nbe stress-free but the radial velocity follows an oscillatory rising-falling equilibrium tide, i.e.\nur=APm\nl(cos\u0012) cos(m\u001e\u0000!t): (4)\nThis boundary condition enforces equilibrium tide at the outer surface to excite dynamical\ntide in the interior. The interior \row driven by the boundary radial \row consists of\nboth tidal \row (equilibrium tide) and waves (dynamical tide). In Equation (4), Ais the\namplitude and in the linear problem it is taken to be 1, Pm\nlis the associate Legendre\npolynomial normalized byq\n(2l+1)(l\u0000m)!\n4\u0019(l+m)!andlandmare taken to be those of the major\ncomponent of tidal potential, namely ( l= 2;m= 2),\u0012and\u001eare respectively colatitude and\nlongitude in the spherical polar coordinates, and !is the tidal frequency in the rotating\nframe, i.e. the Doppler shifted frequency !=!i\u0000m\n (!ibeing the tidal frequency in the{ 8 {\ninertial frame). The method to use equilibrium tide to excite dynamical tide was proposed\nand validated by Ogilvie (2009) and has been numerically implemented in (Favier et al.\n2014).\nCampbell & Papaloizou (1986) studied the magnetic e\u000bect on stellar oscillations and\nfound that a strong magnetic \feld induces a boundary layer to modify the radial \row at\nthe surface. For stellar oscillations (free-oscillation problem) magnetic \feld is signi\fcant\nfor boundary condition whereas for tide (forced-oscillation problem) it is not so signi\fcant\nbecause the tidal frequency is too slow compared to the dynamical frequency at the surface,\ni.e. the low-frequency limit. We now illustrate this point. The normal stress at the\ndisturbed free surface vanishes, i.e.\n\u0000(p0\u0000\u001ag\u0018r)\u00002B\u0001b\n2\u0016+ 2\u001a\u0017@ur\n@r+2Brbr\n\u0016= 0: (5)\nThe terms are successively the Eulerian perturbation of \ruid pressure, the excess pressure\ndue to the disturbed surface, the Eulerian perturbation of magnetic pressure, the viscous\nnormal stress, and the Eulerian perturbation of magnetic normal stress. The round bracket\nconsisting of the \frst two terms is the Lagrangian perturbation of pressure. In the Eulerian\nperturbations of the magnetic pressure and normal stress, the quadratic terms b2=2\u0016and\nb2\nr=\u0016are neglected in the linear regime. In (Ogilvie 2009) it has been veri\fed that in the\nlow-frequency limit !\u001c!d\u0018(g=R)1=2, the Eulerian perturbation of \ruid pressure and the\nviscous normal stress are negligible such that the radial displacement is the equilibrium\ntide with a correction due to self-gravitation. This result is in agreement with (Goldreich\n& Nicholson 1989). Thus the radial \row at the free surface is proportional to the tidal\npotential. We now show that the Eulerian perturbation of magnetic pressure is negligible\nin the low-frequency limit. The ratio of the Eulerian perturbation of magnetic pressure to\nthe excess pressure is\n\f\f\f\f2B\u0001b\n2\u0016\f\f\f\f1\nj\u001ag\u0018rj\u0018B2\n\u001a\u0016(\nR)2b\nB(\nR)2\n(R!2\nd)\u0018r=S2b\nB\u0012\n!d\u00132R\n\u0018r: (6){ 9 {\nThe radial \row urat surface is ur=@\u0018r=@t\u0018!\u0018r, and on the other hand ur\u0018A\nRwhere\nAis the dimensionless factor for the boundary radial \row (4). We are then led to\n\f\f\f\f2B\u0001b\n2\u0016\f\f\f\f1\nj\u001ag\u0018rj\u0018S2b\nB\n!d!\n!d1\nA<S2\n!d!\n!d1\nA: (7)\nIn the usual astrophysical situations, magnetic \feld is not so strong that Sis less than\nunity (in our calculations S2is up to 0.2), rotation frequency \n is much less than dynamical\nfrequency!d, tidal frequency !is also much less than dynamical frequency !d, i.e. the\nlow-frequency limit. Although the factor Awhich measures the strength of equilibrium\ntide is very small, it is still much larger than S2(\n=!d)(!=!d). Therefore this ratio is very\nsmall and the magnetic e\u000bect at surface is negligible. Similarly, the Eulerian perturbation\nof magnetic normal stress is also much less than the excess pressure. Consequently, in the\ntidal problem magnetic \feld is negligible at surface because of the low-frequency limit. The\nradial-\row boundary condition is valid to study the magnetic e\u000bect on tide. However, it\nshould be noted that in some situations this ratio is not so small. For example, a star or\nplanet in binary system rotates so fast that \n =!d\u00180:1, the orbital motion is so fast that\n!=!d\u00180:1 (breaking the low-frequency limit), and the surface displacement is so small that\nA\u00180:01. If magnetic \feld at surface is so strong or density at surface is so low that S2is of\norder of unity, then this ratio will be of order of unity at surface. Therefore, in the situation\nof fast rotation, fast orbital motion, strong surface \feld, and low surface density, magnetic\n\feld will have a signi\fcant e\u000bect at surface and it even deforms the spherical geometry. In\nour calculations we assume this ratio is much less than unity in the low-frequency limit and\nneglect the magnetic e\u000bect at surface.\nThe dimensionless equation of magnetic \feld reads\n@b\n@t=r\u0002(u\u0002^z) +E\nPmr2b; (8)\nwhere the nonlinear term r\u0002(u\u0002b) is neglected and the magnetic Prandtl number\nPm=\u0017\n\u0011(9){ 10 {\nis the ratio of viscosity \u0017to magnetic di\u000busivity \u0011. For the magnetic boundary condition,\nthe magnetic \feld in the insulating exteriors of both the r > Roandr < Riregions is\nrequired to match a potential \feld.\nThe numerical method is the standard spectral method with spherical harmonics used\non the spherical surface and Chebyshev polynomials used in the radial direction, and the\ntoroidal-poloidal decomposition method is employed for divergence-free condition of velocity\nand magnetic \feld (Hollerbach 2000). The time-stepping calculation is used to \fnd the \fnal\n`steady' state (here `steady' is in terms of the volume integrals of energy and dissipation).\nAn alternative method is to separate !(namely@=@t =i!) and then to solve a super-large\nmatrix problem, and we do not use it because it is numerically di\u000ecult in the MHD case. In\nthe longitude direction only one mode m= 2 is necessary for the linear problem, but in the\nlatitude direction the modes much higher than l= 2 should be involved because the Coriolis\nforce, the Lorentz force and the induction term will couple more modes in this direction. In\nour numerical calculations the su\u000ecient resolution for convergence is guaranteed that both\nthe radial and latitude spectra of both kinetic and magnetic energies span more than 10\nmagnitudes.\nIn our calculations we focus on the magnetic e\u000bect on dynamical tide so that we\ninvestigate the two parameters related to magnetic \feld, SandPm, as well as the tidal\nfrequency!but keep all the other parameters constants. The radius ratio Ri=Rois taken to\nbe 1=2, which is the value chosen in (Favier et al. 2014). A larger radius ratio leads to the\nstronger excitation of inertial waves. We choose this value for comparison with the results\nin (Favier et al. 2014) to validate our numerical calculations. The Ekman number is taken\nto beE= 10\u00004which is already su\u000eciently low for the study of rapid rotation (too low\nEkman number is numerically demanding). landmare taken to be 2. The frequency of\ninertial waves is lower than 2\n such that the tidal frequency higher than 2\n cannot excite{ 11 {\ninertial waves in rotating \row. However, in the presence of magnetic \feld the frequency of\nmagneto-inertial waves can exceed 2\n (Wei 2016a), and so the regime of tidal frequency\nhigher than 2\n will be studied. In our calculations the dimensionless tidal frequency !\nranges from 0.1 to 3 with the spacing 0.1, i.e. 30 values for !in our calculations. We\nwill calculate 6 values for S, i.e.S2= 0, 0:01, 0:02, 0:05, 0:1 and 0:2.S= 0 is the case\nof rotating \row without magnetic \feld. At S=p\n0:2\u00190:45 magnetic \feld is already\nsu\u000eciently strong to be comparable to rotation (see Equation (3)). We will calculate 4\nvalues forPm, i.e. 1, 0:5, 0:2 and 0:1, i.e. spanning one magnitude. In the astrophysical\nsituationPmis very low, e.g. Pmis of the order of 10\u00006in the Earth's liquid-iron core\nor in the Jupiter's conducting region or in the Sun's convection zone. Too low Pmis\nnumerically demanding and so we study the range of one magnitude to try to \fnd scaling\nlaws. Altogether we have 30 \u0002(6 + 4) = 300 calculations that will be presented in the next\nsection.\n3. Results\nBy numerically solving Equations (1) and (8) with the boundary condition (4), we\nobtain velocity uand induced \feld b. Kinetic energy indicates the tidal response and\nthe kinetic and magnetic dissipations are responsible for the orbital evolution of a binary\nsystem. Then we calculate the volume integrals of kinetic energy and the two dissipations\nover the spherical shell. In the dimensionless expressions, kinetic energy is normalized with\n\u001aR5\no\n2and both dissipations with \u001a\u0017R3\no\n2, and thus\nkinetic energy =1\n2Z\njuj2dV;\nviscous dissipation = 2Z\nSijSijdV; (10)\nmagnetic dissipation =S2\nPmZ\njr\u0002bj2dV;{ 12 {\nwhereSijis the strain tensor. In the next of this section we will use the dimensionless\nexpressions (10) as output.\nWe also calculate the total angular momentum and it conserves in our linear model.\nIt should be noted that in (Favier et al. 2014) angular momentum conserves in the linear\nmodel of rotating \row but does not conserve in the nonlinear model, and in our linear\nmodel of rotating MHD it conserves as well. It may be inferred that angular momentum\ndoes not conserve in the nonlinear model of rotating MHD, which we do not study in this\npaper. We need to remind readers that this method to use equilibrium tide at the outer\nsurface to excite dynamical tide in the interior is applicable in the linear regime but not in\nthe nonlinear regime.\n3.1. Investigation of S\nFirstly, we investigate the strength of imposed magnetic \feld, namely the dimensionless\nnumberS. Figure 1 shows the kinetic energy as the tidal response versus frequency at\nvariousS2withPm\fxed to be 1. The black curve at S= 0 denotes the kinetic energy\nin rotating \row. The peak values at the tidal frequencies != 0:2, 0:8, 1:1, 1:4 are the\nones near the resonance of inertial waves and tidal force. It should be noted that the\ntidal frequencies at these peaks are not exactly but near the eigen-frequencies of inertial\nwaves. To obtain the accurate eigen-frequenices, as illustrated in Section 2, a super-large\nmatrix problem should be solved, which is numerically di\u000ecult in the MHD case. It is\nwell-known that the frequency of inertial waves is lower than 2\n, and therefore, on the\nblack curve at the frequencies higher than 2 there does not exist any tidal response of\ninertial waves. When magnetic \feld is present and its strength ( S) increases, some new\npeaks appear, e.g. != 1:3 atS2= 0:02,!= 0:3 and 1:8 atS2= 0:05,!= 0:4 and 1:9 at\nS2= 0:1, etc. This is because magnetic \feld modi\fes the dispersion relation of waves, i.e.{ 13 {\ninertial waves in rotating \row become magneto-inertial waves in rotating MHD, such that\nthe eigen-frequencies are changed and the tidal resonances appear at di\u000berent frequencies.\nMoreover, with a strong magnetic \feld, e.g. at S2= 0:1 and 0:2, there exist the tidal\nresponses at the tidal frequencies higher than 2. These results indicate that magnetic \feld\nbroadens the range of tidal resonance in the global spherical geometry.\nFigure 2(a) shows the viscous dissipation. For the tidal frequency higher than 2\n(! > 2), the viscous dissipation is very small in rotating \row ( S= 0) because no inertial\nwave can be excited in this range of tidal frequency, but it is moderate and increases with\nthe \feld strength increasing in this range. In the presence of a strong \feld, e.g. S2= 0:2,\nthe viscous dissipation for ! > 2 can be even higher than that for ! < 2. Figure 2(b)\nshows the magnetic dissipation. Its dependence on the tidal frequency is similar to that of\nviscous dissipation. Figure 2(c) shows the total dissipation, namely viscous dissipation +\nmagnetic dissipation. Not surprisingly, it is very small for ! >2 in rotating \row but does\nnot vary signi\fcantly with tidal frequency in the presence of a strong \feld. Figure 2(d)\nshows the ratio of magnetic dissipation to viscous dissipation. The dashed line indicates\nthe equipartition of the two dissipations. When the tidal frequency increases the two\ndissipations tend to be close to each other and in the high-frequency range (approximately\n! >2) the two dissipations are almost equal to each other. But in the low-frequency range\n(approximately !<2) magnetic dissipation wins out viscous dissipation with a strong \feld\nand viscous dissipation wins out magnetic dissipation with a weak \feld.\nThe result of Figures 1 and 2 can be applied to the astrophysical situation. Magnetic\n\feld broadens the range of tidal resonance such that the tidal force with its frequency\nhigher than 2\n which cannot lead to signi\fcant viscous dissipation without magnetic \feld\ncan lead to signi\fcant viscous and magnetic dissipations with magnetic \feld. For example,\nin a binary system, if the orbital frequency in the rotating frame is faster than twice of the{ 14 {\nrotational frequency then inertial waves cannot be excited by the tidal force, but in the\npresence of a strong magnetic \feld the tidal force can excite magneto-inertial waves and\nenergy can dissipate very quickly through both viscous and magnetic dissipations.\nTo better understand the mechanism of tidal dissipations with magnetic \feld, we plot\nthe contours of kinetic and magnetic energies in a certain meridional plane at \u001e= 90\u000e. Here\nmagnetic energy is in terms of the induced \feld, but does not include the contribution of\nimposed \feld. Figure 3 shows the energy contours at the tidal frequency != 1:0. Figure\n3(a) is for rotating \row and Figures 3(b)-3(f) are for rotating MHD with the \feld strength\ngradually increasing. In Figure 3 both tidal \row (equilibrium tide) and waves (dynamical\ntide) are included. Inertial waves in rotating \row propagate at an angle arcsin( !=2\n) = 30\u000e\ninclined to the rotational axis and Figure 3(a) exhibits the 30\u000eoblique internal shear layers\nbuilt by the propagation of inertial waves. However, in the presence of magnetic \feld as\nshown in Figures 3(b)-3(f), these oblique internal shear layers disappear. With the \feld\nstrength gradually increasing, the contours of kinetic energy become more vertical and\nthose of magnetic energy concentrate on the cylinder tangent to the inner sphere. This is\nbecause the magneto-inertial waves become more Alfv\u0013 en-like than inertial-like when the\n\feld strength increases. In the meanwhile, the kinetic energy spreads outside the tangent\ncylinder whereas the magnetic energy concentrates more in a thin internal shear layer\non the tangent cylinder. This di\u000berence between the \row and \feld structures explains\nwhy magnetic dissipation wins out viscous dissipation in the low-frequency range with a\nstrong \feld because the smaller length scale of magnetic \feld leads to the higher magnetic\ndissipation. It should be noted that with a very weak magnetic \feld, e.g. S= 10\u00004, the\nmagnetic e\u000bect is negligible and the internal shear layers cannot be destroyed by the \feld\n(Lin & Ogilvie 2017). So magnetic \feld wins out rotation at a moderate or stronger \feld\nin the sense that the Alfv\u0013 en velocity is at least of the order of 0.1 of the surface rotational\nvelocity. A more accurate scaling law for this competition was obtained by Lin & Ogilvie{ 15 {\n(2017).\nFor comparison with the low tidal frequency, we plot the energy contours at a high tidal\nfrequency!= 3:0, as shown in Figure 4. Figure 4(a) shows that the contours of kinetic\nenergy of rotating \row consisting of both equilibrium tide and dynamical tide at the high\ntidal frequency are almost vertical but do not exhibit the structure of oblique internal shear\nlayers at the low tidal frequency because the tidal frequency higher than 2\n cannot excite\ninertial waves to build the oblique internal shear layers. When magnetic \feld is present as\nshown in Figures 4(b)-4(f), both kinetic and magnetic energies exhibit similar distributions\nand tend to concentrate inside the tangent cylinder when the \feld strength increases, which\nexplains why viscous and magnetic dissipations are comparable at the high tidal frequency.\nThe dependence of dissipations on tidal frequency is irregular because the resonance in\nspherical geometry is complicated, but the dependence of dissipations on magnetic \feld is\nnot so irregular. Figure 5 shows the log-log relation of kinetic energy and dissipations versus\nS2at the di\u000berent tidal frequencies which are almost a geometric sequence. Figure 5(a)\nshows the kinetic energy versus S2. It is irregular because the resonance occurs irregularly\nat certain frequencies with certain \feld strengths. For dissipations shown in Figures 5(b)\nand 5(c), we cannot \fnd accurate scaling laws, i.e. straight lines in the log-log diagram,\nbecause certain tidal frequencies are near resonance at certain Svalues as shown in Figure\n5(a). However, we can perceive that both viscous and magnetic dissipations increase with\nthe \feld strength increasing. It is not surprising because stronger \feld provides higher\nmagnetic energy to damp. Then we \ft the slopes of this `big trend' for all the curves by\nappropriately removing few points far away from this `big trend' and take the average of\nthese slopes. Interestingly, both viscous and magnetic dissipations obey an identical scaling\nlaw,\nviscous and magnetic dissipations /S2: (11){ 16 {\nThe dashed lines in Figures 5(b) and 5(c) denote this scaling law. This scaling law about\nthe relation of the two dissipations and magnetic \feld reveals that there exists self-similarity\nfor the magnetic e\u000bect on dynamical tide.\n3.2. Investigation of Pm\nIn this subsection we investigate Pm. Similar to Figure 2, Figure 6 shows viscous\ndissipation, magnetic dissipation, total dissipation and ratio of magnetic to viscous\ndissipations versus tidal frequency at various PmwithS2\fxed to be 0.1, namely in a\nstrong-\feld regime. The \frst three sub\fgures show that for all the tidal frequencies a larger\nPmleads to higher viscous dissipation and lower magnetic dissipation but total dissipation\nis almost independent of Pm. This is reasonable because Pmis the ratio of viscosity to\nmagnetic di\u000busivity (see Equation (9)) but cannot in\ruence the total di\u000busivity. More\ninterestingly, Figure 6(d) shows that magnetic dissipation wins out viscous dissipation for\nPm< 1 and this ratio is approximately equal to Pm\u00001. In the astrophysical situation Pm\nis very small, and it can be inferred that magnetic dissipation dominates over viscous\ndissipation with a moderate or stronger magnetic \feld . It should be noted again\nthat with a very weak \feld, e.g. S\u001410\u00004, viscous dissipation wins out magnetic dissipation\neven at a small Pm, e.g.Pm= 10\u00004(Lin & Ogilvie 2017).\nTo \fnd the scaling laws about Pm, we plot the two dissipations versus Pmas shown in\nFigure 7. The dependence on Pmlooks more regular than the dependence on Sin Figure 5\nbecause the \feld strength in\ruences the resonant frequencies whereas di\u000busivities cannot.\nBy taking the average of the slopes for di\u000berent tidal frequencies we obtain the scaling laws\nviscous dissipation /Pm0:6;\nmagnetic dissipation /Pm\u00000:3: (12){ 17 {\nThe dashed lines in Figures 7(a) and 7(b) denote respectively these two scaling laws. Then\nthe ratio of magnetic to viscous dissipations is indeed close to Pm\u00001as shown in Figure\n6(d). It should be noted that our parameter regime is not the real regime, e.g. Eand\nPmare not too small, such that these scaling laws might not work in the real situation.\nWith the lack of the knowledge about the physical properties in stars and planets, we\ndo not know the real parameters, and even the estimation for the order of magnitude is\nnot convincing. These scaling laws obtained in the moderate parameter regime at least\ngive us some qualitative results that a moderate or stronger magnetic \feld improves both\ndissipations and the magnetic dissipation wins out viscous dissipation at a low Pm.\n3.3. A major result\nNow let us come back to the equations governing the system of magnetic tide. In\nequations (1), (4) and (8) there are four parameters, E,S,PmandA. In the linear\nregime we studied, the tidal dissipations should be proportional to A2. Since we focus on\nthe magnetic e\u000bect but not the rotational e\u000bect, Ekman number is not investigated but\n\fxed to be a small value. Through the numerical studies we know the dependences on the\ntwo magnetic parameters SandPm, i.e. viscous dissipation is proportional to S2Pm0:6,\nmagnetic dissipation to S2Pm\u00000:3, and total dissipation to S2but independent of Pm.\nTranslating to the dimensional expression, we obtain the scaling law for the total dissipation\nabout the strength of magnetic \feld,\ntotal dissipation/B2: (13)\nEquation (13) is a major result of this paper which has the astrophysical applications. For\nexample, it can be used to estimate the strength of internal magnetic \feld of the\nastronomical object of a binary system by observing the orbital evolution of the binary\nsystem. It should be noted that this scaling law is about magnetic \feld, and in addition{ 18 {\nto magnetic \feld, some other factors such as di\u000busivities, radius ratio, and so on can also\nin\ruence the tidal dissipation. Lin & Ogilvie (2017) discussed the other factors in detail,\nand in the next section we will brie\ry discuss these other factors.\n4. Summary\nIn this paper we numerically investigated the magnetic e\u000bect on the dynamical tide of\na binary system. We tuned the tidal frequency, the \feld strength, and the two di\u000busivities\nto calculate both viscous and magnetic dissipations which are important for the orbital\nevolution of the binary system. It is found that a moderate or stronger magnetic \feld (in the\nsense that the Alfv\u0013 en velocity is at least of the order of 0.1 of the surface rotational velocity)\ndestroys the internal shear layers built by the propagation of inertial waves. Magnetic \feld\nmodi\fes not only the \row structure but also the dispersion relation of waves excited by the\ntidal force such that the tidal resonance in rotating MHD is quite di\u000berent from that in\nrotating \row, namely the resonance range is broadened by magnetic \feld to be out of 2\n.\nMagnetic dissipation wins out viscous dissipation at the low tidal frequencies with a strong\nmagnetic \feld but the two dissipations are comparable at the high tidal frequencies. The\nratio of magnetic to viscous dissipations is almost inversely proportional to Pmsuch that\nin the astrophysical situation at very low Pmmagnetic dissipation dominates over viscous\ndissipation with a moderate or stronger \feld, but the total dissipation does not depend on\nPm. We obtained three scaling laws, viscous dissipation /S2Pm0:6, magnetic dissipation\n/S2Pm\u00000:3, and total dissipation /S2. The last scaling law in its dimensional expression,\ntotal dissipation/B2, can be applied to the estimation of the strength of internal magnetic\n\feld in the astronomical object of a binary system.\nDue to the limitation of our computational power, we do not compute the frequency-\naveraged dissipation which requires to scan a huge range of tidal frequency, especially in{ 19 {\nthe case of a strong magnetic \feld. Wei (2016a) con\frmed in a periodic box that the\nfrequency-averaged dissipation is constant and interpreted this result with a simple damped\nharmonic oscillator. Ogilvie (2013) con\frmed this result in a spherical shell for rotating\n\row and Lin & Ogilvie (2017) for rotating MHD.\nTo end this paper, we discuss some aspects which we did not consider in this work and\ncan be studied in the future. Firstly, we focus on the two magnetic parameters SandPm\nbut did not investigate the other parameters, i.e. Ekman number Emeasuring rotation and\nthe radius ratio Ri=Romeasuring the stellar or planetary structure. The dependence on Eis\nnot very clear so far (Ogilvie 2014). The dependence of viscous dissipation in rotating \row\non the core size was discussed in (Goodman & Lackner 2009) with the WKB analysis based\non the critical latitude of inertial waves but this dependence is not clear in rotating MHD\nbecause magnetic \feld removes the critical latitude. These two parameters, EandRi=Ro,\nneed to be further studied. Secondly, the imposed magnetic \feld in our model is uniform\nand vertical, which has the simplest geometry, and more complex \feld geometries should\nbe further considered, e.g. dipolar and quadrupolar \felds. In (Wei & Hollerbach 2010)\nvarious \feld geometries were studied in spherical Couette \row, namely \row between two\ndi\u000berentially rotating spheres, and it was found that \row tends to be along \feld lines due\nto Alfv\u0013 en's frozon-in theorem, and it can be inferred that the structure of tidal \row with\nvarious magnetic \feld geometries will be similar, namely \row along the \feld lines. Thirdly,\nour model is linear with the neglect of all the quadratic terms, i.e. u\u0001ru, (r\u0002b)\u0002b\nandr\u0002(u\u0002b). In (Wei 2016b) it was found that the nonlinear inertial force suppresses\nthe tidal response and dissipation near resonance. In (Favier et al. 2014) it was also found\nthat the dynamical tide in the nonlinear regime is quite di\u000berent from that in the linear\nregime. So the nonlinear regime should also be further studied for comparison with the\nlinear regime.{ 20 {\nAcknowledgements\nProf. Gordon Ogilvie and Dr. Yufeng Lin gave me valuable discussions during this\nwork. An anonymous referee gave me good comments and suggestions and provided\nthe information about the online published paper (Lin & Ogilvie 2017). The work was\nsupported by the grant of 1000 youth talents of the Chinese government.{ 21 {\n0 0.5 1 1.5 2 2.5 310−0.810−0.710−0.610−0.510−0.410−0.310−0.210−0.1100\ntidal frequencykinetic energykinetic energy (or tidal response) versus tidal frequency at Pm=1.0\n  \nS2=0\nS2=0.01\nS2=0.02\nS2=0.05\nS2=0.1\nS2=0.2\nFig. 1.| Kinetic energy (or tidal response) versus tidal frequency at various S2.Pm= 1:0.{ 22 {\n0 0.5 1 1.5 2 2.5 310−210−1100101102\ntidal frequencyviscous dissipationviscous dissipation versus tidal frequency at Pm=1.0\n  \nS2=0\nS2=0.01\nS2=0.02\nS2=0.05\nS2=0.1\nS2=0.2\n(a)\n0 0.5 1 1.5 2 2.5 310−1100101102\ntidal frequencymagnetic dissipationmagnetic dissipation versus tidal frequency at Pm=1.0\n  \nS2=0.01\nS2=0.02\nS2=0.05\nS2=0.1\nS2=0.2 (b)\n0 0.5 1 1.5 2 2.5 310−210−1100101102103\ntidal frequencytotal dissipationtotal dissipation versus tidal frequency at Pm=1.0\n  \nS2=0\nS2=0.01\nS2=0.02\nS2=0.05\nS2=0.1\nS2=0.2\n(c)\n0 0.5 1 1.5 2 2.5 30.20.40.60.811.21.41.61.82\ntidal frequency(magnetic dissipation)/(viscous dissipation)(magnetic dissipation)/(viscous dissipation) versus tidal frequency at Pm=1.0\n  \nS2=0.01\nS2=0.02\nS2=0.05\nS2=0.1\nS2=0.2 (d)\nFig. 2.| (a) Viscous dissipation, (b) magnetic dissipation, (c) total dissipation and (d) ratio\nof magnetic to viscous dissipations versus tidal frequency at various S2.Pm= 1:0.{ 23 {\n(a)\n(b)\n (c)\n (d)\n(e)\n (f)\nFig. 3.| (a) shows the contours of kinetic energy at S= 0, and (b)-(f) show the contours\nof kinetic (left panel) and magnetic (right panel) energies at, respectively, S2= 0:01, 0:02,\n0:05, 0:1 and 0:2. The tidal frequency is != 1:0.{ 24 {\n(a)\n(b)\n (c)\n (d)\n(e)\n (f)\nFig. 4.| The same as Figure 3 but for the tidal frequency != 3:0.{ 25 {\n0.01 0.02 0.05 0.1 0.210−0.810−0.710−0.610−0.5\nS2kinetic energykinetic energy versus S2 at Pm=1.0\n  \nω=0.1\nω=0.2\nω=0.4\nω=0.8\nω=1.6\nω=3.0\n(a)\n0.01 0.02 0.05 0.1 0.210−1100101102\nS2viscous dissipationviscous dissipation versus S2 at Pm=1.0\n  \nω=0.1\nω=0.2\nω=0.4\nω=0.8\nω=1.6\nω=3.0\n(b)\n0.01 0.02 0.05 0.1 0.210−1100101102\nS2magnetic dissipationmagnetic dissipation versus S2 at Pm=1.0\n  \nω=0.1\nω=0.2\nω=0.4\nω=0.8\nω=1.6\nω=3.0 (c)\nFig. 5.| (a) kinetic energy (b) viscous dissipation and (c) magnetic dissipation versus S2\nat various!.Pm= 1:0. The two dashed lines in 5(b) and 5(c) denote the scaling law /S2.{ 26 {\n0 0.5 1 1.5 2 2.5 3100101102\ntidal frequencyviscous dissipationviscous dissipation versus tidal frequency at S2=0.1\n  \nPm=1.0\nPm=0.5\nPm=0.2\nPm=0.1\n(a)\n0 0.5 1 1.5 2 2.5 3100101102\ntidal frequencymagnetic dissipationmagnetic dissipation versus tidal frequency at S2=0.1\n  \nPm=1.0\nPm=0.5\nPm=0.2\nPm=0.1 (b)\n0 0.5 1 1.5 2 2.5 3100101102\ntidal frequencytotal dissipationtotal dissipation versus tidal frequency at S2=0.1\n  \nPm=1.0\nPm=0.5\nPm=0.2\nPm=0.1\n(c)\n0 0.5 1 1.5 2 2.5 3024681012\ntidal frequency(magnetic dissipation)/(viscous dissipation)(magnetic dissipation)/(viscous dissipation) versus tidal frequency at S2=0.1\n  \nPm=1.0\nPm=0.5\nPm=0.2\nPm=0.1 (d)\nFig. 6.| (a) Viscous dissipation, (b) magnetic dissipation, (c) total dissipation and (d) ratio\nof magnetic to viscous dissipations versus tidal frequency at various Pm.S2= 0:1.{ 27 {\n0.1 0.2 0.5 1100101102\nPmviscous dissipationviscous dissipation versus Pm at S2=0.1\n  \nω=0.1\nω=0.2\nω=0.4\nω=0.8\nω=1.6\nω=3.0\n(a)\n0.1 0.2 0.5 1100101102\nPmmagnetic dissipationmagnetic dissipation versus Pm at S2=0.1\n  \nω=0.1\nω=0.2\nω=0.4\nω=0.8\nω=1.6\nω=3.0 (b)\nFig. 7.| (a) viscous dissipation and (b) magnetic dissipation versus Pm at various !.\nS2= 0:1. The two dashed lines denote the scaling laws /Pm0:6in (a) and/Pm\u00000:3in (b).{ 28 {\nREFERENCES\nBarker, A. J., & Ogilvie, G. I. 2010, MNRAS, 404, 1849\nBu\u000bett, B. A. 2010, Nature, 468, 952\nCampbell, C. G., & Papaloizou, J. C. B. 1986, MNRAS, 220, 577\nEssick, R., & Weinberg, N. N. 2016, ApJ, 816, 18\nFavier, B., Barker, A. J., Baruteau, C., & Ogilvie, G. I. 2014, MNRAS, 439, 845\nFuller, J., & Lai, D. 2012, MNRAS, 421, 426\nGoldreich, P., & Nicholson, P. D. 1989, ApJ, 342, 1079\nGoodman, J., & Dickson, E. S. 1998, ApJ, 507, 938\nGoodman, J., & Lackner, C. 2009, ApJ, 696, 2054\nGoodman, J., & Oh, S. P. 1997, ApJ, 486, 403\nHollerbach, R. 2000, Int. J. Numer. Meth. Fluids, 32, 773\nKumar, P., & Goodman, J. 1996, ApJ, 466, 946\nLehnert, B. 1954, ApJ, 119, 647\nLin, Y., & Ogilvie, G. I. 2017, MNRAS, doi:10.1093/mnras/stx2764\nOgilvie, G. I. 2005, Journal of Fluid Mechanics, 543, 19\n|. 2009, MNRAS, 396, 794\n|. 2013, MNRAS, 429, 613\n|. 2014, ARA&A, 52, 171{ 29 {\nOgilvie, G. I., & Lin, D. N. C. 2004, ApJ, 610, 477\n|. 2007, ApJ, 661, 1180\nSavonije, G. J., & Papaloizou, J. C. B. 1983, MNRAS, 203, 581\nSouchay, J., Mathis, S., & Tokieda, T., eds. 2013, Tides in Astronomy and Astrophysics\n(Berlin Springer Verlag)\nWei, X. 2016a, ApJ, 828, 30\n|. 2016b, J. Fluid Mechanics, 796, 306\nWei, X., & Hollerbach, R. 2010, Acta Mechanica, 215, 1\nWeinberg, N. N., Arras, P., Quataert, E., & Burkart, J. 2012, ApJ, 751, 136\nWu, Y. 2005a, ApJ, 635, 674\n|. 2005b, ApJ, 635, 688\nZahn, J.-P. 1970, A&A, 4, 452\n|. 1975, A&A, 41, 329\n|. 1977, A&A, 57, 383\nThis manuscript was prepared with the AAS L ATEX macros v5.2."    },    {        "title": "1706.03421v2.The_contribution_of_kinetic_helicity_to_turbulent_magnetic_diffusivity.pdf",        "content": "arXiv:1706.03421v2  [physics.flu-dyn]  24 Aug 2017The contribution of kinetic helicity to turbulent magnetic diffusivity\nA. Brandenburg1,2,3,4,⋆, J. Schober3,andI. Rogachevskii5,1,3\n1Laboratory for Atmospheric and Space Physics, University o f Colorado, Boulder, CO 80303, USA\n2JILA and Department of Astrophysical and Planetary Science s, University of Colorado, Boulder, CO 80303, USA\n3Nordita, KTH Royal Institute of Technology and Stockholm Un iversity, 10691 Stockholm, Sweden\n4Department of Astronomy, AlbaNova University Center, Stoc kholm University, SE-10691 Stockholm, Sweden\n5Department of Mechanical Engineering, Ben-Gurion Univers ity of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel\nNovember 11, 2021, Revision: 1.32\nKey words magnetic fields – magnetohydrodynamics (MHD)\nUsing numerical simulations of forced turbulence, it is sho wn that for magnetic Reynolds numbers larger than unity, i.e .,\nbeyond the regime of quasilinear theory, the turbulent magn etic diffusivity attains an additional negative contribut ion that\nis quadratic in the kinetic helicity. In particular, for lar ge magnetic Reynolds numbers, the turbulent magnetic diffu sivity\nwithout helicity is about twice the value with helicity. Suc h a contribution was not previously anticipated, but, as we\ndiscuss, it turns out to be important when accurate estimate s of the turbulent magnetic diffusivity are needed.\n1 Introduction\nLarge-scale magnetic fields in the turbulent convection\nzones of stars or in supernova-driven turbulence of the in-\nterstellar medium of galaxies evolve according to the equa-\ntions of mean-field electrodynamics and in particular the\nmean-field induction equation. This equation is similar to\nthe original induction equation for the actual magnetic fiel d,\nwhich includes the fluctuations around the mean magnetic\nfield. The presence of turbulence leads to enhanced effec-\ntive magnetic diffusion, which is often orders of magnitude\nlarger than the microphysical value, although this is usual ly\nnot the case in numerical simulations and no restriction con -\ncerning this ratio will be made in this paper. If the velocity\nfield is helical, there is, in addition to ordinary turbulent dif-\nfusion, also the αeffect, which can destabilize an initially\nweak large-scale magnetic field and lead to its exponential\ngrowth.\nMathematically, the evolution of the mean magnetic\nfieldBis described by the equation,\n∂B\n∂t=∇×/bracketleftbig\nαB−(ηt+η)µ0J/bracketrightbig\n, (1)\nwhereJ=∇×B/µ0is the mean current density, µ0is\nthe vacuum permeability, ηis the microphysical magnetic\ndiffusivity, and overbars denote spatial averaging, which we\nwill later specify to be horizontal averaging over two spa-\ntial coordinates xandy. For the purpose of this discussion,\nand throughout this paper, we assume the turbulence to be\nisotropic; otherwise, αandηtwould have to be replaced by\ntensors.\n⋆Corresponding author: brandenb@nordita.orgThe relative importance of turbulent diffusion to micro-\nphysical diffusion is measured by the magnetic Reynolds\nnumber,\nRm=urms/ηkf, (2)\nwhereurmsis the rms velocity of the turbulence and kfis the\nwavenumber of the energy-carrying eddies. The magnetic\ndiffusivity is inversely proportional to the electric cond uc-\ntivity, so in the low conductivity limit, i.e., Rm≪1, we\nhave (Krause & R¨ adler 1980)\nηt=−1\n3η/parenleftBig\nψ2−φ2/parenrightBig\nandα=−1\n3ηψ·u, (3)\nwhereu=∇×ψ+∇φis the turbulent velocity expressed\nin terms of a vector potential ψand a scalar potential φ.\nIn the following, we perform averaging over two coordinate\ndirections.\nOne often considers the limiting case of incompress-\nible turbulence, so φ= 0 andψ2=u2/k2\nfas well as\nψ·u=ω·u/k2\nf, wherekfis the wavenumber of the\nenergy-carrying eddies and ω=∇×uis the vorticity.\nIn that case, we can write\nηt=1\n3τu2andα=−1\n3τω·u (4)\nwith\nτ= (ηk2\nf)−1(5)\nbeing the microphysical magnetic diffusion time based on\nthe wavenumber kf. We reiterate that this expression applies\nonly to isotropic conditions. Indeed, simple anisotropic\nflows can be constructed, where ω·u= 0, butψ·u/ne}ationslash= 0,\nand so they do yield an αeffect (R¨ adler & Brandenburg\n2003). Furthermore, in the compressible case, there is a neg -\native contribution to ηt, so that it can even become negative,\nas has been demonstrated by R¨ adler et al. (2011).\nc/circlecopyrt2 A. Brandenburg, J. Schober, & I. Rogachevskii: Contributi on of helicity to turbulent magnetic diffusivity\nBy contrast, in the high conductivity limit, Rm≫1,\nEq. (4) still applies (Krause & R¨ adler 1980), but now\nτ≈(urmskf)−1(Rm≫1) (6)\nbeing the correlation time. This was also confirmed numer-\nically using the test-field method (Sur et al. 2008), althoug h\nour new results discussed below will show a slight twist to\ntheRm-dependence of their result.\nEquation (4) is also motivated by dimensional argu-\nments. In particular, since αis a pseudoscalar, it is clear that\nin the present case, where the only pseudoscalar in the sys-\ntem isω·u, there can be no other contribution to α. This\nis, however, not the case for ηt, which is just an ordinary\nscalar. Thus, in the present case, there may well be an ad-\nditional contribution proportional to (ω·u)2, for example.\nThe purpose of this paper is to show that this is indeed the\ncase.\nA particularly useful diagnostics is the ratio ηt/α, be-\ncause it is expected to be independent of τand equal to\nu2/ω·uin the limit of small magnetic Reynolds numbers,\nwhere Eq. (4) is obeyed exactly. In this paper, we shall con-\nfirm that this is indeed the case when Rm≪1, but we\nfind a departure from this simple result as Rmis increased.\nWe shall use the test-field method (Schrinner et al. 2005,\n2007), which has been highly successful in measuring tur-\nbulent transport coefficients in isotropic turbulence (Sur et\nal. 2008, Brandenburg et al. 2008b), shear flow turbulence\n(Brandenburg 2005, Brandenburg et al. 2008a, Gressel et\nal. 2008, Gressel 2010, Madarassy & Brandenburg 2010),\nas well as magnetically quenched turbulence (Brandenburg\net al. 2008c, Karak et al. 2014).\n2 Test field method in turbulence simulations\nAs in a number of previous cases (e.g., Brandenburg 2001),\nwe reconsider isotropically forced turbulence either with or\nwithout helicity using an isothermal equation of state. Sin ce\nthe magnetic field is assumed to be weak, there is no back-\nreaction of the magnetic field on the flow. Furthermore, in-\nstead of solving for the magnetic field, we just solve for the\nfluctuations of the magnetic field that arise from a set of\ngiven test fields. This equation is given by\n∂bT\n∂t=∇×/parenleftBig\nu×BT+U×bT+u×bT−u×bT/parenrightBig\n+η∇2bT. (7)\nHere,U+u≡Uis the time-dependent flow, which we\ntake to be the solution to the momentum and continuity\nequations with constant sound speed cs, a random forcing\nfunctionf, densityρ, and the traceless rate of strain tensor\nSij=1\n2(Ui,j+Uj,i)−1\n3δij∇·U(commas denote partial\ndifferentiation),\n∂U\n∂t=−U·∇U−c2\ns∇lnρ+1\nρ∇·(2νρS)+f, (8)\n∂lnρ\n∂t=−U·∇lnρ−∇·U. (9)The following four test fields, BT, are used:\n\ncosk1z\n0\n0\n,\nsink1z\n0\n0\n,\n0\ncosk1z\n0\n,\n0\nsink1z\n0\n.(10)\nFor eachBT, the solutionsbTallow us to compute the mean\nelectromotive force, ET=u×bT, and relate it to BTand\nµ0JT≡∇×BTvia\nEi=αijBT\nj−ηijµ0JT\nj. (11)\nThe four independent test fields constitute eight scalar equ a-\ntions for the xandycomponents of Eiwithi= 1 and 2,\nthat can be solved for the eight unknown relevant compo-\nnents ofαijandηijwithi,j= 1,2. Thei= 3component\ndoes not enter, because we use averaging over xandy, so\nB3= const = 0 owing to ∇·B= 0and the absence of a\nuniform imposed field.\nFor isotropically forced turbulence, we expect α12=\nα21=η12=η21= 0,α11=α22=α, andη11=η22=\nηt. This is, however, only true in a statistical sense, and sinc e\nαandηtare still functions of zandt, we must average over\nthese two coordinates, so we compute\nα=1\n2/an}b∇acketle{tα11+α22/an}b∇acket∇i}htzt, ηt=1\n2/an}b∇acketle{tη11+η22/an}b∇acket∇i}htzt, (12)\nwhere/an}b∇acketle{t·/an}b∇acket∇i}htztdenotes averaging over zandt.\nWe use the forcing function fthat consists of ran-\ndom, white-in-time, plane waves with a certain average\nwavenumber kf(Brandenburg 2001),\nf(x,t) = Re{N˜f(k,t)exp[ik·x+iφ]}, (13)\nwherexis the position vector. We choose N=f0/radicalbig\nc3s|k|,\nwheref0is a nondimensional forcing amplitude. At each\ntimestep, we select randomly the phase −π < φ≤πand\nthe wavevector kfrom many possible discrete wavevectors\nin a certain range around a given value of kf. The Fourier\namplitudes,\n˜f(k) =R·˜f(k)(nohel)withRij=δij−iσǫijkˆk√\n1+σ2,(14)\nwhere the parameter σcharacterizes the fractional helicity\noff, and\n˜f(k)(nohel)= (k׈e)//radicalbig\nk2−(k·ˆe)2 (15)\nis a nonhelical forcing function. Here, ˆeis an arbitrary unit\nvector not aligned with k,ˆkis the unit vector along k, and\n|˜f|2= 1.\nWe will consider both σ= 0andσ= 1, corresponding\nto nonhelical and maximally helical cases. We vary Rm, de-\nfined in Eq. (2), by changing ηwhile keeping ν=ηin all\ncases. We use the P ENCIL CODE1with a numerical resolu-\ntion of up to 2883meshpoints in the case with Rm≈120,\nwhich is the largest value considered here.\n1https://github.com/pencil-code\nc/circlecopyrtAstron. Nachr. / (0000) 3\n3 Results\n3.1 Dependence of αandηtonRm\nAs theoretically expected (Moffatt 1978, Krause & R¨ adler\n1980), and previously demonstrated using the test-field\nmethod (Sur et al. 2008), αandηincrease linearly with Rm\nforRm<1; see Figs. 1 and 2 for nonhelical and helical\ncases. Here, error bars have been evaluated as the maxi-\nmum departure from the averages for any one third of the\nfull time series. In the helical case, both αandηsaturate\naround unity, but in the non-helical case, ηovershoots the\nhelical value by almost a factor of two; see Fig. 2.\n3.2 Ratio of αtoηt\nIn Fig. 3, we plot the ratio ηt/α, normalized by ηt0/α0,\nwhereα0=−urms/3andηt0=urms/3kf. The minus sign\nin our expression for α0takes into account that we are forc-\ning with positive helicity, which then leads to a negative α\neffect (Moffatt 1978, Krause & R¨ adler 1980). For small val-\nues ofRm, this ratio is unity, but it reaches a value of about\ntwo when Rm≈50.\nFig. 1 Dependence of αonRmfor the models with max-\nimum helicity.\n3.3 Difference between nonhelical and helical cases\nIt turns out that the difference between ηtin the nonhelical\nand helical cases increases quadratically in Rm; see Fig. 4.\nThis shows first of all that the difference vanishes for small\nRm, but it also suggests that there is a correction to ηtdue\nto the presence of helicity that is not captured by the second\norder correlation approximation, which is exact for Rm≪\n1. It should be possible, however, to capture this effect of\nhelicity on ηtusing a higher order approximation, which\nhas not yet been attempted, however.Fig. 2 Dependence of ηtonRmfor models with max-\nimum helicity (dashed blue) and with zero helicity (solid\nblack).\nFig. 3 Ratio ofηt/α.\nFig. 4 Rmdependence of the difference between ηtfor\nmodels with zero helicity an maximum helicity.\n3.4 Relation to earlier results\nA similar situation has been encountered previously in the\ncase of the Galloway–Proctor flow (Galloway & Proctor\n1992), where, in addition to an αeffect and turbulent dif-\nc/circlecopyrt4 A. Brandenburg, J. Schober, & I. Rogachevskii: Contributi on of helicity to turbulent magnetic diffusivity\nfusion, also a turbulent pumping effect was found (Cour-\nvoisier et al. 2006). This result was not obtained under the\nsecond order correlation approximation (R¨ adler & Bran-\ndenburg 2009). Using the test-field method, they showed,\nhowever, that the value of γ, which quantifies the turbulent\npumping velocity, does indeed vanish for Rm≪1, but it\nwas found to increase with RmasR5\nm; see R¨ adler & Bran-\ndenburg (2009), who interpreted this as a higher order effec t\nthat should be possible to capture with a six order approxi-\nmation. Our present result therefore suggests that the diff er-\nence between nonhelical and helical cases can also be de-\nscribed as a result of a higher order approximation, which,\nin this case, would be a fourth order approximation.\n4 Conclusions\nOur present results have demonstrated that, at least for int er-\nmediate values of Rmin the range between 1 and 120, there\nis a contribution to the usual expression for the turbulent\nmagnetic diffusivity ηt=τu2/3that depends on (ω·u)2.\nThis is somewhat surprising in the sense that such a result\nhas not previously been reported, but it is fully compatible\nwith all known constraints: no correction for Rm≪1and\nno dependence on the sign of ω·u. On the other hand, our\nresults may still be compatible with the τapproximation\nin the high conductivity limit, if the difference between th e\nturbulent diffusivity in the nonhelical and helical cases v an-\nishes forRm→ ∞ . However, our numerical results do not\nclearly confirm this, because our largest value of Rmwas\nonly about 120.\nThere is a practically relevant application to this phe-\nnomenon, at least in the case of forced turbulence, where its\neffect on the large-scale magnetic field evolution can now\nbe quantified to high accuracy. A factor of nearly two in the\nvalue ofηtis clearly beyond the acceptable accuracy for\nthis case. This was noticed in recent studies of αeffect and\nturbulent diffusion in the presence of the chiral magnetic\neffect (Schober et al. 2017). Our present result therefore r e-\nmoves an otherwise noticeable discrepancy relative to the\ntheoretical predictions. Future applications hinge obvio usly\non the overall accuracy of analytic approximations to par-\nticular circumstances. In most cases, naturally driven flow\nturbulence will be anisotropic, so we expect more compli-\ncated tensorial results for turbulent diffusion.\nAcknowledgements. Support through the NSF Astrophysics and\nAstronomy Grant Program (grant 1615100), and the Research\nCouncil of Norway (FRINATEK grant 231444), are gratefully\nacknowledged. We acknowledge the allocation of computing r e-\nsources provided by the Swedish National Allocations Commi ttee\nat the Center for Parallel Computers at the Royal Institute o f Tech-\nnology in Stockholm. This work utilized the Janus supercomp uter,\nwhich is supported by the National Science Foundation (awar d\nnumber CNS-0821794), the University of Colorado Boulder, t he\nUniversity of Colorado Denver, and the National Center for A t-\nmospheric Research. The Janus supercomputer is operated by the\nUniversity of Colorado Boulder.References\nBrandenburg, A. 2001, ApJ, 550, 824\nBrandenburg, A. 2005, Astron. Nachr., 326, 787\nBrandenburg, A., R¨ adler, K.-H., Rheinhardt, M., & K¨ apyl¨ a, P. 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Fluid Dyn.,\n101, 81\nSur, S., Brandenburg, A., & Subramanian, K. 2008, MNRAS, 385 ,\nL15\nc/circlecopyrt"    },    {        "title": "2011.10215v1.An_Investigation_of_Commercial_Iron_Oxide_Nanoparticles__Advanced_Structural_and_Magnetic_Properties_Characterization.pdf",        "content": " 1 An Investigation of Commercial  Iron Oxide Nanoparticles : \nAdvanced Structural and Magnetic Properties Characterization  \n \nKai Wu†,⊥,*, Jinming Liu†,⊥, Renata Saha†,⊥, Chaoyi Peng†, Diqing Su‡, Andrew Yongqiang Wang§,*, and Jian -\nPing Wang†,* \n†Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455, \nUSA  \n‡Department of Chemical Engineering and Material Science, University of Minnesota, Minneapolis, Minnesota \n55455, USA  \n§Ocean NanoTech, LLC, San Diego, California 92126, USA  \n \nABSTRACT : Magnetic nanoparticles  (MNPs)  have been extensively used as tiny heating sources in magnetic \nhyperthermia therapy, contrast agents in magnetic resonance ima ging (MRI), tracers in magnetic particle imaging \n(MPI), carriers for drug/gene delivery , etc. There have emerged  many  magnetic nanoparticle/microbeads \nsuppliers since the last decade, such as Ocean NanoTech, Nanoprobes, US Research Nanomaterials, Miltenyi \nBiotec, micromod Partikeltechnologie GmbH , and nanoComposix, etc. In this paper , we report the physical and \nmagnetic characterization s on iron oxide nanoparticle products from Ocean NanoTech. Standard characterization \ntools such as Vibrating -Sample Magneto meter (VSM), X -Ray Diffraction (XRD), Dynamic Light Scattering \n(DLS), Transmission Electron Microscopy (TEM), and Zeta Potential Analyzer  are used to provide magnetic \nnanoparticle customers and researchers with an overview  of these iron oxide nanoparticle products. In addition, \nthe dynamic magnetic responses of these iron oxide nanoparticles in aqueous solutions are investigated under low \nand high frequency alternating  magnetic fields, giving a  standard ized operating procedure for characterizing the \nMNPs  from Ocean Nano Tech, thereby yielding the best of magnetic nanoparticles for different applications.  \nKEYWORDS : magnetic nanoparticle, iron oxide, magnetic particle imaging, dynamic magnetic response  \n \n1. INTRODUCTION  \nMagnetic nanoparticles (MNPs) are nanomaterials with sizes between 1 nm and 100 nm. Due to their large \nsurface -to-volume ratio and tunable magnetic properties, MNPs have emerged as one of the most important \nnanomaterials in magnetic, chemical and biomedical applicati ons. The surface of the MNPs can be functionalized \nwith various coatings from inorganic coatings such as silica1 and carbon2 to organic coatings such as polyethylene \nglycol (PEG)3 and dopamine (DP A)4. Compared to non -magnetic particles, MNPs can be manipulated by an \nexternal magnetic field without any physical contact, which leads to various applications such as drug delivery5 \nas well as the separation and concentration of certain molecules6. Under an alternating  magnetic field, MNPs can  2 induce localized temperature increase at the target spot, which makes them promising candidates for hyperthermia \napplications7. Under an external magnetic field, MNPs ca n generate stray fields. By integrating with various \nmagnetic sensors such as magnetoresistance sensors8,9, hall sensors10,11, nuclear magnetic resonance (NMR) \nsensors11, magnetic resonance imaging (MRI)12, and magnetic particle spectroscopy  (MPS)13, MNPs can also \nserve as magnetic marker s in diagnostic applications.  \n    To date, MNPs with various sizes and surface coatings have been successfully commercialized and are \navailable in many co mpanies such as Ocean NanoTech  (San Diego, USA) , Nanoprobes  (New York City, USA), \nUS Research Nanomaterials (Houston, USA), Miltenyi Biotec  (Bergisch Gladbach, Germany) , micromod \nPartikeltechnologie GmbH  (Rostock, Germany)  and nanoComposix  (San Diego, USA) , etc . For those \naforementioned applications , the quest for high magnetic moment, uniform size distribution, and colloidal \nstability MNPs has pushed the development of various nanoparticle manufacturers. In this paper , we first \ncharacterized the magnetic and physical properties of single -core, differently -sized  iron oxide nanoparticle \nproducts from Ocean NanoTech  using Vibrating -Sample Magnetometer (VSM), X -Ray Diffraction (XRD), \nDynamic Light Scattering (DLS), Transmission Electron Microscopy (TEM),  and Zeta Potential Analyzer  \n(summarized in Table 1 ). In addition, we give application -oriented assessment s on these MNP products using a \nhome -built MPS system. Practical suggestions on the applications of these iron oxide nanoparticle with varying \ncore sizes are given at the end of this paper to maxi mize the use of them.  \n \n2. MATERIALS AND METHODS  \n    2.1. Materials.  The SHA series MNPs are provided by the Ocean NanoTech. Six SHA series MNPs with \naverage magnetic core sizes of 5 nm, 10 nm, 15 nm, 20 nm, 25 nm, and 30 nm are characterized  in this paper  \n(denoted as SHA -5, SHA -10, SHA -15, SHA -20, SHA -25, and SHA -30, respectively . Photographs of SHA series \nMNPs used in this work can be found in Supporting Information S1 ). The SHA series MNPs are a group of water -\nsoluble  iron oxide nanoparticles coated with  amphiphilic polymer and functionalized amine reactive groups. They \nare very stable in most buffers in the pH range of 4 – 10 and can be readily conjugated to protein, peptide and \nother carboxylic acid containing molecules.  \n    2.2. Vibrating  Sample Magnetometer  (VSM) Measurement . 25 uL  of SHA  series MNP suspension is \npipetted  onto filter paper and air -dried before the VSM measurements. Three independent magnetization curves \nof each sample are obtained at 20 °C, with the external magnetic field swept f rom -5000 to +5000 Oe ( field step \nof 10 Oe and averaging time of 200 ms ), -500 to +500 Oe  (field step of 2 Oe and averaging time of 200 ms) , and \n-200 to +200 Oe  (field step of 1 Oe and averaging time of 200 ms) , respectively.  \n    2.3. X-Ray Diffraction  (XRD)  Measurement . 50 µL SHA series MNPs suspension is pipetted onto a Si/SiO 2 \nslide and air -dried before the XRD characterization. Cobalt radiation source ( wavelength ~1.79 Å) is used for the \nXRD characterization since it has lower fluorescence especially for magnetite and maghemite14. For a convenient  3 comparison, the characterized XRD patterns are converted to copper radiati on. The crystal structure of SHA series \nMNPs are characterized via the x -ray diffraction (XRD, Bruker D8 Discover 2D).   \n    2.4. Dynamic Light Scattering  (DLS) Measurement . The hydrodynamic size distribution of the SHA series \nMNPs  are characterized using DLS Particle Tracking Analyzer (Model: Microtac Nanoflex). 100 µL of the SHA \nseries MNP  suspension is diluted in  1.4 mL of DI water, reaching a total sample volume of 1.5 mL mixture and \nfollowed by  ultra-sonicat ion for 30 min utes befor e the DLS characterization.  \n    2.5. Transmission Electron Microscopy  (TEM) Analysis. The morpholog ies of these SHA series  MNPs are \ncharacterized by a TEM  system  (FEI T12 120 kV). Each  TEM sample is prepared by putting a drop let (~10 µL) \nof MNP suspension  onto a TEM grid (copper mesh coated with amorphous carbon film). These samples are ready \nfor TEM characterization when the solutions are  fully evaporated  at room temperature in air .  \n    2.6. Zeta Potential Measurement.  Zeta Potential Analyzer (Model: Sta bino) is used to characterize the particle \ncharge distribution or the zeta potential of the SHA series MNPs in DI water. 100 µL of SHA series MNP  is \ndiluted in 4.9 mL of DI water,  reaching  a total sample volume of 5 mL , followed by ultra-sonication  for 30 \nminutes and then used for zeta potential characterization. This particle charge characterization help s analyze the \nsurface binding capabilities of these SHA series MNPs.  \n    2.7. Magnetic Particle Spectroscopy (MPS) Measurement.  The dynamic magnetic responses of SHA series \nMNPs are characterized by a home built  MPS system (see the schematic view and photographs of MPS system \nin S2 and S3  from  Supporting Information ). 200 µL  of SHA series MNP sample is sealed in a plastic vial \n(maximum capacit y of 300 µL). Two sets of copper coils are used to generate sinusoidal magnetic fields with \ntunable frequencies  and magnitudes . One pair of differentially wound pick -up coils (600 windings clockwise and \n600 windings counter -clockwise) collect s the induced voltage signals due to the dynamic magnetic responses of \nMNPs under driving magnetic fields. A laptop with LabVIEW controls  the frequency and magnitude of driving \nmagnetic field through a data acquisition card (DAQ, NI USB -6289).  The analog voltage signals  are sent back \nfrom pick -up coils to DAQ, sampled at 500 kHz, and converted to frequency domain after discrete Fourier \ntransform (DFT). For each MPS measurement, the MPS system run s for 10 seconds to collect the baseline signal \n(noise) followed by insertin g the vial containing MNP sample for another 10 seconds’ signal (total) collection. \nThe induced voltage due to dynamic magnetic responses of MNPs is recovered from the total signal by the phasor \ntheory (see S4 from  Supporting Information ). The higher harmonics specific to dynamic magnetic responses of \nMNPs are extracted for analysis  (see S7 from Supporting Information ). \n \n3. RESULTS AND DISCUSSION  \n    3.1. Magnetic Properties of SHA Series MNPs.  The hysteresis curves of SHA series MNPs are recorded by \nVSM, under field ranges of -5000 Oe – 5000 Oe, -500 Oe – 500 Oe, and -200 Oe – 200 Oe. The magnetic moment \nper microliter is averaged over 25 µL MNP sample and plotted in Figure 1(a) – (c). Under fie ld strength of 500  4 Oe, the saturation magnetizations from highest to lowest are:  SHA -15 > SHA -20 > SHA -30 > SHA -25 > SHA -\n10 > SHA -5. With SHA -5, SHA -10, SHA -15, and SHA -20 being superparamagnetic. We also observed hysteresis \nloops from SHA -25 and SHA -30. The SHA -5 MNPs show very large coercivity, which may due to the surface \nspin-canting effect 15. Due to the varying particle concentrations in SHA series MNP products (listed in Table 1 ), \nthe magnetic moment per microliter is not comprehensive to represent the magnetic property of each MNP. In \naddition, magnetic moment per particle is also summarized in Figure 1(d) – (f), with SHA -30 showing the highest \nmagnetic moment per particle followed by SHA -25, SHA -20, SHA -15, SHA -10, and SHA -5 showing the lowest \nmagnetic moment per particle.  \n \nFigure 1. Magnetization curves of SHA series MNPs obtained by VSM at 20 °C. External field sweeps from (a)  \n& (d) -5000 to +5000 Oe, (b)  & (e) -500 to +500 Oe, and (c) & (f) -200 to +200 Oe.  Magnetization units are \nrepresented by emu/µL and emu/particle for (a) – (c) and (d) – (f), respectively.  \n \nThe crystal structure of SHA series MNPs are characterized via  the x-ray diffraction (XRD, Bruker D8 \nDiscover 2D) as shown in Figure 2 . It is observed that Fe3O4 and γ-Fe2O3 are the main phases in SHA series \nMNPs . There are also several diffraction peaks from the solution denoted by the blue dashed lines in Figure 2 . \nThe sharp diffraction peaks  (labeled by black diamonds)  come  from the chemicals in the MNP buffer (NaCl, KCl, \nNa2HPO 4, KH 2PO 4, etc), and the peaks labeled by black rounds come from the Si/SiO 2 substrate . The full width  5 at half  maximum (FWHM) of the diffraction peaks are wider for the MNPs compared to their bulk counterparts. \nThis is due to the peak broadening effects for the nanoscale materials based on Scherrer equation.  \n \nFigure 2. XRD patterns of SHA series  MNPs . The powder diffraction file s (PDF s) of FeO, Fe 3O4, α-Fe2O3, and \nγ-Fe2O3 are added for references.  \n \nTable 1 . Physical and magnetic properties of SHA series MNPs.  \nSample  Average Size  \n(nm) ± SD  Zeta Potential  \n(mV) ± SD  Saturation \nMagnetization \n(emu/g Fe)a) Saturation \nMagnetization \n(emu/g)b) Magnetic \nMoment \n(emu/particle)  Material  \nSHA -5 10.46 ± 3.88  -0.03 ± 0.005  63.84  ~44.69  1.54×10-17 γ-Fe2O3, Fe3O4 \nSHA -10 18.07 ± 4.72  5.03 ± 0.07  42.64  ~29.85  8.23×10-17 γ-Fe2O3, Fe3O4 \nSHA -15 20.69 ± 6.31  7.66 ± 0.05  83.44  ~58.41  5.13×10-16 γ-Fe2O3, Fe3O4 \nSHA -20 27.56 ± 11.29  -0.41 ± 0.33  78.08  ~54.66  1.18×10-15 γ-Fe2O3, Fe3O4 \nSHA -25 28.28 ± 10.38  1.15 ± 0.49  55.28  ~38.70  1.58×10-15 γ-Fe2O3, Fe3O4 \nSHA -30 32.60 ± 12.17  -0.69 ± 0.04  51.12  ~35.78  2.50×10-15 γ-Fe2O3, Fe3O4 \na) Magnetic moment per gram of Fe  (emu/ g Fe) vs. field can be found in  S5 from  Supporting Information . \nb) Magnet ic moment per gram of iron oxide nanoparticle is calculated by assuming iron holds 70% of nanoparticle weight.  \n  6     3.2. Hydrodynamic Size and Morphological Characterization s on SHA Series MNPs.  Figure 3(a) – (f) \nshows the hydrodynamic size distribution s of samples SHA -5, SHA -10, SHA -15, SHA -20, SHA -25 and SHA -30 \nwith mean values of 10.46 nm, 18.07 nm, 20.69 nm, 27.56 nm, 28.28 nm and 32.6 nm, re spectively. The SHA \nseries MNPs have two organic coating  layers : one monolayer of oleic acid and another monolayer of amphiphilic \npolymer. The total thickness of the organic laye r coating is a bout 4 nm. This causes the hydrodynamic size of the \nnanoparticles to be about 8  - 10 nm larger than their inorganic core size measured by TEM  (see Figure 4 ). In this \nrespect, t he mean values for the DLS in the SHA series show qui te a satisfactory trend  and good agreement with \nthe TEM results ; the results ar e arranged in ascending order of their sizes in the Figure 3 .  \n \nFigure 3. Statistical distribution s of the hydrodynamic size of the samples (a) SHA -5, (b) SHA -10, (c) SHA -15, \n(d) SHA -20, (e) SHA -25 and (e) SHA -30 as characterized by DLS . In each figure, the solid green lines are the  7 fitted log -normal distribution curves  and t he solid red lines are the cumulative distribution curves. µ  values \nrepresent the statistical mean of the hydrodynamic sizes of the samples. The standard deviation a nd R -square \nvalues are represented by σ and R2, respectively for each case.  \n \n    The magnetic core morphologies of SHA series MNPs are  shown in Figure 4 . Some MNPs are agglomerated \nduring the  evaporation process of the MNP suspensions . For the MNPs with smaller size such as the sample s \nSHA -5 and SHA -10, the magnetic core shape s are irregular. However, larger MNPs show  spherical  magnetic \ncores . The contrast of different MNPs from one TEM image is due to the different crystal orientations. When the \ncrystal zone axis is close to the incident electron beam, the MNPs show  darker  color . \n \nFigure 4. The TEM images of SHA series  MNPs. (a) - (f) correspond s to SHA -5, SHA -10, SHA -15, SHA -20, \nSHA -25, and  SHA -30, respectively.  Scale bars represent 20 nm. TEM images of SHA series MNPs under \ndifferent magnifications are given in S6 from Supporting Information . \n \n    3.3. Zeta Potential  of SHA Series MNPs . The SH A series  MNPs  have a neutral to slightly alkaline pH \nbetween 7.2 and 7.6. The measured zeta potential values for SHA -5, SHA -10, SHA -15, SHA -20, SHA -25 and \nSHA -30 are -0.03 mV, + 5.03 mV, + 7.66 mV, -0.41 mV, +1 .15 mV and -0.69 mV, respectively.  \n    3.4. Dynamic M agnetic Responses of SHA Series MNPs under  Low Frequency Driving Field . The \ndynamic magnetic responses of SHA series MNPs under mono -frequency driving field  are investigated . The \ndriving field frequency is varied from 50 Hz to 2850 Hz  and the  field amplitude is set at 170 Oe (Gauss).16,16 –23 \nEach plastic vial containing  200 µL SHA series MNPs in 10 nM PBS and 0.03% NaN 3 is placed under  the \nalternating  magnetic  field for MPS measurements. For MNPs suspended in liquid solution under external \nmagnetic field, they undergo two distinct relaxation mechanisms by which the magnetic moments rotate in  8 response to the field: the  Néel relaxation is the rotation of magnetic  moment inside a stationary MNP and, on the \nother hand, the Brownian relaxation is the physical rotation of the entire MNP along with its magnetic moment. \nIn principle, both relaxation mechanisms play important roles in determining the dynamic magnetic res ponses of \nMNPs in suspension when subject ed to alternating magnetic field. Depending on the magnetic properties (such \nas effective anisotropy constant, saturation magnetization)  24,25, the physical properties (magnetic core size, the \nhydrodynamic size including the polymer coating s and an chored biological compounds such as protein, peptide, \ncells, etc.) of MNPs  15,17,26 –31, the nanoparticle volume fraction of suspension (i.e., dipolar interactions)  15,32 –34, \nthe physical properties of the suspension (temperature, viscosity)  18,19,35 –42, MNPs could undergo either Néel or \nBrownian process -dominated relaxa tion. It has been  reported that for a system of non-interacting iron oxide \nnanoparticles with negligible  polymer coatings, the magnetic dynamics will be dominated by Brownian process \nwhen the core size is above 15 nm and Néel process dominates when the core size is below 15 nm 35,43 –45 (see S8 \nfrom Supporting Information ). \nUnder low frequency driving field (f < 500 Hz) , magnetic moments of SHA series MNPs with diameters from \n5 nm to 30 nm are able to follow the time -varying magnetic field. As shown in  Figure S9 from Supporting \nInformation , all of the six SHA series MNPs show similar phase angles to the driving field  (f < 500 Hz) , as the \nfield frequency increases, the differences in the phase angles between six samples increase. Larger MNPs with \nlarger effective relaxation time show larger phase lag to the driving field.  \nAs is summarized in Figure 5, under low frequency driving field, the dynamic magnetic responses of six SHA \nseries MNPs from strongest to weakest are: SHA -30 > SHA -20 > SHA -15 > SHA -25 > SHA -10 > SHA -5. Figure \n5(a), (b) & (c)  summarize the amplitudes  measured  at the 3rd, the 5th, and the 7th harmonics, respectively. Figure \n5(d), (e) & (f)  highlight the corresponding harmonic amplitudes under driving field frequencies of 350 Hz, 650 \nHz, 1250 Hz, and 1850 Hz.   9  \nFigure  5. Harmonics generated by SHA series MNPs under low frequency driving field. (a) – (c) summarize the \n3rd, the 5th, and the 7th harmonic amplitudes of SHA series MNPs under different driving field frequencies. (d) – \n(f) highlight the harmonic amplitudes at driving field frequency of 350 Hz, 650 Hz, 1250 Hz, and 1850 Hz.  \n \n    Figure 6 summarizes the real -time voltage signal collected from pick -up coils at driving field frequencies of \n350, 950, and 1850 Hz. The extracted harmonics are plotted along with the total signal in real time. MNPs with \nstronger dynamic magnetic responses to the driving field generate  larger harmonic signals, thus are able to cause \nthe distortions in the total signal (the highlighted dark areas in Figure 6). It is observed that SHA -30 and SHA -20 \nshow the strongest dynamic magnetic responses to the low frequency dr iving field, followed by SHA -15 and \nSHA -25. SHA -5 and SHA -10 shows negligible dynamic magnetic responses compared to the former SHA series \nMNPs. Which is mainly due to the low magnetic moments and linear magnetization curves as show in  Figure 1 .  \n 10  \nFigure 6. Recorded r eal-time dynamic magnetic responses of SHA series MN Ps under low frequency driving \nfield. The higher harmonics are extracted and plotted in parallel with the total signal collected from the pick -up \ncoils.  \n \n    3.5. Dynamic Magnetic Responses of SHA Series MNPs under High  Frequency Driving Field . In this \nsection, we report the dynamic magnetic responses of SHA series MNPs under high f requency driving fields. A \ndual-frequency method is used herein, o ne excitation  field is set at 10 Hz and magnitude of 170 Oe, the other high \nfrequency driving field is set at varying frequencies (from 1 kHz to 20 kHz) and magnitude of 17 Oe.13,26,35,43,46 –\n48  \n    Under high frequency driving field, larger MNPs (i.e., SHA -30) are unable to rotate their magnetic moments \nto the fast-switching  magnetic field, thus, their dynamic magnetic responses are weakened. As shown in Figure \nS10 from Supporting Information , there is a constant difference of 50° harmonic phase differ ences between SHA -\n10 and SHA -30 MNPs. As a result, under high frequency driving field, the dynamic magnetic responses of six \n 11 SHA series MNPs from strongest to weakest are: SHA -15 > SHA -20 > SHA -30> S HA-25 > SHA -10 > SHA -5 \n(as shown in Figure 7).  \n \nFigure  7. Harmonics generated by SHA series MNPs under high frequency driving field. (a) – (c) summarize the \n3rd, the 5th, and the 7th harmonic amplitudes of SHA series MNPs under different driving field frequencies. (d) – \n(f) highlight the harmonic amplitudes at driving field frequency of 3 kHz, 5 kHz, 10 kHz, and 20 kHz.  \n \nAlthough recent in origin, MNPs of different core sizes have fou nd their application s in various fields of science.  \nThis section of the paper is dedicated in identifying the utili ty of the different sized and surface functionalized  \nMNPs in rea listic  applications . The SHA series particles are amine functionalized  MNPs. As the amine groups \nare less selective and less specific for antibodies and proteins, they capture a varied range of bacterial pathogens \nand allow purification of water, food and urine samples  49. The VSM characterization of the SHA series i n Figure \n1(a) - (f) show that SHA -5, SHA -10, SHA -15 and SHA -20 are superparama gnetic. Although  SHA -25 & SHA -30 \nshow  higher  magnetic moment s, they show hysteresis  loops . For magnetic biosensing , higher moment particles \n 12 are preferred in order to generate higher magnetic signal per particle.  In the meantime,  the MNPs are required to \nbe superparamagnetic  to prevent aggregations . For the SHA series, SHA -25 exhibits the second highest magnetic \nmoment /particle  with a remanent magnetization of 1 .28%M s, where M s is the saturation magnetization . Although \nSHA -30 has  a higher magnetic moment /particle  compared to SHA -25, a much larger remanence magnetization \nof 10.93 %Ms is observed  from  SHA -30. Taking both magnetic moments and superparamagnetism into \nconsideration, SHA -25 is the optimum candidate from  SHA series for biosensing applications.  On a different note, \nfor cell separation & sorting and drug /gene  deliv ery, as the property of superparamagnetism is not essential  and \nhigher magnetic moment ensures larger magnetic torque (force) , the highest magnetic moment MNP, SHA -30 is \nprobably a better candidate.  For magnetic hyperthermia therapy , the area of magnetic hysteresis loop, 𝐴, \ncorresponds to the dissipated energy or specific absorption rate (SAR), which is evaluated by the equation 𝑆𝐴𝑅=\n𝐴∙𝑓. Since the maximum SAR achievable is directly proportional to the Ms of MNPs  50–54. Hence the SHA -30 \nMNPs are best for hyperthermia treatments.  Furthermore, magn etic resonance imaging (MRI) techniques require \nthe MNPs to be injected into the body fluids which then  accumulate in the target tissues . Hence, for MRI \napplications it is extremely essential for the MNPs to be small  as larger MNPs have greater tendency to block the \narteries . In that case, SHA -5 & SHA -10 MNPs will be quit e useful  55. The dynamic magnetic responses of SHA \nseries MNPs are compared  in this paper  using  a home -built MPS system . The harmonics are induced  under \ndifferent driving magnetic fields , which is a result of the joint effects of relaxation mechanisms and the magnetic \nmoment  of each MNP. For MPI and MPS -based bioassays, larger dynamic magnetic responses (higher harmonic \namplitudes) ensur e higher signal -to-noise ratio and sensitivity. Thus, SHA -30 MNPs are suggested for MPI and \nMPS -based bioassays where the driving field frequencies are below 2 kHz, while SHA -15 MNPs are suggested \nfor these applications where the driving field frequencies a re above 2 kHz.  \n \n4. CONCLUSION S \nIn this paper , we characterized the magnetic and physical properties of SHA series MNPs from Ocean NanoTech , \nusing standard characterization tools . The VSM results show that SHA -5, SHA -10, SHA -15, and SHA -20 MNPs \nare superparam agnetic and, on the other hand, SHA -25 and SHA -30 are not superparamagnetic. With SHA -30 \nshow ing the highest magnetic moment per particle, followed by SHA -25, SHA -20, SHA -15, SHA -10, and SHA -\n5. Thus, SHA series iron oxide nanoparticles with larger core siz es are preferred for magnetic biosensing and \ndrug delivery where high moment MNPs are desired  for higher magnetic signals and higher magnetic  torques. \nHowever, SHA -25 and SHA -30 show remnant magnetizations upon the removal of magnetic field (non -\nsuperparamagnetic), thus they are not applicable for applications where superparamagnetism is required. The \nXRD results show that all SHA series MNPs a re composed of γ-Fe2O3, Fe3O4. The dynamic magnetic responses \nof these iron oxide nanoparticles are investigated by a home -built MPS system, where both the responses under \nlow and high driving field frequencies are summarized. It is observed that under low  driving field frequencies,  13 the dynamic magnetic responses of SHA series MNPs from strongest to weakest are: SHA -30 > SHA -20 > SHA -\n15 > SHA -25 > SHA -10 > SHA -5. However, under high driving field frequencies, due to the larger phase lags of \nlarger MNPs, the  dynamic magnetic responses from strongest to weakest are modified: SHA -15 > SHA -20 > \nSHA -30> SHA -25 > SHA -10 > SHA -5. These results give hints on designing MPI and MPS -based bioassays to \nmaximize the use of different MNPs of different core sizes. At the e nd of this paper, based on the requirements \nand goals of MNP -based applications, we suggested different SHA MNPs for each application.  \n \nASSOCIATED CONTENT  \nSupporting Information  \nPhotographs of SHA series iron oxide nanoparticles; Magnetic Particle Spectroscopy (MPS) system; Magnetic \nParticle Spectroscopy (MPS) system setups; Phasor theory; Magnetic moment per gram of Fe; TEM images of \nSHA series MNPs captured at various magnificat ions; Magnetic dynamic responses and higher harmonic models; \nBrownian and Néel relaxations; Phase angles of higher harmonics monitored under low frequency driving field \nfor SHA series samples; Phase angles of higher harmonics monitored under high frequency  driving field for SHA \nseries samp les. \n \nAUTHOR INFORMATION  \nCorresponding Authors  \n*E-mail: wuxx0803@umn.edu  (K.W.)  \n*E-mail: jpwang@umn.edu  (J.-P.W.)  \nORCID  \nKai Wu: 0000 -0002 -9444 -6112  \nJinming Liu: 0000 -0002 -4313 -5816  \nRenata Saha: 0000 -0002 -0389 -0083  \nChaoyi Peng: 0000 -0003 -1608 -3886  \nDiqing Su: 0000 -0002 -5790 -8744  \nJian-Ping Wang: 0000 -0003 -2815 -6624  \nAuthor Contributions  \n⊥K.W. , J.L., and R.S. contributed equally to this  work.  \nNotes  \nThe authors declare no conflict of interest.  \n \nACKNOWLEDGMENTS   14 This study was financially supported by the Institute of Engineering in Medicine of the University of Minnesota  \nthrough FY18 IEM Seed Grant Funding Program . Portions of this work were conducted in the Minnesota Nano \nCenter, which is supported by the National Science Foundation through the National Nano Coordinated \nInfrastructure Network (NNCI) under Award Number ECCS -1542202. Portions of this work were carried out in \nthe Characteri zation Facility, University of Minnesota, a member of the NSF -funded Materials Research Facilities \nNetwork (www.mrfn.org) via the MRSEC program.  \n \nREFERENCES  \n(1)  Crespo, P.; de la Presa, P.; Marin, P.; Multigner, M.; Alonso, J. M.; Rivero, G.; Yndurain, F.; Gonzalez -\nCalbet, J. M.; Hernando, A. Magnetism in Nanoparticles: Tuning Properties with Coatings. J. Phys. -\nCondens. Matter  2013 , 25 (48). https://doi.org/10.1088/0953 -8984/25/48/ 484006.  \n(2)  Albert, E. L.; Abdullah, C. A. C.; Shiroshaki, Y. Synthesis and Characterization of Graphene Oxide \nFunctionalized with Magnetic Nanoparticle via Simple Emulsion Method. Results Phys.  2018 , 11, 944 –\n950. \n(3)  Torchilin, V. 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Magnetic Nanoparticles in MR Imaging and Drug Delivery. Adv. Drug Deliv. \nRev. 2008 , 60 (11), 1252 –1265.  \n \nTOC Graphic  \n \n \n \n S1 Supporting Information  \nAn Investigation of Commercial  Iron Oxide Nanoparticles : \nAdvanced Structural and Magnetic Properties Characterization  \n \nKai Wu†,⊥,*, Jinming Liu†,⊥, Renata Saha†,⊥, Chaoyi Peng†, Diqing Su‡, Andrew Yongqiang Wang§,*, and Jian -\nPing Wang†,* \n†Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455, \nUSA  \n‡Department of Chemical Engineering and Material Science, University of Minnesota, Minneapolis, Minnesota \n55455, USA  \n§Ocean NanoTech, LLC, San Diego, California 92126, USA  \n*Correspondence and requests for materials should be addressed to K.W. (email: wuxx0803@umn.edu ), A.W. \n(email:  awang@oceannanotech.com ), and J.-P.W. (email: jpwang@umn.edu ). \n⊥These authors  contributed equally to this work.  \n \n   S2 S1. Photographs of SHA series iron oxide nanoparticles.  \n \nFigure S1 . Photographs of SHA series iron oxide nanoparticles. Each p ipette tip  contains 100 µL liquid sample \nof SHA -X (X=5, 10, 15, 20, 25, and 30). The color gradually changes from dark brown to dark black (left to right) \nwith the increase of nanoparticle size.  \n  \n S3 S2. Magnetic Particle Spectroscopy (MPS) system.  \nSince the pioneering work of Gleich and Weizenecker in 2005, magnetic particle imaging (MPI) has emerged as \na new 3D tomographic technique for clinical diagnosis, vascular imaging, and therapy1–10. Unlike the \nnanoparticle -enhanced magnetic resonance imaging (MRI) where MNP s are used as supportive contrast agents, \nthe MPI exploits the magnetic response signals directly from MNPs and thus the only visualized elements11–16. \nMagnetic particle spectroscopy (MPS) -based bioassay, a deri vative technique of MPI, has gained a lot of attention \nin the area of magnetic immunoassays in recent years10,17–27. The MPS -based bioassay can be interpreted as a 0D \nMPI scanner where a sinusoidal magnetic field is applied to MNPs, which periodically drives MNPs into \nmagnetically saturat ed region. The nonlinear magnetic responses of MNPs are monitored by a pair of specially \ndesigned pick -up coils and the frequency domain signals are extracted. Those higher harmonics that are specific \nto MNPs are interpreted as indicators of the binding st atus of MNPs to target analytes19,20,22,28 –30. \n \nFigure S 2. (a) Schematic view of mono - and dual -driving field MPS system. (b) Frequency domain \nrepresentations of driving fields and magnetic responses of MNPs: i) mono -frequency driving field; ii) dynamic \nmagnetic responses of MNPs to the mono -frequency field are  picked up by the MPS system, higher harmonics \ncontaining 3f (denoted as the 3rd harmo nic in this work), 5f (denoted as the 5th harmonic in this work), 7f (denoted \nas the 7th harmonic in this work), … are observed due to the nonlinear magnetic responses of MNPs; iii) dual -\nfrequency driving field; iv) dynamic magnetic responses of MNPs to th e dual -frequency fields are  picked up by \nthe MPS system, higher harmonics containing 3f L, 5f L, 7f L, …, f H±2f L (denoted as the 3rd harmonic in this work), \nfH±4f L (denoted as the 5th harmonic in this work), f H±6f L (denoted as the 7th harmonic in this work),  …, 3f H, 5f H, \n7fH, … are observed due to the nonlinear magnetic responses of MNPs.   \n S4 S3. Magnetic Particle Spectroscopy (MPS) system setups.  \n \nFigure S 3. (a) Photograph of home -built MPS system setup. (b) Photograph of copper coils that generate driving \nmagnetic fields and pick -up coils for signal collection. (i) DAQ; (ii) Amplifier; (iii) PC with LabVIEW; (iv) One \nset of assembled coils; (v)&(vi) are coil s for the generation of driving magnetic fields; (vii) One pair of \ndifferentially wound pick -up coils; (viii) Plastic vial with maximum volume capacity of 300 µL.  \n   S5 S4. Phasor theory.  \nThe voltage and phase generated from MNPs at specific frequencies are re presented by a phasor: 𝐴∙𝑒𝑗(𝜔𝑡+𝜑) (or \nexpressed as 𝐴∠𝜑), where 𝜔 is the angular frequency of driving field, 𝜑 is the phase lag, and 𝑗=√−1. \nFor each MPS measurement, the MPS system is run for 10 seconds to collect the baseline signal (noise) \nfollowed by inserting the vial containing MNP sample for another 10 seconds’ signal (total) collection. The \nbackground noise can be expressed as 𝐴𝑁𝑜𝑖 𝑠𝑒𝑒𝑗𝜑𝑁𝑜𝑖𝑠𝑒. The total signal is expressed as 𝐴𝑇𝑂𝑇𝑒𝑗𝜑𝑇𝑂𝑇. This signal is \nthe sum of two phasors: the background noise and the signal generated by MNPs (namely, 𝐴𝑀𝑁𝑃 𝑒𝑗𝜑𝑀𝑁𝑃).  \nSo, \n𝐴𝑁𝑜𝑖𝑠𝑒 𝑒𝑗𝜑𝑁𝑜𝑖𝑠𝑒 +𝐴𝑀𝑁𝑃 𝑒𝑗𝜑𝑀𝑁𝑃 =𝐴𝑇𝑂𝑇𝑒𝑗𝜑𝑇𝑂𝑇, \nwhich reduces to an equation set:  \n{𝐴𝑁𝑜𝑖𝑠𝑒 ×𝑐𝑜𝑠𝜑𝑁𝑜𝑖𝑠𝑒 +𝐴𝑀𝑁𝑃 ×𝑐𝑜𝑠𝜑𝑀𝑁𝑃 =𝐴𝑇𝑂𝑇 ×𝑐𝑜𝑠𝜑𝑇𝑂𝑇\n𝐴𝑁𝑜𝑖𝑠𝑒 ×𝑠𝑖𝑛𝜑𝑁𝑜𝑖𝑠𝑒 +𝐴𝑀𝑁𝑃 ×𝑠𝑖𝑛𝜑𝑀𝑁𝑃 =𝐴𝑇𝑂𝑇 ×𝑠𝑖𝑛𝜑𝑇𝑂𝑇 \nBy solving the equation set above, we can  get the harmonic amplitude 𝐴𝑀𝑁𝑃 and phase 𝜑𝑀𝑁𝑃 of each type \nof MNPs at different driving field frequencies.  \n   S6 S5. Magnetic moment per gram of Fe.  \n \nFigure S 5. Magnetization curves of SHA series MNPs obtained by VSM.  \n  \n S7 S6. TEM images of SHA series  MNPs captured at various magnifications.  \n \nFigure S 6-1. TEM images of SHA series MNPs. Scale bar represents 100 nm. (a) – (f) are SHA -5, SHA -10, \nSHA -15, SHA -20, SHA -25, and SHA -30, respectively.  \n \n \nFigure S 6-2. TEM images of SHA series MNPs. Scale bar represents 50 nm. (a) – (f) are SHA -5, SHA -10, \nSHA -15, SHA -20, SHA -25, and SHA -30, respectively.  \n  \n S8 S7. Magnetic dynamic responses and higher harmonic models . \nIn the presence of alternating magnetic fields, MNPs a re magnetized and their magnetic moments tend to align \nwith the fields. For a monodispersed, non -interacting MNP system, the static magnetic response obeys the \nLangevin model:  \n𝑀𝐷(𝑡)=𝑚𝑠𝑐𝐿(𝜉), \nwhere,  \n𝐿(𝜉)=coth 𝜉−1\n𝜉,𝜉=𝑚𝑠𝐻(𝑡)\n𝑘𝐵𝑇 \nThe MNPs are characterized by magnetic core diameter 𝐷, saturation magnetization 𝑀𝑠 and particle \nconcentration 𝑐, assuming MNPs are spherical without mutual interactions. The magnetic moment of each particle \nis expressed as 𝑚𝑠=𝑀𝑠𝑉𝑐=𝑀𝑠𝜋𝐷36⁄, where 𝑉𝑐 is volume of the magnetic core, 𝜉 is the ratio of magnetic \nenergy over thermal energy, 𝑘𝐵 is Boltzmann constant, and 𝑇 is the absolute temperature in Kelvin. 𝐻(𝑡) is the \nexternal  magnetic  driving fields represented in Figure S 2. \n    However, a rtificially synthesized MNPs do not yield identical diameters. Another reasonable and commonly \nused approach is the log -normal size distribution model. Here, to simpl ify the model, we are assuming the MNPs \nare with identical core diameter of 𝐷. \n    Harmonics model of mono -frequency MPS system:  \n    Taylor expansion of 𝑀𝐷(𝑡) shows the major  harmonic  components:  \n𝑀𝐷(𝑡)\n𝑚𝑠𝑐=𝐿(𝑚𝑠𝐻(𝑡)\n𝑘𝐵𝑇) \n=1\n3(𝑚𝑠\n𝑘𝐵𝑇)𝐻(𝑡)−1\n45(𝑚𝑠\n𝑘𝐵𝑇)3\n𝐻(𝑡)3+2\n945(𝑚𝑠\n𝑘𝐵𝑇)5\n𝐻(𝑡)5+⋯ \n=⋯+[1\n180𝐴3(𝑚𝑠\n𝑘𝐵𝑇)3\n−1\n1512𝐴5(𝑚𝑠\n𝑘𝐵𝑇)5\n+⋯]×𝑠𝑖𝑛[2𝜋∙3𝑓∙𝑡] \n+[1\n7560𝐴5(𝑚𝑠\n𝑘𝐵𝑇)5\n+⋯]×𝑠𝑖𝑛[2𝜋∙5𝑓∙𝑡] \n+⋯ \n    The higher odd harmonics  are expressed as : \n𝑀𝐷(𝑡)|3𝑟𝑑≈𝑚𝑠𝑐\n180𝐴3(𝑚𝑠\n𝑘𝐵𝑇)3\n×𝑠𝑖𝑛[2𝜋∙3𝑓∙𝑡] \n𝑀𝐷(𝑡)|5𝑡ℎ≈𝑚𝑠𝑐\n7560𝐴5(𝑚𝑠\n𝑘𝐵𝑇)5\n×𝑠𝑖𝑛[2𝜋∙5𝑓∙𝑡] \n \n    Harmonics model of dual -frequency MPS system:  \n    Taylor expansion of 𝑀𝐷(𝑡) shows the major frequency mixing components:   S9 𝑀𝐷(𝑡)\n𝑚𝑠𝑐=𝐿(𝑚𝑠𝐻(𝑡)\n𝑘𝐵𝑇) \n=1\n3(𝑚𝑠\n𝑘𝐵𝑇)𝐻(𝑡)−1\n45(𝑚𝑠\n𝑘𝐵𝑇)3\n𝐻(𝑡)3+2\n945(𝑚𝑠\n𝑘𝐵𝑇)5\n𝐻(𝑡)5+⋯ \n=⋯+[−1\n60𝐴𝐻𝐴𝐿2(𝑚𝑠\n𝑘𝐵𝑇)3\n+⋯]×𝑐𝑜𝑠[2𝜋(𝑓𝐻±2𝑓𝐿)𝑡] \n+[1\n1512𝐴𝐻𝐴𝐿4(𝑚𝑠\n𝑘𝐵𝑇)5\n+⋯]×𝑐𝑜𝑠[2𝜋(𝑓𝐻±4𝑓𝐿)𝑡] \n+⋯ \n    The mixing frequency components are found at odd harmonics exclusively:  \n𝑀𝐷(𝑡)|3𝑟𝑑≈−𝑚𝑠𝑐\n60𝐴𝐻𝐴𝐿2(𝑚𝑠\n𝑘𝐵𝑇)3\n×𝑐𝑜𝑠[2𝜋(𝑓𝐻+2𝑓𝐿)𝑡] \n𝑀𝐷(𝑡)|5𝑡ℎ≈𝑚𝑠𝑐\n1512𝐴𝐻𝐴𝐿4(𝑚𝑠\n𝑘𝐵𝑇)5\n×𝑐𝑜𝑠[2𝜋(𝑓𝐻+4𝑓𝐿)𝑡] \n    According to the Faraday's law of induction, the induced voltage in a pair of pick -up coils is expressed as:  \n𝑢(𝑡)=−𝑆0𝑉𝑑\n𝑑𝑡𝑀𝐷(𝑡) \nwhere 𝑉 is volume of MNP suspension. Pick -up coil sensitivity 𝑆0 equals to the external magnetic fiel d strength \ndivided by current.  \n    The static magnetic response mode (the Langevin model) discussed above is unable to describe the dynamic \nresponses of MNPs suspended in solution. Herein, Néel and Brownian relaxation models are introduced to \ncomplete the model.    S10 S8. Brownian and Néel relaxations.  \nFigure S 8 shows that small MNP s relax via Néel process whereas larger MNP s relax via Brownian process. The \ncut off size is around 13  nm. \n \nFigure S 8. Simulated Brownian, Néel, and effective relaxation time as function of MNP core diameters. The \ncrystal asymmetry on the surface of nanoparticles (also called “magnetically dead layer”) yields a smaller \nsaturation magnetization 𝑀𝑠 and a larger anisotrop y constant 𝐾𝑒𝑓𝑓 than the bulk materials  31. Due to this surface \nspin-canting effect, 𝐾𝑒𝑓𝑓 and 𝑀𝑠 are calculated for different sizes of MNPs, polymer coating layer thickness is \nassum ed to be 𝑑= 4 𝑛𝑚, solution viscosity is assumed to be  𝜂=1 𝑐𝑝. \n   S11 S9. Phase angles of higher harmonics monitored under low frequency driving field for SHA series samples . \n \nFigure  S9. Phase angles of higher harmonics monitored under low frequency driving field for SHA series samples. \n(a), (b) and (c) summarizes the phase angles of the 3rd, the 5th, and the 7th harmonics under different driving field \nfrequencies.  \n  \n S12 S10. Phase angles of higher harmonics monitored under high frequency driving field fo r SHA series \nsamples.  \n \nFigure S10. Phase angles of higher harmonics monitored under high frequency driving field for SHA series \nsamples. (a), (b) and (c) summarizes the phase angles of the 3rd, the 5th, and the 7th harmonics under different \ndriving field frequencies.  \n \n  \n S13 References  \n(1)  Reeves, D. 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Phys.  2017 , 50 (8), 085005.  \n "    },    {        "title": "0704.3440v2.Dynamical_models_and_the_phase_ordering_kinetics_of_the_s_1_spinor_condensate.pdf",        "content": "arXiv:0704.3440v2  [cond-mat.stat-mech]  27 Apr 2007Dynamical models and the phase ordering kinetics of the s= 1spinor condensate\nSubroto Mukerjee,1,2Cenke Xu,1and J. E. Moore1,2\n1Department of Physics, University of California, Berkeley , CA 94720\n2Materials Sciences Division, Lawrence Berkeley National L aboratory, Berkeley, CA 94720\n(Dated: October 26, 2018)\nThes= 1 spinor Bose condensate at zero temperature supports ferr omagnetic and polar phases\nthat combine magnetic and superfluid ordering. We investiga te the formation of magnetic domains\nat finite temperature and magnetic field in two dimensions in a n optical trap. We study the general\nground state phase diagram of a spin-1 system and focus on a ph ase that has a magnetic Ising\norder parameter and numerically determine the nature of the finite temperature superfluid and\nmagnetic phase transitions. We then study three different dy namical models: model A, which\nhas no conserved quantities, model F, which has a conserved s econd sound mode and the Gross-\nPitaevskii (GP) equation which has a conserved density and m agnetization. We find the dynamic\ncritical exponent to be the same for models A and F ( z= 2) but different for GP ( z≈3). Externally\nimposed magnetization conservation in models A and F yields the value z≈3, which demonstrates\nthat the only conserved density relevant to domain formatio n is the magnetization density.\nPACS numbers: 03.75.Mn, 03.75.Kk, 64.60.Ht, 75.40.Gb\nI. INTRODUCTION\nThe field of cold atomic gases has witnessed an explo-\nsion of experimental and theoretical research in the last\nten years. The studyofthesesystemshascombinedideas\nfromvariousdisciplinesofphysicssuchasatomicphysics,\ncondensed-matter physics, optics etc. Cold atomic sys-\ntems haveprovideda testing groundfor some of the most\nfundamental principles of collective quantum behavior\nlike Bose-Einstein Condensation. Of particular interest\nis the study of spinor condensates, which are condensates\nof atoms with non-zero spin and have been the focus of\nintenseexperimental1,2,3,4andtheoretical5,6,7,8studiesin\nrecent years. The spin degree of freedom opens up the\npossibility of interesting collective magnetic behavior in\nthese systems in addition to the phenomenon of Bose-\nEinstein condensation. It has alreadybeen demonstrated\nthat the presence of spin greatly modifies the nature of\nthe condensate and superfluid transition in spinor con-\ndensates compared to those without spin5,9.\nSpinor condensates have over the last few years been\nrealized in both magnetic and optical traps. The lat-\nter are more interesting from the point of view of spin\nordering, since the spin degree of freedom is not frozen\nout. The most widely studied atomic systems are those\nof the spin-1 alkali atoms23Na and87Rb. These sys-\ntems differ from each other in the nature of the effective\ntwo-body interaction, which is antiferromagnetic in the\nformer and ferromagnetic in the latter. The condensates\nwith antiferromagnetic interactions are also called polar.\nRecent advances have made it possible to image ferro-\nmagnetic domains in optical traps of87Rb, making it\npossibletostudytheinterestingphysicsofdomainforma-tion in them3. This technique requires the application of\na magnetic field, an additional tunable parameter which\nmakes the phase diagram of these systems interesting.\nMoreover, these atoms have also been trapped in two di-\nmensional geometries, where the physics of collective be-\nhavior is often more exotic than in higher dimensions3,4.\nThe importance of this experiment for basic condensed\nmatterphysicsistwofold: it probesboth ourunderstand-\ningofphase-orderingkineticsatfinitetemperature(when\nobserved at the longest times) and, as the temperature is\nlowered or the observation time is shortened, our under-\nstanding of dynamics across quantum phase transitions.\nIn this paper we will investigate magnetic domain for-\nmation in spin-1 systems at finite temperature and mag-\nnetic field. The main purpose of this study is to compare\nand contrast various plausible dynamical models with re-\nspect to coarsening of a magnetic order parameter. The\nquantity of primary interest, will be the dynamic critical\nexponentzwhich determines the rate of domain forma-\ntion at large times: the domain size Lgrows with time\nasL∼tz. We will examine the general phase diagram\nof spin-1 condensates in the presence of a magnetic field\nin an optical trap and comment on the broken symme-\ntries of the various ordered phases. We will then choose\nthe phase that is most convenient to a study of magnetic\ndomain formation and elucidate the similarities and dif-\nferences between dynamic models, highlighting the im-\nportance of different conservation laws in the dynamics.\nWe will compare our results with existing ones wherever\npossible.\nA natural question is how the stochastic time-\ndependent Ginzburg-Landau (TDGL) approach in this\npaper is related to previous studies using deterministic\nequations of motion, such as the Gross-Pitaevskii equa-2\ntion for the condensate, plus quantum kinetic theory for\nexcited states8,10,11. The answer is that the correct de-\nscription depends on experimental parameters such as\nthe time scale of observation and the normal-state pop-\nulation. The time scale at which stochastic processes\nresulting from interaction with the normal cloud become\nimportant can be increased by decreasing the tempera-\nture of the system. The initial instability in a finite trap\nis likely to be described correctly by the deterministic\ntheories in the literature; coupling to the many degrees\nof freedom in the normal cloud is irrelevantfor the imme-\ndiate dynamics of the condensate. However, the longer\ntimes accessed in current and future experiments are ex-\npected to be described by the theory developed here. In\nother words, the universal dynamical properties in the\nsense of critical phenomena are described by the theo-\nries presented here at any finite temperature, as long as\nthe system is observed for a sufficiently long time. We\nbelieve that current experiments may already be in the\nregime where the theory presented here is valid. How-\never, even if they are not, increases in observation time\nwillsoonenableaprecisecomparisonbetweentheoryand\nexperiment.\nOur main results on phase ordering of spinor conden-\nsates are contained in sections VII and VIII. We argue\nin the final discussion that one specific dynamical model\n(“modelF”dynamics, inthenotationofthe reviewpaper\nof Hohenberg and Halperin12) is expected to describe the\nlong-time dynamics of spinor condensates. This dynami-\ncalmodelisamorecomplicatedversionofthemodelused\nin earlier studies of superfluids13,14,15,16, and reproduces\nthe known propagating modes of the spinor condensate\nat zero temperature. All parameters in the dynamical\nmodel can be determined from measurements of the con-\ndensate, as explained in the appendix.\nII. THE MAGNETIC PHASE DIAGRAM OF\nSPIN-1 BOSONS IN AN OPTICAL TRAP\nSpin-1 condensates are theoretically more complex\nthanthosewith zerospin5,17in thatthe condensateorder\nparameter is a three component complex vector\nΨ =\nψ+1\nψ0\nψ−1\n, (1)\nwithψαbeing the order parameter in the spin state of\neigenvalueαalong some arbitrarily chosen direction. If\none assumes that the condensate state is a single particle\nzeromomentumstate,thetotalenergyforagivendensity\nof atoms in an optical trap with a magnetic field Bin the\nzdirection can be written as\nE=c2∝an}b∇acketle{t/vectorS∝an}b∇acket∇i}ht2+g2∝an}b∇acketle{tS2\nz∝an}b∇acket∇i}ht. (2)Here/vectorS=Sxˆx+Syˆy+Szˆz, where\nSx=1√\n2\n0 1 0\n1 0 1\n0 1 0\n (3)\nSy=i√\n2\n0−1 0\n1 0−1\n0 1 0\n\nSz=\n1 0 0\n0 0 0\n0 0−1\n\nare the generators of SU(2) in the spin-1 representa-\ntion and ∝an}b∇acketle{tA∝an}b∇acket∇i}ht= Ψ†AΨ.c2is the spin-spin interaction\nwhich can be antiferromagnetic ( c2>0) or ferromag-\nnetic (c2<0).g2∝B2and the second term is just the\nquadratic Zeeman term. The absence of a linear term is\ndue to the fact that the time for the relaxation of mag-\nnetization in optical traps is less than the lifetime of the\ncondensate itself. The ground state manifolds can be\nobtained by minimizing the free energy with respect to\n{ψ∗\nα}. It has already been shown that in the absence of\na magnetic field, the ground state manifolds in the po-\nlar and ferromagnetic cases are isomorphic to the spaces\nU(1)×S2\nZ2andSO(3) respectively9. The phase diagram in\nthe presence of a magnetic field is given below.\nPolar\nU(1)Polar\nU(1) U(1)X\n2Z\nFerro\nU(1)X Z2Ferro\nU(1)XU(1)g> 02g< 02> 0\n< 0c\nc2\n2\nFIG. 1: The ground state phase diagram of a spin-1 conden-\nsate in an optical trap in the presence of a magnetic field\nthat couples through a quadratic Zeeman term. The different\nquadrants have different phases with various types of in-pla ne\nand out-of-plane ordering. This figure has been taken from\nMukerjee et. al9.\nFig. 1 has four quadrants labelled by signs of c2and\ng2. In the polar ( c2>0) case, the magnetic ordering is\neither in plane or out of plane depending on the sign of\ng2. “Magneticordering”here refersto the orderingofthe3\nspin-quantization axis ( ˆn). The ground state is always a\nmacroscopicallyoccupiedsingleparticlestateofzerospin\nprojection in this case. For g2>0, the only symmetry\nthat is broken in the ordered state is that of the U(1)\nphase (θ) of the condensate. For g2<0, however there\nis an additional U(1) due to the in-plane ordering of the\nspin-quantization axis. The phase and spin are coupled\nthrough aZ2identification, which denotes symmetry un-\nderθ→θ+πandˆn→ −ˆn. The vortices corresponding\nto the spinand phasearethuscoupled andcan leadtoin-\nteresting finite-temperature physics in two dimensions18.\nThelowerpartofthephasediagramcorrespondstothe\nferromagnetic case ( g2<0) and will be of primary inter-\nest tous. The lowerleft quadrantcorrespondstothe case\ng2<0. The ground state now breaks a U(1) symmetry\ncorresponding to the phase and an Ising Z2symmetry\ncorresponding to the spin. Physically this means that in\nthe condensate, the bosons are either in a state of spin\nprojection 1 or -1. It is this Ising degree of freedom,\nwe exploit to study domain formation in two dimensions.\nThe reason is that since long range Ising order is possi-\nble in two dimensions (as opposed to U(1) order), it is\neasier to define and measure the sizes of large magnetic\ndomains required to investigate long time behavior. It is\nthus this quadrant that will be the focus of the rest of\nout studies. For the sake of completeness we note that\nthe lower right quadrant, which corresponds to the case\ng2>0 is divided into two parts by a straight line with\nequationg2= 2c2. To the left of this line, one has in-\nplane ferromagnetic ordering with the spins pointing in\nsomeU(1) direction in the plane. The ground state thus\nbreaks two U(1) symmetries, one corresponding to the\nphase and the other corresponding to the spin. To the\nright of the line, it is energetically favorable for the sys-\ntem to be in a polar out of plane state. This suggests\nthe interesting possibility of a quantum phase transition\nin these systems tuned by the magnetic field.III. 2D FINITE TEMPERATURE PHASE\nTRANSITIONS\nSince we are interested in studying finite temperature\ncoarsening dynamics in the 2D system with c2<0 and\ng2<0, it is important for us to locate the position of\nthe superfluid and magnetic transitions. This problem is\nalso interesting in its own right since such situations also\ncome up in the study of classical frustrated spin systems,\nlike the fully frustrated XYantiferromagnet (with πflux\nper plaquette) on asquarelattice orthe triangularlattice\nXYantiferromagnet, where the Z2corresponds to a chi-\nrality. The U(1) andZ2transitionsarein closeproximity\nto each other in these cases. The situation is our partic-\nular case is not very different. We find that the U(1)\ntransition is of the Kosterlitz-Thouless (KT) type and\ntheZ2transition of the 2D Ising type. Furthermore, we\nfind that for a certain range of parameters, TZ2>TU(1)\nwhich is also what is observed in the fully frustrated XY\nmodel on the square lattice19and for others the order\nof the transitions appears to be reversed. This depends\non the magnitude of the ratio of the parameters g2/c2.\nFor small values of this ratio, TZ2>TU(1). There is pre-\nsumably also a point where the two transitions occur at\nexactly the same temperature, where the combined tran-\nsition is in a different universality class from 2D Ising\nand KT. We present here numerical data on just one set\nof parameters where TZ2> TKTand illustrate how the\ntwo transitions can be accurately determined despite be-\ning reasonably close to each other in temperature. The\nmethod used is due to Olsson19and we employ a nu-\nmerical Monte-Carlo simulations that uses the following\nGinzburg-Landau free energy functional\nF=/integraldisplay\ndr/bracketleftBig\nα∇ψ∗\na∇ψa+a0(T−TMF\nc)ψ∗\naψa+c0\n2ψ∗\naψ∗\nbψbψa+c2\n2ψ∗\naψ∗\na′Sab.Sa′b′ψb′ψb+g2ψ∗\na/parenleftbig\nS2\nz/parenrightbig\nabψb/bracketrightBig\n.(4)\nwith the following set of parameters, {α= 0.5,a0=\n5.5,c0= 7.0,c2=−2.4,g2=−1.3}.\nThe Kosterlitz-Thouless (KT) transition is detected\nby observing the temperature dependence of the helicity\nmodulusY. The helicity modulus for a discrete system\nofNlattice points is defined as\nΓ =1\nN∂2<F >\n∂δ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nδ=0(5)whereδis a flux twist applied along a particular direc-\ntion. The helicity modulus undergoes a jump of magni-\ntude2Tc\nπat the location of the transition. This is shown\nin Fig. 2, where the transition temperature is seen to be\nTc≈0.44.\nThe standard method to determine the location of the\nZ2transition using fourth order cumulants of the mag-\nnetic order parameter fails here for the same reason that\nit does in the case of the fully frustrated XYmodel,\nwhich is the proximity to the KT transition19. The cu-4\n0.2 0.3 0.4 0.5 0.600.10.20.30.40.5\nTΓ\n  \n8 × 8\n24 × 24\n72 × 72\n2T/πTc = 0.44\nFIG. 2: The helicity modulus as function of Γ for the param-\neter set{}for different system sizes. A clear jump is visible\nof 2Tc/πis visible at Tc≈0.44.\n0.52 0.54 0.56 0.58 0.6 0.6200.050.10.150.20.250.30.35\nT1/ξ100 × 100\nTZ\n2=0.53\nFIG. 3: The magnetic correlation length as a function of tem-\nperature fitted to the 2D Ising form. Tc≈0.53 as estimated\nthis way.\nmulant method assumes that the only relevant length\nscale at the transition is the system size which is not\ntrue here because of the large correlation length corre-\nsponding to the closely situated KT transition. Thus, a\nmore accurate method is to look at the critical exponent\nof the correlationlength of the magnetic order parameterand determine Tcby fitting it to the expected 2D Ising\nform. The magnetization M(r) is given by\nM(r) =|ψ+1(r)|2−|ψ−1(r)|2. (6)\nThe correlation length ξ(T) can be extracted from the\nmagnetic autocorrelation function\ng(r) =∝an}b∇acketle{tM(r)M(0)∝an}b∇acket∇i}ht=e−r/ξ(T). (7)\nIf the transition is 2D Ising like,\nξ(T)∼1\nT−Tc. (8)\nThe numerical result is shown is Fig. 3, which shows that\nthe correlation length fits the 2D Ising form fairly well.\nThe obtained transition temperature is Tc≈0.53. A\nmore careful finite-size scaling analysis can be done to\ndetermine the two transition temperatures, but even at\nthis level of analysis it is clear that TZ2>TU(1).\nIV. DYNAMICAL MODELS\nThe study of the formation of domains of the order pa-\nrameter requires careful consideration of the dynamical\nmodes of the system. Dynamical models are often con-\nstrained by conservation laws that are present as a con-\nsequence of symmetries or otherwise in the system. It is\nwellknownthat the presenceofconservationlawsusually\naffects the rate of formation of domains, since the phase\nspace of states that the system can pass through in the\napproach to the ordered state is constrained by the con-\nservation laws. However, not all conservation laws affect\ndomain formation in the same way and some might be\nmore important than others. In this section we consider\nsome dynamical models appropriate for the description\nof our system and comment on the conservation laws and\nthe dynamical modes obtained from them.\nThe most commonly used dynamical model to describe\nspinor condensates is the Gross-Pitaevskii (GP) equa-\ntion17. This is the model that has been extensively used\nto study domain formation in these systems. The model\nconsists of treating the condensate as a classical field\nat zero temperature whose dynamics are given by the\nHamilton equations of motion of the appropriate Hamil-\ntonian. In our case, the Hamiltonian is\nH=/integraldisplay\ndr/bracketleftbigg/planckover2pi12\n2m∇ψ∗\na∇ψa+U(r)ψ∗\naψa+c0\n2ψ∗\naψ∗\nbψbψa+c2\n2ψ∗\naψ∗\na′Sab.Sa′b′ψb′ψb+g2ψ∗\na/parenleftbig\nS2\nz/parenrightbig\nabψb/bracketrightbigg\n,(9)5\nwith the dynamical equation of motion\ni/planckover2pi1∂ψa\n∂t=−δH\nδψ∗a(10)\nIt should be noted that the condensate density ψ∗\naψa\nand magnetization are both conserved by this equation.\nHowever, the GP equation cannot correctly describe the\napproach to equilibrium at finite temperatures, since\nthe dynamics is only precessional and not relaxational.\nWhile this equation might be appropriate for the de-\nscription of the dynamics once the condensate has been\nformed, it is inappropriate for the study of dynamic phe-\nnomena in other cases, for example quenches from high\ntemperature where the energy of the condensate is not\nconserved.\nThe effect of finite temperature on the dynamics in\nspinor systems has thus far been taken into account\nthrough phenomenological rate equations, which too do\nnot describe the approach to equilibrium. A simple\nmodel which is more appropriate is the so-called “model\nA” of the Hohenberg and Halperin classification12. This\nmodel uses the Ginzburg-Landau free energy Eqn. 4 with\nsimple Langevin dynamics. Operationally, this means\nthe dynamical equation\n∂ψa\n∂t=−Γ0δF\nδψ∗a+ζa(r,t). (11)\nThermal fluctuations due to finite temperature are con-\ntained in the noise variable ζa(r,t), which has the follow-\ning autocorrelation function\n∝an}b∇acketle{tζ∗\na(r,t)ζb(r′,t′)∝an}b∇acket∇i}ht= 2Re(Γ 0)kBTδ(r−r′)δ(t−t′)δab(12)\nconsistent with the fluctuation-dissipation theorem, that\ndrives the system to equilibrium from a non-equilibrium\nstate. This model like the GP model only considers the\ncondensate as a classical field but unlike the GP model\ndoes not possess any conservation laws. It is relaxational\nin nature with the rate of relaxation of the order param-\neter set by Re Γ 0and can be a complex number. The\ncondensate density is no longer conserved and neither is\nthe magnetization. The condensate is exchanging parti-\ncles and energy with the “normal fluid” in this model.\nThe non-conservation of energy of this model can be rec-\ntified by implicitly including the normal fluid through\na conserved “second sound” mode ( m), which is a real\nscalar field. Based on previous experience with the su-\nperfluid transition in helium, one expects that a correct\ndescription of the dynamics near the transition requires\nthis additional field and in the notation of Hohenberg\nand Halperin, this model “model F”, with random forces\nζaandθfor the fields ψaandmrespectively. The free\nenergyFSSwith this second-sound mode is\nFSS=F+/integraldisplay\ndr/parenleftbigg\nγ0mψ∗\naψa+1\n2C0m2/parenrightbigg\n,(13)whereFis given by Eqn. 4. The dynamics are given by\n∂ψ\n∂t=−Γ0δFSS\nδψ∗a−ig0ψδFSS\nδm+ζa(r,t) (14)\n∂m\n∂t=λm\n0∇2δFSS\nδm+2g0Im/parenleftbigg\nψ∗\naδFSS\nδψ∗a/parenrightbigg\n+τ(r,t)\nThese equations conserve the second sound density m\nand the noise correlator for θ, consistent with the\nfluctuation-dissipation theorem is\n∝an}b∇acketle{tτ(r,t)τ(r′,t′)∝an}b∇acket∇i}ht=−2λm\n0∇2δ(r−r′)δ(t−t′) (15)\nThe free energy FSShas extra terms compared to F; a\nterm that couples mand the condensate density and\nothers that contain the energy of the mode m. The\ndynamical equations also incuding coupling terms as\na consequence of the non-vanishing Poisson brackets\n{m,ψa}20,21,22. Asidefromtermsthatresultfromderiva-\ntives of the Ginzburg-Landau free energy, there could\nin principle be additional magnetic terms analogous to\nthoseintheHeisenbergferromagnet(e.g., S×∇2S, where\nSis the local spin density). Since this is even in S, it\nwill not be obtained as the Sderivative of any free en-\nergy, but will originate in the microscopic Hamiltonian.\nA check that no such additional terms are necessary is\nthat the above equations reproduce the previously ob-\ntained modes in the GP equation. Such a calculation\nwas carried out by Hohenberg and Halperin for the case\nof Helium and we extend that to the case of spinor con-\ndensates in the next section.\nThe three different dynamical models, GP, model A\nand model F are the ones we will use to investigate do-\nmain formation in spinor condensates at finite temper-\nature and magnetic fields. We will in addition to the\ndynamical equations above also impose the conservation\nof magnetization on models A and F, to investigate the\neffect of that conservation law on the dynamics. As can\nbe seen from the above discussion, model F contains\nmany more parameters than model A and the GP equa-\ntion. Theparametersofthismodelarerelatedtopossible\nexperimentally-measurable quantities in the appendix.\nV. DYNAMICAL MODES IN THE ORDERED\nSTATE\nModelFcontainsinitboththeGPequationandmodel\nA, which can be seen by setting the appropriate param-\neters in it to zero. However, it is important that model\nF produces all the dynamical modes that the GP equa-\ntion does even when these parameters are not zero, in\norder for this treatment to be valid. There will also be\nadditional modes produced (for example in m), that are\nabsent in the GP equation. We explicitly demonstrate\nthis in this section.6\nWe begin by setting the temperature and magnetic\nfield to zero, to enable comparison with the GP equa-\ntion. The idea is to check that the introduction of the\nextra parameters of model F does not alter the modes\nthat have already been calculated5. It is known that\nthere are three linearly dispersing mode in the polar case\nand one gapped, linear and quadratic mode each in the\nferromagnetic case. The dynamical equations describing\nthe modes (either propagating or diffusion) are\n∂ψα\n∂t=−Γ0δFSS\nδψ∗α−ig0ψδFSS\nδψm(16)\n∂m\n∂t=λm\n0∇2∂FSS\n∂m+2g0Im(ψ∗\nαδFSS\nδψ∗α)\nFor brevity of notation, we also set T=−a∇2,µ=\na0TMF\ncand explicitly write Γ 0= Γ1+iΓ2, where both\nΓ1and Γ2are real.\nA. The polar case\nWe assume that\nΨ =N+Φ, (17)\nwhere\nN=√n0\n0\n1\n0\n, (18)\nis the value of the order parameter in the ordered polar\nstate and Φ is a perturbation on it. The number of parti-\ncles within the condensate, is related to the value µand\nc0by minimizing the free energy FSS\n√n0=/radicalbiggµ\nc0(19)\n1. Polar state with Γ1= 0,λ0= 0,γ= 0\nLet us first ignore the coupling γmψ∗\nαψα, as well as all\nthe dissipation terms in the equations, like the term with\ncoefficient Γ 1andλ0. We will put them back in later.\nIn this case, all the modes are propagating, since there\nis no dissipation. After expanding the equations around\nN, the linearized equations we have are\n∂φ0\n∂t=iΓ2(Tφ0+2c0n0(φ0+φ∗\n0))+im√n0\nCg0\n∂m\n∂t=−ig0√n0∇2(φ0−φ∗\n0)\n∂φ1\n∂t=−iΓ2(Tφ1+n0c2(φ1+φ∗\n−1))\n∂φ−1\n∂t=−iΓ2(Tφ−1+n0c2(φ∗\n1+φ−1)) (20)Notice that in order to get these equations, we need µto\ntake exactly the value in (19))\nThe equations for φ1andφ−1have the same form as\nfor the GP equation5, so the spin wave modes are the\nsame. Both M+=φ+1+iφ−1andM−=φ+1−iφ−1\ndisperse linearly with k, with velocity cM= Γ2√n0c2.\nThe step by step solution for the coupled equation be-\ntweenφ0,φ∗\n0andmis tedious, so we only write down\nthe result here. Basically, the second sound mode and\ndensity fluctuation δn0=φ∗\n0+φ0couple and form two\nmodes with linear dispersion relations, the velocity is\ncs=/radicalbigg\ng2\n0n0\nC+2aΓ2\n2c0n0 (21)\nNotice that the first term in the square root in the\nabove equation is the square of the second sound veloc-\nity12, with Γ 2= 0 and ignoring the propagating mode of\nψ. The second term in the square root is the one that\nappears as the density fluctuation mode in Ref. 5, where\nthethe secondsoundmodewasignored. Hereweseethat\nif we take into account both densities, the second sound\nmode and the density fluctuation mode couple into a new\nmode with velocity cs.\n2. Polar state with Γ1= 0,λ0/ne}ationslash= 0,γ= 0\nHere the dissipation λ0term is added back into the\nequations. We will not consider the case with finite Γ 1,\nsince we assume that within the condensate, the dissipa-\ntion of the modes is very small.\nThe coupled equations between mandφ0now become\n∂φ0\n∂t=iΓ2(Tφ0+2c0n0(φ0+φ∗\n0))+im√n0\nCg0(22)\n∂m\n∂t=−ig0√n0∇2(φ0−φ∗\n0)+λ0\nC∇2m (23)\nThe detailed solution is again tedious and we will solve\nthe equation based on following approximation that the\nhigher order terms of spatial derivatives are small, since\nwe are only interested in the limit kgoing to zero. Under\nthis approximation the dispersion relation is\nω=csk+iλ0\nCk2(24)\nThe mode gets a propagating part, which is linear in\nk, and a damping part, which is proportional to k2.csis\ngiven by (21).7\n3. Polar state with Γ1= 0,λ0/ne}ationslash= 0,γ/ne}ationslash= 0\nTurning on γ, changesonly two terms in the equations.\nFirst,\n∂φ0\n∂t=iΓ2(Tφ0+2c0n0(φ0+φ∗\n0))+ig0γn0(φ0+φ∗\n0)+im√n0\nCg0.\n(25)\nWe can redefine\nc′\n0=c0+g0γ\nΓ2(26)\nand make the equation look exactly like (20), except for\nreplacingc0byc′\n0.γalso modifies (23), by adding a term\nto the right hand side\n∂φ0\n∂t=iΓ2(Tφ0+2c0n0(φ0+φ∗\n0))+im√n0\nCg0\n∂m\n∂t=−ig0√n0∇2(φ0−φ∗\n0)+λ0\nC∇2m\n+λ0γ\nC√n0∇2(φ0+φ∗\n0) (27)\nSolving this modified equation, we see that the term pro-\nportional to γonly contributes higher order momentum\nterms. So, up to linear order in k, the dispersion rela-\ntion is not changed. Therefore all the results here are the\nsame as case 2, if we replace c0byc′\n0.\nB. The ferromagnetic case\nThe solution of the ferromagnetic case is very similar\nto the polar case. The general formalism and effective\naction Eqn. 13 still apply. The difference is in how we\nlinearize the equations. In the ferromagnetic case, we\nshould linearize the equations around the state\nΨ =N+Φ, (28)\nwith\nN=√n0\n1\n0\n0\n (29)\nHere the density of the condensate is not only related to\nthe coefficient c0, but also to the coefficient c2.\n√n0=/radicalbiggµ\nc0+c2(30)\nWe now obtain the following linearized equations\n∂φ1\n∂t=iΓ2(Tφ0+2(c0+c2)n0(φ1+φ∗\n1))+im√n0\nCg0\n∂m\n∂t=−ig0√n0∇2(φ1−φ∗\n1)+λ0\nC∇2m∂φ0\n∂t=−iΓ2Tφ0\n∂φ−1\n∂t=−iΓ2Tφ−1+2c2n0φ−1 (31)\nThe dispersion relation for the second sound mode is\nω=csk+iλ0\nCk2(32)\ncs=/radicalbigg\ng2\n0n0\nC+2aΓ2\n2(c0+c2)n0,\nwhere we have kept terms only to lowest order in the\nmomentum in every step of the calcultion. For the den-\nsity fluctuation mode δn=√n0(φ1+φ1∗), the dispersion\nrelation is\nω=csk (33)\nThe spin wave mode δM−=√n0φ∗\n0has the same dis-\npersion relation as the one obtained in the GP case5,\nω=ak2. Again, turning on the interaction γdoes not\nchange the result much. It causes a redefinition of c0in\nthe propagating part of the mode, and only contributes\nhigher order momentum terms in the diffusion or damp-\ning part of the mode, which are not important when the\nmomentum is small.\nThus, we see that in both the polar and ferromagnetic\ncases, the dynamic modes obtained in the presence of the\nextraparametersofmodelFareconsistentwith thoseob-\ntained from the GP equations. The nature of the density\nmode changes because of coupling with the second sound\nmode but the spin wave modes remain unaffected.\nVI. DOMAIN FORMATION\nA typical experiment or numerical simulation of\ncoarsening involves starting the system off at a high-\ntemperature (usually disordered) state and rapidly\nquenching it to a temperature below the ordering tran-\nsition to observe the growth of domains of the ordered\nstate. At the heart of the theoretical analysis of this\nprocess is the scaling hypothesis23. The equal-time cor-\nrelationfunction ofthe orderparameter m(r,t) is defined\nas\nC(r,t) =∝an}b∇acketle{tm(x+r,t)m(r,t)∝an}b∇acket∇i}ht. (34)\nThe scaling hypothesis states that\nC(r,t) =f/parenleftbiggr\nL(t)/parenrightbigg\n(35)\nL(t) isacharacteristiclength scale, the domainsize. Fur-\nther, at long times L∝t1/z, wherez, the dynamical\ncritical exponent. The dynamical critical exponent is8\ndependent on the model used to describe the ordering\ndynamics of the system, the symmetry of the order pa-\nrameter and the nature of defects present in the initial\nstate. For instance, it is known that z= 2 for the Ising\nmodel with dynamics which do not conserve the total\nmagnetization, after a high temperature quench. For an\nXYmodel on the other hand z= 2 but with a loga-\nrithmic correction as a function of time23. The difference\nfrom the Ising case can be attributed to the different bro-\nken symmetry and consequently the topological defects\npresent in the high temperature state. The Ising model\nwith a conserved order parameter on the other hand pro-\nduces a different dynamic critical exponent z= 323,24.\nThe growth of domains is slower in this case compared\nto the case with no magnetization conservation since the\nconservation law places constraints on the phase space\navailable during domain growth.\nVII. DETAILS OF THE NUMERICAL\nSIMULATION\nIn this work we study the dynamics of domain growth\nin 2D after a magnetic field quench and not a temper-\nature quench. The motivation is a similar approach\nadopted in recent experiments in optical traps. To be\nspecific, we study a ferromagnetic condensate in two di-\nmensions whose initial state is a polar out-of-plane state\ninthefourthquadrantofFig.1andquenchittoavalueof\nthe field, where the ordered state is a ferromagnetic out-\nof-plane state (in quadrant 3 of Fig. 1). Operationally,\nwe sweep the parameter g2from a large positive value\nto a negative value. For models A and F, this is done\nat finite temperature and the order parameter eventually\nrelaxes to a uniform value consistent with the ferromag-\nnetic out-of-plane state. The Gross-Pitaevskii equation\non the other hand does not cause the system to relax\nbut rather to oscillate between different concentrations\nof the three spinor components. Further, it does not al-\nlowaninitial state which iscompletely polarout-of-plane\nto form magnetic domains of the +1 and -1 components\nat any value of the time. In this case, we start with an\ninitial state, which has 90% of the atoms in the 0 (polar\nout-of-plane) state and the other 10%, divided equally\namong the +1 and -1 states. The phase of each spinor\ncomponent in the initial state is chosen to be a random\nnumber between 0 and 2 πallowing for spatial inhomo-\ngeneity which leads to domain formation.\nThe equations of motion corresponding to each model\nare integrated numerically using a first order Euler\nmethod with the noise functions drawn from a Gaussian\ndistribution. The size of the numerical grid ranges from\n50×50 to 200 ×200. The time step is adjusted depend-\ning on the values of the other parameters and varied to\ncheck for consistency. The number of parameters is large(especially for model F) and we present results only for\na fixed set of parameters. However, we have explored\nother parts of the parameter space consistent with fer-\nromagnetic out-of-plane order and not found any qual-\nitative and wherever appropriate (like for the value of\nz) quantitative difference in the results. The set of pa-\nrameters for which we report results are those in section\nIII with the additional model F parameters, {Re(Γ0) =\n1.30,Im(Γ0) = 0.26,g0= 0.35,λm\n0= 0.84,γ0= 1.5},\nwherever applicable.\nThe domain size L(t) is measured using the relation\nL(t) =/radicalBigg\nS0(t)\nS2(t), (36)\nwhereS0(t) andS2(t) are respectively the zeroth and\nsecond moment of the structure function\nS(k,t) =/integraldisplay\ndr∝an}b∇acketle{tM(r,t)M(0,t)∝an}b∇acket∇i}hteik.r,(37)\nwhich is the Fourier transform of the order parameter\ncorrelation function. The domain size is also calculated\nby measuring the size of domain boundaries directly in\nthe simulation grid. This method serves as a consistency\ncheckonthefirstmethod. Itshouldbementionedthough\nthat the second method is useful and consistent with the\nfirst one only when there are very few small bubbles of\none value of the order parameter inside large islands of\nthe other value. This method essentially ignores these\nbubbles by looking for large closed domain walls and\nworks best when the domains are large in size.\nVIII. RESULTS\nA. Finite temperature without conservation of\nmagnetization\nWe present results for the domain size as a function of\ntime for models A and F in Fig 4. The results presented\nare for the set of parameters mentioned in the preceding\nsection, with a magnetic field quench and have been ob-\ntained on a grid of size 200 ×200. It can be seen that\ndomain formation is faster for model A than for model F,\nwhich can be attributed to the presence of the extra con-\nservation law. This certainly appears to be the case over\nthe range of parameters that we have explored, but may\nnot be the case elsewhere in parameter space. Whether\nor not this is a universal feature requires more careful\nanalysis. The curve for L(t) as a function of tfor model\nA dynamics seems to yield a z= 2±0.15 over the entire\nrange of values of time we have presented. Further, the\nvalue ofzobtained at different values of time seems to\nbe fairly constant. This is the value of z, one would ex-\npect forahightemperaturequenchinapureIsingmodel.9\n1 2 3 4 52345678\nlog10tL(t)\n  \nmodel A\nmodel F\nFIG. 4: L(t) as a function of log10tfor model F with the\nparameter set R, with no magnetization conservation .\n0.1 0.2 0.3 0.4 0.50.460.470.480.490.50.51\n1/L(t)1/z(t)\nFIG. 5: 1 /z(t) as a function of 1 /L(t) for model F and no\nmagnetization conservation with the parameter set Rdemon-\nstrating the drift of z(t) as a function of tand thus increasing\nL(t) towards the value z= 2.\nModel F also yields z≈2. Unlike in model A dynam-\nics, there is a small drift in the value of zobtained at\ndifferent values of t. A similar drift (of a larger magni-\ntude) has been seen in the case of the Ising model with\ndynamics that conserve magnetization and it has been\nargued by Huse24that this is due to excess transport in\ndomain interfaces. It then follows that the effective dy-\nnamic critical exponent z(t) drifts in the following wayto first order in the domain size\n1\nz(t)=1\nz(t=∞)/bracketleftbigg\n1−L0\nL(t)/bracketrightbigg\n(38)\nThis suggests that z(t) approaches its infinite time value\nfrom above, which appears to be the case here as well as\ncan be seen from Fig. 5 , which is a plot of 1 /z(t) vs.\n1/L(t). However, we emphasize that the above analy-\nsis is strictly applicable only to the case where the or-\nder parameter is conserved , which is not the case here.\nThe quantity that is conservedis the second sound mode.\nNevertheless, it is possible that the drift can be explained\nby some mechanism similar to the above.\nTo conclude this part, we remark that both models A\nand F without any explicit magnetization conservation\nboth yield the same dynamic critical exponent z= 2 for\ncoarsening with a magnetic field quench.\nB. Finite temperature with conserved\nmagnetization\n5.566.577.588.59345678\nlog10tL(t)\n  \nmodel A\nmodel F\nFIG. 6: L(t) as a function of log10tfor models A and F and\nconserved magnetization density with the parameter set R.\nWe now present results for models A and F with con-\nserved magnetization. The magnetization conservation\nis implemented in terms of a local continuity equation\nin the magnetization density and a magnetization cur-\nrent. We illustrate how we do this for model A and\nthe implementation for model F proceeds along simi-\nlar lines. We first note that the magnetization density\nM=|ψ+1(r,t)|2− |ψ−1(r,t)|2only involves the ampli-\ntudes ofthe componenets ofthe condensate orderparam-\neter. We first write down model A dynamics in terms of10\n0.10.150.20.250.30.350.40.230.240.250.260.270.28\n1/L(t)1/z(t)\n  \nmodel A\nmodel F\nFIG. 7: 1 /z(t) as a function of 1 /L(t) for models A and F\nand conserved magnetization density with the parameter set\nRdemonstrating the drift of z(t) as a function of tand thus\nincreasing L(t) towards the value z= 3.\nseparate dynamical equations for the phase and ampli-\ntude of each component of the condensate order param-\neter. These turn out to be\n∂|ψa|\n∂t=−1\n2Re(Γ0)δF\nδ|ψa|+1\n2|ψa|Im(Γ0)δF\nδθa+µa(r,t),\n(39)|ψa|∂θa\n∂t=−1\n2|ψa|Re(Γ0)δF\nδθa+1\n2Im(Γ0)δF\nδ|ψa|+νa(r,t),\n(40)\nwhere the noise correlators are\n∝an}b∇acketle{tµa(r,t)µb(r′,t′)∝an}b∇acket∇i}ht=∝an}b∇acketle{tνa(r,t)νb(r′,t′)∝an}b∇acket∇i}ht (41)\n= Re(Γ 0)kBTδ(r−r′)δ(t−t′)δab.\nNote that every quantity in the above equations is now\nreal. If we were interested in conservingthe density |ψa|2\nofeachcomponent individually , wewouldmodifyEqn.39\nto\n∂|ψa|2\n∂t=−Re(Γ0)∇2/parenleftbigg\n|ψa|δF\nδ|ψa|/parenrightbigg\n+Im(Γ 0)∇2/parenleftbiggδF\nδθa/parenrightbigg\n+µa(r,t), (42)\nwith the correlator for µnow given by\n∝an}b∇acketle{tµa(r,t)µa(r′,t′)∝an}b∇acket∇i}ht= 4Re(Γ 0)∇2[|ψa(r,t)|2δ(r−r′)]δ(t−t′).\n(43)\nThis ensures there is a conservation equation of the sort\n∂|ψa(r,t)|2\n∂t=−∇.Ja(r,t) (44)\nfor each component. We are however not interested in\nconserving the density of each component, but only the\ncombination M=|ψ+1|2−|ψ−1|2. To this end, proceed-\ning as above, we obtain the following set of equations.\n∂M\n∂t=−Re(Γ0)∇2/parenleftbigg\n|ψ+1|δF\nδ|ψ+1|−|ψ−1|δF\nδ|ψ−1|/parenrightbigg\n+Im(Γ 0)∇2/parenleftbiggδF\nδθ+1−δF\nδθ−1/parenrightbigg\n+µM(r,t) (45)\n∂N\n∂t=−Re(Γ0)/parenleftbigg\n|ψ+1|δF\nδ|ψ+1|+|ψ−1|δF\nδ|ψ−1|/parenrightbigg\n+Im(Γ 0)/parenleftbiggδF\nδθ+1+δF\nδθ−1/parenrightbigg\n+µN(r,t) (46)\n∂|ψ0|\n∂t=−1\n2Re(Γ0)δF\nδ|ψ0|+1\n2|ψ0|Im(Γ0)δF\nδθ0+µ0(r,t) (47)\n|ψa|∂θa\n∂t=−1\n2|ψa|Re(Γ0)δF\nδθa+1\n2Im(Γ0)δF\nδ|ψa|+νa(r,t) (48)\nHereN=|ψ+1|2+|ψ−1|2and the noise correlators are given by\n∝an}b∇acketle{tµM(r,t)µM(r′,t′)∝an}b∇acket∇i}ht= 4Re(Γ 0)∇2[{|ψ+1(r,t)|2+|ψ+1(r,t)|2}δ(r−r′)]δ(t−t′)\n∝an}b∇acketle{tµN(r,t)µN(r′,t′)∝an}b∇acket∇i}ht= 4Re(Γ 0){|ψ+1(r,t)|2+|ψ+1(r,t)|2}δ(r−r′)δ(t−t′)\n∝an}b∇acketle{tµ0(r,t)µ0(r′,t′)∝an}b∇acket∇i}ht= Re(Γ 0)kBTδ(r−r′)δ(t−t′)\n∝an}b∇acketle{tνa(r,t)νb(r′,t′)∝an}b∇acket∇i}ht= Re(Γ 0)kBTδ(r−r′)δ(t−t′)δab (49)\nThe noise functions µN,µM,µ0andνaare mutually un- correlated. The above equations for MandNcan be11\nused to generate equations for |ψ+1|and|ψ−1|, which\nis the way the numerical calculation is performed. Note\nthat the full set of dynamical equations written above\nhas no conservation law except the one for M. We nowuse the same procedure to impose magnetization conser-\nvation on model F. The dynamical equations in this case\nare\n∂M\n∂t=−Re(Γ0)∇2/parenleftbigg\n|ψ+1|δFss\nδ|ψ+1|−|ψ−1|δFss\nδ|ψ−1|/parenrightbigg\n+Im(Γ 0)∇2/parenleftbiggδFss\nδθ+1−δFss\nδθ−1/parenrightbigg\n+µM(r,t) (50)\n∂N\n∂t=−Re(Γ0)/parenleftbigg\n|ψ+1|δFss\nδ|ψ+1|+|ψ−1|δFss\nδ|ψ−1|/parenrightbigg\n+Im(Γ 0)/parenleftbiggδFss\nδθ+1+δFss\nδθ−1/parenrightbigg\n+µN(r,t) (51)\n∂|ψ0|\n∂t=−1\n2Re(Γ0)δFss\nδ|ψ0|+1\n2|ψ0|Im(Γ0)δFss\nδθ0+µ0(r,t) (52)\n|ψa|∂θa\n∂t=−1\n2|ψa|Re(Γ0)δFss\nδθa+1\n2Im(Γ0)δFss\nδ|ψa|−g0|ψa|∂Fss\n∂m+νa(r,t) (53)\n∂m\n∂t=λm\n0∇2δFSS\nδm+g0Im/parenleftbigg\n|ψa|δFSS\nδ|ψa|/parenrightbigg\n+τ(r,t) (54)\nThe noise correlators are the same as for model A with\nmagnetization conservation with the additional correla-\ntor\n∝an}b∇acketle{tτ(r,t)τ(r′,t′)∝an}b∇acket∇i}ht=−2λm\n0∇2δ(r−r′)δ(t−t′),(55)\nas in the case of model F without magnetization con-\nservation. It should be noted that the coefficient g0ap-\npears only in the dynamical equation for the phases of\nthe different components of the condensate thus making\nits identification as a precessional term obvious. Further,\nthe above dynamical equations conserve both the mag-\nnetizationMand the second sound mode mand thus\nrepresent perhaps the most realistic dynamical model for\naBECatfinite temperature andfield; onewherethe con-\ndensate can exchange charge and energy with the “nor-\nmal cloud” but not magnetization.\nOnce again, the results presented are for a 200 ×200\nsimulation grid. We have checked that the additional\nmagnetization conservation law does not affect the static\nproperties of the model. As in the previous case, we\nagain see that domain formation is faster for model A\nand than model F. This time, however, the dynamic crit-\nical exponent obtained is not equal to 2. As can be seen\nfrom Fig. 7, which is a plot of 1 /z(t) vs. 1/L(t) for both\nmodels, there is a significant drift of z(t) as a function of\ntime. Once again the direction of the drift is consistent\nwith Eqn. 38 and in this case, the analysis mentioned in\nthe previous subsection is directly applicable, since it is\nthe order parameter (the magnetization) that is directly\nconserved. However, the noise levels of the simulations\ndo not permit a fit to Eq. 38. It should be noted though\nthat to the extent observable in the numerical simula-\ntion, there is a drift in the direction of z= 3 in the data.\nThis is what is observed in a pure Ising model with con-servedmagnetization in a high temperature quench. One\ninteresting observation is that the value of 1 /z(t) seems\nto deviate more strongly from Eq. 38 at large times for\nmodel F than model A. This deviation has also been ob-\nserved for the simple Ising model with conserved magne-\ntization, where it was attributed to finite-size effects and\ncorrelated noise in the simulations. That could well be\nthe case here as well, although it is not clear why these\neffects should be more pronounced in one model than in\nthe other. It should be noted that in the simulations on\nthe Ising model24, the values of z(t) observed for com-\nparable simulation times are roughly close to what we\nobserve.\nTheconclusionofthispartisthatmodelsAandFwith\nexplicit magnetization conservation yield a dynamic crit-\nicalexponent z≈3, differentfromtheexponentobtained\nwithout magnetizationconservation. Thus, the models A\nand F seem to be identical as far as long-time coarsen-\ning behavior of the magnetization is concerned and the\nbehavior is truly determined by whether or not the mag-\nnetization is conserved, which it is not explicitly in either\nmodel. The difference between these two models will be-\ncome apparent, when the coarsening of domains related\nto the conserved second mode is investigated.\nC. The Gross-Pitaevskii equation\nWe finally investigate domain formation in the Gross-\nPitaevskii equation. As has been remarked earlier, this\nformalism assumes that the dynamics of the condensate\nis completely determined by the precessional(and not re-12\nlaxational) dynamics of the classical order parameter. It\nis thus a formalism that on the one hand is valid strictly\nat zero temperature, but on the other hand ignoresquan-\ntum fluctuations. The initial state chosen for models A\nand F considered earlier does not evolve in this formal-\nism and hence we choose a slightly different initial state\nas mentioned in section VII with 90% of the condensate\ndensityinthe0stateand5%eachinthe+1and-1states.\nThe precessional nature of the GP equation implies that\nthere is never any true relaxation to a state with only\ndomains of +1 and -1 and the amplitudes of these two\ncomponents oscillate together, π/2 out of phase with the\namplitude of the 0 component. The dynamical critical\nexponentzin these simulations is extracted by looking\nat the time interval when the amplitudes of the +1 and\n-1 components are growing with time and that of the 0\ncomponent falling. It is important that a sizeable win-\ndow be identified within this interval where the domain\nsizeL(t) is indeed growing as L(t)∝t1/z. The oscil-\nlatory nature of the dynamics ensures that in this case,\nboth the magnetization and condensate density are con-\nserved. The former is manifested in the fact the +1 and\n-1 components always have the same amplitude and the\nlatter in the fact that these two components are always\nπ/2 out of phase with the 0 component.\n4 5 6 7 8 92345678\nlog10tL(t)\nFIG. 8:L(t) as a function of log10tin the GP equation with\nthe parameter set R. The data is from a time interval during\nwhich the amplitudes of the ±1 components are increasing\nwith time.\nThe domain size here is obtained only using Eqn. 36\nsince the presence of bubbles of the 0 state renders the\nmethod of measuring the domain boundaries directly un-\nreliable. The data for the domain size L(t) as a function\noftis shown in Fig. 8. The data as in the previous\ntwo cases has been obtained over a range of about four\ndecades. The dynamic critical exponent z(t) as extracted\nis shown as a function of 1 /L(t). Once again, there ap-0.1 0.2 0.3 0.4 0.50.260.2650.270.2750.280.2850.290.2950.3\n1/L(t)1/z(t)\nFIG. 9: 1 /z(t) as a function of 1 /L(t) in the GP equation\nwith the parameter set R. The exponent z(t) as a function\noftand thus increasing L(t) seems to drift towards the value\nz= 3 like with models A and F with conserved magnetization\ndensities.\npears to be a drift towards the value z= 3 at infinite\ntime, although in this case, it appears (to within the\nnoise) that the drift is more consistent with Eqn. 38 (i.e.\nlinear in 1/L(t)) than for models A and F.\nIt appearsthat the GPmodelgivesadifferentdynamic\ncritical exponent z= 3 from models A and F ( z= 2),\nunless magnetization is conserved explicitly in the latter.\nThis should be compared and contrasted with the case\nof spinless bosons, where it has been hypothesized that\nmodel F and the GP equation give the same dynamic\ncritical exponent for phase ordering25. Further, this ex-\nponent was numerically found to be equal to 1 (different\nfrom model A, which has z= 2) from numerical studies\nof the GP equation in this case. The situation we ana-\nlyze is different from the case of spinless bosons in that\nwe study the magnetization. The second sound mode of\nmodel F arises from total energy and number conserva-\ntionbetweenthecondensateandthe“normalstate”. The\nGP equation too conserves both these quantities. How-\never,forbosonswithspin, theGPequationalsoconserves\nmagnetization, which is not present in model F unless in-\ncluded by hand. Thus the value of zfor the GP equation\nis different from that of model F and agreement is ob-\ntained only when magnetization is explicitly conserved\nin the latter. It is interesting to note however that the\nvalue ofzobtained from the GP equation is larger than\nfrom model A in our study whereas for spinless bosons it\nis smaller.13\nIX. CONCLUSIONS\nTo conclude, we have studied the statics and dynamics\nof spin-1 condensates at finite temperature in the pres-\nence of a magnetic field. We have obtained a ground\nstate phase diagram for this system and have focussed\non the phase that is most amenable to a numerical study\nof magnetic domain formation in 2D, namely the Ferro-\nmagnetic out-of-plane phase and have numerically deter-\nmined the nature and order of the superfluid and mag-\nnetic phase transitions. We have argued that the “cor-\nrect” dynamical model for spinor condensates at finite\nfield and temperature is model F in the Halpering and\nHohenbergclassificationandhavedemonstratedthatthis\nmodel contains all the modes in the standard GP equa-\ntion. We have numerically studied magnetic domain for-\nmation in the GP model and models A and F with and\nwithoutmagnetizationconservation,andhavefoundthat\nit is the only when magnetization is explicitly conserved\nin models A and F that the dynamic critical exponenent\nzobtained from the GP equations agrees with the zob-\ntained from them. In the absence of this conservation,\nmodels A and F yield z= 2.\nWhile we have focussed only on one part of the phase\ndiagram of spinor condensates in this study to highlight\nthe difference between various dynamical models, similar\nstudies can be performed in the other parts of the phase\ndiagramaswell. Thiswillbereportedelsewhere. Itisour\nbeliefthat model F is fundamentallyamorecomplete dy-\nnamical model to describe spinor condensates than the\nGP model. In addition to studies of the dynamics far\naway from the critical point, as presented here, this dy-\nnamical model could be used to obtain dynamical criti-\ncal exponents, for comparison to dynamical experiments\nnear the static phase transition of the spinor condensate.\nAPPENDIX A: RELATION OF THE MODEL F\nPARAMETERS TO MEASURABLE QUANTITIES\nAs remarked earlier, Model F has many more param-\neters than the GP equation. Here we comment on how\nthese parameters can be related to experimentally mea-\nsurable quantities. There are 9 important real param-\neters, which are Γ 1, Γ2,g0,λ0,µ,c0,c2,C,γ. Some\nof them are directly measurable. For example, λ0is the\nthermal conductivity, and Cis the specific heat. We now\ndiscuss how to measure all the other parameters.1.g0\ng0only exists in the dynamical equations and does not\nappearinthe staticfreeenergy. Inthegeneralformalism,\nat the operator level, it appears in the Poisson bracket\nbetweenmandψaas\n[ψ†\na,m] =g0ψ†\na (A1)\nThis means if we create a particle through ψ†, the expec-\ntation value of min the system increases by g0. Sincem\nis effectively the“heat” in the system (this can be seen\nfrom the physical meaning of Candλ),g0is effectively\nthe “heat” per particle. Writing,\ng0=Tσ, (A2)\nwhereσis the entropy per particle, the value of g0can\nnow be obtained from measuring the specific heat\ng0=T/integraldisplayT\n0dTc\nT. (A3)\nHerecis the specific heat per particle.\n2.c0+Cγ2\nThe reason we discuss c0andγtogether is that c0\nandγcan only appear in the combination c0+γ2in\nall static quantities. This can be seen from mean field\ntheory:mappears in a quadratic term 1 /(2C)m2and a\nlinear term γmn0: the expectation value of mis thusCγ.\nPlugging this value of minto Eqn. 13, the m4interaction\nterm becomes c0+Cγ2, which we denote as c′. How\ndo we measure c′? At low temperature, almost all the\natomsare in the condensate. All the terms in Eqn. 13are\nproportional to the density of the condensate, except the\nc′term which is proportionalto the square ofthe density.\nThis term will therefore contribute to the compressibility\nκof the condensate, where\nκ−1=−VdP\ndV=Vd2E\ndV2= 2c′n2\n0 (A4)\nWhen the temperature is low enough, the dominant con-\ntribution to the compressibility will be from the conden-\nsate. By measuring the compressibility, we can measure\nthe parameter c′.\n3.µ\nKnowingthevalueof c′, thevalueof µisquite straight-\nforward to measure. It can be related to the density of14\natoms within the condensate, using the relation\nn0=/radicalbiggµ\n2c′(A5)\n4.c2\nc2is a static parameter and should be measurable\nfrom static properties, for example, the spin suscepti-\nbility. The terms in the free energy involving the mag-\nnetization can be rewritten as c2M2\nz+hMz, wherehis\nthe magnetic field along z. The spin susceptibility is ap-\nproximately h/c2, from which we can deduce c2. In most\npractical cases at low temperature, the value of c2is not\nvery different from that obtained from the atomic swave\nscattering rates.\n5.Γ2\nThe modeφ+1+φ−1will have an oscillatory compo-\nnent and also a part that is decaying. The value of theoscillation frequency can be shown to be (1+√\n5)Γ2c2n0.\nIf we know the value of c2from static experiments, we\ncan obtain the value of Γ 2.\n6.c0andγ\nWe have shown that c0+Cγ2can be determined by\nmeasuringc′. To disentangle the two quantities, we look\natthesecondsoundvelocity. Theequationforthe second\nsound velocity is\ncs=/radicalBigg\ng2\n0n0\nC+2Γ2\n2(c′+g0γ\nΓ2)n0 (A6)\nIf we know cs, the only unknown variable is γ. Having\nobtainedγfrom this equation, we can calculate c0, from\nthe known value of c′.\nThe authors wish to acknowledge conversations with\nD. A. Huse, A. Lamacraft, S. Leslie, D. Podolsky, L.\nSadler, D. M. Stamper-Kurn, M. 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A\n54, 5037 (1996)."    },    {        "title": "1907.09833v2.Transverse_magnetic_field_influence_on_wakefield_in_complex_plasmas.pdf",        "content": "Transverse magnetic \feld in\ruence on wake\feld in\nplasmas\nSita Sundar1and Zhandos A. Moldabekov2;3\n1Department of Aerospace Engineering, Indian Institute of Technology Madras,\nChennai - 600036, India\n2Institute for Experimental and Theoretical Physics, Al-Farabi Kazakh\nNational University, 71 Al-Farabi str., 050040 Almaty, Kazakhstan\n3Institute of Applied Sciences and IT, 40-48 Shashkin Str., 050038 Almaty,\nKazakhstan\nE-mail: sitaucsd@gmail.com\nApril 2019\nAbstract.\nWe present the results of an investigation of the wake\feld around a stationary\ncharged grain in an external magnetic \feld with non-zero transverse component\nwith respect to the ion \row direction. The impact of the orientation of magnetic\n\feld on the wake behavior is assessed. In contrast to previously reported\nsigni\fcant suppression of the wake oscillations due to longitudinal magnetic\n\feld applied along \row, in the presence of transverse to \row magnetic \feld the\nwake\feld exhibits a long range recurrent oscillations. Extensive investigation for\na wide range of parameters reveal that in the sonic and supersonic regimes the\nwake has strong dependence on the direction of the magnetic \feld and exhibits\nsensitivity to even a meager deviation of magnetic \feld from the longitudinal\norientation. The tool obtained with the study of impact of transverse component\nof magnetic \feld on the wake around grain in streaming ions can be used\nto potentially maneuver the grain-grain interaction to achieve controlled grain\ndynamics.arXiv:1907.09833v2  [physics.plasm-ph]  26 Jul 2019Transverse magnetic \feld in\ruence on wake\feld in plasmas 2\n1. Introduction\nWake\feld in a plasma manifests itself in a variety\nof physical phenomena and has gained tremendous\ninterest in the recent years [1, 2, 3]. To understand\nthese physical phenomena exhibited by complex\nplasma, e.g. dust acoustic waves, wake features,\nsolitary structures etc, a number of schemes has been\ndevised. One of the fundamental research problems\ncomplex plasma physicists have dealt with is to acquire\nthe knowledge about controlled behavior of dusty\nplasma. This controlled behavior of grain-plasma\ndynamics has often been achieved by applying external\nelectric [4, 5, 6, 7, 8] and magnetic \felds [9, 10, 11, 12,\n13, 14, 15].\nStudy of dusty plasmas in vertical magnetic \feld\nwas studied \frst by Fujiyama et al. [16]. They\ninvestigated experimentally the transport of silicon\nparticles by modulated magnetic \feld. Salimullah\net al [17] explored the inequalities of charge and\nnumber densities of electron, ion, and dust particle\nand presented the low-frequency dust-lower-hybrid\nmodes in a dusty plasma. Low frequency waves and\nmagnetoacoustic modes in magnetized dusty plasmas\nwere investigated in Ref. [18]. This was taken further\nby Sato et al. [19]. They discussed the drastic e\u000bect of\nmagnetic \feld on levitating dust cloud shape control\nand rotation for strongly coupled dusty plasmas due\nto ion drag on \fne particles. They also reported the\ndependence of rotation on plasma density variation.\nIn another work, a di\u000berent perspective for impact of\nlongitudinal magnetic \feld on complex plasmas was\npresented by Samsonov et.al. [20]. They depicted the\nagglomeration and levitation of magnetic grains. They\nalso proposed the possibility of magnetic \feld induced\ncrystal formation which promoted the interest in inter-\ndisciplinary scienti\fc interaction. Contemporary work\nfor the same regime was carried out by Yaroshenko\net.al. [21]. Their explanation for the mutual interaction\nwas based on dipole theory and they delineated\nthat \feld-aligned individual particle containing chains\nobserved in experiments is due to the dipole short-\nrange force [22, 23].\nWith the progress in understanding of complex\nplasma dynamics and transport, invesigation of wake\nformation behind grain started concurrently [24, 11,\n12]. Observation of wake oscillations behind grain\ndue to ion focusing is now one of the fascinating\nfeatures displayed by complex plasmas and it has\ngained the reputation of one of the most important\nphenomena observed in dusty plasma simulations. In\nrecent works, impact of magnetic \feld aligned along\n\row on these wakes has been investigated in great\ndetail [11, 12, 25, 26, 27], but, surprisingly, as far\nas the impact on wake is concerned, the in\ruence\nof magnetic \feld perpendicular to the \row has notreceived signi\fcant attention yet. Moreover, the\nwake\feld in an external magnetic \feld with non-zero\nboth longitudinal and transverse components has not\nbeen explored so far. Therefore, it is the aim of this\nwork to develop fundamental insight on the e\u000bect of\ntransverse component of an external magnetic \feld on\nthe wake\feld in complex plasmas.\nMagnetic \feld in\ruences the behavior of charged\nplasma particles, i.e., ions and electrons, and hence\na\u000bects the overall dynamics of the system. Dust\nparticulates are heavy and it takes comparatively\nhigher strength of magnetic \feld to make the grain\nmagnetized. It is pertinent to ask about the role of\nmagnetic \feld on dusty plasma phenomena especially\nthe exciting wake \feld features reported for the case\nof grains in moving ions in the sheath region. Nambu\net al. [28] pioneered the study of grain in magnetized\nion \row. Their main analytical result was damping of\nwake features for grain in \rowing ions in the presence\nof magnetic \feld applied along the ion \row. They\nillustrated the reduction in ion overshielding around\ngrain eventually resulting in suppression of wake\feld.\nA decrease in the interaction force with increasing\nlongitudinal magnetic \feld strength was also recently\nreported by Carstensen et al. [26].\nWhen the magnetic \feld is directed along the \row\nor along the direction of electric \feld, the dynamics of\ngrain can be studied in a comparatively simple manner,\nas in the case of EkB, the Lorentz force related\nto streaming velocity vanishes. However, when the\ndirection of Bis perpendicular to the \row, E\u0002Bdrift\ncomplicates the dynamics (with Ebeing an external\nelectric \feld). We have applied the magnetic \feld with\nnon-zero transverse component to explore this very\ncomplicated dynamics with ion \row direction along\nz. We present here the detailed numerical exploration\nfor three di\u000berent \row regimes (subsonic, sonic, and\nsupersonic) and for a wide range of magnetization.\nFurthermore, we compare our results with the case of\nlongitudinal magnetic \feld as well as the case without\nmagnetic \feld. The investigation has been performed\nwith Maxwellian ion distribution which should be\nchanged to non-Maxwellian for pressures greater than\n10 Pa and other pertinent situations.\nThe outline of the paper is as follows. In Sec. 2\nwe introduce the numerical simulation scheme used\nand present the description of the method. In Sec. 3\nwe present the results regarding the impact of the\nmagnetic \feld with non-zero transverse component on\nthe wake\feld. We start, in Sec. 3.1, by considering\npurely transverse magnetic \feld case (i.e., with zero\nlongitudinal component). Then, in Sec. 3.2, we explore\nthe intermediate regime with non-zero transverse and\nlongitudinal components of magnetic \feld induction.\nFinally, we present a summary and conclusion inTransverse magnetic \feld in\ruence on wake\feld in plasmas 3\nSec. IV .\n2. Simulation details and plasma parameters\n2.1. Particle-in-cell simulation\nThe wake features around a grain were explored\nwithCOPTIC particle-in-cell simulations [29]. The\nsimulated system consists of a stationary charged grain\nin the presence of streaming ions under the in\ruence\nof an external magnetic \feld. Besides providing\ncomparatively accurate solutions for even non-linear\nregime, the importance of the code lies in the fact\nthat it can be customized to resolve the particle in\nthe near neighborhood of grain by orders of magnitude\nthan elsewhere by imposing a non-uniform grid in\nthe neighborhood of the grain. The simulation set-\nup is similar to the one considered in the recent\nworks [25, 12, 7, 30] except that we have magnetic\n\feld component perpendicular as well as parallel (few\ncases) to the \row direction. Further numerical detail\nand fundamentals can be gleaned from the paper with\ndescriptions about coptic [29, 31].\nBesides \fnite-sized objects, coptic facilitates the\nincorporation of point-charge grains also. The grain\nconsidered in the present work is point-charge and\nis solved using particle-particle particle-mesh ( P3M)\ntechnique. The ion dynamics in six-dimensional phase\nspace in the presence of the self-consistent electric\n\feld\u0000r\u001e, an optional external force D[31] (this\nextra force Dis zero in our simulations for the shifted\nMaxwellian distribution ) and an external magnetic \feld\nBis delineated by the equation\nmidv\ndt=q[\u0000r\u001e+v\u0002B] +D; (1)\nhereqdenotes the ion charge.\nIn order to solve the Poisson equation, a second-\norder accurate \fnite-di\u000berence-scheme equivalent to\nShortley-Weller approximation has been adopted. The\nsolution of this Poisson equation, with given electron\nand ion number densities neandnirespectively,\nr2\u001e= (ene\u0000qni)=\u000f0; (2)\nendows us with the resultant potential. The\ninterpolation of the electric \feld and the solution of the\nPoisson equation is incorporated with higher precision\nand convergence as it uses compact di\u000berence stencil\nwhich is specialized in dealing with arbitrary oblique\nboundaries for Cartesian mesh as well. On the outer\nmesh-edge, the potential gradient along ^ zis set to\nzero. For the present work, implementation of the\ndynamics of the lighter electron species is governed by\nthe Boltzmann description,\nne=ne1exp(e\u001e=T e): (3)\nEB\nIon  focusingαIon flow, VE Vc VdDustFigure 1. Schematic depicting the system of grain in streaming\nions in the presence of electric and magnetic \felds.\nThe value of the Boltzmann constant is taken as unity\nfor the present work. It is important to mention here\nthat the Boltzmann approximation of electrons restrict\nus in capturing all kinetic e\u000bects. The present work is a\ntrade-o\u000b between the e\u000bects we propose to look forward\nand the computational cost incurred by implementing\nthe particle treatment of electrons.\nIn the code, collisions are included according to\nthe constant velocity-independent collision frequency\nPoisson statistical distribution which is similar to the\nBGK-type collisions. Predominant collision in the\nsystem is the ion-neutral charged-exchange collision.\nThis charge-exchange collision is implemented in the\ncode by the mutual exchange of the ion velocity with\nthe neutral velocity randomly drawn from neutral\nvelocity distribution.\nWe considered a cell grid of 64 \u000264\u0002128 with\nmore than 60 million ions and grid side length of\n15\u000215\u000220 Debye lengths for simulation purposes.\nWe also performed few simulations with grids of even\nhigher resolution and non-uniform mesh spacing to\nresolve the dynamics in the vicinity of the grain [29].\nSimulation is progressed in time for 1000-2000 time\nsteps (with units discussed below) by which it usually\nreaches steady-state.\n2.2. Dimensionless quantities and plasma parameters\nThe most important dimensionless parameters are as\nfollows:\n(i) The orientation of the magnetic \feld is charac-\nterized by the angle \u000bbetween magnetic \feld in-\nduction vector and streaming (an external elec-\ntric \feld) direction (which is along zaxis) as il-\nlustrated in Fig. 1. In general we have 0 \u0014\u000b\u0014Transverse magnetic \feld in\ruence on wake\feld in plasmas 4\n\u0019=2. Clearly, the purely transverse magnetic \feld\nand purely longitudinal magnetic \feld cases cor-\nrespond to \u000b=\u0019=2 and\u000b= 0, respectively.\n(ii) The strength of the magnetic \feld is conveniently\nand conventionally characterized by the parameter\n\f=!ci=!pide\fned as the ratio of cyclotron\nfrequency of an ion to the plasma frequency of\nions. Dimensionless parameters \u000band\fare\nsu\u000ecient to completely describe the magnetic\n\feld.\n(iii) Mach number Mdescribing ionic streaming speed\nhas been de\fned as M=vd=cs, wherecs=p\nTe=miis the ion sound speed and vdis\nionic streaming velocity. Thermal Mach number\nMthshares the relation with Mach number\nMaccording to the relation M=vd=cs=p\nTi=TeMth.\nBesides, we follow the standard normalization,\nas described in [29], i.e., the space coordinate is\nnormalized as r!r=r0, velocity as v!v=cs, and\npotential as \u001e!\u001e=(Te=e), wherer0= (\u0015De=5) is the\nnormalizing scale length and csis unity in normalized\nunits. Time units as \u0017=(cs=r0)\u00180:2(\u0017=!pi) is used\nto normalize collision frequency \u0017, where!piis the ion\nplasma frequency. The normalized grain charge, Qd, is\nrepresented as \u0016Qd=Qde=(4\u0019\u000f0\u0015DeTe), wheree0is the\nunit electron charge.\nA summary of plasma parameters are presented\nin Table 1. For typical dusty plasma parameters\n(e.g. an electron Debye length of \u0015de= 845\u0016m and\nan electron temperature of 2 :585 eV), the normalized\ngrain charge \u0016Qd= 0:01 redacted in units of electron\ncharge is approximately 7 :5\u0002103e0and\f\u00140:35 is\nthe magnetization parameter corresponding to B<\n50 mT.\nTable 1. List of plasma and simulation parameters.\nPhysical property Parameter range\nMagnetization \f 0.0-1.0\nTemperature ratio Te=Ti 100\nMach Number M 0.5 - 1.5\nCollision frequency \u0017=!pi 0.002\nNormalized grain charge \u0016Qd0.01\n3. Results\nStudy of grain under streaming ions in the presence of\ncross-\row magnetic \feld serves the twofold objective\nof (a) investigation of the in\ruence of magnetic \feld\napplied perpendicular to the \row and (b) a study\nof the intermediate case with no-zero transverse and\nlongitudinal components of magnetic \feld. In allpresented \fgures a dust particle is located at the origin\nx=y=z= 0 (in Cartesian co-ordinate) and is\npost-processed to provide the result in cylindrical co-\nordinate (with grain located at z= 0 andr= 0).\n3.1. Impact of a transverse magnetic \feld on the dust\nwake\nLet us start from the illustration of the wake\feld at\nM= 1 in an external magnetic \feld (with \f= 1:0)\ncompared to that of in the absence of magnetic \feld\n(i.e.\f= 0). Accordingly, in Fig. 2 the contour plots\nof the potential in the cases: (a) without magnetic\n\feld, (b) with magnetic \feld applied along and (c)\nin transverse direction with respect to the \row are\nshown. Fig. 2 leads us to the \frst insight about the\nrole played by the magnetic \feld and its direction for\nthe case of grain in streaming plasmas. We clearly\nsee from Fig. 2 that the considered three cases have\ndi\u000berent pattern compared to each other. In the\nwell studied magnetic \feld free case, Fig. 2 (a), we\nsee a V-shaped wake\feld with interchanging maxima\nand minima along \row direction. The wake\feld with\nan external magnetic \feld applied along streaming\nvelocity|another in detail investigated situation|\nshows no oscillatory picture with \\candle \rame\"\nshape instead (see Fig. 2 (b)). The rotation of\nthe magnetic \feld orientation from longitudinal to\ntransverse direction restores oscillatory pattern of the\nwake \feld as illustrated in subplot (c) of Fig. 2.\nThe noticeable di\u000berences of the latter case from the\nmagnetic \feld free case are a stronger localization of\nthe ion focusing and depletion regions around maxima\nand minima accompanied by a weaker manifestation\nof the V-shape, and a weakened damping of these\noscillations. We also notice the sustained long range\noscillatory structures with reduced e\u000bective wake\nwavelength in the case of grain in transverse to \row\nmagnetic \feld (cf. subplot (c)) and will be discussed\nlater. In dimensional units, for typical dusty plasma\nparameters, e.g. Te= 2:585,\f= 1:0, andM= 1,\nthe \frst wake peak would correspond to a potential\nof 60:62 mV for magnetic \feld transverse to \row and\n25:98 mV for magnetic \feld along \row [12].\nNext, in Fig. 3, we consider the wake\feld in the\ntransverse magnetic \feld with di\u000berent magnetization\nparameter at subsonic, sonic, and supersonic cases.\nThe three rows of the \fgure denote the cases with\nthree di\u000berent Mach numbers M= 0:5,M= 1,\nandM= 1:5 (from top to bottom), while the three\ncolumns show the cases with increasing strengths of\nmagnetic \felds \f= 0:07,\f= 0:14, and\f= 0:35\n(from left to right). For M= 0:5 case (top row),\nmagnetic \feld decreases the oscillations amplitude.\nThe amplitude of wake oscillation is so small that\nno explicit trend is observed. Further increasing theTransverse magnetic \feld in\ruence on wake\feld in plasmas 5\n(a)(b)(c)\nFigure 2. Wake potential contours e\u001e=T e, averaged over the azimuthal angle, with the ion \row velocity M= 1:0 for: (a) without\nmagnetic \feld ( \f\u00180:0), (b) with magnetic \feld applied along streaming direction ( \f= 1:0 and\u000b= 0), and (c) with magnetic \feld\nin transverse direction with respect to the \row( \f= 1:0 and\u000b=\u0019=2).\nmagnetic \feld strength to \f= 0:3535, we see an\nextended positive potential region at subsonic \rows.\nAs we increased the \row speed to M= 1 (middle row)\nandM= 1:5 (bottom row), we see pronounced wake\noscillations behind the grain. With the increase in \f,\nthe e\u000bect of magnetic \feld manifests in the increase of\nthe number of oscillations and decrease in the distance\nbetween subsequent potential maxima and minima.\nAdditionally, the aforementioned stronger localization\nof the wake pattern around maxima and minima is\ndistinctly visible.\nTo better understand the impact of the transverse\nmagnetic \feld, we show in Fig. 4 the wake potential\nalong the streaming axis for M= 0:5;1, and 1:5, along\nwith alomost unmagnetized or very feeble magnetic\n\feld case (\f= 0:07). It was found that \f= 0:07 case\nprovides almost the same data for the wake potential\nas the magnetic \feld free case already reported in\nprevious works [12, 7, 11, 24]. At subsonic and\nsonic regimes (top and middle panel, respectively),\nthe transverse magnetic \feld with \f= 0:14 leads to\nthe deviation of the potential from the magnetic \feld\nfree case. However, in the supersonic regime (bottom\npanel), data for \f= 0:14 case weakly di\u000bers from \f=\n0:07\u00190 case. With further increase in the magnetic\n\feld strength to \f= 0:35, we see a strong impact of\nthe transverse magnetic \feld in subsonic, sonic andsupersonic regimes. In this case, we understand that\nthe transverse magnetic \feld decreases the oscillations\namplitude as well as the distance between maxima and\nminima of the wake\feld. It is important to emphasize\nthat while the oscillation amplitude is reduced, the\ndamping of these oscillations become weaker.\nIn Fig. 5, the potential pro\fle along transverse\ndirection with z= 0 is shown. This \fgure shows\nthat the transverse magnetic \feld with \f\u00140:35\ndoes not change the potential pro\fle in perpendicular\ndirection to zaxis withz= 0. This is an important\ninformation as in experiments often the dust particles\nare located on a single plane perpendicular to the\n\row direction. Moreover, this behavior is in contrast\nto the longitudinal magnetic \feld case reported in\nprevious works [12, 25, 11], where it was shown that\nthe longitudinal magnetic \feld has a strong impact on\nthe potential pro\fle in perpendicular direction to zaxis\n(streaming direction) with z= 0.\nFor completeness, for the case of transverse\nmagnetic \feld, the wake peak height dependence on\n\fis given in Fig. 6 for M= 1:0. In agreement\nwith the above presented wake potential data, the\nwake peaks are substantially smaller for the case of\nM= 0:5 compared to sonic and supersonic cases. With\nincrease in the \row speed, the wake peak amplitude\nincreases (M= 1) and then decreases ( M= 1:5). ThisTransverse magnetic \feld in\ruence on wake\feld in plasmas 6\n0\n0-2.0\n-2.0(a)\n(i)(h)(g)(f)(e)(d)(c)(b)ß≈0ß=0.14ß=0.35M=0.5M=1.0M=1.5\nFigure 3. Wake potential contours e\u001e=T efor various strengths of magnetic \feld. The left column corresponds to \f= 0:07, the\nmiddle column is for \f= 0:14, and the right column is for \f= 0:35. The ion \row velocity M= 0:5 (top row), M= 1:0 (middle\nrow), andM= 1:5 (bottom row).Transverse magnetic \feld in\ruence on wake\feld in plasmas 7\n0.140.35\nFigure 4. Wake potential along the streaming axis for\ntransverse to \row magnetic \feld ( \u000b=\u0019=2) with\f\u00180:0 (red\nsolid line), \f= 0:14 (green dashed line), and \f= 0:35 (blue\ndoted line) for M= 0:5 (top panel), M= 1:0 (middle panel),\nandM= 1:5 (bottom panel).\nbehavior is due to the competition between increase\nin the number of in\rux ions and higher escape ability\nof ions with increase in streaming speed. Magnetic\n\feld applied perpendicular to the \row does not change\nthis non-monotonic behavior with respect to change\nin the \row speed. The data for stronger magnetic\n\feld exhibits the smaller wake peak amplitude in\ncoherence with that obtained for magnetic \feld applied\nalong the \row [12]. However, the relative damping\nof the oscillations is weaker in downstream direction\ncompared to the longitudinal magnetic \feld case.\n0.140.35Figure 5. Wake potential in transverse to \row direction (with\nz= 0,\u000b=\u0019=2 andM= 1:0) for magnetic \feld with \f=\u00180\n(red solid line), \f= 0:14 (green dashed line), and \f= 0:35 (blue\ndoted line).\n0.07        0.14       0.21         0.28          0.35\nFigure 6. Variation of the maximum of the peak amplitudes of\nthe wake potential as a function of \fwith\u000b=\u0019=2.\nNoteworthy, the ion density contour pro\fle\nsupports the discussed wake\feld variation in the\npresence of transverse magnetic \feld as illustrated in\nFig. 7 for \row speed M= 1:5 for\f= 0:07;0:14;0:21,\nand 0:35. We know from the work by Sundar et\nal.[12] that the presence of longitudinal magnetic\n\feld induces the ion density \ructuation. Here, we\nsee a magnetic \feld applied perpendicular to the\n\row direction also induces \ructuation in ion density,\nhowever, the \ructuation in the two cases exhibit\na stark di\u000berence. In the case of magnetic \feld\nperpendicular to the \row, there is no `candle \rame'\nlike protruding structure in the density contour as\nobtained for the parallel to \row magnetic \feld case [12].\nAdditionally, transverse magnetic \feld, makes the\ndensity \ructuation propagate farther from the grain.Transverse magnetic \feld in\ruence on wake\feld in plasmas 8\n10\n010203040z(a)\n0.00.51.01.5(b)\n0.00.51.01.5\n10\n 0 10\nr10\n010203040z(c)\n0.00.51.01.5\n10\n 0 10\nr(d)\n0.00.51.01.5\nFigure 7. Spatial pro\fles of the ion density (normalized to the distant unperturbed ion density) for for M= 1:5,\u000b=\u0019=2, and (a)\n\f= 0:07, (b)\f= 0:14, (c)\f= 0:21, and (d) \f= 0:35.\n3.2. E\u000bect of deviation of magnetic \feld orientation\nfrom longitudinal direction\nIt is shown above that in the case of longitudinal\nmagnetic \feld (i.e., when Bis directed along \row)\nthe wake behavior is completely di\u000berent from that\nof magnetic \feld applied perpendicular to the \row.\nTherefore, it is interesting to inquire the intermediate\ncase, i.e. when both longitudinal and transverse\ncomponents of the magnetic \feld are non-zero.\nThe consideration of the intermediate con\fguration\nprovides picture about the in\ruence of deviation\nof magnetic \feld orientation from the longitudinal\ndirection on the wake features.\nFigure 8, depicts the impact of magnetic \feld\norientation on the wake in the subsonic (top row),\nsonic (middle row), and supersonic (bottom row) cases.\nHere, we keep the strength of the overall magnetic\n\feld intact (with \f= 1:0) and vary BkandB?\ncomponents. In Fig. 8, from left to right, the strength\nof the magnetic \feld component Bk(along the ion\n\rux) reduces and the strength of the component B?\nincreases. Accordingly, the angle between the ion \rux\ndirection and magnetic \feld induction direction, \u000b,\nchanges from 0\u000e[the left column] to 36\u000e(0:64 rad)[the\nright column] with intermediate value corresponding\nto 10\u000e(0:2 rad ) [the middle column]. The change\nin the orientation of magnetic \feld modi\fes the\nwake behavior substantially. It is seen from Fig. 8that, in the sonic and supersonic cases, a small\ndeviation of the magnetic \feld direction from the\npurely longitudinal case has strong impact on the\nwake\feld (compare the left pair of columns). This is\nespecially strongly manifested in the supersonic case\n(M= 1:5, see the bottom row), where 10\u000edeviation\nof the magnetic \feld induction from the longitudinal\ndirection creates absolutely di\u000berent pattern of the\nwake \feld in comparison with the purely longitudinal\ncase (i.e., B=Bk). In contrast to the sonic and\nsupersonic cases, in the subsonic regime the wake \feld\nmodi\fes smoothly with the rotation of the magnetic\n\feld induction direction from transverse to longitudinal\ncon\fguration.\n4. Discussion and conclusion\nThe present work provides a comprehensive picture\nof the in\ruence of the transverse magnetic \feld\non the grain wake\feld and reveals the important\nrole played by the orientation of the magnetic\n\feld. The investigation has been performed with\nMaxwellian ion distribution and the e\u000bect of non-\nMaxwellian ions in transverse magnetic \feld is topic\nfor forthcoming studies. We considered weak and\nmoderate magnetization of ions with \f\u00140:35 in\nsubsonic, sonic, and supersonic regimes with Mach\nnumberM\u00141:5.Transverse magnetic \feld in\ruence on wake\feld in plasmas 9\n-2.0\n-2.0(a)\n(d)(b)(c)\n(i)(g)(f)\n(h)(e)M=0.5M=1.0M=1.5α=0α=0.2α=0.64\nFigure 8. Wake potential contours e\u001e=T ewith the ion \row velocity M= 1:0 and magnetization parameter \f= 1:0 for various\norientation angle of magnetic \feld \u000b, where\u000bis the angle between magnetic \feld direction and ionic streaming direction [i.e., z\naxis]. From left to right column we have \u000b= 0 [i.e. B=BkwithB?= 0 ],\u000b= 0:2, and\u000b= 0:64.Transverse magnetic \feld in\ruence on wake\feld in plasmas 10\nWe revealed several new features of wake\feld due\nto transverse magnetic \feld. First of all, in stark\ncontrast to the case of magnetic \feld along \row [12, 11],\nunder the in\ruence of the transverse magnetic \feld the\nnumber of wake oscillations increases in comparison to\nthe magnetic \feld free case. Secondly, the damping\nof the amplitude of wake\feld oscillations is not as\nstrong as that observed for parallel to the \row magnetic\n\feld case. Another important result we demonstrated\nherein is that the deviation of the orientation of\nmagnetic \feld induction from the parallel to \row\ndirection leads to a signi\fcant wake\feld modi\fcation\nin downstream region.\nMagnetic \feld in\ruences the wake amplitude as\nwell as overall structure of the grain plasma dynamics\n(see e.g. [32, 33, 34]). Magnetic \feld along \row has\nbeen known to provide shape control of dust cloud.\nWake formation has also been attributed to vertical\nalignment of paired grains. Essentially, B\feld could\nbe of utmost importance for future research for dust\nshape control. Therefore, the presented results are\nrelevant for accurate description of the complex plasma\nexperiments in weak to moderate external magnetic\n\feld [15].\nFurthermore, our results can also be useful to\nunderstand better other phenomena involving the\nmotion of particles in cross \felds, e.g. the dynamics\nof scramjet mixing and charge focusing with electric or\nmagnetic \felds.\nOn a \fnal note, we mention that the work studied\nherein should be of interest for open challenges in\nastrophysics, e.g. the dynamics of trapped particles in\nVan Allen Belts [35], the Saturn ring spokes formation\n[36] as well as other astrophysical phenomena wherein\nthe motion of dust particles encounter magnetic \feld\nand streaming plasmas.\n5. Acknowledgments\nS. Sundar would like to acknowledges the support\nand hospitality of IIT Madras India. This work\nwas supported by the DRDO project via project no.\nASE1718144DRDOASAM. Zh. Moldabekov thanks\nthe funding from the Ministry of Education and\nScience of the Republic of Kazakhstan via the grant\nBR05236730 \\Investigation of fundamental problems\nof Phys. Plasmas and plasma-like media\". Our\nnumerical simulations were performed at the HPC\ncluster of IIT Madras.[1] Mor\fll G E and Ivlev A V 2009 Rev. Mod. Phys. 81(4)\n1353{1404\n[2] Bonitz M, Becker K, Lopez J and (Eds) H T 2014 Springer\nSeries\n[3] Veeresha B M, Das A and Sen A 2005 Physics of Plasmas\n12044506\n[4] Bastykova N K, Kov\u0013 acs A Z, Korolov I, Kodanova S K,\nRamazanov T S, Hartmann P and Donk\u0013 o Z 2015\nContributions to Plasma Physics 55671{676\n[5] Bastykova N K, Donk Z, Kodanova S K, Ramazanov T S\nand Moldabekov Z A 2016 IEEE Transactions on Plasma\nScience 44545{548\n[6] Donk\u0013 o Z, Derzsi A, Korolov I, Hartmann P, Brandt S,\nSchulze J, Berger B, Koepke M, Bruneau B, Johnson\nE, La\reur T, Booth J P, Gibson A R, O'Connell D and\nGans T 2017 Plasma Physics and Controlled Fusion 60\n014010\n[7] Sundar S, K ahlert H, Joost J P, Ludwig P and Bonitz M\n2017 Physics of Plasmas 24102130\n[8] Sundar S and Moldabekov Z A 2019 Phys. Rev. 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E 98(2) 023206\n[13] Kodanova S K, Bastykova N K, Ramazanov T S, Nigmetova\nG N, Maiorov S A and Moldabekov Z A 2019 IEEE\nTransactions on Plasma Science 1{5\n[14] Kaw P K, Nishikawa K and Sato N 2002 Physics of Plasmas\n9387{390\n[15] Abdirakhmanov A R, Moldabekov Z A, Kodanova S K,\nDosbolayev M K and Ramazanov T S 2019 IEEE\nTransactions on Plasma Science 473036{3040 ISSN\n0093-3813\n[16] Fujiyama H, Kawasaki H, Yang S C and Matsuda Y 1994\nJapanese Journal of Applied Physics 334216{4220\n[17] Salimullah M 1996 Physics Letters A 215 296 { 298 ISSN\n0375-9601\n[18] Rao N N 1995 Journal of Plasma Physics 53317334\n[19] Sato N, Uchida G, Kaneko T, Shimizu S and Iizuka S 2001\nPhysics of Plasmas 81786{1790\n[20] Samsonov D, Zhdanov S, Mor\fll G and Steinberg V 2003\nNew Journal of Physics 524\n[21] Yaroshenko V V, Mor\fll G E, Samsonov D and Vladimirov\nS V 2003 New Journal of Physics 518\n[22] Tskhakaya D 2004 Journal of Experimental and Theoretical\nPhysics 9853\n[23] Ramazanov T S, Moldabekov Z A and Gabdullin M T 2016\nPhys. Rev. E 93(5) 053204\n[24] Ludwig P, Miloch W J, K ahlert H and Bonitz M 2012 New\nJournal of Physics 14053016\n[25] Ludwig P, Jung H, K ahlert H, Joost J P, Greiner F,\nMoldabekov Z A, Carstensen J, Sundar S, Bonitz M and\nPiel A 2017 EPJ D 52124004\n[26] Carstensen J, Greiner F and Piel A 2012 Phys. Rev. Lett.\n109(13) 135001\n[27] Miloch W J, Jung H, Darian D, Greiner F, Mortensen M\nand Piel A 2018 New Journal of Physics 20073027\n[28] Nambu M, Salimullah M and Bingham R 2001 Phys. Rev.\nE63(5) 056403\n[29] Hutchinson I H 2011 Physics of Plasmas 18032111\n[30] Sundar S 2018 Phys. Sci. Tech. 2(5)\n[31] Hutchinson I H and Haakonsen C B 2013 Physics of\nPlasmas 20083701\n[32] Konopka U, Samsonov D, Ivlev A V, Goree J, Steinberg V\nand Mor\fll G E 2000 Phys. Rev. E 61(2) 1890{1898Transverse magnetic \feld in\ruence on wake\feld in plasmas 11\n[33] Puttscher M and Melzer A 2014 Physics of Plasmas 21\n123704\n[34] Kodanova S K, Ramazanov T S, Bastykova N K and\nMoldabekov Z A 2015 Physics of Plasmas 22063703\n[35] Ukhorskiy A Y and Sitnov M I 2014 Dynamics of Radiation\nBelt Particles (Boston, MA: Springer US) pp 545{578\n[36] Goertz C K 1989 Reviews of Geophysics 27271{292"    },    {        "title": "1405.2347v1.Magnetization_dynamics_and_damping_due_to_electron_phonon_scattering_in_a_ferrimagnetic_exchange_model.pdf",        "content": "Magnetization Dynamics and Damping due to Electron-Phonon Scattering in a\nFerrimagnetic Exchange Model\nAlexander Baral,\u0003Svenja Vollmar, and Hans Christian Schneidery\nPhysics Department and Research Center OPTIMAS,\nKaiserslautern University, P. O. Box 3049, 67663 Kaiserslautern, Germany\n(Dated: June 4, 2018)\nWe present a microscopic calculation of magnetization damping for a magnetic \\toy model.\"\nThe magnetic system consists of itinerant carriers coupled antiferromagnetically to a dispersionless\nband of localized spins, and the magnetization damping is due to coupling of the itinerant carriers\nto a phonon bath in the presence of spin-orbit coupling. Using a mean-\feld approximation for\nthe kinetic exchange model and assuming the spin-orbit coupling to be of the Rashba form, we\nderive Boltzmann scattering integrals for the distributions and spin coherences in the case of an\nantiferromagnetic exchange splitting, including a careful analysis of the connection between lifetime\nbroadening and the magnetic gap. For the Elliott-Yafet type itinerant spin dynamics we extract\ndephasing and magnetization times T1andT2from initial conditions corresponding to a tilt of the\nmagnetization vector, and draw a comparison to phenomenological equations such as the Landau-\nLifshitz (LL) or the Gilbert damping. We also analyze magnetization precession and damping for\nthis system including an anisotropy \feld and \fnd a carrier mediated dephasing of the localized spin\nvia the mean-\feld coupling.\nPACS numbers: 75.78.-n, 72.25.Rb, 76.20.+q\nI. INTRODUCTION\nThere are two widely-known phenomenological ap-\nproaches to describe the damping of a precessing mag-\nnetization in an excited ferromagnet: one introduced\noriginally by Landau and Lifshitz1and one introduced\nby Gilbert,2which are applied to a variety of prob-\nlems3involving the damping of precessing magnetic mo-\nments. Magnetization damping contributions and its in-\nverse processes, i.e., spin torques, in particular in thin\n\flms and nanostructures, are an extremely active \feld,\nwhere currently the focus is on the determination of novel\nphysical processes/mechanisms. Apart from these ques-\ntions there is still a debate whether the Landau-Lifshitz\nor the Gilbert damping is the correct one for \\intrin-\nsic\" damping, i.e., neglecting interlayer coupling, inter-\nface contributions, domain structures and/or eddy cur-\nrents. This intrinsic damping is believed to be caused\nby a combination of spin-orbit coupling and scattering\nmechanisms such as exchange scattering between s and d\nelectrons and/or electron-phonon scattering.4{6Without\nreference to the microscopic mechanism, di\u000berent macro-\nscopic analyses, based, for example, on irreversible ther-\nmodynamics or near equilibrium Langevin theory, prefer\none or the other description.7,8However, material param-\neters of typical ferromagnetic heterostructures are such\nthat one is usually \frmly in the small damping regime so\nthat several ferromagnetic resonance (FMR) experiments\nwere not able to detect a noticeable di\u000berence between\nLandau Lifshitz and Gilbert magnetization damping. A\nrecent analysis that related the Gilbert term directly to\nthe spin-orbit interaction arising from the Dirac equa-\ntion does not seem to have conclusively solved this dis-\ncussion.9\nThe dephasing term in the Landau-Lifshitz form isalso used in models based on classical spins coupled\nto a bath, which have been successfully applied to\nout-of-equilibrium magnetization dynamics and magnetic\nswitching scenarios.10The most fundamental of these\nare the stochastic Landau-Lifshitz equations,10{13from\nwhich the Landau-Lifshitz Bloch equations,14,15can be\nderived via a Fokker-Planck equation.\nQuantum-mechanical treatments of the equilibrium\nmagnetization in bulk ferromagnets at \fnite temper-\natures are extremely involved. The calculation of\nnon-equilibrium magnetization phenomena and damp-\ning for quantum spin systems in more than one dimen-\nsion, which include both magnetism and carrier-phonon\nand/or carrier-impurity interactions, at present have to\nemploy simpli\fed models. For instance, there have been\nmicroscopic calculations of Gilbert damping parameters\nbased on Kohn-Sham wave functions for metallic ferro-\nmagnets16,17and Kohn-Luttinger p-dHamiltonians for\nmagnetic semiconductors.18While the former approach\nuses spin density-functional theory, the latter approach\ntreats the anti-ferromagnetic kinetic-exchange coupling\nbetween itinerant p-like holes and localized magnetic\nmoments originating from impurity d-electrons within a\nmean-\feld theory. In both cases, a constant spin and\nband-independent lifetime for the itinerant carriers is\nused as an input, and a Gilbert damping constant is ex-\ntracted by comparing the quantum mechanical result for\n!!0 with the classical formulation. There have also\nbeen investigations, which extract the Gilbert damping\nfor magnetic semiconductors from a microscopic calcula-\ntion of carrier dynamics including Boltzmann-type scat-\ntering integrals.19,20Such a kinetic approach, which is of\na similar type as the one we present in this paper, avoids\nthe introduction of electronic lifetimes because the scat-\ntering is calculated dynamically.arXiv:1405.2347v1  [cond-mat.mtrl-sci]  9 May 20142\nThe present paper takes up the question how the spin\ndynamics in the framework of the macroscopic Gilbert\nor Landau-Lifshitz damping compare to a microscopic\nmodel of relaxation processes in the framework of a rel-\natively simple model. We analyze a mean-\feld kinetic\nexchange model including spin-orbit coupling for the itin-\nerant carriers. Thus the magnetic mean-\feld dynamics is\ncombined with a microscopic description of damping pro-\nvided by the electron-phonon coupling. This interaction\ntransfers energy and angular momentum from the itin-\nerant carriers to the lattice. The electron-phonon scat-\ntering is responsible both for the lifetimes of the itiner-\nant carriers and the magnetization dephasing. The lat-\nter occurs because of spin-orbit coupling in the states\nthat are connected by electron-phonon scattering. To be\nmore speci\fc, we choose an anti-ferromagnetic coupling\nat the mean-\feld level between itinerant electrons and\na dispersion-less band of localized spins for the magnetic\nsystem. To keep the analysis simple we use as a model for\nthe spin-orbit coupled itinerant carrier states a two-band\nRashba model. As such it is a single-band version of the\nmulti-band Hamiltonians used for III-Mn-V ferromag-\nnetic semiconductors.18,21{24The model analyzed here\nalso captures some properties of two-sublattice ferrimag-\nnets, which are nowadays investigated because of their\nmagnetic switching dynamics.25,26The present paper is\nset apart from studies of spin dynamics in similar mod-\nels with more complicated itinerant band structures19,20\nby a detailed comparison of the phenomenological damp-\ning expressions with a microscopic calculation as well as\na careful analysis of the restrictions placed by the size\nof the magnetic gap on the single-particle broadening in\nBoltzmann scattering.\nThis paper is organized as follows. As an extended\nintroduction, we review in Sec. II some basic facts con-\ncerning the Landau-Lifshitz and Gilbert damping terms\non the one hand and the Bloch equations on the other.\nIn Sec. III we point out how these di\u000berent descriptions\nare related in special cases. We then introduce a micro-\nscopic model for the dephasing due to electron-phonon\ninteraction in Sec. IV, and present numerical solutions\nfor two di\u000berent scenarios in Secs. V and VI. The \frst\nscenario is the dephasing between two spin subsystems\n(Sec. V), and the second scenario is a relaxation process\nof the magnetization toward an easy-axis (Sec. VI). A\nbrief conclusion is given at the end.\nII. PHENOMENOLOGIC DESCRIPTIONS OF\nDEPHASING AND RELAXATION\nWe summarize here some results pertaining to a single-\ndomain ferromagnet, and set up our notation. In equilib-\nrium we assume the magnetization to be oriented along\nits easy axis or a magnetic \feld ~H, which we take to\nbe thezaxis in the following. If the magnetization\nis tilted out of equilibrium, it starts to precess. As\nillustrated in Fig. 1 one distinguishes the longitudinal\nFIG. 1. Illustration of non-equilibrium spin-dynamics in pres-\nence of a magnetic \feld without relaxation (a) and within\nrelaxation (b).\ncomponent Mk, inzdirection, and the transverse part\nM?\u0011q\nM2\u0000M2\nk, precessing in the x-yplane with the\nLarmor frequency !L.\nIn connection with the interaction processes that re-\nturn the system to equilibrium, the decay of the trans-\nverse component is called dephasing. There are three\nphenomenological equations used to describe spin de-\nphasing processes:\n1. The Bloch(-Bloembergen) equations27,28\n@\n@tMk(t) =\u0000Mk(t)\u0000Meq\nT1(1)\n@\n@tM?(t) =\u0000M?(t)\nT2(2)\ndescribe an exponential decay towards the equilib-\nrium magnetization Meqinzdirection. The trans-\nverse component decays with a time constant T2,\nwhereas the longitudinal component approaches its\nequilibrium amplitude with T1. These time con-\nstants may be \ft independently to experimental\nresults or microscopic calculations.\n2. Landau-Lifshitz damping1with parameter \u0015\n@\n@t~M(t) =\u0000\r~M\u0002~H\u0000\u0015~M\nM\u0002\u0000~M\u0002~H\u0001\n(3)\nwhere\ris the gyromagnetic ratio. The \frst term\nmodels the precession with a frequency !L=\rj~Hj,\nwhereas the second term is solely responsible for\ndamping.\n3. Gilbert damping2with the dimensionless Gilbert\ndamping parameter \u000b\n@\n@t~M(t) =\u0000\rG~M\u0002~H+\u000b\u0010~M\nM\u0002@t~M\u0011\n(4)\nIt is generally accepted that \u000bis independent of\nthe static magnetic \felds ~Hsuch as anisotropy\n\felds,18,29and thus depends only on the material\nand the microscopic interaction processes.3\nThe Landau-Lifshitz and Gilbert forms of damping are\nmathematically equivalent2,7,30with\n\u000b=\u0015\n\r(5)\n\rG=\r(1 +\u000b2) (6)\nbut there are important di\u000berences. In particular, an in-\ncrease of\u000blowers the precession frequency in the dynam-\nics with Gilbert damping, while the damping parameter\n\u0015in the Landau-Lifshitz equation has no impact on the\nprecession. In contrast to the Bloch equations, Landau-\nLifshitz and Gilbert spin-dynamics always conserve the\nlengthj~Mjof the magnetization vector.\nAn argument by Pines and Slichter,31shows that there\nare two di\u000berent regimes for Bloch-type spin dynamics\ndepending on the relation between the Larmor period and\nthe correlation time. As long as the correlation time is\nmuch longer than the Larmor period, the system \\knows\"\nthe direction of the \feld during the scattering process.\nStated di\u000berently, the scattering process \\sees\" the mag-\nnetic gap in the bandstructure. Thus, transverse and\nlongitudinal spin components are distinguishable and the\nBloch decay times T1andT2can di\u000ber. If the correlation\ntime is considerably shorter than the Larmor period, this\ndistinction is not possible, with the consequence that T1\nmust be equal to T2. Within the microscopic approach,\npresented in Sec. IV D, this consideration shows up again,\nalbeit for the energy conserving \u000efunctions resulting from\na Markov approximation.\nThe regime of short correlation times has already been\ninvestigated in the framework of a microscopic calcula-\ntion by Wu and coworkers.32They analyze the case of\na moderate external magnetic \feld applied to a non-\nmagnetic n-type GaAs quantum well and include di\u000ber-\nent scattering mechanisms (electron-electron Coulomb,\nelectron-phonon, electron-impurity). They argue that\nthe momentum relaxation rate is the crucial time scale\nin this scenario, which turns out to be much larger than\nthe Larmor frequency. Their numerical results con\frm\nthe identity T1=T2expected from the Pines-Slichter\nargument.\nIII. RELATION BETWEEN\nLANDAU-LIFSHITZ, GILBERT AND BLOCH\nWe highlight here a connection between the Bloch\nequations (1, 2) and the Landau-Lifshitz equation (3).\nTo this end we assume a small initial tilt of the mag-\nnetization and describe the subsequent dynamics of the\nmagnetization in the form\n~M(t) =0\n@\u000eM?(t) cos(!Lt)\n\u000eM?(t) sin(!Lt)\nMeq\u0000\u000eMk(t)1\nA (7)\nwhere\u000eM?and\u000eMjjdescribe deviations from equilib-\nrium. Putting this into eq. (3) one gets a coupled set ofequations.\n@\n@t\u000eM?(t) =\u0000\u0015HMeq\u0000\u000eMk(t)\nj~M(t)j\u000eM?(t) (8)\n@\n@t\u000eMk(t) =\u0000\u0015H1\nj~M(t)j\u000eM2\n?(t) (9)\nEq. (8) is simpli\fed for a small deviation from equilib-\nrium, i.e.,\u000eM(t)\u001cMeqandj~M(t)j\u0019Meq:\n\u000eM?(t) =Cexp(\u0000\u0015Ht) (10)\n\u000eMk(t) =C2\n2Meqexp(\u00002\u0015Ht) (11)\nwhereCis an integration constant. For small excitations\nthe deviations decay exponentially and Bloch decay times\nT1andT2result, which are related by\n2T1=T2=1\n\u0015H: (12)\nOnly this ratio of the Bloch times is compatible with a\nconstant length of the magnetization vector at low exci-\ntations. By combining Eqs. (12) and (5) one can connect\nthe Gilbert parameter \u000band the dephasing time T2\n\u000b=1\nT2!L: (13)\nIf the conditions for the above approximations apply, the\nGilbert damping parameter \u000bcan be determined by \ft-\nting the dephasing time T2and the Larmor frequency !L\nto computed or measured spin dynamics. This dimen-\nsionless quantity is well suited to compare the dephasing\nthat results from di\u000berent relaxation processes.\nFigure 2 shows the typical magnetization dynamics\nthat results from (3), i.e., Landau-Lifshitz damping. As\nan illustration of a small excitation we choose in Fig. 2(a)\nan angle of 10\u000efor the initial tilt of the magnetization,\nwhich results in an exponential decay with 2 T1=T2.\nFrom the form of Eq. (3) it is clear that this behavior\npersists even for large !Land\u0015. Obviously the Landau-\nLifshitz and Gilbert damping terms describe a scenario\nwith relatively long correlation times (i.e., small scat-\ntering rates), because only in this regime both decay\ntimes can di\u000ber. The microscopic formalism in Sec. IV\nworks in the same regime and will be compared with\nthe phenomenological results. For an excitation angle\nof 90\u000e, the Landau-Lifshitz dynamics shown in Fig. 2(b)\nbecome non-exponential, so that no well-de\fned Bloch\ndecay times T1,T2exist.\nIV. MICROSCOPIC MODEL\nIn this section we describe a microscopic model that in-\ncludes magnetism at the mean-\feld level, spin-orbit cou-\npling as well as the microscopic coupling to a phonon\nbath treated at the level of Boltzmann scattering inte-\ngrals. We then compare the microscopic dynamics to4\n0 5000.51δM⊥/Meq\ntime (ps)0 5000.51\ntime (ps)δM/bardbl/Meq0 5000.010.02δM/bardbl/Meq\ntime (ps)0 5000.10.2\ntime (ps)δM⊥/Meq\n  \nT1= 5.02 ps(a)\n(b)T2= 10.04 ps\nFIG. 2. Dynamics of \u000eM?and\u000eMkcomputed using to\nLandau-Lifshitz damping ( !L= 1 ps\u00001,H= 106A\nm\u0019\n1:26\u0001104Oe,\u0015= 10\u00007m\nA ps). (a) An angle of 10\u000eleads to\nexponential an exponential decay with well de\fned T1andT2\ntimes. (b). For an angle of 90\u000e, the decay (solid line) is not\nexponential as comparison with the exponential \ft (dashed\nline) clearly shows.\nthe Bloch equations (1), (2), as well as the Landau-\nLifshitz (3) and Gilbert damping terms (4). The mag-\nnetic properties of the model are de\fned by an anti-\nferromagnetic coupling between localized magnetic im-\npurities and itinerant carriers. As a prototypical spin-\norbit coupling we consider an e\u000bectively two-dimensional\nmodel with a Rashba spin-orbit coupling. The reason\nfor the choice of a model with a two-dimensional wave\nvector space is not an investigation of magnetization dy-\nnamics with reduced dimensionality, but rather a reduc-\ntion in the dimension of the integrals that have to be\nsolved numerically in the Boltzmann scattering terms.\nSince we treat the exchange between the localized and\nitinerant states in a mean-\feld approximation, our two-\ndimensional model still has a \\magnetic ground state\"\nand presents a framework, for which qualitatively dif-\nferent approaches can be compared. We do not aim at\nquantitative predictions for, say, magnetic semiconduc-\ntors or ferrimagnets with two sublattices. Finally, we\ninclude a standard interaction hamiltonian between the\nitinerant carriers and acoustic phonons. The correspond-\ning hamiltonian reads\n^H=^Hmf+^Hso+^He\u0000ph+^Haniso: (14)\nOnly in Sec. VI an additional \feld ^Haniso is included,\nwhich is intended to model a small anisotropy.A. Exchange interaction between itinerant carriers\nand localized spins\nThe \\magnetic part\" of the model is described by the\nHamiltonian\n^Hmf=X\n~k\u0016~2k2\n2m\u0003^cy\n~k\u0016^c~k\u0016+J^~ s\u0001^~S: (15)\nwhich we consider in the mean-\feld limit. The \frst term\nrepresents itinerant carriers with a k-dependent disper-\nsion relation. In the following we assume s-like wave\nfunctions and parabolic energy dispersions. The e\u000bective\nmass is chosen to be m\u0003= 0:5me, wheremeis the free\nelectron mass, and the ^ c(y)\n~k\u0016operators create and annihi-\nlate carriers in the state j~k;\u0016iwhere\u0016labels the itinerant\nbands, as shown in Fig. 3(a).\nThe second term describes the coupling between itiner-\nant spins~ sand localized spins ~Svia an antiferromagnetic\nexchange interaction\n^~ s=1\n2X\n~kX\n\u0016\u00160h~k;\u00160j^~ \u001bj~k;\u0016i^cy\n~k\u0016^c~k\u00160 (16)\n^~S=1\n2X\n\u0017\u00170h\u00170j^~ \u001bj\u0017iX\n~K^Cy\n~K\u0017^C~K\u00170 (17)\nHere, we have assumed that the wave functions of the lo-\ncalized spins form dispersionless bands, i.e., we have im-\nplicitly introduced a virtual-crystal approximation. Due\nto the assumption of strong localization there is no or-\nbital overlap between these electrons, which are therefore\nconsidered to have momentum independent eigenstates\nj\u0017iand a \rat dispersion, as illustrated in Fig. 3(a). The\ncomponents of the vector ^~ \u001bare the Pauli matrices ^ \u001biwith\ni=x;y;z , and ^C(y)\n~K\u0017are the creation and annihilation op-\nerators for a localized spin state.\nWe do notinclude interactions among localized or itin-\nerant spins, such as exchange scattering. For simplicity,\nwe assume both itinerant and localized electrons to have\na spin 1=2 and therefore \u0016and\u0017to run over two spin-\nprojection quantum numbers \u00061=2. In the following we\nchosse an antiferromagnetic ( J > 0) exchange constant\nJ= 500 meV, which leads to the schematic band struc-\nture shown in Fig. 3(b).\nIn the mean \feld approximation used here, the itiner-\nant carriers feel an e\u000bective magnetic \feld ^Hloc\n~Hloc=\u0000J\u0016B\u0016\ng~S (18)\ncaused by localized moments and vice versa. Here \u0016B\nis the Bohr magneton and g= 2 is the g-factor of the\nelectron. The permeability \u0016is assumed to be the vac-\nuum permeability \u00160. This time-dependent magnetic\n\feld~Hloc(t) de\fnes the preferred direction in the itiner-\nant sub-system and therefore determines the longitudinal\nand transverse component of the itinerant spin at each\ntime.5\nr#k (a) (b) E(k) \nk \n\u0010\nk \n\u000e\u0010,k\n\u000e,kE(k) \nEF \nFIG. 3. Sketch of the band-structure with localized (\rat\ndispersions) and itinerant (parabolic dispersions) electrons.\nAbove the Curie-Temperature TCthe spin-eigenstates are de-\ngenerate (a), whereas below TCa gap between the spin states\nexists.\nB. Rashba spin-orbit interaction\nThe Rashba spin-orbit coupling is given by the Hamil-\ntonian\n^Hso=\u000bR(^\u001bxky\u0000^\u001bykx) (19)\nA Rashba coe\u000ecient of \u000bR= 10 meV nm typical for semi-\nconductors is chosen in the following calculations. This\nvalue, which is close to the experimental one for the\nInSb/InAlSb material system,33is small compared to the\nexchange interactions, but it allows the exchange of an-\ngular momentum with the lattice.\nC. Coherent dynamics\nFrom the above contributions (15) and (19) to the\nHamiltonian we derive the equations of motion contain-\ning the coherent dynamics due to the exchange interac-\ntion and Rashba spin-orbit coupling as well as the inco-\nherent electron-phonon scattering. We \frst focus on the\ncoherent contributions. In principle, one has the choice\nto work in a basis with a \fxed spin-quantization axis or\nto use single-particle states that diagonalize the mean-\n\feld (plus Rashba) Hamiltonian. Since we intend to use\na Boltzmann scattering integral in Sec. IV D we need to\napply a Markov approximation, which only works if one\ndeals with diagonalized eigenenergies. In our case this is\nthe single-particle basis that diagonalizes the entire one-\nparticle contribution of the Hamiltonian ^Hmf+^Hso. In\nmatrix representation this one-particle contribution for\nthe itinerant carriers reads:\n^Hmf+^Hso= \n~2k2\n2m\u0003+ \u0001loc\nz(\u0001loc\n++R~k)\u0003\n\u0001loc\n++R~k~2k2\n2m\u0003\u0000\u0001loc\nz!\n(20)\nwhere we have de\fned \u0001loc\ni=J1\n2h^SiiandR~k=\n\u0000i\u000bRkexp(i'k) with'k= arctan(ky=kx). The eigenen-\nergies are\n\u000f\u0006\n~k=~2k2\n2m\u0003\u0007q\nj\u0001loczj2+jR~k+ \u0001loc\n+j2: (21)and the eigenstates\nj~k;+i=\u0012\n1\n\u0018~k\u0013\n;j~k;\u0000i=\u0012\u0000\u0018\u0003\n~k\n1\u0013\n(22)\nwhere\n\u0018~k=\u0001loc\n++R~k\n\u0001locz+q\nj~\u0001locj2+jR~kj2(23)\nIn this basis the coherent part of the equation of mo-\ntion for the itinerant density matrix \u001a\u0016\u00160\n~k\u0011 h^cy\n~k\u0016^c~k\u00160i\nreads\n@\n@t\u001a\u0016\u00160\n~k\f\f\f\ncoh=i\n~\u0000\n\u000f\u0016\n~k\u0000\u000f\u00160\n~k\u0001\n\u001a\u0016\u00160\n~k: (24)\nNo mean-\feld or Rashba terms appear explicitly in these\nequations of motion since their contributions are now hid-\nden in the time-dependent eigenstates and eigenenergies.\nSince we are interested in dephasing and precessional\ndynamics, we assume a comparatively small spin-orbit\ncoupling, that can dissipate angular momentum into the\nlattice, but does not have a decisive e\u000bect on the band-\nstructure. Therefore we use the spin-mixing only in the\ntransition matrix elements of the electron-phonon scat-\nteringM~k0\u00160\n~k\u0016(31). For all other purposes we set R~k= 0.\nIn particular, the energy-dispersion \u000f\u0006\n~kis assumed to be\nuna\u000bected by the spin-orbit interaction and therefore it\nis spherically symmetric.\nWith this approximation the itinerant eigenstates are\nalways exactly aligned with the e\u000bective \feld of the local-\nized moments ~Hloc(t). Since this e\u000bective \feld changes\nwith time, the diagonalization and a transformation of\nthe spin-density matrix in \\spin space\" has to be re-\npeated at each time-step. This e\u000bort makes it easier\nto identify the longitudinal and transverse spin compo-\nnents with the elements of the single-particle density\nmatrix: The o\u000b-diagonal entries of the density matrix\n\u001a\u0006\u0007\n~k, which precess with the k-independent Larmor fre-\nquency!L= 2\u0001loc=~, always describe the dynamics of\nthe transverse spin-component. The longitudinal compo-\nnent, which does not precess, is represented by the diag-\nonal entries \u001a\u0006\u0006\n~k. Since both components change their\nspatial orientation continuously, we call this the rotating\nframe. The components of the spin vector in the rotating\nframe are\nh^ski=1\n2X\n~k\u0000\n\u001a++\n~k\u0000\u001a\u0000\u0000\n~k\u0001\n(25)\nh^s?i=X\n~k\f\f\u001a+\u0000\n~k\f\f (26)\nThe components in the \fxed frame are obtained from\nEq. (16)\nh^~ si=1\n2X\n~kX\n\u0016\u00160h~k;\u00160j^~ \u001bj~k;\u0016i\u001a\u0016\u00160\n~k(27)6\nIn this form, the time-dependent states carry the infor-\nmation how the spatial components are described by the\ndensity matrix at each time step. No time-independent\n\\longitudinal\" and \\transverse\" directions can be identi-\n\fed in the \fxed frame.\nIn a similar fashion, the diagonalized single-particle\nstates of the localized spin system are obtained. The\neigenenergies are\nE\u0006=\u0007\f\f~\u0001itin\f\f (28)\nwhere \u0001itin\ni=J1\n2h^siiis the localized energy shift caused\nby the itinerant spin component si. The eigenstates are\nagain always aligned with the itinerant magnetic mo-\nment. In this basis the equation of motion of the localized\nspin-density matrix \u001a\u0017\u00170\nloc\u0011P\n~Kh^Cy\n~K\u0017^C~K\u00170iis simply\n@\n@t\u001a\u0017\u00170\nloc=i\n~(E\u0017\u0000E\u00170)\u001a\u0017\u00170\nloc (29)\nand does not contain explicit exchange contributions.\nEqs. (25), (26), and (27) apply in turn to the components\nhSkiandhS?iof the localized spin and its spin-density\nmatrix\u001a\u0017\u00170\nloc.\nD. Electron-phonon Boltzmann scattering with\nspin splitting\nRelaxation is introduced into the model by the interac-\ntion of the itinerant carriers with a phonon bath, which\nplays the role of an energy and angular momentum sink\nfor these carriers. Our goal here is to present a derivation\nof the Boltzmann scattering contributions using stan-\ndard methods, see, e.g., Refs. 34 and 36. However, we\nemphasize that describing interaction as a Boltzmann-\nlike instantaneous, energy conserving scattering process\nis limited by the existence of the magnetic gap. Since we\nkeep the spin mixing due to Rashba spin-orbit coupling\nonly in the Boltzmann scattering integrals, the resulting\ndynamical equations describe an Elliott-Yafet type spin\nrelaxation.\nThe electron-phonon interaction Hamiltonian reads34\n^He\u0000ph=X\n~ q~!ph\nq^by\n~ q^b~ q\n+X\n~k~k0X\n\u0016\u00160\u0000\nM~k0\u00160\n~k\u0016^cy\n~k\u0016^b~k\u0000~k0^c~k0\u00160+ h.c.\u0001(30)\nwhere ^b(y)\n~ qare the bosonic operators, that create or an-\nnihilate acoustic phonons with momentum ~ qand linear\ndispersion!ph(q) =cphj~ qj. The sound velocity is taken\nto becph= 40 nm/ps and we use an e\u000bectively two-\ndimensional transition matrix element35\nM~k0\u00160\n~k\u0016=Dq\nj~k\u0000~k0jh~k;\u0016j~k0;\u00160i (31)\nwhere the deformation potential is chosen to be D=\n60 meVnm1=2. The scalar-product between the initialstatej~k0;\u00160iand the \fnal state j~k;\u0016iof an electronic\ntransition takes the spin-mixing due to Rashba spin-orbit\ncoupling into account.\nThe derivation of Boltzmann scattering integrals for\nthe itinerant spin-density matrix (24) leads to a memory\nintegral of the following shape\n@\n@t\u001aj(t)\f\f\f\ninc=1\n~X\nj0Zt\n\u00001ei(\u0001Ejj0+i\r)(t\u0000t0)Fjj0[\u001a(t0)]dt0;\n(32)\nregardless whether one uses Green's function36or\nequation-of-motion techniques.34Since we go through a\nstandard derivation here, we highlight only the impor-\ntant parts for the present case and do not write the equa-\ntions out completely. In particular, for scattering process\nj0=j\u00160;~k0i!j=j\u0016;~ki, we useFjj0[\u001a(t0)] as an abbre-\nviation for a product of dynamical electronic spin-density\nmatrix elements \u001a, evaluated at time t0<t, and equilib-\nrium phononic distributions. The corresponding energy\ndi\u000berence is denoted by \u0001 Ejj0=Ej\u0000Ej0\u0006~!ph(j~k\u0000~k0j),\nand\rdescribes the decay of the exponential function due\nto dissipation and/or higher order correlation functions.\nIn general, the integral has to be evaluated numerically\nand contains memory e\u000bects. To apply the Markov ap-\nproximation one needs to compare two time scales: the\n\\memory depth\" 1 =\r, i.e., the time scale on which the\nexp[\u0000\r(t\u0000t0)] factor essentially cuts o\u000b the integral, and\nthe typical time scale on which the Fterm changes. In\nthis paper we deal with relaxation processes not too far\naway from equilibrium, so that the typical time scale of\nthe components of the spin-density matrix contained in\nFis set by the Bloch times T1andT2. We can thus\napproximate Fjj0[\u001a(t0)] byFjj0[\u001a(t)], for all transitions\nlabeled byjandj0, if the memory depth is shorter than\nthe Bloch time(s), or\n\r\u001d1\nT1: (33)\nProvided condition (33) holds, the integral (32) can\nbe done using the Markov approximation Fjj0[\u001a(t0)]'\nFjj0[\u001a(t)]\n@\n@t\u001aj(t)\f\f\f\nincoh=i\n~X\nj0Fjj0[\u001a(t)]1\n\u0001Ejj0+i~\r:(34)\nAs it is customary, we neglect in the following the real\npart of the complex energy denominator, which results in\nshifts of the single-particle energies. While these shifts\nmay play an important role in non-Markovian problems\nwith discrete energy levels37, the imaginary parts yield\nthe relaxation contributions that are important for the\npresent paper\n@\n@t\u001aj(t)\f\f\f\nincoh=X\nj0Fjj0[\u001a(t)]~\r\n(\u0001Ejj0)2+ (~\r)2(35)\nAll transitions are thus weighted by a Lorentzian peaked\nat resonant transitions (\u0001 Ejj0= 0) with a broadening7\nof~\rthat may be interpreted as an energy uncertainty.\nFor relaxation processes in a system with a spin-splitting\n(due to internal \felds and/or spin-orbit coupling), this\nbroadening must not be so large as to blur the distinc-\ntion between the split bands. Consequently, only if the\nbroadening \ris smaller than the magnetic splitting, i.e,,\nif\r\u001c!L, it is possible to distinguish between longi-\ntudinal and transverse components of the spin-density\nmatrix. With Eq. (33) the inequality \r\u001c!Lyields the\ncondition\n!L\u001d1\nT1(36)\nfor the Larmor frequencies and Bloch times, for which\nit is permissible to replace the Lorentzian by an energy\nconserving \u000efunction\n~\r\n(\u0001Ejj0)2+ (~\r)2\r!0\u0000!\u0019\u000e(\u0001Ejj0): (37)This reduces the numerical e\u000bort very considerably, be-\ncause it allows one to eliminate an integration from the\nscattering term, and the energy conserving \u000efunction is\ntherefore often used without explicitly checking its valid-\nity.\nThe considerations leading to the connection between\nEqs. (36) and (37) are a microscopic version of an argu-\nment due to Pines and Slichter,31according to which T1\nandT2can di\u000ber only for correlation times that long in\ncomparison to a Larmor period. The microscopic Boltz-\nmann scattering terms, which contain the energy con-\nserving\u000efunctions and will be used in the following,\ndo not apply in a regime outside of condition (36). If\n!L'1=T1, a \fnite broadening \rhas to be taken into\naccount. Together the full equation of motion in the\nregime (36) for the itinerant-carrier spin density matrix\nthus reads\n@\n@t\u001a\u0016\u00160\n~k=i\n~\u0000\n\u000f\u0016\nk\u0000\u000f\u00160\nk\u0001\n\u001a\u0016\u00160\n~k\n+\u0019\n~X\nk0X\n\u00161\u00162\u00163M~k\u0016\n~k0\u00161M~k0\u00162\n~k\u00163\u000e\u0000\n\u0001E~k0\u00162~k\u00163\u0001h\u0000\n1 +Nph\nj~k0\u0000~kj\u0001\n\u001a\u00161\u00162\n~k0\u0000\n\u000e\u00163\u00160\u0000\u001a\u00163\u00160\n~k\u0001\n\u0000Nph\nj~k0\u0000~kj\u001a\u00163\u00160\n~k\u0000\n\u000e\u00161\u00162\u0000\u001a\u00161\u00162\n~k0\u0001i\n\u0000\u0019\n~X\n~k0X\n\u00161\u00162\u00163M~k\u0016\n~k0\u00161M~k0\u00162\n~k\u00163\u000e\u0000\n\u0001E~k\u00163~k0\u00162\u0001h\u0000\n1 +Nph\nj~k\u0000~k0j\u0001\n\u001a\u00163\u00160\n~k\u0000\n\u000e\u00161\u00162\u0000\u001a\u00161\u00162\n~k0\u0001\n\u0000Nph\nj~k\u0000~k0j\u001a\u00161\u00162\n~k0\u0000\n\u000e\u00163\u00160\u0000\u001a\u00163\u00160\n~k\u0001i\n+\u0019\n~X\n~k0X\n\u00161\u00162\u00163M~k0\u00161\n~k\u00160M~k\u00163\n~k0\u00162\u000e\u0000\n\u0001E~k0\u00162~k\u00163\u0001h\u0000\n1 +Nph\nj~k0\u0000~kj\u0001\n\u001a\u00162\u00161\n~k0\u0000\n\u000e\u0016\u00163\u0000\u001a\u0016\u00163\n~k\u0001\n\u0000Nph\nj~k0\u0000~kj\u001a\u0016\u00163\n~k\u0000\n\u000e\u00162\u00161\u0000\u001a\u00162\u00161\n~k0\u0001i\n\u0000\u0019\n~X\n~k0X\n\u00161\u00162\u00163M~k0\u00161\n~k\u00160Mk\u00163\n~k0\u00162\u000e\u0000\n\u0001E~k\u00163~k0\u00162\u0001h\u0000\n1 +Nph\nj~k\u0000~k0j\u0001\n\u001a\u0016\u00163\n~k\u0000\n\u000e\u00162\u00161\u0000\u001a\u00162\u00161\n~k0\u0001\n\u0000Nph\nj~k\u0000~k0j\u001a\u00162\u00161\n~k0\u0000\n\u000e\u0016\u00163\u0000\u001a\u0016\u00163\n~k\u0001i\n(38)\nHere, \u0001E~k\u0016~k0\u00160=\u000f\u0016\nk\u0000\u000f\u00160\nk0\u0000~!ph\nj~k\u0000~k0j, andNph\nqis the\noccupation function of a thermalized phonon bath, given\nby a Bose-Einstein distribution\nNph\nq=1\ne\f~!ph\u00001(39)\nwhere\f= 1=(kBTph). The numerical results for the mi-\ncroscopic dynamics in the following sections are obtained\nby numerically solving the equations of motion (29) and\n(38). In the numerical calculations the spin-density ma-\ntrix is transformed to the single-particle basis of the in-\nstantaneous, diagonalized eigenstates (22).\nV. DEPHASING BETWEEN LOCALIZED AND\nITINERANT SPINS: NUMERICAL RESULTS\nSince we are interested in this paper in a comparison\nof the model described above with Landau-Lifshitz and\nGilbert damping, we investigate magnetization dynam-\nics with an initial spin-density matrix that correspondsto a tilting of the spins out of their equilibrium posi-\ntion without changing the kinetic energy of the carriers,\nbecause we need initial conditions that lead to generic\nmagnetization dephasing without carrier heating and the\ncorresponding demagnetization dynamics.\nA. Initial state and spin dynamics\nThus we take as the equilibrium initial state the\nsteady-state that is reached for the coupled spins inter-\nacting with the phonon bath at low temperature ( Tph=\n1 K), as shown in Fig. 4(a). In this equilibrium state\nthe spin density matrix is characterized by shifted Fermi\nfunctions for the distributions \u001a\u0016\u0016\n~k=f(\u000f\u0016\nk\u0000EF;Tph)\nand vanishing coherences \u001a+\u0000\n~k= 0. In particular, the\nsteady-state calculation determines the equilibrium mag-\nnetic gap.\nWe then change the itinerant density matrix to that\ncorresponding to an itinerant spin tilted by \f= 10\u000e\nout of equilibrium, see Fig. 4(b). This initial condition8\nFIG. 4. (a) Localized spin h~Siand itinerant spin h~ siin ther-\nmal equilibrium. (b) Itinerant spins tilted out of equilibrium\nby an angle \f.\n0 1 2 300.51\nk(nm−1)Occupations  \n0 1 2 3−0.500.5\nk(nm−1)Coherences\n  ρ|k|− −\nρ|k|+ +\nℜ(ρ|k|− +)\nℑ(ρ|k|− +)\nFIG. 5. Initial spin density-matrix (occupations and coher-\nences) for itinerant electrons corresponding to a tilt \f= 50\u000e.\nThe deviation from equilibrium occurs only between the Fermi\nwave-vectors of the \\+\" and \\ \u0000\" bands.\nachieves a tilting of the spins without heating and avoids\ngeneric de- and remagnetization dynamics. Microscop-\nically the tilted spin corresponds to the spin density-\nmatrix shown in Fig. 5. The perturbation for distribu-\ntions and coherences exists only between the two Fermi\nwave vectors for the \u0016= + and\u0016=\u0000bands. For smaller\ntilt angles the deviation is much less pronounced.\nFrom this initial condition, both spins start to precess\naround the instantaneous direction, along which the ex-\nchange interaction tries to align them. This direction\nis determined for the itinerant carriers by the localized\nspins, and vice versa. A return back into equilibrium re-\nquires the scattering of itinerant electrons with phonons.\nIf we switch o\u000b spin mixing, the dynamics shown in\nFig. 6(a) result: No angular momentum is exchanged\nwith the phonon bath, the excited system cannot relax\ninto equilibrium and the precession goes on inde\fnitely.\nFig. 6(b) shows the same result including spin-mixed itin-\nerant states. Now angular momentum can be transferred\nfrom the itinerant sub-system into the lattice and the to-\ntal spinh~Si+h~ sichanges. In the presence of spin-orbit\ncoupling, electron-phonon scattering, which is by itself\nspin-diagonal, can return the spin system into equilib-\nrium, characterized by aligned spins and vanishing trans-\nverse components. Since we consider here a small Rashba\n(a) (b) \n5&:P; 5&:P; \nO&:P; O&:P; FIG. 6. Non-equilibrium dynamics of localized (red) and itin-\nerant (blue) spins including electron-phonon scattering with-\nout spin-mixing (a) and within spin-mixing (b).\n0 1 2 30.120.130.14\ntime (ps)|s|0 1 2 3−0.0400.04\ntime (ps)sx\n0 1 2 3−0.0400.04\ntime (ps)sy\n0 1 2 3−0.14−0.13−0.12\ntime (ps)sz\nFIG. 7. Relaxation dynamics of itinerant spins in the \fxed\nframe.\ncoupling, the interaction with the phonon bath removes\nenergy much faster than angular momentum. The car-\nrier temperature therefore stays practically equal to the\nphonon temperature Tphduring the entire relaxation pro-\ncess, and no heat-induced demagnetization processes oc-\ncur. The \fnal magnetization is, however, not necessarily\noriented in the zdirection of the \fxed frame, because\nin the results discussed in the this section there is no\nexternal \feld or anisotropy to induce such an alignment.\nWe plot the resulting dynamics of the itinerant spins\nduring the dephasing process in Fig. 7, which shows that9\n00.511.522.53−0.133−0.132−0.131\ntime(ps)s/bardbl\n00.511.522.5300.010.020.03\ntime(ps)s⊥\nFIG. 8. Relaxation dynamics of itinerant spins in the rotating\nframe. An exponential \ft determines the Bloch decay times\ntoT1= 0:5 ps andT2= 1:0 ps.\nall itinerant spatial components precess, and no spatially\n\fxed component can be considered to be longitudinal.\nFirst, the localized spin turns away from the zdirec-\ntion due to the tilted itinerant spin and subsequently\nboth spin-systems precess around each other because the\nquantization axis of each system changes continuously\ndue to the mutual interaction. During the entire relax-\nation process the absolute value of the itinerant spin is\nconserved to better than 1%. Fig. 8 shows the same\ndynamics in the rotating frame, where the longitudinal\nand transverse dynamics can be seen clearly. Both itin-\nerant components show exponential dynamics, which are\ntherefore well described by decay times T1andT2. If well-\nde\fned decay times exist, one expects a ratio T2=T1= 2\nas long as the length of the spin is conserved. The \ft for\nour numerical results indeed gives 2 T1'T2.\nFigure 9 plots the T1andT2values extracted from the\ndynamics as a function of the strength of the electron-\nphonon coupling, or deformation potential, D. The de-\npendence of the decay times on Dcan be \ft extremely\nwell by a 1=D2relation, which demonstrates the propor-\ntionality of the Bloch decay times T1;2/1=D2. Further,\nthe ratioT2=T1stays equal to 2 for all coupling strengths.\nAs discussed in Sec. IV D about the Markov approxima-\ntion, our microscopic description cannot reach regimes\nwhereT1andT2are indistinguishable and therefore\nequal. However, these results show that for small tilting\nangles, not even a pronounced electron-phonon coupling\nleads to a noticeable deviation from the T2= 2T1be-\nhavior. Because this relation between T1andT2holds,\nthe dynamics in Fig. 8 can be equally well described by\nan Landau-Lifshitz or Gilbert damping term. By \ftting\nthe dephasing time T2\u00191 ps and the Larmor-frequency\n!L\u0019281 ps\u00001, equation (13) yields the corresponding\nGilbert damping parameter \u000biso\u00193:6\u000110\u00003.\n204060801001201401.822.2\nD(meV√nm)T2/T1\n  2040608010012014010−2100102\nD(meV√nm)T1,2(ps)\n  FIG. 9. Top: Longitudinal (black squares) and the transverse\n(blue circles) Bloch decay times vs. electron-phonon coupling\nstrengthD. Two \ft curves /1=D2show thatT1;2/1=D2.\nBottom:The ratio of both decay times remains almost con-\nstantT2=T1\u00192 over the entire range of coupling.\n369  \n0 0.5 1279279.5280280.5281\n1/T2(ps−1)ω(ps−1)\n  \nLL\nGilbert\nMicroscopic\nFIG. 10. Precession frequency with respect to the damping\nstrength in terms of the decay rate 1 =T2.\nB. Precession-frequency shift due to dephasing\nIn the Landau-Lifshitz equation the contributions de-\nscribing, respectively, the precession and the damping are\ncompletely independent, so that the damping constant \u0015\nhas no impact on the precession. By contrast, an increase\nof\u000bin the Gilbert equation does not only increase the\ndephasing rate, it lowers the precession frequency as well.\nIn this section we investigate the change of the preces-\nsion frequency in the microscopic calculation and com-\npare it with the macroscopic descriptions. To this end,\nwe use the o\u000b-diagonal components of the itinerant den-\nsity matrix \u001a+\u0000(t) =P\n~k\u001a+\u0000\n~k(t), which describe the dy-\nnamics of the transverse spin components in the rotating\nframe. The modulus of its Fourier transform j\u001a+\u0000(!)j\nshows a distinct peak, which is exactly at the precession\nfrequency.\nTo compare the precession frequency for di\u000berent\ndamping parameters, we use the dependence on T2, be-\ncause all dephasing parameters can be related to T2for10\nsmall excitations. Fig. 10 plots the Larmor frequency vs.\nthe transverse relaxation rate 1 =T2. The precession pa-\nrameters of the Landau-Lifshitz and the Gilbert equation\nare chosen such that the Larmor frequency in the un-\ndamped limiting case is equal to that of the microscopic\nsimulation. In order to stay within the bounds set by con-\ndition (36), we do not extend the plot in Fig. 10 to higher\ndephasing rates. Fig. 10 shows that the microscopic cal-\nculation yields a reduction of the precession frequency\nwith the damping rate. Although the Gilbert dynamics\nalso show such a reduction, it occurs only at shorter T2.\nAs mentioned above, for the Landau-Lifshitz damping,\nthe frequency is independent of the damping parame-\nter. Even though the change of precession frequency in\nthe microscopic calculation is small, the Landau-Lifshitz\ndamping completely fails to include this e\u000bect. While\nGilbert damping does show a reduction of precession fre-\nquency, it is not at all close to the microscopic calcula-\ntion on the frequency scale considered here. Both phe-\nnomenological damping expressions thus do not repro-\nduce the dependence of the precession frequency on T2.\nEven though the numerical di\u000berences are small, these\ndi\u000berences already occur in the small-excitation regime,\nand may perhaps be detectable.\nC. Dephasing at larger excitation angles\nThe phenomenological Landau-Lifshitz and Gilbert\ndamping contributions describe an exponential decay\nonly for small excitation angles, as studied in the pre-\nvious section. In this section we investigate the e\u000bect of\nlarger excitation angles ( >10\u000e) on the spin dynamics in\nthe microscopic calculation. Apart from this the initial\ncondition of the dynamics is the same as before, in par-\nticular, the itinerant spin is tilted such that the absolute\nvalue of the spin is unchanged.\nFigure 11 shows the time development of the skand\ns?components of the itinerant spin in the rotating frame\nfor an initial tilt angle \f= 140\u000e. While the transverse\ncomponent s?in the rotating frame can be well described\nby an exponential decay, the longitudinal component sk\nshows a di\u000berent behavior. It initially decreases with a\ntime constant of less than 1 ps, but does not reach its\nequilibrium value. Instead, the eventual return to equi-\nlibrium takes place on a much longer timescale, during\nwhich the s?component is already vanishingly small.\nThe long-time dynamics are therefore purely collinear.\nFor the short-time dynamics, the transverse component\ncan be \ft well by an exponential decay, even for large ex-\ncitation angles. This behavior is di\u000berent from Landau-\nLifshitz and Gilbert dynamics, cf. Fig. 2, which both ex-\nhibit non-exponential decay of the transverse spin com-\nponent.\nIn Fig. 12 the dependence of T2on the excitation an-\ngle is shown. From small \fup to almost 180\u000e, the decay\ntime decreases by more than 50%. This dependence is\nexclusively due to the \\excitation condition,\" which in-\n0 1 2 3 4−0.100.1\ntime (ps)s/bardbl\n0 1 2 3 400.050.1\ntime (ps)s⊥FIG. 11. Dynamics of the longitudinal and transverse itiner-\nant spin components in the rotating frame (solid lines) for a\ntilt angle of \f= 140\u000e, together with exponential \fts toward\nequilibrium (dashed lines). The longitudinal equilibrium po-\nlarization is shown as a dotted line.\nvolves only spin degrees of freedom (\\tilt angle\"), but no\nchange of temperature. Although one can \ft such a T2\ntime to the transverse decay, the overall behavior with\nits two stages is, in our view, qualitatively di\u000berent from\nthe typical Bloch relaxation/dephasing picture.\nTo highlight the similarities and di\u000berences from the\nBloch relaxation/dephasing we plot in Fig. 13 the mod-\nulus of the itinerant spin vector j~ sjin the rotating\nframe, whose transverse and longitudinal components\nwere shown in Fig. 11. Over the 2 ps, during which the\ntransverse spin in the rotating frame essentially decays,\nthe modulus of the spin vector undergoes a fast initial\ndecrease and a partial recovery. The initial length of ~ s\nis recovered only over a much larger time scale of several\nhundred picoseconds (not shown). Thus the dynamics\ncan be seen to di\u000ber from a Landau-Lifshitz or Gilbert-\nlike scenario because the spin does not precess toward\nequilibrium with a constant length. Additionally they\ndi\u000ber from Bloch-like dynamics because there is a com-\nbination of the fast and slow dynamics that cannot be\ndescribed by a single set of T1andT2times. We stress\nthat the microscopic dynamics at larger excitation angles\nshow a precessional motion of the magnetization with-\nout heating and a slow remagnetization. This scenario is\nsomewhat in between typical small angle-relaxation, for\nwhich the modulus of the magnetization is constant and\nwhich is well described by Gilbert and Landau-Lifshitz\ndamping, and collinear de/remagnetization dynamics.\nVI. EFFECT OF ANISOTROPY\nSo far we have been concerned with the question\nhow phenomenological equations describe dephasing pro-\ncesses between itinerant and localized spins, where the11\n0 50 100 1500.40.60.81\nβ(◦)T2(ps)\nFIG. 12.T2time extracted from exponential \ft to s?dynam-\nics in rotating frame for di\u000berent initial tilting angles \f.\n0 0.5 1 1.5 20.040.060.080.10.120.14\ntime (ps)|s|\n  \n10°\n50°\n90°\n140°\nFIG. 13. Dynamics of the modulus j~ sjof the itinerant spin\nfor di\u000berent initial tilt angles \f. Note the slightly di\u000berent\ntime scale compared to Fig. 11.\nmagnetic properties of the system were determined by a\nmean-\feld exchange interaction only. Oftentimes, phe-\nnomenological models of spin dynamics are used to de-\nscribe dephasing processes toward an \\easy axis\" deter-\nmined by anisotropy \felds.29\nIn order to capture in a simple fashion the e\u000bects of\nanisotropy on the spin dynamics in our model, we sim-\nply assume the existence of an e\u000bective anisotropy \feld\n~Haniso, which enters the Hamiltonian via\n^Haniso =\u0000g\u0016B\u0016^~ s\u0001~Haniso (40)\nand only acts on the itinerant carriers. Its strength is\nassumed to be small in comparison to the \feld of the\nlocalized moments ~Hloc. This additional \feld ~Haniso has\nto be taken into account in the diagonalization of the\ncoherent dynamics as well, see section IV C.\nFor the investigation of the dynamics with anisotropy,\nwe choose a slightly di\u000berent initial condition, which is\nshown in Fig. 14. In thermal equilibrium, both spins\nare now aligned, with opposite directions, along the\nanisotropy \feld ~Haniso, which is assumed to point in the\nzdirection. At t= 0 they are both rigidly tilted by an\n5&:P; \nO&:P; U T \nV E *_lgqm FIG. 14. Dynamics of the localized spin ~Sand itinerant spin\n~ s. Att= 0, the equilibrium con\fguration of both spins is\ntilted (\f= 10\u000e) with respect to an anisotropy \feld ~Haniso.\nThe anisotropy \feld is only experienced by the itinerant sub-\nsystem.\n01002003004005006000.490.4950.5\ntime(ps)Sz(t)\n010020030040050060000.050.1\ntime(ps)/radicalBig\nS2x(t)+S2y(t)\n  \nFIG. 15. Relaxation dynamics of the localized spin toward the\nanisotropy direction for longitudinal component Szand the\ntransverse componentp\nS2x+S2y. An exponential \ft yields\nBloch decay times of Taniso\n1 = 67:8 ps andTaniso\n2 = 134:0 ps.\nangle\f= 10\u000ewith respect to the anisotropy \feld.\nFigure 14 shows the time evolution of both spins in the\n\fxed frame, with zaxis in the direction of the anisotropy\n\feld for the same material parameters as in the previous\nsections and an anisotropy \feld ~Haniso =\u0000108A\nm\u0001~ ez.\nThe dynamics of the entire spin-system are somewhat\ndi\u000berent now, as the itinerant spin precesses around the\ncombined \feld of the anisotropy and the localized mo-\nments. The localized spin precesses around the itinerant\nspin, whose direction keeps changing as well.\nFigure 15 contains the dynamics of the components\nof the localized spin in the rotating frame. Both com-\nponents show an exponential behavior that allows us to\nextract well de\fned Bloch-times Taniso\n1 andTaniso\n2. Again\nwe \fnd the ratio of 2 Taniso\n1\u0019Taniso\n2, because the abso-\nlute value of the localized spin does not change, as it is\nnot coupled to the phonon bath.\nIn Fig. 16 the Larmor-frequency !aniso\nL, which is the\nprecession frequency due to the anisotropy \feld, and the\nBloch decay times Taniso\n2 are plotted vs. the strength of\nthe anisotropy \feld ~Haniso. The Gilbert damping pa-12\n0 5 10 150510\nHaniso(107A/m)ωaniso\nL (ps−1)\n0 5 10 1505001000\nHaniso(107A/m)Taniso\n2 (ps)\n0 5 10 1501020\nHaniso(107A/m)αaniso (10−4)\nFIG. 16. Larmor frequency !aniso\nL and Bloch decay time Taniso\n2\nextracted from the spin dynamics vs. anisotropy \feld Haniso,\nas well as the corresponding damping parameter \u000baniso.\nrameter\u000baniso for the dephasing dynamics computed via\nEq. (13) is also presented in this \fgure.\nThe plot reveals a decrease of the dephasing time Taniso\n2\nand a almost linear increase of the Larmor frequency\n!aniso\nL with the strength of the anisotropy \feld Haniso.\nThe Gilbert damping parameter \u000baniso shows only a neg-\nligible dependence on the anisotropy \feld Haniso. This\ncon\frms the statement that, in contrast to the dephas-\ning rates, the Gilbert damping parameter is independent\nof the applied magnetic \feld. In the investigated range\nwe \fnd an almost constant value of \u000baniso'9\u000210\u00004.\nThe Gilbert damping parameter \u000baniso for the de-\nphasing toward the anisotropy \feld is about 4 times\nsmaller than \u000biso, which describes the dephasing between\nboth spins. This disparity in the damping e\u000eciency\n(\u000baniso< \u000b iso) is obviously due to a fundamental di\u000ber-\nence in the dephasing mechanism. In the anisotropy case\nthe localized spin dephases toward the zdirection with-\nout being involved in scattering processes with itinerant\ncarriers or phonons. The dynamics of the localized spins\nis purely precessional due to the time-dependent mag-\nnetic moment of the itinerant carriers ~Hitin(t). Thus,\nonly this varying magnetic \feld, that turns out to be\nslightly tilted against the localized spins during the en-\ntire relaxation causes the dephasing, in presence of the\ncoupling between itinerant carriers and a phonon bath,\nwhich acts as a sink for energy and angular momentum.\nThe relaxation of the localized moments thus occurs only\nindirectly as a carrier-meditated relaxation via their cou-\npling to the time dependent mean-\feld of the itinerant\nspin.\nNext, we investigate the dependence of the Gilbert pa-\nrameter\u000baniso on the bath coupling. Fig. 17 shows that\n0 50 100 150 20000.0040.0080.012\nD(meV√nm)αaniso\n  FIG. 17. Damping parameter \u000baniso vs. coupling constant D\n(black diamonds). The red line is a quadratic \ft, indicative\nof\u000baniso/D2.\n\u000baniso increases quadratically with the electron-phonon\ncoupling strength D.\nSince Fig. 9 establishes that the spin-dephasing rate\n1=T2for the fast dynamics discussed in the previous sec-\ntions, is proportional to D2, we \fnd\u000baniso/1=T2. We\nbrie\ry compare these trends to two earlier calculations\nof Gilbert damping that employ p-dmodels and assume\nphenomenological Bloch-type rates 1 =T2for the dephas-\ning of the itinerant hole spins toward the \feld of the\nlocalized moments. In contrast to the present paper, the\nlocalized spins experience the anisotropy \felds. Chovan\nand Perakis38derive a Gilbert equation for the dephasing\nof the localized spins toward the anisotropy axis, assum-\ning that the hole spin follows the \feld ~Hlocof the localized\nspins almost adiabatically. Tserkovnyak et al.39extract\na Gilbert parameter from spin susceptibilities. The re-\nsulting dependence of the Gilbert parameter \u000baniso on\n1=T2in both approaches is in qualitative accordance and\nexhibits two di\u000berent regimes. In the the low spin-\rip\nregime, where 1 =T2is small in comparison to the p-dex-\nchange interaction a linear increase of \u000baniso with 1=T2\nis found, as is the case in our calculations with micro-\nscopic dephasing terms. If the relaxation rate is larger\nthan thep-ddynamics,\u000baniso decreases again. Due to\nthe restriction (36) of the Boltzmann scattering integral\nto low spin-\rip rates, the present Markovian calculations\ncannot be pushed into this regime.\nEven though the anisotropy \feld ~Haniso is not cou-\npled to the localized spin ~Sdirectly, both spins precess\naround the zdirection with frequency !aniso\nL. In analogy\nto Sec. V B we study now the in\ruence of the damping\nprocess on the precession of the localized spin around\nthe anisotropy axis and compare it to the behavior of\nLandau-Lifshitz and Gilbert dynamics. Fig. 18 reveals a\nsimilar behavior of the precession frequency as a function\nof the damping rate 1 =Taniso\n2 as in the isotropic case. The\nmicroscopic calculation predicts a distinct drop of the\nLarmor frequency !aniso\nL for a range of dephasing rates\nwhere the precession frequency is unchanged according\nto the Gilbert and Landau-Lifshitz damping models. Al-\nthough Gilbert damping eventually leads to a change in\nprecession frequency for larger damping, this result shows\na qualitative di\u000berence between the microscopic and the13\n0 0.02 0.04 0.06 0.087.127.167.27.24\n1/Taniso\n2(ps−1)ωaniso(ps−1)\n  \nGilbert\nLL\nMicroscopic\nFIG. 18. Precession frequency of the localized spin around\nthe anisotropy \feld vs. Bloch decay time 1 =Taniso\n2.\nphenomenological calculations.\nVII. CONCLUSION AND OUTLOOK\nIn this paper, we investigated a microscopic descrip-\ntion of dephasing processes due to spin-orbit coupling\nand electron-phonon scattering in a mean-\feld kinetic\nexchange model. We \frst analyzed how spin-dependent\ncarrier dynamics can be described by Boltzmann scat-\ntering integrals, which leads to Elliott-Yafet type relax-\nation processes. This is only possible for dephasing rates\nsmall compared to the Larmor frequency, see Eq. (36).\nThe microscopic calculation always yielded Bloch times\n2T1=T2for low excitation angles as it should be due\nto the conservation of the absolute value of the mag-\nnetization. A small decrease of the e\u000bective precession\nfrequency occurs with increasing damping rate, which is\na fundamental di\u000berence to the Landau-Lifshitz descrip-\ntion and exceeds the change predicted by the Gilbert\nequation in this regime.We modeled two dephasing scenarios. First, a relax-\nation process between both spin sub systems was studied.\nHere, the di\u000berent spins precess around the mean-\feld of\nthe other system. In particular, for large excitation an-\ngles we found a decrease of the magnetization during the\nprecessional motion without heating and a slow remag-\nnetization. This scenario is somewhat in between typi-\ncal small angle-relaxation, for which the modulus of the\nmagnetization is constant and which is well described\nby Gilbert and Landau-Lifshitz damping, and collinear\nde/remagnetization dynamics. Also, we \fnd important\ndeviations from a pure Bloch-like behavior.\nThe second scenario deals with the relaxation of the\nmagnetization toward a magnetic anisotropy \feld expe-\nrienced by the itinerant carrier spins for small excitation\nangles. The resulting Gilbert parameter \u000baniso is inde-\npendent of the static anisotropy \feld. The relaxation of\nthe localized moments occurs only indirectly as a carrier-\nmeditated relaxation via their coupling to the time de-\npendent mean-\feld of the itinerant spin.\nTo draw a meaningful comparison with Landau-\nLifshitz and Gilbert dynamics we restricted ourselves\nthroughout the entire paper to a regime where the elec-\ntronic temperature is equal to the lattice temperature Tph\nat all times. In general our microscopic theory is also ca-\npable of modeling heat induced de- and remagnetization\nprocesses. We intend to compare microscopic simulations\nof hot electron dynamics in this model, including scat-\ntering processes between both types of spin, with phe-\nnomenological approaches such as the Landau-Lifshitz-\nBloch (LLB) equation or the self-consistent Bloch equa-\ntion (SCB)40.\nWe \fnally mention that we derived relation (13) con-\nnecting the Bloch dephasing time T2and the Gilbert\ndamping parameter \u000b. 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Phys. 113, 163911 (2013)."    },    {        "title": "1308.3989v1.Dynamical_bar_mode_instability_in_rotating_and_magnetized_relativistic_stars.pdf",        "content": "Dynamical bar-mode instability in rotating and magnetized relativistic stars\nLuca Franci,1Roberto De Pietri,1Kyriaki Dionysopoulou,2and Luciano Rezzolla2, 3\n1Universit `a di Parma and INFN gruppo collegato di Parma, Italy\n2Max-Planck-Institut f ¨ur Gravitationsphysik, Albert-Einstein-Institut, Golm, Germany\n3Institut f ¨ur Theoretische Physik, Frankfurt am Main, Germany\n(Dated: October 15, 2018)\nWe present three-dimensional simulations of the dynamical bar-mode instability in magnetized and differ-\nentially rotating stars in full general relativity. Our focus is on the effects that magnetic fields have on the\ndynamics and the onset of the instability. In particular, we perform ideal-magnetohydrodynamics simulations\nof neutron stars that are known to be either stable or unstable against the purely hydrodynamical instability,\nbut to which a poloidal magnetic field in the range of 1014–1016G is superimposed initially. As expected,\nthe differential rotation is responsible for the shearing of the poloidal field and the consequent linear growth\nin time of the toroidal magnetic field. The latter rapidly exceeds in strength the original poloidal one, leading\nto a magnetic-field amplification in the the stars. Weak initial magnetic fields, i.e., .1015G, have negligible\neffects on the development of the dynamical bar-mode instability, simply braking the stellar configuration via\nmagnetic-field shearing, and over a timescale for which we derived a simple algebraic expression. On the other\nhand, strong magnetic fields, i.e., &1016G, can suppress the instability completely, with the precise threshold\nbeing dependent also on the amount of rotation. As a result, it is unlikely that very highly magnetized neutron\nstars can be considered as sources of gravitational waves via the dynamical bar-mode instability.\nPACS numbers: 04.25.Dm, 04.40.Dg, 95.30.Lz, 97.60.Jd\nI. INTRODUCTION\nMain-sequence stars with masses greater than about 8M\f\nfollow two evolutionary paths; either they form a degenerate\ncore of O/Ne/Mg, or a degenerate Fe core, which, after un-\ndergoing a Type II supernova core collapse, forms a proto-\nneutron star [1, 2]. Neutron stars (NSs) are also expected\nto form through the accretion-induced collapse of a white\ndwarf [3, 4]. At birth, NSs are rapidly and differentially ro-\ntating, which makes them subject to various types of instabili-\nties. Among these, the dynamical bar-mode instability and the\nshear-instability are particularly interesting because of their\npotential role as sources of gravitational waves (GWs).\nIndeed, a newly born NS may develop a dynamical bar-\nmode instability when the rotation parameter \f:=T=jWj,\nwithTthe rotational kinetic energy and Wthe gravitational\nbinding energy, exceeds a critical value \fc(see, for instance,\n[5, 6] for some reviews). Under these conditions, the rapidly\nrotating NS can become severely deformed, leading to a\nstrong emission of GWs in the kHz range. Analytic inves-\ntigations of the conditions under which these dynamical insta-\nbilities develop in self-gravitating rotating stars can be found\nin [7, 8], but these are inevitably restricted to Newtonian grav-\nity or to simple shell models. To improve our understanding\nof these instabilities also in the nonlinear regimes, and to be\nable to extract useful physical information from the potential\nGW emission, it is clear that a general-relativistic numerical\nmodeling is necessary. This has been the focus of a number\nof recent works, e.g., [9–12], which have provided important\nclues about the threshold for the instability and its survival\nunder realistic conditions. As an example, for a polytropic\nrelativistic star with polytropic index \u0000 = 2 , the calculations\nreported in [10] revealed that the critical value is \fc\u00180:254,\nand that a simple dependence on the stellar compactness can\nbe used to track this threshold from the Newtonian limit overto the fully relativistic one [11]. Furthermore, numerical sim-\nulations have also revealed that the instability is in general\nshort-lived and that the bar-deformation is suppressed over a\ntimescale of a few revolutions (this was first pointed out in\nRef. [10] and later confirmed in Ref. [13], where it was inter-\npreted as due to a Faraday resonance).\nOne aspect of the bar-mode instability that so far has not\nreceived sufficient attention is about the occurrence of the in-\nstability in magnetized stars. This is not an academic question\nsince NSs at birth are expected to be quite generically magne-\ntized, with magnetic fields that have strengths up to '1012G\nin ordinary NSs and reaching strengths in excess of 1015G in\nmagnetars, if instabilities or dynamos have taken place in the\nproto-neutron star phase [14, 15].\nMagnetic fields of this strength can affect both the struc-\nture and the evolution of NSs [16–19], and it is natural to ex-\npect that they will influence also the development of the in-\nstability when compared to the purely hydrodynamical case.\nA first dynamical study in this direction has been carried out\nrecently in Ref. [20], where the development of the dynami-\ncal bar-mode instability has been studied for differentially ro-\ntating magnetized stars in Newtonian gravity and in the ideal-\nmagnetohydrodynamics (MHD) limit (i.e., with a plasma hav-\ning infinite conductivity). Not surprisingly, this study found\nthat magnetic fields do have an effect on the development of\nthe instability, but that this is the case only for very strong\nmagnetic fields. We here consider the same problem, but ex-\ntend the analysis to a fully general-relativistic framework, as-\nsessing the impact that the results have on high-energy astro-\nphysics and GW astronomy.\nOur investigation of the dynamics of highly-magnetized\nand rapidly rotating NSs is also part of a wider study of this\ntype of objects to explain the phenomenology associated with\nshort gamma-ray bursts. These catastrophic phenomena, in\nfact, are normally thought to be related to the merger of aarXiv:1308.3989v1  [gr-qc]  19 Aug 20132\nbinary system of NSs [21–24], which could then lead to the\nformation of a long-lived hypermassive NS (HMNS) [25–28].\nIf highly magnetized, the HMNS could then also lead to an\nintense electromagnetic emission [29, 30]. This scenario has\nrecently been considered in Refs. [31, 32], where numerical\nsimulations of an axisymmetric differentially rotating HMNS\nwere carried out. The HMNS had initially a purely poloidal\nmagnetic field, which eventually led to a magnetically driven\noutflow along the rotation axis.\nA similar setup has also been considered in a number\nof works, either in two-dimensional (2D) [33] or in three-\ndimensional (3D) simulations [34], with the goal of determin-\ning whether or not the conditions typical of a HMNS can lead\nto the development of the magnetorotational (MRI) instability\n[35]. Although this type of simulations in 3D still stretches\nthe computational resources presently available, the very high\nresolutions employed in Ref. [34], and the careful analysis\nof the results, provided the first convincing evidence that the\nMRI can develop from 3D configurations. This has of course\nimportant consequences on much of the phenomenology as-\nsociated with HMNSs, as it shows that very strong magnetic\nfields, up to equipartition, will be produced in the HMNS if\nthis survives long enough for the MRI to develop.\nIn the simulations reported here we necessarily adopt much\ncoarser resolutions and hence we will not be able to concen-\ntrate our attention on the development of the MRI. Rather,\nwe will here extend our previous work on the dynamical bar-\nmode instability [10, 11, 36] also to the case of magnetized\nstars, determining when and how magnetic fields can limit\nthe development of the dynamical bar-mode instability. Our\ninitial models correspond to stationary equilibrium configu-\nrations of axisymmetric and rapidly rotating relativistic stars.\nMore precisely, our initial models are described by a poly-\ntropic EOS with adiabatic index \u0000 = 2 and are members of\na sequence with a constant rest-mass of M'1:5M\fand a\nconstant amount of differential rotation.\nInterpreting the results of our simulations can be rather\nstraightforward. Because we work in the ideal-MHD limit,\nthe magnetic field lines are “frozen” in the fluid and follow its\ndynamics (see [37] for a recent extension of the code to re-\nsistive regime). As a consequence, differential rotation drives\nthe initial purely poloidal magnetic field into rotation, wind-\ning it up and generating a toroidal component. At early times,\nthe toroidal magnetic field grows linearly with time, tapping\nthe NS’s rotational energy. At later times, the growth starts\ndeviating from the linear behavior and the magnetic tension\nproduced by the very large magnetic-field winding, alters the\nangular velocity profile of the star. Depending on the models\nadopted and the initial magnetic field strength, the magnetic\nwinding could become the most efficient mechanisms for re-\ndistributing angular momentum, with the MRI being the dom-\ninant one when the Alfv ´en timescale becomes comparable to\nthe magnetic winding timescale.\nOverall, we find that if the initial magnetic fields are .1015\nG, then they have a negligible effect on the occurrence of the\ndynamical bar-mode instability, which develops in close anal-\nogy with the purely hydrodynamical case. On the other hand,\nif the initial magnetic fields are &1016G, they can suppressthe instability completely. Note that the precise threshold\nmarking the stability region depends not only on the strength\nof the magnetic field, but also on the amount of rotation. We\ntrace this threshold by performing a number of simulations\nof a number of sequences having the same parameter \fbut\ndifferent magnetizations. An important consequence of our\nresults is that because the instability is suppressed in strongly\nmagnetized NSs, these can no longer be considered as po-\ntential sources of GWs, at least via the dynamical bar-mode\ninstability.\nThe organization of the paper is as follows. In Sect. II we\ndescribe the numerical methods and the setup employed in our\nsimulations, as well as the full set of equations we solve. In\nSect. III we mention briefly the main properties of the stellar\nmodels adopted as initial data, together with the simplifica-\ntions and assumptions we make. In Sect. IV we examine in\ngreat detail the effects of the presence of an initial poloidal\nmagnetic field on differentially rotating stars covering a wide\nrange in the parameter space. We further discuss the qualita-\ntive and quantitative features of the evolution of models with\nthe same total rest mass but different rest-mass density and an-\ngular momentum profiles that are known to be unstable to the\npurely hydrodynamic bar-mode instability. Finally, we inves-\ntigate whether magnetic fields affect the stellar evolution even\nwhen the bar-mode instability does not develop. Our conclu-\nsions are drawn in Sect. V and two appendices discuss the\ninfluence of symmetries on the development of the instability\nand the convergence of our results. Unless stated differently,\nwe adopt geometrized units in which c= 1,G= 1,M\f= 1.\nII. MATHEMATICAL AND NUMERICAL SETUP\nThe simulations have been carried out using the general-\nrelativistic ideal-MHD (GRMHD) code WhiskyMHD [24, 38,\n39]. The code provides a 3D numerical solution of the full set\nof the GRMHD equations in flux-conservative form on a dy-\nnamical background in Cartesian coordinates. It is based on\nthe same high-resolution shock-capturing (HRSC) techniques\non domains with adaptive mesh refinements (AMR) [40, 41])\nas discussed in [42]. The reconstruction method adopted is the\none discussed in the piecewise parabolic (PPM) [43], while\nthe Harten-Lax-van Leer-Einfeldt (HLLE) approximate Rie-\nmann solver [44] has been employed in order to compute the\nfluxes. The divergence of the magnetic field is enforced to\nstay within machine precision by employing the flux-CD ap-\nproach as implemented in [39], but with the difference that\nwe adopt as evolution variable the vector potential instead of\nthe magnetic field. This method ensures the divergence-free\ncharacter of the magnetic field since the magnetic field is com-\nputed as the curl of the evolved vector potential using the same\nfinite-differencing operators as the ones for computing the di-\nvergence of the magnetic field.\nBecause of the gauge invariance of Maxwell equations, a\nchoice needs to be made and we have opted for the simplest\none, namely, the algebraic Maxwell gauge. This choice can\nintroduce some spurious oscillations close to the AMR bound-\naries in highly dynamical simulations, but this has not been3\nthe case for the simulations reported here. On the other hand,\nit has allowed us to keep the divergence of the magnetic field\nessentially nearly at machine precision. A more advanced pre-\nscription has been also introduced recently in Ref. [45]; this\napproach requires a certain amount of tuning for optimal per-\nformance and will be considered in future works. Additional\ninformation on the code can also be found in Refs. [38, 39].\nFurthermore, to remove spurious post-shock oscillations in\nthe magnetic field we add a fifth-order Kreiss-Oliger type\nof dissipation [46] to the vector potential evolution equation\nwith a dissipation parameter of 0:1. Finally, the evolution of\nthe gravitational fields is obtained through the CCATIE code,\nwhich provides the solution of the conformal traceless formu-\nlation of the Einstein equations [47]. The time integration of\nthe evolution equations is achieved through a third-order ac-\ncurate Runge-Kutta scheme. Essentially all of the simulations\npresented in this paper use a 3D Cartesian grid with four re-\nfinement levels and with outer boundaries located at a distance\n\u0018150km from the center of the grid. The finest resolution is\n\u0001x'0:550km (between 40and60points across the stellar\nradius, depending on the model) and the coarsest extends up\nto about\u0018150km, namely more than five times the stellar ra-\ndius. Unless stated differently, all of the simulations discussed\nhereafter have been performed imposing a bitant symmetry,\ni.e., a reflection symmetry across the z= 0plane.\nFor convenience we report here the full set of the evolution\nequations we solve numerically which consists in the coupled\nsystems of Einstein and MHD equations, i.e.,\nR\u0016\u0017\u00001\n2g\u0016\u0017R= 8\u0019T\u0016\u0017; (2.1)\nr\u0016T\u0016\u0017= 0; (2.2)\nr\u0016(\u001au\u0016) = 0; (2.3)\nr?\n\u0016F\u0016\u0017= 0; (2.4)\nr\u0016F\u0016\u0017= 4\u0019J\u0017; (2.5)\nwhereR\u0016\u0017;g\u0016\u0017andRare the Ricci tensor, the metric ten-\nsor and the Ricci scalar, respectively. On the electromagnetic\nside,F\u0016\u0017is the Maxwell tensor, dual of the Faraday tensor\n\u0003F\u0016\u0017,J\u0016is the current four-vector, and on the matter side\n\u001ais the rest-mass density, u\u0016is the 4-velocity of the fluid\nsatisfying the normalization condition u\u0016u\u0016=\u00001. The total\nenergy-momentum tensor T\u0016\u0017is the linear combination of the\ncontributions coming from a perfect fluid, i.e., T\u0016\u0017\n\r, and from\nthe electromagnetic fields, i.e., T\u0016\u0017\nem\nT\u0016\u0017=T\u0016\u0017\nem+T\u0016\u0017\n\r:\nwhere\nT\u0016\u0017\nfluid:=\u001ahu\u0016u\u0017+pg\u0016\u0017; (2.6)\nT\u0016\u0017\nem:=F\u0016\u001bF\u0017\n\u001b\u00001\n4g\u0016\u0017F\u000b\fF\u000b\f\n=\u0012\nu\u0016u\u0017+1\n2g\u0016\u0017\u0013\nb2\u0000b\u0016b\u0017; (2.7)\nIn the expressions above we recall that h= 1 +\u000f+p=\u001ais\nthe specific enthalpy, \u000fthe specific internal energy. Hence,the energy density in the rest-frame of the fluid is just e=\n\u001a(1 +\u000f). At the same time, the four-vector b\u0016represents the\nmagnetic field as measured in the comoving frame, so that the\nMaxwell and Faraday tensors are expressed as (see [38, 39]\nfor details)\nF\u0016\u0017=\u000f\u0016\u0017\u000b\fu\u000bb\f=n\u0016E\u0017\u0000n\u0017E\u0016+\u000f\u0016\u0017\u000b\fB\u000bn\f;\n(2.8)\n\u0003F\u0016\u0017=b\u0016u\u0017\u0000b\u0017u\u0016=n\u0016B\u0017\u0000n\u0017B\u0016\u0000\u000f\u0016\u0017\u000b\fE\u000bn\f;\n(2.9)\nwhere the second equalities introduce the electric and mag-\nnetic fields measured by an observer moving along a normal\ndirectionn\u0017. We further note that thep4piterms appearing in\nEqs. (2.4),(2.5) are absorbed in the definition of the magnetic\nfield.\nIn the interest of compactness, we will not discuss here\nthe detailed formulation of the Eqs. (2.1)–(2.3) we use in\nthe numerical solution and refer the interested reader to the\nfollowing works where these aspects are discussed in detail:\nRef. [47] for the formulation of the Einstein equations and the\ngauge conditions used, Refs. [38, 39] for the formulation of\nthe MHD equations and the strategy for enforcing a zero di-\nvergence of the magnetic field, Refs. [27, 48] for the computa-\ntional infrastructure and the numerical methods used. What is\nhowever important to remark here is that we employ an “ideal-\nfluid” (or Gamma-law) equation of state (EOS) [48]\np=\u001a\u000f(\u0000\u00001); (2.10)\nwhere \u0000is the adiabatic exponent, which we set to be \u0000 = 2 .\nMore realistic EOS could have been used, as done for instance\nin Ref. [49], and this will indeed be the focus of future work.\nAt this stage, however, and because this is the first study of\nthis type, the simpler analytic EOS (2.10) is sufficient to col-\nlect the first qualitative aspects of the development of the in-\nstability.\nIII. INITIAL DATA AND DIAGNOSTICS\nThe initial data of our simulations are computed as station-\nary equilibrium solutions of axisymmetric and rapidly rotat-\ning relativistic stars in polar coordinates and without magnetic\nfields [50]. In generating these equilibrium models we adopt a\n“polytropic” EOS [48], p=K\u001a\u0000, withK= 100 and\u0000 = 2 ,\nand assume the line element for an axisymmetric and station-\nary relativistic spacetime to have the form\nds2=\u0000e\u0016+\u0017dt2+e\u0016\u0000\u0017r2sin2\u0012(d\u001e\u0000!dt)2\n+e2\u0018(dr2+r2d\u00122);(3.1)\nwhere\u0016,\u0017,!and\u0018are space-dependent metric functions.\nTo reach the large angular momentum needed to trigger the\ndynamical bar-mode instability, a considerable amount of dif-\nferential rotation needs to be introduced and we do so follow-\ning the traditional constant specific angular momentum law4\nModel\u001acrp=reAbReM0M M=R eJ J=M2\nc \neT W \f \f mag\n(10\u00004) [km] [M\f] [M\f] [rad/s] [rad/s] (10\u00002) (10\u00002) (10\u00006)\nU13 0:599 0:200 1:85\u000210635:9 1:505 1:462 0:0601 3.747 1:753 3647 1607 2:183 7:764 0:2812 5:3\nU11 1:092 0:250 1:46\u000210634:4 1:507 1:460 0:0627 3.541 1:661 3997 1747 2:284 8:327 0:2743 4:7\nU3 1:672 0:294 8:74\u000210532:4 1:506 1:456 0:0664 3.261 1:538 4434 1916 2:352 9:061 0:2596 3:5\nS1 1:860 0:307 6:94\u000210531:6 1:512 1:460 0:0682 3.191 1:497 4593 1976 2:384 9:388 0:2540 3:0\nS6 2:261 0:336 4:50\u000210530:0 1:505 1:449 0:0713 2.965 1:412 4901 2093 2:369 9:859 0:2403 2:3\nS7 2:754 0:370 2:01\u000210528:1 1:506 1:447 0:0760 2.741 1:309 5284 2234 2:360 10:56 0:2234 1:0\nS8 3:815 0:443 5:96\u000210426:7 1:506 1:439 0:0862 2.322 1:121 5995 2482 2:255 11:96 0:1886 0:4\nTABLE I. Main properties of the stellar models evolved in the simulations. In the first column we report the model name, while in the next\nthree the parameters we used to generate the initial models, namely the central rest-mass density \u001ac, the ratio between the polar and the\nequatorial coordinate radii rp=reand the parameter Abof Eq. (3.10) that would generate a magnetic field whose initial maximum value in\nthe (x;y) plane is 1\u00021015G. In the remaining columns we report, from left to right, the proper equatorial radius Re, the rest mass M0, the\ngravitational mass M, the compactness M=R e, the total angular momentum J,J=M2, the angular velocities at the axis \nc= \n(r= 0) and\nat the equator \ne= \n(r=Re), the rotational kinetic energy Tand the gravitational binding energy W, their ratio\f=T=jWj(instability\nparameter) and finally the ratio between the total magnetic energy and the sum of the rotational energy and the gravitational binding energy\n(\fmag=Emag=(T+jWj)). Unless explicitly stated, all these quantities are expressed in geometrized units in which G=c=M\f= 1.\nFIG. 1. Initial profiles of the rest-mass density \u001a(left panel), the angular velocity \n(center panel) and of the z-component of the magnetic\nfield (right panel) for models S8,S7,S6,S1,U3,U11 andU13. The profiles of the stable models are here drawn with blue solid lines, while\nthose for the unstable models with red dot-dashed lines.\n(“j-constant”) of differential rotation, in which the angular ve-\nlocity distribution takes the form [51, 52]\n\nc\u0000\n =1\n^A2R2e\u0014(\n\u0000!)r2sin2\u0012e\u00002\u0017\n1\u0000(\n\u0000!)2r2sin2\u0012e\u00002\u0017\u0015\n;(3.2)\nwhereReis the coordinate equatorial stellar radius and the\ncoefficient ^Ais a measure of the degree of differential rota-\ntion, which we set to ^A= 1 in analogy with other works\nin the literature. Once imported onto the Cartesian grid and\nthroughout the evolution, we compute the angular velocity \n(and the period P) on the (x;y)plane as\n\n :=u\u001e\nu0=uycos\u001e\u0000uxsin\u001e\nu0p\nx2+y2; P =2\u0019\n\n:(3.3)\nOther characteristic quantities of the system, such as the\nbaryon mass M0, the gravitational mass M, the angular mo-\nmentumJ, the rotational kinetic energy T, and the gravita-tional binding energy Ware calculated as in [53]\nM:=Z\nd3x\u000bp\r\u0002\n\u00002(T\r)0\n0+ (T\r)\u0016\n\u0016\u0003\n; (3.4)\nM0:=Z\nd3xp\rD; (3.5)\nEint:=Z\nd3xp\rD\u000f; (3.6)\nJ:=Z\nd3x\u000bp\r(T\r)0\n\u001e; (3.7)\nT:=1\n2Z\nd3x\u000bp\r\n(T\r)0\n\u001e; (3.8)\nW:=T+Eint+M0\u0000M; (3.9)\nwhere\u000fis the specific internal energy, Dis the conserved\nrest-mass density, \ris the determinant of the three-metric and\n(T\r)\u0016\n\u0017corresponds to the fluid contributions to the stress-\nenergy tensor. A couple of important caveats need to be\nmade about the definitions above. First, we note that we have5\nFIG. 2. Representation of the initial models in a (\f;\f mag)plane.\nBlue and red symbols mark models that are respectively bar-mode\nstable and bar-mode unstable at zero magnetizations, while the ver-\ntical red dashed line marks the stability threshold for zero magnetic\nfields. The red-shaded area collects as a function of their magne-\ntization models that the evolutions reveal to be bar-mode unstable;\nhence, red squares refer to initial models that develop a bar-mode in-\nstability, while red triangles refer to potentially bar-unstable models\nthat are stabilized by the strong magnetic fields.\ndefined the gravitational mass and angular momentum tak-\ning into account only the fluid part of the energy-momentum\ntensor and thus neglecting the electromagnetic contributions.\nThis is strictly speaking incorrect, but tolerable given that the\nrelative electromagnetic contributions to the mass and angu-\nlar momentum are .10\u00005. Second, the definitions above for\nJ,T,Wand\fare meaningful only in the case of stationary\naxisymmetric configurations and should therefore be treated\nwith care once the rotational symmetry is lost.\nThe main properties of all the stellar models we have used\nas initial data are reported in Table I, where we have intro-\nduced part of our notation to distinguish the different mod-\nels. In particular models indicated as U*and as S*refer to\nNSs that are unstable and stable to the purely dynamical bar-\nmode instability, respectively (this result was determined in\nRefs. [10, 11]). Figure 1 shows the initial profiles of the rest-\nmass density \u001a(left panel), of the rotational angular velocity \n(central panel), and of the z-component of the magnetic field\n(right panel) for all the models we have evolved. The profiles\nfor the models that are unstable in the unmagnetized case are\ndrawn with blue solid lines, while we use red dot-dashed lines\nfor stable models. Note that the position of the maximum of\nthe rest-mass density coincides with the center of the star only\nfor models with low \f; for those with a larger \f, the maxi-\nmum of the rest-mass density resides, instead, on a circle on\nthe equatorial plane.All the equilibrium models are members of a sequence hav-\ning a constant rest-mass M0'1:5M\fand are stable to grav-\nitational collapse on the basis of the results of [54]. An initial\npoloidal magnetic field is added as a perturbation to the ini-\ntial equilibrium models by introducing a purely toroidal vector\npotentialA\u001egiven by\nA\u001e=Ab(max(p\u0000pcut;0))2; (3.10)\nwherepcutis 4 % of the maximum pressure, while Abis cho-\nsen in a way to have the chosen value for the maximum of\nthe initial magnetic field B. The Hamiltonian and momen-\ntum constraint equations are not solved after superimposing\nthe magnetic field, but we have verified that for the magnetic-\nfield strength considered here, this perturbation introduces\nonly negligible additional violations of the constraints.\nThe strength of the initial magnetic field can be character-\nized by the value of the ratio between the total magnetic en-\nergy\nEmag:=Z\nd3x\u000b2p\rT00\nem; (3.11)\nand the sum of the rotational kinetic energy Tand of the grav-\nitational binding energy W, which we indicate as \fmag:=\nEmag=(T+jWj), in analogy with the instability parameter\n\f:=T=jWj. This parameter should not be confused with\nwhat is usually defined as the \fparameter of a plasma, i.e., the\nratio of the fluid pressure to the magnetic pressure.\nIn Table I we also report the values of the coefficient Ab[see\nEq. (3.10)] and the parameter \fmagcorresponding to an initial\npoloidal magnetic field strength equal to 1015G. All the initial\nmodels are also reported in Fig. 2 according to the values of\ntheir parameters \fand\fmag. The models that are known to\nbe stable against the bar-mode instability in the unmagnetized\ncase are here drawn in blue ( S1,S6,S7andS8), while the un-\nstable ones are drawn in red ( U3,U11 andU13). The different\nsymbols used in this figure will be further discussed in Sect.\nIV when illustrating the results of our work; it is sufficient to\nsay for now that squares and triangles refer to unstable mod-\nels with unmodified and modified growth times, respectively.\nHereafter we will also extend our notation and denote a partic-\nular magnetized model by marking it by the maximum initial\nvalue of the z-component of the magnetic field on the (x;y)\nplane, i.e.,Bz\nmaxjt;z=0, expressed in Gauss. As an example, the\nbar-mode unstable model with initial Bz\nmaxjt;z=0= 1:0\u00021015\nG will be indicated as U11-1.0e15 .\nIn order to analyze better the effects of magnetic fields\non the dynamics of the bar-mode instability, we have intro-\nduced additional diagnostic variables to quantify and describe\nthe evolution of the magnetic field itself. For axisymmetric\nconfigurations one usually decomposes the magnetic field in\ntoroidal and poloidal components, studying their dynamics\nseparately. When axisymmetry is lost, however, this nice de-\ncomposition is no longer available. Nevertheless, there exists\na decomposition that can be defined even if axisymmetry is\nnot preserved, which is reduced to the usual poloidal-toroidal\none in the axisymmetric stationary case. The main idea of this\ndecomposition is to separate the magnetic field in a compo-\nnent in the direction of the fluid motion and hence parallel to6\nthe fluid three-velocity and in a component that is orthogonal\nto it. We therefore split the magnetic field measured by an\nEulerian observer as\nBi=Bkvi\np\n\rijvivj+Bi\n?; (3.12)\nwhere we define the “perpendicular” part of the magnetic field\nfrom the condition Bi\n?vi= 0, while the “parallel” part is\na scalar and is trivially defined as Bk:=Bjvj=(vivi)1=2.\nInitially, when the flow is essentially azimuthal, Bi\n?corre-\nsponds to the poloidal component of the magnetic field, while\nBkvi=(vjvj)1=2to the toroidal component. Hereafter we will\nrefer loosely to these as the “poloidal” and “toroidal” compo-\nnents, respectively.\nWithin this decomposition, we can then define the electro-\nmagnetic energy contributions associated to the “toroidal” and\n“poloidal” magnetic-field components as\nEtor\nmag:=Z\nd3xp\r1\n2BkBk; (3.13)\nEpol\nmag:=Z\nd3xp\r1\n2\rijBi\n?Bj\n?(1 +\rrsvrvs):(3.14)\nNote that the total electromagnetic energy satisfies the condi-\ntionEmag=Etor\nmag+Epol\nmag, since the electric field Eiprovides\na contribution to the energy, EiEi= (vivi)(BiBi\u0000B2\nk), that\nis already included in the definitions (3.13) and (3.14). An-\nother important set of diagnostic quantities focuses instead on\nthe detection of a bar deformation, which can be conveniently\nquantified in terms of the distortion parameters [55]\n\u0011+:=Ixx\u0000Iyy\nIxx+Iyy; (3.15)\n\u0011\u0002:=2Ixy\nIxx+Iyy; (3.16)\n\u0011:=q\n\u00112\n++\u00112\n\u0002; (3.17)\nwhere the quadrupole moment of the matter distribution can\nbe computed in terms of the conserved density Das in [10, 56]\nIjk=Z\nd3xp\rDxjxk: (3.18)\nNote that all quantities in Eqs. (3.15)–(3.17) are expressed in\nterms of the coordinate time tand do not represent therefore\ninvariant measurements at spatial infinity. However, for the\nsimulations reported here, the length-scale of variation of the\nlapse function at any given time is always larger than twice\nthe stellar radius at that time, ensuring that the events on the\nsame time-slice are also close in proper time.\nIn addition, \u0011+can be conveniently used to quantify both\nthe growth time \u001cbarof the instability and the oscillation fre-\nquencyfbarof the unstable bar once the instability is fully\ndeveloped. In practice, we obtain a measurement of \u001cbarand\nfbarby performing a nonlinear least-square fit of the com-\nputed distortion \u0011+(t)with the trial function\n\u0011+(t) =\u00110et=\u001cBcos(2\u0019fBt+\u001e0): (3.19)\nFIG. 3. Evolution of the distortion parameters \u0011+and\u0011for model\nU11 with four different values of the initial poloidal magnetic field:\nBz\nmaxjt;z=0= 1:0\u00021014,2:0\u00021015,4:0\u00021015, and 1:0\u00021016G.\nIV . RESULTS\nA. Effects of the magnetic field on unstable models\nWe start by discussing in detail the results relative to\nmodel U11 when evolved for different values of the ini-\ntial poloidal magnetic field. The dynamics of this unstable\nmodel are very clear and allow us to show a full qualita-\ntive and quantitative picture of what happens as the bar-mode\ninstability develops. We will therefore focus our attention\non models U11-1.0e14 ,U11-2.0e15 ,U11-4.0e15\nandU11-1.0e16 , which, as discussed before, have ini-\ntial poloidal magnetic field such that Bz\nmaxjt;z=0is equal to\n1:0\u00021014,2:0\u00021015,4\u00021015and1:0\u00021016G, respec-\ntively.\nIn Fig. 3 we show the evolution of the distortion param-\neters\u0011+(top panel) and \u0011(bottom panel) for these models.\nIn the least magnetized model (i.e., U11-1.0e14 ),\u0011+starts\noscillating after about 10 ms of evolution with an amplitude\nthat almost reaches unity, and it keeps oscillating for about 20\nms. At the same time, \u0011undergoes an exponential growth,\nincreasing its value by about three orders of magnitude un-\ntil it reaches a saturation level, which persists for about 10\nms and then decays. This is exactly the behavior we expect\nfrom a stellar model which is unstable against the dynamical\nbar-mode instability, as model U11 is known to be in the un-\nmagnetized case (cf., Refs. [10, 56]).\nHowever, when the initial poloidal magnetic field is two or-\nders of magnitude stronger (i.e., as for model U11-1.0e16 ),\nthe dynamics shows a very different behavior. The amplitude7\nU11-1.0e14\nU11-4.0e15\nU11-1.0e16\nFIG. 4. Snapshots of the rest-mass density on the (x;y)plane for model U11-1.0e14 (top row), U11-4.0e15 (central row) and\nU11-1.0e16 (bottom row) at different times during the evolution, namely, t= 15:0ms (left column), t= 22:5ms (central column) and\nt= 30 ms (right column). Additionally, isodensity contours are shown for \u001a= 106;1011;1012;1012:5;1013;1013:5, and 1014g cm\u00003.\nof the oscillations in \u0011+is negligible and \u0011does not grow\nexponentially, being two orders of magnitude lower than it is\nfor model U11-1.0e14 during the whole evolution. This\nindicates that although the model is unstable in the absence\nof magnetic fields, no bar-mode deformation develops in this\ncase over a timescale of \u001835ms of evolution and for this\nmagnetic-field strength.\nFor intermediate initial poloidal magnetic fields, we find\na significant change in the dynamics by simply varying thefield strength by a factor of two, which corresponds to a\nchange of a factor of four in the magnetic energy. Moreover,\nin model U11-2.0e15 the bar-mode instability still devel-\nops, even though it takes a little longer to grow, while model\nU11-4.0e15 is stable and the bar-mode instability is sup-\npressed, since \u0011does not show an exponential growth. As a\nresult, we can bracket the stability threshold for the develop-\nment of the bar-mode instability between these two models in\nthe presence of strong magnetic fields (cf., Fig. 2).8\nTo better illustrate the different behavior of the matter evo-\nlution for different initial poloidal magnetic field strengths,\nin Fig. 4 we show three snapshots of the evolution of\nthe rest-mass density on the (x;y)plane for three of the\nabove models (i.e., models U11-1.0e14 ,U11-4.0e15\nandU11-1.0e16 ) at timest= 15:0;22:5;30:0ms. In\nparticular, in the top row of Fig. 4 we show the evolution of\nmodel U11-1.0e14 , which as discussed previously is bar-\nmode unstable as also its un-magnetized counterpart. After\n15 ms we can already observe a small deformation with re-\nspect to the initial axisymmetric configuration, which is then\namplified until a bar is fully formed after about 10 ms of the\nfirst oscillations observed in \u0011+. In the central row we show\nthe evolution of model U11-4.0e15 , which as mentioned\nbefore is instead stable against bar-mode deformations due to\nthe presence of the strong magnetic field. In this case, after 15\nms the density profile has already changed, turning from an\ninitial toroidal profile (cf., Fig. 1) to an oblate profile with its\nmaximum residing on the z-axis. Later in the evolution, we\nobserve an increase in the central density and the outer layers\nexpanding well beyond the borders of the finest grid. Finally,\non the bottom row we show snapshots of the density for model\nU11-1.0e16 , which refers to the strongly magnetized case\nand which is also stable and shows a similar behavior to the\nprevious model. The only important difference is the larger\nincrease of the central rest-mass density and the more signifi-\ncant expansion of the outer layers of the star. Indeed, after the\nfirst 15 ms of evolution, matter has been shed already beyond\nthe edges of the finest grid.\nA deeper insight in the matter dynamics in the three differ-\nent cases discussed above can be gained through the spacetime\ndiagrams shown Fig. 5, and that are reminiscent of similar\nones first presented in Ref. [28]. In particular, the left col-\numn of Fig. 5 shows the rest-mass density profile along the\nx-axis for the three models U11 using both a colormap (see\nthe right-edge of the different panels) and some representative\ncontour lines; note that the colorcode and the color ranges\nare the same in the three cases. It is worth mentioning that\nthe low-magnetic-field model U11-1.0e14 (top panel in left\ncolumn) shows the evolution we expect from a bar-mode un-\nstable model, since the bar deformation is clearly visible after\nabout 20 ms. The highly magnetized model U11-4.0e15\n(middle panel in left column), on the other hand, shows no bar\ndeformation and exhibits instead a transition from a toroidal\nconfiguration to an oblate one as is evident in Fig. 4. In addi-\ntion, a small amount of matter is shed on the equatorial plane\nafter about 15 ms of evolution. Finally, for the very highly\nmagnetized model U11-1.0e16 (bottom panel in left col-\numn), the expansion of the outer layers is much more rapid\nand the stellar material reaches a size of about 100 km (not\nshown in the figure), which is almost twice as large as for\nmodel U11-4.0e15 . The ejected material creates an ex-\ntended and flattened envelope of high-density matter1, with\nrest-mass densities as high as 1012g cm\u00003.\n1It is tempting and sometimes encountered in the literature to refer to the\nenvelope as “disk” or “torus”; however, we find this is very misleading asTo determine whether the ejected matter is gravitationally\nbound or not, we look at the time component of the fluid four-\nvelocityut(central column of Fig. 5) since the local condi-\ntionut>\u00001provides a necessary although not sufficient\ncondition for a fluid element to be unbound [28]. We recall\nthat this condition is exact only in an axisymmetric and sta-\ntionary spacetime. These requirements are not matched dur-\ning the matter-unstable phase, but the conditions can be used\nnevertheless as a first approximation to determine whether\npart of the material is actually escapes to infinity during the\nevolution. As is evident from Fig. 5, this condition is ful-\nfilled throughout the whole evolution for the highly mag-\nnetized models U11-1.0e16 andU11-4.0e15 not only\non the finest refinement level shown in Fig. 5, but on the\nwhole computational domain. However, this is not the case\nfor model U11-1.0e14 at the time the bar-mode instabil-\nity is fully developed. In fact, in this case we observe that a\ncertain amount of unbound matter is shed in correspondence\nwith the spiral arms of the bar. The ejection of matter oc-\ncurs only in very low-density regions around the star, where\n\u001a'1010g cm\u00003'10\u00004\u001ac. Overall, the total amount of\nmatter (both bound and unbound) escaping from the outer grid\nafter 20ms of evolution is less than 0:2%of the total initial\nrest mass of the NSs.\nWe complete the description of the dynamics of these three\nU11 models by reporting in the right column of Fig. 5 the\nspacetime diagram relative to the angular velocity \nalong the\nx-axis. We recall that all models have the maximum of the\n\nat the stellar center (cf., Fig. 2) and this remains the case\nalso for the low-magnetic-field and bar-mode unstable model\nU11-1.0e14 , modulo the variations brought in by the de-\nvelopment of the instability. On the other hand, for models\nU11-4.0e15 andU11-1.0e16 , the angular velocity at the\nstellar center first increases, then reaches a maximum and later\ndecreases again; at the same time, the outer layers of the star\nexpand and the maximum of the angular velocity occurs at\nlarger radii. By the time an extended flattened envelope has\nbeen produced near the equatorial plane, much of the differ-\nential rotation has been washed out and the NS has acquired a\ncentral angular velocity that is smaller but mostly uniform.\nWe can summarize the main features described in detail\nabove for the three magnetized U11 models as follows:\n\u000fmodel U11-1.0e14 is still bar-mode unstable and no\neffects are evident on the onset and development of the\ninstability; a very small fraction of the rest-mass is shed\nat the edges of the bar-deformed object.\n\u000fmodel U11-4.0e15 is bar-mode stable for the\ntimescales considered here and after about 25 ms of\nevolution it settles into a more compact configuration;\nthe new equilibrium structure has an almost uniform an-\ngular velocity and is surrounded by a differentially and\nflattened envelope.\nthe envelope is not disjoint from the star but rather an integral part of it\nwhich should not be discussed separately.9\nU11-1.0e14\nU11-4.0e15\nU11-1.0e16\nFIG. 5. Spacetime diagrams of the evolution of the rest-mass density \u001a(left column), of the time-component of the fluid four-velocity ut\n(central column), and of the angular velocity \n(right column) along the x-axis. The models considered here are U11-1.0e14 (top row),\nU11-4.0e15 (central row), and U11-1.0e16 (bottom row). The color code is indicated to the right of each plot. In addition, all diagrams\nalso report isodensity contours of the rest-mass density \u001a= 106;1011;1012;1012:5;1013;1013:5, and 1014g cm\u00003.\n\u000fmodel U11-1.0e16 is also bar-mode stable with a\ndynamics that resembles that of model U11-4.0e15 ;\nthe main differences are the shorter timescales re-\nquired to reach equilibrium and the flattened envelope\nwith larger mean rest-mass densities present in model\nU11-4.0e15 .\nAltogether, the behavior summarized above is consistent with\nwhat we would expect for highly magnetized and differen-\ntially rotating fluids. Under these conditions, in fact, mag-\nnetic braking transfers angular momentum from the core tothe outer layers, changing the rest-mass density and the ro-\ntation profiles of the star. Because during this process part\nof the rotational energy of the star is tapped, the onset of the\ninstability is inhibited.\nWe next discuss the dynamics of the magnetic fields, in\nFig. 6, and show for comparison representative snapshots of\nthe total electromagnetic energy density T00\nem(shown with a\ncolorcode) as measure in the Eulerian frame and the magnetic\nfield lines (shown as white solid lines), as measured on a hor-\nizontal plane at z'1:5km, corresponding to the three mag-10\nU11-1.0e14\nU11-4.0e15\nU11-1.0e16\nFIG. 6. Snapshots of the total electromagnetic energy density T00\nem, as measured in the Eulerian frame, on a horizontal plane at z'1:5km for\nmodel U11-1.0e14 (top row), U11-4.0e15 (central row), and U11-1.0e16 (right row), at different times during the evolution, namely,\nt= 15:0ms (left column), t= 22:5ms (central column), and t= 30 ms (right column). The magnetic field lines are shown with white solid\nlines.\nnetized U11 models studied before. Note that all the panels\nhave the same color ranges but that the colormap is different\nat different times (i.e., in different columns) in order to bet-\nter highlight the internal structure of the electromagnetic field.\nThe various columns refer to different times and coincide with\nthose already reported in Fig. 4.\nAs expected under the ideal-MHD approximation, with the\nmagnetic field being frozen into the fluid, the field lines are\ndragged along with the fluid in differential rotation and rapidly\nwind on a timescale of very few milliseconds, leading to a sud-den formation and rapid linear growth of a toroidal magnetic\nfield component. This component is soon amplified far above\nthe initial poloidal one. The winding of the field lines and the\nlinear growth of the toroidal field are present in all three mod-\nels and are independent of the initial poloidal magnetic field\nstrength. The reason is that they only depend on the angu-\nlar velocity profile, or equivalently on the differential rotation\nlaw, which is the same for all U11 models in the first part of\nthe evolution. It interesting to note in the first row of the figure\n(i.e., the unstable model U11-1.0e14 ), that the distortion of11\nthe magnetic field lines also mimics the bar-mode deformation\nas the star undergoes the development of the instability.\nA more quantitative assessment of the influence of the mag-\nnetic fields on the unstable models has been obtained after\nperforming a number of simulations of models U3,U11 and\nU13, with initial poloidal magnetic fields varying between\nthe two extreme cases presented in Figs. 4–6. More specif-\nically, we have performed 27 simulations with initial maxi-\nmum magnetic fields in the range Bz\nmaxjt;z=0= 1:0\u00021014\nand1:0\u00021016G. The results of this extensive investigation\nare collected in Figs. 2 and 7, as well as in Table II, which\nreports the measured growth time of the instability \u001cbarand its\nfrequencyfbar. In particular, Fig. 2 reports the initial models\nwithin a (\f; \f mag)diagram and allows one to easily distin-\nguish the ranges of rotational and magnetic energies that give\nrise to the development of a dynamical bar-mode instability.\nIt is, in fact, easy to distinguish models that are bar-mode sta-\nble (blue symbols) from those that are unstable (red symbols)\nat zero magnetizations; of course, models that are stable at\nzero magnetizations are also stable at all magnetizations (this\nis marked with the vertical red dashed line). Equally simple\nis to distinguish models that although unstable in the absence\nof magnetic fields (red squares), become stable with sufficient\nmagnetization (red triangles). As an example, for models U3\nthe threshold between squares and triangles appears for initial\nmaximum magnetic fields Bz\nmaxjt;z=0>6:0\u00021014G, while\nfor models U11 andU13 the threshold is at about 2:0\u00021015\nand2:4\u00021015G respectively. As a result, only the light-red\nshaded area in Fig. 2 collects stellar models that are bar-mode\nunstable. Outside this region, either the rotational energy is\ninsufficient, or the magnetic tension is too strong to allow for\nthe development of the instability.\nSimilarly, Fig. 7 reports the measured growth time of the\ninstability\u001cbar(and the corresponding error bars) for the three\ndifferent classes of unstable models ( U3,U11 andU13) as a\nfunction of the magnetization parameter \fbar. Taking the hor-\nizontal dashed lines as references for the unmagnetized mod-\nels, it is easy to realize that as the magnetization increases, so\ndoes the growth time for the instability. This behavior can be\nphysically interpreted as due to the fact that as the magnetic\nfield strength increases, so does the timescale over which the\nmagnetic tension needs to be won to develop a bar deforma-\ntion2.\nWe can next focus on the growth of the magnetic-field\nstrength in bar-mode unstable models as this also offers the\nopportunity for a number of useful considerations. More\nspecifically, we show in Fig. 8 the evolution of the total\nelectromagnetic energy Emagnormalized to the initial val-\nues for models U11 (left panel), U13 (middle panel) and U3\n(right panel), and for different initial poloidal magnetic-field\nstrengths. The first obvious thing to notice in Fig. 8 for all the\nmagnetizations considered for model U11 is that the growth of\nthe magnetic energy is linear in time initially. This is not sur-\nprising and is indeed the mere manifestation of the “frozen-in”\n2Note that the error bars are larger for model U3because this is closer to the\nstability threshold (cf., Table I).\nFIG. 7. Growth time of the bar-mode instability for the three unstable\nmodels U3(blue), U11 (red) and U13 (black), shown as a function\nof the initial magnetization. The horizontal dashed lines report the\ngrowth times in the absence of magnetic fields, while the dotted lines\nrepresent the corresponding error bars.\ncondition of the magnetic field within the ideal-MHD approx-\nimation. Using the induction equation it is, in fact, straightfor-\nward to show that in a linear regime the differential rotation\nwill generate toroidal magnetic field at a rate which is linear\nin time. This is because as long as the stellar configuration\nremains essentially axisymmetric the poloidal magnetic field\nis not affected by the newly produced toroidal field, and the\ntotal electromagnetic energy can only grow linearly with time\ntapping part of the rotational energy of the star.\nAs a result of this growth, the toroidal component becomes\nrapidly larger than the initial poloidal one and an amplification\nof the total electromagnetic energy takes place for all mod-\nels that reaches a higher value of about two orders of magni-\ntude over a timescale of \u001810ms. After this initial phase, the\ntoroidal field keeps growing at a slower rate, reaching a sat-\nuration with the maximum amplification being almost inde-\npendent of the initial poloidal magnetic field strength and of\nthe rotation of the stellar model. The only exceptions to this\nbehavior appear in models with ultra-strong magnetic fields,\nin which cases the saturation occurs at values that are about\ntwo orders of magnitude smaller (cf., blue solid lines in the\ndifferent panels of Fig. 8).\nInterestingly, for models U11 andU13, that is for the un-\nstable models with small growth rates and far from the thresh-\nold of the dynamical bar-mode instability, the linear growth\nof the magnetic field is accompanied also by a rather short\nexponential growth of the magnetic field. While this behav-\nior is very similar to the one seen in Ref. [34], where it was12\nFIG. 8. Left panel: evolution of the total magnetic energy Emag, normalized to its initial value, for models U11 and different values of the\ninitial poloidal magnetic field. The black solid line refers to the less magnetized case, the blue solid line for the most magnetized case, and a\nred solid line for the last unstable model, before excessive magnetic tension suppresses the instability. Middle and right panels: the same as in\nthe left panel, but for model U13 and model U3, respectively.\nModel \fmagt1t2\u0011max\u001cbarfbar\n[ms] [ms] [ms] [Hz]\nU11-0.0e00 0:0 16:2 18:30:7841:10+0:04\n\u00000:05490+1\n\u00004\nU11-1.0e14 4:7\u000210\u0000814:7 16:80:7871:09+0:05\n\u00000:02491+3\n\u00005\nU11-2.0e14 1:9\u000210\u0000715:0 17:00:7781:11+0:02\n\u00000:01488+1\n\u00001\nU11-4.0e14 7:5\u000210\u0000715:1 17:70:7731:12+0:03\n\u00000:01488+2\n\u00002\nU11-8.0e14 3:0\u000210\u0000614:8 18:20:7541:15+0:03\n\u00000:04490+2\n\u00005\nU11-1.0e15 4:7\u000210\u0000614:2 16:80:7511:17+0:04\n\u00000:05491+2\n\u00004\nU11-1.4e15 9:2\u000210\u0000613:9 16:20:7141:22+0:04\n\u00000:03491+1\n\u00002\nU11-1.6e15 1:2\u000210\u0000514:5 17:30:6811:32+0:07\n\u00000:07489+2\n\u00001\nU11-1.8e15 1:5\u000210\u0000513:2 16:70:6391:34+0:08\n\u00000:08490+2\n\u00001\nU11-2.0e15 1:9\u000210\u0000514:8 17:30:5321:49+0:09\n\u00000:11489+4\n\u00002\nU13-0.0e00 0:0 11:6 14:70:8650:94+0:01\n\u00000:01449+1\n\u00003\nU13-1.0e14 5:3\u000210\u0000812:2 15:30:8660:94+0:02\n\u00000:01450+2\n\u00002\nU13-4.0e14 8:5\u000210\u0000712:7 15:80:8510:94+0:02\n\u00000:01450+2\n\u00002\nU13-8.0e14 3:3\u000210\u0000612:7 15:80:8420:95+0:01\n\u00000:01451+1\n\u00002\nU13-1.0e15 5:3\u000210\u0000614:1 16:70:8330:96+0:01\n\u00000:02451+3\n\u00001\nU13-1.6e15 1:3\u000210\u0000511:6 14:80:8130:98+0:02\n\u00000:01456+1\n\u00002\nU13-2.4e15 3:0\u000210\u0000513:0 15:90:7341:09+0:04\n\u00000:06461+1\n\u00001\nU3-0.0e00 0:0 24:8 26:40:4862:55+0:28\n\u00000:34540+2\n\u00002\nU3-1.0e14 3:5\u000210\u0000824:9 27:10:4722:38+0:59\n\u00000:18537+5\n\u000010\nU3-2.0e14 1:4\u000210\u0000726:1 28:00:4562:47+0:21\n\u00000:04536+5\n\u00003\nU3-4.0e14 5:6\u000210\u0000724:0 26:30:4212:81+0:20\n\u00000:13537+2\n\u00003\nU3-6.0e14 1:2\u000210\u0000624:2 25:70:3003:12+0:31\n\u00000:10535+5\n\u00006\nTABLE II. Main properties of the initial part of the instability for\nmodel U11,U13 andU3for different values of the initial poloidal\nmagnetic field. Here we report the representative times t1andt2\nbetween which the maximum values of the distortion parameter \u0011,\nthe growth times \u001cbarand the frequencies fbarare computed.\nattributed to the development of the MRI, a similar conclu-\nsion cannot be drawn with confidence here. On the one hand,\nthere are a number of combined elements that seem to sup-\nport the suggestion that the exponential growth is the result\nof the development of an MRI: (i) the instability disappearswith decreasing resolution (the smallest wavelength needs to\nbe properly resolved); (ii) the growth rate does not depend on\nthe initial poloidal magnetic field (in the simplest description\nthe growth rate depends only on the local angular velocity);\n(iii) the exponential growth is followed by a rapid decay pos-\nsibly caused by reconnection processes (this behavior was also\nfound in Ref. [34]); (iv) the exponential growth disappears for\nsufficiently strong magnetic fields (the bar-mode deformation\nis no longer the lowest energy state energetically because of\nthe large magnetic-field contribution). However, our resolu-\ntions here are considerably coarser than those employed in\nRef. [34], and it is therefore difficult to see the appearance\nof channel-flow structures typical of the MRI and hence to\nmake robust measurements of the wavelengths of the fastest-\ngrowing modes. One important feature of models U11 and\nU13 is that they develop pronounced bar-mode deformations\n(they are further away from the stability threshold in Fig. 2)\nand it is therefore possible that these large deviations from\naxisymmetry act as an additional trigger, favouring the devel-\nopment of the MRI3. This could explain why an exponential\ngrowth is seen in these models despite the coarse resolution.\nAt the moment this is just a conjecture, which however, if\nconfirmed, could shed light on the sufficient conditions for\nthe development of the MRI and in particular on the degree of\naxisymmetry needed by the system. Additional simulations\nat much higher resolutions will be necessary to address this\npoint in the future.\nInterestingly, no exponential growth has been measured in\nthe dynamics of model U3for all the different magnetizations\nconsidered (cf., right panel of Fig. 8). Although the angular\n3We recall that the assumption of axisymmetry is a fundamental one in all\nperturbative calculations on the MRI and that it is exactly the absence of\naxisymmetry that allows for the development of dynamos against the limi-\ntations of the Cowling theorem [57].13\nFIG. 9. Evolution of the rotation parameter \f:=T=jWj(top panel)\nand of the total magnetic energy normalized to its initial value (bot-\ntom panel) for models S1,S6,S7andS8which are stable against\nthe bar-mode deformation in the un-magnetized case. In both panels\nthe solid lines refer to models with Bz\nmaxjt;z=0= 1015G, while the\ndash-dot lines to models with Bz\nmaxjt;z=0= 1016G.\nfrequency of these models is larger than that of U11 andU13\nand hence the timescale for the development of the MRI \u001cMRI\nwould be correspondingly shorter ( \u001cMRI\u0018\n\u00001). The evolu-\ntions have been carried out on sufficiently long timescales to\nallow for the potential appearance of the MRI. This behavior\nis indeed consistent with the conjecture discussed above, since\nthis class of models is very close to the threshold for the devel-\nopment of the bar-mode instability. As a result, these models\nexperience much smaller bar-mode deformations and main-\ntain a configuration which is more axisymmetric than those\nfound in models U11 andU13. Because these conditions are\nmore similar to those assumed by perturbative MRI analysis,\nthe corresponding predictions are expected to be more accu-\nrate. Hence, it is not surprising that no MRI is observed in\nthis case simply because no MRI can be seen for these quasi-\naxisymmetric objects at these resolutions.\nB. Effects of the magnetic field on stable models\nAfter having discussed in detail the properties of the dy-\nnamics of bar-mode unstable models, we now turn to illustrat-\ning how magnetic fields affect the dynamics of bar-mode sta-\nblemodels. Although these are comparatively simpler config-\nurations, they provide a number of interesting considerations,\nas we will see.\nWe recall that using the same EOS adopted here, Ref. [10]has determined the threshold for the development of a dy-\nnamical bar-mode instability to be \f'0:255(cf., Fig. 2).\nWe have therefore considered a number of stable models,\nnamely S1,S6,S7andS8, that are increasingly more dis-\ntant from the threshold. For each of these classes we have\nthen added two different magnetic-field strengths, namely,\nBz\nmaxjt;z=0= 1:0\u00021015G andBz\nmaxjt;z=0= 1:0\u00021016\nG, and performed simulations to record the different impact\nof the magnetic fields on the dynamics.\nOf course, since these models are already stable in the ab-\nsence of magnetic fields, they will remain stable also with\nthe additional magnetic tension. However, while models with\nBz\nmaxjt;z=0= 1:0\u00021015G do not show in their dynamics any\nsignificant deviation from a purely hydrodynamical evolution,\nmodels with Bz\nmaxjt;z=0= 1:0\u00021016G do quite the opposite.\nThis is shown in the top panel of Fig. 9, which reports the evo-\nlution of the rotation parameter \ffor all these stable models.\nSolid lines of different color refer to the different models but\nall having an initial magnetic field Bz\nmaxjt;z=0= 1:0\u00021015\nG. On the other hand, dot-dashed lines of different color refer\nto models with Bz\nmaxjt;z=0= 1:0\u00021016G. Note that for com-\nparatively “low” magnetic fields, the rotation parameter does\nnot show any significant variation from the initial value over\na timescale of around 25ms, with changes that are .0:4%\nfor model S1and.1:0%for model S8. On the other hand,\nfor magnetic fields that are one order of magnitude larger, the\nrotation parameter changes significantly, decaying almost lin-\nearly with time. This is obviously due to the combined ac-\ntion of the differential rotation and of the magnetic winding,\nwhich increases the magnetic tension and drives the NS to-\nwards a configuration that is uniformly rotating. This is also\nvery clearly shown in the bottom panel of Fig. 9, which re-\nports the evolution of the normalized magnetic energy. It is\nthen rather clear that while the energy increases (linearly) with\ntime in the case of comparatively small magnetic fields (solid\nlines), it stops growing and saturates in the case of large mag-\nnetic fields (dot-dashed lines). Over the timescale of the simu-\nlations,\u001825ms, the magnetic energy has increased of almost\nthree orders of magnitude in the former case and of only one\nin the latter case.\nWe can use the results in the top panel of Fig. 9 to obtain an\nimportant estimate on the rate at which the stellar rotational\nenergy is completely tapped by the generation of a toroidal\nmagnetic field. In particular, using the numerical data it is\npossible to express \fas\n\f(t)'\f0+aexp\u0012\n1\u0000b\nt\u0013\n\u0000ct; (4.1)\nwhere\f0:=\f(t= 0) anda;b;c are three constant coeffi-\ncients to be computed from a fit to the numerical data. Using\nthis expression is possible to compute the “braking timescale”\n\u001cbr, that is, the timescale needed for an axisymmetric and dif-\nferentially rotating configuration to lose all of its rotational en-\nergy via magnetic-field shearing and thus be brought to have\n\f(\u001cbr) = 04. The numerical fits show that the parameter bwe\n4It is perfectly plausible that a nonrotating configuration is never reached14\nobtain is of the order of the simulation time and indeed the ex-\npression in Eq. (4.1) is used only for time t<b . Considering\nthe limitt\u001dbin the expression above and after a little bit of\nalgebra we can show that the timescale is\n\u001cbr'\f0+ae\nc: (4.2)\nInterestingly, for the EOS and the magnetic field of 1016G\nconsidered here, the coefficients aandchave a simple depen-\ndence on\f05, i.e.,\na'k1+k2\f0; c'k3+k4\f0; (4.3)\nwherek1'\u00000:0589 ,k2'0:332,k3'\u00000:017ms\u00001and\nk4'0:092 ms\u00001. As a result, the general expression for\nthe braking time is now just a function of the initial rotation\nparameter\n\u001cbr'[\f0+e(k1+k2\f0)]\nk3+k4\f0; (4.4)\nand of the four coefficients k1;k2;k3andk4. As an example,\nwe can use the expression (4.4) to estimate the braking time\nfor models S1andS8, readily obtaining \u001cbr\u00180:050s and\n\u001cbr\u00180:56s, respectively.\nMuch of what is discussed above can also be deduced when\nanalyzing the structural changes in the NSs. These are shown\nin Fig. 10, where we report the initial (i.e., at t= 0 ms with\nblack lines) and the final (i.e., at t= 25 ms with blue lines)\nnormalized profiles along the x-direction of the angular ve-\nlocity (solid lines; cf., left y-axis of the figure) and of the\nrest-mass density (dashed lines; cf., right y-axis of the fig-\nure) for model S1and the two different values of the mag-\nnetic field (Bz\nmaxjt;z=0= 1:0\u00021015G in the top panel and\nBz\nmaxjt;z=0= 1:0\u00021016G in the bottom one).\nModulo of course the fact that the star will have shed some\nmatter and produced a more extended, very low-density outer\nmantle, it is clear from the top panel of Fig. 10 that the an-\ngular velocity and rest-mass density in the stellar core hardly\nchange. This is to be contrasted with what shown in the bot-\ntom panel, which clearly shows a very large increase of the\nrest-mass density in the inner regions of the star and a corre-\nsponding decrease in the outer ones. At the same time, the\nangular velocity profile has flattened considerably and indeed\nthe star is essentially axisymmetric and in uniform rotation\nwithin a coordinate radius of '15km.\nAs a final remark, we note that the considerations made\nhere and in particular the braking timescale estimated in\nEq. (4.2), are of course of more general validity than the bar-\nmode instability context considered here. Compact, axisym-\nmetric and differentially rotating magnetized configurations\nbecause the system will first collapse to a black hole. Of course this is pos-\nsible only for initial stellar configurations that are supermassive (see [58]\nfor the exploration of this possibility).\n5The coefficient bdescribes the transient and is not relevant to study the\nasymptotic solution.\nFIG. 10. Initial (i.e., shown with black lines at t= 0 ms) and the\nfinal (i.e., shown with blue lines at t= 25 ms) normalized profiles\nalong thex-direction of the angular velocity (solid lines) and of the\nrest-mass density (dashed lines) for model S1and the two different\nvalues of the magnetic field, i.e., 1015G (top panel) and 1016G\n(bottom panel).\ncan in fact be produced in a number of scenarios, from stellar-\ncore collapse to the merger of binary NSs. Determining the\ntimescales over which uniform rotation is established under\ncontrolled setups in terms of differential-rotation laws and\nmagnetizations is of course of great importance and will be\naddressed in a future work.\nV . CONCLUSIONS\nWe have presented a study of the dynamical bar-mode in-\nstability in differentially rotating and magnetized NSs in full\ngeneral relativity and investigated how the presence of mag-\nnetic fields affects the onset and the development of the insta-\nbility. In order to do that, we have performed 3D ideal-MHD\nsimulations of a large number of stellar models that were al-\nready studied in the absence of magnetic fields [10, 11, 36], by\nadding an initial purely poloidal magnetic field with strengths\nbetween 1014and1016G. In this way, we were able to ex-\nplore quite extensively the parameter space (\f; \f mag)from\n\f= 0:1886 to0:2812 , determining a threshold for the on-\nset of the instability both in terms of the rotation parameter\n\f=T=jWjand of the magnetization parameter \fmag =\nEmag=(T+jWj). In all cases considered, the differential ro-\ntation shears the poloidal magnetic field, generating a toroidal\ncomponent that grows linearly in time, and which soon pro-\nvides the largest contribution to the total electromagnetic en-15\nergy.\nWhen considering initial stellar models that are bar-mode\nunstable in the absence of magnetic fields, we found that no\neffects are present on the dynamics of the bar-mode defor-\nmation for initial poloidal magnetic fields that are .1015G,\nwith the exact threshold depending on the rotational proper-\nties of the initial model and being higher for slower rotat-\ning models. This is not particularly surprising given that in\nthese cases the magnetic energy, even the one produced via\nmagnetic-field shearing, is only a small contribution to the to-\ntal energy of the system. For initial magnetic fields that are\ninstead &1016G or larger, the corrections introduced by the\nmagnetic tension become quite large. In particular, below a\ncritical\fmag, the development of the instability is modified,\nshowing growth rates and bar-mode distortions that become\nsmaller with increasing magnetic fields, and possibly exhibit-\ning an exponential growth of the toroidal component at later\ntimes. Above a critical \fmag, on the other hand, the instability\nis totally suppressed as the enormous magnetic tension cannot\nbe overcome by the differential rotation. Under these condi-\ntions, the star sheds its outer layers leading to an extended,\naxisymmetric object with a high, uniform-density core and a\nlow-density, slowly rotating envelope.\nOn the basis of the phenomenology discussed above, and\nafter carrying-out a large number of simulations, we were able\nto locate in the (\f; \f mag)diagram the regions in which the\nvalues of the rotational and magnetic energies are sufficient to\ngive rise to the development a dynamical bar-mode instability.\nIn this sense, our study confirms the Newtonian results of [20]\nand extends them to a general-relativistic framework and to a\nmore generic range of initial conditions.\nWe have complemented our investigation by considering\nalso initial stellar models that are bar-mode stable in the ab-\nsence of magnetic fields. While these are comparatively sim-\npler configurations, also in this case the magnetic fields can\nprovide structural changes if sufficiently strong. More specif-\nically, for magnetic fields &1016G, the stellar models are\nbraked considerably in their rotation and evolve into configu-\nrations that have uniformly rotating extended cores with large\nrest-mass densities when compared to the initial values. A\nsimple algebraic expression has been derived to estimate the\ntimescale over which an axisymmetric and differentially ro-\ntating configuration will lose all of its rotational energy via\nmagnetic-field shearing.\nAs a final remark we note that although we have restricted\nour attention to a simplified EOS, our results also point out\nthat it is unlikely that very highly magnetized NSs can develop\nthe dynamical bar-mode instability and hence be considered\nas strong sources of GWs.\nACKNOWLEDGMENTS\nWe have benefited from discussions with several colleagues\nand friends and we are particularly grateful to Alessandra Feo,\nSebastiano Bernuzzi, Niccol ´o Bucciantini, Riccardo Ciolfi,\nLuca Del Zanna, Filippo Galeazzi, and Frank L ¨offler. This\nwork has been partially supported by the “CompStar”, a\nFIG. 11. Evolution of the distortion parameters \u0011+(top panel) and\n\u0011(bottom panel) for model U11-1.0e15 when imposing a bitant\nsymmetry (black solid line) or when using the full domain (blue\ndashed line).\nResearch Networking Programme of the European Science\nFoundation and the DFG grant SFB/Transregio 7. The calcu-\nlations have been performed using the HPC resources of the\nINFN “Theophys” cluster, the PRACE allocation (6th-call)\n“3dMagRoI” on CINECA’s supercomputer “Fermi”, the clus-\nter ‘at the Albert-Einstein Institute, and the PRACE allocation\n“pr32pi” on “SuperMUC” at the “Leibniz-Rechenzentrum”.\nAppendix A: The role of symmetries\nAs discussed in Sect. II, all of the results presented here\nwere achieved with a spatial resolution \u0001x= 0:375M\f'\n0:550 km on the finest grid and exploiting a “bitant sym-\nmetry”, i.e., a reflection symmetry with respect to the (x;y)\nplane. While this choice obviously reduces the computational\ncosts by a factor two, it is important to verify that it does not\nintroduce systematic effects and that all the results would be\nunchanged if this symmetry was suppressed.\nAlthough the WhyskyMHD code employed here has been\ntested in a number of different scenarios and its accuracy\nhas already been explicitly reported in various works [24,\n38, 39]; nevertheless, we have performed additional tests to\ncheck that the specific settings used are sufficient to capture\nthe main properties of the evolved systems. To this scope\nwe have evolved the bar-mode unstable model U11 when\nthreaded by an initially moderate magnetic field, i.e., model\nU11-1.0e15 , both when imposing the bitant symmetry and\nwhen evolving the equations in the full domain. Further-16\nFIG. 12. Initial growth of the square root of the toroidal component\nof the magnetic energy Etor\nmag [Eq. (3.13)] for different resolutions\nof the finest grid. The reference resolution, \u0001x='0:550km, is\nshown with a black solid line. The inset shows instead the growth rate\n\ras a function of the resolution and its fit with a quadratic function\n(dashed line). Note that the expected value \r= 1 is approached in\nthe limit of \u0001x!0.\nmore, we have varied the resolution of more than a factor of\ntwo, that is, with the finest grid having resolutions between\n\u0001x= 0:370km and 0:920km.\nFor all these runs we computed the growth rate, \u001cbar, and\nthe frequency, fbar, of the bar-mode instability. The results of\nthis extensive series of tests are reported in Table III) and show\nthat these quantities do not depend on resolution within the\naccuracy of our estimate. Hence, we conclude that all of the\nresults have been achieved at sufficient resolution to extract\nphysically significant information.\nIn addition, we have also verified that no systematic effects\nhave been introduced by the use of a bitant symmetry and this\nis shown in Fig. 11, where we report the evolution of the dis-\ntortion parameters \u0011+(top panel) and \u0011(bottom panel) for\nmodel U11-1.0e15 . The simulations have been performed\nat the reference resolution of \u0001x= 0:550km on the finest\ngrid, and the figure offers a comparison between a simulation\nusing the bitant symmetry (blue dot-dashed line) and one us-\ning the full domain (black solid line). Clearly, no significant\ndifferences can be observed between the two simulations dur-\ning the first 25ms of evolution. The same conclusion holds\nfor all quantities related to the magnetic field and they have\nbeen also monitored.\u0001x\u0001xsymmetry\u0011max\u001cbarfbar\n[M\f][km] symmetry [ms] [Hz]\n0.250 0.370 bitant 0:7431:15+0:01\n\u00000:01491+1\n\u00001\n0.350 0.445 bitant 0:7461:16+0:03\n\u00000:03492+1\n\u00001\n0.375 0.520 bitant 0:7511:17+0:04\n\u00000:05491+2\n\u00004\n0.375 0.520 full 0:7531:14+0:01\n\u00000:01490+3\n\u00002\n0.450 0.665 bitant 0:7451:18+0:03\n\u00000:05489+2\n\u00002\n0.540 0.800 bitant 0:7541:19+0:05\n\u00000:05487+3\n\u00005\n0.625 0.920 bitant 0:7431:20+0:11\n\u00000:05484+2\n\u00007\nTABLE III. Main properties of the bar-mode instability for model\nU11-1.0e15 at different resolutions. Here we report the resolution\nin terms of solar masses and kilometers, the symmetry we imposed\nto the computational domain, the maximum value of the distortion\nparameter\u0011, the growth times \u001cbarand the frequencies fbarof the\nbar-mode deformation.\nAppendix B: The role of resolution and convergence\nDetermining the convergence properties of our simulations\nis of course an essential validation of the results presented and\na considerable effort has been put into performing these mea-\nsures within the numerical setup used here. Lacking an ana-\nlytic solution that describes the fully nonlinear development\nof the bar, we can only perform self-convergence tests at this\nstage. The results will be discussed below.\nHowever, there is a regime in our calculations in which\nwe can exploit the knowledge of an analytic solution and this\nrefers to the initial shearing of the poloidal magnetic field by\nthe differentially rotating star. It is in fact not difficult to show\nthat within an ideal-MHD framework the induction equation\npredicts a growth of the toroidal magnetic field which is lin-\near in time (see, for instance, [59] for a pedagogic presen-\ntation of the perturbed induction equation). To explore this\nregime we have performed a large number of simulations of\nmodel U11-1.0e15 with varying resolution and monitored\nthe growth of the square root of the toroidal magnetic energy\nEtor\nmag [cf., Eq. (3.13]; we recall that the poloidal magnetic\nfield is not expected to grow during this stage (cf., Sect. IV A).\nFigure 12 reports the results of these simulations relatively\nto the first\u00187ms, with different curves referring to different\nresolutions. It is then evident that the curves are getting closer\nand closer to straight lines as the resolution increases. To\nmeasure whether a linear-in-time-growth is actually reached\nwe have actually computed the growth rate “ \r” by fitting\nthe square root of the magnetic energy with a trial function\nwhich is a power-law in time with undetermined growth rate,\ni.e., with\nq\nEtormag(t) =y(t) =y0+mt\r; (B1)\nwhere the time interval has been selected to be between 0:2to\n5ms.\nAlso reported in the inset of Fig. 12 are the values of \r\n(colored symbols) as a function of the resolution \u0001x, as well\nas a fit for \r(\u0001x)(dashed line) when assuming a second-\norder convergence with resolution, i.e., assuming \r(\u0001x) =\n\rj\u0001x=0+k\u0001x2(the point for \u0001x= 0:920km has been ex-17\nFIG. 13. Top panel: evolution of the toroidal component of the mag-\nnetic energy Etor\nmagfor three different resolutions. Top panel: order\nof the self-convergence test, \rc, shown as a function of time. Note\nthat a convergence order around 2is measured before the bar-mode\ninstability develops and shocks are produced (gray-shaded area).\ncluded from the fit). Having made this assumption, we do find\nthat the growth rate is in very good agreement with the one\nexpected in this linear regime, with \rj\u0001x=0= 1\u00060:005. Of\ncourse this result does not prove directly that we have second-order convergence over this period of time. However, what it\ndoes prove is that if a second-order convergence is assumed,\nthen our solution matches the expected perturbative one.\nNext we consider a more general calculation of the con-\nvergence order by performing again simulations of model\nU11-1.0e15 for a range of resolutions. This time our re-\nsults for the convergence are obtained by taking into account\nthe data corresponding to the whole timescale of the simula-\ntions, i.e.,\u001825ms. Also in this case we monitor the growth\nof the toroidal magnetic energy Etor\nmag and report in the top\npanel of Fig. 13 its evolution for three runs at resolutions:\n\u0001x= 0:370;0:550, and 0:665km, respectively. The bottom\npanel of the same figure reports instead the convergence order\n\rc, computed via a self-convergence test [48], when shown as\na function of time.\nIn this case it is then possible to recognize that the code\ndoes indeed converge at around second order during the linear\ngrowth stage (i.e., for t.5ms), in agreement with the results\nfound in purely hydrodynamical simulations [60], or with the\nnew resistive code [37]. However, as the bar-mode instability\ndevelops, the second-order convergence is lost and the conver-\ngence order reduces to one. 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Ney1,∗\n1Institute of Semiconductor and Solid State Physics,\nJohannes Kepler University Linz, 4040 Linz, Austria\n2Faculty of Physics and Center for Nanointegration Duisburg -Essen (CENIDE),\nUniversity of Duisburg-Essen, 47057 Duisburg, Germany\n3Paul Scherrer Institut, 5232 Villigen PSI, Switzerland\n4Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Ger many\n5Max-Planck-Institut f¨ ur Intelligente Systeme, 70569 Stu ttgart, Germany\n6Stanford Synchrotron Radiation Laboratory, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA\nUsing a time-resolved detection scheme in scanning transmi ssion X-ray microscopy (STXM) we\nmeasured element resolved ferromagnetic resonance (FMR) a t microwave frequencies up to 10GHz\nand a spatial resolution down to 20nm at two different synchro trons. We present different methods\nto separate the contribution of the background from the dyna mic magnetic contrast based on the\nX-ray magnetic circular dichroism (XMCD) effect. The relati ve phase between the GHz microwave\nexcitation and the X-ray pulses generated by the synchrotro n, as well as the opening angle of the\nprecession at FMR can be quantified. A detailed analysis for h omogeneous and inhomogeneous\nmagnetic excitations demonstrates that the dynamic contra st indeed behaves as the usual XMCD\neffect. The dynamic magnetic contrast in time-resolved STXM has the potential be a powerful tool\nto study the linear and non-linear magnetic excitations in m agnetic micro- and nano-structures with\nunique spatial-temporal resolution in combination with el ement selectivity.\nI. INTRODUCTION\nIn spintronics and magnonics it is imporant to under-\nstand the magnetization dynamics on the micro- and\nnano-scale e.g. to be able to control the propagation of\nspin waves. A well-established technique to measure the\ndynamic magnetic behavior of a system is ferromagnetic\nresonance (FMR). Yet classical resonator based FMR\nmeasurements are not able to detect single micro- or\nnano-sized objects due to their detection limit of around\n1011spins [1]. This sensitivity limit has been overcomein\nrecent years by the development of lithographically fab-\nricated micro-resonators [2] which are capable of mea-\nsuring down to 106spins [3], corresponding to a single\nFe-nanocube with dimensions of 30 ×30×30nm3. Due\nto the lack of spatial resolution below the diameter of\nthe micro-resonator of typically a few tens of microns\nit is impossible to separate the FMR signal of a single\nnano-particle from the resonance signal of the whole en-\nsemble during the homogeneous excitation of the micro-\nresonator cavity.\nTo facilitate spatial resolution other measurement tech-\nniques have been combined with FMR excitation in or-\nder to measure a single nano-sized object in an ensemble.\nThese measurement techniques include but are not lim-\nited to: magneto optic Kerr effect (MOKE) [4], Brillouin\nlight scattering (BLS) [5], magnetic force microscopy\n(MFM) [6], scanning thermal microscopy (SThM) [7],\nscanning electron microscopy with polarization analy-\n∗Electronicaddress: andreas.ney@jku.at; Phone: +43-732- 2468-9642; FAX: -9696sis (SEMPA) [8], and X-ray photoemission electron mi-\ncroscopy (X-PEEM) [9]. For most of these measurement\ntechniques it is not possible to measure with element\nselectivity (MOKE, BLS, MFM, SThM and SEMPA),\nwhile other measurement techniques like X-PEEM can\nonly probe the surface of the sample with element selec-\ntivity. In recentyearsthe X-raymagneticcirculardichro-\nism(XMCD) [10–12]effect hasbeencombinedwith FMR\ninordertoprobethe dynamicmagneticexcitation, the so\ncalled X-ray detected ferromagnetic resonance (XFMR)\n[13], utilizing the element selectivity of the X-rays. A\nspatialresolutionofdownto20nmcanbeachievedbyus-\ning a scanning transmission X-ray microscope (STXM);\nhowever, initially the time-resolution was restricted to\nbelow 1 GHz [14]. By combining the micro-resonator\nFMR with STXM (STXM-FMR) within a synchroniza-\ntion scheme for the exciting microwaves and the probing\nX-ray photons of the synchrotron it is possible to detect\nFMR with a high temporal (ps-regime) as well as spa-\ntial resolution (nm regime) [7, 15, 16]. Combining these\nfeatures STXM-FMR measurements bare the potential\nto significantly deepen our understanding of, e.g., mag-\nnetic domain wall dynamics [17], spin wave emitters [18],\nthe dynamic magnetization of ferromagnetic heterosys-\ntems containing different chemical elements [19] as well\nas non-ferromagnets with induced magnetization[20].\nIn order to be able to draw valid conclusions from the\ndynamic magnetic contrast in STXM-FMR, it is neces-\nsary to perform a range of control-experiments in the\nfirst place as well as testing the robustness of the evalu-\nation of the raw data to establish that STXM-FMR in-\ndeed provides significant information about the dynamic\nmagnetic behavior of a given magnetic specimen based2\non the XMCD effect. In this paper a range of control\nexperiments will be presented as well as a detailed anal-\nysis of the separation of the true magnetic contrast from\nbackground effects. The obtained results allow to reli-\nably image homogeneous and inhomogeneous magnetic\nexcitations in magnetic micro-stuctures with very high\nspatio-temporal resolution. Furthermore, it is possible\nto obtain quantitative information about the local pre-\ncessionangleinFMRanditsrelativephasewithinagiven\nSTXM-FMR experiment.\nII. EXPERIMENTAL DETAILS\n100mma)\n b)\nc)5mm\n2mm\nFIG. 1: Scanning electron microscope (SEM) image of the\nstrip-line resonator on top of a SiN-membrane. For this work\ntwo different sample systems were chosen. In b) two perpen-\ndicular Py stripes (”T”-sample) are shown, while in c) the\nPy-Co disk stripe sample can be seen.\nThe magnetic specimen is placed inside a micro-\nresonator and microwaves are used to excite the FMR.\nThe micro-resonator is fabricated on a 200nm thick,\n250×250µm2large silicon nitride membrane suspended\nby a 5×10mm2silicon frame of high resistivity. In a first\nstep the magnetic specimen is fabricated on the SiN-\nmembrane using electron beam lithography (EBL). Two\ndifferent designs for the magnetic specimen were made.\nThe first one consists of two perpendicular permalloy\n(Py) stripes with dimensions of 5 ×1×0.03µm3(see\nFig.1b)), which are deposited using magnetron sputter-\ning at room temperature and capped with aluminum.\nThe second sample system is a combination of a Py\ndisk with a Co stripe. For this in a first step a Py disk\nwith a diameter of 2.6 µm and a thickness of 30nm is\nfabricated. With a second EBL step a Co stripe with\nlateral dimensions of 2 ×0.6µm2and a thickness of 30nm\nis placed on top of the Py disk (see Fig.1c)). Finally,\nthe micro-resonator is patterned around the magnetic\nspecimen using optical lithography (OL), leaving the\nsample inside the Ω shaped resonator loop in Fig.1a).\nThe gold used to produce the micro-resonator has athickness of 600nm and an additional 5nm of titanium is\nused as an adhesion layer. Both materials are deposited\nby thermal evaporation.\nTo measure the STXM-FMR at the Stanford Syn-\nchrotron Radiation Lightsource (SSRL), beamline 13.1,\nthe FMR excitation needs to be synchronized to the\nbunch freuqency of the synchrotron, thus enabling us\nto measure the FMR precession with time resolution.\nBy utilizing the STXM it is possible to measure with\na spatial resolution, 35nm at SSRL and 20nm at\nthe MAXYMUS beamline at BESSY II, achieved by\nfocusing the X-rays onto the magnetic specimen using\na zone-plate. The stroboscopic time resolution for the\nSTXM-FMR measurement is achieved by phase locking\nthe GHz microwavefrequency to the 476.315MHz bunch\nfrequency of the SSRL synchrotron. Furthermore, a\nPIN-diode was installed to switch the microwave on\nand off with the synchrotron revolving frequency of\n1.28MHz. This comparison of X-ray transmission de-\ntected with and without applied microwave power allows\nto detect very small changes in the x-ray transmission\nas a result of the magnetization precession. A fast\navalanche photo diode (APD) detects the transmitted\nX-ray photons behind the sample. The APD signal is\nfinally stored in 12 different channels. Each of these\nchannels corresponds to the signal of a specific group of\nX-ray pulses. The first 6 channels are used for the APD\nsignal of the transmitted X-rays with applied microwave,\nwhile the second 6 channels are used to measure the\nX-ray transmission without applied microwave. For the\nfirst 6 channels the magnetization inside the sample\nis precessing, while the magnetization is static for the\nsecond 6 channels. Each of these channels corresponds\nto a specific relative phase of the FMR precession\nwith respect to the microwave excitation. The latter\nnon-precessing channels are crucial to eliminate the\ninfluence of the filling pattern of the bunch train of\nelectron buckets on the resulting STXM images. The\n6 different channels correspond to 6 specific bunches\nwhich are phase shifted each by 60◦, with respect to the\nmicrowave frequency of up to 9.6GHz. Therefore, the\n6 phases correspond to time resolved snap-shot images\nwhich are separated by 17.4ps, and comprise one full\nprecession cycle of the magnetization. One should note,\nthat each X-ray flash has a pulse duration of 50ps, which\nfundamentally limits the attainable temporal resolution.\nAdditional information regarding the synchronization\nscheme and the X-ray microscope can be found in [7, 15].\nA similar approach for measuring the dynamic magneti-\nzation in an FMR experiment with spatial and temporal\nresolution has been implemented at the Maxymus\nendstation at BESSY II [16]. There are two moderate\ndifferences with respect to the SSRL experiment: One\nis that the BESSYII operation frequency is appx. 500\nMHz, correspondingto a repetition period of the probing\nx-ray flashes of 2 ns. Secondly, the signal is recorded\nonly for the microwave on (precessing) case. Therefore,3\non the one hand side, it is not necessary to excite\nthe sample at direct higher harmonic frequencies of\nthe synchrotron and thus the exciting frequencies can\nbe chosen more freely to f=500MHz*M/N, depending\non the number of detection channels used (N) and a\nselectable integer multiplier M.[18] Here N for most\ncases is also equal to the number of simultaneously\nacquired excitation phases (not limited to 6). Since the\nnot-precessing magnetization (microwave off) cannot be\nused as a baseline for comparison, on the other hand, for\nnormalization purposes only the transmitted intensity\nratio of each channel I(t)/ ¡ I ¿t with respect to the\ntemporal average state can be evaluated in order to\nextract the dynamic magnetic contrast originating from\nthe precession of magnetization\nIII. CONTRAST MECHANISM\nFor a better understanding of the measured STXM-\nFMR data, we briefly discuss the underlying physical ef-\nfect which yields the dynamic magnetic contrast images.\nA. X-ray absorption\nThe transmission of electromagnetic radiation through\nany material is described by Beer-Lambert‘s law [21]. In\na STXM the transmitted X-ray intensity Iis detected.\nThis transmitted intensity is comprised of the X-ray ab-\nsorption (XA) coefficient of the entire sample (magnetic\nspecimen and SiN-membrane). Tuning the photon en-\nergy to any characteristic core-level excitation resulting\nin the well-known element selectivity of XA measure-\nments. However, the above mentioned law only consid-\ners a single layer system. In an STXM-FMR experiment\nthe sample consists at least of a two layer system since\nany magnetic specimen is suspended on a SiN-membrane\nthrough which the X-rays need to be transmitted as well.\nIn order to include this second layer Beer-Lambert‘s law\nneeds to be modified [21]:\nIs/m=I0e−(µsts+µmtm)(1)\nwherets,tmare the thicknesses and µs,µmare the ab-\nsorption coefficients of the magnetic specimen and SiN-\nmembrane respectively, and I0is the incoming inten-\nsity. In any sample one can find areas where the X-\nrays only transmit through the SiN-membrane while in\notherregionstheX-raysaretransmittedthroughtheSiN-\nmembrane plus the magnetic specimen, which can also\nconsist of more than one layer which would be added to\nthe exponent in Eq.1. Therefore, from a single STXM-\nFMR image one can separate the dynamic magnetic\ncontrast from the background transmission of the SiN-\nmembranebydefining respectiveregionsofinterest(RoI)\nfrom the time-averagedz-contrastimages like in Fig.1c).IV. ANALYSIS OF STXM-FMR\nMEASUREMENTS\nIn the light of the preceding discussion, evaluation\nmethods for the extraction of quantitative information\nfrom the STXM-FMR data will be presented, with spe-\ncial attention to how to extract the dynamic contribution\nof the magnetic specimen. Additionally it is possible to\nquantify the opening angle of the magnetization preces-\nsion in FMR directly from the change in absorption co-\nefficient during a full precession cycle.\nA. Raw data treatment\nI\nt2=18.4ps t1=0.0ps t3=36.8pst4=55.2pst5=73.6ps t 6=92ps\nIIb\nIIcIIa2mm\nFIG. 2: The top shows the chemical contrast of the disk stripe\nsample measured at the Ni- L3-edge. Representation of the\ndifferent evaluation methods for the 6 phases of the magne-\ntization precession: In I the absorption coefficient differen ce\nis shown obtained the background corrected microwave on\nand off measurement. IIa is the ratio between the not back-\nground corrected microwave on and off measurement. Apply-\ning a background correction to IIa the images labelled IIb ar e\ngenerated. The images labelled IIc show the absorption co-\nefficient change between microwave power on and off for the\ndifferent phases.\nToeliminate thesecondabsorptioncoefficient µ2in equa-\ntion 1 the raw data needs to be corrected by the SiN-\nmembrane background. Thus the absorption coefficient\nof the magnetic specimen alone can be investigated. For\nthat we average the transmission signal over the area\nof only the SiN-membrane for each of the 12 images\n(6 phases with microwave on Ion\nmand 6 phases with4\nmicrowave off Ioff\nm) separately. Each individual image\nis then divided by its respective averaged transmission\nvalue of the SiN-membrane. The resulting transmission\nIonandIoffthen contains exclusively the information\nabout the absorption coefficient µ1of the magnetic spec-\nimen:\nIon\ns=Ion\ns/m\nIonm=eµon\ns·tsIoff\ns=Ioff\ns/m\nIoff\nm=eµoff\ns·ts(2)\nNote, that this also eliminates the dependence on the\nindividual incoming intensity I0for each phase. Sub-\nsequently, the dynamic magnetic contrast is derived by\ntaking the natural logarithm of the ratio of the precess-\ning(microwaveon)versusnon-precessing(microwaveoff)\ncase to obtain ∆ µcorresponding to the difference in ab-\nsorption coefficient equivalent to the usual definition of\nthe XMCD effect:\nln(Ion\ns\nIoff\ns) = (µoff\ns−µon\ns)·t= ∆µ·t(3)\nwheretis the thickness of the magnetic specimen. The\nresulting dynamic magnetic contrast ∆ µ·tof the Py disk\nrecorded at the Ni L3-edge at 9.04GHz is shown for all\nsix phases in Fig.2, row I. It is clearly visible that only\nthe contrastofthe Pydiskreversesduringafull measure-\nment cycle representing the perpendicular component of\nthe high-frequency magnetization, while the background\nstays constant.\nHowever,one canchangethe sequenceofextracting∆ µ·t\nand take a closer look at each individual step. First, the\nratio of microwave on and microwave off is taken and\nall 6 phases are displayed in Fig.2 row IIa. It is obvious,\nthat the backgroundcorrespondingto the SiN-membrane\noscillates as well, which will be discussed further below.\nIn a second step, the influence of the oscillating back-\nground is corrected as mentioned before by dividing each\nphase with the respective averaged transmission of the\nSiN-membrane. The resulting 6 phases are shown in Fig.\n2, rowIIb and alreadycomparewell with the full analysis\nof I revealing no visible background oscillation.\nHowever, the images of IIb do not directly reflect the nu-\nmerical values of ∆ µ·t. Taking the natural logarithm\nof IIb one obtains the 6 phases as shown in Fig.2, row\nIIc. A direct comparison between IIb and IIc reveals,\nthat the qualitative behavior of the dynamic magnetic\ncontrast is identical. However, only IIc shall be mathe-\nmatically equivalent to the full analysis in I. Both evalu-\nations depend on the selection of the RoI from which the\nbackground of the SiN-membrane is derived.\nTo verify if the sequence of the evaluation steps indeed\nyield the same results, the quantitative outcome of meth-\nods I and IIc are compared in Fig.3. In the STXM-FMR\nimage the area outside the blue box defines the RoI used\nfor determining the background of the SiN-membrane.\nThe red box indicates the RoI which is used for deter-\nmining the average ∆ µ·tof the magnetic specimen. To\nderive the absorption coefficient ∆ µat NiL3-edge theresulting averaged value has to be divided by the effec-\ntive thickness t= 24nm, since the Py film is 30nm thick\nand contains 80% nickel, note that the non-resonant XA\nof the iron can be excluded due to the ratio between\nthe measurements with and without applied microwave\npower. The two panels show the averaged values (sym-\nbols) of the 6 phases for method I (right) and IIc (left)\nreflecting the dynamic magnetic contrast of the homoge-\nneous excitation, i.e., uniform mode of the Py disk. The\nsine fits (solid lines) are done for the fixed frequency of\nthe exciting microwave of 9.04GHz while amplitude A\nand phase ϕare fitting parameters. Indeed, both meth-\nods reveal identical numerical values for A= ∆µand\nϕof (340±31)cm−1and−39◦±5◦, respectively. The\nquantitative numerical values will be discussed in the fol-\nlowing.\nB. Precession angle\n2mm\nFIG. 3: By averaging over the sample in the six different\nphases for evaluation method IIc (left side) and I (right sid e)\nfrom Fig.2 one can obtain the shown curves. Both were fitted\nwith a sine fit due to the sinusoidal behavior of the exciting\nmicrowave. The frequency of this sinusoidal was given by the\nmicrowave frequency applied to the system which in this case\nwas 9.04GHz.\nThe first quantity which can be extracted from a STXM-\nFMR experiment is the amplitude Acorresponding to\nthe dynamic magnetic contrast ∆ µ. For a known thick-\nnesstof the magnetic specimen one can extract the\nopening angle θof the precessing magnetization. Other\nthan for the phase ϕ, the amplitude Aand thus ∆ µ\ncan be compared between different samples. For that\na usual XMCD experiment is carried out on a specimen\nof known thickness dwhere ∆ µXMCDis derived as the\ndifference in absorptionwith the magnetizationfully par-\nallel and antiparallel to the kvector of the X-rays, yield-\ning ∆µabs= ∆µXMCD/d. One should keep in mind that\nin an XMCD experiment the magnetization is fully re-\nversedwhile in the STXM-FMR measurementmicrowave\noff corresponds to the fully perpendicular case. There-\nfore, 2Ahas to be taken when comparing with ∆ µabs.\nIn addition, from geometrical considerations this method\nyields the full opening angle of the precession cone cor-\nresponding to 2 θ, therefore yielding:\nsin(2θ) =2A\n∆µabs(4)5\nHere 2Ais (680±31)cm−1and ∆µabs≈200000cm−1,\nwhich yields an opening angle of θ= 0.10◦±0.01◦. As\nalready pointed out before [15], one has to consider the\neffect of the pulse length of the X-rays on the measured\nintensity. A pulse length of 50ps yields a reduction fac-\ntor of 1.5 for this measurement at 9.04GHz, due to the\naveraging over part of the dynamic magnetic response of\nthe system. Therefore, the actual opening angle for this\nFMR measurement is θ= 0.15◦±0.02◦which is of the\nsame order as the previously reported opening angle of\n0.1◦for a Co-stripe [15]. It has to be taken into consid-\neration that the obtained opening angle of the FMR is\nonly the out-of-plane angle, which in turn can differ from\nthe in plane angle due to the magnetic anisotropy of the\nthin film sample.\nC. Origin and influence of the background signal\nDue to the background oscillation observed in Fig.2\nIIa it is possible that small changes in the dynamic mag-\nnetization can not be measured properly. In Fig.4 a mea-\nsurement on a Py ”T”-sample is shown, where the polar-\nization of the X-ray photons is switched from σ+toσ−.\nFig.4a) shows the chemical contrast image of the mea-\nb)\na)\nBext1mm\nFIG. 4: a) shows the chemical contrast images of the mea-\nsured sample, while the direction of the external magnetic\nfield (Bext) is indicated by the blue arrow. b) shows the\naveraged transmission intensity of the stripe parallel to t he\nexternal magnetic field for two different X-ray polarization s\nσ+andσ−.\nsured sample. The direction of the external magnetic\nfield is indicated by the blue arrow. In Fig.4b) the av-\neraged transmission intensity of the stripe parallel to the\nexternal magnetic field is shown for the two X-ray polar-\nizationsσ+(blackcurve)and σ−(redcurve). The result-\ning relative phases for the individual measurements are\n1◦±3◦forthemeasurementusing σ+and74◦±11◦forthe\nmeasurement using σ−. As can be seen the phase of the\nSTXM-FMR measurement does not change by 180◦as it\nwould be expected for the XMCD effect, when switching\nthe polarization of the incoming X-rays. To understandthe reason of this non reversal of the dynamic magneti-\nzation the background signal observed in Fig.2 IIa needs\nto be discussed.\nThe output signal of the avalanche photo diode is ampli-\nfiedbyafactorof1000(60dB)tobedetected. Therefore,\nit is very sensitive to issues with the pre-amplification.\nThe cables inside the STXM (power supply for the APD\nand signal output of the APD) can act as antennas for\nstanding waves generated by the microwave excitation of\nthe sample. This can cause false positives/negatives de-\npending of the phase of the microwavewith regard to the\nphoton arrival time, which can be misinterpreted as bulk\n(low spatial frequency) dynamics. This is an issue since\nmicrovolts of induced voltage by the microwave can be\namplified to a ”photon” level in the signal output of the\nAPD.\nAdditionally common detection methods can only detect\none photon per bunch. Multi photon events only reg-\nister as single events. This creates a non-linear detec-\ntor response that gets more pronounced for higher count\nrates, and can interfere with normalization of dynamic\ncontrast when imaging samples with big static contrast.\nWhile the signal in dark areas (magnetic specimen) is\nlinear, the signal in bright areas (SiN-membrane) is com-\npressed, thus the normalization algorithms that work by\naveraging obtain a skewed response that can create false\ndynamic contrast proportional to the static contrast.\nHowever, if the dynamic contrastreverseswhen changing\nthe polarization is switched, it can be concluded that the\nobserved signal is a consequence of the dynamic mag-\nnetic response of the system to the external excitation\ngenerated by the microwave. In order to do so the phase\nbetween the microwave excitation and the X-ray pulses\ncan be obtained from the STXM-FMR measurements.\nD. Absolute vs. relative phase\nThe absolute phase should be measured between the\nprecessing magnetization and the arrival of the X-ray\npulse. However, this is complicated due to several issues.\nFirst, as in any resonance experiment, there is a phase\ndifference between the microwave excitation and the pre-\ncessing magnetization. Second, the phase of the X-ray\npulse cannot be determined directly since only the driv-\ning frequency of the rf-cavity of the storage ring is acces-\nsible. Therefore, the travel time of the electron bunches\nfrom the cavity to the undulator as well as the travel\ntime of the X-ray pulse from the undulator to the sam-\nple have to be taken into account. These are in principle\nknownandshouldbefixedvaluesforagivensynchrotron.\nIn addition, the length of the used cabling has an influ-\nence on the phase as well and this changes when the\nmicrowave set-up including sample and micro-resonator\nis physically changed. In a practical experiment the fit-\nted phase ϕis only a relative number and comprises all\nthe above factors. Therefore, it can only be compared\nas long as the entire microwave set-up as well as the ex-6\ncitation frequency of the STXM-FMR experiment is not\nchanged. This implies, that it is only possible to compare\nrelative phases within the same sample and not between\ndifferent samples. In other words, the obtained phase\nϕ=−39◦±5◦is basically meaningless for comparing dif-\nferent sample measured in the STXM-FMR. However,\nthe relative phase of different measurements using the\nsame parameters and sample upon, e.g., the reversal of\nthe helicity of the light can be compared and - according\nto the XMCD effect - should be 180◦.\nV. EXPERIMENTAL VERIFICATION OF THE\nCONTRAST BEHAVIOUR\nHaving discussed the small variation of the absorption\ncoefficient during precession with a small opening angle\ntogether with the presence of a rather pronounced back-\nground signal of the SiN-membrane, it is important to\ninvestigate the behavior of the dynamic magnetic con-\ntrast upon reversal of the helicity of the light to assure\nthat a true XMCD effect is indeed observed.\nA. Contrast reversal with helicity\nFIG. 5: a) Chemical contrast picture of the Py stripe. In\nb) two STXM-FMR measurement with different X-ray polar-\nization measured at the Ni- L3-edge are shown as well as the\ndifference between the two measurements. c) shows the av-\nerage transmission intensity of the X-rays through the stri pe\nsample for the three cases shown in b). The averaged data\nwas fitted with a microwave frequency of 9.61GHz.\nAs a first test experiment two STXM-FMR measure-\nments at 9.61GHz were done using σ+andσ−polarized\nX-rays at the Ni L3-edge of the horizontal Py stripe of\nthe T sample shown in Fig.1b). In Fig.,5a) the chemical\ncontrast is shown while the individual 6 phases with σ+\n(blue) and σ−(red) light are displayed in Fig.5b), top\ntwo rows. All contrast variations in b) are shown on\nthe same scale in order to emphasize the difference in\n∆µ·tfor the different measurements. Fig.5c) collates\nthe averaged dynamic magnetic contrast for all 6 phases\nderived by averaging over the respective marked areas.\nThe RoIs were identified using the chemical contrastimage in Fig.5a) as indicated by the red box. The same\ncolour scheme was used for the averaged intensities of\nthe two measurements shown in Fig.5c).\nAs one can clearly see in the individual phase images\nthere is a phase difference of −27◦±2◦forσ−and\n+99◦±6◦forσ+light. Note, that the value for the\nphases in Fig. 3 and 5 differ because the samples and\nthus the microwave setup are different. The XMCD\neffect suggests that by reversing the X-ray polarization\nthe relative phase should change by 180◦, i.e., an ideal\nreversal of the contrast in all 6 images. However, the\nrelative phase difference between the two measurement\nis only 127◦±8◦, which is significantly smaller. This\nis most likely due to the experimental constraint that\nonly 6 phases can be resolved because of the X-ray\npulse length of 50ps, while the time difference between\nthe individual phases is only 17.4ps. Therefore, the\nexperimental uncertainty is larger, than the errors from\nthe fitting procedure, especially considering the small\noverall size of the dynamic contrast change. In turn,\none can increase the magnetic contrast by taking the\ndifference between the two experiments with σ+andσ−\nlight according to usual XMCD experiments. The result\nis shown in Fig. 5b) and c) (green) and it is obvious\nthat the dynamic contrast is enhanced significantly.\nNevertheless, as visible in Fig.5 c) the amplitude is not\nincreased by a factor of 2 as expected which is due to\nthe non-ideal reversal of the contrast as reflected by the\nbehavior of the relative phases. Nevertheless, this is\na first indication that a homogeneous FMR excitation\nbehaves the same way as previously observed spin wave\nexcitations [15].\nB. Helicity versus field direction\nIn Fig.6 a second control STXM-FMR measurement\ndone at the Maxymus endstation at BESSY II is shown\nwhere the STXM-FMR was measured with a slightly in-\ncreased phase resolution of 7 images for one full preces-\nsion cycle. In addition to reversing the helicity of the\nlight one helicity was also measured for two different ex-\nternal magnetic field orientations Bext, i.e., (σ+,B+),\n(σ+,B−), and (σ−,B+). The relative orientation of Bext\nis indicated together with the image of the chemical con-\ntrast in Fig.6a), top. The RoI where the contrast is\nspatially averaged is indicated by the red boxes in all\nimages to ensure that the observed averaged signal only\noriginates from the stripe and does not contain the SiN-\nmembranebackground(see above). Thetime-normalized\nspatial average intensity for each phase of the 3 different\nmeasurementsis shownin Fig.6b)-d). The measurement\nin a positive magnetic field B+ and circular polarization\nσ+is shown on the left side of Fig.6a). The resulting\naveraged normalized X-ray transmission can be found in\nFig.6b), where it was fitted using a sine function with\nthe microwave frequency of 6.785GHz. This fit yields a7\ns+;B+ s+;B- s-;B+\nt1=21ps\nt2=42ps\nt3=63ps\nt4=84ps\nt5=105ps\nt6=126pst0=0psB+ B\u0000\nFIG. 6: STXM-FMR measurement done at the Maxymus\nbeamline at the Fe- L3-edge at B+=60mT and B-=-60mT\nand different X-ray polarization σ+, σ−. The applied mi-\ncrowave frequency for all measurements shown in this figure\nwas 6.785GHz. The left hand side shows the chemical con-\ntrast image together with the different directions for B- and\nB+. The averaged area is indicated by the red box. Below,\nthe normalized intensity (with respect to the time average\nstate) for the 7 different excitation phases (or delay times)\nis shown for different field directions and X-ray polarisatio ns.\nThe spatially averaged intensity for each of these boxes can\nbe found on the right hand side with their respective colour\ncoding.\nrelative phase between the X-ray pulses and the magne-\ntization precession of 102◦±12◦.\nAs mention above the sign of the static XMCD effect re-\nverses when the external magnetic field along an axis of\nsensitivity is reversed [11, 12]. At first glance it could be\nexpected that this leads to a contrast reversal for the dy-\nnamic magnetic contrast in STXM-FMR as well. How-\never, as can be seen in Fig.6a) the contrast does not\nreverse when the external field is reversed (middle col-\numn). This can be explained due to the fact that the\nSTXM-FMR inthe presentconfigurationisonlysensitive\nto the transversal dynamic component of the magnetiza-\ntionprecessionandnotthedirectionofthemagnetization\nitself. Due to the field reversal the magnetization still\nprecesses around the external field with the same phase\nrelation as before. The dynamic magnetic contrast is\nonly dependent on the projection of the dynamic magne-\ntization onto the X-ray k-vector. This projection in turn\nexhibits a cosine behaviour, thus does not depend on the\nsense of rotation regarding the X-ray k-vector. This is\nevidenced by comparing the averaged dynamic magnetic\ncontrastfor σ+,B+in Fig.6b) and σ+,B−in c). The re-\nsulting relativephaseis 102◦±12◦and 100◦±16◦, respec-\ntively, i.e., identical within error bars. In contrast, com-\nparingσ+withσ−polarization for B+in Fig.6b) and\nd), respectively, a clear contrast reversal is seen which is\nreflectedbythe resultingrelativephasesof102◦±12◦and\n−94◦±7◦. The resulting phase change is thus 196◦±19◦\nwhich agrees within error bars with the expected value\nof 180◦for the ideal XMCD effect.\nC. Contrast reversal for inhomogeneous excitations\nUntil now all measurements were done for a uniform\nmagneticexcitationofthemagneticspecimen. Therewasa drastic difference in conventional XA for areas where\nonly background effects were observed to regions where\nthe dynamic magnetic contrast was measured. Since we\nhave already attributed the oscillating contrast of the\nbackground to an interaction of the microwave with the\nAPD, non-linearities of the APD can also have a non-\nnegligible influence on the extracted dynamic magnetic\ncontrast. This is especially relevant for homogeneous ex-\ncitations of the magnetic specimen. In order to exclude\nthis, an inhomogeneous excitation of the magnetic spec-\nimen is the method of choice.\ns+\ns-a)\nb)\n2.5mm\nFIG. 7: Comparison of the two circular polarizations for an\ninhomogeneous excitation of a T stripe. The left part shows\nthe chemical contrast pictures for the different polarizati on\nmeasurements respectively. For contrast maximization the\noppositephases ofthesame measurementaresubtracted. The\nred and green boxes indicate the position of the Py stripe\n(extracted from the chemical contrast) for each of the two\nmeasurements to better visualize the excitation.\nIn Fig.7 a STXM-FMR experiment, measured at the\nSSRL, of an inhomogeneous excitation of the stripe\nparallel to the external magnetic field of a Py ”T”-\nsample is shown. Details on these types of excitations\ngo beyond the scope of this paper and will be discussed\nelsewhere; integral FMR measurements together with\nmicromagnetic simulations have already been published\n[3]. Figure 7 a) shows the measurement with σ+light,\nwhereas in b) the σ−case can be seen. On the left-hand\nside the chemical contrast images are provided while\non the right-hand side a single image of the dynamic\nmagnetic contrast is displayed. In order to maximize the\ncontrast, the difference between the same two opposite\nphases has been taken for both polarizations. An\nadditional smoothing as in Ref. [7] has been carried out\nto better visualize the inhomogeneous excitation. One\ncan clearly see that there are regions with a pronounced\nmagnetic contrast to either end of the stripe while\nthe center shows a much weaker contrast with zero\ncontrast in between. Importantly, the regions of strong\nmagnetic contrast clearly reverse upon reversal of the\nhelicity of the light while in other regions the contrast\nremains unaffected. Therefore, the contrast reversal\nis also observable with respect to a non-reversing re-\ngion where the overall XA does not change, underlining\nthat the contrastmechanismis indeed ofmagneticorigin.8\nVI. CONCLUSION\nWe have shown a way to correctly separate the quan-\ntitative pure dynamic magnetic contrast from the back-\nground signal in STXM-FMR. The opening angle of the\nFMR excitation of Py was determined at the Ni-L 3-edge\nby evaluating the amplitude of the dynamic magnetic\ncontrast yielding an opening angle of 0.15◦which corre-\nsponds well with previously reported values for Co[15].\nFurthermore, by switching the polarization of the X-\nray photons from σ+toσ−the dynamic magnetic con-\ntrast switches its sign for the STXM-FMR measurement.\nHowever, contrary to static classical XMCD a reversal\nof the external magnetic field does not change the dy-\nnamic magnetic contrast of the STXM-FMR because of\nthe transversalgeometry. Enhancement of the signal can\nbe achieved by measuring the STXM-FMR with different\npolarizations ( σ+andσ−). Finally, the contrast reversal\nupon reversalofthe helicitywasobservablefortwodiffer-\nent STXM-FMR setups, at twodifferent synchrotrons,as\nwell as for an inhomogeneous excitation. This evidencesthat the contrast in STXM-FMR behaves similar under\nreversal of the X-ray helicity to the static XMCD effect\nand one can take advantagefrom the unique combination\nof element selectivity and spatio-temporal resolution in\nfuture studies of magnetically excited micro- an nano-\nstructures.\nAcknowledgments\nThis research was funded the Austrian Science Foun-\ndation (FWF), project No I-3050 as well as the Ger-\nman Research Foundation (DFG), project No OL513/1-\n1Use of the Stanford SynchrotronRadiation Lightsource,\nSLAC National Accelerator Laboratory, is supported by\nthe U.S. Department of Energy, Office of Science, Of-\nfice of Basic Energy Sciences under Contract No. DE-\nAC02-76SF00515. Part of the measurements were car-\nried out at the MAXYMUS endstation at BESSY II at\nthe Helmholtz-Zentrum Berlin. We thank HZB for the\nallocation of synchrotron radiation beamtime.\n[1] Poole, Ch. P. 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Appl.\nPhys.200597, 10E704\n[15] Bonetti, S., Kukreja, R., Chen, Z., Spoddig, D., Ollefs ,\nK., Sch¨ oppner, C., Meckenstock, R., Ney, A., Pinto, J.,\nHouanche, R., Frisch, J., St¨ ohr, J., D¨ urr, H. and Ohldag,\nH. Microwave soft x-raymicroscopy for nanoscale magne-\ntization dynamics in the 5-10 GHz frequency range. Rev.\nSci. Instrum. 201586, 093703\n[16] Weigand, M., Realization of a new Magnetic Scanning X-\nray Microscope and Investigation of Landau Structures\nunder Pulsed Field Excitation. (Dissertation), Cuvillier\nVerlag, G¨ ottingen (2014)\n[17] Stein, F. U., Bocklage, L., Weigand, M. and Meier, G.\nTime-resolved imaging of nonlinear magnetic domain-\nwall dynamics in ferromagnetic nanowires. Sci. Rep.\n2013 3, 1737\n[18] Wintz, S., Tiberkevich, V., Weigand, M., Raabe, J., Lin -\nder, J., Erbe, A., Slavin A. and Fassbender, Magnetic9\nvortex cores as tunable spin-wave emitters. Nat. Nan-\notech.201611948\n[19] Feggeler, T. et al. 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Powsta ´nc´ow Warszawy 6, 35-959 Rzesz ´ow, Poland\n3Department of Physics and CFIF, Instituto Superior T ´ecnico, TU Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal\n4Martin-Luther-Universit ¨at Halle-Wittenberg, Heinrich-Damerow-Str. 4, 06120 Halle, Germany\nKey words: Domain walls, magnetic dynamics, spintronics\n\u0003Corresponding author: e-mail sedlmayr@physik.uni-kl.de , Phone: +49-631-205 3159\nWe consider the dynamics of polarization of a single domain wall in a magnetic nanowire, which is strongly pinned\nby impurities. In this case the equation of motion for the polarization parameter does not include any other dynamical\nvariables and is nonlinear due to magnetic anisotropy. We calculated numerically the magnetization dynamics for\ndifferent choices of parameters under short current pulses inducing polarization switching. Our results show that the\nswitching is most effective for very rapid current pulses. Damping also enhances the switching probability.\nCopyright line will be provided by the publisher\n1 Introduction Since the possibility of moving mag-\nnetic domain walls (DWs) in nanowires due to electric cur-\nrents was realized, the behaviour of DWs in wires sub-\nject to a variety of pulses, currents, fields and pinning\nforces has been extensively studied [1,2,3,4,5,6]. Elec-\ntrons passing a DW transfer momentum and spin with the\nDW, which results in a modification of the resistivity of\nthe wire and the motion of the DW [7,8,9]. In simple sce-\nnarios the equations governing the DW motion can be set\nup and solved [10,11,12,13,14,15,16,17]. The ability to\nmove domain walls around raises possible technological\napplications, in particular the possibility of constructing\na solid state memory with magnetic domains playing the\nrole of bits [18,19]. In these works the DW dynamics was\nmostly investigated for the translational motion along the\nwire.\nThen in an experiment performed on a Permalloy\nnanowire it was found that the polarity of a transverse\nDW can be controlled with current pulses [20,21]. Ad-\nditionally, it was shown that the DW polarity is mainly\ndetermined by the direction of the current. Since then at-\ntention has also been attracted to the DW dynamics of\ntranslationally non-invariant systems [22,23].\nIn this work we concentrate on the dynamics of a\nstrongly pinned DW, which is controlled by short currentpulses. In this case the possible motion along the wire is\nlimited to a very short distance. We use a model with a\nsingle DW pinned to the potential just like a particle in\nan oscillatory potential. This model allows us to study in\na simple way how the rotation of the DW polarization is\nrelated to the oscillatory longitudinal DW motion.\n2 Model Let us start from the Lagrangian density de-\nscribing the spin dynamics of a one dimensional domain\nwall (DW) [4,13,14,17]\nL=~S_\u001e(cos\u0012\u00001)\u0000S2\n2n\nJ[(r\u0012)2+ sin2\u0012(r\u001e)2]\n+ sin2\u0012(K+K?sin2\u001e)o\n\u0000Vpin; (1)\nwhere the angles \u0012=\u0012(x;t)and\u001e(x;t)determine the spin\norientation in the DW:\n^n(x;t) =0\nB@cos[\u001e(x;t)] cos[\u0012(x;t)]\nsin[\u001e(x;t)] cos[\u0012(x;t)]\nsin[\u0012(x;t)]1\nCA: (2)\nSis the localized spin, Jthe exchange coupling between\nspins andKandK?the magnetic anisotropy. We have\nintroduced a pinning field Vpin\u0011SV0a2=2whereais the\nposition of the DW. In the following we set ~= 1\nCopyright line will be provided by the publisherarXiv:1309.3687v1  [cond-mat.mes-hall]  14 Sep 20132 N. Sedlmayr et al.: Dynamics of a pinned domain wall\nFigure 1 A schematic of the two stable in plane DW polar-\nizations, a) and b) in the wire. The polarization dynamics\nflips the system between these two orientations.\nAssuming that a single DW is strongly pinned at the\npointx= 0, we can use the following parametrization with\ntwo functions a(t)andf(t)\n\u0012(x;t) =\u00120(x\u0000a(t));\n\u001e(x;t) = 0(x\u0000a(t))f(t); (3)\nwhere we assume that \u00120(x)and 0(x)are known func-\ntions describing the DW shape. The time dependent func-\ntionsa(t)andf(t)describe the fluctuation of the DW loca-\ntion and the polarization dynamics around the pinning site\nx= 0, respectively. We are ineterested in the dynamics of\nthe polarization between two stable in plane configurations\nshown schematically in Fig. 1.\nWe use the profile of a static DW[24,25] \u00120(x) =\ncos\u00001tanh(x=\u0015). As we are interested only in strongly\npinned DWs it is sufficient to consider a constant DW\nwidth\u0015. 0(x)is chosen to impose a similar characteristic\nDW width\u0015\u001eto the variation of angle \u001ewithin the wall.\nIf we assume \u0015\u001c\u0015\u001ethen 0(x)\u00191. The Lagrangian\ndensity Eq. (1) becomes\nL'~S 0_f[cos\u00120\u00001]\u0000SV0a2\n2(4)\n\u0000S2\n2n\nJ(r\u00120)2+ sin2\u00120\u0002\nK+K?sin2\u0000\nf 0\u0001\u0003o\n:\nWe obtain dynamical equations for a(t)andf(t)by vari-\nation of the Lagrangian L=R\ndxLovera(t)andf(t).\nNoting that \u00120(x)is an odd function of xwe obtain the\ncoupled equations\n2_f=\u0000V0a; (5)\n2_a=SK?Zsin(f 0) cos(f 0) 0dx\ncosh2(x=\u0015); (6)\nwhere we have used that for \u0015\u001e\u001d\u0015\n\u0000Z\ndx 0sin\u00120\u00120\n0\u00192;and (7)\nZ\ndxsin2\u00120(r 0)2\u00190: (8)\nWe can now add damping, \u000b, and torque, T, to the\nequations of motion[17]:\n\u00002\u0012\n_f+\u000b_a\n\u0015\u0013\n=V0a+Tt, and (9)\n\u00002\u0010\n_a\u0000\u000b\u0015_f\u0011\n=SK?\u0015sin(2f) +Tl: (10)\n-1-0.500.51Mi(x=a)\n0 50 100 150 200t-0.6-0.4-0.200.2a(t)Figure 2 The DW motion for Tt=~T(t\u000020)+ ~T(t\u0000120)\nwithT0=\f= 1and\u000b= 0:1, see Eq. (14). Shown is the\nmotion of the centre of the domain wall a(t)and the DW\norientation at the centre: Mi(x=a(t);t)fori=x(blue,\nstraight),i=y(red, dashed), and i=z(black, dotted).\nTt(l)is the transverse (longitudinal) torque.\nTo find the dynamical equation for f(t)we differenti-\nate Eqs. (9) and (10) with respect to time resulting in\n2f+2\u000ba\n\u0015=\u0000V0_a\u0000_Tt; (11)\nand\na=\u000b\u0015f\u0000SK?\u0015_fcos(2f)\u0000_Tl\n2: (12)\nThen we find\n(1 +\u000b2)f\u0000\u0012\nS\u000bK?cos(2f)\u0000V0\u000b\u0015\n2\u0013\n_f\n\u0000V0SK?\u0015sin(2f)\n4\u0000\u000b_Tl\n2\u0015\u0000V0Tl\n4+_Tt\n2= 0: (13)\nThis is the equation of motion for the polarization param-\neterfof the pinned DW. Here the dynamics depends on\nthe magnitude and the time derivative of longitudinal and\ntransverse components of the spin torque.[26] In the limit-\ning case of strong pinning V0\u0015\u001dSK?and Eq. (13) could\nbe further simplified.\n3 Results We now solve Eq. (13) for particular sce-\nnarios. In the following numerical calculations we set\nSK?= 0:2and\u0015V0= 2, which satisfies the strong pin-\nning condition. Furthermore we only apply a transverse\ntorque so that Tl= 0. As there is no deformation of the\ndomain wall the exchange strength itself does not appear\nas a parameter of the dynamics. One can then define a\nGaussian shaped pulse:\n~T(t\u0000t0) =T0e\u0000(t\u0000t0)2\f; (14)\nwhich starts the DW dynamics. We are interested in how\nchanges on the torque strength and shape affect the polar-\nization dynamics of the DW. The DW is always started in\nan equilibrium position with f(t= 0) =\u0000\u0019=2.\nCopyright line will be provided by the publisherpss header will be provided by the publisher 3\n-1-0.500.51Mi(x=a)\n0 50 100 150 200\nt-0.4-0.200.2a(t)\nFigure 3 The DW motion for Tt=~T(t\u000020)+ ~T(t\u0000120)\nwithT0= 1,\f= 0:25and\u000b= 0:2, see Eq. (14). Shown\nis the motion of the centre of the domain wall a(t)and the\nDW orientation at the centre: Mi(x=a(t);t)fori=x\n(blue, straight), i=y(red, dashed), and i=z(black,\ndotted).\nSolving Eq. (13) for the case of two successive pulses\nwe find the distribution of magnetization orientation and\nthe position of DW centre as a function of time under cur-\nrent pulses of intermediate duration: Tt=~T(t\u000020) +\n~T(t\u0000120) . The results are shown in Figs. 2 and 3. Each\ncurrent pulse induces some damped oscillations of the DW\nnearx= 0. These are correlated with the rotation and\nmovement of the DW orientation. For short pulses with\nweak damping the polarization is not changed, see Fig. 2.\nIf we increase the damping then we find the polarization\nis changed by the first pulse and then changed back by the\nsecond pulse, see Fig. 3. If we then increase the length of\nthe pulse then there is no change in polarization again. Fur-\nther increasing the pulse strength will change the polariza-\ntion again.\nAs Eq. (13) depends also on the time derivative of the\ntorque, we can effectively move the DW with a smaller\ntorque if it changes very quickly. To see this we plot the\nfinal DW position for a variety of pulse strengths T0with\ndifferent\f. As\fis decreased the necessary pulse strength\nto make a change in the polarization of the DW is also\ndecreased, see the inset of Fig. 4. Shown is the angle\nf(t! 1 ), i.e. the final value of the DW polarization.\nIt should be noted that for increasing pulse strength this\nis not just a monotonically increasing function. All though\nwe find a\f-dependent minimum pulse strength which will\nrotate the DW, the final polarity of the DW changes almost\nrandomly as a function of increasing pulse strength, see\nFig. 4.\nTo relate the results of our calculations to some ma-\nterial parameters, one can choose for example K?=\n10\u00004J/m2, corresponding to Co [27], and S= 10\u000016m2\nas for a wire with a width of 10nm. This gives us the\nenergy unit E0= 10\u000020J\u001960meV . In our numerical\n0 50 100 150 200t-1-0.500.51My(t)\n0 2 4 6\nT0-2-1012f(t->∞)/πT0=6\nT0=8Figure 4 The DW motion for Tt=~T(t\u000020)with\u000b= 0:2,\nsee Eq. (14). The main figure shows My(t)for different T0\nand\f= 0:01. In the inset are results for \f= 0:01(black\ncircles),\f= 0:008(red squares), and \f= 0:006(blue\ndiamonds).\ncalculations we take SK = 0:2E0which is close to the\nparameters of Co or Fe. The other parameter, the pinning\nfield, was taken to be V0\u0015= 2E0. The estimation of the\nDW width with these parameters is \u0015\u0019p\nJ=K\u001910nm.\nThe corresponding time unit, as used in Figs. 2-4, is then\n~=E0\u001910\u000014s= 10 fs. Hence, the dynamics of interest\nhere are in the picosecond regime.\n4 Conclusion We considered the dynamics of a\nstrongly pinned DW in magnetic nanowire under short\ncurrent pulses. For this purpose we derived the equation\nof motion for the polarization parameter of the DW. This\nequation includes longitudinal and transverse components\nof the spin torque. The essential point is that the time\nderivative of the transverse torque also acts on the DW,\nwhich makes it possible to enhance the effect by using\nrapidly changing, i.e. short, pulses. Our numerical calcula-\ntions allow us to visualize the dynamics when one changes\nthe parameters of damping, pinning and anisotropy. This\nfact points to the possibility of optimal control of DW\nmotion in the spirit of Ref.[28]. For this purpose torque\npulses, generated possibly with laser-induced current\npulses, should be in the picosecond regime.\nAcknowledgements This work is supported by the Na-\ntional Science Center in Poland as a research project in years\n2011 – 2014, by the DFG contract BE 2161/5-1, and by the Grad-\nuate School of MAINZ (MATCOR).\nReferences\n[1] C. H. Marrows, Advances in Physics 54, 585 (2005).\n[2] A. Thiaville, Y . Nakatani, J. Miltat, and N. Vernier, Journal\nof Applied Physics 95, 7049 (2004).\n[3] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[4] H. B. Braun and D. Loss, Phys. Rev. B 53, 3237 (1996).\nCopyright line will be provided by the publisher4 N. Sedlmayr et al.: Dynamics of a pinned domain wall\n[5] N. Sedlmayr, V . K. Dugaev, and J. Berakdar, Phys. Rev. B\n79, 174422 (2009); N. Sedlmayr, V . K. Dugaev, and J. Be-\nrakdar, Phys. Rev. B 83, 174447 (2011).\n[6] N. Sedlmayr, V . K. Dugaev, M. Inglot, and J. Berakdar,\nphysica status solidi (RRL) 5, 450 (2011).\n[7] M. A. 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Thomas, Science 320,\n190 (2008).\n[19] L. Thomas, R. Moriya, C. Rettner, and S. S. Parkin, Sci-\nence330, 1810 (2010).\n[20] A. Vanhaverbeke, A. Bischof, and R. Allenspach, Phys.\nRev. Lett. 101, 107202 (2008).\n[21] M. Kl ¨aui, Physics 1, 17 (2008).\n[22] E. Martinez, L. Torres, and L. Lopez-Diaz, Phys. Rev. B\n83, 174444 (2011).\n[23] O. A. Tretiakov, Y . Liu, and A. Abanov, Phys. Rev. Lett.\n108, 247201 (2012).\n[24] A. Thiaville, J. M. Garcia, and J. Miltat, Journal of Mag-\nnetism and Magnetic Materials 242, 1061 (2002).\n[25] A. Thiaville and Y . Nakatani, Spin Dynamics in Confined\nMagnetic Structures III, Topics in Applied Physics V ol.\n101 (Springer-Verlag, Berlin, 2006).\n[26] L. Bocklage, et al. , Phys. Rev. Lett. 103, 134433 (2009).\n[27] M. T. Johnson, P. J. H. Bloemen, F. J. A˙ den Broeder, and\nJ. J. Vries, Rep. Prog. Phys. 59, 1409 (1996).\n[28] A. Sukhov, and J. Berakdar, Phys. Rev. Lett. 79, 197204\n(2009).\nCopyright line will be provided by the publisher"    },    {        "title": "1009.1723v1.Closed_magnetic_geodesics_on_closed_hyperbolic_Riemann_surfaces.pdf",        "content": "arXiv:1009.1723v1  [math.DG]  9 Sep 2010CLOSED MAGNETIC GEODESICS ON CLOSED\nHYPERBOLIC RIEMANN SURFACES\nMATTHIAS SCHNEIDER\nAbstract. We prove the existence of Alexandrov embedded closed\nmagneticgeodesics onclosedhyperbolicsurfaces. Closedm agneticgeodesics\ncorrespond to closed curves with prescribed geodesic curva ture.\n1.Introduction\nLet (M,g) be a compact, two dimensional, oriented manifold equipped\nwith a smooth metric gandk:M→Ra smooth positive function. We\nconsider the following two equations for curves γonM:\nDt,g˙γ=k(γ)Jg(γ)˙γ, (1.1)\nand\nDt,g˙γ=|˙γ|gk(γ)Jg(γ)˙γ, (1.2)\nwhereDt,gis the covariant derivative with respect to g, andJg(x) is the\nrotation by π/2 inTxMmeasured with gand the given orientation.\nEquation (1.1) describes the motion of a charge in a magnetic field corre-\nsponding to the magnetic form kdVgand solutions to (1.1) will be called\n(k-)magnetic geodesics (see [1,11]). Equation (1.2) corresponds to the prob-\nlem ofprescribing geodesic curvature , as its solutions γare constant speed\ncurves with geodesic curvature kg(γ,t) given by k(γ(t)) (see [15]).\nIt is easy to see that a nonconstant magnetic geodesic γlies in a fixed energy\nlevelEc, i.e. there is c >0, such that\n(γ,˙γ)∈Ec:={(x,V)∈TM:|V|g=c}.\nFor fixed kandc >0 the equations (1.1) and (1.2) are equivalent in the\nfollowing sense: If γis a nonconstant solution of (1.2) with kreplaced by\nk/c, then the curve γc(t) :=γ(ct/|˙γ|g) is ak-magnetic geodesic in Ec, and a\nk-magnetic geodesic in Ecsolves (1.2) with kreplaced by k/c. We emphazise\nthatk-magnetic geodesics on different energy levels are not repara meteriza-\ntions of each other.\nWe study the existence of closed curves with prescribed geod esic curvature\nor equivalently the existence of periodic magnetic geodesi cs on prescribed\nDate: September 9, 2010.\n2000Mathematics Subject Classification. 53C42, 37J45, 58E10.\nKey words and phrases. prescribed geodesic curvature, periodic orbits in magneti c\nfields, closed magnetic geodesics.\n12 MATTHIAS SCHNEIDER\nenergy levels Ec.\nThere is a vast literature on the existence of closed magneti c geodesics. We\nlimit ourselves to quote [12,18] for the approach via Morse- Novikov theory\nfor (possibly multi-valued) variational functionals, [1, 5] for the application\nof the theory of dynamical systems and symplectic geometry, [4] concerning\nAubry-Mather’s theory, and [15], wherethe theory of vector fields on infinite\ndimensional manifolds is applied to (1.2). We refer to [4,6, 7,19] for a survey\nand additional references.\nFrom the example of the horocycle flow below, closed magnetic geodesics\nneed not exist on a fixed energy level in general. However, fro m [6,17,18],\nthere are always closed magnetic geodesics in high and low en ergy levels, i.e.\ninEcwithc≥c0andc≤(c0)−1, wherec0>0 depends on ( M,g) andk(in\ncase of a flat torus and high energy levels kis assumed not to vanish). If the\nmagnetic form is exact, i.e. [ kdVg] = 0 inH2\ndR(M), then there is a periodic\nmagnetic geodesic in every energy level (see [4]). Concerni ngnon exact mag-\nnetic forms positive functions kare of special interest, since the magnetic\nform is symplectic in this case. For k >0 a closed magnetic geodesic exists\nin every energy level, if ( M,g) is a flat torus [2,10] or if ( M,g) is a sphere S2\nwith nonnegative curvature [16]. The (essentially) only no nexistence result\nfor closed magnetic geodesics is based on an old result of Hed lund [8].\nExample (Horocycle flow [6]) .Let(M,g)be a compact hyperbolic surface\nof constant curvature Kg≡ −1andk≡1.\n(1)If0< c <1, thenEccontains a contractible closed magnetic geo-\ndesic.\n(2)There are no closed magnetic geodesics in E1.\n(3)Ifc >1, there are no contractible closed magnetic geodesics in Ec,\nbut any non trivial free homotopy class of closed curves can b e rep-\nresented by one.\nThe existence question for closed magnetic geodesics on hyp erbolic sur-\nfaces for non constant functions kis poorly understood. We shall show: If\n(M,g) is a compact hyperbolic surface with Gaussian curvature Kg≥ −1\nandk≥1 a positive function, then there is a contractible closed ma gnetic\ngeodesics in Ecfor all 0 < c <1. The example of the horocycle flow shows\nthat this existence result is sharp.\nWe consider curves, that are Alexandrov embedded.\nDefinition 1.1. (oriented Alexandrov embedded) Let B⊂R2denote the\nopen ball of radius 1centered at 0∈R2. An immersion γ∈C1(∂B,M)\nwill be called oriented Alexandrov embedded , if there is an immersion F∈\nC1(B,M), such that F|∂B=γandFis orientation preserving in the sense\nthat\n/a\\}b∇acketle{tDF|xx,Jg(γ(x))˙γ(x)/a\\}b∇acket∇i}htTγ(x)S2,g>0\nfor allx∈∂B.CLOSED MAGNETIC GEODESICS 3\nWe shall prove\nTheorem 1.2. Let(M,g)be a smooth, compact, orientable surface with\nnegative Euler characteristic and k∈C∞(M)a positive function. Assume\nthere isK0>0such that kand the Gaussian curvature Kgof(M,g)satisfy\nk >(K0)1\n2andKg≥ −K0.\nThen there is an oriented Alexandrov embedded curve γ∈C2(S1,M)that\nsolves(1.2)and the number of such solutions is at least −χ(M)provided\nthey are all nondegenerate.\nThe equivalence between (1.1) and (1.2) leads to\nCorollary 1.3. Let(M,g)be a smooth, compact, orientable surface with\nnegative Euler characteristic and k∈C∞(M)a positive function. Assume\nthere isK0>0such that kand the Gaussian curvature Kgof(M,g)satisfy\nk≥(K0)1\n2andKg≥ −K0.\nThen every energy level Ecwith0< c <1contains an oriented Alexandrov\nembedded closed magnetic geodesic and the number of such clo sed magnetic\ngeodesics in Ecis at least −χ(M)provided they are all nondegenerate.\nThe proof of our existence results is organized as follows. W e consider\nsolutions to (1.2) as zeros of the vector field Xk,gdefined on the Sobolev\nspaceH2,2(S1,M): Forγ∈H2,2(S1,M) we letXk,g(γ) be the unique weak\nsolution of/parenleftbig\n−D2\nt,g+1/parenrightbig\nXk,g(γ) =−Dt,g˙γ+|˙γ|gk(γ)Jg(γ)˙γ (1.3)\ninTγH2,2(S1,M). The uniqueness implies that any zero of Xk,gis a weak\nsolution of (1.2) which is aclassical solution in C2(S1,M) applyingstandard\nregularity theory.\nAfter setting up notation in Section 2 and introducing the cl asses of maps\nand spaces needed for our analysis we recall in Section 3 the d efinition and\nproperties of the S1-equivariant Poincar´ e-Hopf index defined in [15],\nχS1(Xk,g,MA)∈Z,\nwhereMAis the set of oriented Alexandrov embedded regular curves in\nH2,2(S1,M).\nFrom theuniformization theorem ( M,g) is isometric to( H/Γ,eϕg0), whereΓ\nis a group of isometries of the standard hyperbolicplane ( H,g0) acting freely\nand properly discontinuously and ϕis a function in C∞(H/Γ,R). Since the\nproblem of prescribing geodesic curvature is invariant und er isometries we\nmay assume without loss of generality that\n(M,g) = (H/Γ,eϕg0).\nInSection 4we analyze the unperturbed problem withk≡k0>0andg=g0:\nWe compute the set of oriented Alexandrov embedded zeros of Xk0,g0and\nthe image and kernel of the corresponding linearizations. T he perturbative4 MATTHIAS SCHNEIDER\nanalysis in Section 5, which carries over from [15], is used t o compute the\ndegree of the unperturbedproblem in Section 6: For large pos itive constants\nk0and the standard metric g0we shall show that\nχS1(Xk0,g0,MA) =−χ(M),\nwhereχ(M) denotes the Euler characteristic of M.\nSection 7 contains the apriori estimate whichimplies that u ndertheassump-\ntions of Theorem 1.2 the set of solutions to (1.2) is compact i nMA. The\nhomotopy invariance of the S1-equivariant Poincar´ e-Hopf index then leads\nto the identity\nχS1(Xk,g,MA) =χS1(Xk0,gcan,MA) =−χ(M).\nThe resulting proof of Theorem 1.2 is given in Section 8.\n2.Preliminaries\nIt is convenient for the functional analytic setting to assu me that Mis\nembeddedin some RqM. We consider for m∈N0theset of Sobolev functions\nHm,2(S1,M) :={γ∈Hm,2(S1,RqM) :γ(t)∈Mfor a.e.t∈S1.}\nForm≥1 the set Hm,2(S1,M) is a sub-manifold of the Hilbert space\nHm,2(S1,RqM) and is contained in Cm−1(S1,RqM). Hence, if m≥1 then\nγ∈Hm,2(S1,M) satisfies γ(t)∈Mfor allt∈S1. In this case the tangent\nspace at γ∈Hm,2(S1,M) is given by\nTγHm,2(S1,M) :={V∈Hm,2(S1,RqM) :V(t)∈Tγ(t)Mfor allt∈S1}.\nForm= 0 the set H0,2(S1,M) =L2(S1,M) fails to be a manifold. We\ndefine for γ∈H1,2(S1,M) the space TγL2(S1,M) by\nTγL2(S1,M) :={V∈L2(S1,RqM) :V(t)∈Tγ(t)Mfor a.e.t∈S1}.\nA metric gonMinduces a metric on Hm,2(S1,M) form≥1 by setting for\nγ∈Hm,2(S1,M) andV, W∈TγHm,2(S1,S2)\n/a\\}b∇acketle{tW,V/a\\}b∇acket∇i}htTγHm,2(S1,S2),g:=/integraldisplay\nS1/angbracketleftBig/parenleftbig\n(−1)/rightangleswm\n2/rightanglese(Dt,g)m+1/parenrightbig\nV(t),\n/parenleftbig\n(−1)/rightangleswm\n2/rightanglese(Dt,g)m+1/parenrightbig\nW(t)/angbracketrightBig\nγ(t),gdt,\nwhere/rightangleswm/2/rightanglesedenotes the largest integer that does not exceed m/2.\nSincegandkare smooth, Xk,gis a smooth vector field (see [15,20, Sec. 6])\non the set H2,2\nreg(S1,M) of regular curves,\nH2,2\nreg(S1,M) :={γ∈H2,2(S1,M) : ˙γ(t)/\\e}atio\\slash= 0 for all t∈S1}.CLOSED MAGNETIC GEODESICS 5\nFrom [15] there holds\n/parenleftbig\n−D2\nt,g+1/parenrightbig\nDgXk,g|γ(V)\n=−D2\nt,gV−Rg/parenleftbig\nV,˙γ/parenrightbig\n˙γ+|˙γ|−1\ng/a\\}b∇acketle{tDt,gV,˙γ/a\\}b∇acket∇i}htgk(γ)Jg(γ)˙γ\n+|˙γ|g/parenleftbig\nk′(γ)V/parenrightbig\nJg(γ)˙γ+|˙γ|gk(γ)/parenleftBig/parenleftbig\nDgJg|γV/parenrightbig\n˙γ+Jg(γ)Dt,gV/parenrightBig\n.(2.1)\nWe note that (see also [21, Thm. 6.1])\n/parenleftbig\n−D2\nt,g+1/parenrightbig\nDgXk,g|γ(V) = (−D2\nt,g+1)V+T(V),\nwhereTis a linear map from TγH2,2(S1,M) toTγL2(S1,M) that depends\nonlyonthefirstderivativesof Vandisthereforecompact. Takingtheinverse\n(−D2\nt,g+1)−1we deduce that DgXk,g|γis the form identity+compact and\nthus a Rothe map (see [15]).\nThe vector field Xk,gas well as the set of solutions to (1.2) is invariant\nunder a circle action: For θ∈S1=R/Zandγ∈H2,2(S1,M) we define\nθ∗γ∈H2,2(S1,M) by\nθ∗γ(t) =γ(t+θ).\nMoreover, for V∈TγH2,2(S1,M) we let\nθ∗V:=V(·+θ)∈Tθ∗γH2,2(S1,M).\nThenXk,g(θ∗γ) =θ∗Xk,g(γ) for any γ∈H2,2(S1,M) andθ∈S1. Thus,\nany zero gives rise to a S1-orbit of zeros. We call γaprimecurve, if the\nisotropy group {θ∈S1:θ∗γ=γ}ofγis trivial.\nForm≥1 the exponential map Expg:THm,2(S1,M)→Hm,2(S1,M) is\ndefined for γ∈Hm,2(S1,M) andV∈TγHm,2(S1,M) by\nExpγ,g(V)(t) :=Expγ(t),g(V(t)),\nwhereExpz,gdenotes the exponential map on ( M,g) atz∈M. Due to its\npointwise definition\nθ∗Expγ,g(V)(t) =Expθ∗γ,g(θ∗V)(t).\nWe shall findsolutions to (1.2) in the class of oriented Alexa ndrov embedded\ncurves. Let γ∈H2,2(S1,M) be an oriented Alexandrov embedded curves\nwith corresponding oriented immersion FfromBtoM. If we equip Bwith\nthe metric F∗ginduced by F, then the outer normal NB(x) atx∈∂Bwith\nrespect to F∗gsatisfies\nDF|xNB(x) =Nγ(x)\nwhereNγ(x) denotes the normal to the curve γatx∈∂Bdefined by\nNγ(x) :=|˙γ(x)|−1Jg(γ(x))˙γ(x).\nIn [16] the following two basic properties of oriented Alexa ndrov embedded\ncurves are shown.\nLemma 2.1.6 MATTHIAS SCHNEIDER\n(1)Let(γn)inC2(∂B,M)be a sequence of immersions, which are ori-\nented Alexandrov embedded, such that (γn)converges to an immer-\nsionγ0inC2(∂B,M)with strictly positive geodesic curvature. Then\nγ0is oriented Alexandrov embedded.\n(2)The set of regular, oriented Alexandrov embedded curves is op en in\nH2,2(S1,M).\nProperty (1) and (2) are given in [16] for closed curves in S2. Since the\nanalysis in the proof of (1) and (2) is done in tubular neighbo rhoods of\nclosed curves, properties (1) and (2) continue to hold if S2is replaced by a\ngeneral surface M.\n3.TheS1-Poincar ´e-Hopf index\nIn [15] a S1-equivariant Poincar´ e-Hopf index or S1-degree is introduced\nfor equivariant vector fields on subsets of H2,2(S1,S2). TheS1-degree is\nbased on an equivariant version of the Sard-Smale lemma [15, Lem 3.9],\nwhich depends on an appropriate change of a vector field local ly around its\ncritical orbits. It’s merely a matter of form to extend this l ocal argument,\nwhenS2is replaced by a general surface M. We give a short account of\nthe definition and properties of the S1-degree for equivariant vector fields\non subsets of H2,2(S1,M).\nWe define a C2equivariant vector field WgonH2,2(S1,M) by\nWg(γ) = (−(Dt,g)2+1)−1˙γ,forγ∈H2,2(S1,M).\nWe will compute the S1-Poincar´ e-Hopf index for thefollowing class of vector\nfields.\nDefinition 3.1. LetMbe an open S1-invariant subset of prime curves in\nH2,2(S1,M). AC2vector field XonMis called (M,g,S1)-admissible, if\n(1)XisS1-equivariant, i.e. X(θ∗γ) =θ∗X(γ)for all(θ,γ)∈S1×M.\n(2)Xis proper in M, i.e. the set {γ∈M:X(γ) = 0}is compact,\n(3)Xis orthogonal to Wg, i.e./a\\}b∇acketle{tX(γ),Wg(γ)/a\\}b∇acket∇i}htTγH2,2(S1,M)= 0for all\nγ∈M.\n(4)Xis a Rothe field, i.e. if X(S1∗γ) = 0thenDgX|γand Proj/an}bracketle{tWg(γ)/an}bracketri}ht⊥◦\nDgX|γare Rothe maps in L(TγH2,2(S1,M))andL(/a\\}b∇acketle{tWg(γ)/a\\}b∇acket∇i}ht⊥), re-\nspectively.\n(5)Xis elliptic, i.e. there is ε >0such that for all finite sets of charts\n{(Expγi,g,B2δi(0)) :γi∈H4,2(S1,M)for1≤i≤n},\nand finite sets\n{Wi∈TγiH4,2(S1,M) :/ba∇dblWi/ba∇dblTγiH4,2(S1,M)< εfor1≤i≤n},CLOSED MAGNETIC GEODESICS 7\nthere holds: If α∈n∩\ni=1Expγi,g(Bδi(0))⊂H2,2(S1,M)satisfies\nX(α) =n/summationdisplay\ni=1Proj/an}bracketle{tWg(α)/an}bracketri}ht⊥◦DExpγi,g|Exp−1\nγi,g(α)(Wi)\nthenαis inH4,2(S1,M).\nIt is shown in [15] that Xk,gsatisfies properties (3) −(4). Hence, Xk,g\nis (M,g,S1)-admissible if and only if Xk,gis proper in M. Note that the\nregularity property (5), taking Wi= 0, shows that any zero of Xbelongs\ntoH4,2(S1,M). Furthermore, for γ∈H4,2(S1,M) the map θ/ma√sto→θ∗γisC2\nfromS1toH2,2(S1,M). Hence, if X(γ) = 0 then\n0 =Dθ(X(θ∗γ))|θ=0=DgX|γ(˙γ),\nsuch that the kernel of DgX|γis nontrivial. If Xis a vector field orthogonal\ntoWgandX(γ) = 0, then\n0 =D/parenleftbig\n/a\\}b∇acketle{tX(α),Wg(α)/a\\}b∇acket∇i}htTαH2,2(S1,M),g/parenrightbig\n|γ=/a\\}b∇acketle{tDgX|γ,Wg(γ)/a\\}b∇acket∇i}htTγH2,2(S1,M),g\nwhere the various curvature terms and terms containing deri vatives of Wg\nvanish as X(γ) = 0. Thus, X(γ) = 0 implies\nDgX|γ:TγH2,2(S1,M)→ /a\\}b∇acketle{tWg(γ)/a\\}b∇acket∇i}ht⊥, (3.1)\nand the projection Proj/an}bracketle{tWg(γ)/an}bracketri}ht⊥in (4) is unnecessary.\nDefinition 3.2. LetMbe an open S1-invariant subset of prime curves in\nH2,2(S1,M),S1∗γ⊂M, andXa(M,g,S1)-admissible vector field on M.\nThe orbit S1∗γis called a critical orbit ofX, ifX(γ) = 0.\nThe orbit S1∗γis called a nondegenerate critical orbit ofX, ifX(γ) = 0\nand\nDgX|γ:/a\\}b∇acketle{tWg(γ)/a\\}b∇acket∇i}ht⊥− → /a\\}b∇acketle{tWg(γ)/a\\}b∇acket∇i}ht⊥\nis an isomorphism.\nNote that if γ∈H4,2(S1,M)⊂H2,2(S1,M) then ˙γ/\\e}atio\\slash∈ /a\\}b∇acketle{tWg(γ)/a\\}b∇acket∇i}ht⊥.\nDefinition 3.3. Let{gt:t∈[0,1]}be a family of smooth metrics on\nM, which induces a corresponding family of metrics on H2,2(S1,M), still\ndenoted by gt. LetMbe an open S1-invariant subset of prime curves in\nH2,2(S1,M)andX0,X1two vector-fields on Msuch that Xiis(M,gi,S1)-\nadmissible for i= 0,1. AC2family of vector-fields X(t,·)onMfort∈[0,1]\nis called a (M,gt,S1)-homotopy between X0andX1, if\n•X(0,·) =X0andX(1,·) =X1,\n• {(t,γ)∈[0,1]×M:X(t,γ) = 0}is compact,\n•Xt:=X(t,·)is(M,gt,S1)-admissible for all t∈[0,1].\nWe write (M,g,S1)-homotopy, if the family of metrics {gt}is constant.8 MATTHIAS SCHNEIDER\nNote that, if {kt∈C∞(M,R) :t∈[0,1]}is aC2family of smooth\nfunction, then t/ma√sto→Xkt,gtis a (M,gt,S1)-homotopy, if and only if the set\n{(t,γ)∈[0,1]×M:Xkt,gt(γ) = 0}\nis compact.\nWe letMbe an open S1-invariant subset of prime curves in H2,2(S1,M)\nandXa (M,g,S1)-admissible vector field on M. The local S1-degree of an\nisolated, nondegenerate critical orbit S1∗γ0is defined by\ndegloc,S1(X,S1∗γ0) := sgnDgX|γ0,\nwhere sgn DgX|γ0is the sign of the Rothe map DgX|γ0inL(/a\\}b∇acketle{tWg(γ)/a\\}b∇acket∇i}ht⊥).\nSinceDgXk,g|γ0is of the form identity +compact, in the above situation\nsgnDgXk,g|γ0is given by the usual Leray-Schauder degree.\nUsing an equivariant version of the Sard-Smale lemma a S1-equivariant\nPoincar´ e-Hopf index\nχ(X,M)∈Z\nis defined in [15] with the following properties.\nLemma 3.4.\n(1)IfXis(M,g,S1)-admissible with only finitely many critical orbits,\nthat are all nondegenerate, then\nχS1(X,M) :=/summationdisplay\n{S1∗γ⊂M:X(S1∗γ)=0}degloc,S1(X,S1∗γ).\n(2)IfX0andX1are(M,gt,S1)-homotop, then χ(X0,M) =χ(X1,M).\n4.The Unperturbed Problem\nLetH⊂R3be the standard hyperbolic plane\nH:={(ξ1,ξ2,τ)∈R3:τ2−|ξ|2= 1 and τ >0}\nwith metric g0induced by the Minkowski metric gm,\ngm:= (dξ1)2+(dξ2)2−(dτ)2=/a\\}b∇acketle{t·,·/a\\}b∇acket∇i}htm.\nWe choose the orientation on Hsuch that Jg0(y) is given for y∈Hby\nJg0(y)(v) :=y×mvfor allv∈TyH,\nwhere×mdenotes the twisted cross product in R3,\n\nv1\nv2\nv3\n×m\nw1\nw2\nw3\n:=\nv3w2−v2w3\nv1w3−v3w1\nv1w2−v2w1\n.CLOSED MAGNETIC GEODESICS 9\nThe twisted cross product ×mis related to the usual cross product ×inR3\nbyv×mw=I2,1v×I2,1w, whereI2,1is given by\nI2,1:=\n1 0 0\n0 1 0\n0 0−1\n,\nand satisfies for a, b, c, d ∈R3\n/a\\}b∇acketle{t(a×mb),a/a\\}b∇acket∇i}htm= 0 =/a\\}b∇acketle{t(a×mb),b/a\\}b∇acket∇i}htm,\na×m(b×mc) =−b/a\\}b∇acketle{ta,c/a\\}b∇acket∇i}htm+c/a\\}b∇acketle{ta,b/a\\}b∇acket∇i}htm,\n/a\\}b∇acketle{t(a×mb),(c×md)/a\\}b∇acket∇i}htm=−/a\\}b∇acketle{ta,c/a\\}b∇acket∇i}htm/a\\}b∇acketle{tb,d/a\\}b∇acket∇i}htm+/a\\}b∇acketle{tb,c/a\\}b∇acket∇i}htm/a\\}b∇acketle{ta,d/a\\}b∇acket∇i}htm\nWe fix a compact, orientable Riemannian surface ( M,g0),\nM:=H/Γ,\nwhere Γ ⊂SO(2,1)+is a group of oriented isometries acting freely and\nproperly discontinuously on H. Concerning the metric we will be sloppy\nand denote by g0the metric on Has well as the induced metric on H/Γ.\nThe unperturbed problem on Mis given by\nDt,g0˙γ=|˙γ|g0k0Jg0(γ)˙γ, (4.1)\nwherek0is a positive constant.\nWe shall compute the S1-degree of the unperturbed equation (4.1) in three\nsteps.Step 1:We compute explicitly the set ZMof Alexandrov embedded\nsolutions in H2,2(S1,M) to (4.1) and show that ZMis a finite dimensional,\nnondegenerate manifold, in the sense that we have for all ˜ α∈ ZM\nT˜αZM= kernel( Dg0Xk0,g0|˜α),\nT˜αH2,2(S1,M) =T˜αZM⊕R(Dg0Xk0,g0|˜α).\nStep 2:In Section 5 we perform a finite dimensional reduction of a sli ghtly\nperturbed problem: We consider for k1∈C∞(M,R), which will be chosen\nlater, and ε∈R, which is assumed to be very small, the perturbed vector\nfieldXg0,εdefined by\nXg0,ε(γ) := (−D2\nt,g0+1)−1/parenleftbig\n−Dt,g0˙γ+|˙γ|g0(k0+εk1(γ))Jg0(γ)˙γ/parenrightbig\n=Xk0,g0(γ)+εK1(γ),\nwhere the vector field K1is given by\nK1(γ) := (−D2\nt,g0+1)−1|˙γ|g0/parenleftbig\nk1(γ)Jg0(γ)˙γ/parenrightbig\n.\nWe show that if S1∗˜α0⊂ ZMis a nondegenerate critical orbit of the vector\nfield ˜α/ma√sto→P1(˜α)◦K1(˜α) onZM, whereP1(˜α) is a projection onto T˜αZM\ndefined below, then for any 0 < ε << 1 there is a unique nondegenerate\ncritical orbit S1∗˜γ(ε) ofXg0,εsuch that ˜ γ(ε) converges to ˜ α0asε→0+and\ndegloc,S1(Xg0,ε,S1∗γ(ε)) =−degloc(P1(·)◦K1(·),S1∗˜α0).10 MATTHIAS SCHNEIDER\nStep 3:InSection 6we choose aMorsefunction k1∈C∞(M,R) withcritical\npoints\n{˜wi∈M: 1≤i≤n}.\nWe show that if k0>>1 is large, then P1(·)◦K1(·) has exactly ncritical\norbits{S1∗˜αi,k0: 1≤i≤n}such that for 1 ≤i≤n\ndegloc(P1(·)◦K1(·),S1∗˜αi,k0) = degloc(∇k1,˜wi).\nThis yields the formula χS1(Xk0,g0,MA) =−χ(M), where MAis the subset\nofH2,2(S1,M) consisting of Alexandrov embedded, regular curves.\nStep 1:The prescribed geodesic curvature equation with k≡k0on (H,g0)\nis given by\nProjγ⊥,gm¨γ=|˙γ|mk0γ×m˙γ, (4.2)\nwhereγ∈H2,2(S1,H), ˙γand ¨γare the usual derivatives of γconsidered as\na curve in R3,|˙γ|mis the Minkowski norm of ˙ γin (R3,gm).\nIfk0>1 then there is a unique r=r(k0)>0 such that\nk0=√\n1+r2\nr.\nWe call a triple of vectors {v0,v1,w}inR3a positive oriented orthonormal\nsystem with respect to gm, if\n/a\\}b∇acketle{tv0,v1/a\\}b∇acket∇i}htm=/a\\}b∇acketle{tv0,w/a\\}b∇acket∇i}htm=/a\\}b∇acketle{tv1,w/a\\}b∇acket∇i}htm= 0,\n/a\\}b∇acketle{tv0,v0/a\\}b∇acket∇i}htm=/a\\}b∇acketle{tv1,v1/a\\}b∇acket∇i}htm=−/a\\}b∇acketle{tw,w/a\\}b∇acket∇i}htm= 1,\nv0×mv1=w.\nWe define for λ >0 and a positive oriented orthonormal system {v0,v1,w}\nthe function α∈C∞(R,H) by\nα(t,λ,v0,v1,w) :=/radicalbig\n1+r2w+rcos(λr−1t)v1+rsin(λr−1t)v0(4.3)\nAdirectcalculation showsthat α(·,λ,v0,v1,w)solves (4.2). Wefix( γ0,˜v0)∈\nTHwith ˜v0/\\e}atio\\slash= 0 and define the parameter λ:=|˜v0|mand the positive\noriented orthonormal system ( v0,v1,w) by\nv0:=λ−1˜v0, v1:=−rγ0−/radicalbig\n1+r2(v0×mγ0), w:=v0×mv1.\nThenα(·,λ,v0,v1,w) satisfies the initial conditions\nα(0,λ,v0,v1,w) =γ0,˙α(0,λ,v0,v1,w) = ˜v0,\nand we deduce that all non constant solutions of (4.2) are obt ained in this\nway. Since we are only interested in solutions in H2,2(S1,H) we get an extra\ncondition on λ, i.e. the 1-periodicity leads to\nλ∈2πNr.CLOSED MAGNETIC GEODESICS 11\nLemma 4.1. The oriented Alexandrov embedded solutions in H2,2(S1,H)\nof equation (4.2) are given by the set of simple solutions\nZH:=/braceleftbig\nα(·,2πr,v0,v1,w) :\n{v0,v1,w}is a pos. orth. system in (R3,gm)/bracerightbig\n.\nProof.From the analysis above the periodic solutions to (4.2) are g iven by\n/braceleftbig\nα(·,2πnr,v0,v1,w) :n∈Nand\n{v0,v1,w}is a pos. orth. system in ( R3,gm)/bracerightbig\n.\nWe fixn∈Nand a positive orthonormal system {v0,v1,w}and write\nγn:=α(·,2πnr,v0,v1,w).\nAssumeγnisorientedAlexandrovembeddedandlet Fnbethecorresponding\nimmersion. Since Hisdiffeomorphicto R2wemay assumethat γ1isasimple\ncurve in the plane ( R2,δ) with standard metric δ. If we apply the Gauß-\nBonnet formula to ( B,F∗\nnδ) and the embedded curve γ1in the plane, we\nobtain\n2π=/integraldisplay\n∂BkF∗nδdSF∗nδ+/integraldisplay\nBKF∗nδdAF∗nδ\n=/integraldisplay\nγnkδdSδ=n/integraldisplay\nγ1kδdSδ=n2π,\nwhich is only possible for n= 1.\nThe curve γ1is oriented Alexandrov embedded using polar coordinates an d\n[0,2π]×[0,1]∋(t,s)/ma√sto→/radicalbig\n1+s2r2w+srcos(t)v1+srsin(t)v0.\n/square\nThe Lorentz transformations S0(2,1)+of (R3,gm),\nSO(2,1)+:={A∈O(2,1) :A(H)⊂Hand detA= 1},\ncorrespond to the oriented isometries of ( H,g0) and act on solutions: if γ\nsolves (4.2) so does A◦γfor anyA∈SO(2,1)+. We have\nA◦α(·,λ,v0,v1,w) =α(·,λ,A(v0),A(v1),A(w)).\nMoreover, there holds,\nα(·,2πr,v0,v1,w) =θ∗α(·,2πr,v′\n0,v′\n1,w′) (4.4)\nfor some θ∈S1if and only if w=w′. Consequently, the critical orbits of\n(4.2) inH,{S1∗γ:γ∈ ZH},are parametrized by w∈Hand correspond\nto “circles” with radius raround the center winH.\nWe letπMbe the natural projection, πM:H→H/Γ. Any point z∈H\nadmits a neighborhood U=Bδ(z) such that πM|U:U→πM(U) is an\nisometry. From (4.3) there is Ck0>1 such that if k0≥Ck0then any\nsolution to (4.2) on Hpassing through zremains in U. ForMis compact\nCk0=Ck0(Γ) andδ >0may bechosen independentlyof z. Equation (1.2) is12 MATTHIAS SCHNEIDER\ninvariant under isometries, hence the set of solutions to (4 .1) with k0≥Ck0\nis given by\n/braceleftbig\nπM◦α(·,2πr,v0,v1,w) :\n{v0,v1,w}is a pos. orth. system in ( R3,gm)/bracerightbig\n.\nMoreover, we have\nLemma 4.2. Ifk0≥Ck0, then the oriented Alexandrov embedded solutions\ninH2,2(S1,M)of equation (4.1)are given by the set of simple solutions\nZM:=/braceleftbig\n˜α=πM◦α(·,2πr,v0,v1,w) :\n{v0,v1,w}is a pos. orth. system in (R3,gm)/bracerightbig\n.\nProof.We fixn∈Nand a positive orthonormal system {v0,v1,w}and write\nγn:=πM◦α(·,2πnr,v0,v1,w)\nFrom the above analysis any periodic solution to (4.1) on ( M,g0) is of this\nform. Hence, it is enough to show that γnis oriented Alexandrov embedded,\nif and only if n= 1.\nConcatenating the immersion in the proof of Lemma 4.1 with πMwe deduce\nthatγ1is oriented Alexandrov embedded. Suppose γnis oriented Alexan-\ndrov embedded with an immersion Fn:B→M. From the homotopy\nlifting property of the covering πM:H→Mwe may lift Fnto see that\nα(·,2πnr,v0,v1,w) is oriented Alexandrov embedded in H. From Lemma\n4.1 this is only possible for n= 1. /square\nFrom (4.4) we find\nπM◦α(·,2πr,v0,v1,w) =θ∗πM◦α(·,2πr,v′\n0,v′\n1,w′)\nfor some θ∈S1if and only if πM(w) =πM(w′), such that the critical orbits\nof (4.2) in Mare parametrized by w∈Mand correspond to projections on\nMof “circles” in H.\nIn the following we always assume that\nk0≥Ck0.\nWe denote by Xk0,g0,Hthe vector fieldon H2,2(S1,H) correspondingto equa-\ntion (4.2). We fix a solution α=α(·,2πr,v0,v1,w) of (4.2) and note that\nforV∈TαH2,2(S1,H)\nRg0(V,˙α)˙α=−V|˙α|2\nm+/a\\}b∇acketle{tV,˙α/a\\}b∇acket∇i}htm˙α.\nBy (2.1) a vector field Wis contained in the kernel of Dg0Xk0,g0,H|αif and\nonly ifWis a periodic solution of\n0 =−D2\nt,g0W+W|˙α|2\nm−/a\\}b∇acketle{tW,˙α/a\\}b∇acket∇i}htm˙α\n+|˙α|−1\nm/a\\}b∇acketle{tDt,g0W,˙α/a\\}b∇acket∇i}htmk0(α×m˙α)+|˙α|mk0(α×mDt,g0W).(4.5)CLOSED MAGNETIC GEODESICS 13\nDue to the geometric origin of equation (4.2) and the SO(2,1)+invariance\nwe find that\nW0(t,v0,v1,w) :=t˙α, (4.6)\nW1(t,v0,v1,w) := ˙α= 2πr(−sin(2πt)v1+cos(2πt)v0),\nW2(t,v0,v1,w) := (1+ r2)1\n2v1+rcos(2πt)w,\nW3(t,v0,v1,w) := (1+ r2)1\n2v0+rsin(2πt)w,\nsolve (4.5). In the sequel, we will omit the dependence of Wion (v0,v1,w),\nif there is no possibility of confusion. The initial values o fW0,...,W 3\nW0(0,v0,v1,w) = 0, Dt,g0W0(0,v0,v1,w) = 2πrv0,\nW1(0,v0,v1,w) = 2πrv0, Dt,g0W1(0,v0,v1,w) =−4π2r3k0(k0v1+w),\nW2(0,v0,v1,w) =rk0v1+rw, Dt,g0W2(0,v0,v1,w) = 0,\nW3(0,v0,v1,w) =rk0v0, Dt,g0W3(0,v0,v1,w) =−2πr3(k0v1+w).\nare a basis of/parenleftbig\nTα(0)H/parenrightbig2, such that any solution to (4.5) is a linear combina-\ntion ofW0,...,W 3. As only W1,...,W 3are periodic, we obtain\nkernel(Dg0Xk0,g0,H|α) =/a\\}b∇acketle{tW1, W2, W3/a\\}b∇acket∇i}ht. (4.7)\nWe fix a neighborhood Uofα(0) as above, where πM:U→πM(U) is an\nisometry. Then α∈H2,2(S1,U) andπMinduces isomorphisms\nπM:H2,2(S1,U)→H2,2(S1,πM(U)), α/ma√sto→πM◦α,\n(πM)∗:TαH2,2(S1,H)→TπM◦αH2,2(S1,M), V/ma√sto→dπM|αV,\nwhere (πM)∗is an isometry. Moreover, there holds on H2,2(S1,U)\n(πM)∗◦Xg0,k0,H=Xg0,k0◦πM,\n(πM)∗◦Dg0Xg0,k0,H|α=Dg0Xg0,k0|πM◦α◦(πM)∗. (4.8)\nSinceZHandZMare three dimensional submanifolds of H2,2(S1,H) and\nH2,2(S1,M), respectively, we have for α∈ ZHand ˜α=πM◦α∈ ZM\nTαZH= kernel( Dg0Xk0,g0,H|α) =/a\\}b∇acketle{tW1, W2, W3/a\\}b∇acket∇i}ht,\nT˜αZM= kernel( Dg0Xk0,g0|˜α)\n=/a\\}b∇acketle{t˜Wi:= (πM)∗◦Wi: 1≤i≤3/a\\}b∇acket∇i}ht.\nTo compute the image of Dg0Xk0,g0,H|αwe note that {˙α,α×m˙α}is an\northogonal system in TαHfor anyt∈S1. Thus any V∈TαH2,2(S1,H) may\nbe written as\nV=λ1˙α+λ2(α×m˙α)\nfor some functions λ1, λ2∈H2,2(S1,R). Using the fact that\nDt,g0˙α=|˙α|mk0(α×m˙α) andDt,g0(α×m˙α) =−|˙α|mk0˙α,14 MATTHIAS SCHNEIDER\nwe obtain\nDg0Xk0,g0,H|α(V) = (−D2\nt,g0+1)−1/parenleftbig\n(−λ′′\n1+2π/radicalbig\n1+r2λ′\n2)˙α\n+(−λ′′\n2−(2π)2λ2)(α×m˙α)/parenrightbig\n. (4.9)\nConcerning W1,...,W 3andWg0we find\nW1(t) = ˙α(t),\nW2(t) =−1\n2πr/parenleftbig/radicalbig\n1+r2sin(2πt)˙α(t)+cos(2 πt)(α×m˙α)/parenrightbig\n,\nW3(t) =−1\n2πr/parenleftbig\n−/radicalbig\n1+r2cos(2πt)˙α(t)+sin(2πt)(α×m˙α)/parenrightbig\nWg0(α) = (1+ |˙α|2\nmk2\n0)−1˙α= (1+4π2(1+r2))−1W1. (4.10)\nLemma 4.3. Ifr/\\e}atio\\slash= (2π)−1, then we have for α∈ ZH\n{0}=/a\\}b∇acketle{tW1,W2,W3/a\\}b∇acket∇i}ht∩R/parenleftbig\nDg0Xk0,g0,H|α/parenrightbig\n,\n/a\\}b∇acketle{tW1/a\\}b∇acket∇i}ht⊥=/a\\}b∇acketle{tW2,W3/a\\}b∇acket∇i}ht⊕R/parenleftbig\nDg0Xk0,g0,H|α/parenrightbig\nProof.Forλ1,λ2∈H2,2(S1,R) we have\n(−D2\nt,g0+1)/parenleftbig\nλ1˙α+λ2(α×m˙α)/parenrightbig\n=/parenleftbig\n−λ′′\n1+4π/radicalbig\n1+r2λ′\n2+(4π2(1+r2)+1)λ1/parenrightbig\n˙α\n+/parenleftbig\n−λ′′\n2−4π/radicalbig\n1+r2λ′\n1+(4π2(1+r2)+1)λ2/parenrightbig\nα×m˙α\nHence we get by direct calculations\n(−D2\nt,g0+1)(W1) = (4π2(1+r2)+1)˙α,\n(−D2\nt,g0+1)(−2πrW2) =/radicalbig\n1+r2(4π2r2+1)sin(2 πt)˙α\n+(−4π2r2+1)cos(2 πt)(α×m˙α),(4.11)\n(−D2\nt,g0+1)(−2πrW3) =−/radicalbig\n1+r2(4π2r2+1)cos(2 πt)˙α\n+(−4π2r2+1)sin(2 πt)(α×m˙α).(4.12)\nConsequently, by(3.1), (4.10), andtheabove computations W1isorthogonal\nto/a\\}b∇acketle{tW2,W3/a\\}b∇acket∇i}htand toR/parenleftbig\nDg0Xk0,g0,H|α/parenrightbig\ninTαH2,2(S1,H). As in L2(S1,R)\nλ′′\n2+(2π)2λ2⊥L2/a\\}b∇acketle{tcos(2πt),sin(2πt)/a\\}b∇acket∇i}ht,/a\\}b∇acketle{tλ′′\n1,λ′\n2/a\\}b∇acket∇i}ht ⊥L2const,\nwe get from (4.9) and the fact that 1 −4π2r2/\\e}atio\\slash= 0\n{0}= (−D2\nt,g0+1)/parenleftbig\n/a\\}b∇acketle{tW1,W2,W3/a\\}b∇acket∇i}ht/parenrightbig\n∩(−D2\nt,g0+1)Dg0Xk0,g0,H|α(TαH2,2(S1,H))\nandtheclaim followsfor Dg0Xk0,g0,H|αisaFredholmoperatorofindex0. /squareCLOSED MAGNETIC GEODESICS 15\nMoreover, we see for α∈ ZH\nR(Dg0Xk0,g0,H|α) =/braceleftbig\n(−D2\nt,g0+1)−1/parenleftbig\n(−λ′′\n1+2π/radicalbig\n1+r2λ′\n2)˙α\n−(λ′′\n2+(2π)2λ2)(α×m˙α)/parenrightbig\n:λ1, λ2∈H2,2(S1,R)/bracerightbig\n=/braceleftbig\n(−D2\nt,g0+1)−1/parenleftbig\nλ1˙α+λ2(α×m˙α)/parenrightbig\n:λi∈L2(S1,R),\nλ1⊥L21, λ2⊥L2/a\\}b∇acketle{tcos(2πt),sin(2πt)/a\\}b∇acket∇i}ht/bracerightbig\n=/a\\}b∇acketle{t(α×m˙α)/a\\}b∇acket∇i}ht⊕E+, (4.13)\nwhereE+is given by\nE+=/braceleftbig\n(−D2\nt,g0+1)−1/parenleftbig\nλ1˙α+λ2(α×m˙α)/parenrightbig\n:\nλi∈L2(S1,R), λ1⊥L21, λ2⊥L2/a\\}b∇acketle{t1,cos(2πt),sin(2πt)/a\\}b∇acket∇i}ht/bracerightbig\nWe have for V=λ1˙α+λ2(α×m˙α) inTαH2,2(S1,H)\nDg0Xk0,g0,H|α(V)∈E+⇐⇒λ2⊥L21⇐⇒V⊥L2(α×m˙α).\nWe fix\nV= (−D2\nt,g0+1)−1(λ1˙α+λ2(α×m˙α))∈E+.\nThen/integraldisplay\nS1/a\\}b∇acketle{tV,α×m˙α/a\\}b∇acket∇i}htm\n=/integraldisplay\nS1/a\\}b∇acketle{t(−D2\nt,g0+1)−1(λ1˙α+λ2(α×m˙α)),α×m˙α/a\\}b∇acket∇i}htm\n=/integraldisplay\nS1/a\\}b∇acketle{tλ1˙α+λ2(α×m˙α),(−D2\nt,g0+1)−1(α×m˙α)/a\\}b∇acket∇i}htm\n= (4π2(1+r2)+1)−1/integraldisplay\nS1/a\\}b∇acketle{tλ1˙α+λ2(α×m˙α),α×m˙α/a\\}b∇acket∇i}htm= 0.\nConsequently, Dg0Xk0,g0,H|α(E+) =E+.\nE+isL2-orthogonal to α×m˙αand ˙α, we may thus write\nV= (ν1+f1)˙α+(ν2+f2)(α×m˙α),\nwhere\nν1,ν2⊥L2/a\\}b∇acketle{t1,sin(2π·),cos(2π·)/a\\}b∇acket∇i}htandf1,f2∈ /a\\}b∇acketle{tsin(2π·),cos(2π·)/a\\}b∇acket∇i}ht.\nThen\n/a\\}b∇acketle{t(−D2\nt,g0+1)Dg0Xk0,g0,H|α(V),V/a\\}b∇acket∇i}htL2\n=/integraldisplay\nS1(ν′\n1)2−2π/radicalbig\n1+r2ν′\n1ν2+(ν′\n2)2−4π2(ν2)2\n/integraldisplay\nS1(f′\n1)2−2π/radicalbig\n1+r2f′\n1f2. (4.14)16 MATTHIAS SCHNEIDER\nSinceν2⊥L2/a\\}b∇acketle{t1,cos(2π·),sin(2π·)/a\\}b∇acket∇i}htwe have\n/integraldisplay\nS1(ν′\n2)2−4π2(ν2)2≥/integraldisplay\nS116π2(ν2)2\nand for 0 < r≤1\n/integraldisplay\nS1(ν′\n1)2−2π/radicalbig\n1+r2ν′\n1ν2+(ν′\n2)2−4π2(ν2)2\n≥/integraldisplay\nS1(ν′\n1)2−1\n4(ν′\n1)2−4π2(1+r2)(ν2)2+(ν′\n2)2−4π2(ν2)2\n≥/integraldisplay\nS13\n4(ν′\n1)2+4π2(ν2)2.\nConcerning the remaining term in (4.14) we note that as ( −D2\nt,g0+1) maps\n/braceleftbig\nλ1˙α+λ2(α×m˙α) :λ1,λ2∈ /a\\}b∇acketle{tsin(2π·),cos(2π·)/a\\}b∇acket∇i}ht/bracerightbig\ninto itself and V∈E+there holds\nf1˙α+f2(α×m˙α)∈(−D2\nt,g0+1)−1/angbracketleftbig\ncos(2π·)˙α,sin(2π·)˙α/angbracketrightbig\n.\nHence, by explicit computations there are x,y∈Rsatisfying\nf1(t) =xcos(2πt)+ysin(2πt),\nf2(t) =8π2√\n1+r2\n4π2(2+r2)+1/parenleftbig\nycos(2πt)−xsin(2πt)/parenrightbig\n,\nsuch that\n/integraldisplay\nS1(f′\n1)2−2π/radicalbig\n1+r2f′\n1f2=2π2(1−4π2r2)\n4π2(2+r2)+1(x2+y2).\nThis shows that if r <(2π)−1, then\n/a\\}b∇acketle{t(−D2\nt,g0+1)Dg0Xk0,g0,H|α(V),V/a\\}b∇acket∇i}htL2>0 for all V∈E+\\{0},\nand the homotopy\n[0,1]∋s/ma√sto→(1−s)/parenleftbig\nDg0Xk0,g0,H|α/parenrightbig\n|E++sid|E+\nis admissible. We use the decomposition in (4.13) and\nDg0Xk0,g0,H|α(α×m˙α) =−4π2\n4π2(1+r2)+1(α×m˙α)\nto see that under the assumption r <(2π)−1\n/parenleftbig\nDg0Xk0,g0,H|α/parenrightbig\n|R(Dg0Xk0,g0,H|α)∼/parenleftbigg\n−1 0\n0id|E+/parenrightbigg\n.\nConsequently, for r <(2π)−1\nsgn/parenleftbig\nDg0Xk0,g0,H|α/parenrightbig\n|R(Dg0Xk0,g0,H|α)=−1. (4.15)CLOSED MAGNETIC GEODESICS 17\nWe remark that the formula for the degree continues to hold fo rr >(2π)−1.\nFrom (4.8) and the fact that ( πM)∗is an isometry we obtain for ˜ α∈ ZM\n{0}=/a\\}b∇acketle{t˜W1,˜W2,˜W3/a\\}b∇acket∇i}ht∩R/parenleftbig\nDg0Xk0,g0|˜α/parenrightbig\n,\n/a\\}b∇acketle{t˜W1/a\\}b∇acket∇i}ht⊥=/a\\}b∇acketle{t˜W2,˜W3/a\\}b∇acket∇i}ht⊕R/parenleftbig\nDg0Xk0,g0|˜α/parenrightbig\n,\n−1 =sgn/parenleftbig\nDg0Xk0,g0|˜α/parenrightbig\n|R(Dg0Xk0,g0|˜α). (4.16)\nWe fix ˜α0∈ ZMand a parametrization ϕofZM, which maps an open\nneighborhood of 0 in /a\\}b∇acketle{t˜W1(˜α0),˜W2(˜α0),˜W3(˜α0)/a\\}b∇acket∇i}htintoZM, such that\nϕ(0) = ˜α0andDϕ|0=id.\nAsZMconsists of smooth functions, ZMis a sub-manifold of Hm,2(S1,M)\nfor 1≤m <∞. We define a map Φ from an open neighborhood Uof 0 in\nT˜α0H2,2(S1,M) =/a\\}b∇acketle{t˜W1(˜α0),˜W2(˜α0),˜W3(˜α0)/a\\}b∇acket∇i}ht⊕Range(Dg0Xk0,g0|˜α0)\ntoH2,2(S1,M) by\nΦ(W,U) :=Exp˜α0,g0/parenleftbig\nExp−1\n˜α0,g0(ϕ(W))+U/parenrightbig\n. (4.17)\nThen (Φ ,U) is a chart of H2,2(S1,M) around ˜ α0such that Uis an open\nneighborhood of 0 in T˜α0H2,2(S1,M), Φ(0) = ˜ α0, and\nDΦ|0=id,Φ−1/parenleftbig\nZM∩Φ(U)/parenrightbig\n=U ∩/a\\}b∇acketle{t˜W1(˜α0),˜W2(˜α0),˜W3(˜α0)/a\\}b∇acket∇i}ht.\nFrom the properties of Exp˜α0,g0the map Φ is a chart of Hk,2(S1,M) around\n˜α0for any 1 ≤k≤4 and shrinking Uwe may assume that\nTΦ(V)H1,2(S1,M) =/a\\}b∇acketle{td\ndtΦ(V)/a\\}b∇acket∇i}ht⊕DΦ|V(/a\\}b∇acketle{t˙˜α0/a\\}b∇acket∇i}ht⊥,H1,2), (4.18)\nTΦ(V)H2,2(S1,M) =/a\\}b∇acketle{tWg0(Φ(V))/a\\}b∇acket∇i}ht⊕DΦ|V(/a\\}b∇acketle{tWg0(˜α0)/a\\}b∇acket∇i}ht⊥),(4.19)\nProj/an}bracketle{tWg0(Φ(V)/an}bracketri}ht⊥◦DΦ|V:/a\\}b∇acketle{tWg0(˜α0)/a\\}b∇acket∇i}ht⊥∼=− → /a\\}b∇acketle{tWg0(Φ(V)/a\\}b∇acket∇i}ht⊥, (4.20)\nand the norm of the projections in (4.18) and (4.19) as well as the norm of\nthe map in (4.20) and its inverse are uniformly bounded with r espect to V.\n5.The perturbative analysis\nFor ˜α0∈ ZMthe vectors ˜W1(˜α0) andWg0(˜α0) are collinear and we use\n/a\\}b∇acketle{t˜W1(˜α0)/a\\}b∇acket∇i}htinstead of /a\\}b∇acketle{tWg0(˜α0)/a\\}b∇acket∇i}htin the analysis below.\nWe define a S1-invariant vector bundle SH2,2(S1,M) by\nSH2,2(S1,M) :={(γ,V)∈TH2,2(S1,M) :γ/\\e}atio\\slash= const, V∈ /a\\}b∇acketle{tWg(γ)/a\\}b∇acket∇i}ht⊥}.\nAs in [15, Sec. 4] we obtain a chart Ψ for the bundle SH2,2(S1,M) around\n(˜α0,0) by,\nΨ :U ×U ∩/a\\}b∇acketle{t ˜W1(˜α0)/a\\}b∇acket∇i}ht⊥→SH2,2(S1,M),\nΨ(V,U) :=/parenleftbig\nΦ(V),Proj/an}bracketle{tWg0(Φ(V))/an}bracketri}ht⊥◦DΦ|V(U)/parenrightbig\n.18 MATTHIAS SCHNEIDER\nWe define\nXΦ\ng0,ε:U ∩/a\\}b∇acketle{t˜W1(˜α0)/a\\}b∇acket∇i}ht⊥→ /a\\}b∇acketle{t˜W1(˜α0)/a\\}b∇acket∇i}ht⊥\nby\nXΦ\ng0,ε(V) :=Proj2◦Ψ−1/parenleftbig\nΦ(V),Xg0,ε(Φ(V))/parenrightbig\n.\nAs in [15, Lem. 3.5] it is easy to see that\nV∈ U ∩/a\\}b∇acketle{t˜W1(˜α0)/a\\}b∇acket∇i}ht⊥is a (nondegenerate) zero of XΦ\ng0,εif and only if\nS1∗Φ(V) is a (nondegenerate) critical orbit of Xg0,ε,\n(5.1)\nand ifXΦ\ng0,ε(V) = 0, then\nDg0XΦ\ng0,ε|V=A−1\nV◦Dg0Xg0,ε|Φ(V)◦DΦ|V, (5.2)\nwhere the isomorphism AV:/a\\}b∇acketle{t˜W1(˜α0)/a\\}b∇acket∇i}ht⊥→ /a\\}b∇acketle{tWg0(Φ(V))/a\\}b∇acket∇i}ht⊥is given by\nAV=Proj/an}bracketle{tWg0(Φ(V))/an}bracketri}ht⊥◦DΦ|V.\nFrom Lemma 4.3 we may assume\nU ∩/a\\}b∇acketle{t˜W1(˜α0)/a\\}b∇acket∇i}ht⊥=U1×U2,\nwhereU1andU2are open neighborhoods of 0 in /a\\}b∇acketle{t˜W2(˜α0),˜W3(˜α0)/a\\}b∇acket∇i}htand\nR/parenleftbig\nDg0Xk0,g0|˜α0/parenrightbig\n. We denote for ˜ α∈ ZMbyP2(˜α) the projection onto\nR(Dg0Xg0,0|˜α) with respect to the decomposition\n/a\\}b∇acketle{t˜W1(˜α)/a\\}b∇acket∇i}ht⊥=/a\\}b∇acketle{t˜W2(˜α),˜W3(˜α)/a\\}b∇acket∇i}ht⊕R/parenleftbig\nDg0Xk0,g0|˜α/parenrightbig\n,\nand byP1(˜α) the projection onto /a\\}b∇acketle{t˜W2(˜α),˜W3(˜α)/a\\}b∇acket∇i}ht. Moreover, for W∈ U1\nwe define for i= 1,2\nPΦ\ni(W) := (AW)−1◦Pi(Φ(W))◦AW.\nThe projections PΦ\n1(W) andPΦ\n2(W) correspond to the decomposition\n/a\\}b∇acketle{t˜W1(˜α0)/a\\}b∇acket∇i}ht⊥=/a\\}b∇acketle{t˜W2(˜α0),˜W3(˜α0)/a\\}b∇acket∇i}ht⊕R/parenleftbig\nDg0XΦ\ng0,0|W/parenrightbig\n,(5.3)\nas we have for W∈ U1\nDg0XΦ\ng0,0|W=A−1\nW◦Dg0Xg0,0|Φ(W)◦AW.\nMoreover, for ˜ α∈ ZMthe vector field K1(˜α) is orthogonal to ˜W1(˜α) and we\nmay define a vector field on ZMby\nZM∋˜α/ma√sto→P1(˜α)◦K1(˜α)∈ /a\\}b∇acketle{t˜W2(˜α),˜W3(˜α)/a\\}b∇acket∇i}ht.\nNote that P1(·)◦K1(·) isS1-equivariant, i.e.\nθ∗/parenleftbig\nP1(˜α)◦K1(˜α)/parenrightbig\n=P1(θ∗˜α)◦K1(θ∗˜α) for all ( θ,˜α)∈S1×ZM.\nIfP1(˜α0)◦K1(˜α0) = 0 for some ˜ α0∈ ZMdifferentiating the identity\n0≡ /a\\}b∇acketle{tP1(˜α)◦K1(˜α),˜W1(˜α)/a\\}b∇acket∇i}htCLOSED MAGNETIC GEODESICS 19\nwe find that the covariant derivative DZM/parenleftbig\nP1(·)◦K1(·)/parenrightbig\n|˜α0maps\nT˜α0ZM=/a\\}b∇acketle{t˜W1(˜α0),˜W2(˜α0),˜W3(˜α0)/a\\}b∇acket∇i}ht\nto/a\\}b∇acketle{t˜W2(˜α0),˜W3(˜α0)/a\\}b∇acket∇i}htand theS1equivariance leads to\nDZM/parenleftbig\nP1(·)◦K1(·)/parenrightbig\n|˜α0/parenleftbig˜W1(˜α0)/parenrightbig\n= 0.\nConsequently, we say that S1∗˜α0∈ ZMis a nondegenerate zero orbit of\nP1(·)◦K1(·), ifP1(˜α0)◦K1(˜α0) = 0 and\nDZM/parenleftbig\nP1(·)◦K1(·)/parenrightbig\n|˜α0:/a\\}b∇acketle{t˜W2(˜α0),˜W3(˜α0)/a\\}b∇acket∇i}ht → /a\\}b∇acketle{t˜W2(˜α0),˜W3(˜α0)/a\\}b∇acket∇i}ht\nis invertible.\nUsing the above notation the perturbative analysis done in [ 15] carries over\nandwe state thefollowing fourresults withoutproof (see [1 5, Lem. 5.2-5.5]).\nLemma 5.1. For˜α0∈ Zafter possibly shrinking Uthere are ε0>0and\nU∈C2([−ε0,ε0]×U1,/a\\}b∇acketle{t˜W1(˜α0)/a\\}b∇acket∇i}ht⊥),\nR∈C2([−ε0,ε0]×U1,/a\\}b∇acketle{t˜W2(˜α0),˜W3(˜α0)/a\\}b∇acket∇i}ht),\nsuch that for all (ε,W)∈[−ε0,ε0]×U1\nR(ε,W) =XΦ\ng0,ε(W+U(ε,W)),\n0 =PΦ\n1(W)◦U(ε,W),\nO(ε)ε→0=/ba∇dblU(ε,W)/ba∇dbl+/ba∇dblDWU(ε,W)/ba∇dbl+/ba∇dblR(ε,W)/ba∇dbl+/ba∇dblDWR(ε,W)/ba∇dbl,\nR(ε,W) =εPΦ\n1(W)◦KΦ\n1(W)+o(ε)ε→0,\nU(ε,W) =−ε(Dg0XΦ\ng0,0|W)−1◦PΦ\n2(W)◦KΦ\n1(W)+o(ε)ε→0.\nMoreover, the functions U(ε,W)andR(ε,W)are unique, in the sense that,\nif(ε,W,U,R )in[−ε0,ε0]×U1×U ∩/a\\}b∇acketle{t˜W1(˜α0)/a\\}b∇acket∇i}ht⊥×U1satisfies\nXΦ\ng0,ε(W+U) =RandPΦ\n1(W)/parenleftbig\nU/parenrightbig\n= 0,\nthenU=U(ε,W)andR=R(ε,W).\nLemma 5.2. Under the assumptions of Lemma 5.1 we have as ε→0\nXΦ\ng0,ε(W+U(ε,W)) =εPΦ\n1(W)◦KΦ\n1(W)+O(ε2)ε→0,\nwhereKΦ\n1is the vector-field K1in the coordinates Φ, i.e.\nKΦ\n1=XΦ\ng0,1−XΦ\ng0,0.\nLemma 5.3. Under the assumptions of Lemma 5.1 suppose 0is a nondegen-\nerate zero of the vector-field PΦ\n1(·)◦KΦ\n1(·), in the sense that PΦ\n1(0)◦KΦ\n1(0) =\n0and\nDW(PΦ\n1(·)◦KΦ\n1(·))|0∈ L(/a\\}b∇acketle{tW2(˜α0),W3(˜α0)/a\\}b∇acket∇i}ht)20 MATTHIAS SCHNEIDER\nis an isomorphism. Then, after possibly shrinking ε0andU, for any 0<\nε≤ε0there is a unique W(ε)∈ U1such that\nXΦ\ng0,ε(W(ε)+U(ε,W(ε))) = 0,\nW(ε)→0asε→0.\nMoreover, V(ε) :=W(ε) +U(ε,W(ε))is the only zero of XΦ\ng0,εinU ∩\n/a\\}b∇acketle{t˜W1(˜α0)/a\\}b∇acket∇i}ht⊥and is nondegenerate with\nsgn(DXΦ\ng0,ε|V(ε)) =−det(DW(PΦ\n1(·)◦KΦ\n1(·))|0).\nLemma 5.4. Under the assumptions of Lemma 5.1 suppose ˜α0is a non-\ndegenerate zero of the vector-field P1(·)◦K1(·)onZM, in the sense that\nP1(˜α0)◦K1(˜α0) = 0and\nDZ(P1(·)◦K1(·))|˜α0∈ L(/a\\}b∇acketle{tW2(˜α0),W3(˜α0)/a\\}b∇acket∇i}ht)\nis an isomorphism. Then for any 0< ε < ε 0there is˜γ(ε)∈Φ(U)satisfying\nXg0,ε(˜γ(ε)) = 0and˜γ(ε)→˜α0asε→0.\nMoreover, S1∗˜γ(ε)is the unique critical orbit of Xg0,εinΦ(U)and is\nnondegenerate with\ndegloc,S1(Xg0,ε,S1∗˜γ(ε)) =−det(DZM(P1(·)◦K1(·))|˜α0).\n6.The computation of the degree\nIn order to compute the S1-degree of Xg0,εwe choose a smooth Morse\nfunction k1∈C∞(M,R). Thecorrespondingvector-field K1onH2,2(S1,M)\nis given by\nK1(γ) = (−D2\nt,g0+1)−1(|˙γ|g0k1(˜γ)Jg0(γ)˙γ).\nWe note that for ˜ α=πM◦α(·,2π|r|,v0,v1,w)∈ ZMandr >0 small enough\nwe have\nK1(˜α) = 2πr(−D2\nt,g0+1)−1/parenleftbig\nk1(˜α)(ϕM)∗(α×m˙α)/parenrightbig\n.\nConsequently, from (4.11), (4.12), and (4.13)\nP1(˜α)◦K1(˜α) =σ2(˜α)˜W2(˜α)+σ3(˜α)˜W3(˜α), (6.1)\nwhereσ2(˜α), σ2(˜α)∈Rare defined by the condition that\n2πrk1(˜α)−σ2(˜α)\n2πr(1−4π2r2)cos(2π·)−σ3(˜α)\n2πr(1−4π2r2)sin(2π·)\nisL2-orthogonal to /a\\}b∇acketle{tcos(2π·),sin(2π·)/a\\}b∇acket∇i}ht. Hence,\nσ2(˜α) =8π2r2\n1−4π2r2/integraldisplay1\n0k1◦˜α(t)cos(2πt)dt,\nσ3(˜α) =8π2r2\n1−4π2r2/integraldisplay1\n0k1◦˜α(t)sin(2πt)dt.CLOSED MAGNETIC GEODESICS 21\nIn the following we are interested in the asympotics of σ2andσ3asr→0+\nor equivalently as k0→ ∞. There holds\n1−4π2r2\n8π2r2σ2(˜α)\n=/integraldisplay1\n0/parenleftBig\nk1◦πM(w)+rdk1|πM(w)cos(2πt)(πM)∗v1\n+rdk1|πM(w)sin(2πt)(πM)∗v0+O(r2)/parenrightBig\ncos(2πt)dt\n=1\n2rdk1|πM(w)(πM)∗v1+O(r2), (6.2)\nand analogously we find\n1−4π2r2\n8π2r2σ3(˜α) =1\n2rdk1|πM(w)(πM)∗v0+O(r2). (6.3)\nFrom the above expansion we easily deduce\nLemma 6.1. For allδ >0there isr0>0such that for all 0< r≤r0and\n˜α=πM(/radicalbig\n1+r2w+rcos(2πt)v1+rsin(2πt)v0)∈ ZM\nsatisfying P1(˜α)◦K1(˜α) = 0there holds\nπM(w)∈n∪\ni=1Bδ(˜wi),\nwhere{˜wi: 1≤i≤n}denotes the set of critical points of k1inM.\nFixw0∈Hand a positive orthonormal system {v0,v1,w0}in (R3,m)\nsuch that πM(w0) is a critical point of k1inM. We choose δ >0, a\nparametrization\nw:B1(0)⊂R2→Bδ(w0)⊂H,(x,y)/ma√sto→w(x,y),\nand smooth maps v0,v1:B1(0)→R3such that {v0(x,y),v1(x,y),w(x,y)}\nis orthonormal for all ( x,y)∈B1(0) and\n(v0(0,0),v1(0,0),w(0,0)) = (v0,v1,w0),∂w\n∂x|(0,0)=v1,∂w\n∂y|(0,0)=v0.\n(6.4)\nShrinking δ >0 we may assume that πM◦ϕwparametrizes Mand that\n(x,y)/ma√sto→˜α(x,y) is an injective immersion from B1(0) toZM, where\n˜α(x,y) :=πM/parenleftbig/radicalbig\n1+r2w(x,y)+rcos(2π·)v1(x,y)+rsin(2π·)v0(x,y)/parenrightbig\n.\nFrom (4.6) and (6.4) we get as r,δ→0+\n∂\n∂x˜α|(x,y)=˜W2(˜α(x,y))+O(r)+O(δ)\n∂\n∂y˜α|(x,y)=˜W3(˜α(x,y))+O(r)+O(δ). (6.5)22 MATTHIAS SCHNEIDER\nDefineH:B1(0)→R2by\nH(x,y) :=/parenleftbig\nσ2(˜α(x,y)),σ3(˜α(x,y))/parenrightbig\n.\nBy (6.2) and (6.3) we have as r→0+\nH(x,y)\n8π2r3:=/parenleftbig\ndk1|πM(w(x,y))(πM)∗v1(x,y),dk1|πM(w(x,y))(πM)∗v0(x,y)/parenrightbig\n+O(r).\nSince\nd\ndxπM◦w|0,0= (πM)∗v1(0,0),d\ndyπM◦w|0,0= (πM)∗v0(0,0)\nwe find for small values of δ >0 andr >0\ndeg(H,B1(0),0) = deg( ∇(k1◦πM◦w),B1(0),0)\n= deg(∇k1,Bδ(πM(w0)),0) = degloc(∇k1,πM(w0)),\nand the set of zeros of HinB1(0) is non-empty. Fix a zero ( x0,y0)∈B1(0)\nofH. Then\ndH|(x0,y0)=/parenleftBigg∂\n∂x(σ2◦˜α)|(x0,y0)∂\n∂y(σ2◦˜α)|(x0,y0)\n∂\n∂x(σ3◦˜α)|(x0,y0)∂\n∂y(σ3◦˜α)|(x0,y0)/parenrightBigg\n.\nFrom (6.2), (6.3), and the fact that H(x0,y0) = 0 we get\ndk1|πM(w(x0,y0))=O(r).\nThus, we have as r→0+\ndk1|˜α(x0,y0)(t)∂\n∂x˜α|(x0,y0)(t)\n=dk1|πM(w(x0,y0))(πM)∗∂\n∂xw|(x0,y0)\n+r/parenleftBig\n∇∂\n∂r˜α(x0,y0)(t)|r=0dk1|πM(w(x0,y0))(πM)∗∂\n∂xw|(x0,y0)\n+dk1|πM(w(x0,y0))∇∂\n∂r˜α(x0,y0)(t)|r=0∂\n∂x˜α(x0,y0)(t)|r=0/parenrightBig\n+O(r2)\n=dk1|πM(w(x0,y0))(πM)∗∂\n∂xw|(x0,y0)\n+r(∇dk1)|πM(w(x0,y0))/parenleftBig\n(πM)∗∂\n∂xw|(x0,y0),\ncos(2πt)(πM)∗v1(x0,y0)+sin(2πt)(πM)∗v0(x0,y0)/parenrightBig\n+O(r2).CLOSED MAGNETIC GEODESICS 23\nUsing (6.4), this leads to, as r,δ→0+\n∂\n∂x(σ2◦˜α)|(x0,y0)\n=/integraldisplay1\n0dk1|˜α(x0,y0)(t)∂\n∂x˜α|(x0,y0)(t)cos(2πt)dt\n=r\n2(∇dk1)|πM(w(x0,y0))((πM)∗∂\n∂xw|(x0,y0),(πM)∗v1(x0,y0))+O(r2)\n=r\n2(∇dk1)|πM(w0)/parenleftBig\n(πM)∗v1,(πM)∗v1/parenrightBig\n+O(r2)+O(rδ).\nAnalogously, we may compute the remaining partial derivati ves ofHand\nwe find for small values of δ >0 andr >0\nsgndet(dH|(x0,y0)) = sgndet( ∇dk1|πM(w0)) = degloc(∇k1,πM(w0)),(6.6)\nsuch that ( x0,y0) is the unique zero of HinB1(0). From (6.1) we see that\nP1(˜α(x0,y0))◦K1(˜α(x0,y0)) = 0,\nby (6.5) we obtain as r,δ→0+\n∇˜W2/parenleftbig\nP1(·)◦K1(·)/parenrightbig\n|˜α(x0,y0)\n=/parenleftBig\ndσ2|˜α(x0,y0)˜W2(˜α(x0,y0))/parenrightBig\n˜W2(˜α(x0,y0))\n+/parenleftBig\ndσ3|˜α(x0,y0)˜W2(˜α(x0,y0))/parenrightBig\n˜W3(˜α(x0,y0))\n=∂\n∂x(σ2◦˜α)|(x0,y0)˜W2(˜α(x0,y0))\n+∂\n∂x(σ3◦˜α)|(x0,y0)˜W3(˜α(x0,y0))+O(r)+O(δ).\nConcerning the covariant derivative of P1(·)◦K1(·) in direction ˜W3we have\nto replace∂\n∂xby∂\n∂yin the above formula. Consequently, from (6.6)\nsgndet(DZM(P1(·)◦K1(·))|˜α(x0,y0)) = degloc(∇k1,πM(w0)).\nThus we arrive at the following\nLemma 6.2. Let{˜wi: 1≤i≤n}denote the set of critical points of k1\ninM. Then there is r0>0such that for all 0< r≤r0the set of critical\norbits of P1(·)◦K1(·)is given by {S1∗˜αi,r: 1≤i≤n}, where\n˜αi,r=πM(/radicalbig\n1+r2wi,r+rcos(2πt)v1,i,r+rsin(2πt)v0,i,r)∈ ZM.\nMoreover, we have for 1≤i≤n\nπM(wi,r)→˜wiasr→0+,\nsgndet(DZM(P1(·)◦K1(·))|˜αi,r) = degloc(∇k1,˜wi).\nProof.From Lemma 6.1 and the analysis of Hwe may choose δ >0 and\nr0>0 such that the union ∪n\ni=1Bδ(˜wi) is disjoint and for every iand 0<\nr≤r0there is a unique πM(wi,r)∈Bδ(˜wi) corresponding to a critical orbit24 MATTHIAS SCHNEIDER\nS1∗˜αi,r. Moreover, if r→0+we may shrink δ >0, which yields together\nwith the uniqueness of πM(wi,r) the claimed asymptotic. /square\nLemma 6.3. LetMAbe the set of oriented Alexandrov embedded regular\ncurves in H2,2(S1,M). There is Ck0>0such that for all k0≥Ck0we have\nχS1(Xk0,g0,MA) =−χ(M),\nwhereχ(M)denotes the Euler characteristic of M.\nProof.We choose a Morse function k1onMwith nondegenerate critical\npoints{˜wi: 1≤i≤n}. FromLemma6.2weobtain Ck0>0suchthat forall\nk0≥Ck0thecritical orbitsof P1(·)◦K1(·)aregivenby {S1∗˜αi,k0: 1≤i≤n}\nsatisfying\nsgndet(DZM(P1(·)◦K1(·))|˜αi,k0) = degloc(∇k1,˜wi).\nWe fixk0≥Ck0. By Lemma 5.3 there is ε >0 such that for any 0 < ε < ε 0\nand 1≤i≤nthere is ˜γi(ε)∈Φ(Ui) satisfying\nXg0,ε(˜γi(ε)) = 0 and ˜ γ(ε)→˜αi,k0asε→0.\nMoreover, S1∗˜γi(ε) is the unique critical orbit of Xg0,εin Φ(Ui) and is\nnondegenerate with\ndegloc,S1(Xg0,ε,S1∗˜γ(ε)) =−degloc(∇k1,˜wi). (6.7)\nTo show that there is an open neighborhood UofZMandε0>0 such\nthat for all 0 < ε < ε 0the critical orbits of Xg0,εinUare given exactly\nby{˜γi(ε) : 1≤i≤n}we argue by contradiction. Suppose there are\nεn→0+and a sequence (˜ αn) of zeros of Xg0,εnthat converges to ZMbut\n˜αn/∈ {˜γi(ε) : 1≤i≤n}. Up to a subsequence we may assume\n˜αn→˜α0∈ ZM\nasn→ ∞. For large nwe use the chart Φ around ˜ α0given in (4.17). From\nthe existence of a slice of the S1-action (see [15, Lem. 3.1]) we get sequences\nθn∈R/ZandVn∈ /a\\}b∇acketle{t˜W1(˜α0)/a\\}b∇acket∇i}ht⊥converging to 0 such that\nθn∗˜αn= Φ(Vn).\nNote that from the S1-invariance and by construction\nXg0,εn(θn∗˜αn) = 0 and XΦ\ng0,εn(Vn) = 0.\nWe consider the map\nΛ :/a\\}b∇acketle{t˜W1(˜α0)/a\\}b∇acket∇i}ht⊥=/a\\}b∇acketle{t˜W2(˜α0),˜W3(˜α0)/a\\}b∇acket∇i}ht⊕R/parenleftbig\nDg0XΦ\ng0,0|0/parenrightbig\n→ /a\\}b∇acketle{t˜W1(˜α0)/a\\}b∇acket∇i}ht⊥,\ndefined by\nΛ(W,V) :=W+Dg0XΦ\ng0,0|W(V).\nFrom (5.3) the map Λ is a diffeomorphism locally around (0 ,0), hence we\nmay decompose\nVn= Φ−1(θn∗αn) =Wn+Un,CLOSED MAGNETIC GEODESICS 25\nwhereWn∈ /a\\}b∇acketle{t˜W1(α0)/a\\}b∇acket∇i}ht⊥andUn∈R(Dg0XΦ\nk0,g0|Wn) converge to 0 as n→ ∞.\nFrom the uniqueness part of Lemma 5.1, as XΦ\ng0,εn(Wn+Un) = 0, we get\nUn=U(εn,Wn). By Lemma 5.2 we see that\nP1(˜α0)◦K1(˜α0) = 0.\nConsequently, S1∗α0∈ {S1∗˜αi,k0: 1≤i≤n}. From the uniqueness part\nin Lemma 5.3 we finally arrive at the contradiction\nS1∗˜αn∈ {S1∗˜γi(εn) : 1≤i≤n}.\nFrom the definition of the S1-equivariant Poincar´ e-Hopf index, the classi-\nfication of Alexandrov embedded zeros of Xk0,g0, and (6.7) there holds for\nsmallε >0\nχS1(Xk0,g0,MA) =χS1(Xk0,g0,U) =χS1(Xg0,ε,U)\n=n/summationdisplay\ni=1degloc,S1(Xg0,ε,S1∗˜γi(ε))\n=−n/summationdisplay\ni=1degloc(∇k1,˜wi) =−χ(M).\n/square\n7.The apriori estimate\nWe fix a continuous family of metrics {gt:t∈[0,1]}onMand a con-\ntinuous family of positive continuous function {kt:t∈[0,1]}onM. We\nassume that there is K0>0, such that the Gaussian curvature Kgtof each\nmetricgtonMand the functions {kt}satisfy\nKgt≥ −K0, (7.1)\nkinf:= inf{kt(x) : (x,t)∈M×[0,1]}>(K0)1\n2. (7.2)\nWe letXtbe the vector field on H2,2(S1,M) defined by\nXt:=Xkt,gt.\nWe denote by MA⊂H2,2(S1,M) the set\nMA:={γ∈H2,2\nreg(S1,M) :γis prime and oriented Alexandrov embedded. }.\nWe shall show that the set\nX−1(0) :={(γ,t)∈MA×[0,1] :Xt(γ) = 0}\nis compact in MA×[0,1]. Fix ( γ,t)∈X−1(0). Then there is an oriented\nimmersion F:B→MwithF|∂B=γ. We denote by F∗gtthe induced\nmetric on B.26 MATTHIAS SCHNEIDER\nLemma 7.1. For any (γ,t)∈X−1(0)there isϕ∈C2(B,R)satisfying\n−∆F∗gtϕ+KF∗gt+K0eϕ= 0inB,\n∂νϕ= 0on∂B, (7.3)\nwhereνdenotes the unit normal oriented to the outside.\nMoreover, there is C0>0, which may be chosen independently of (γ,t)∈\nX−1(0), such that\n0≥ϕ≥ −C0.\nProof.To show the existence of a solution ϕwe use the method of upper\nand lower solutions (see also [9]). The function ϕ+≡0 satisfies\n−∆F∗gtϕ++KF∗gt+K0eϕ+=KF∗gt+K0≥0,\nfrom (7.1) and the fact that Fis a local isometry. Hence, ϕ+is a superso-\nlution of (7.3). To find a subsolution, we let ϕ1∈C∞(M,R) be defined as\nthe solution to the linear equation\n−∆gtϕ1+Kgt−2πχ(M)vol(M,gt)−1= 0 inM,/integraldisplay\nMϕ1dgt= 0.\nBy standard elliptic estimates using a Green’s function on ( M,gt) (see [3,\nThm 4.13]) we have\nsup\nM|ϕ1| ≤C(gt)/parenleftbig\nsup\nM|Kgt|−2πχ(M)vol(M,gt)−1/parenrightbig\n≤C1,\nbecause{gt:t∈[0,1]}is a compact set of smooth metrics. We may choose\nC2>1 such that we have for all t∈[0,1]\n−C2≤ln/parenleftbig\n−2πχ(M)K0vol(M,gt)−1/parenrightbig\n.\nSinceFis a local isometry, there holds\n∆F∗gt(ϕ1◦F) = (∆ gtϕ1)◦F.\nWe define ϕ−∈C2(B,R) by\nϕ−:=ϕ1◦F−C1−C2\nand get\n−∆F∗gtϕ−+KF∗gt+K0eϕ−= 2πχ(M)vol(M,gt)−1+K0eϕ1◦F−C1−C2≤0.\nHenceϕ−is a subsolution of (7.3) satisfying\n−C0:=−(2C1+C2)≤ϕ−< ϕ+.\nUsing a version of the method of upper and lower solutions giv en in [14] we\nfind a solution ϕto (7.3) satisfying ϕ−≤ϕ≤ϕ+. /squareCLOSED MAGNETIC GEODESICS 27\nWeconsider Bequippedwiththemetric ht:=eϕF∗gt. ThentheGaussian\ncurvature Khtand the geodesic curvature khtof∂Bwith respect to ( B,ht)\nare given by (see [3, Sec 5.8.2])\nKht≡ −K0andkht=kF∗gte−ϕ\n2.\nConsequently, since 0 ≤ϕ,\ninf\n∂Bkht≥inf\n∂BkF∗gt≥kinf.\nThe Gauss-Bonnet formula applied to ( B,ht) gives\n2π=−/integraldisplay\nBK0dht+/integraldisplay\n∂BkhtdSht≥ −K0A(B,ht)+kinfL(∂B,ht),\nwhereA(B,ht) denotes the area of BandL(∂B,ht) the length of ∂Bwith\nrespect to ht. The isoperimetric inequality (see [13, Thm 4.3]) yields\nL(∂B,ht)2≥4πA(B,ht)+K0A(B,ht)2≥K0A(B,ht)2.\nThus we arrive at\n2π≥ −(K0)1\n2L(∂B,ht)+kinfL(∂B,ht).\nThis yields\nL(γ,gt) =L(∂B,F∗gt)≤eC0L(∂B,ht)≤eC02π\nkinf−(K0)1\n2.\nUsing again the Gauss-Bonnet formula we see\n2π=−/integraldisplay\nBK0dht+/integraldisplay\n∂BkhtdSht\n≤eC0\n2/parenleftbig\nsup{kt(x) : (x,t)∈M×[0,1]}/parenrightbig\nL(∂B,ht)\n≤e2C0/parenleftbig\nsup{kt(x) : (x,t)∈M×[0,1]}/parenrightbig\nL(γ,gt).\nConsequently, there is C >0, such that\nC≤L(γ,gt)≤C−1, (7.4)\nfor all (γ,t)∈X−1(0).\nFix a sequence ( γn,tn)n∈NinX−1(0). As a solution each γnis parameter-\nized proportional to its arc-length. From (7.4), ( γn) is uniformly bounded\ninC1(S1,M). Using the equation (1.2) we obtain a uniform bound of ( γn)\ninC3(S1,M), such that we may extract a subsequence, still denoted by\n(γn,tn)n∈N, which converges in C2(S1,M)×[0,1] to (γ0,t0). The conver-\ngence inC2(S1,M) and the lower boundin (7.4) imply that Xt0(γ0) = 0 and\nthatγ0is an immersion. By Lemma 2.1 the curve γ0is oriented Alexandrov\nembedded and hence ( γ0,t0)∈X−1(0). This shows that\nLemma 7.2. Under the assumptions (7.1)and(7.2)the setX−1(0)is com-\npact.28 MATTHIAS SCHNEIDER\n8.Existence results\nWe give the proof of our main existence result.\nProof of Theorem 1.2. From the uniformization theorem ( M,g) is isometric\nto (H/Γ,eϕg0), where Γ ⊂O(2,1)+is a group of isometries acting freely\nand properly discontinuously and ϕis a function in C∞(H/Γ,R). Due to\nthe invariance of (1.2) under isometries we may assume witho ut loss of\ngenerality that\n(M,g) = (H/Γ,eϕg0).\nWe consider the family of metrics {gt:=etϕg0:t∈[0,1]}and choose a\nlarge constant k0>>1, such that\nk0>/parenleftbig\n−inf{Kgt(x) : (x,t)∈M×[0,1]}/parenrightbig1\n2+inf\nMk+Ck0,\nwhereKgtdenotes the Gaussian curvature of the metric gtgiven by\nKgt=e−tϕ/parenleftbig\n−t∆g0(ϕ)+2/parenrightbig\n.\nFrom Lemma 7.2 the homotopy\n[0,1]∋t/ma√sto→Xk0,gt\nis (MA,gt,S1)-admissible. By Lemma 6.3 and the homotopy invariance of\ntheS1-equivariant Poincar´ e-Hopf index we obtain\n−χ(M) =χS1(Xk0,g0,MA) =χS1(Xk0,g,MA).\nFort∈[0,1] we define kt∈C∞(M,R) by\nkt(x) := (1−t)k0+tk(x).\nThen\ninf{kt(x) : (x,t)∈M×[0,1]}= inf\nMk >/parenleftbig\n−inf\nMKg/parenrightbig1\n2.\nFrom Lemma 7.2 the homotopy\n[0,1]∋t/ma√sto→Xkt,g\nis (MA,g,S1)-admissible and there holds\nχS1(Xk,g,MA) =χS1(Xk0,g,MA) =−χ(M).\nThis gives the claim. /square\nReferences\n[1] V. I. Arnold. The first steps of symplectic topology. Uspekhi Mat. Nauk , 41(6(252)):3–\n18, 229, 1986.\n[2] V. I. Arnold. On some problems in symplectic topology. In Topology and geometry—\nRohlin Seminar , volume 1346 of Lecture Notes in Math. , pages 1–5. Springer, Berlin,\n1988.\n[3] Thierry Aubin. Some nonlinear problems in Riemannian geometry . Springer Mono-\ngraphs in Mathematics. Springer-Verlag, Berlin, 1998.CLOSED MAGNETIC GEODESICS 29\n[4] Gonzalo Contreras, Leonardo Macarini, and Gabriel P. Pa ternain. Periodic orbits for\nexact magnetic flows on surfaces. Int. Math. Res. Not. , (8):361–387, 2004.\n[5] V. L. Ginzburg. New generalizations of Poincar´ e’s geom etric theorem. Funktsional.\nAnal. i Prilozhen. , 21(2):16–22, 96, 1987.\n[6] Viktor L. Ginzburg. On closed trajectories of a charge in a magnetic field. An applica-\ntion of symplectic geometry. In Contact and symplectic geometry (Cambridge, 1994) ,\nvolume 8 of Publ. Newton Inst. , pages 131–148. Cambridge Univ. Press, Cambridge,\n1996.\n[7] Viktor L. Ginzburg and Ba¸ sak Z. G¨ urel. Periodic orbits of twisted geodesic flows and\nthe Weinstein-Moser theorem. Comment. Math. Helv. , 84(4):865–907, 2009.\n[8] Gustav A. Hedlund. Fuchsian groups and transitive horoc ycles.Duke Math. J. ,\n2(3):530–542, 1936.\n[9] Jerry L. Kazdan and F. W. Warner. Curvature functions for compact 2-manifolds.\nAnn. of Math. (2) , 99:14–47, 1974.\n[10] V. V. Kozlov. Calculus of variations in the large and cla ssical mechanics. Uspekhi\nMat. Nauk , 40(2(242)):33–60, 237, 1985.\n[11] S. P. Novikov. The Hamiltonian formalism and a multival ued analogue of Morse\ntheory.Uspekhi Mat. Nauk , 37(5(227)):3–49, 248, 1982.\n[12] S. P. Novikov and I. A. Taimanov. Periodic extremals of m ultivalued or not every-\nwhere positive functionals. Dokl. Akad. Nauk SSSR , 274(1):26–28, 1984.\n[13] Robert Osserman. The isoperimetric inequality. Bull. Amer. Math. Soc. , 84(6):1182–\n1238, 1978.\n[14] Klaus Schmitt. Revisiting the method of sub- and supers olutions for nonlinear elliptic\nproblems. In Proceedings of the Sixth Mississippi State–UBA Conference on Differ-\nential Equations and Computational Simulations , volume 15 of Electron. J. Differ.\nEqu. Conf. , pages 377–385, San Marcos, TX, 2007. Southwest Texas State Univ.\n[15] Matthias Schneider. Closed magnetic geodesics on S2. Preprint, arXiv:0808.4038\n[math.DG] , 2008.\n[16] Matthias Schneider.Alexandrovembeddedclosed magne tic geodesics on S2.Preprint,\narXiv:0903.1128 [math.DG] , 2009.\n[17] I. A. Taimanov. Non-self-intersecting closed extrema ls of multivalued or not-\neverywhere-positive functionals. Izv. Akad. Nauk SSSR Ser. Mat. , 55(2):367–383,\n1991.\n[18] I. A. Taimanov. Closed extremals on two-dimensional ma nifolds.Uspekhi Mat. Nauk ,\n47(2(284)):143–185, 223, 1992.\n[19] I. A. Taimanov. The type numbers of closed geodesics. Regul. Chaotic Dyn. , 15(1):84–\n100, 2010.\n[20] A. J. Tromba. A general approach to Morse theory. J. Differential Geometry ,\n12(1):47–85, 1977.\n[21] A. J. Tromba. The Euler characteristic of vector fields o n Banach manifolds and a\nglobalization of Leray-Schauder degree. Adv. in Math. , 28(2):148–173, 1978.\nRuprecht-Karls-Universit ¨at, Im Neuenheimer Feld 288, 69120 Heidelberg,\nGermany,\nE-mail address :mschneid@mathi.uni-heidelberg.de"    },    {        "title": "1804.10503v1.Excitation_and_coherent_control_of_magnetization_dynamics_in_magnetic_tunnel_junctions_using_acoustic_pulses.pdf",        "content": "1 \n Excitation  and c oherent control of magnetization dynamics \nin magnetic tunnel junctions  using acoustic  pulses  \n \nH. F. Yang,1 F. Garcia -Sanchez,1,2 X. K. Hu,1 S. Sievers,1 T. Böhnert,3 J. D. Costa,3* \nM. Tarequzzaman,3 R. Ferreira,3 M. Bieler,1+ and H. W. Schumacher1 \n1Physikalisch -Technische Bundesanstalt, Bundesallee 100,  \nD-38116 Braunschweig, Germany  \n2Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91,  \n10135 Torino, Italy  \n3International Iberian Nanotechnology Laboratory, Av. Mestre José Veiga,  \n4715 -330 Braga, Portugal  \nWe experimentally study  magnetization dynamics in magnetic tunnel junctions  driven \nby femtosecond -laser -induced surface acoustic waves . The acoustic pulses induce a \nmagnetization precession in the  free layer of the  magnetic tunnel junction through \nmagnetoelastic coupling. The frequency and amplitude of the precession shows a \npronounced dependence on the applied magnetic field  and the las er excitation position . \nComparing the acoustic -wave -induced precession frequencies with precession induced \nby charge currents and with micromagnetic simulations we identify spatially non -\nuniform magnetization modes  localized close  the edge regions as being responsible for \nthe optically induced magnetization dynamics . The experimental scheme even allows us \nto coherent ly control the magnetization precession using two acoustic pulses . This \nmight prove important for future applications requiring ultrafa st spin manipulation . \nAdditionally, our results  directly pinpoint the importance of  acoustic pulses since  they \ncould be relevant when  investigating optically -induced temperature effects in magnetic \nstructures .  \n_____________________________  \n* Present address: IMEC, 3000 Leuven, Belgium  \n+ Electronic address: mark.bieler@ptb.de   2 \n Manipulating the spin through external  stimulus is a key issue in the field of spintronic \nwith the aim to boost logic and memory applications. Such a manipulation  can be \nachieved  by different physical effects, employing photon s, electron s, heat  flux, THz \nradiation , as well as phonon s.1–8 In particular , the interaction of acoustic pulses with \nspin structures provides interesting prospects . This is because acoustic pulses can be \neasily generated on picosecon d time scales  and the magnetoelastic effect (the change of \nmagnetic properties due to elastic deformation) governing the interaction may lead to \nsignificant magnetization changes . Recently, the influence of surface acoustic wave s \n(SAW s) on certain nano elements and magnetic bubbles was studied.6,9 Additionally, \nlaser -induced acoustic pulses  were  used to excite  magnetization dynamics in \nferro magn etic layers.10,11 It has been found that the acoustic -pulse -induced precession \ncan be enhanced when being resonantly driven,12–14 yet, an identification of the \nprecession  mode s is difficu lt, even in cases with out acoustic perturbation .15–17 Based \nupon the previous  studies  on magnetoelastic effects ,6,9–16 the next logical step would be \nto extend these studies to magnetic devices having important industrial relevance. \nMagnetic tunnel junction s (MTJ s) are certainly among these devices as they are used in \ndata storage, magnetic sensor  and other spintronic applications. So far manipulation  of \nthe magnetization  in MTJs  has been realized by charge current s or heat current s \nthrough spin transfer  torque (STT) .18,19 Yet, no experiment on SAW -induced \nmagnetization dynamics in MTJs has been reported .  \nHere, we study  the excitation  of magnetization dynamics in  MTJs by femtosecond -laser -\ninduced SAW  taking advantage of the magnetoelastic coupling. Due to the dependence \nof the tunnel resistance on magnetization orientation of the free magnetic  layer in the \nMTJs, our  technique can directly measure small spin precession  angle s. Using a time \nresolved detection technique we are able to pinpoint acoustic pulses as being \nresponsible for the spin manipulation and exclude  other effect s resulting from laser \npulse excitation.  Moreover, the magnetization  mode driven by the SAW s in the free \nlayer of the MTJ can be determined by comparison to magnetization modes triggered by \ncharge current pulses and micromagnetic simulations . So far, the identification of the 3 \n exact magnetization mode  in previous magnetoelastic experiments has not been \naccomplished. Taking advantage of the coherent nature  of SAW s, we also show that our \nscheme allows coherent control of the magnetization . Using two separate acoustic \npulses we can either  enhance or s witch off the precession  in the MTJ . \nThe experiment s were carried out on a rectangular MTJ nanopillar  stack with lateral \ndimension s of 100  nm × 550  nm. The stack  was deposited on a Si wafer  and consists \n(from bottom to top) of an antiferromagnet (20 nm IrMn),  a synthetic antiferromagnet \n(2 nm Co 70Fe30, 0.85  nm Ru, 2.6 nm Co 40Fe40B20), the tunnel junction (~0.8  nm MgO , \ncorresponding to a resistance -area product of 1.8  Ωµm2), and the free layer (2.6  nm \nCo40Fe40B20). Between the Si wafer and the MTJ stack a  100 nm thick Al2O3 layer and a \nCuN layer serve as isolation layer and bottom contact, respectively.  On top of the MTJ a \n30 nm thick (Ti 10W90)100-xNx layer and a 300  nm thick Al layer were deposited and \npatterned, serving as top contact and transducer to convert ultrafast  laser pulses  into \nacoustic phonon pulses, see Fig.  1(a). The tunnel magnetoresistance of the MTJ is \napproximately 100% and its  magnetic anisotropy field is 100  Oe as being determined \nfrom  a measured Stoner -Wohlfarth astroid . The laser pulses were obtaine d from a \nfemtosecond laser (500  fs pulse w idth, 300  kHz repetition rate, 1040  nm center \nwavelength, ~15  nJ pulse energy) and focused to a 1/e2 diamete r of ~8  µm on the top Al \nlayer , see Fig. 1(c). A more detailed description of the sample properties can found in a \nprevious study .20 Although the measurements detailed in this paper have been \nconducted on the nanopillar stack described above, samples with different nanopillar \ndimensions show ed similar  results.  \nTo obtain information about  the acoustic  pulse s generated in the Al layer by the \nfemtosecond laser pulse s, we performed a pump and probe reflect ometry experiment.21 \nTherefore , the pump laser pulse  was focused on th e Al layer generating  acoustic  pulses \nvia thermoelastic coupling . The probe pulse , which can be time delayed with respect to \nthe pump pulse, was focused on the Al layer next to the pump beam and the reflection \nchange of the probe beam was recorded versus time delay between pump and probe  \nbeams. In principle, both, picosecond strain  pulse s (propagating into the sample) and 4 \n SAWs  (propagating along the surface) can be detected. We will show below that f or the \nexcitation of magnetization of precession only SAWs are relevant.  The time traces of \nSAWs for three different  distances  (5.3 µm, 8.9 µm, and 13 .4 µm) between pump and \nprobe  pulses are shown in Fig. 1(b).  The SAW s have a bipolar shape which is similar to \nthe shape of a typical picosecond strain pulse propagating into the sample but with \nmuch longer duration  of several ns. The duration difference between  the picosecond  \nstrain pulses and the nanosecond SAW  is related to the thermal distribution in the Al \nlayer after laser excitation  (which extends several µm along the surfac e but only several \n10 nm normal to the surface) . Comparing the time delay between the measured SAWs  \nwith the distances between pump and probe , we estimate the SAW velocity to be \n(3.3±0.5)  µm/ns.  This value com pares well with literature data  for SAW s in Al  of \n2.95  µm/ns .22 \nThe tunnel resistance  of MTJs depends on the angle 𝜙 between the magnetization \norientation of the free layer and the fixed layer23:  \n𝑅(𝜙)=𝑅⊥[1+𝐵cos(𝜙)]−1,  (1) \nWhere  𝐵=𝑅AP−𝑅P\n𝑅AP+𝑅P, 𝑅⊥=2𝑅AP𝑅P\n𝑅AP+𝑅P and 𝑅P and 𝑅AP are the resistance  values for 𝜙=0° \n(parallel alignment , P) and 𝜙=180°  (antiparallel alignment , AP), respectively.  Due to \nthe magnetoelastic effect , a phonon pulse leads to an angular excursion of \nmagnetization of the free la yer, which in turn cause s a change of the tunnel resistance . \nWe neglect  the magnetoelastic effect  in the fixed layer , since its magnetization is well \npinned by the synthetic antiferromagnet . The measurement of SAW -induced \nmagnetization dynamics in the  MTJ is  realized by time resolved measur ements of tunnel \nresistance  changes . It is worth to mention that the penetration depth of SAW into the \nsubstrate is close to its wave length  𝜆≈ several µm.24 The MTJ is located just 150 nm \nbelow the top Al layer  and, thus, well within the pen etration depth of the SAWs . \nTo study the SAW induced magnetization  dynamic s in the MTJ, we measured the time \nresolved voltage change  under constant bias current  due to tunnel resistance  changes 5 \n by using a sampling oscilloscope with 50   input impedance. The trigger signal for the \noscilloscope was obtained from the femtosecond laser system such that the oscilloscope \ntime axis is synchronized with the laser pulses. A  small  current ( IDC = ±400 µA) was \napplied to the MTJ through a bias tee ,16 see Fig.  1(a).  To separate signals due to tunnel \nresistance  changes from unwanted  back ground signals, two oscilloscope traces  taken \nfor +IDC and -IDC were subtracted from each other  after averaging over 2000 individual \ntraces. Static magnetic fields up to 300  Oe at various in -plane angles θ were  applied as  \nindicated  in Fig . 1(c).  \nWe now comment on the experimental results. Figure s 2(a) and (b) show  the measured \noscilloscope  traces for two different magnetic field amplitudes and two different laser \nexcitation positions . While the magnetic field was either Hθ=85° = 120 Oe (red, lower \ncurves ), corresponding to an AP state  of the MTJ, or Hθ=85° = -150 Oe (black , upper \ncurves ), corresponding to a P state  of the MTJ, the excitation spots were either right \nabove the MTJ (a) or 10  µm away  (b). In all cases, an  oscillatory behavior due to \nprecession is observed.  The magnetization precession is approximately a factor of three \nlarger  in the AP state than in the P state and it maintains several nanoseconds due to \nthe long SAW duration , see Fig.  1(b). The difference between the AP and P state is linked \nto the depende nce on the applied magnetic field angle, which will be discussed below. \nComparing the precession obtained for the different laser excitation spots it is obvious \nthat for excitation  10 µm away from the MTJ nanopillar, the precession starts about 3  ns \nlater as compared to an excitation right above the MTJ. This time delay agrees very well \nwith the propaga tion time of the SAWs, see Fig.  1(b), underlining  that it is indeed the \nSAW which induces the magnetization dynamics.  We can exclude the  strain pulse \npropaga ting into the substrate as being responsible for the magnetization precession , \nsince it only takes about 80  ps for the strain pulse to propagate from the top Al layer to \nthe MTJ.   \nFor laser excitation right above the MTJ, the  AP signal experiences a slow decay  with in \nthe first ns after the laser pulse hits the Al contact, see Fig.  2(a). This is not observed in  \nFig. 2(b) where the red curve is shifted along the y axis for clarity . The slowly decaying 6 \n signal in Fig.  2(a) is due to the heat diffusion from the  Al surface to the buried MTJ  and \nresult s from temperature dependence of spin polarized tunneling . Using a previously \npublished method,20 we estimate the time -dependent  temperature rise in the MTJ to be \napproximately 3.2  K. We checked the temperature dependence of the pre cess ion \nfrequency obtained from STT experiments with step -like voltage pulses using an electric \nheating stage  below the MTJ sample . A temperature increase  up to 30  K has almost no \ninfluence on the precession.  Due to this dependence and because  the slow ly decay ing \ncontribution  of Fig.  2(a) vanishes  for an excitation position ~10 µm away  from the MTJ , \nwe can safely exclude  the laser -induced temperature rise and, thus , a thermal STT  as the \norigin of the observed magnetization dynamics.  \nTo further study  the SAW induced magnetization  dynamic s, the applied magnetic field  \namplitude and angle was  systematic ally varied for laser excitation right above the MTJ \nWe find that the largest precession amplitude occurs for a m agnetic field of \napproximately 150  Oe (AP state) , see Fig.  2(c). The magnetic field dependence is \nattributed to a  resonance between the induced magnetoelastic mode  and the SAW \nfrequency .12,25,26 In addition to the dependence on magnetic field  amplitude we also \nfind a pronounced dependence on the magnetic field angle, see Fig. 2(d). The largest \nprecession occurs for an  applied field close to  the hard axis of the MTJ  (θ = 85°). The \namplitude of pre cession gradually decreases  when the magnetic field is changed  from \nθ = 85° to θ = 0°. At θ  = 0°, only small magnetization pre cession can be found in a small \nmagnetic field range which is close to  the switching  field. We believe that this angle \ndependence of the precession signal mainly indicates the existence of non -uniform \nmodes localized close to the edges of a ferromagnetic stripe. These modes typically \noccur for external magnetic fields applied perpendicular to an anisotropy field.17,27 We \nwill comment on the existence of non -uniform modes in more detail further below .  \nWe now analyze the dependence of the precession f requency on the SAW stimulus. This \nanalysis is important to identify the type of magnetization modes being excited by the \nSAW. In Fig.  3(a) we have plotted the precession frequency of the magnetoelastic mode \nversus applied magnetic field amplitude for θ = 85° and two optical excitation energies 7 \n (blue squares and red triangles for 7.5 nJ and 15  nJ, respectively ). The right -hand -side \ninset of Fig.  3(a) shows the precession frequency spectrum versus magnetic field for \n15 nJ excitation pulse energy. In general , the precession frequency increases with \napplied field amplitude and does not depend on the optical excitation power. The latter \ndependence further demonstrates  that the SAW induced precession  is not the result of  \noptical heating of the sample. In such a c ase we would expect a pronounced \ndependence on optical excitation power.  \nIn previous studies on magnetoeleastic effects in thin magnetic layers, precession \nsignals were read out using optical techniques6,11,14,25,28 and it was difficult to determine \nthe spin wave mode  of the unperturbed  and perturbed  system (with respect to elastic \nperturbations). In our work  we measure the precession using fast electrical read out of \nthe MTJ resistance . Since its magnetization dynamic s driven by charge current  pulses  \nhas been well studied16,18 our experimental  scheme all ows for the comparison of the \nunperturbed magnetization modes with the SAW induced modes . With this comparison \nwe are able to assign the SAW -induced magnetization mode to a spatially non -uniform \nspin wave mode being mainly localized at the edges of the fre e layer as explained in the \nfollowing.   \nWe applied  180-ps-long current pulses with a n amplitude of approximately 8  mA and a  \nrepetition rate of 100  kHz to the MTJ. The f ree magnetization precession induced  by the \ncurrent pulse through STT was measured after the pulse decay using a fast sampling \noscilloscope16,20 with the magnetic fi eld applied along the hard axis . The precession \nfrequencies obtained from this experiment  are visual ized in Fig.  3(a) as black circles \nversus magnetic field amplitude . The left -hand -side inset of Fig.  3(a) shows the \nprecession frequency spectrum versus magnetic field. While the magnetic field \ndependence of the free precession qualitatively resembles the magnetic field \ndependence of the SAW induced pre cession, the SAW frequencies are always larger \nthan the free precession frequencies . Most likely, this frequency difference results from \nthe transition of pure spin waves to magnetoelastic waves. Calculating the dispersion \nrelation of magnetoelastic waves one finds that in the crossover region between pure 8 \n spin waves and pure elastic waves two magnetoelastic branches exist, having larger and \nsmaller frequencies than the pure spin wave.29 We believe  that the SAW mainly excites \nthe larger frequency branch of the magnetoelastic mode. We can rule out that the \nmagnetization change induced by the magnetoelastic effect causes a significant shift of \nthe precession frequency of a certain magnetization mode. If this were true , we would \nhave observed a dependence of the precession frequency on the optical excitation \nenergy in Fig.  3(a).  \nSince  the magnetic field dependence of the SAW induced precession closely resembles \nthe free precession, it is very likely tha t the magnetization modes are equal. It is \ntherefore possible to simulate  the free precession in the free layer to obtain qualitative \ninformation about the magnetization mode  being induced by the SAW. A detailed \ncalculation of  the magnetoelastic mode is beyond  the scope of this paper. For the \nsimulation of free precession we have employed the micr omagnetic simulation tool \nmumax30. The following parameters have been used for the simulation : saturation \nmagnetisation  Ms = 796 kA/m , perpendicular magnetic anisotropy Ku = 7.96  kJ/m3, and \nexchange constant A ex = 20 pJ/m. The simulated sample, which has the  same  nominal  \ndimensions as the in the experiment , is discretized  in elements of 2  nm × 2 nm × 2.6 nm. \nFigure 3(b) shows the simulated free precession frequency versus  applied magnetic field \nalong the y direction for an excitation of the free layer with a sinc function having  a cut -\noff frequency of 15  GHz and a total simulation time  of 50  ns. The simulated behavior \nqualitatively agrees with the measurements . However, the simulated frequencies are \nhigher than experimental ly observed . We attribute this difference to certain param eters \nof the simulation which are not exactly know n such as the exact shape of the free layer \n(e.g., deviation f rom the no minal  shape after lithograph y). The upper inset of Fig.  3(b) \nshows the y component of the static magnetization vector  𝐦𝑒𝑞(𝑥,𝑦) of the \nmagnetization mode at an applied magnetic field of 225  Oe. The lower inset shows the y \ncomponent of the dynamical part 𝛅𝐦(𝑥,𝑦) of the magnetization mode, whic h is \nobtained from the magnetization 𝐦 at a certain time instant using 𝛅𝐦=𝐦−\n(𝐦∙𝐦𝑒𝑞)𝐦𝑒𝑞. Both, the static and the dynamical parts clearly show that the free 9 \n precession mode is  confined clo se to the edges of the  free layer. This, in turn, strongly  \nsuggests that  also the SAW induced magnetization dynamics in the free layer of the MTJ \nis linked to a  spatially non -uniform  mode  being localized close to the edges .  \nFinally, t he coherent nature of  SAW -induced magnetization dynamics also enables the \ncoherent manipulation of the magnetizat ion by means of two SAW pulses.  Figure 4(a) \nshows time -resolved magnetization traces (again obtained from tunnel resistance  \nmeasurements using a fast sampling oscilloscope) employing two laser pulses, which \nwere focused onto the same position  10 µm away from MTJ and can be time delayed \nwith respect to each other . A clear p eriodic dependence of destructive and constructive  \ninterferences  of magnetic oscillation s on the delay between two laser pulse s is observed, \nreveal ing that we can either amplify or quench the precession by coherent control . It \nshould be noted that coherent control of the magnetization precession can also be \nachieved when keeping the time delay between the two laser pulses constant and \nvarying the position of one laser  spot with respect to the other . The measured \ninterference pattern ag rees very well with a calculated  superposition of two separately \nmeasured mag netization traces;  see Fig . 4(b). The coherent control study directly \nextends previous coherent control experiments on magnetization dynamics2,4,31 –33 to an \nindustrially relevant  device and, thus, might prove useful for future application.   \nIn summary, we have employed MTJs to study  magnetization dynamic s driven  by \nfemtosecond -laser -pulse -induced SAW s. We could identify a spatially non -uniform \nmagnetization mode as being excited b y the SAWs and demonstrated coherent control \nof magnetization in MTJs using acoustic pulses. Our results open prospects  for future \napplications, in which magnetization has to be controlled on ultrafast time scales. \nAdditionally, they provide valuable infor mation for spincaloritronic studies in which \ntemperature and temperature gradients are generated by excitation with ultrafast \noptical pulses. Our time -resolved experiments  directly show that the optically generated \nacoustic pulses must not be neglected and , under certain experimental conditions, even \nfully determine the optically induced magnetization dynamics.  10 \n The authors thank Piet Schmidt for the loan of the laser system used in the present work \nand Erik Benkler for technical assistance . Funding by the European Metrology Research \nProgramme (EMRP, Joint Research Project EXL04 , RMG  15SIB06 -RMG2 ) is gratefully \nacknowledged . The EMRP is jointly funded by the EMRP participating countries within \nEURAMET and the European Union.  J.D.C. is thankful f or the support of FCT  grant \nSFRH/BD/7939/2011.   \n  11 \n References  \n1 Y. Otani, M. Shiraishi, A. Oiwa, E. Saitoh, and S. Murakami, Nat. Phys. 13, 829 (2017).  \n2 A. V. Kimel, A. Kirilyuk, P.A. Usachev, R. V. Pisarev, A.M. Balbashov, and T. Rasing, Nature 435, \n655 (2005).  \n3 N. Bergeard, M. Hehn, S. Mangin, G. Lengaigne, F. Montaigne, M.L.M. Lalieu, B. Koopmans, and \nG. Malinowski, Phys. Rev. Lett. 117, 147203 (2016) . \n4 K. Yamaguchi, M. Nakajima, and T. Suemoto, Phys. Rev. Lett. 105, 237201 (2010).  \n5 K. Shen and G.E.W. Bauer, Phys. Rev. Lett. 115, 197201 (2015).  \n6 M. Foerster, F. Macià, N. Statuto, S. Finizio, A. Hernández -Mínguez, S. Lendínez, P. V. Santos, J. \nFontcu berta, J.M. Hernàndez, M. 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Wadley, V. Holy, S.A. Cavill, \nA. V. Akimov, A.W. Rushforth, and M. Bayer, Appl. Phy s. Lett. 103, 32409 (2013).  \n29 E. Schlömann, J. Appl. Phys. 31, 1647 (1960).  13 \n 30 A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia -Sanchez, and B. Van \nWaeyenberge, AIP Adv. 4, 107133 (2014).  \n31 S. Garzon, L. Ye, R.A. Webb, T.M. Crawford, M. Covington, and S. Kaka, Phys. Rev. B - Condens. \nMatter Mater. Phys. 78, 180401 (2008).  \n32 D. Bossini, S. Dal Conte, Y. Hashimoto, A. Secchi, R. V. Pisarev, T. Rasing, G. Cerullo, and A. V. \nKimel, Nat. Commun.  7, 10645 (2016).  \n33 J.W. Kim, M. Vomir, and J.Y. Bigot, Sci. Rep. 5, 8511 (2014).  \n \n  14 \n Figure 1  \n \nFig. 1. (a) Schematic sketch of the experimental setup . (b) Time -domain reflectance \nmeasurements of  SAW s induced by femtosecond laser pulses for three different \ndistances (5.3 µm, 8.9  µm, and 13.4  µm) between pump and probe  pulse . The dashed \nvertical lines denote the minima of the SAWs used to calculate the SAW velocity. (c) \nMicroscope image of the electrical contacts above the MTJ nanopillar with two di fferent  \nlaser heating positions  (orange dots) and orientation of the MTJ’s easy axis as well as \nthe external ly applied in-plane magnetic field.  \n  \n15 \n Figure 2  \n \nFig. 2. Time -resolved voltage change across the MTJ for different excitation conditions \nand magnetic fields. (a) Optical excitation right above the MTJ nanopillar and (b) 10 µm \naway from the nanopillar. The black and red  traces correspond to parallel  (P, Hext=-\n150 Oe) and antiparallel (AP, Hext=120  Oe) alignment of the MTJ for θ = 85°. In (b) t he \nred curve is shifted by -0.2 mV for clarity. (c) Optical excitation right above the MTJ and \ndifferent magnetic field amplitudes for θ = 85°. All curves are shifted along the y axis for \nclarity. (d) Optical excitation right above the MTJ and a magnetic fie ld amplitude of \nHext=120  Oe but for different field angle s θ. The black and blue curves are shifted along \nthe y axis for clarity.   \n  \n16 \n Figure 3  \n \nFig. 3. (a) Precession frequency versus applied magnetic field amplitude along the hard \naxis for two different optical excitation energies  (blue squares for 7.5 nJ and red \ntriangles for 15 nJ) and for excitation with 180 -ps-long current pulses (black dots ). The \ninsets show the corresponding frequency spectra versus magnetic  field . (b) Simulated \nfrequency of the free precession in the free layer versus applied magnetic field \namplitude . Upper inset: Simulated equilibrium configuration of the magnetization \ncomponent in the y direction in the free layer for an  applied magnetic field amplitude of \n225 Oe. Lower inset: Dynamical changes of the magnetization component in the y \ndirection  in the free layer for an applied magnetic  field amplitude of 225 Oe. \n  \n17 \n Figure 4  \n \nFig. 4. (a) Coherent control of magnetization dynamics using two time -delayed laser \npulses.  (b) Calculated magnetization dynamics from superposition of  two separatel y \nmeasured magnetization traces, where the time delay of one laser pulse  is kept fix and  \nthe other is subsequently shifted in time .  \n"    },    {        "title": "1001.1716v1.Nonlocal_effects_on_magnetism_in_the_diluted_magnetic_semiconductor_Ga__1_x_Mn__x_As.pdf",        "content": "arXiv:1001.1716v1  [cond-mat.str-el]  11 Jan 2010Nonlocal effects on magnetism in the diluted magnetic semico nductor Ga 1−xMnxAs\nUnjong Yu,1,2Abdol-Madjid Nili,1Karlis Mikelsons,1,3Brian Moritz,4Juana Moreno,1and Mark Jarrell1\n1Department of Physics and Astronomy & Center for Computatio n and Technology,\nLouisiana State University, Baton Rouge, LA 70803, USA\n2Department of Applied Physics, Gwangju Institute of Scienc e and Technology, Gwangju 500-712, Korea\n3Department of Physics, University of Cincinnati, Cincinna ti, Ohio 45221, USA\n4Stanford Institute for Materials and Energy Science,\nSLAC National Accelerator Laboratory and Stanford Univers ity, Stanford, CA 94305, USA\n(Dated: October 30, 2018)\nThe magnetic properties of the diluted magnetic semiconduc tor Ga 1−xMnxAs are studied within\nthe dynamical cluster approximation. We use the k·pHamiltonian to describe the electronic struc-\nture of GaAs with spin-orbit coupling and strain effects. We s how that nonlocal effects are essential\nfor explaining the experimentally observed transition tem perature and saturation magnetization.\nWe also demonstrate that the cluster anisotropy is very stro ng and induces rotational frustration\nand a cube-edge direction magnetic anisotropy at low temper ature. With this, we explain the\ntemperature-driven spin reorientation in this system.\nPACS numbers: 75.50.Pp, 75.30.Gw, 78.55.Cr\nThe discovery of high temperature ferromagnetism in\ndilutedmagneticsemiconductors(DMS)hasstimulateda\ngreatdealofattention [1]. The interest in these materials\nis due to possible applications in spintronics [2] as the\nsource of a spin polarized current or as the base material\nforachipthatcansimultaneouslystoreandprocessdata.\nIn spite of extensive studies, our understanding of fer-\nromagnetism in these systems is far from complete [3].\nThere are a few serious difficulties in the theoretical\nstudy of DMS: (i) The magnetic interaction between lo-\ncal magnetic moments and itinerant carrier spin, which\nis responsible for the high transition temperature ( Tc), is\nstrong and outside the Ruderman-Kittel-Kasuya-Yosida\n(RKKY) regime. (ii) There exists strong disorder from\nthe random distribution of magnetic ions. (iii) Nonlo-\ncal effects are expected to be crucial judging from the\nspatially oscillating and anisotropic magnetic interaction\npredicted in theory[4–7] and observedin experiments [8].\nThe mean-field study by Dietl et al.[9] captures the\nmain features of DMS systems qualitatively and some\neven quantitatively. However, it ignores strong corre-\nlations, disorder effects, and spatial fluctuations, and\nfails to describe some DMS materials, such that subse-\nquent studies have brought their approach into question\n[10, 11]. Studies [12] based on the dynamical mean-field\ntheory (DMFT) [13] have made considerable improve-\nments by including strong correlation and disorder ef-\nfects. However, the local nature of the DMFT presents\nseverelimitations when studying this system. The effects\nof short-range fluctuations and spatial correlations were\nshown to be important in the classical Heisenberg model\n[14]. In this letter, we show that nonlocal effects may be\nequally important for the itinerant carriers, which me-\ndiate the effective interaction between local moments in\nDMS.\nThe dynamical cluster approximation (DCA) [15] sys-\ntematically incorporates nonlocal effects as the cluster\nsize (Nc) increases while retaining strong correlations.WhenNc= 1, the DCA is equivalent to the DMFT,\nand exact results are approached as Nc→ ∞. Since all\nthe possible disorder configurations are considered in a\ncluster, the DCA is alsoa better approximationfordisor-\nderaveragethan the coherentpotentialapproximationor\nDMFT by includingmulti-impurity scatteringterms[16].\nThus, the DCAis an ideal method forstudying DMS sys-\ntems. In this Letter, we study the magnetic properties\nof the prototypical DMS system Ga 1−xMnxAs using the\nDCA and the k·pmethod, which describes the non-\ninteracting band structure of pure GaAs. We show that\nnonlocaleffectsareveryimportantforproperlycapturing\nthe magnitude of Tc, the saturation magnetization, and\nthe magnetic anisotropy of this material. In particular,\nwe show that the strong cluster anisotropy is responsible\nforthemagneticanisotropyalongthecube-edgedirection\nand the spin reorientation at low temperature.\nThe model Hamiltonian we adopt is\nH=Hk·p+Jc/summationdisplay\nIS(RI)·J(RI), (1)\nwhere the first term describes the electronic structure\nof the host material (GaAs) in the k·papproximation\nand the second term introduces a magnetic interaction\nbetween the carrier spin ( J) and the local magnetic mo-\nment (S) of Mn at position RI. The large magnitude of\nthe Mn magnetic moment ( S= 5/2) allows us to treat it\nclassically. This model is generally accepted to describe\nDMS [4, 5, 9, 12, 17, 18], since a mean-field treatment\nof the Hamiltonian [9, 17, 18] is able to explain many\nphysical properties of the system. For the k·pHamilto-\nnian (Hk·p), we adopt a 4 ×4 Luttinger-Kohn model de-\nscribing heavy and light hole bands with spin-orbit cou-\npling, but ignoring the conduction and split-off bands.\nWe use the Luttinger parameters γ1= 6.98,γ2= 2.06,\nandγ3= 2.93 [19]. Biaxial strain is included in Hk·p\nthrough the strain tensor εxx=εyy=ε0= ∆a/aand\nεzz= (−2c12/c11)ε0with the ratio of elastic stiffness2\n0.00.20.40.60.81.0\n 0 50 100 150 200 250 300Magnetization\nTemperature [K](a)Nc= 1 \nNc=16\nNc=22\nNc=24\n050100150200250\n 0 0.01 0.02 0.03 0.04 0.05 0.06Tc [K]\np(b) Nc= 1 \nNc=16\nKu et al.\nWang et al.\nSùrensen et al.\nFIG. 1: (Color online) (a) Magnetization of Ga 1−xMnxAs cal-\nculated by DMFT ( Nc= 1) and DCA ( Nc= 16,Nc= 22,\nandNc= 24) with Mn doping x= 0.05 and hole concen-\ntrationp= 0.025. No strain effect is considered. (b) Fer-\nromagnetic transition temperature ( Tc) as a function of hole\nconcentration ( p) with DMFT ( Nc= 1) and DCA ( Nc= 16)\nforx= 0.05. Experimental results [26–28] are also shown.\nThe Mn concentration in experiments is x=0.085 (Ku et al.),\nx=0.017-0.09 (Wang et al.), andx=0.05 (Sørensen et al.).\nconstants c12/c11= 0.46. Parameter ais the lattice con-\nstant of Ga 1−xMnxAs, and ∆ ais the difference between\nthe lattice constants of Ga 1−xMnxAs and the substrate.\nWe use the hydrostatic deformation potential av= 1.16\neV and the shear deformation potential b=−2.0 eV [19].\nIn addition to the parameters of the k·pHamiltonian,\nwe must determine the value of the exchange coupling\nJc. It can be obtained from photoemission [20], infrared\n[21, 22], and resonant tunneling [23] spectroscopy and\nmagneto-transportexperiments[24], whichgive Jc= 0.6-\n1.5 eV. We adopt Jc= 1 eV throughout this Letter.\nFigure 1(a) shows the magnetization per Mn ion of the\nGa1−xMnxAs system as a function of temperature with\nDMFT ( Nc= 1) and DCA ( Nc= 16,Nc= 22, and\nNc= 24). We chose three fcc clusters that are perfect\naccording to Betts et al.[25]. The difference between\nDCA and DMFT stems from nonlocal effects, not cap-\ntured in DMFT. The Tcwith DCA is far lower than that\nobtained with DMFT, approaching the regime of experi-\nmental values [26–28] [see Fig. 1(b)]. Another important\npoint is the reduction of the saturation magnetization\nat low temperature, consistent with experiments [28–30].\nThis behavior is a product of the rotational frustration\n[4], to be discussed in detail later. This effect also re-0ë  15ë  30ë  \n      φ\n  ε0 = −0.2% z (b) M\nθ\nφNc=16\nNc=22\nNc=240ë 30ë 60ë 90ë\n      θ\n   ε0 = −0.2%\nε0 = +0.2%\n(a)Nc=16\nNc=22\nNc=24\n0ë  15ë  30ë  \n00.20.40.60.81φ\nT/Tc(c)Welp et al.\nSawicki et al.\nMasmanidis et al.\nWang et al.\nFIG. 2: (Color online) (a) Polar angle ( θ) of the magneti-\nzation as a function of the normalized temperature ( T/Tc)\nfor two strain values. Compressive ( ε0=−0.2%) and tensile\n(ε0= +0.2%) strain induce in-plane and perpendicular mag-\nnetic anisotropy, respectively. (b) Azimuthal angle ( φ) of the\nmagnetization with respect to the [110] direction with com-\npressive strain. Experimental results [31–34] are provide d in\n(c) to compare with (b).\nducesTc. The dependence of Tcon hole concentration\n(p) is shown in Fig. 1(b). Tcattains a maximum value\nwhen holeconcentrationishalfofMnconcentration, con-\nsistent with previous DMFT studies [12].\nNext, we studied the magnetic anisotropy of\nGa1−xMnxAs. The magnetic anisotropy of this system\ndepends on strain, hole concentration, and temperature\nin a complicated manner, but generally it has in-plane\nanisotropywith compressivestrainandperpendicular-to-\nplane anisotropy with tensile strain [1]. With compres-\nsive strain, the magnetization changes direction within\nplane from [110] or [1 ¯10] at high temperature to [100] or\n[010] at low temperature [31–34] [see Fig. 2(c)]. As is\nshown in Fig. 2(a) and 2(b), the DCA reproduces experi-\nmental results on the dependence of magnetic anisotropy\non strain and temperature remarkably well.\nWhile the strain dependence of the magnetic\nanisotropy was explained within the mean-field the-\nory [17, 18], the spin reorientation within the plane\nhas not been explained yet. In the absence of strain,\nGa1−xMnxAs has diagonal magnetic anisotropy within\nthe mean-field theory because the heavy holes, which\ndominate at low carrierdensity, havelargerdensity along3\n0ë 30ë 60ë 90ë\n   ∠ (R, Mdimer)\n(c)Nc = 16\nNc = 22\nNc = 24\n0ë 30ë 60ë  \n   ∠ (Mtot, Mdimer)\n(d)NN NNN 3rdNN(a) (b)●\n●● ●●\n●● ●\nFIG. 3: (Color online) (a) DCA results at low temperature\nshow that the dimer magnetization ( Mdimer) is prefered to be\nperpendicular to the vector connecting the two Mn ions ( R)\nfor nearest-neighbor (NN) Mn-dimers. (b) For larger dimers ,\nMdimeraligns with the total magnetization ( Mtot) irrespec-\ntive ofR. The dotted and solid arrows represent Mtotand\nMdimer, respectively. (c) and (d) Angle between Mdimerand\nRand between MdimerandMtot, respectively, at T=23.2 K,\nx=0.05,p=0.025, and ε0=−0.2%. The left panel is for the\n12 NN dimers [ e.g., when the two Mn ions are at (0 ,0,0)\nand (a/2,a/2,0)], the middle panel is for the 6 next-nearest-\nneighbor (NNN) dimers [ e.g., when the two Mn ions are at\n(0,0,0) and ( a,0,0)], and the right panel is for the third-\nnearest-neighbor (3rdNN) dimers [ e.g., when the two Mn ions\nare at (0 ,0,0) and (a,a/2,a/2)].\nthe diagonal direction in k-space [17]. With compressive\nstrain, it has the anisotropy in [110] or [1 ¯10]. This ex-\nplains the magnetic anisotropy at high temperature.\nThe [100]or [010]anisotropyat low temperature is due\nto the cluster anisotropy originating from the anisotropic\ninteraction between neighboring Mn ions. Because Mn\nionsaredistributedrandomlythroughoutthesystem, the\nnumber of Mn ions within a cluster varies between zero\nandNc. We call a cluster a monomer (dimer) when it\nincludes one Mn ion (two Mn ions). All possible dis-\ntributions of Mn ions are considered effectively in this\ncalculation, but at low doping, the magnetic properties\nare dominated by Mn-monomers and Mn-dimers. Since\nthere is no cluster anisotropy in monomers, we investi-\ngate magnetization of Mn-dimers in detail. Because of\ntranslational symmetry, we need to consider only Nc−1\ndimers. The two Mn ions are nearest-neighbors in 12\ndimers, next-nearest-neighbors in 6 dimers, and third-\nnearest-neighbors in Nc−19 dimers. Figure 3 shows\nthe magnetization direction of each dimer obtained by\nthe Monte-Carlo method at low temperature. The mag-\nnetization of the nearest-neighbor Mn-dimer is always× 2 (a) × 8 × 2\n× 8 (b) × 40.74 0.76 0.78 0.80 \n0ë  15ë  30ë  45ë〈 MNNdimer 〉\nφ(c)\nFIG. 4: (Color online) (a) Magnetic configurations of the 12\nnearest-neighbor (NN) Mn-dimers ( MNN\ndimer) that maximize\nthe total magnetization ( Mtot) whenφ= 0◦(i.e.Mtotis\nalong [110] or [1 ¯10]). The dotted and solid arrows represent\nMtotandMNN\ndimer, respectively. The numbers below each dia-\ngram indicate the degeneracy of the Mn-dimer configurations .\nThe dimer configuration with MNN\ndimerperpendicular to the\nvector connecting the two Mn ions ( R) is energetically fa-\nvored. (b) Same as (a) but when φ= 45◦(i.e.Mtotis along\n[100] or [010]). (c) Maximum value of the average MNN\ndimervs.\nφ.\nperpendicular to the vector connecting the two Mn ions.\nThis cluster anisotropy prevents the magnetic moment\nof some dimers from aligning parallel to the total mag-\nnetization and leads to the rotational frustration [4, 5].\nWhen the two Mn ions are farther apart, this anisotropy\nis very weak and magnetization of the dimer aligns par-\nallel to the total magnetization.\nThe cluster anisotropy can also explain the enhance-\nment of Tcup to 260 K in the quasi-two-dimensional\nδ-doped systems [35]. When the Mn ions are within\none plane, all the nearest-neighbor Mn-dimers can point\nin the same direction (perpendicular-to-plane direction)\nwithout rotational frustration. This may induce a larger\nsaturation magnetization and higher Tcin these systems.\nNotably, due to this cluster anisotropy, the maximum\ntotal magnetization depends on the magnetization di-\nrection, and this dependence introduces another type of\nmagnetic anisotropy. When we assume the magnetic mo-\nment is perpendicular to the vector connecting the two\nMn ions and in-plane magnetization, the maximum value\nof the average magnetization of the 12 nearest-neighbor\ndimers is calculated to be\n/angbracketleftMNN\ndimer/angbracketright=1\n6/bracketleftBig\ncosφ+sinφ+/radicalbig\n3+sin(2 φ)\n+/radicalbig\n3−sin(2φ)/bracketrightBig\n(2)\nfor the fcc lattice. This becomes maximal when the mag-\nnetization is along [100] or [010], as shown in Fig. 4(c).\nSince larger magnetization leads to lower magnetic en-\nergy, it introduces magnetic anisotropy along [100] or\n[010]. We note that this effect becomes unimportant\nat high temperature, where the total magnetization is4\nsmall. Thus, the spin reorientation from [110] or [1 ¯10]\nat high temperature to [100] or [010] at low tempera-\nture is captured within our calculation. This behavior\narises from the multi-impurity scattering and cannot be\nobtained within the mean-field theory or the DMFT.\nIn summary, we investigated the magnetic properties\nof the prototypical DMS system Ga 1−xMnxAs by DCA\ntogether with the k·pmethod. We showed that nonlo-\ncal effects, not included in the mean-field theory or the\nDMFT but included in the DCA for Nc>1, are very im-\nportant to quantitatively explain the Tc, the saturation\nmagnetization, and the magnetic anisotropy of this ma-\nterial. We find that the edge-direction anisotropy at low\ntemperature is due to the cluster anisotropy, and this al-\nlows us to reproduce the spin reorientationin this system\nremarkably well.\nWe acknowledge helpful discussions with Z. Xu. This\nwork was supported by the National Science Founda-\ntion through OISE-0730290, DMR-0548011, and DMR-\n0706379. BM acknowledges support from the U.S. De-\npartment of Energy, Office of Basic Energy Sciences, un-\nder contract DE-AC02-76SF00515. Portions of this re-\nsearch were conducted with high performance computa-\ntional resources provided by the Louisiana Optical Net-\nwork Initiative (http://www.loni.org).\n[1] H. Ohno, Science 281, 951 (1998).\n[2]Spintronics , edited by T. Dietl, D. D. Awschalom,\nM. Kaminska, and H. Ohno (Academic Press, New York,\n2008).\n[3] For a review, see T. Jungwirth, J. Sinova, J. Maˇ sek,\nJ. Kuˇ era, A. H. MacDonald, Rev. Mod. Phys. 78, 809\n(2006).\n[4] G. Zar´ and and B. Jank´ o, Phys. Rev. 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Lett. 95, 017201 (2005)."    },    {        "title": "2105.14721v1.Theoretical_development_in_the_viscosity_of_ferrofluid.pdf",        "content": "1 \n Theoretical d evelopment in the viscosity of ferrofluid  \nAnupam Bhandari  \nDepartment of Mathematics, School of Engineering  \nUniversity of Petroleum & Energy Studies (UPES)  \nEnergy Acres Building, Bidholi  \nDehradun - 248007, Uttarakhand, India  \nE-mail: pankaj.anupam6@gmail.com  \nhttps://www.upes.ac.in  \nAbstract:  \nThe v iscosity of ferrofluid ha s an important role in liquid sealing of the hard disk drives, biomedical \napplications as drug delivery, hyperthermia, and magnetic resonance imaging. In the absence of a magnetic \nfield, the viscosity of ferrofluid depends on the volume concentration of magnetic nanoparticles including  \nsurfactant layers. However, under the influence of  a stationary magnetic field, the viscosity of ferrofluid \ndepends on the angle between the applied magnetic field and vorticity in the flow. If this angle is 90o, then \nthere is a maximum increase in the vi scosity . If the magnetic field and the vorticity in the flow  are parallel \nto each other , then there is no change in the viscosity  since  the applied magnetic field does not change the \nspeed of the rotation of magnetic nanoparticles in the fluid.  \nThe viscos ity of ferrofluid in the presence of an alternating magnetic field demonstrates interesting \nbehavior.  When field frequency matches with the relaxation time, known as resonance cond ition, then there \nis no impact of an alternating magnetic field in the visco sity of ferrofluid. If the frequency of an alternating \nmagnetic field is less than resonance frequency, then an alternating magnetic field increases the viscosity \nof ferrofluid. Using higher frequency than resonance condition reduces the visc osity of ferro fluid and \nresearchers reported this incident as the negative viscosity effect. If the frequency of an alternating magnetic \nfield tends to infinite, then ferrofluid ceases to feel a magnetic field. In this case, there is no impact of an \nalternating magnetic  field on the viscosity of ferrofluid.    \nKeywords: Ferrofluids, viscosity, magnetic field, volume concentration, negative viscosity, \nferrohydrodynamics.   \n1. Introduction to Ferrofluids  \n1.1. Formation of ferrofluid  2 \n Ferrofluid is also  known as magnetic fluid. Ferrofluids are not directly available  in nature [1]. These fluids \nare artificially synthesized of colloidal mixtures of carrier liquid, typically water or oil, and magnetic \nnanoparticles [1], [2] . Surfactants are used in the colloidal mixtures to ensure the stability of the \nferrofluid [1], [3] . Surfactants prevent the agglomeration of the magnetic particles [4], [5] . Ferrofluids , which \nwork at zero gravity region,  are recognized as colloidal suspension of superparamagneti c materials [1], [4], \n[5]. In ferrofl uid preparation, we consider the size of the magnetic particles  5-15 nanometers  in diameter \nand volume fraction up to approximately 10% [1], [3] .  We select water -based carrier liquid for medical \npurpose s, mineral oil and silicon organic -based carrier liquid for lubrication  and sealing system , and \nhydrocarbon -based carrier liquids for printing devices [4], [5] . Properties of the ferrofluid depend on the \nsize of the magnetic particles and their magnetization [1], [3] . The s tability  of ferrofluid is ensured by the \nthermal motion which prevents the agglomeration and precipitation [4], [5] . Thermal motion increases with \ndecreasing the size of the particles. The magnetic properties dis appear if the size of magnetic particles is \nless than 1 -2nanometers [4], [5] . Surfactant is important for the stability of ferrofluid and long -chain \nmolecules (e.g., OOH, H 2OH, H 2NH 2, and so on) are used for surfactants [4], [5] . Surfactants produce the \nchemical reaction in the colloidal mix ture and this reaction reduces the size of the magnetic particles [5]. \nReduction of the size of magnetic particles loses magnetic properties [5].  \nFor the application of ferrofluid, it is essential ly required that ferrofluid should be very stable concerning  \nthe temperature in the presence of a magnetic field [6]. Therefore, agglomeration of the magnetic particles \nmust be avoided for proper commercial use [6]. Kilkuchi et al. described experimentally that the reaction \ntemperature from 20 0 to 250 oC, the size of the magnetic particles increases from 5 t o 11 nanometers [7]. \nConsidering the nonhom ogeneous distribution of pH and dielectric constant, the microemulsion  method is \nuseful to prepare stable ferrofluid [8]. Imran et al. synthesized highly stable ferrofluid using  motor oil as \nbase fluid and found  13 nanometers  average particle  size of  γ-Fe2O3 [9]. The stability of ferrofluid remains \nprotective for small particles size since their concentrations and dipolar couplin g energies are too low for \nfield-induced dipolar structure formation [10]. D. P. Lalas  and S. Carmi  investigated the stability of \nmotionless ferrofluid using the concept of Rayleigh number [11].  Tari et al. have investigated the r ole of \nmagnetization and temperature for the stability of diester -based Fe 3O4  ferrofluid [12]. In the transi tion from \nthe laminar to turbulence motion of ferrofluid, we need to check the accuracy of the numerical solution \nalong with the stability and uniqueness [13]. Thermal and magnetic stress is an important class of surface \ninteractions  of magnetic particles near Curie temperature [14].  \nStability of ferrofluid in the presence of non -uniform magnetic field investigated through stability \ncoefficients using microsensors [15]. Internal structures and macroscopic physical properties [16] and \narrangements of the mag netic particles in special structures [17] can serve in the development of the 3 \n applications of the magnetic fluid. There is no suitable procedure to explain the thermodynamical and \ndynamical properties of magnetic fluids with developed microstructure [18]. There is a critical chain  number \nfor phas e transition  in ferrofluid [19] and the magnetic field exhilarate s the formation of chains from Ferro -\nparticles [20]. Wiedenmann investigated the stability of nanoparticles in ferrofluid against electrostatic \nrepulsion or surfacta nts[21]. Gazeau et al. demonstrated the Brownian motion nanoparticles in ferrofluid \nunder applied magnetic field [22]. Sousa  et al. investigated the surface magnetic properties of \nNiFe 2O4 nanoparticles [23]. Raikher et al. demonstrated a magneto -optical way to analyze the internal and \nexternal magnetic relaxation in magnetic fluids [24].  Raikher et al. explained the particle orientation \ndynamics using the general Fokker -Planck equation [25]. The synthesis of ferrofluid, stability , and \ncharacteristics with magnetic properties have been investigated for different types of magnetic \nnanoparticles [26]–[28].  \n1.2.  Real-life a pplications of ferrofluids   \nIn the mid -1960s, ferrofluid  was developed by NASA as a method for con trolling fluid in space since the \nflow of ferrofluid  can be controlled by the external magnetic field. Conventional ferrofluids are useful in \nliquid seals, shock absorbers, controlling heat in loudspeakers, printing for paper money , and lubrication \nbearings [1]–[5]. The commercial use of ferrofluids  has been available in the literature [29], [30] . Ferrofluids \ncan be used in heat transfer and damping problems [31]. The viscosity of ferrofluid has an important role in \nthese applications [32]. Ferrof luids ha ve an important role in biomedical applications for diagnostic and \ntherapy. Drug delivery, hyperthermia treatments , and magnetic reso nance imaging are some of the major \napplication s in the field of biomedical engineering [33]–[35]. Fundamental and applied research in the \nferrofluid, now adays, researchers are trying to develop magneto -optical devices and ferrofluidic \nsensors [36], [37] . Even, researchers have shown the application of ferrofluid in environmental \nengineering [38].  \n2. Governing equations  in ferrohydrodynamic flow  \nTo describe the behavior of ferrofluid  in different flow domains, the researchers use the following set of \nequations [39]–[42]: \nThe equation of continuity [1], [43], [44]  \n𝛁.𝒗=0                                                                                                            (1) \nThe equation of motion [39], [44] –[46] \n𝜌𝑑𝒗\n𝑑𝑡=−𝛁𝑝+𝜇∇2𝒗+𝜇0(𝑴.𝛁)𝑯+𝐼\n2𝜏𝑠 𝛁×(𝝎p −𝜴)                                     (2) 4 \n The equation of magnetization [3], [40]  \n𝑑𝑴\n𝑑𝑡=𝝎p×𝑴−1\n𝜏𝐵(𝑴−𝑴𝟎)                                                                            (3) \nThe equation of rotational motion [39], [44], [47]  \n𝐼𝑑𝝎p\n𝑑𝑡=𝑴×𝑯−𝐼\n𝜏𝑠(𝝎p−𝜴)                                                                           (4) \nThe equation of instantaneous magnetization [4][1]–[3], [5]  \n𝑴𝟎=𝑛𝑚𝐿 (𝜉)𝑯\n𝐻,       𝜉=𝑚𝐻(𝑡)\n𝑘𝐵𝑇,              𝐿(𝜉)=coth 𝜉−𝜉−1                                      (5) \nThe Energy Equation [4], [5], [48], [49]  \n𝜌𝑐𝑝[𝜕𝑇\n𝜕𝑡+(𝑽.∇)𝑇]=𝑘∇2𝑇−𝜇0𝑇𝜕𝑀\n𝜕𝑇𝑽.𝛁𝐻+𝜇Φ        (6) \nWhere  𝒗 denotes the velocity, 𝜌 denotes the density, 𝜇 denotes the viscosity, 𝜇0 denotes the permeability \nof free space, 𝑴 denotes the magnetization, 𝐻 denotes the magnetic field intensity , 𝝎p  denotes the angular \nvelocity of magnetic particles in the flow , 𝜴 denotes the vorticity in the flow , 𝜏𝐵 Brownian relaxation time, \n𝐼 denotes the moment of inertia, 𝜏𝑠 denotes the rotational relaxation time, 𝑴𝟎 denotes the equilibrium \nmagnetization, 𝑚 denotes the magnetic moment, 𝑛 denotes the number of particles, 𝐿(𝜉) denotes the \nLanggevin function for paramagnetism, 𝑘𝐵 denotes the Boltzmann constant and 𝑇 denotes the temperature, \n𝑐𝑝  denotes the specific heat at constant pressure, 𝑘 denotes the thermal conductivity and Φ denotes the \nviscous dissipation term.  \nTwo mechanism s of ferrofluid Neel relaxation and Brown ian relaxation time have an important role in the \nstudy of ferrofluid [50], [51] . This mechanism shows that the magnetization in ferrofluid can relax after \nchanging the strength of the magnetic field [52]. In Brown ian relaxation time occurs due to nanoparticles \nrotation of the colloidal mixture and Neel relaxation time occurs due to rotation of the magnetic vector \nwithin the particle [1], [3] –[5]. \nA Brownian relaxation time time 𝜏𝐵 is given by [1] \n                        𝜏𝐵=3𝜇𝑉\n𝑘𝐵𝑇                                                                                     (7) \nwhere  𝑉=𝜋(𝑑+2𝑠)3\n6 denotes the hydrodynamic volume of the particle including surfactant layers. Here 𝑑  \ndenotes the diameter of the particle and 𝑠 denotes the thickness of the surfactant layer.  5 \n Under certain material conditions, the magnetic moment may rotate inside the particle relative to crystal \nstructure [39], [53] . This kind of relaxation of the colloidal particles can take place if the thermal energy is \nhigh enough to overcome the energy barrier provided by the crystallographic anisotropy of the magnetic \nmaterial [4], [5] .  The height of this energy barrier is given by 𝐾𝑉, where 𝐾 is the anisotropy constant of the \nmaterial [3], [5] . For the case  𝐾𝑉≪𝑘𝐵𝑇, the thermal energy is large enough t o induce fluctuations of the \nmagnetization inside the grain with a characteristic time 𝜏𝑁:  \n                          𝜏𝑁=1\n𝑓0exp(𝐾𝑉\n𝑘𝐵𝑇)                                                                                  (8) \nwhere 𝑓0 is a frequency hav ing the approximate value 109 𝑠−1. \nWhen 𝜏𝑁≪𝜏𝐵, relaxation occurs by the Neel mechanism, and the material is called to possess intrinsic \nsuperparamagnetism [2], [5] . When 𝜏𝐵≪𝜏𝑁, the Brownian mechanism is determined and the material \nexhibits extrinsic superparamagnetism [3], [5] . However, if the smaller time constant is much greater in \ncomparison to the time scale of the experiment, then the same may be regarded as ferromagnetic [2], [53] . \nAn effective relaxation time combined from Neel and Brown times for the relevant particle diameter can \nbe calculated as [1], [3], [39] : \n                        𝜏𝑒𝑓𝑓=𝜏𝐵𝜏𝑁\n(𝜏𝐵+𝜏𝑁)                                                                                        (9) \nFor the specific types of ferrofluid, the researchers are using the thermophysical properties of ferrofluid. \nThe follow ing mathematical  equations are being used by the researchers [54]–[58]: \n𝜌𝑛𝑓=𝜌𝑓[(1−𝜑)+𝜑(𝜌𝑠\n𝜌𝑓)]           (10) \n(𝜌𝑐𝑝)𝑛𝑓=(𝜌𝑐𝑝)𝑓[(1−𝜑)+𝜑(𝜌𝑐𝑝)𝑠\n(𝜌𝑐𝑝)𝑓]        (11) \n𝜇𝑛𝑓=𝜇𝑓\n(1−𝜑)2.5             (12) \n𝑘𝑛𝑓\n𝑘𝑓=𝑘𝑠+2𝑘𝑓−2𝜑(𝑘𝑓−𝑘𝑠)\n𝑘𝑠+2𝑘𝑓+𝜑(𝑘𝑓−𝑘𝑠)           (13) \nWhere 𝜌𝑓 the density of the base fluid, 𝜑 denotes the volume concentration of nanoparticles, (𝜌𝑠,𝜌𝑓  ) \ndenotes the density of nanoparticles and base fluid, respectively, ((𝜌𝑐𝑝)𝑠,(𝜌𝑐𝑝)𝑓 ) denotes the heat \ncapacitance  of solid and base fluid, respectively, 𝜇𝑓 dynamic viscosity of the base fluid, (𝑘𝑠,𝑘𝑓 ) denotes \nthe thermal conductivity of nanoparticles and base fluid respectively.  \n3. Viscosity of ferrofluid  \n3.1. Viscosity in the absence of the magnetic field  6 \n In the absence of the magnetic field, the viscosity of ferrofluid depends on the volume concentration. T he \nmathematical expression for the viscosity of ferrofluid is given as [3], [59] –[61]: \n𝜇(𝐻=0)=𝜇𝑐(1+5\n2𝜑̃)           (14) \n𝜇(𝐻=0) denotes  the viscosity of ferrofluid in the absence of the magnetic fluid, 𝜇𝑐 denotes the viscosity of \nthe base fluid, 𝜑̃ denotes the volume concentration of magnetic nanoparticles including the surfactant layer.  \nThe volume concentration 𝜑̃ of the suspended mat erial in the colloidal suspension can be expressed as:  \n𝜑̃=𝜑(𝑑𝑚+2𝑠\n𝑑𝑚)3\n             (15) \nWhere 𝜑 denotes the volume concentration of the magnetic nanoparticles, 𝑑𝑚 denotes the diameter of the \nmagnetic core and 𝑠 denotes the thickness of the surfacta nt layers.  \nIn 1970, the  first improvement Eq. (14 ) was given as [3], [51], [62] : \n𝜇(𝐻=0)=𝜇𝑐(1+5\n2𝜑̃+31\n5𝜑̃2)            (16) \nIn 1985, Rosensweig has modified the expression for the viscosity of ferrofluid as[1]: \n𝜇(𝐻=0)=𝜇𝑐\n(1−5\n2𝜑̃+𝑏𝜑̃2)            (17) \nWhere 𝑏=(5\n2𝜑̃𝑐−1)\n𝜑̃𝑐2 and 𝜑̃𝑐 denotes the cr itical volume fraction of the suspended material.  \n  Recently for the viscosity of the solution, the researchers use the following expressions [63]–[66]: \n𝜇𝐻=0=𝜇𝑐\n(1−𝜑)2.5            (18) \n3.2. Viscosity of ferrofluid in the presence of magnetic field  \nIn the presence of a magnetic field, the rotation of the magnetic particle was also considered in the viscosity \nof ferrofluid. In 1969, researchers have introdunced the theortical expressions for the viscosity of ferrofuid \nunde the influence of external  magnetic field [67]. This expression of viscosity depends on the strength and \ndirection of the magnetic field. This expression is [3]–[5], [68] : \n 𝜇𝐻=𝜇𝑠(1+5\n2𝜑̃+3\n2𝜑′𝑠𝑖𝑛2𝜀1)         (19) 7 \n where  𝜇𝑠 denotes the viscosity of the solvent, 𝜑′ denotes the volume fraction of the particles including the \nsurfactant layer, 𝜑̃ denotes the volume fraction of all suspended material including dispersants or free \nsurfactants molecules. The term 𝑠𝑖𝑛2𝜀1 includes the magnetic part in the following form [3]–[5]: \n𝑠𝑖𝑛2𝜀1=1\n2(1+1\n𝜉𝑟2)−[1\n4(1+1\n𝜉𝑟2)2\n−1\n𝜉𝑟2𝑠𝑖𝑛2𝜃]1\n2        (20) \nwhere 𝜃 denotes the angle between the vorticity of the flow and the magnetic field direction and 𝜉𝑟 denotes \nthe ratio of the magnetic torque and the viscous torque acting on a particle. The relation between 𝜉𝑟 and \nmagnetic field can be written as [4], [5] : \n1\n𝜉𝑟=𝜇0𝑚𝐻\n4𝜋𝜇𝑠𝑑3𝛾𝑟              (21) \nwhere, 𝛾𝑟 denotes the shear rate.   \nIn a planer Cou ette flow, the additional viscosity due to magnetic field is given as [3], [69] –[73]: \n∆𝜇=𝜇0𝜏𝐵𝑀0𝐻\n4(1+𝜇0𝜏𝑠𝜏𝐵𝑀0𝐻\n𝐼)                (22) \nwhere 𝜏𝑠, 𝜏𝐵 and 𝐼 are defined as:  \n𝜏𝑠=𝑑12𝜌1\n45𝜇 ,  𝜏𝐵=𝜋𝑑13𝜇\n2𝑘𝐵𝑇,    𝐼=1\n10𝑑12𝜌1𝜑1          (23) \nWhere 𝑑1 denote the mean diameter of the magnetic particles, 𝜌1 denotes the mean density of the whole \nsolid fraction and 𝜑1 denotes the volume concentration including surfactant layer. Using Eq. (20), the \nexpress ion for the viscosity in Eq. (22 ) becomes [3], [47], [74], [75] : \n∆𝜇=3\n2𝜑1𝜇𝜉−tanh 𝜉\n𝜉+tanh 𝜉             (24) \nEq. (24 ) shows the expression for the rotational viscosity due to the magnetic field when the magnetic field \nis perpendicular to the vorticity in the flow. For arbitrary angle between magne tic field and vorticity, Eq. \n(24) can be written as [4], [39], [75] : \n∆𝜇=3\n2𝜑1𝜇𝜉−tanh 𝜉\n𝜉+tanh 𝜉 𝑠𝑖𝑛2𝛽            (25) \nMagnetic torque 𝑴×𝑯 and viscous torque (𝝎𝒑−𝜴) in ferrofluid flow generates the rotational viscosity \nin ferrofluids [1], [44], [73] .  In the presence of the magnetic field, the fluid and particles in the colloidal \nsuspensions rotate  with different angular velocities  and this difference of angular velocituies  creates an \nadditional resistance in the flow. The equilibrium of these two torques give:  8 \n 𝜇0𝑴×𝑯=6𝜇𝜑1(𝝎𝒑−𝜴)           (26) \nUsing Eq. (2 6), Bacari et al. demonstrated the theoretical expression for the viscosity [39], [44], [76]   \n∆𝜇=3\n2𝜇𝜑1𝛺−𝜔𝑝\n𝛺            (27) \nThe relative viscosity can  be presented as [3], [42], [77] : \n𝑅=𝜇(𝐻)−𝜇(𝐻=0)\n𝜇(𝐻=0)=∆𝜇\n𝜇(𝐻=0)            (28) \nIn the presence of strong the magnetic field [3]–[5]:  \n𝑅(𝐻→∞)=3\n2𝜑1 𝑠𝑖𝑛2𝛽            (29) \nIf the magnetic field is perpendicular to the vorticity in the flow,  then the maximum relative viscosity is [3], \n[78] \n𝑅𝑚𝑎𝑥=3\n2𝜑1             (30) \nIf size of the particle is 10 nanometers including the surfactant layer, the maximum increase in the viscosity \nof ferrofluid is approximately 40%.  In a weak magnetic field, we consider  tanh 𝜉=𝜉−1\n3𝜉3+0(𝜉5), \ntherefore,  \n𝜉−tanh 𝜉\n𝜉+tanh 𝜉≈1\n6𝜉2             (31) \nA weak magnetic field represents the following expression of  relative viscosity of ferrofluid:  \n𝑅≈1\n4𝜑1𝜉2              (32) \nResearches have used these expressions of viscosity in ferrohydrodynamic flow in different regimes [79]–\n[84].  \n3.3. Negative viscosity effects in ferrofluid  \nIn the presence of a stationary  magnetic field, the viscosity of ferrofluid  always increases. However, in the \npresence of an alternating magnetic field , the viscosity of ferrofluid depends not only on the strength but \nalso on the frequency of the alternating magnetic field. In the presence of the alternating magnetic field, the \nrelation between the angular velocity of the particle and field frequency is given by [39], [85], [86] : \n𝜔𝑝=𝛺(1−𝜉2\n6(1−𝜔02𝜏𝐵2)\n(1+𝜔02𝜏𝐵2)2)           and        𝜉=𝜇0𝑚𝐻\n𝑘𝐵𝑇           (33) 9 \n Where 𝜔0 denotes the frequency of an alternating magnetic field.  \n For the weak field (𝑚𝐻 ≤𝑘𝐵𝑇) the viscosity of ferrofluid is [39], [87], [88] : \n∆𝜇=1\n4𝜇𝜑1𝜉2(1−𝜔02𝜏𝐵2)\n(1+𝜔02𝜏𝐵2)2           (34) \nHere, 𝜔0𝜏𝐵  denotes the dimensionless field frequency. The condition 𝜔0𝜏𝐵=1 is known as resonance \ncondition. This condition can be achieved when the frequency of alternating magnetic field s matches w ith \nrelaxation time. In this case, there is no impact of rotational viscosity due to the magnetic field [44]. A case \n𝜔0𝜏𝐵<1, the expressions in Eq. (3 4) remains positive. This case always enhances the viscosity of \nferrofluid due to applied magnetic field. For the case, 𝜔0𝜏𝐵>1, the expressions in Eq. (3 4) becomes \nnegative [39], [44] . In other  words, afte r applying a magnetic field, the viscosity of ferrofluid becomes less \nthan without magnetic field . This viscosity reduction is known as the negative viscosity effect. If we take, \n𝜔0𝜏𝐵→∞, the impact of Eq. (34 ) in the viscosity of ferrofluid becomes negl igible [39].  \nThe viscosity of ferrofluid for arbitrary amplitude is [39]: \n ∆𝜇=1\n4𝜇𝜑1𝜉2(2−tanh 𝜖−2tanh 𝜖\n𝜖); 𝜖=𝜋\n2𝜔0𝜏𝐵        (35) \nConsidering limit  𝜖→∞, Eq. (32) becomes ∆𝜇=1\n4𝜇𝜑1𝜉2 (viscosity due to stationary magnetic field).   \n4. Conclusions  \nThis review on the viscosity of ferrofluid has presented the recent fundamental theor etical development of \nviscosity of ferrofluid. The viscosity of ferrofluid has a major role in the application of ferrofluid in sealing, \nbiomedical engineering , and heat transfer analysi s. 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Zvezdin1, 2,y\n1Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991 Moscow, Russia\n2Moscow Institute of Physics and Technology (State University), 141700 Dolgoprudny, Russia\n3Moscow Technological University (MIREA), 119454 Moscow, Russia\n4Institute for Molecules and Materials, Radboud University, Nijmegen 6525 AJ, The Netherlands\nWe derive an e\u000bective Lagrangian in the quasi-antiferromagnetic approximation that allows to\ndescribe the magnetization dynamics for uniaxial f-d(rare-earth - transition metal) ferrimagnet\nnear the magnetization compensation point in the presence of external magnetic \feld. We perform\ncalculations for the parameters of GdFeCo, a metallic ferrimagnet with compensation point that is\none of the most promising materials in ultrafast magnetism. Using the developed approach, we \fnd\nthe torque that acts on the magnetization due to ultrafast demagnetization pulse that can be caused\neither by ultrashort laser or electrical current pulse. We show that the torque is non-zero only in\nthe non-collinear magnetic phase that can be acquired by applying external magnetic \feld to the\nmaterial. The coherent response of magnetization dynamics amplitude and its timescale exhibits\ncritical behavior near certain values of the magnetic \feld corresponding to a spin-\rop like phase\ntransition. Understanding the underlying mechanisms for these e\u000bects opens the way to e\u000ecient\ncontrol of the amplitude and the timescales of the spin dynamics, which is one of the central problems\nin the \feld of ultrafast magnetism.\nPACS numbers: 79.20.Ds, 75.50.Gg, 75.30.Kz, 75.78.-n\nINTRODUCTION\nMost of the prominent advances in the \feld of ultrafast magnetism have been achieved by using thermal mechanism\nof magnetization control [1{7]. These studies rooted from the pioneering work by Beaurepaire et al. [8] on ultrafast\nlaser-induced demagnetization of Ni. In this experiment, partial destruction of magnetic order was found at much\nfaster rates that were believed to be possible prior to that publication. Since then, the \feld of ultrafast magnetism has\nbeen rapidly growing and the possible channels of ultrafast angular momentum transfer have been studied extensively\n[9]. Ultrafast demagnetization can be achieved by applying ultrashort laser pulses [1{7], or, alternatively, by using\nshort pulses of electric currents [10, 11].\nIn the last decades, GdFeCo and other rare-earth - transition metal compounds (RE-TM) have been in the center\nof attention in this regard [12]. For example, all-optical switching has been demonstrated for the \frst time in GdFeCo\n[13]. It was found that the switching is possible due to di\u000berent rates of sublattice demagnetization, which enables\nultrafast magnetization reversal to occur because of the angular momentum conservation [1, 2].\nIn many RE-TM compounds, GdFeCo and TbFeCo being part of them, realization of the magnetization compen-\nsation point is possible. At this point, the magnetizations of the two antiferromagnetically coupled sublattices with\ndi\u000berent dependencies on temperature become equal and the total magnetization of the material turns to zero. In\nthe presence of the external magnetic \feld, a record-breaking fast subpicosecond magnetization switching was found\nin GdFeCo across the compensation point [13]. In addition, a number of anomalies in the magnetic response was\nobserved near this point [14{16], which has never been explained theoretically. All said above illustrates the impor-\ntance of understanding the role of the compensation point in the dynamics and working out an appropriate tool for\nits description.\nE\u000ecient control of the amplitude and the timescales of the response of the magnetic system to an ultrafast de-\nmagnetizing impact on a medium is one of the most important issues in the area of ultrafast magnetism nowadays\n[9, 14, 15]. Understanding of the mechanisms and of the exhaustive description of the subsequent spin dynamics is\nalso a long-standing goal that will help to promote the achievements of this area towards practical applications in\nmagnetic recording [3, 17], magnonics [18] and spintronics [19]. In this work, we expand the understanding of response\nof magnetic system of a uniaxial f-dferrimagnet near the compensation point in the external magnetic \feld to an\nultrafast demagnetizing pulse, which can be induced either by a femtosecond laser or an electric current pulse. We\npresent a theoretical model and calculations, which allow to describe the ultrafast response of the system that resides\nin an angular phase before the impact. We show that in this magnetic phase the coherent precessional response is\npossible and the subsequent magnetization dynamics may become greatly nonlinear and is governed by large inter-\nsublattice exchange \feld [20]. We derive the e\u000bective Lagrangian that governs the dynamics of the system near thearXiv:1901.03072v2  [cond-mat.mtrl-sci]  28 Jan 20192\ncompensation point and obtain the torque acting on the magnetizations of the two sublattices due to demagnetiza-\ntion. In ref. [15], the critical response of the amplitude and the time of the signal rise have been found in GdFeCo\nin external magnetic \feld along the easy axis. At given laser pump \ruences, the response was found to be negligible\nin collinear phase, but it was dramatically large in angular one. We elaborate on this example and show that the\ncritical behavior of the response is the consequence of the second-order magnetic phase transition from collinear to an\nangular in the external magnetic \feld. We \fnd that these e\u000bects are pronounced in the vicinity of the compensation\npoint, where the phase transitions cross each other[21{23]. Thus, the proposed model explains a range of important\nexperimental observations as well as allows for developments of methods and tools of magnetization control by setting\nthe temperature near the compensation point and applying magnetic \feld. Moreover, by changing the composition\nof the alloy, the [24], the position of the magnetization compensation point can be tuned arbitrary close to the room\ntemperature. Our results might open new ways for technologies for ultrafast optical magnetic memory.\nEFFECTIVE LAGRANGIAN AND RAYLEIGH DISSPATION FUNCTION\nOur approach is based on Landau-Lifshitz-Gilbert equations for a two-sublattice (RE-TM) ferrimagnet. These\nequations are equivalent to the following e\u000bective Lagrangian and Rayleigh dissipation functions:\nL=Mf\n\r(1\u0000cos\u0012f)@'f\n@t+Md\n\r(1\u0000cos\u0012d)@'d\n@t\u0000\b(Mf;Md;H); (1)\nR=Rf+Rd;Rf;d=\u000bMf;d\n2\r\u0010\n_\u00122\nf;d+ sin2\u0012f;d_'2\nf;d\u0011\n(2)\nwhere\ris the gyromagnetic ratio, MdandMfare the magnetizations, \u0012d(TM) and\u0012f(RE) are the polar, 'dand\n'fare the azimuthal angles of d- andf- sublattices correspondingly in the spherical system of coordinates with z-axis\naligned along the external magnetic \feld H. \b(Mf;Md;H) is the thermodynamic potential for the system that we\ntake in the following form:\n\b =\u0000MdH+\u0015MdMf\u0000MfH\u0000Kf(Mfn)2\nM2\nf\u0000Kd(Mdn)2\nM2\nd; (3)\nwhere\u0015is the intersublattice exchange constant, nis the direction of the easy axis and Kf;dare the anisotropy\nconstants for f- andd- sublattices, respectively.\nNext, we transfer to description in terms of the antiferromagnetic L=MR\u0000Mdand the total magnetization\nM=MR+Mdvectors. In the vicinity of the compensation point the di\u000berence between the sublattice magnetizations\njMR\u0000Mdj\u001cLis small. The two vectors are parametrized using the sets of angles \u0012;\"and';\f, which are de\fned\nas:\n\u0012f=\u0012\u0000\"; \u0012d=\u0019\u0000\u0012\u0000\";\n'f='+\f; 'd=\u0019+'\u0000\f:(4)\nIn this case the antiferromagnetic vector is naturally de\fned as L= (Lsin\u0012cos';Lsin\u0012sin';Lcos\u0012).\nFollowing the work [25] we use the quasi-antiferromagnetic approximation to describe the dynamics near the mag-\nnetization compensation point. F In this approximation the canting angles are small \"\u001c1,\f\u001c1, and we can\nexpand the Lagrangian (1) and the corresponding thermodynamic potential up to quadratic terms in small variables:\nL=\u0000m\n\r_'cos\u0012\u0000M\n\rsin\u0012\u0010\n_'\"\u0000\f_\u0012\u0011\n\u0000\b;\n\b =\u0000K(l;n)2\u0000Hmcos\u0012\u0000\"MH sin\u0012+\u000e\n2\u0000\n\"2+ sin2\u0012\f2\u0001\n:(5)\nHerem=MR\u0000Md,M=MR+Md,K=KR+Kdis the e\u000bective uniaxial anisotropy constant, l=L=Lis\nthe unit antiferromagnetic vector \u000e= 2\u0015MdMRand we assume the anisotropy to be weak K\u001c\u0015M. For GdFeCo\nwith 24% Gd and compensation point near 283 K, we assume the following values of parameters: M\u0019800 emu/cc,\nK= 1:5\u0002105erg/cc,\u0015= 18:5 T/\u0016B,\u000e\u0019109erg/cc and mchanges in the range between 150 emu/cc and \u000050\nemu/cc at \felds H\u0019H\u0003\u00194 T. The characteristic values of small angles \u000fand\fare of the order of 10\u00002.3\nNext, we exclude the variables \"and\fby solving the Euler-Lagrange equations. Substituting them into the\nLagrangian (5), we obtain the e\u000bective Lagrangian, which describes the dynamics of a uniaxial ferrimagnet in the\nvicinity of the compensation point:\nLeff=\u001f?\n2 _\u0012\n\r!2\n+mcos\u0012\u0012\nH\u0000_'\n\r\u0013\n+\u001f?\n2sin2\u0012\u0012\nH\u0000_'\n\r\u00132\n+K(l;n)2; (6)\n\beff(H) =\u0000mHcos\u0012\u0000\u001f?\n2H2sin2\u0012\u0000K(l;n)2; (7)\nReff=\u000bM\n2\r\u0010\n_\u00122+ sin2\u0012_'2\u0011\n(8)\nwhere\u001f=2M2\n\u000e. In GdFeCo \u001f\u00191:6\u000210\u00003and\u000b\u00190:05. In the derivation above we assumed the gyrotropic factor\n\rand Gilbert damping constant \u000bto be the equal for both sublattices. Taking into account the di\u000berence between\nthese values for di\u000berent sublattices will lead to the angular momentum compensation e\u000bect at certain temperature.\nThe Lagrangian, Rayleigh function and equations of motion preserve the same form in this case if we substitute the\nparameters \rand\u000bwith temperature-dependent factors e\rande\u000bde\fned as:\n1\ne\r=1\n\u0016\r\u0012\n1 +M\nm\rf\u0000\rd\n\rf+\rd\u0013\n=Md\n\rd\u0000Mf\n\rf\n(Md\u0000Mf);1\n\u0016\r=1\n2\u00121\n\rd+1\n\rf\u0013\n;e\u000b=(\u000bd\rf+\u000bf\rd)\n(\rf+\rd)1\n1 +M\nm\rf\u0000\rd\n\rf+\rd(9)\nThis allows to reproduce the angular moment compensation phenomenon, which was studied experimentally in ref.\n[14].\nEXCITATION OF THE SPIN DYNAMICS\nThe proposed approach presents a powerful tool allowing analyzing coherent magnetization dynamics in ferrimagnets\nthat occurs under a broad range of conditions. Let us consider the following example that poses an important problem\nin the \feld of ultrafast magnetism. An femtosecond laser pulse strikes the uniaxial ferrimagnet (for instance, of\nGdFeCo, TbFeCo type) in the presence of external static magnetic \feld. The impact of the laser pulse leads to the\ndemagnetization of one or both of the sublattices. What coherent magnetization dynamics will occur as a consequence\nof this impact? The proposed model can be further developed in order to answer to this question and is applicable\nfor small values of demagnetization \u000eM.\nIn our framework the spin dynamics in ferrimagnet is described by Euler-Lagrange equations of the formd\ndt@L\n@_q\u0000@L\n@q=\n\u0000@R\n@_q, whereq=\u0012; ' are the polar and azimuthal angles describing the orientation of the antiferromagnetic vector\nL, correspondingly. Let us consider a particular case when the easy magnetization axis is aligned with the external\nmagnetic \feld, which leads to the presence of azimuthal symmetry in the system. In this case n= (0;0;1). In this\nparticular case the Euler-Lagrange equations can be rewritten as:\n\u001f?\n\r2\u0012=@Leff\n@\u0012\u0000@Reff\n@_\u0012;d\ndt@Leff\n@_'=\u0000@Reff\n@_'(10)\nThe nonlinear equations that are similar to Eqs. (10) and describe the spin dynamics of two-sublattice ferrimagnets\nwere obtained in the work [26] under the conditions H= 0 andReff= 0. Over the short time of demagnetization the\nsecond equation can be approximately treated as a conservation law and the conserving quantity (angular momentum\nof magnetization precession J) stays approximately constant as @L=@'= 0 due to the Noether theorem:\nJ=@Leff\n@_'=\u00001\n\r\u0014\nmcos\u0012+\u001f?sin2\u0012\u0012\nH\u0000_'\n\r\u0013\u0015\n=const (11)\nLet the moment of time t= 0\u0000denote the moment before the laser pulse impact and system initially is in the\nground state de\fned by the ground state angles \u0012(0\u0000) =\u00120,'(0\u0000) ='0, and their derivatives _ '(0\u0000) = 0, _\u0012(0\u0000) = 0.4\nDepending on the external parameters and preparation of the sample, the system might reside in one of the two\npossible antiferromagnetic collinear phases or in angular phase, which are separated by the magnetic phase transition\nlines [27]. If the demagnetization due to the laser pulse action is small, it produces the changes in the values of Mf,Md\nandMof the order of percent or less, whereas the change of m(which is approximately equal to total magnetization\nnear the compensation point) may be of several orders of magnitude, as its value is almost compensated. In what\nfollows, we assume that the demagnetization is associated only with change of m, namelym=m0+ \u0001m(t). As we\nwill see below, the change in this quantity already leads to several drastic e\u000bect in dynamics.\nTherefore, the conservation law (11) leads to the emergence of azimuthal dynamics _ '(t) at the demagnetization\ntimescales (\u0001 t) due to demagnetization pulse \u0001 m(t):\n_'(t) =\r\u0001m(t)\n\u001f?cos\u00120\nsin2\u00120(12)\nWe see that the torque is non-zero only in the angular phase, where 0 <\u00120<\u0019. Emergence of the azimuthal spin\nprecession as a result of demagnetization of the medium is similar to the well-known Einstein-de-Haas e\u000bect, where\nthe demagnetization leads to azimuthal precession of the body. Subsequently, this azimuthal spin dynamics leads to\nthe emergence of polar dynamics \u0012(t), which is most commonly measured in pump-probe experiments of ultrafast\nmagnetism, by acting as an e\u000bective \feld Heff=H\u0000_'\n\rin the Lagrangian (5). We can then view the Lagrangian\nas depending only on variable \u0012and the e\u000bective \feld Heff. At demagnetization \u000em\u00180:01Min GdFeCo the value\nof _'can reach up to 1 THz, and the corresponding e\u000bective magnetic \feld is of the order of 10 T. Note that initial\nstate of the system corresponds to the condition@\b\n@\u00120(Heff=H) = 0. We can rewrite the Euler-Lagrange equation\nfrom eq. (10) for polar angle as follows:\n\u001f?\n\r2\u0012+@\b(Heff)\n@\u0012=\u0000\u000bM\n\r_\u0012: (13)\nOr, alternatively:\n\u001f?\n\r2\u0012+msin\u0012Heff\u0000\u001f?sin\u0012cos\u0012\u0012\nH2\neff\u00002K\n\u001f?\u0013\n=\u0000\u000bM\n\r_\u0012 (14)\nBy integrating this equation over the short demagnetization pulse duration \u0001 twe obtain the state of system after\nthe laser pulse impact at t= 0+, which is characterized by the initial conditions\n\u0012(0+) =\u00120;_'(0+); '(0+) =Z\u0001t\n0_'(t)dt;_\u0012(0+) =Z\u0001t\n0\u0012(t)dt\nThe value \u0001 tis of the order of the optical pulse length. It may also include the time of restoration of the magnetization\nlength (or the value of m). After the moment of time 0+ free magnetization precession occurs in the model. Analysis\nof the spin dynamics under laser pump excitation will lead to emergence of critical dynamics near the second-order\nphase transitions to the collinear phases where \u0012= 0;\u0019, as is already seen from (12). We will discuss this behavior\nbelow.\nCRITICAL DYNAMICS\nIn a simple case of a quick decay of demagnetization (at the exciton relaxation timescales) with \u0001 m(t) = \u0001m,\n0<t< \u0001t; \u0001m(t) = 0,t>\u0001t, we obtain the initial condition from (14):\n_\u0012(+0)\u0019\u0014\n\u0000\u0012\n2cos2\u00120\nsin\u00120+ sin\u00120\u0013\nH+m0\n\u001f?cos\u00120\nsin\u00120+\u0001m\n\u001f?cos\u00120\nsin3\u00120\u0015\r2\n\u001f?\u0001m\u0001t=B(\u00120)\u0001m+O(\u0001m2): (15)\nThis quantity de\fnes the initial angular momentum of the polar spin precession that is induced in the system due to\nthe optical spin torque created by the femtosecond laser pulse. The amplitude of oscillations is proportional to the\ninitial condition (15). Its dependence on the external magnetic \feld is illustrated in Fig. 1 for di\u000berent temperatures\nfor magnetic parameters of GdFeCo uniaxial ferrimagnet. At low values of external magnetic \felds there is only\ncollinear ground state in the ferrimagnet and above certain \feld Hsfthe transition to an angular state occurs [21].\nThe schematic of the magnetic phase diagram for GdFeCo is shown in insertions in Fig. 1. At T= 275 K and T= 2885\nFIG. 1. The amplitude of the magnetization precessional response after the demagnetization due to the femtosecond laser\npulse action in GdFeCo ferrimagnet near the compensation point at di\u000berent temperatures. Insertions: the schematic of the\nmagnetic phase diagram. There are two antiferromagnetic collinear phases with Mddirected along(opposite) to the external\nmagnetic \feld above(below) the compensation temperature TM. They are separated by the \frst-order phase transition line\n(blue). Above them, an area where the angular phase exists, which is \flled with gray color. The black solid lines are the\nsecond-order phase transition lines. The dashed lines corresponds to the \fxed temperature in the plot. The red dot is the point\nof phase transition for this temperature.\nK the phase transitions are of the second order, which corresponds to a smooth transition from angle \u00120= 0 to\u00120>0,\nand the divergence of the response occurs at Hsf. Immediately above the compensation temperature the transition is\nof the \frst order and the behavior of the response above the is more complex; however, there is no critical divergence.\nThe critical behavior of the signal amplitude was observed experimentally for GdFeCo in ref. [15].\nAnother feature in the dynamics described by the proposed model is the critical behavior of the characteristic\ntimescales that occurs in the vicinity of the second-order phase transitions. To demonstrate this e\u000bect analytically,\nwe assume small deviations of \u0012during oscillations: \u0012(t) =\u00120+\u000e\u0012(t). We obtain:\n\u000e\u0012+!2\nr(\u00120)\u000e\u0012=\u0000\u000b!ex\u000e_\u0012; (16)\nwhere!2\nr(\u00120) =\r2h\nm\n\u001f?Hcos\u00120+\u0010\n2K\n\u001f?\u0000H2\u0011\ncos 2\u00120i\n,!ex=\rM\n\u001f?. The initial conditions are \u0012(0) =\u00120and eq.\n(15). In the limit of small oscillations and !r<\u000b!ex=2 (is ful\flled near the second-order transition) the solution has\nthe form\u000e\u0012(t) =Ae\u0000\ftsinh!t, where\f=\u000b!ex=2,!2=\f2\u0000!2\nr,A=B(\u00120)=!. The rise time can be estimated from\nthe condition _\u0012(\u001crise) = 0:\n\u001crise\u0019atanh!\n\f\n!=atanhp\n\f2\u0000!2r\n\fp\n\f2\u0000!2r(17)\nThe time of the oscillations decay (relaxation time) is proportional to the imaginary part of eigenfrequency and can\nbe estimated by the following expression:\n\u001crelax\u00194\u0019\f\n!2r: (18)\nNear second-order phase transition the mode softening occurs and the eigenfrequency turns to zero: !r!0,\nand we observe growth of the both timescales. The critical behavior of the rise time has been observed in GdFeCo\nexperimentally [15] and the typical values of \u001crisewere of the order of 10 ps.6\nCONCLUSIONS\nTo sum up, the developed theoretical model based on quasi-antiferromagnetic Lagrangian formalism proved to\nbe suitable for description of the coherent ultrafast response of RE-TM ferrimagnets near the compensation point\ndue to an ultrashort pulse of demagnetization in the presence of external magnetic \feld. We have found that the\ntorque acting on magnetizations is non-zero in the noncollinear phase only. We have explained the experimentally\nobserved critical behavior of the response amplitude and characteristic timescales as the consequence of the second-\norder magnetic phase transition from collinear to an angular in the external magnetic \feld and the mode softening\nnear it. These e\u000bects are vivid in the vicinity of the compensation point in external magnetic \feld. Understanding\nthe ultrafast response to demagnetizing optical or electrical pulses and subsequent spin dynamics can facilitate future\ndevelopments in the \felds of ultrafast energy-e\u000ecient magnetic recording, magnonics and spintronics.\nACKNOWLEDGMENTS\nThis research has been supported by RSF grant No. 17-12-01333.\n\u0003davydova@phystech.edu\nyzvezdin@gmail.com\n[1] T. Ostler, J. Barker, R. Evans, R. Chantrell, U. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui, L. Le Guyader, E. Men-\ngotti, L. 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Phys. 82, 2731 (2010).\n[10] R. B. Wilson, J. Gorchon, Y. Yang, C.-H. Lambert, S. Salahuddin, and J. Bokor, Phys. Rev. B 95, 180409 (2017).\n[11] Y. Yang, R. B. Wilson, J. Gorchon, C.-H. Lambert, S. Salahuddin, and J. Bokor, Science Advances 3(2017), 10.1126/sci-\nadv.1603117, http://advances.sciencemag.org/content/3/11/e1603117.full.pdf.\n[12] R. Chimata, L. Isaeva, K. K\u0013 adas, A. Bergman, B. Sanyal, J. H. Mentink, M. I. Katsnelson, T. Rasing, A. Kirilyuk,\nA. Kimel, O. Eriksson, and M. Pereiro, Phys. Rev. B 92, 094411 (2015).\n[13] C. D. Stanciu, A. Tsukamoto, A. V. Kimel, F. Hansteen, A. Kirilyuk, A. Itoh, and T. Rasing, Phys. Rev. Lett. 99, 217204\n(2007).\n[14] C. D. Stanciu, A. V. Kimel, F. Hansteen, A. Tsukamoto, A. Itoh, A. Kirilyuk, and T. Rasing, Phys. Rev. B 73, 220402\n(2006).\n[15] J. Becker, A. Tsukamoto, A. Kirilyuk, J. C. Maan, T. Rasing, P. C. M. Christianen, and A. V. Kimel, Phys. Rev. Lett.\n118, 117203 (2017).\n[16] Z. Chen, R. Gao, Z. Wang, C. Xu, D. Chen, and T. Lai, Journal of Applied Physics 108, 023902 (2010),\nhttps://doi.org/10.1063/1.3462429.\n[17] A. Moser, K. Takano, D. T. Margulies, M. Albrecht, Y. Sonobe, Y. Ikeda, S. Sun, and E. E. Fullerton, Journal of Physics\nD: Applied Physics 35, R157 (2002).\n[18] B. Lenk, H. Ulrichs, F. Garbs, and M. Mnzenberg, Physics Reports 507, 107 (2011).\n[19] J. Walowski and M. Mnzenberg, Journal of Applied Physics 120, 140901 (2016), https://doi.org/10.1063/1.4958846.\n[20] M. Liebs, K. Hummler, and M. F ahnle, Phys. Rev. B 46, 11201 (1992).\n[21] B. Goransky and A. Zvezdin, JETP Lett. 10, 196 (1969).\n[22] A. Zvezdin and V. Matveev, JETP 35, 140 (1972).\n[23] C. K. Sabdenov, M. Davydova, K. Zvezdin, D. Gorbunov, I. Tereshina, A. Andreev, and A. Zvezdin, J. Low. Temp. Phys\n43, 551 (2017).\n[24] M. Ding and S. J. Poon, Journal of Magnetism and Magnetic Materials 339, 51 (2013).\n[25] A. Zvezdin, JETP Lett. 28, 553 (1979); arXiv preprint arXiv:1703.01502 (2017).\n[26] B. Ivanov and A. Sukstanskii, Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 84, 370 (1983).7\n[27] A. Zvezdin, Handbook of Magnetic Materials 9, 405 (1995)."    },    {        "title": "1608.04733v1.Power_Spectral_Density_of_Magnetization_Dynamics_Driven_by_a_Jump_Noise_Process.pdf",        "content": "1\nPower Spectral Density of Magnetization Dynamics Driven by a\nJump-Noise Process\nA. Lee1, G. Bertotti2, C. Serpico3, and I. Mayergoyz1\n1University of Maryland, College Park, MD 20740 USA\n2INRIM, Torino, Italy\n3University of Naples Federico II, Naples, Italy\nRandom magnetization dynamics driven by a jump-noise process is reduced to stochastic magnetic energy dynamics on specific\ngraphs using an averaging technique. An approach to analyzing stochastic energy dynamics on graphs is presented and applied\nto the calculation of power spectral density of random magnetization dynamics. An eigenvalue technique for computing the power\nspectral density under specific cases is also presented and illustrated by numerical results.\nIndex Terms —Magnetization Dynamics on graphs, Jump-Noise Process, Power Spectral Density\nI. I NTRODUCTION\nIN recent years, stochastic magnetization dynamics has been\nthe focus of considerable research due to its scientific im-\nportance and promising technological applications. It has been\nrecently proposed [1,2] that a jump-noise process can be used\nto describe thermal bath effects on magnetization dynamics.\nA distinct advantage of this approach is that both the damping\nand fluctuation effects emerge from the nature of the jump-\nnoise process. This jump-noise process term in the equation\nfor magnetization dynamics is usually small in comparison\nwith the precessional term. This term leads to damping and\nfluctuation effects that occur on a much longer time-scale than\nmagnetic precessions. For this reason, an averaging technique\ncan be applied to the random magnetization dynamics to\nreduce it to stochastic magnetic energy dynamics on graphs.\nIn this paper, an approach to the analysis of stochastic energy\ndynamics on graphs is presented and applied to the calculation\nof power spectral density of random magnetization dynamics.\nBy using the differential equation for transition probability\ndensity, formulas for the autocovariance function and the\npower spectral density are derived and illustrated by numerical\ncomputations.\nII. T ECHNICAL DISCUSSION\nRandom magnetization dynamics driven by a jump-noise\nprocess is governed by the equation,\ndM\ndt=\u0000\r(M\u0002Heff) +Tr(t); (1)\nwhere Mis the magnetization vector, \ris the gyromagnetic\nratio,Heffis an effective magnetic field, and Tr(t)is a jump-\nnoise torque that describes the effects of the thermal bath. The\nrandom process Tr(t)is described by the formula\nTr(t) =X\ni=1mi\u000e(t\u0000ti); (2)\nwhere miare random jumps in magnetization occurring at\nrandom time instances ti. The statistics of the random jumps\nCorresponding author: I. Mayergoyz (email: isaak@umd.edu)miand random times tican be fully specified (see [1,2]) by\nintroducing the transition probability rate S(Mi;Mi+1)with\nMi=M(t\u0000\ni)andMi+1=M(t+\ni) =Mi+miwhere M(t\u0000\ni)\nandM(t+\ni)are the magnetization immediately before and after\na jump. Using the transition probability rate S(M;M0), the\nstatistics of random times tiand random jumps miare defined\nby the formulas\nProbability (ti+1\u0000ti>\u001c) = exp\u0014\n\u0000Zti+1\nti\u0015(M(t))dt\u0015\n;\n(3)\n\u0015(M(t)) =I\n\u0006S(M(t);M0)d\u0006M0; (4)\n\u001f(mijMi) =S(Mi;Mi+mi)\n\u0015(Mi): (5)\nHere:\u0015(M)is the scattering rate, \u001f(mijMi)is the probability\ndensity for magnetization jumps miat timetifrom the states\nM(t) =Mi, and \u0006is the spherejM0j=Ms=const .\nThe following formula has been suggested in [1] for tran-\nsition probability rate\nS(M;M0) =Ae\u0000jM\u0000M0j2\n2\u001b2eg(M)\u0000g(M0)\n2kT; (6)\nwhere A is an empirically derived constant characterizing the\nstrength of the jump-noise process Tr(t).\nIt is important to note that the stochastic magnetization\ndynamics (1) can also be equivalently described in terms of a\nKolmogorov-Fokker-Planck equation for transition probability\ndensityw(M;t):\n@w\n@t=\u0000\rr\u0006\u0001[(M\u0002r \u0006g)w] (7)\n+Z\n\u0006[S(M0;M)w(M0)\u0000S(M;M0)w(M)]d\u0006M0\nwheregis the micromagnetic energy.\nThe last equation is linear and deterministic with respect to\nw. This is the main advantage of equation (7) in comparison\nwith equation (1), which is nonlinear stochastic differential\nequation.arXiv:1608.04733v1  [cond-mat.mes-hall]  16 Aug 20162\nUsually, damping and fluctuations caused by the thermal\nbath occur on a much longer time-scale than the fast mo-\ntion of magnetization precessions. If one’s interest lies in\nanalyzing only the thermal effects, then the magnetization\ndynamics described by equation (1) can be averaged over\nprecessional trajectories, which are uniquely defined by their\nmicromagnetic energy g. This averaging leads to a description\nof stochastic dynamics for gthat is defined on specific energy\ngraphs. These graphs reflect the energy landscapes on \u0006of\nmagnetic particles.\nBy using equation (7) and averaging both sides over pre-\ncessional trajectories in the manner discussed in [6], it can be\nshown that in terms of the transition probability density \u001a(g;t),\nthe stochastic energy dynamics is described by the following\nequation\nd\u001ak(g;t)\ndt=X\nnZ\nLn[Kn;k(g0;g)\u001an(g0;t) (8)\n\u0000Kk;n(g;g0)\u001ak(g;t)]dg0;(k= 1;2;:::N );\nwhereLnis an edge of the graph corresponding to the region\nRnof the sphere \u0006with the property that there exists only\none precessional trajectory Cn(g)corresponding to energy g,\n\u001an(g;t)is the probability density on edge Ln, and function\nKn;k(g0;g)is related to S(M0;M)by the formula\nKn;k(g0;g) = (9)\n1\n\u001cn(g)I\nCk(g)I\nCn(g0)S(M0;M)\njr\u0006g(M0)jjr\u0006g(M)jdlM0dlM;\nwhere\n\u001cn(g) =I\nCn(g)dlM\njr\u0006g(M)j: (10)\nIn equation (8), the summation is performed over all N edges\nof the graph.\nThe effect of spin-torque can be included by modifying\nequation (8) as follows\n@\n@t\u001ak(g;t) =@\n@g[\bk(g)\u001ak(g;t)]\n+X\nnZ\nLn[Kn;k(g0;g)\u001an(g0;t)\n\u0000Kk;n(g;g0)\u001ak(g;t)]dg0; (11)\nwhere \bk(g)is a function that describes the effect of spin-\ntorque as discussed in [8].\nNow, we proceed to the discussion of computation of the\npower spectral density of random micromagnetic energy g.\nIn control theory, the power spectral density is computed for\nlinear time-invariant systems. For such systems, the power\nspectral density of the output is equal to the power spectral\ndensity of the input multiplied by the squared magnitude of the\ntransfer function. However, the magnetization dynamics de-\nscribed by the stochastic differential equation (1) is nonlinear.\nNevertheless, it turns out that the power spectral density can\nbe computed by using linear techniques. This can be done by\nexploiting the linearity of equations (7) and (8) for transition\nprobability densities wand\u001a, respectively. This approach is\nused in our subsequent discussion.The power spectral density is defined by the formula\n^Sf(!) =Z1\n\u00001^Cf(\u001c)e\u0000j!\u001cd\u001c; (12)\nwhere ^Cf(\u001c)is the autocorrelation function. This autocorre-\nlation function is given by the formula\n^Cf(\u001c) =Z\nLZ\nLf(g)f(g0)\u001a(g;t0;g0;t0\u0000\u001c)dg0dg\n\u0000Z\nLZ\nLf(g)f(g0)\u001aeq(g)\u001aeq(g0)dg0dg: (13)\nwhere\u001aeq(g)is the equilibrium probability density.\nFor a stationary Markovian process, the joint probability\ndensity can be expressed by the formula\n\u001a(g;t0;g0;t0\u0000\u001c) =\u001a(g;\u001cjg0;0)\u001aeq(g0): (14)\nUsing the formula (14), equation (13) can be written as\n^Cf(\u001c) =Z\nLf(g)\u0014Z\nLf(g0)[\u001a(g;\u001cjg0;0)\u0000\u001aeq(g)]\u001aeq(g0)dg0\u0015\ndg:\n(15)\nThe expression within the brackets in equation (15) is\n f(g;\u001c) =Z\nLf(g0)[\u001a(g;\u001cjg0;0)\u0000\u001aeq(g)]\u001aeq(g0)dg0:(16)\nTherefore, formula (13) can be expressed as\n^Cf(\u001c) =Z\nLf(g) f(g;\u001c)dg: (17)\nUsing equation (17), the power spectral density can be written\nas follows\n^Sf(!) = 2Re\u001aZ\nLf(g)\u0014Z1\n0 f(g;\u001c)e\u0000j!\u001cd\u001c\u0015\ndg\u001b\n:(18)\nThe expression in the inner bracket of formula (18) can be\nseen as the following Fourier Transform\n\tf(g;!) =Z1\n0 f(g;\u001c)e\u0000j!\u001cd\u001c: (19)\nTherefore, the power spectral density can be written as\n^Sf(!) = 2Re\u001aZ\nLf(g)\tf(g;!)dg\u001b\n: (20)\nWhen we are interested in the spectral density of g, equation\n(17) is reduced to\n^Cg(\u001c) =Z\nLg g(g;\u001c)dg; (21)\nwhere\n g(g;\u001c) =Z\nLg0[\u001a(g;\u001cjg0;0)\u0000\u001aeq(g)]\u001aeq(g0)dg0:(22)\nIt is clear now that equation (11) can be used to compute the\npower spectral density of g.\nIn formula (11), \u001a(g;t)is the simplified notation for\n\u001a(g;t0+\u001cjg0;t0). For a stationary process,\n\u001a(g;t0+\u001cjg0;t0) =\u001a(g;\u001cjg0;0): (23)3\nUsing equation (23), equation (11) can be written as\n@\n@\u001c\u001ak(g;\u001cjg0;0) =@\n@g[\bk(g)\u001ak(g;\u001cjg0;0)]\n+X\nnZ\nLn[Kn;k(g0;g)\u001an(g0;\u001cjg0;0)\n\u0000Kk;n(g;g0)\u001ak(g;\u001cjg0;0)]dg0: (24)\nAtt= 0, the following initial condition is valid:\n\u001a(g0;0;g;\u001c)j\u001c=0=\u000e(g\u0000g0): (25)\nAt equilibrium, the stationary probability distribution \u001aeq\nsatisfies the equation\n@\n@g[\bk(g)\u001aeq\nk(g)]+X\nnZ\nLn[Kn;k(g0;g)\u001aeq\nn(g0)\n\u0000Kk;n(g;g0)\u001aeq\nk(g)]dg0= 0: (26)\nTaking the difference between equations (24) and (26) leads\nto the formula\n@\n@\u001c[\u001ak(g;\u001cjg0;0)\u0000\u001aeq\nk(g)]\n=@\n@g[\bk(g)[\u001ak(g;\u001cjg0;0)\u0000\u001aeq\nk(g)]]\n+X\nnZ\nLn[Kn;k(g0;g)[\u001an(g;\u001cjg0;0)\u0000\u001aeq\nn(g)]\n\u0000Kk;n(g;g0)[\u001ak(g;\u001cjg0;0)\u0000\u001aeq\nk(g)]]dg0:\n(27)\nNow, we introduce the function\n\u0010k(g;\u001cjg0;0)\u0011\u001ak(g;\u001cjg0;0)\u0000\u001aeq\nk(g): (28)\nThe initial condition for this function is:\n\u0010k(g;\u001cjg0;0)j\u001c=0=\u000e(g\u0000g0)\u0000\u001aeq\nk(g): (29)\nUsing formula (28), equation (27) can be written in the form\n@\n@\u001c\u0010k(g;\u001cjg0;0) =@\n@g[\bk(g)\u0010k(g;\u001cjg0;0)]\n+X\nnZ\nLn[Kn;k(g0;g)\u0010n(g0;\u001cjg0;0)\n\u0000Kk;n(g;g0)\u0010k(g;\u001cjg0;0)]dg0: (30)\nBy solving for \u0010, function g(g;\u001c)in equation (22) can also\nbe found:\n k(g;\u001c)\u0011Z\nLg0\u001aeq\nk(g0)\u0010k(g;\u001cjg0;0)dg0: (31)\nUsing formula (31), equation (30) can be transformed as\nfollows:\n@\n@\u001c k(g;\u001cjg0;0) =@\n@g[\bk(g) k(g;\u001cjg0;0)]\n+X\nnZ\nLn[Kn;k(g0;g) n(g0;\u001cjg0;0)\n\u0000Kk;n(g;g0) k(g;\u001cjg0;0)]dg0: (32)\nIt follows from equation (29) and (31) that the initial condition\nfor kis:\n k(g;\u001cjg0;0)j\u001c=0=\u001aeq\nk[g\u0000hgi]: (33)Applying the one-sided Fourier transform in (19) to equation\n(32) results in:\nj!\tk(g;!jg0)\u0000 (g;\u001cjg0;0)j\u001c=0=\n@\n@g[\bk(g)\tk(g;!jg0)]\n+X\nnZ\nLn[Kn;k(g0;g)\tn(g0;!jg0)\n\u0000Kk;n(g;g0)\tk(g;!jg0)]dg0: (34)\nThis equation can now be used to compute \tk. The power\nspectral density can then be found as:\n^Sg(!) = 2Re\u001aZ\nLg\t(g;!)dg\u001b\n: (35)\nIn the case of no applied spin-torque, equation (34) can be\nsimplified to\nj!\tk(g;!jg0)\u0000 k(g;\u001cjg0;0)j\u001c=0\n=X\nnZ\nLn[Kn;k(g0;g)\tn(g0;!jg0)\n\u0000Kk;n(g;g0)\tk(g;!jg0)]dg0: (36)\nBy numerically solving equation (34) or the last equation,\nfunctions \tk(g;!)can be found and then they can be used\nin equation (35) to compute the power spectral density Sg(!).\nIn the case of no applied spin-torque, a special technique of\nsolving equation (36) can be useful. The right-hand side of\nthis equation has the form of the collision integral and, for\nthis reason, it can be solved using an eigenvalue approach:\n^K\u001ei=\u0015i\u001ei (37)\nwhere ^Kis the collision integral operator in (36).\nFunction \tk(g;!)can be decomposed with respect to\neigenfunctions of ^K:\n\tk(g;!) =X\niai(!)\u001ei(g): (38)\nSimilarly,\n k(g;\u001cjg0;0)j\u001c=0=X\nibi\u001ei(g); (39)\nwhere\nbi=h\u001aeq\nk[g\u0000hgi];\u001ei(g)i: (40)\nNow, equation (36) can be reduced to\nj!ai(!)\u001ei(g)\u0000bi\u001ei(g) =\u0015iai(!)\u001ei(g): (41)\nConsequently,\nai=bi\nj!\u0000\u0015i: (42)\nTherefore, we have\n\t(g;!) =X\ni\u0015ih\u001aeq\nk[g\u0000hgi];\u001ei(g)i\nj!\u0000\u0015i\u001ei(g): (43)\nThis eigenvalue approach allows for very fast calculations\nof power spectral density:\n^Sg(!) = 2Re(X\ni\u0015i\nj!\u0000\u0015iZ\nLgh\u001aeq\nk[g\u0000hgi];\u001ei(g)i\u001ei(g)dg)\n:\n(44)4\nIII. N UMERICAL RESULTS\nThe techniques described in the previous section have been\nnumerically implemented for the case of uniaxial particles\nwith and without the presence of the spin-torque effect. Some\nsample numerical results are presented below in figures 1\nthrough 9 for the same effective anisotropy coefficient Keff\nand applied magnetic field Hazbut different parameters of the\nthermal noise.\nFig. 1. Uniaxial particle with Keff= 0:5;Haz=\u00000:7;\u001b2= 0:001;A=\n1012.\nFig. 2. Uniaxial particle with Keff= 0:5;Haz=\u00000:7;\u001b2= 0:001;A=\n1015.\nAs seen in figures 1, 2, and 3, the power spectral density has\na flat intensity response at lower frequencies before reaching\na ‘knee’ and decreasing as the frequency increases. The\nplacement of the ‘knee’ is controlled by the strength of thermal\nnoise and the roll-off appears to follow a 1=f2dependence.\nIndeed, by comparing figures 1 and 2 where the intensity\nof the noise is increased from A= 1012toA= 1015,\nthe placement of the knee is seen to have shifted to higher\nfrequencies. This is due to the fact that higher intensities of\nthermal noise cause more frequent jumps in magnetization.\nLikewise, by comparing figures 1 and 3 where the variance\nof the noise is increased from \u001b2= 0:001 to\u001b2= 0:01, the\nFig. 3. Uniaxial particle with Keff = 0:5;Haz=\u00000:7;\u001b2= 0:01;A=\n1012.\nplacement of the knee also shifts to higher frequencies. This\nis due to larger \u001b2allowing for larger jumps in magnetization\nand therefore an overall larger transition probability rate and\nscattering rate. This reasoning is consistent with formula (6).\nFig. 4. PSD using the eigenvalue approach for an uniaxial particle with\nKeff= 0:5;Haz=\u00000:7;\u001b2= 0:001;A= 1012.\nFigures 4, 5, and 6 represent the same power spectral\ndensities as in Figures 1, 2, and 3 but calculated using\nthe eigenvalue approach. As seen from these figures, the\npower spectral densities exactly match their respective plots\nin Figures 1, 2, and 3.\nFigures 7, 8, and 9 show the effect of spin-torque on the\npower spectral density. The power spectral density is now seen\nto have two separate ‘knees’. By comparing figure 7 with the\ncorresponding power spectral density in the case of no spin-\ntorque, we clearly see that the additional knee is a spin-torque\neffect. This is also evident by comparing Figures 7, 8, and 9\nwhere the strength of the spin torque ( \f) is increased from\n\f= 107to\f= 109and\f= 1012. The change in strength of\nthe spin-torque shifts the location of the knee.5\nFig. 5. PSD using the eigenvalue approach for an uniaxial particle with\nKeff= 0:5;Haz=\u00000:7;\u001b2= 0:001;A= 1015.\nFig. 6. PSD using the eigenvalue approach for an uniaxial particle with\nKeff= 0:5;Haz=\u00000:7;\u001b2= 0:01;A= 1012.\nFig. 7. Uniaxial particle with Keff= 0:5;Haz=\u00000:7;\u001b2= 0:001;A=\n1015and spin-torque characterized by epz=\u00001;cp= 0:5;\f= 107.\nACKNOWLEDGMENTS\nThis work was partially supported by Progetto Premi-\nale MIUR-INRIM Nanotecnologie per la metrologia elettro-\nFig. 8. Uniaxial particle with Keff= 0:5;Haz=\u00000:7;\u001b2= 0:001;A=\n1015and spin-torque characterized by epz=\u00001;cp= 0:5;\f= 109.\nFig. 9. Uniaxial particle with Keff= 0:5;Haz=\u00000:7;\u001b2= 0:001;A=\n1015and spin-torque characterized by epz=\u00001;cp= 0:5;\f= 1012.\nmagnetica, by MIUR-PRIN Project No. 2010ECA8P3 Dy-\nNanoMag, and by NSF.\nREFERENCES\n[1] I. Mayergoyz, G. Bertotti, and C. Serpico, Phys. Rev. B 83, 020402 (R)\n(2011).\n[2] I. Mayergoyz, G. Bertotti, and C. Serpico, J. Appl. Phys. 109, 07D312\n(2011).\n[3] H.A. Kramers, Physica (Amsterdam) 7, 284 (1940).\n[4] W. F. Brown, Phys. Rev. 130, 1677 (1963).\n[5] C. E. Korman and I. D. Mayergoyz, Phys. Rev. B 54, 17620 (1996).\n[6] G. Bertotti, I. Mayergoyz, and C. Serpico, (Elsevier, Beford, 2009).\n[7] C. Serpico, I.D. Mayergoyz, and G. Bertotti, J. Appl. Phys. 89, 6991\n(2001).\n[8] Liu, Z.; Lee, A.; McAvoy, P.; Bertotti, G.; Serpico, C.; Mayergoyz, I.,\nIEEE Transactions on Magnetics, vol.49, no.7, pp.3133, July 2013"    },    {        "title": "2009.13440v1.Magnetic_domains_and_domain_wall_pinning_in_two_dimensional_ferromagnets_revealed_by_nanoscale_imaging.pdf",        "content": "Magnetic domains and domain wall pinning in two-dimensional\nferromagnets revealed by nanoscale imaging\nQi-Chao Sun1,8, Tiancheng Song2,8, Eric Anderson2, Tetyana Shalomayeva1, Johaness\nF orster3, Andreas Brunner1, Takashi Taniguchi4, Kenji Watanabe4, Joachim Gr afe3,\nRainer St ohr1,5,*, Xiaodong Xu2,6and J org Wrachtrup1,7\n13. Physikalisches Institut, University of Stuttgart, 70569 Stuttgart, Germany\n2Department of Physics, University of Washington, Seattle, Washington 98195, USA\n3Max Planck Institute for Intelligent Systems, 70569 Stuttgart, Germany\n4National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan\n5Center for Applied Quantum Technology, University of Stuttgart, 70569 Stuttgart,\nGermany\n6Department of Materials Science and Engineering, University of Washington, Seattle,\nWashington 98195, USA\n7Max Planck Institute for Solid State Research, 70569 Stuttgart, Germany\n8These authors contributed equally\n*e-mail: q.sun@pi3.uni-stuttgart.de; rainer.stoehr@pi3.uni-stuttgart.de\nSeptember 29, 2020\nMagnetic-domain structure and dynamics play an important role in understanding and con-\ntrolling the magnetic properties of two-dimensional magnets, which are of interest to both\nfundamental studies and applications[1{5]. However, the probe methods based on the spin-\ndependent optical permeability[1, 2, 6] and electrical conductivity[7{10] can neither provide\nquantitative information of the magnetization nor achieve nanoscale spatial resolution. These\ncapabilities are essential to image and understand the rich properties of magnetic domains.\nHere, we employ cryogenic scanning magnetometry using a single-electron spin of a nitrogen-\nvacancy center in a diamond probe to unambiguously prove the existence of magnetic domains\nand study their dynamics in atomically thin CrBr 3. The high spatial resolution of this technique\nenables imaging of magnetic domains and allows to resolve domain walls pinned by defects. By\ncontrolling the magnetic domain evolution as a function of magnetic \feld, we \fnd that the\npinning e\u000bect is a dominant coercivity mechanism with a saturation magnetization of about\n26\u0016B/nm2for bilayer CrBr 3. The magnetic-domain structure and pinning-e\u000bect dominated\ndomain reversal process are veri\fed by micromagnetic simulation. Our work highlights scan-\nning nitrogen-vacancy center magnetometry as a quantitative probe to explore two-dimensional\nmagnetism at the nanoscale.\nTwo-dimensional (2D) magnetism is of fundamental interest for the study of long-range\nmagnetic order in the presence of the enhanced spin \ructuation at reduced dimensionality[11].\nRecent developments in van der Waals magnetic materials have greatly enriched the variety\nand controllability of 2D magnets, and have magnetic properties that can be modi\fed via\nelectron doping[6, 12] and by changing layer number[1, 2] or stacking order[10, 13]. Due to\ntheir micrometer size and atomic thickness, the magnetic signal of 2D magnets is too weak\nto be detected by conventional magnetometry. Several probe techniques, such as magneto-\noptical Kerr e\u000bect microscopy[1, 2], magnetic circular dichroism microscopy[6, 14], anomalous\n1arXiv:2009.13440v1  [cond-mat.mtrl-sci]  28 Sep 2020hall e\u000bect[7, 8], and electron tunneling[9, 10] have been used in previous studies. Although\nthese methods can reveal phase transitions, quantitative magnetization information is di\u000ecult\nto extract from spin-related signals. Moreover, only micrometer-scale spatial resolution can\nbe achieved by these methods due to the laser di\u000braction limit or the size of the electrode.\nQuantitatively study of 2D magnets at the nanoscale would allow accurate analysis of their\nmagnetic properties, which is crucial to understand and control new phases.\nThe negatively charged nitrogen-vacancy (NV) center in diamond exhibits a spin-1 triplet\nelectronic ground state. The triplet state has energy levels sensitive to surrounding magnetic\n\felds, and its spin states can be easily accessed via optical initialization/readout, and coher-\nent microwave manipulation. This atomic-sized magnetometer is suitable to probe features of\nmost known 2D magnets with a dynamic range of magnetic \feld measurements spanning DC\nto several GHz and operational temperatures from below one to several hundreds of Kelvin[15].\nScanning magnetometry combining atomic force microscopy and NV center magnetometer al-\nlows for quantitative nanoscale imaging of magnetic \felds and has been well established in room\ntemperature measurements[16{21]. While cyrogenic implementation of this technique is more\nchallenging[22, 23], it is crucial for the study of 2D magnets such as the chromium trihalides\n(CrX 3, X=Cl, Br, and I) as their magnetic ordering exists only at low temperature[24]. Re-\ncently, this technique has been implemented successfully to image the magnetization in layered\nCrI 3samples[25].\nCrBr 3, for which ferromagnetic order persists from bulk crystal down to the monolayer, is\na unique platform to study the ferromagnetism and spin \ructuation in the 2D limit[14, 26, 27].\nAlthough magnetic domains in layered CrBr 3have been predicted from its anomalous hysteresis\nloop in magneto-photoluminescence and micromagnetometry measurements[26, 27], the mag-\nnetic domain structure and its evolution has not been detected in real-space. Additionally,\nZhang et al. detect magnetic domains only in multi-layer samples[27], while Kim et al. show\nsigns of domains even in the monolayer case[26].This ambiguity cannot be resolved with mi-\ncroscale measurements. In this work, we overcome this limitation using a single NV center in a\ndiamond probe to map the stray magnetic \feld of the sample. We image the magnetic domain\nstructures in a CrBr 3bilayer, determine its magnetization, and study the magnetic domain evo-\nlution. In the main text, we focus on measurements of a bilayer CrBr 3sample conducted with a\ndiamond probe. Additional data obtained using a diamond probe with a di\u000berent NV axis ori-\nentation, as well as the results from another CrBr 3sample, are provided in the Supplementary\nInformation.\nFig.1 (a) shows a schematic of the scanning NV magnetometry setup. The NV center is\nimplanted in the apex of the pillar etched from a diamond cantilever, which is attached to the\ntuning fork of an atomic force microscope (AFM). The CrBr 3sample, which is encapsulated with\nhexagonal boron nitride (hBN) on both sides, is transferred on a SiO 2/Si substrate in a glove-\nbox \flled with pure nitrogen (see Methods for details of sample preparation). The microscope\nhead is suspended in an insertion tube \flled with Helium bu\u000ber gas, which is dipped in the\nliquid Helium cryostat equipped with a set of vector superconducting coils. The NV spin state\nis optically measured through the cantilever supporting the diamond tip (see Methods). In\nthe measurement, the AFM operates in a frequency modulation mode with a tip oscillation\namplitude of about 1.5 nm. The topography of the sample is obtained from the AFM readout\nwith a lateral spatial resolution of about 200 nm, which is limited by the diameter of the pillar\napex. A typical in-situ AFM image of part of the CrBr 3sample is shown in Fig.1 (b). Fig.1 (c)\nshows the height of the step along the vertical direction by an average of the data in the dashed\nbox in Fig.1 (b). The 2 nm step height indicates a bilayer sample.\nThe stray magnetic \feld is mapped by taking the electron spin resonance spectrum via the\npulsed optically detected magnetic resonance (ODMR) scheme[28] at each pixel. The measure-\nment sequence is shown in Fig.1 (d). This pulsed scheme signi\fcantly decreases microwave\nheating compared with continuous wave ODMR[25]. In our experiment, the sample tempera-\n2Figure 1: Cryogenic scanning magnetometry with a single NV center in the diamond\ntip. a , Schematic of the experiment. The stray magnetic \feld of the CrBr 3bilayer is measured\nusing a single NV center in a diamond probe attached to the tuning fork of the AFM. The\nsystem is placed in a liquid Helium bath cryostat to maintain a temperature below 5 K during\nthe measurement. bandc, A typical topography image of the measured area and the step at\nthe edge of the sample, respectively. The scale bar is 1 \u0016m.d, Measurement sequence of the\npulsed optically detected magnetic resonance (ODMR).\n3Figure 2: Magnetic domains and saturation magnetization. a , Stray magnetic \feld and\nb, the reconstructed magnetization of a CrBr 3bilayer under an external \feld of 2 mT along the\nNV axis. c, Magnetization image at external magnetic \feld of 11 mT. The dashed boxes in b\nandcdenote the common sample area in the two images. Scale bar is 1 \u0016m for all images. d\nande, Histograms of the magnetization values in images bandc, respectively.\nture only increases by a few hundred milli-Kelvin from the base temperature of about 4.2 K\nduring measurement. Moreover, the magnetic \feld is measured via microwave pulses, which are\napplied 600 ns after the laser beam has been switched o\u000b. Thus, the measured stray magnetic\n\feld is not disturbed by laser-induced excitations such as spin-waves[29, 30]. Fig.2 (a) shows\na typical stray magnetic \feld image of the sample under a 2 mT external magnetic \feld after\nbeing cooled down under zero \feld. The external magnetic \feld is used to split the energy levels\njms=\u00061iso that the direction of the stray magnetic \feld can be determined. The external\nmagnetic \feld direction is set parallel to the NV axis in all measurements in this work to avoid\nthe mixing of spin states due to an o\u000b-axis magnetic \feld component[31]. The axis of all the\nNV centers in the (100)-oriented diamond cantilevers we used here is about 54 :7\u000ewith respect\nto the vertical direction, as shown in Fig.1 (a). The pixel size in this work is set to 30 nm and\nthe data accumulation time is 2 s at each pixel. The resulting stray magnetic \feld map clearly\nshows magnetic domains with prominent positive and negative values of the magnetic \feld and\ndomain walls with nearly zero \felds. To reveal further details, we reconstruct the magnetization\nfrom the stray magnetic \feld using reverse-propagation protocol[21, 25, 32, 33].\nAs discussed in more detail in the supplementary information, to uniquely determine the\nmagnetization, we need some initial knowledge of the sample such as the direction of spin po-\nlarization. It has been reported that few-layer CrBr 3has an out-of-plane easy axis and can be\npolarized by a small external magnetic \feld of about 4 mT, while a relatively high external\n\feld (Bc\nk\u00190:44 T) is required to polarize the spins in the in-plane direction[24]. The in-plane\nexternal \feld component in this work is much lower than the critical \feld Bc\nk, allowing us to use\nthe assumption of out-of-plane magnetization in the magnetization reconstruction. In addition,\nwe neglect the \fnite thickness of the domain walls. Fig.2 (b) presents the magnetization image\nreconstructed from the stray magnetic \feld image in Fig.2 (a). It clearly shows the magnetic\ndomain structure, with positive (negative) values indicating the magnetization direction parallel\n(anti-parallel) to the external magnetic \feld. The sample can be polarized by increasing the ex-\nternal magnetic \feld. Fig.2 (c) shows a magnetization image taken at 11 mT external magnetic\n\feld. The common areas in Figs.2 (b) and (c) are marked with dashed boxes. The saturation\n4Figure 3: Magnetic domain evolution upon increasing the external magnetic \feld.\na-g, Magnetization images taken successively at external magnetic \felds of 2, 2.5, 3, 3.5, 4, 5,\nand 6 mT along the NV axis, respectively. The sample is thermally demagnetized by heating\nto 45 K and then cooling down under zero \feld. h(i), Magnetization image of the sample area\nindicated by the dashed box in e(g) during another thermal cycle at external magnetic \felds\nof 4 and 6 mT, respectively. The solid and dashed yellow circles denote the positions of two\nrepresentative pinning sites. See the supplementary information for the other magnetization\nimages. Scale bar is 1 \u0016m for all images. j, Initial magnetization curves extracted from the\nmagnetization images in a-g. The blue bars are the ratios of magnetization to external magnetic\n\feld.\nmagnetization can be estimated using the magnetization statistics of the two magnetization\nimages, as shown by the histograms in Figs.2 (d) and (e). Due to sample imperfection, mea-\nsurement error, and truncation error in the reconstruction, the reconstructed magnetization is\ndistributed in a range around the zero-magnetization and the saturation magnetization values.\nThe near-zero-magnetization pixels are mostly in domain walls, defects, and the non-sample\narea on the left part of the images. The saturation magnetization values are \u001826(\u000028) and\n\u001826\u0016B/nm2, respectively, with \u0016Bthe Bohr magneton. These values are close to the 3 \u0016B\nsaturation moment per Cr3+ion in CrBr 3at 0 K, i.e.,\u001832\u0016B/nm2for a CrBr 3bilayer[34].\nIn addition to elucidating the magnetic domain structure of 2D magnets, our scanning\nmagnetometry measurements enable a more detailed study of coercivity mechanisms in these\nsystems. A multi-domain ferromagnet typically reverses its magnetization direction through\nprocesses such as nucleation of reverse domains and their growth through domain wall motion[5,\n35]. Defects in the material alter the energy of the magnetic domain walls and hence a\u000bect\ndomain wall motion. This behaviour can be demonstrated by taking magnetization images\nwhile varying the external magnetic \feld. Figs.3 (a)-(g) show magnetization images obtained\nwhile increasing the \feld from 2 mT to 6 mT after the sample is thermally demagnetized\nand cooled down under zero \feld. The area of positive (negative) domains grows (shrinks)\nwith increasing \feld, as the domain walls move toward the negative domains. Before entirely\ndisappearing, the negative domain size becomes very small, and only near-zero-magnetization\nspots of several tens of nanometer diameter are revealed in the magnetization images. As we\nare limited by the spatial resolution of the NV center (about 80 nm for the results shown in the\nmain text, determined by the distance between the NV center and the sample for the diamond\nprobe), we cannot obtain the detailed magnetization pattern inside the spots. These spots\n5are usually associated with defects, which increase the local switching \feld. Domain walls are\npinned by these defects (see Figs.3 (a)-(i), solid and dashed yellow circles). To con\frm the\npinning sites, we compare the magnetization images at 4 and 6 mT for di\u000berent thermal cycles\n(see dashed box in Figs.3 (e) and (g), and compare to Figs.3 (h) and (i)). Though the magnetic\ndomain structures are di\u000berent upon successive thermal cycles, the positions of pinning sites\nare reproducible.\nTo verify that the pinning e\u000bect is a dominant coercivity mechanism, we extract the initial\nmagnetization curve of the thermally demagnetized sample by estimating its average magneti-\nzation asM=M sat=N+\u0000N\u0000\nN++N\u0000, whereMsatis the saturation magnetization and N+(N\u0000) is the\nnumber of pixels with evident positive (negative) magnetization (absolute value greater than 10\n\u0016B/nm2, according to the Fig.2 (e)). When pinning e\u000bects are negligible, the magnetic domain\nwalls can move freely, resulting in a high initial magnetization even with a small external mag-\nnetic \feld. Defects, however, increase the energy barrier for the displacement of domain walls.\nIn samples with a large defect density, the magnetization increases slowly until the external\nmagnetic \feld is large enough to overcome the pinning energy. The initial magnetization curve\nshown in Fig.3 (j) is measured at external magnetic \felds above 2 mT. We extend the curve\nto the origin using B-spline interpolation, assuming zero magnetization in a thermally demag-\nnetized sample. The average permeability is very low under \feld of 2 mT and it signi\fcantly\nincreases when the \feld is above 2 mT (see the blue bars in Fig.3 (j)), which is consistent with\nthe behaviour of a pinning e\u000bect dominated initial magnetization.\nWe use a similar approach to measure the magnetic hysteresis loop. Fig.4 (a) shows the\nmagnetization image at an external magnetic \feld of 2 mT after thermal demagnetization. We\nchoose the area marked by the dashed box (#1) to analyse the magnetization as we cycle the\nexternal magnetic \feld to saturate the magnetization in both the positive and negative z di-\nrection. Representative magnetization images are shown in Figs.4 (b)-(i) (see supplementary\ninformation for all images). The hysteresis loop of this area is shown by the black curve in\nFig.4 (j). Remarkably, a magnetic domain with the same irregular shape but opposite magne-\ntization direction is observed in Figs.4 (e) and (i). The border of the magnetic domain (#2)\nis denoted by the dashed line. We note that there are several defects (shown in Figs.4 (c) and\n(g)) around the #2 region, leading to a very hard inverted domain. The magnetic domain wall\nis strongly pinned by the defects, and the magnetization is inverted abruptly when the external\n\feld is larger than \u00185 mT. This can be interpreted as an increased local switching \feld due\nto a strong pinning e\u000bect, which is consistent with the higher coercive \feld indicated by the\nnear-rectangular local hysteresis loop of this domain, as shown by the red curve in Fig.4 (j).\nThe shape of the hysteresis loop is thus highly dependent on the area used to estimate the\nmagnetization and on the local pinning sites. The rectangular hysteresis loop in bilayer CrBr 3\nreported in a previous study[27] could be attributed to this local pinning dependence.\nIn conclusion, we study magnetic domains in few-layer samples of CrBr 3by quantitatively\nmapping the stray magnetic \feld with a cryogenic scanning magnetometer based on a single\nNV center in a diamond probe. The magnetization of bilayer CrBr 3is determined, and the\nmagnetic domain evolution is observed in real space. We show that pinning is the dominant\ncoercivity mechanism by observing the evolution of both the individual magnetic domains and\nthe average magnetization with changing external magnetic \feld. To verify the observations, we\nreproduce similar magnetic-domain structures using micromagnetic simulation with parameters\nwithin the ranges estimated from measurement of CrBr 3bulk[36] (see Methods and Supple-\nmentary Information). We \fnd the hysteresis loop cannot be reproduced due to the limited\ncapability of the micromagnetic simulation to account for the structure defects and pinning\ne\u000bect. This also support our conclusion that pinning e\u000bect is a dominant mechanism in the\ndomain reversal of CrBr 3bilayer. We note that our approach is also compatible with other\npulsed measurement sequences which can be used to detect electron spin resonance[37], nuclear\nmagnetic resonance[19] and spin waves[38] in the 2D magnetic materials.\n6Figure 4: Magnetic hysteresis loop. a , Magnetic domains at 2 mT external magnetic\n\feld along the NV axis after being thermally demagnetized and cooled down under zero \feld.\nThe dashed box denotes the area #1 used to analyze the hysteresis loop. b-i, Representative\nmagnetization images of area #1 on the hysteresis loop with the corresponding labels in j, and\nthe dashed lines denote the area #2. Scale bar is 1 \u0016m for all the images. j, Hysteresis loop\nextracted from the magnetization images of area #1 and #2, which are denoted by the black\nstars and red dots, respectively. The solid curve connects measured values, while the dashed\ncurve is an extension to demagnetized and saturated states.\n7References\n[1] Huang, B. et al. Layer-dependent ferromagnetism in a van der Waals crystal down to the\nmonolayer limit 546, 270 (2017).\n[2] Gong, C. et al. Discovery of intrinsic ferromagnetism in two-dimensional van der Waals\ncrystals 546, 265 (2017).\n[3] Burch, K. S., Mandrus, D. & Park, J.-G. Magnetism in two-dimensional van der Waals\nmaterials 563, 47{52 (2018).\n[4] Mak, K. F., Shan, J. & Ralph, D. C. Probing and controlling magnetic states in 2D layered\nmagnetic materials. Nat Rev Phys 1, 646{661 (2019).\n[5] Hubert, A. & Sch afer, R. 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Commun. 6, 7886 (2015).\n[39] Vansteenkiste, A. et al. The design and veri\fcation of MuMax3 4, 107133 (2014).\n9Acknowledgements\nThe authors thank Dr. Thomas Oeckinghaus for his support with the experiment. J.W. ac-\nknowledges the Baden-W urttemberg Foundation, the European Research Council (ERC) (SMel\ngrant agreement No. 742610). R.S. thanks the EU ASTERIQS. The work at U. Washington\nis mainly supported by DOE BES DE-SC0018171. Device fabrication is partially supported by\nAFOSR MURI program, grant no. FA9550-19-1-0390. The authors also acknowledge the use\nof the facilities and instrumentation supported by NSF MRSEC DMR-1719797.\n10Methods\nSample fabrication The hBN \rakes of 10-30 nm were mechanically exfoliated onto 90\nnm SiO2/Si substrates and examined by optical and atomic force microscopy under ambient\nconditions. Only atomically clean and smooth \rakes were used for making samples. A V/Au\n(10/200 nm) microwave coplanar waveguide was deposited onto an 285 nm SiO 2/Si substrate\nusing standard electron beam lithography with a bilayer resist (A4 495 and A4 950 polymethyl\nmethacrylate (PMMA)) and electron beam evaporation. CrBr 3crystals were exfoliated onto\n90 nm SiO2/Si substrates in an inert gas glovebox with water and oxygen concentration less\nthan 0.1 ppm. The CrBr 3\rake thickness was identi\fed by optical contrast and atomic force\nmicroscopy. The layer assembly was performed in the glovebox using a polymer-based dry\ntransfer technique. The \rakes were picked up sequentially: top hBN, CrBr 3, bottom hBN.\nThe resulting stacks were then transferred and released in a gap of the pre-patterned coplanar\nwaveguide. In the resulting heterostructure, the CrBr 3\rake is fully encapsulated on both sides.\nFinally, the polymer was dissolved in chloroform for less than \fve minutes to minimize the\nexposure to ambient conditions.\nConfocal microscope The optics of the confocal microscope consists of the low-temperature\nobjective (Attocube LT-APO/VISIR/0.82) with 0.82 numerical aperture and home-built optics\nhead (see supplementary information). The 515 nm excitation laser generated by an electrically\ndriven laser diode is transmitted to the optics head through a polarization maintaining single-\nmode \fber and collimated by an objective lens. A pair of steering mirrors are used to align the\nbeam for perpendicular incidence to the center of the objective. The NV center's \ruorescence\nphotons are also collected by the objective and transmitted through the same free-beam path\nto the optics head. The collected \ruorescence photons are separated from the green laser beam\nvia a dichroic mirror and then passed through a band-pass \flter to further decrease background\nphotons. Finally, the photons are coupled to a single-mode \fber and detected by a \fber-coupled\nsingle-photon detector.\nStray magnetic \feld measurement With an external magnetic \feld applied along the\nNV-axis,Bk, the spin statesjms= 0iandjms=\u00061iexhibit Zeeman splitting of f=Ds\u0006\reBk,\nwhereDs\u00192:87 GHz is the zero \feld splitting between levels jms=\u00061iandjms= 0iand\n\re= 28 GHz/T is the electronic spin gyromagnetic ratio. In the pulsed ODMR measurement,\nthe NV center is optically initialized in spin state jms= 0i. After a delay of \u001c= 600 ns, a \u0019-\npulse (about 80 ns) of microwave radiation is applied. If the microwave frequency is in resonance\nwith one of the transitions, the NV center is driven to spin state jms=\u00061i. The population\ndi\u000berence between the spin states jms= 0iandjms=\u00061ican then be optically read out via\n\ruorescence contrast. In our experiment, the laser pulse duration is 600 ns, and only the \frst\n400 ns of the \ruorescence photon signal is used in the data analysis. The ODMR curve is\nobtained by sweeping the microwave frequency, which is achieved by modulating a microwave\nsignal with a pair of sinusoidal signals in quadrature via an IQ mixer. All the control signals\nof the measurement sequences are generated using an arbitrary waveform generator (AWG) so\nthat the measurement sequence is well synchronized, and fast microwave frequency sweeping is\nrealized by altering the frequency of the sinusoidal signals in each unit segment.\nMicromagnetic simulation To complement the experimental results micromagnetic sim-\nulations of the systems magnetic ground state were conducted using \\MuMax3\"[39]. Saturation\nmagnetization was set to Ms=270 kA/m, which is in accordance to the values measured here\nand reported in Ref[[36]]. Uniaxial magnetic anisotropy constant was assumed to be Ku=86\nkJ/m3along the normal axis, which has been reported for bulk material[36]. A global exchange\nsti\u000bness constant Aexwas \frst roughly estimated from Curie temperature and experimentally\nobserved domain wall width to lie within 10\u000012to 10\u000014J/m. Magnetization was initialized in\na random con\fguration and then relaxed to the minimum energy state at zero external \feld.\nThis was done for varying values of Aexfrom the interval estimated above.\nThe results for the normal magnetization component are shown in the Supplementary In-\n11formation for a system trying to locally approximate the irregular shape of the real sample.\nThis simulation assumed Aex= 3\u000210\u000013J/m, which resulted in the closest match for the\nexperiment. The simulation shows a domain structure that is qualitatively very similar to the\nexperimental observations, which are further supported by this result. However, due to the\nlimited capacity of the simulation to account for structural defects and pinning e\u000bects, the hys-\nteresis behavior observed experimentally could not accurately be recreated by it. This further\nsupports the conclusion that pinning is a major factor in this materials hysteresis.\n12"    },    {        "title": "1006.4075v2.Dynamics_of_magnetic_charges_in_artificial_spin_ice.pdf",        "content": "arXiv:1006.4075v2  [cond-mat.mtrl-sci]  24 Aug 2010Dynamics of magnetic charges in artificial spin ice\nPaula Mellado, Olga Petrova, Yichen Shen, and Oleg Tchernyshyov\nDepartment of Physics and Astronomy, The Johns Hopkins Univ ersity, Baltimore, Maryland 21218, USA\nArtificial spinice hasbeenrecentlyimplementedintwo-dim ensional arrays ofmesoscopic magnetic\nwires. We propose a theoretical model of magnetization dyna mics in artificial spin ice under the\naction of an applied magnetic field. Magnetization reversal is mediated by domain walls carrying\ntwo units of magnetic charge. They are emitted by lattice jun ctions when the the local field exceeds\na critical value Hcrequired to pull apart magnetic charges of opposite sign. Po sitive feedback from\nCoulomb interactions between magnetic charges induces ava lanches in magnetization reversal.\nSpin ice [1] shares some remarkable properties with\nwater ice [2]: both possess a very large number of low-\nenergy, nearly degenerate configurations satisfying the\nBernal-Fowler ice rules. In water ice, an O2−ion has\ntwo protons nearby and two farther away; in spin ice,\ntwo spins point into and two away from the center of ev-\nery tetrahedron of magnetic ions. Because the ice rules\nare satisfied by a large fraction of states, the system re-\ntains much entropy down to very low temperatures [3].\nLow-frequencydynamicsin ice is associatedwith the mo-\ntion of defects violating the ice rules. In water ice, these\ndefects carry fractional electric charges of ±0.62e(ionic\ndefects) and ±0.38e(Bjerrum defects) [2]. Fractionaliza-\ntion takes an even more surprising form in spin ice: while\nthe original degrees of freedom are magnetic dipoles, the\ndefects are magnetic monopoles [4–8].\nThe charge of an ice defect is defined in terms of the\nnet flux of electric field Eor magnetic field Hemerging\nfrom the defect. On the atomic scale, the flux is obscured\nby the fields of background ionic charges or magnetic\ndipoles. Coarse graining is required to reveal the field\nflux of a defect on longer length scales [5]. An alternative\napproach is to alter the model by stretching point-like\nspin dipoles into dumbbell magnets until they touch one\nanother, while keeping their dipole moments fixed [4]. At\nthe expense of a slight change in the Hamiltonian, the\nmagnetic charge of a defect becomes well defined even on\nthe microscopic scale. It equals ±2q≡ ±2µ/a, whereµ\nis the dipole moment and ais the length of a dumbbell.\nThe dumbbell model is realized in artificial spin ice, a\nnetwork of submicron ferromagnetic islands [9] or wires\n[10–12]. Each element represents a spin whose mag-\n+\n(a) (c)−++−+\n−H\n(b)−+++−\nFIG. 1: (a) A configuration of square spin ice with no mag-\nnetic charges. (b) Honeycomb spin ice always has magnetic\ncharges. (c) Magnetized honeycomb spin ice.netic dipole moment is aligned with the wire by shape\nanisotropy, Fig. 1. The magnetostatic energy is a posi-\ntive definite quantity Edip= (1/8π)/integraltextH2dV, where the\nintegral is taken over the entire space. It is minimized\nwhen the magnetic field H= 0.Edipcan be expressed as\nthe Coulomb interaction of magnetic charges with den-\nsityρ(r)≡ ∇·H/4π=−∇·M. The field is zero, and the\nenergyisminimized, when therearenomagneticcharges.\nThis yields the ice rule: a network node with zero mag-\nnetic charge has zero influx of magnetization. The zero-\nflux rule can be satisfied in square ice, Fig. 1(a), but not\nin honeycomb ice, Fig. 1(b), also known as kagome ice,\nwhere the allowed values of magnetic charge Qon a site\nare±qand±3qin units of q≡MA, whereMis the\nmagnetization of the magnetic wire and Ais its cross\nsection. Minimization of magnetic charge restricts Qto\nthe values of ±q, yielding the modified ice rule for this\nlattice: two arrows in and one out, or vice versa [10, 11].\nThe presence of residual magnetic charges in honey-\ncomb ice even at low temperatures may result in a se-\nquence of two phase transitions as its temperature is\nlowered: magnetic charge order appears first, spin order\narises later [13, 14]. Unfortunately, thermal fluctuations\nare virtually absent in artificial spin ice: reversing the\ndirection of magnetization in a single wire requires going\nover an energy barrier of a few million kelvins [9]. Left to\nitself, the system remains forever in the same magnetic\nmicrostate. Wang et al.suggested a way to introduce\nmagnetization dynamics into artificial spin ice by plac-\ning the system in a rotating magnetic field of an oscillat-\ning magnitude [15, 16], the analog of fluidizing granular\nmatter through vibration. It has been suggested [17, 18]\nthat such induced dynamics of magnetization effectively\ncreate a thermal ensemble with an effective temperature.\nIn this Letter we present an entirely different approach\nto the dynamics of artificial spin ice that incorporates\nthe physics of magnetization reversal in ferromagnetic\nnanowires, a process mediated by the creation, propaga-\ntion, and annihilation of magnetic domain walls [19, 20].\nIt is inherently dissipative [21, 22]: as a domain wall\npropagates, magnetic energy is transferred to the lattice.\nLike fluidized granular matter, artificial spin ice is a sys-\ntem far out of equilibrium and it is not obvious that it\ncan be described in the framework of equilibrium ther-2\nmodynamics [23]. Mesoscopic degrees of freedom of spin\nice tend to move downhill in the energy landscape until\nthey come to rest at a local energy minimum. We use\nthis approach to describe the dynamics of magnetization\nobserved in honeycomb spin ice [11] in an applied field.\nIn static equilibrium, artificial spin ice is fully de-\nscribed by specifying the direction of the magnetization\nvector in every link of the lattice. These are Ising vari-\nables because magnetization is aligned with the wire.\nSites of the lattice carry magnetic charge of ±qor±3qas\nexplained above. Site charges can be deduced from mag-\nnetization variables because the magnetic charge equals\nthe net influx of magnetization. The converse is not true\nbecause the number of links exceeds the number of sites\nby a factor of 3/2, so the magnetic state of artificial spin\nice cannot be described in terms of charges alone [11].\nSpin variables must be specified for a complete descrip-\ntion.\nTransitions between static states, triggered by the ap-\nplication of an external magnetic field, involve intermedi-\natestatesinwhichthe magnetizationofoneormorelinks\nis being reversed. At the mesoscopic level of our theory,\nsuch links are pictured as having two sections uniformly\nmagnetized in opposite directions separated by a domain\nwall of magnetic charge Q=±2q[21]. The reversal of\nmagnetization in a link begins with the creation of a do-\nmain wall at one of the link ends. The process conserves\nmagnetic charge: when a site with magnetic charge −q\nemits a domain wall of charge −2q, the charge of the site\nchanges to + q, Fig. 2. The Zeeman force −2qµ0Hthen\npushes the domain wall to the opposite end of the link.\nThe critical field required to initiate the reversal can\nbe estimated as follows. A site of charge + qand a do-\nmain wall −2qattract each other with a Coulomb force\nF∼µ02q2/(4πr2) at distances rexceeding the charac-\nteristic size of the charges a. The attraction weakens for\nshort distances r<∼awhen the two charges merge. The\nmaximum attraction is thus Fmax≈µ02q2/(4πa2). To\npull the charges apart, the Zeeman force 2 qBfrom the\napplied field must exceed Fmax, giving the critical field\nHc=q/(4πa2) =Mtw/(4πa2). (1)\nDomain walls in nanowires of submicron width whave\nthe characteristic size a≈0.6w[24]. For the permalloy\nhoneycomb network of Qi et al.[11] with magnetization\nM= 8.6×105A/m, width w= 110 nm and thickness\nt= 23 nm, µ0Hc≈50 mT.\nWhen the magnetic field is applied at an angle θto a\nlink, the Zeeman force comes from the longitudinal com-\nponentHcosθ. For this reason we expected the reversal\nto occur at a higher field H(θ) =Hc/cosθ. A similar\nangular dependence has been observed in magnetic wires\nwith submicron width [25].\nTotestthisphenomenologicalmodel, weperformednu-\nmerical simulations of magnetization reversal in a single(g)+1 +1H\n−2 (b) +1 +1H\n−2 (c)\n+1 −1H\n(d) +1 +1H\n(e)−2−1 +1H\n(a)\n+1 −1H\n−2 (h) +1 +1H\n−2 (i)\n−2\n+1 +1H\n(j) −2 +1H\n(k) −1+1 +1H\n(f)\n+1 −1H\n−2\nFIG. 2: Magnetization reversal in honeycomb spin ice. (a-\nd) A domain wall is emitted at one end of a link, travels to\nthe other end, and gets absorbed at the junction. (e-f) If\nthe applied field is sufficiently strong, a new domain wall can\nbe emitted into an adjacent link triggering its magnetizati on\nreversal. (g-k) When a domain wall encounters a site with\nlike magnetic charge, it induces the emission of a new domain\nwall into an adjacent link.\nµ0H, mT\n 90\n 80\n 70\n 60\n 50\nθ, deg 30 15 0−15−30−45−60−75Link 1\nLink 2\nbest fit\nFIG. 3: The reversal field Hof two out of three magnetic\nwires forming a junction vs. the angle between the field and\nthe axis of the wire whose magnetization is being reversed.\nThe lines are fits to Eq. (2) with µ0Hc= 52.0 and 55.3 mT.\njunction of three ferromagnetic nanowires using micro-\nmagnetics software package OOMMF [26] with the cell\nsize of 2 nm ×2 nm×23 nm. The dependence H(θ) is\nnot symmetric, Fig. 3, and is fit well by the function\nH(θ) =Hc/cos(θ+α), (2)\nwhere the offset α= 19◦reflects an asymmetric distribu-\ntion of magnetization at the junction, as we will discuss\nelsewhere [27]. The critical-field parameter Hcvaried\nslightly between links reflecting small random variations\nof the width caused by lattice discretization. Two links\nof the same junction exhibited slightly different critical\nfieldsHc, Fig. 3.\nWe use these phenomenologicalconsiderations and mi-\ncromagnetics simulations to build a discrete mesoscopic3\nmodel of magnetization dynamics in artificial spin ice.\nWe start with a fully magnetized state in which links of\nthe same orientation have the same direction of magne-\ntization and magnetic charges form a staggered pattern.\nSuch a state can be obtained by placing the system in a\nstrong magnetic field, Fig. 1(c). In this state, each mag-\nnetic wire has uniform magnetization pointing along the\nwire’s axis and each junction contains a magnetic charge\nof±1 in the units of q=Mtwdetermined by the flux\nof magnetization into the junction. The external field is\nthenappliedintheoppositedirectionwithagraduallyin-\ncreasing magnitude. Magnetization reversal begins when\nthe net field Hnetat one of the junctions exceeds a crit-\nical value determined by Eq. 2. The net magnetic field\nHnetis a superposition of the applied field Happand of\nthe demagnetizing field of the sample Hdem. The latter\nis computed as a sum of Coulombic fields of individual\njunctions, H=Qr/(4πr3). The junction, initially con-\ntaining charge ±1, emits a domain wall with charge ±2\nand changes its own charge to ∓1. The emitted domain\nwall is pushed by the magnetic field to the other end\nof the link, reversing the link magnetization in the pro-\ncess, Fig. 2 (b-c). Quenched disorder, inevitably present\nin real samples, is modeled by setting at random slightly\ndifferent critical fields Hcin individual wires with a mean\n¯Hcand a distribution width ∆ Hc.\nAs the domain wall with charge ±2 reaches the other\nend of the link, its further fate depends on sign of the\nmagnetic charge it meets at the junction. If the charge\nis of opposite sign, ∓1, then the domain wall is absorbed\nby the junction, Fig. 2(c-d), whose charge reverts to ±1.\nIf the net field is strong enough to stimulate the emission\nof a new domain wall of charge ±2 out of this junction,\nFig. 2(e), one of the adjacent links reverses its magneti-\nzation, Fig. 2(f). Otherwise, the evolution stops at the\nstage shown in Fig. 2(d).\nAlternatively, if the domain wall comes to a junction\nwith the same sign of charge, Fig. 4(a-b), it stops dis-\ntanceashort of the junction thanks to magnetostatic\nrepulsion. While this could be a new equilibrium posi-\ntion, the charged domain wall creates a field of strength\n2Hcat the junction, so that the net field at the junc-\ntion is close to 3 Hc. Its projection onto an adjacent link,\n1.5Hc, is sufficient to stimulate the emission of a new\ndomain wall of charge ±2 into that link, Fig. 4(c). The\njunction, now carrying charge of the opposite sign, ∓1,\npulls in the original domain wall and settles down in a\nstate with charge ±1, Fig. 4(d).\nThe sequence illustrated in Fig. 4 explains why ice rule\nviolations are hard to find in honeycomb ice of Qi et al.\n[11]. Unless variations of the critical field are so strong\nthatHcat some junctions exceeds 1 .5¯Hc, triply charged\njunctions, Fig. 4(b), are unstable and decay via the emis-\nsion of a new domain wall, Fig. 4(c-d). Permalloy sam-\nples of Qi et al.exhibit a Gaussiandistribution ofcritical\nfields with a standard deviation ∆ Hc= 0.04¯Hc[28], so(g)H\n+1\n−2+1(c)H\n+1 −1(a)H\n(b)−3 +1 +2\nH\n+1 +1\n(d)H\n+1+2−1\n(e)H\n−1+2\n+2−1\n(f)H\n+1 −1\nFIG. 4: Magnetization reversal in uniformly magnetized spi n\nice. (a-b)Inthebulk, thereversal inalinkmagnetized agai nst\nthe field would lead to the formation of triple charge, which\ncan only happen when the field is of order 3 Hc. (c) Instead,\nthe reversal occurs first in links magnetized at 120◦to the\nfield when H≈2Hc. (d-g) At the edge, the reversal begins\nwhenH≈Hcand propagates into the bulk.\nthat states with charge ±3 are only transients. Much\nstronger disorder exists in cobalt samples of Ladak et\nal.[12] who observed magnetization reversal in a field\nrange between H= 50 and 75 mT. Thus some of the\ndomain walls encounter junctions whose critical field ex-\nceeds 1.5H, which explains the presence of charges ±3.\nIn the limit of weak disorder, ∆ Hc≪¯Hc, there is\nanother characteristic scale of the field that becomes im-\nportant. The new scale set by the demagnetizing field\nof the sample Hdem, is the strength of the field created\nby a unit magnetic charge, Q=Mtw, at a neighboring\njunction distance Laway,H0=Mtw/(4πL2). When\n∆Hc≫H0, the reversal of magnetization is controlled\nmostly by the effects of quenched disorder, with links re-\nversinginalargelyindependent fashionintheorderofin-\ncreasing critical field Hc. Conversely, when ∆ Hc≪H0,\nthe reversal proceeds in a correlated fashion because of\na positive feedback: the reversal of magnetization in one\nlink redistributes magnetic charges at its ends, which in\nturnincreasesthe netfield at adjacentjunctions andthus\ntriggers the emission of domain walls there. In samples\nof Qiet al.,H0= 0.87 mT, which is comparable to the\nwidth of their reversal region, ∆ Hc= 2 mT.\nWe simulated magnetization reversal in this model\nwith the critical fields uniformly distributed in an in-\nterval of width ∆ Hc= 5 mT around the mean ¯Hc= 50\nmT and the Coulomb field scale H0= 0.87 mT. For sim-\nplicity we set the offset angle α= 0. A sample contain-\ning 937 links was initially magnetized along one subset\nof links, Fig. 1(c). Subsequently, the field was switched\noff and a reversal curve M(H) was measured in field ro-\ntatedthroughangle θfromtheinitialdirection. For120◦,\nquenched disorder dominates so that magnetization re-\nversals occur largely independently, in two stages. Links\nmagnetizedagainstthefieldswitchwhentheappliedfield\nis within the range ¯Hc±∆Hc/2, whereas links magne-\ntized at 120◦to the field switch in the range 2 ¯Hc±∆Hc.\nThe net magnetization Mxgrows in an approximately\nlinear fashion in both ranges, Fig. 5, as expected for4\nMx\n 600\n 400\n 200\n 0\n−200\n−400\n−600\nH, mT 100 90  80  70  60  50θ = 180°\nθ = 120° 0.001 0.01 0.1\ns 30 20 10 0θ = 180°\nθ = 120°\nFIG. 5: Simulated magnetization reversals. A sample is ini-\ntially magnetized in a strong field directed as in Fig. 1(c).\nSubsequently, the field is switched off and reapplied at angle s\nθ= 120◦and 180◦to the initial direction. Vertical dashed\nlines are at ¯Hc±∆Hc/2 and 2¯Hc±∆Hc. Inset: Distribution\nof avalanche lengths D(s) in the range of fields near ¯Hc. De-\nviations near the bottom of the graph are due to statistical\nnoise.\nlinks with a uniform distribution of Hc. Links do not\nreverse completely independently from one another: as\nnoted previously, the redistribution of magnetic charges\ninduced by the reversal of magnetization in one link may\ntrigger another reversal nearby. We observed that re-\nversals often involves small groups of links. As can be\nseen in the inset of Fig. 5, the distribution of the num-\nber of links sreversing in a single event is Gaussian,\nD(s)∝exp(−s2/2ξ2), withξ= 4.6.\nAn entirely different process is observed when the field\nis rotated through θ= 180◦. In this case, Coulomb in-\nteractions play a major role and the reversal proceeds\nthrough avalanches evidenced by steps in Mx(H), Fig. 5.\nWhen the field is near ¯Hc, the reversal cannot begin in\nthe bulk because links parallel to the applied field have\nthe wrong sign of magnetic charges at the ends and will\nreverse only in a much higher field (of order 3 Hc). Links\nat the edges have no such problem and the reversalstarts\nwhen a site at the edge emits a domain wall, Fig. 4(d-e).\nWhen the domain wall reaches the other end of the link,\nit encounters a site with like magnetic charge and trig-\ngersthe emission ofa new domain wall, Fig. 4(f), and the\nreversal of magnetization in an adjacent link, Fig. 4(g).\nThis triggers an avalanche of reversals that stops when\nthe traveling domain wall is absorbed by a junction with\na large critical field Hcorruns into already reversedlinks\n[12, 29]. The distribution of avalanche lengths (Fig. 5)\nfits a power law, D(s)∝s−τ, with the exponent τ= 1.6,\nindicative of self-organized criticality [30]. Chain rever-\nsals involving 3 links have been reported by Ladak et al.\n[12] in this geometry; avalanches involving up to 39 links\nhave been observed by Daunheimer et al.[28].We havepresentedadiscretemodel ofartificialspinice\nwhere magnetization dynamics is mediated by domain\nwalls carrying magnetic charge. Interactions between\nmagnetic charges compete with the effects of quenched\ndisorder. In samples with low disorder, positive feed-\nback from charge redistribution is responsible for mag-\nnetic avalanches that have been observed in some exper-\nimental situations.\nWe thank John Cumings for numerous discussions and\nfor sharing his unpublished data. This work was sup-\nported in part by the NSF Grant No. DMR-0520491.\n[1] S. T. Bramwell and M. J. P. Gingras, Science 294, 1495\n(2001).\n[2] V. F. Petrenko and R. W. Whitworth, Physics of ice\n(Oxford University Press, 1999).\n[3] A. P. Ramirez, A. Hayashi, R. J. Cava, R. Siddharthan,\nand B. S. Shastry, Nature 399, 333 (1999).\n[4] C. Castelnovo, R. Moessner, and S. L. Sondhi, Nature\n451, 42 (2008).\n[5] L. D. C. Jaubert and P. C. W. Holdsworth, Nat. Phys.\n5, 258 (2009).\n[6] D. Morris et al., Science 326, 411 (2009).\n[7] T. Fennell, P. Deen, A. Wildes, K. Schmalzl, D. Prab-\nhakaran, A. Boothroyd, R. Aldus, D. Mcmorrow, and\nS. Bramwell, Science 326, 415 (2009).\n[8] S. T. Bramwell, S. R. Giblin, S. Calder, R. Aldus,\nD. Prabhakaran, and T. Fennell, Nature 456, 956 (2009).\n[9] R. F. Wang et al., Nature 439, 303 (2006).\n[10] M. Tanaka, E. Saitoh, H. Miyajima, T. Yamaoka, and\nY. Iye, Phys. Rev. B 73, 052411 (2006).\n[11] Y. Qi, T. Brintlinger, and J. Cumings, Phys. Rev. B 77,\n094418 (2008).\n[12] S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and\nW. R. Branford, Nat. Phys. 6, 359 (2010).\n[13] G. M¨ oller and R. Moessner, Phys. Rev. B 80, 140409\n(2009).\n[14] G.-W. Chern, P. Mellado, and O. Tchernyshyov (unpub-\nlished), arXiv:0906.4781.\n[15] R. P. Cowburn, Phys. Rev. B 65, 092409 (2002).\n[16] X. Ke, J. Li, C. Nisoli, P. E. Lammert, W. McConville,\nR. F. Wang, V. H. Crespi, and P. Schiffer, Phys. Rev.\nLett.101, 037205 (2008).\n[17] C. Nisoli, R. Wang, J. Li, W. F. McConville, P. E. Lam-\nmert, P. Schiffer, and V. H. Crespi, Phys. Rev. Lett. 98,\n217203 (2007).\n[18] L. A. M´ ol, R. L. Silva, R. C. Silva, A. R. Pereira,\nW. A. Moura-Melo, and B. V. Costa, J. Appl. Phys. 106,\n063913 (2009).\n[19] A. Thiaville and Y. Nakatani, in Spin Dynamics in Con-\nfined Magnetic Structures III (Springer, 2006), vol. 101\nofTopics in applied physics , pp. 161–205.\n[20] M. Hayashi, L. Thomas, C. Rettner, R. Moriya, and\nS. S. P. Parkin, Nat. Phys. 3, 21 (2007).\n[21] O. A. Tretiakov, D. Clarke, G.-W. Chern, Y. B. Baza-\nliy, and O. Tchernyshyov, Phys. Rev. Lett. 100, 127204\n(2008).\n[22] C. H. Wong and Y. Tserkovnyak, Phys. Rev. B 81,\n060404 (2010).5\n[23] G. M¨ oller and R. Moessner, Phys. Rev. Lett. 96, 237202\n(2006).\n[24] A. Kunz, J. Appl. Phys. 99, 08G107 (2006).\n[25] W. Wernsdorfer, B. Doudin, D. Mailly, K. Hasselbach,\nA. Benoit, J. Meier, J. P. Ansermet, and B. Barbara,\nPhys. Rev. Lett. 77, 1873 (1996).\n[26] M. J. Donahue and D. G. Porter, Tech. Rep. NISTIR\n6376, National Institute of Standards and Technology,Gaithersburg, MD (1999), http://math.nist.gov/oommf.\n[27] P. Mellado et al., manuscript in preparation.\n[28] S. Daunheimer, Y. Qi, and J. Cumings (unpublished).\n[29] O. Tchernyshyov, Nat. Phys. 6, 323 (2006).\n[30] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. A 38,\n364 (1988)."    },    {        "title": "1205.0795v1.Dynamical_Friction_in_a_magnetized_gas.pdf",        "content": "arXiv:1205.0795v1  [astro-ph.GA]  3 May 2012Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 4 July 2018 (MN L ATEX style file v2.2)\nDynamical Friction in a magnetized gas\nMohsen Shadmehri⋆and Fazeleh Khajenabi †\nSchool of Physics, Faculty of Science, Golestan University , Gorgan, Iran\n4 July 2018\nABSTRACT\nWhen a gravitating point mass moves subsonically through a magnetiz ed and isother-\nmal medium, the dynamical structure of the flow is studied far from the mass using a\nperturbation analysis. Analytical solutions for the first-order de nsity and the velocity\nperturbations are presented. Validity of our solutions is restricte d to the cases where\nthe Alfven velocity in the ambient medium is less than the accretor’s ve locity. The\ndensity field is less dense because of the magnetic effects according to the solutions\nand the dynamical friction force becomes lower as the strength of the magnetic field\nincreases.\nKey words: magnetohydrodynamics - ISM: general - galaxies: kinematics and d y-\nnamics - stars: kinematics\n1 INTRODUCTION\nIn many astronomical systems when an object (e.g., a star\n) travels through a collisionless or a collisional system, i t\nis decelerated in the direction of its motion because of the\ngravitational interaction of the perturber (i.e, the objec t)\nand the wakes it excites in the ambient medium. Theoretical\nefforts to study this problem have been started after a\npioneering work by Chandrasekhar (1943). He considered\na collisionless background and the drag force that the per-\nturber experiences was found analytically. Since then it ha s\nbeen applied to many astronomical systems, as in migration\nof protoplanets in protoplanetary accretion discs (e.g.,\nMuto, Takeuchi & Ida 2011; Villaver & Livio 2009), motion\nof supermassive black holes within gas-rich galaxies (e.g. ,\nSijacki, Springel & Haehnelt 2011; Ruderman & Spiegel\n1971), orbital evolution of embedded binary stars (e.g.,\nStahler 2010), motion of compact stars around super-\nmassive black holes (Narayan 2000), and the heating of\nthe galaxy clusters (Kim, El-Zant & Kamionkowski 2005;\nEl-Zant, Kim & Kamionkowski 2004) and even studies at\nthe cosmological scales (Tittley et al. 2001).\nThe standard approach based on the linearized hy-\ndrodynamics equations for describing the excited small\namplitude perturbations and calculating their gravitatio nal\ninteraction with the object moving in a straight-line\ntrajectory (e.g., Dokuchaev 1964; Rephaeli & Salpeter\n1980; Ostriker 1999). Numerical simulations confirmed\nresults of the linear analysis for the drag force (e.g.,\nS´ anchez-Salcedo & Brandenburg 1999; Kim & Kim 2009).\n⋆Email:m.shadmehri@gu.ac.ir\n†E-mail: f.khajenabi@gu.ac.ir;The standard approach has been extended to the cases\nwith circular-orbit perturber (Kim 2010), double per-\nturbers (Kim et al. 2008) or accelerated motion in a\nstraight-line (Namouni 2010). In calculating the drag\nforce, the role of the relativistic effects has also been stud -\nied by Barausse (2007). A modified Newtonian approach\nis used by S´ anchez-Salcedo (2009) to analysis the drag forc e.\nRecently, Lee & Stahler (2011) (hereafter LS) argued\nthat the physical size of the perturber is much smaller than\nthe accretion radius in the astronomical systems such as\nsupermassive black holes within galaxies or giant planets\ninside protoplanetary discs. Under these circumstances, o ne\ncan not neglect the mass accretion and the amplitude of\nthe wakes is not small as authors have assumed in their lin-\near approach for calculating the drag force. Thus, to a large\nextend it is the mass accretion which provides the linear mo-\nmentum transfer from the ambient medium to the object.\nLS showed analytically that the steady-sate friction force is\nequal to the accretion rate onto the object multiply by its\nvelocity. If the accretion rate is prescribed by the Bondi-\nHoyle rate (Bondi & Hoyle 1944), the drag force does not\nrise monotonically but instead peaks around Mach number\n0.68 and then begin to decline (LS). It is an interesting re-\nsult that may find application in a number of astronomical\nsystems. However, there are many other important physical\nfactors that may modify the drag force, and, we think, the\napproach of LS has this possibility to be generalized to in-\ncludethose physicalingredientssuchas amagnetized and/o r\nisentropic ambient medium. Just recently, Khajenabi & Dib\n(2012) extended the approach of LS to the case of an isen-\ntropic medium and found a set of analytical solutions for\nthe structure of the flow far from object. They proved the\nc/circlecop†rt0000 RAS2M. Shadmehri & F. Khajenabi\nproportionality between the friction force and the accreti on\nrate also holds in the isentropic case.\nThe purpose of this paper is to calculate the dynami-\ncal friction force when the perturber moves with constant\nvelocity through a magnetized medium. In most of the as-\ntronomical systems, the ambient medium is actually mag-\nnetized, and one may expect the excited wakes are modified\nbecause of the magnetic effects. In performing this analy-\nsis, we also assume the physical size of the object is much\nsmaller than the accretion radius and the approach of LS\nis followed, but including the magnetic fields. Basic equa-\ntions are presented in the next section. Introducing pertur -\nbation expansions in section 3, we will find analytical so-\nlutions for the first-order perturbed variables in section 4 .\nHowever, second-order solutions and the dynamical frictio n\nforce are obtained numerically in section 5. We conclude by\nsome physical implications of the results in section 6.\n2 BASIC EQUATIONS\nThe basic equation of our problem in the steady state are\nwritten as\n∇·(ρU) = 0, (1)\nρ(U·∇)U=−∇P−ρGM\nR2+1\n4πJ×B, (2)\n∇×(U×B) = 0, (3)\n∇·B= 0, (4)\nwhereρ,U,Pare the density, the velocity and the pressure,\nrespectively. Here, Ris the radial distance in the spher-\nical coordinates ( R,θ,ϕ) whose origin is anchored on the\ngravitating mass M(see Figure 1). The current density is\nJ=∇ ×B. Moreover, we also assume the gas is isother-\nmal, i.e.P=ρc2\nswherecsis the sound speed. We ignore\nthe complications from density gradients. Very far from the\nmass which travels in a straight line trajectory with veloc-\nityVnot only the density, but the magnetic fields are both\nassumed to be spatially uniform, i.e. lim R→∞ρ≡ρ0and\nlimR→∞B≡B0ezwhereezis the unit vector along the z\ndirection (Figure 1).\nIntroducing a set of non-dimensional variables as r=\nR/Rs,u=U/cs,̺=ρ/ρ0andb=B/B0, we can write\nour main equations in the non-dimensional forms. Here, the\nsonic radius is defined as Rs≡GM/c2\ns. Therefore, we have\n∇·(̺u) = 0, (5)\n̺(u·∇)u=−∇̺−̺\nr2+ξ2(∇×b)×b, (6)\n∇×(u×b) = 0, (7)\n∇·b= 0, (8)\nwhere the non-dimensional parameter ξis defined as the ra-\ntio of the Alfven velocity VA=B0/√4πρ0and the sound\nspeed, i.e.ξ=VA/cs. Obviously, in equations (5)-(8) the\nradial derivatives are calculated with the respect to the no n-\ndimensional radial distance r. We also define β=V/cs.\nThus, properties of the flow at the distances far from the\ngravitating object Mareu=βez,̺= 1 and b=ezwhen\nr≫1./s122/s114/s32/s61/s32/s82/s47/s82\n/s115\n/s77/s86\n/s66\n/s48\nFigure 1. This figure shows the assumed magnetic field and the\nflow velocity. The object is fixed at the origin and the magneti c\nfield and the velocity of the gas are both uniform at the large\ndistances from the object.\nNote that in our subsequent calculations, it is assumed\nthat the system is axisymmetric ( ∂/∂ϕ= 0). If we introduce\nthe non-dimensional stream function ψ(r,θ) as\n̺u=∇×(ψ\nrsinθeϕ), (9)\nthen the non-dimensional continuity equation (5) is auto-\nmatically satisfied. Thus, the components of the velocity ca n\nbe written in terms of the stream function as\nur=1\n̺r2sinθ∂ψ\n∂θ, (10)\nuθ=−1\n̺rsinθ∂ψ\n∂r. (11)\nWe can also define the non-dimensional positive-definite\nmagnetic flux function Φ as\nb=∇×(Φ\nrsinθeϕ). (12)\nSo, equation (8) is satisfied and the components of the mag-\nnetic field become\nbr=1\nr2sinθ∂Φ\n∂θ, (13)\nbθ=−1\nrsinθ∂Φ\n∂r. (14)\nHaving the above equations for the components of the\nvelocity in terms of ψand the magnetic field in terms of Φ,\nwe are left only with the momentum equation (6) and the\ninduction equation (7). Thus, we have\n̺/parenleftbigg\nur∂ur\n∂r+uθ\nr∂ur\n∂θ−u2\nθ\nr/parenrightbigg\n=−∂̺\n∂r−̺\nr2−ξ2bθ\nr×\n/bracketleftBig∂\n∂r(rbθ)−∂br\n∂θ/bracketrightBig\n, (15)\n̺/parenleftBig\nur∂uθ\n∂r+uθ\nr∂uθ\n∂θ+uruθ\nr/parenrightBig\n=−1\nr∂ρ\n∂θ+ξ2br\nr×\n/bracketleftBig∂\n∂r(rbθ)−∂br\n∂θ/bracketrightBig\n, (16)\nurbθ−uθbr= 0. (17)\nc/circlecop†rt0000 RAS, MNRAS 000, 000–000Dynamical Friction in a magnetized gas 3\n/s45/s49/s50 /s45/s56 /s45/s52 /s48 /s52 /s56 /s49/s50/s45/s49/s50/s45/s56/s45/s52/s48/s52/s56/s49/s50\n/s45 /s45\n/s32/s32\n/s49/s46/s48/s57/s49/s46/s48/s57\n/s49/s46/s48/s57\n/s49/s46/s49/s51/s49/s46/s49/s51 /s49/s46/s49/s57/s49/s46/s49/s57/s49/s46/s52/s56\n/s45\n/s32/s32/s82\n/s90/s49/s46/s53/s56\nFigure 2. Isodensity curves corresponding to the first-order den-\nsity perturbation for β= 0.5 andξ= 0.4 (solid) andξ= 0\n(dashed). Each contour is labeled by its level. The gray inner cir-\ncle denotes the sonic radius. Obviously, the solutions are m ore\naccurate at the distances far from the inner sonic circle.\nSince we have neglected the dissipative terms in the induc-\ntion equation and the system is steady state, we could say\nbased on the equations (5) and (8) that the magnetic field\nand the velocity vectors are parallel. The simplified induc-\ntion equation (17) also shows this behavior.\n3 PERTURBATION EXPANSIONS\nWe expand the density and the stream function similar to\nLS as\n̺(r,θ) = 1+g−1(θ)r−1+g−2(θ)r−2+g−3(θ)r−3+···,(18)\nψ(r,θ) =f2(θ)r2+f1(θ)r+f0(θ)+f−1(θ)r−1+···,(19)\nand in our magnetized model, the magnetic flux function is\nexpanded as\nΦ(r,θ) =h2(θ)r2+h1(θ)r+h0(θ)+h−1(θ)r−1+···.(20)\nSince the velocity tends to be constant far from the\nobject, one can simply obtain f2(θ) = (1/2)βsin2θ. Also,\nmagnetic fieldlines areuniform farfrom theobjectandthen,\nh2(θ) = (1/2)sin2θ. We can set ψ(r,π) = 0, and so fi(π) =\n0 fori= 1,0,−1,−2 etc., and regularity of the components\nof the velocity on the axis defined by θ= 0 andθ=π\nimpliesf′\ni(π) =f′\ni(0) = 0, for i= 1,0,−1,−2,etc., and\nfi(0) = 0 for i= 1,−1,−2,etc. (see LS for details of their\narguments). Boundary conditions on the geometry of the\nmagnetic field lines also imply hi(π) =h′\ni(π) =h′\ni(0) = 0,\nfori= 1,0,−1,−2,etc., andhi(0) = 0 fori= 1,−1,−2,etc.\nSubstituting the above expansions into the main equations,\na set of ordinary differential equations are obtained, which/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s48/s49/s50/s51/s52/s53/s54\n/s48/s46/s52/s48/s46/s51/s48/s46/s50/s61/s48\n/s32/s32/s32/s32/s32/s32/s32/s103\n/s45/s50\n/s47/s32/s32/s61/s32/s48/s46/s53\n/s48/s46/s50/s61/s48\n/s48/s46/s51\n/s48/s46/s52/s103\n/s45/s50/s32/s40 /s41/s32/s61/s32/s45/s48/s46/s56/s56\n/s32/s32/s32/s32/s32/s32/s32/s102\n/s48\nFigure 3. Profiles of f0(θ) andg−2(θ) for different values of ξ\nwhen the initial value of g−2(π) is -0.88. Each curve is labeled by\nthe corresponding value of ξ.\nare solvable analytically and/or numerically subject to th e\nmentioned boundary conditions.\nAlso, the mass accretion rate is written as\n˙M=−2π/integraldisplayπ\n0ρURR2sinθdθ. (21)\nIf we nondimensionalize the accretion rate ˙Mby 2πρ0csR2\ns\nand using expansion (19), the nondimensonal accretion rate\n˙Mbecomes (see LS for the details)\n˙M=f0(0). (22)\n4 FIRST ORDER EQUATIONS\nWe can now substitute the expansions (18), (19) and (20)\ninto the momentum equations (15) and (16) and the induc-\ntion equation (17). Then, we match the coefficients of each\npower ofr. Like the non-magnetized case (LS), we find the\nhighest power of risr−1, but all the corresponding coeffi-\ncients on both sides of the equations vanish identically. So ,\nwe consider the next power of r(i.e.r−2) and the first-order\nequations are obtained. From the radial component of the\nmomentum equation, we have\n−βf′′\n1−βf1+β2sinθcosθg′\n−1+(β2cos2θ−1)g−1+\n1+ξ2(h′′\n1−cotθh′\n1) = 0, (23)\nc/circlecop†rt0000 RAS, MNRAS 000, 000–0004M. Shadmehri & F. Khajenabi\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s57/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s45/s51/s45/s50/s45/s49/s48/s49/s50\n/s48/s46/s52/s48/s46/s51/s48/s46/s50/s61/s48\n/s32/s32/s103\n/s45/s50\n/s47/s48/s46/s52/s48/s46/s51/s48/s46/s50/s61/s48\n/s32/s61/s32/s48/s46/s53/s103\n/s45/s50/s40 /s41/s32/s61/s32/s48/s46/s49/s50\n/s32/s32/s102\n/s48\nFigure 4. The same as Figure 3, but for g−2(π) = 0.12.\nand theθ−component of the momentum equation gives\n(1−β2sin2θ)g′\n−1−β2sinθcosθg−1+\nξ2cotθ(h′′\n1−cotθh′\n1) = 0, (24)\nand finally the induction equation (17) yields\ncosθf1−βcosθh1+βsinθh′\n1−sinθf′\n1= 0 (25)\nFor the non-magnetized gas, the above first-order equa-\ntions reduce to the equations of LS if we set ξ= 0. We can\nsolve these equations for f1,g−1andh1, and then proper-\nties of the flow to the first order are determined. One can\nintegrate equation (25) simply by re-arranging its terms as\ncosθ(f1−βh1)−sinθd\ndθ(f1−βh1) = 0. (26)\nIntroducing W=f1−βh1, this equation becomes dW/dθ=\nWcotθand its solution is W=W0sinθwhereW0is an\narbitrary constant to be determined from the boundary\nconditions. Thus, equation f1−βh1=W0sinθis valid\nfor the whole range of θincludingθ=π. So, we have\nf′\n1(π)−βh′\n1(π) =W0cosπ=−W0. Considering our im-\nposed boundary conditions f′\n1(π) =h′\n1(π) = 0, we obtain\nW0= 0 and\nf1=βh1. (27)\nWecannowsubstituteequation(27)intoequations(23)\nand (24) and the following equations for f1andg−1are\nobtained/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s49/s46/s56/s45/s49/s46/s50/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54/s49/s46/s50/s45/s56/s45/s54/s45/s52/s45/s50/s48\n/s48/s46/s52/s48/s46/s50/s61/s48\n/s32/s32/s103\n/s45/s50\n/s47/s48/s46/s52/s48/s46/s50/s61/s48\n/s32/s61/s32/s48/s46/s53/s103\n/s45/s50/s40 /s41/s32/s61/s32/s49/s46/s49/s50\n/s32/s32/s102\n/s48\nFigure 5. The same as Figure 3, but for g−2(π) = 1.12.\n(ξ2−β2\nβ)f′′\n1−ξ2\nβcotθf′\n1−βf1+β2sinθcosθg′\n−1+\n(β2cos2θ−1)g−1+1 = 0, (28)\n(1−β2sin2θ)g′\n−1−β2sinθcosθg−1+ξ2\nβcotθf′′\n1−\nξ2\nβcot2θf′\n1= 0. (29)\nThus, equations (28) and (29) are our main equa-\ntions to be solved. Interestingly, these equations are inte -\ngrable, though the mathematical manipulation is cumber-\nsome. Here, we summarize the solutions as\nf1(θ) =1\nβ−1\nβ1√\nΓ/radicalbig\nΓ−β2sin2θ, (30)\ng−1(θ) =1−ξ2/β2\n√\nΓ1/radicalbig\nΓ−β2sin2θ, (31)\nwhere Γ = 1 + ξ2−ξ2/β2. For the non-magnetized case\n(i.e.ξ= 0), we have Γ = 1 and the above solutions reduce\nto equations (25) and (26) of LS. Note that the above first\norder solutions are physically acceptable if Γ −β2sin2θ>0.\nIfforθ=π/2(where thesecond term has amaximum value)\nthe left hand side becomes positive, then for other values of\nθthe left-hand side would be positive (as we want). Having\nΓ = 1+ξ2−ξ2/β2, we can write 1 + ξ2−ξ2/β2−β2>0\nwhich implies β > ξ. Thus, our solutions are applicable to\nthecase where the Alfven speed is less thanthe sound speed.\nc/circlecop†rt0000 RAS, MNRAS 000, 000–000Dynamical Friction in a magnetized gas 5\n/s45/s49/s50 /s45/s56 /s45/s52 /s48 /s52 /s56 /s49/s50/s45/s49/s50/s45/s56/s45/s52/s48/s52/s56/s49/s50\n/s45 /s45\n/s32/s32/s82\n/s90/s45/s49/s46/s48/s53/s49/s46/s50/s49/s46/s50/s56\n/s45/s49/s50 /s45/s56 /s45/s52 /s48 /s52 /s56 /s49/s50/s45/s49/s50/s45/s56/s45/s52/s48/s52/s56/s49/s50\n/s45 /s45\n/s32/s32/s82\n/s90/s45/s49/s46/s48/s53/s49/s46/s49/s54\n/s49/s46/s50\nFigure 6. Density contours for the non-magnetized ( top plot)\nand the magnetized flow ( bottom plot ) withβ= 0.5,ξ= 0.4 and\ng−2(π) =−0.88.\nFigure 2 shows isodensity contours for the first-order\ndensity perturbation, i.e. equation (31). Solid contours a re\nfor a magnetized ambient medium when ξ= 0.4. Nonmag-\nnetized contours are also shown by the dashed curves. In\nboth cases, the Mach number is fixed β= 0.5. A circle with\nunit radius at the center denotes the sonic radius. Obvi-\nously, our solutions are valid beyond the sonic radius and\nall the contours are drawn for the radii larger than 2. Each\ncontour is labeled by the corresponding level. For the non-\nmagnetized solutions, the density varies from 1 .09 to 1.58 at\nthe inner parts and the magnetic effects change this interval\nfrom 1.09 to 1.48. In other words, the density field in the\npresence of the magnetic field is less dense in comparison to\nthe non-magnetic solutions. When magnetic fields are not\nconsidered, the perturbations in the ambient medium grow\neasier and so the wakes have larger amplitudes comparing\nto the magnetic case. It is because of the magnetic pressure/s45/s49/s50 /s45/s56 /s45/s52 /s48 /s52 /s56 /s49/s50/s45/s49/s50/s45/s56/s45/s52/s48/s52/s56/s49/s50\n/s45/s45\n/s32/s32/s82\n/s90/s45\n/s45/s49/s50 /s45/s56 /s45/s52 /s48 /s52 /s56 /s49/s50/s45/s49/s50/s45/s56/s45/s52/s48/s52/s56/s49/s50\n/s45 /s45\n/s32/s32/s82\n/s90/s45\nFigure 7. Streamlines for the non-magnetized ( top plot) and the\nmagnetized flow ( bottom plot ) withβ= 0.5,ξ= 0.4 andg−2(π) =\n−0.88.\nwhich acts as an extra pressure against further growth of\nthe perturbations.\n5 SECOND-ORDER EQUATIONS\nNow, we can equate coefficients of the next power of rin the\ncomponents of the momentum equation and the induction\nequation. From the induction equation, we obtain\nf0=βh0. (32)\nConsidering the above relation between f0andh0, we\nobtain the following equation from the radial component of\nthe momentum equation\n(ξ2\nβ−β)f′′\n0−(β+ξ2\nβ)cotθf′\n0+β2sinθcosθg′\n−2+\n(2β2cos2θ−2)g−2=A1+A2+A3+A4, (33)\nc/circlecop†rt0000 RAS, MNRAS 000, 000–0006M. Shadmehri & F. Khajenabi\nwhere all the terms on the righthand side depend on the first\norder variables as\nA1=f2\n1\nsin2θ−f1f′\n1cosθ\nsin3θ+(f′\n1)2\nsin2θ+f1f′′\n1\nsin2θ, (34)\nA2=βf1g−1−2βf′\n1g−1cotθ−βf1g′\n−1cotθ−\nβf′\n1g′\n−1+βf′′\n1g−1, (35)\nA3= 2(g−1)2−3g−1. (36)\nA4= (ξ2\nβ)(−2f′′\n1g−1+2cotθf′\n1g−1+f′\n1f1cotθ\nβsin2θ−\nf′′\n1f1\nβsin2θ) (37)\nAlso, theθ−component of equation of motion gives\n(ξ2\nβ)cotθf′′\n0−(β+ξ2\nβcot2θ)f′\n0+(1−β2sin2θ)g′\n−2−\n2β2sinθcosθg−2=B1+B2+B3+B4, (38)\nwhere\nB1=f2\n1cotθ\nsin2θ−f1f′\n1\nsin2θ, (39)\nB2=βf1g−1cotθ+βf′\n1g−1+2βf1g′\n−1, (40)\nB3=−2g−1g′\n−1, (41)\nB4=ξ2\nβ2(f′2\n1cotθ\nsin2θ−f′\n1f′′\n1\nsin2θ−2βf′′\n1g−1cotθ+\n2βf′\n1g−1cot2θ). (42)\nBoth righthand sides of equations (33) and (38) are de-\ntermined analytically because we have already found f1and\ng−1, i.e. equations (30) and (31). Although equation (38) is\nintegrable as we will show, it is very unlikely to solve this\nequation andequation (33) for f0(θ) andg−2(θ)analytically.\nSo, one should solve these equations numerically subject to\nthe appropriate boundary conditions. Note that if we set\nξ= 0, these equations reduce to equations (27) and (31)\nof LS. Introducing D= Γ−β2sin2θ, we can re-write the\nsecond-order equation (33) as\n(ξ2\nβ−β)f′′\n0−(β+ξ2\nβ)cotθf′\n0+β2sinθcosθg′\n−2+\n(2β2cos2θ−2)g−2=1\n−Γ(Γ−D)D2(β2−1)2×\n(R1D5/2+R2D2+R3D3/2+R4D+R5), (43)\nwhere\nR1=√\nΓ(1−β2)(β2+2−3Γ), (44)\nR2= 2Γ3−4β2Γ2+(5β2−3)Γ+β2−β4, (45)\nR3= 3Γ3/2(β2−1)(β2−Γ), (46)\nR4= Γ(Γ−β2)[−4Γ2+2(1+2β2)Γ+1−3β2],(47)\nR5= 2Γ2(Γ−1)(β2−Γ)2. (48)\nAlso, equation (38) is written as\n(ξ2\nβ)cotθf′′\n0−(β+ξ2\nβcot2θ)f′\n0+(1−β2sin2θ)g′\n−2−2β2sinθcosθg−2=β2sinθcosθ\n−Γ(Γ−D)2D2(β2−1)2×\n(R6D5/2+R7D2+R8D3/2+R9D+R10D1/2+R11),(49)\nwhere\nR6=√\nΓ(Γ−1)(β2−1), (50)\nR7= 2Γ3−4β2Γ2+(5β2−3)Γ+β2−β4, (51)\nR8= Γ3/2(4β2−1−3Γ)(β2−1), (52)\nR9= Γ(Γ−β2)[−4Γ2+(5β2−1)Γ+2−2β2], (53)\nR10= 2Γ5/2(Γ−β2)(β2−1), (54)\nR11= Γ2(β2−Γ)[−2Γ2+(3β2−1)Γ+1−β2].(55)\nThe lefthand side of the second-order equation (49) is\nperfect derivative. By integrating both side of this equati on\nfromθ=πto 0, we find\n/bracketleftbigg\nξ2\nβcotθf′\n0+(ξ2\nβ−β)f0+Dg−2/bracketrightbigg\nθ=π=\n/bracketleftbigg\nξ2\nβcotθf′\n0+(ξ2\nβ−β)f0+Dg−2/bracketrightbigg\nθ=0. (56)\nSincef′\n0(π) =f′\n0(0) = 0, the above equation gives\nf0(0) =β\nβ2−ξ2[g−2(0)−g−2(π)]. (57)\nIf we setξ= 0, the above equation reduces to equation (46)\nof LS. Here, like to the non-magnetized case, the accretion\nrate is proportional to the difference, upstream and down-\nstream, of the second-order density perturbation. However ,\nthe proportionality constant not only depends on the veloc-\nity of the accretor, but also it depends on the magnetic field\nstrength.\nWe can solve the equations for the same three values\nofg−2(π) as were used by LS, just in order to make easier\ncomparison to the non-magnetized case. Note that our main\nequations (43) and (49) are singular at both endpoints θ= 0\nandθ=π. Figures 3, 4 and 5 show behaviors of f0(θ) and\ng−2(θ) for three fiducial values of g−2(π) =−0.88,0.12,1.12.\nJust to make an easier comparison to the non-magnetized\nprofiles of LS these boundary values for g−2(π) have been\nselected. In each figure, the starting value of g−2(π) is fixed,\nand we play with the ratio ξas a free input parameter.\nProfiles of the second-order coefficients strongly depend on\nthe strength of the magnetic field as shown in these figures.\nThe non-magnetic curves passes through the same point at\nθ=π/2. But magnetic effects change this trend, in partic-\nular, when ξbecomes closer to β. However, the coefficient\ng−2(θ) is flat atθ= 0 andπwhich means the second-order\ndensity perturbation is uniform at both the downstream\nand upstream of the flow irrespective of the existence of\nthe magnetic fields. It is understandable based on our im-\nposed boundary conditions. Density contours and the veloc-\nity streamlines are also shown in Figures 6 and 7 for the\nmagnetized and the non-magnetized case corresponding to\nthe parameters β= 0.5,ξ= 0.4 andg−2(π) =−0.88. Here,\nthe curves are shown for r/greaterorequalslant2. But the inner contours in\nthese figures continue to the region with r<2 which is not\nc/circlecop†rt0000 RAS, MNRAS 000, 000–000Dynamical Friction in a magnetized gas 7\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s32/s61/s32/s48\n/s32/s32/s70/s114/s105/s99/s116/s105/s111/s110/s32/s70/s111/s114/s99/s101/s48/s46/s50\n/s48/s46/s51\n/s48/s46/s52\n/s48/s46/s53\n/s48/s46/s55/s48/s46/s54\nFigure 8. The nondimensional dynamical friction force versus\nthe Mach number βfor different values of ξ. Each curve is limited\nby the inequality ξ < βand the numbers are the corresponding\nvalue of ξ.\nshown and so, are connected vertically artificially. The rea -\nson we exclude an inner part goes to back to the validity\nof our expansions for the radial distances greater than the\naccretion radius.\nLike LS, there is no restriction on g−2(π) in the mag-\nnetized case as long as we do not extend the solutions to\nthe inner part of flow or else adopting a model for the ac-\ncretion and fixing the accretion rate. LS followed the sec-\nond approach which is much easier and avoids complexi-\nties related to the conditions on the sonic surface. Their\naccretion model is the classical Bondi-Hoyle (Bondi 1952;\nBondi & Hoyle 1944) model for which there is an analytical\nrelation for accretion rate as,\n˙M=2(λ2+β2)1/2\n(1+β2)2. (58)\nHavingthe above the relation for the accretion rate, one\nboundary condition is given at θ= 0 asf0(0) =˙M. The rest\nof boundary conditions are at θ=πasf0(π) =f′\n0(π) = 0.\nThus, we can solve the second-order equations as a two-\npoints boundary value problem. As far as we know, for the\nmagnetized accretion there is not a closed analytical for-\nmula for the accretion rate. However, it was shown that\nthe magnetic field effects suppress the accretion rate (e.g.,\nIgumenshchev 2006; Shadmehri 2004; Shcherbakov 2008;\nIgumenshchev & Narayan 2002; Perna et al. 2003). For ex-\nample, when the accretion occurs as a radiatively inefficient\nflow, the magnetic field is amplified as the gas flows in, and\nso, the accretion rate is reduced in comparison to the Bondi\nrate as studied numerically by Igumenshchev & Narayan\n(2002). But there are many uncertainties regarding to the\nmagnetized accretion, in particular about the true physica l\nmechanism of the accretion or when the accretor is in trav-\neling. In order to proceed analytically, we introduce a toy\naccretion rate as˙Mmag=2(λ2+β2)1/2\n(1+ξ2+β2)2, (59)\nwhich gives a reduced rate because of the magnetic field.\nHere, the rate decreases with ξand it gives Bondi-Hoyle ac-\ncretion rate if we set ξ= 0. It is as if we replace the thermal\npressure by the sum of thermal and the magnetic pressures,\ni.e.c2\ns→c2\ns+v2\nA. In fact, the extra pressure provided by\nthe magnetic force lead to the reduction in the accretion\nrate. We emphasize our prescribed accretion rate relation\n(59) isnotan accurately physically confirmed relation, but\nit shows a reduction to the accretion rate with increasing\nthe strength of the magnetic field at least qualitatively.\nHaving equation (59) for the accretion rate, we did a\ndetailed parameter studyby solving the main equations (43)\nand (49) numerically. We also obtained the non-magnetic\nresults of LS by setting ξ= 0. In particular, LS proved\nthat the nondimensional dynamical friction force for a non-\nmagnetized flow is\nF=−/integraldisplayπ\n0/bracketleftbig\n(1−β2)sinθcosθg−2+β(1+cos2θ)f′\n0/bracketrightbig\ndθ,(60)\nand then they showed analytically that the above integral\nis equal to β˙M. They obtained the above relation by inte-\ngrating the stress tensor over sphere surrounding the objec t.\nSince we are considering magnetic effects, it is known that\nthe contribution of the magnetic field to the stress tensor is\n1\n2B2δij−BiBj, i.e. the tress tensor is\nT=ρUU+(P+1\n2B2)I−BB. (61)\nThus, in our non-dimensional notation, the magnetic part\nof the drag force becomes\nFmag=−ξ2\n2/integraldisplay\nb2r2cosθsinθdθ+ξ2/integraldisplay\nbzbrr2sinθdθ.(62)\nAlso, we have\nbzbrr2sinθ=cotθ\nr2/parenleftBig∂Φ\n∂θ/parenrightBig2\n+1\nr/parenleftBig∂Φ\n∂θ/parenrightBig/parenleftBig∂Φ\n∂r/parenrightBig\n, (63)\nand\nb2r2sinθcosθ= cotθ/bracketleftbigg\n1\nr2/parenleftBig∂Φ\n∂θ/parenrightBig2\n+/parenleftBig∂Φ\n∂r/parenrightBig2/bracketrightbigg\n. (64)\nNow, we can substitute corresponding expansions into\nthe above equations to calculate the drag force. However,\nthe resulting integral is not integrable and one should cal-\nculate it numerically. Our numerical solutions show that th e\ndynamical friction force in the magnetized case is approx-\nimately equal to ( β−ξ2/β)f0(0), where the accretion rate\nis replaced by the magnetized accretion rate, i.e. equation\n(59). Thus, we numerically confirm F≃(β−ξ2/β)˙Mmag.\nFigure 8 shows dynamical friction force Fversus parameter\nβfor different values of ξ. Note that our solutions are re-\nstricted to the cases with ξ<β. Each curve is labeled by its\nparameterξand it is truncated due to the inequality ξ <β.\n6 CONCLUSIONS\nWhen a gravitating object move subsonically through a\nmagnetized medium, we studied the structure of the flow\nc/circlecop†rt0000 RAS, MNRAS 000, 000–0008M. Shadmehri & F. Khajenabi\nusing a perturbation analysis. Our study is a direct gener-\nalization of LS to the magnetized case. We found analytical\nrelations for the first-order density and velocity perturba -\ntions. According to these solutions, the structure of the flo w\nin comparison to the non-magnetized medium is less dense.\nHowever, in order to determine the dynamical friction force\nthe second-order variables are needed. The dynamical fric-\ntion force was found to be F≃β˙Mmagwhich implies a\nreduced force because of considering the magnetic effects. I s\nthis relation valid for a physically confirmed magnetized ac -\ncretion rate? This is an open question to be studied further.\nFuture models of the magnetized accretion should give us\nan analytical or numerical relation for the accretion rate a s\na function of βandξ, i.e.˙Mmag(β,ξ). Then, our main equa-\ntions (43) and (49) could be solved easily to calculate the\ndynamical friction force. However, we think the magnetic\neffects will still reduce the dynamical friction force.\nIn the cluster of the galaxies, the motion of\nthe each member of group is affected by the dy-\nnamical friction force and it has been suggested\nthat work done by the force may heat up the in-\ntragalactic medium (El-Zant, Kim & Kamionkowski\n2004; Kim, El-Zant & Kamionkowski 2005). This ex-\ntra source of the heating may resolve the cooling flow\nproblem in the cluster of the galaxies (for a detailed\nanalysis, see Kim, El-Zant & Kamionkowski (2005)).\nKim, El-Zant & Kamionkowski (2005) used the classical\nformula for calculating the drag force which is based\non assumption that the object’s radius is greater than\nthe accretion radius. But as LS argued in such systems,\naccretion radius is comparable to the radius of the accretor .\nThus, it would be interesting to calculate the heating due\nto the friction force using the formula that is presented\nby LS, at least to the regions of the cluster where the\ngalaxies are moving nearly subsonically like at the outer\nparts. But intracluster medium is magnetized, and so, our\nanalysis imply a lower heating rate when magnetic fields\nare taken into account. We finally note that several issues\nof potential importance were not investigated in this paper .\nFor example, we assumed that the ambient medium is\nisothermal. One can relax this assumption and re-do a\nsimilar analysis, but for a magnetized isentropic ambient\nmedium.\nACKNOWLEDGMENTS\nWe are grateful to the referee, Steven Stahler, for his helpf ul\ncomments and suggestions that improved the paper.\nREFERENCES\nBarausse E., 2007, MNRAS, 382, 826\nBondi H., 1952, MNRAS, 112, 195\nBondi H., Hoyle F., 1944, MNRAS, 104, 273\nChandrasekhar S., 1943, ApJ, 97, 255\nDokuchaev V. P., 1964, Soviet Astronomy, 8, 23\nEl-Zant A. A., Kim W.-T., Kamionkowski M., 2004, MN-\nRAS, 354, 169\nIgumenshchev I. V., 2006, ApJ, 649, 361Igumenshchev I. V., Narayan R., 2002, ApJ, 566, 137\nKhajenabi F., Dib S., 2012, Ap& SS, p. 98\nKim H., Kim W.-T., 2009, ApJ, 703, 1278\nKim H., Kim W.-T., S´ anchez-Salcedo F. J., 2008, ApJL,\n679, L33\nKim W.-T., 2010, ApJ, 725, 1069\nKim W.-T., El-Zant A. 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P., 2001, ApJ,\n561, 69\nVillaver E., Livio M., 2009, ApJL, 705, L81\nc/circlecop†rt0000 RAS, MNRAS 000, 000–000"    },    {        "title": "2206.10529v4.Magnetic_Forces_in_Paramagnetic_Fluids.pdf",        "content": "arXiv:2206.10529v4  [physics.chem-ph]  15 Dec 2022Magnetic Forces in Paramagnetic Fluids\nTim A. Butcher1,∗and J. M. D. Coey1\n1School of Physics and CRANN, Trinity College, Dublin 2, Irel and\n(Dated: December 16, 2022)\nAn overview of the effect of a magnetic field gradient on fluids w ith linear magnetic susceptibilities is\ngiven. It is shown that two commonly encountered expression s, the magnetic field gradient force and\nthe concentration gradient force for paramagnetic species in solution are equivalent for incompress-\nible fluids. The magnetic field gradient and concentration gr adient forces are approximations of the\nKelvin force and Korteweg-Helmholtz force densities, resp ectively. The criterion for the appearance\nof magnetically induced convection is derived. Experiment al work in which magnetically induced\nconvection plays a role is reviewed.\nI. INTRODUCTION\nThe classic hallmark of magnetic materials is their mo-\ntion upon magnetisation in a magnetic field gradient. A\nforce density known as the Kelvin force draws param-\nagnetic substances into increasing fields along the field\ngradient, whereasdiamagnetic matter is repelled. A fluid\nconstitutes a state of matter for which this is particularly\napparent, due to the deformations and convective move-\nment that magnetic forces can bring about. Examples\nof paramagnetic fluids range from paramagnetic gases\nto electrolytic solutions of paramagnetic salts, which are\nparticularly interesting for electrochemical experiments.\nParamagnetic salt solutions contain paramagnetic ions\nof transition metals with an unfilled d-subshell (e.g. Cr,\nMn, Fe, Ni and Co) or of rare earths with unpaired 4f\nelectrons (e.g. Gd, Tb, Dy, Ho and Er) [1]. Oxygen\nfound in its paramagnetic molecular form is an example\nof a paramagnetic gas [2]. Air consists of 21% O 2and\nis therefore also paramagnetic. The temperature depen-\ndence of their magnetic susceptibility is given by Curie’s\nlaw, which also directly relates the magnetic properties\nto the electronic structure of the atom [3, 4].\nColloidalsuspensionsofferromagneticorferrimagnetic\nnanoparticlesin anon-magneticcarrierliquid suchaswa-\nter or oil are known as ferrofluids and show a stronger\nresponse to magnetic field gradients than paramagnetic\nfluids based on individual non-interacting magnetic mo-\nments. Ferrimagnetic magnetite (Fe 3O4) or maghemite\n(γ-Fe2O3) nanoparticles with sizes in the order of 10nm\nare most commonly employed in ferrofluids. Due to their\nsmall size the nanoparticles are monodomain and super-\nparamagnetic. A ferrofluid cannot sustain a remanent\nmagnetisation upon removal of an applied magnetic field\nand its magnetisationcurve resemblesthat of a paramag-\nnetic salt solution, albeit with a significantly higher and\nreadily attainable saturation magnetisation. Hence, fer-\nrofluids can be referred to as superparamagnetic liquids.\nThis article provides an overview of the origin of mag-\nnetically induced convection and situations in which ef-\n∗tbutcher@tcd.ie; Present address: Swiss Light Source, Pau l\nScherrer Institut, 5232 Villigen PSI, Switzerlandfects of magnetic forces are relevant for experiments in-\nvolving paramagnetic salt solutions and gases.\nII. MAGNETIC FIELD EFFECTS ON\nMICROSCOPIC DIPOLES\nIn order to understand the Kelvin force it is neces-\nsary to recall the forces on the magnetic moments of the\natoms, ions or molecules that the fluid comprises. The\nmicroscopic force on an individual magnetic moment m\n(in Am2) in a magnetic field gradient ∇H(in Am−2) is\ngiven by [5, 6]:\nfmag=µ0(m·∇)H, (1)\nwith the vacuum permeability µ0= 4π×10−7NA−2.\nThis is the expression for the force on a magnetic dipole\nin the magnetic charge model [7–10]. The expression\ndescribes the situation when a single magnetic dipole is\nunperturbed by collision with other molecules. Thus, the\nmeasurement of this force requires a setup in which a sig-\nnificant free path length of the paramagneticmolecules is\nguaranteed. This was first achieved in the Stern-Gerlach\nexperiment [11–14], which was performed with a charge\nneutral molecular beam [15, 16] of paramagnetic silver\natoms just over a hundred years ago. The Ag atoms\nentered a magnetic field gradient where they were de-\nflected by the magnetic force before their detection. Es-\nsentially, the apparatus constituted a microscope for the\nmomentum of the Ag atoms [13]. A Stern-Gerlach type\nsplitting of charged particles has never been observed,\nalthough proposals for an experimental campaign exist\n[17–23]. An argument based on a combination of the\nLorentzforce andthe uncertaintyprinciple wasoriginally\nmade by Niels Bohr against the observability of such an\neffect [24].\nThe mean free path in a gas is drastically reduced and\ncommon values are in the order of 70nm. Molecules in\nliquids are in close contact and react collectively due to\ninterparticle collisions. What effect does the application\nof a magnetic field have on such an ensemble?\nA uniform magnetic field does not exert a force on the\nmagnetic moments, but causes them to precess. This2\nLarmor precession is irrelevant for monatomic gases, as\nthe collisions are independent of the orientation of the\natoms. However, polyatomic gases in a homogeneous\nmagnetic field show a decrease in thermal conductivity\nandviscosityoforderonepercent. Theeffectinparamag-\nnetic polyatomic gases is known as the Senftleben effect\n[25–27]. The extension to diamagnetic gases is called the\nSenftleben-Beenakker effect [28–31]. The modification of\nthe transport properties relies on the averaging of the\ncollision cross section and change of the mean free path\nby the precession of the magnetic moment in the field\n[32, 33]. The strength of the effect is proportional toH\nP,\nwith the pressure P. No equivalent of the Senftleben ef-\nfect existsinliquids, whereameanfreepath isundefined.\nIn the case of an inhomogeneous field, the microscopic\nforce given by Eq. 1 acts on the individual magnetic\ndipoles in the medium. The microscopic forces are trans-\nferred to the bulk fluid via interparticle forces that are\nmediated by collisions. Thus, a magnetic body force\narises, which can lead to bulk motion of a magnetised\nfluid in a process called magnetic convection. A force\ndensity must be introduced to describe the action of the\nfield gradient, which will be discussed in the next section.\nThe appearance of magnetic convection has been cast\nas a paradox in the past [34–41]. One argument is that\nthe discrepancy between magnetic and thermal energies\nis so large that it makes any magnetic modification of\nthe movement impossible. This reasoning does not stand\nup to scrutiny. Velocities due to Brownian motion aver-\nage out in a portion of fluid that contains an enormous\nnumber of molecules. The collisions due to the magnetic\nforces, on the other hand, do not. Brownian motion does\nnotinfluenceamacroscopichydrodynamicflow, butforce\ndensities do. Fluid dynamics describes the situation for\ncontinuousmediaandthefluidisassignedaveragemacro-\nscopic quantities.\nIn the last decade, several publications claimed to ob-\nserve the enrichment of paramagnetic ions from homo-\ngeneous solutions [42–48]. This came as a surprise, be-\ncause the Brownian motion of the ions enforces a uni-\nform concentration throughout the solvent. Thermody-\nnamically, the magnetic field hardly affects the chemical\npotential and the thermodynamic equilibrium remains\nvirtually unchanged [49]. It later became clear in a set\nof experiments [50–56] that the observed concentration\nchanges in the field gradient could be explained by evap-\noration of the solvent. A completely filled and sealed\ncuvette does not show any inhomogeneity upon exposure\nto a magnetic field gradient [51]. Measurable concen-\ntrations of paramagnetic ions do not settle from homo-\ngeneous solutions in magnetic field gradients commonly\nencountered in laboratories. The density of the param-\nagnetic ions does not change and paramagnetic salt solu-\ntions are incompressible. However, pre-existing concen-\ntrations ofparamagneticions can be readilymanipulated\nwith a magnet in the time window before the system is\nhomogenised by diffusion [57, 58].\nUnlike paramagnetic salt solutions with their individ-ual solvated paramagnetic cations, ferrofluids consist of\nferromagnetic or ferrimagnetic nanoparticles in which all\nthe encapsulated magnetic moments are ordered by ex-\nchangeinteraction. Theresultingtotalmagneticmoment\nofsuch nanoparticlesisin the orderof104µB(Bohrmag-\nneton:µB= 9.274×10−24JT−1, note that JT−1and\nAm2are equivalent units)) [59], which can be compared\nto the effective magnetic moment of Fe3+µeff= 5.9µB\n[4]. According to Eq. 1, the magnitude of the force on\na magnetic nanoparticle in a magnetic field gradient is\napproximately 1000 times higher than that on an indi-\nvidual paramagnetic ion. Ferrofluids are synthesised to\navoid agglomeration in a magnetic field gradient, which\nis ensured by the small size of the nanoparticles that is\nbelow 10nm in ideal ferrofluids [60]. Ideal ferrofluids are\nincompressible. In concentrated ferrofluids, there is an\nadditional risk of agglomeration due to magnetic dipo-\nlar and Van der Waals interactions between individual\nnanoparticles. The effect of these interactions can be\nsuppressed by coating the nanoparticles with a surfac-\ntant [59].\nWithout an applied magnetic field ferrofluids show no\nmagnetisation, because the magnetic moments of the\nnanoparticles fluctuate. There are two reasons: firstly,\nrotational diffusion of the nanoparticles in the carrier liq-\nuid results in random orientations of the magnetic mo-\nments. Secondly, the thermal energy at room tempera-\nture may be high enough to cause the magnetic moment\nto flip randomly as is characteristic for superparamag-\nneticparticles. TheformerisknownasBrownianandthe\nlatter as N´ eel relaxation [59]. When the size of the mag-\nneticparticlesisincreasedtothemicroscale,theresulting\nliquids are known as magnetorheologicalfluids [61]. This\nmaterial class was developed in the 1940s prior to the\ndiscovery of ferrofluids in the mid 1960s. Magnetorheo-\nlogical fluids experience a dramatic increase in their vis-\ncosity upon exposure to magnetic fields, but this comes\nto the detriment of the stability of the fluid. Brownian\nmotion is no longer able to prevent agglomeration of the\nmagnetic particles.\nIII. MACROSCOPIC MAGNETIC FORCE\nDENSITIES\nMagnetic field gradients exert forces on magnetically\npolarisable media. Derivations of expressions for the\nforce density are outlined in many texts [6, 62–70]. In\norder to heuristically transform Eq. 1 into an expression\nfor the force density of non-interacting magnetic dipoles,\nscaling with their number Nper unit volume V(num-\nber density n=N\nV) is necessary. The magnetisation\nM=/summationtextN\ni=1mi\nV=Nm\nV=nmis a macroscopic variable\nthat is defined as the number density of magnetic dipole\nmoments multiplied by the average microscopic dipole\nmoment in direction of the applied field. Replacing m\nwithMin Eq. 1, one immediately obtains the Kelvin\nforce expression for the magnetic force density in Nm−3:3\nFK=µ0(M·∇)H. (2)\nKelvin arrived at this expression by considering the\nmagnetic moment per unit volume and relating it to the\nmechanical force in a dipole model [71]. The Kelvin force\nis often referred to as the magnetic field gradient force\nin magnetoelectrochemistry [3, 72]. Its intensity is pro-\nportional to the magnetic susceptibility χ=M\nHof the\nfluid, which is around 1 for ferrofluids and 10−3for con-\ncentrated paramagnetic salt solutions.\nProvidingthemagnitude ofthe magneticsusceptibility\nis small (χ≪1), the magnetic flux density B(unit: T)\ncanbeapproximatedas B=µ0(H+M) =µ0(H+χH)≈\nµ0H. Consequently, the Kelvin force (Eq. 2) becomes:\nF∇B=χ\nµ0(B·∇)B. (3)\nThe name of Eq. 3 is magnetic field gradient force and\nit is commonly encountered in magnetoelectrochemistry,\nwhich deals with paramagnetic salt solutions, gases or\ndiamagnetic fluids [3]. A version that is not scaled by χ\nis known as the magnetic tension force in magnetohydro-\ndynamics (MHD) [73]. The name is aptly chosen, since\nit straightens out magnetic field lines.\nIn the presence of current densities j(in Am−2), a\nLorentz force density FL=j×Bmust be added to the\nmagnetic force density to account for the interaction of\nmoving chargeswith the magnetic field [74]. The Lorentz\nforce density is of smaller magnitude than the Kelvin\nforce when a high concentration of paramagnetic ions is\npresentin thesolution[3,75]. Inthefollowingdiscussion,\ncurrents are ignored.\nThere is a second expression for the force on a para-\nmagnetic liquid, which incorporates the gradient of the\nmagnetic susceptibility (or permeability):\nFH=−H·H\n2µ0∇χ=−1\n2H2µ0∇χ. (4)\nThis expression predicts a force density at interfaces\nwhere the susceptibility changes. This form of the mag-\nnetic force density was developed first by Diederik Ko-\nrteweg [76], then by Hermann von Helmholtz [77] and\nothers [78, 79] within the framework of the thermody-\nnamic principle of minimum energy. It is known as the\nKorteweg-Helmholtz force [5]. In magnetoelectrochem-\nistry, the Korteweg-Helmholtz force is referred to as the\nconcentration gradient force [72, 80, 81] with the molar\nmagnetic susceptibility χm(in m3mol−1) and approxi-\nmationB≈µ0Hforχ≪1:\nFc=−1\n2H2µ0χm∇c≈ −B2\n2µ0χm∇c. (5)\nOn the face of it, the situation seems bewildering. The\nKelvin force (Eq. 2) and the Korteweg-Helmholtz force(Eq. 4) predict force densities in inhomogeneous mag-\nnetic fields or at interfaces at which the susceptibility\nchanges, respectively. There has been controversy over\nwhich of the expressions is correct ever since they were\nfirstderived. Theargumenthasnotabatedtothepresent\nday [81–84]. Criticism was first levelled at the Korteweg-\nHelmholtz force by followers of Kelvin [71, 85, 86].\nThe apparentparadoxcan be resolvedby re-expressing\nthe Kelvin force density (Eq. 2) with the help of vector\nidentities [5]. The first step is to expand the expression\nwith (H·∇)H= (∇×H)×H+1\n2∇(H·H) after writing\nM=χH:\nFK=µ0χ[(∇×H)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=0×H+1\n2∇(H·H)].(6)\nIn the absence of currents ∇ ×H= 0 and the first\nterm disappears. The remaining dot product of His\nalready redolent of the Korteweg-Helmholtzforce (Eq. 4)\nand can be further transformed by employing ∇(χH2) =\nχ∇H2+H2∇χ:\nFK=∇/parenleftBigµ0\n2χH·H/parenrightBig\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nPmag−µ0\n2H·H∇χ. (7)\nThe term in the bracketson the left is a magneticread-\njustment of the internal pressure in the fluid Pmag. This\nis not present in the Korteweg-Helmholtz version of the\nforce density [70, 87] and instead appears within the def-\ninition of the internal pressure itself. The gradient of the\nmagnetically induced pressure is the difference between\nthe two expressions for the magnetic force density on a\nparamagnetic fluid:\nFK=∇Pmag+FH. (8)\nFormulation in terms of the magnetic field gradient\nforceandtheconcentrationgradientforceisalsopossible:\nF∇B=∇Pmag+Fc. (9)\nWhat are the implications of this algebra for the dy-\nnamics of a paramagnetic fluid element that is exposed\nto a magnetic field gradient? Gradients of the pressure\nare unimportant for the prediction of deformation of a\nportion of incompressible fluid. Only rotational forces\ncan cause fluid movement which deforms the paramag-\nnetic solution in a closed system in which the fluid is\nbounded by solid walls [88]. The gradients of internal\npressure are by definition irrotational and inconsequen-\ntial fordeformationsthat leavethe volumeunchanged. It\nfollows that the Kelvin and Korteweg-Helmholtz expres-\nsions for the magnetic force density on a paramagnetic\nfluid predict the same motion [5, 87, 89, 90], despite the\nfact that they lead to different distributions of the force4\ndensity. Deformations of incompressible fluids, such as\nparamagnetic salt solutions, are equally well described\nbyeitherexpression. For compressible paramagnetic flu-\nids, the magnetic readjustment of the pressureis relevant\nand can cause magnetostriction [68, 70]. In this case, the\nmagnetic contribution to the pressure and density must\nbe accounted for as was shown in dielectric liquids [89–\n91].\nThe futility of the argument about the correct force\ndensity was first pointed out in the 1950s [64, 87, 92].\nThe reasoning was later presented again [65, 89, 90, 93].\nAn interpretationofthe effects ofthe magnetic forceden-\nsities must be sought in fluid dynamics, which is the aim\nof the next section.\nIV. MAGNETICALLY INDUCED\nCONVECTION\nThe Navier-Stokes equation is fundamental for fluid\ndynamics. It relates the acceleration of the fluid, given\nby the material derivative of the fluid velocity (Du\nDt=\n∂u\n∂t+u·∇u), to force fields and pressure gradients. The\nNavier-Stokes equation for a paramagnetic fluid can be\nwritten with the Kelvin force:\nρDu\nDt=−∇P+η∇2u+ρg+µ0(M·∇)H,(10)\nwith the pressure P, dynamic viscosity η(in Nsm−2),\nthe density ρand the gravitational acceleration g. The\ntwo last terms on the right are the gravitational and\nKelvin force densities. Generally, the pressure is un-\nknown and it is impossible to make a direct prediction\nof fluid motion from Eq. 10 without the incompressibil-\nity constraint ∇ ·u= 0. The issue can be sidestepped\nby applying the curl operator ( ∇×) to the Navier-Stokes\nequation, disposing of ∇P[88]. This brings into play the\nvorticityof the flow. Any body force with a non-zerocurl\nis able to deform the fluid and causes it to rotate [94]. If\nthe flow is irrotational, a velocity potential u=∇φcan\nbe introduced and the situation is known as potential\nflow. Any irrotational (potential) force is balanced by\nthe pressure field in the liquid. These pressure changes\nare equilibrated by solid walls in a closed cell and the\nfluid remains static in the absence of free surfaces. The\npressure serves as an assurance that only deformations in\nwhich the volume is maintained are allowed in an incom-\npressible fluid. No appreciable changes in concentration\nare to be expected from commonly encountered pressure\ndifferences. Only rotational body forces can create inter-\nnal flows.\nIn order for paramagnetic fluids to be moved by mag-\nnetic convection, the Kelvin force must be rotational.\nSo what is the condition for the non-potentiality of the\nKelvin force and the appearance of magnetically induced\nconvection? The prerequisites can be obtained by deriv-\ning the condition formechanicalequilibrium from Eq.10.Any magnetised fluid undergoes convective movement\nwith the aim to establish this mechanical equilibrium.\nUnder zero flow ( u= 0) the application of ∇×to Eq. 10\nyields:\n∇×[ρg+µ0(M·∇)H] = 0. (11)\nWith the vectoridentity ∇×(ψA) =ψ∇×A+∇ψ×A\nand using the collinearity of MandHthis becomes:\n∇ρ×g+µ0∇×/parenleftbiggM\nH(H·∇)H/parenrightbigg\n= 0.(12)\nThe vector identity ( H·∇)H= (∇×H)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=0×H+1\n2∇H2\nand the previously used vector identity for the curl lead\nto the criterion of hydrostatic equilibrium for magnetic\nfluids:\n∇ρ×g+µ0∇M×∇H= 0. (13)\nAny departure from this condition leads to magnetic\nconvection [67, 95]. The magnetic field gradient modifies\ndensity difference driven convectionand pulls the param-\nagnetic fluid into the external field gradient. Mathemati-\ncally, the curl of the Kelvin force causes non-potentiality\nof gravity by introducing density gradients orthogonal to\nthe direction of gravity. This proceeds until Eq. 13 is\nsatisfied.\nThe first term ensures stability in the gravitational\nfield. Gravity induces convection if there is a density\ngradient that is non-parallel to the direction of the grav-\nitational acceleration. For this reason, a column of water\ndoes not simply undergo an internal downward flow due\nto gravity.\nThe second term is due to the interaction of the mag-\nnetic field gradient with the paramagnetic fluid. Gra-\ndients in both the magnetisation ∇Mand the external\nmagnetic field ∇Hmust be present to drive convection.\nSimply applying a magnetic field gradient to homoge-\nneousparamagneticfluidshasnoeffect, becausethemag-\nnetic pressure is cancelled by the walls. A gradient in\nthe concentration of the paramagnetic species must be\npresent in the system, which is accompanied by a gradi-\nent in density.\nFurthermore, the gradients of ∇Mand∇Hmust be\nnon-parallel for a flow to appear. This requirement is\nstraightforward to achieve since Gauss’s law for mag-\nnetism dictates a divergence-free magnetic flux density:\n∇·B= 0. (14)\nHence, it is impossible to generate a magnetic field\ngradient exclusively in a single dimension.\nIt is equally valid to derive the criterion for mechan-\nical equilibrium in magnetic fluids with the Korteweg-\nHelmholtz force (Eq. 4) instead of the Kelvin force in5\nEq. 10. This alsoresults in the condition given by Eq.13,\nas the Kelvin force and the Korteweg-Helmholtz force\ndiffer by a gradient of the magnetic pressure (see Eq. 7).\nThegradientofthepressureisirrotationalanddropsout.\nIn magnetoelectrochemistry it is useful to analyse the\ncurl of the magnetic field gradient force (Eq. 3) to deter-\nmine whether convection takes place in a paramagnetic\nsalt solution. An expression with the molar magnetic\nsusceptibility and the concentration can be obtained:\n∇×F∇B=1\n2µ0χm∇c×∇B2, (15)\nwhich is simply a reformulation of the magnetic part of\nEq. 13.\nThefollowingsectionwillnowprovideareviewofstud-\nies in which magnetically induced convection was ob-\nserved. The focus will lie on experiments with param-\nagnetic salt solutions and gases. For discussions of the\napplications of ferrofluids, the reader is referred to exist-\ning review articles [60, 96–99].\nV. OVERVIEW OF EXPERIMENTAL\nOBSERVATIONS OF MAGNETICALLY\nINDUCED CONVECTION\nMagnetic convection in a paramagnetic fluid occurs as\nlongastheconditiongivenbyEq.13isnotsatisfied. Nec-\nessary requirements are a magnetic field gradient and a\nnon-collinear gradient of the magnetisation. The gradi-\nent in the magnetisation is due to an inhomogeneity in\nthe concentration of paramagnetic species. The focus of\nthe first part of this section lies on systems in which this\ninhomogeneity is caused by the input of thermal energy.\nThe second part deals with the introduction of electrical\nenergy to electrolytic solutions under magnetic fields in\nmagnetoelectrochemistry.\nThe classical way to provoke convection is to create\ndensity differences by heating the fluid. Temperature\ngradients in the magnetic fluid can modify both the mag-\nnetisation and the density of paramagnetic fluids. This\nis the case in magnetothermal convection [100–115].\nMagnetothermal convection was first investigated in\ngaseous paramagnetic oxygen [2, 100–103, 111, 112]. It\neven found application in the measurement of O 2levels\nin high altitude flights in the 1940s[100, 116, 117]. When\nO2is heated, the magnetic susceptibility ( χ= 0.145×\n10−6at 20°C [118]) decreases according to T−2. This is\ndue to the combined effect of the Curie law ( χ=C\nT) and\nthe expansion of O 2(ρ∝T−1). Thus, the warm gas\nis pushed out of a field gradient and an inhomogeneous\nmagnetic field increases the thermal conductivity by this\nconvective motion.\nThe magnetic convectionofoxygencanalsolead to the\nphenomenon of “magnetic wind”[2, 101–103]. A related\neffect may be the reported enhancement of the evapora-\ntion rate of water in magnetic field gradients [119, 120],which has been ascribed to the magnetic convection of\ndiamagnetic water vapour in the paramagnetic air above\nthe water surface [119].\nLater experimental studies concentrated on magnet-\nically modified heat transfer in paramagnetic liquids,\nwith gadolinium nitrate solution as a model system in\nthe magnetic field ofsuperconducting magnets[104–108].\nThe susceptibility of an aqueous one-molar solution of\nGd3+ions isχ= 321×10−6[57]. Small permanent\nmagnets are also sufficient to drive magnetothermal con-\nvection of concentrated paramagnetic salt solutions [52].\nIt is possible to magnetically force convection in dia-\nmagneticwater( χ=−9×10−6) [109, 121]. Heat transfer\nin regular diamagnetic water can be modified by intense\nmagnetic field gradients, which can be provided by spe-\ncialised hybrid superconducting magnets [109]. A num-\nberofnumericalstudiesthat describethephenomenaun-\nderpinning the observed heat transfer in non-conducting\nfluids exist [113–115].\nAttempts to use magnetothermal convection to sepa-\nrate rare earth ions were made by Ida and Walter Nod-\ndack in the 1950s [122–124]. They applied a magnetic\nfield gradient to a Clusius-Dickel separation column in\nwhich the liquid is heated by Joule heat generated from\na current-carryingwire [125–131] and reported an ampli-\nfication of the rare earth separation by thermodiffusion\n[123, 124].\nMagnetic field gradients of permanent magnets can\nalso act on paramagnetic salt solutions in which the sol-\nvent is allowed to evaporate and the top layer of the\nfluid gains a higher concentration in a distillation pro-\ncess. Several experimental works showed that placing\na magnet on top of an insufficiently sealed cuvette con-\ntaining paramagnetic salt solutions levitates the layer of\nhigher concentration above the bulk fluid [50–56]. With-\nout a magnet close to the liquid surface, this layer sinks\nin a Rayleigh-Taylor instability and mixes with the bulk\nsolution.\nThe crystallisation of paramagnetic salts from concen-\ntrated solutions can be facilitated by confinement in a\nfield gradient. This was first demonstrated for Mohr’s\nsalt [132] and later for nickel sulfate [133, 134]. Magnetic\nfield gradients have also been reported to enhance the\nseparation of rare earths by crystallisation [135–137].\nInhibition of regular convection by trapping magnetic\nfluids in magnetic field gradients may also become rele-\nvant in the field of microfluidics [138, 139], because the\nmagnetic field gradient can stabilise liquid within liquid\ntubes [57, 140]. These dispense with the friction encoun-\ntered at solid walls. The concept of magnetic control of\nliquid-in-liquidflowwasoriginallydevelopedwithparam-\nagnetic salt solutions [57] and later extended to ferroflu-\nids to maximise the confinement in the magnetic field\ngradient [140]. It is also possible to halt double diffu-\nsive convection in multicomponent systems containing a\nparamagnetic species [58].\nIn electrochemistry, an input of electrical energy to\nelectrolytic solutions via electrodes drives chemical reac-6\ntions and mass transport of ions. The transport of the\nelectrolyte to the electrode surface is strongly influenced\nby bulk movement of the fluid. This is where magnetic\nbody forces can have noticeable effects, because diffusion\nlimited concentration gradients appear close to the elec-\ntrodes.\nThe first explorations of magnetic field effects on elec-\ntrochemical cells were carried out with homogeneous\nfields and focused on the Lorentz force [141–146]. This\nwork laid the foundation for what is now known as the\nmagnetohydrodynamic(MHD)effect, namelythestirring\noftheelectrolytesolutionbytheLorentzforceclosetothe\nelectrode [74]. A magnetic field parallel to the electrode\nsurface is orthogonal to the current and the electrolytic\nsolution rotates. A direct consequence of this is that the\nlimiting current is increased [142, 143, 147–155], as the\nconvective flow helps to replenish the concentration at\nthe electrode [75].\nWhen strong magnetic field gradients ∇B≈\n100Tm−1and paramagnetic species are present in the\nsolution, the Lorentz force is insignificant and the Kelvin\nforce becomes dominant as soon as a gradient of para-\nmagnetic ions is established. The classical and much\nstudied example in which paramagnetic ions can be in-\nfluenced by the Kelvin force is the electrodeposition of\ncopper [156–170]. A one-molar aqueous solution of Cu2+\nhasχ= 7×10−6[57]. One ofthe firstreportsofstructur-\ning copper electrodeposits from CuSO 4solutions with in-\nhomogeneous magnetic fields was reported in 1979 [156].\nSince then there has been an increase in the obtainable\nfield gradient, with experiments relying on small perma-\nnent magnets or the stray field of a magnetised wire to\ndeliver high magnetic field gradients. The electrodeposi-\ntion in a magnetic field gradient is accompanied by the\nformation of both gradients in the density and the mag-\nnetic susceptibility. This means that the interplay of the\ngravitationaland the Kelvin forcedensities re-establishes\nmechanical equilibrium according to Eq. 13 [161].\nThe observation of this structuring is not limited to\nparamagnetic Cu2+solutions. Any paramagnetic ion\nspecies that undergoes reaction at an electrode causes\na concentration gradient with respect to the bulk solu-\ntion. An example of an organic compound that can be\ncaptured in a magnetic field gradient after electrochem-\nical conversion is nitrobenzene [80, 171–174]. It is also\npossible to draw paramagnetic oxygen close to an elec-\ntrode backed by a magnetic field gradient and enhance\nthe current in oxygen reduction reactions [175–179].\nAn important class of fluid dynamical phenomenon is\nflow separationwhere the fluid velocity changes direction\nwith respect to that of the main fluid. This frequently\nhappens in the vicinity of solid obstacles in the way of\nthe flow, but it can also be triggered by magnetic field\ngradients [180]. A necessary requirement for separation\nto occur is that the vorticity of the flow changes sign\n[181]. In the case of a solid wall, the fluid passing in\nimmediate proximity to the obstacle is spun around and\nforms a counterflow. In the magnetic case, changes inthe vorticity can be precipitated by variations in the curl\nof the Kelvin force (Eq. 13).\nA phenomenon due to this magnetically induced sepa-\nration of flow is the inverse structuring of electrodeposits\ninthe intensemagneticfield gradientsofpermanentmag-\nnets or magnetised Fe wires [162–165, 182, 183]. In these\nexperiments, the electrochemical cell was filled with a so-\nlution containing non-magnetic, but electroplatable ion\nspecies (such as Bi3+or Zn2+) and a paramagnetic ion\nspecies that was non-electroplatable under the employed\nexperimental conditions (such as Mn2+or Dy3+). When\nan electrodeposition from this mixture was carried out\nabovean arrayofsmallNd-Fe-B permanentmagnets, the\nnon-magnetic species exhibited much thicker deposits in\nthe area of low magnetic ( B·∇)B. The presence of the\nparamagnetic species caused the formation of zones into\nwhich the transport of electroactive ions from the bulk\nsolution was inhibited. The explanation of this effect was\nprovided by analysing the curl of the magnetic field gra-\ndient force (Eq. 15) [180, 183].\nRecently, theimprovementofwatersplittingbytheap-\nplication of a magnetic field has attracted interest [184–\n188]. Numerous studies report an increase of the mea-\nsuredcurrentwhen amagneticfield isapplied to the elec-\ntrodes. Apopularexplanationforthisobservationisthat\nthe triplet state ( ↑↑) of molecular oxygen is favoured and\ntherefore the magnetic field promotes the oxygen evolu-\ntion reaction. As of yet, there is no clear experimental\nevidence to support this interpretation. The magnetic\nfields in these experiments were usually inhomogeneous.\nFor instance in [184], an increase in current was observed\nwhen a Nd-Fe-B magnet was placed behind the magnetic\nelectrodes. The authors showed that the effect persisted\nwhen the solution was stirred and concluded that it was\nnot a mass transport effect. Past experiments showed\nthat the magnetic field gradient force can act in the in-\nverse oxygen reduction reaction even when the electrode\nis rotated [177, 179]. It is possible that the magnetic field\nfacilitates the removal of gas bubbles from the electrode\ndue to the Lorentz force [189–192] and this is what is\nincreasing the current.\nVI. CONCLUSION\nThe body force caused by magnetic field gradients can\ninduce convection in inhomogeneous systems of param-\nagnetic fluids. This leads to a situation in which the\nKelvin force becomes rotational, which modifies mechan-\nical equilibrium. The two common expressions for the\nforce density in magnetoelectrochemistry are the mag-\nnetic field gradient force and the concentration gradi-\nent force. These are approximations of the Kelvin and\nKorteweg-Helmholtz force, respectively. Both of these\naccurately describe the effect of magnetic field gradients\non incompressible fluids with linear magnetic susceptibli-\nties. 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Hervieux2, G. Manfredi2\n1INRIA Nancy Grand-Est and Institut de Recherche en Math´ ema tiques Avanc´ ees,\n7 rue Ren´ e Descartes, F-67084 Strasbourg, France\n2Institut de Physique et Chimie des Mat´ eriaux de Strasbourg ,\n23 rue du Loess, F-67037 Strasbourg, France\n(Dated: November 8, 2018)\nAbstract\nWe present a dynamical model that successfully explains the observed time evolution of the mag-\nnetization in diluted magnetic semiconductor quantum well s after weak laser excitation. Based on\nthe pseudo-fermion formalism and a second order many-parti cle expansion of the exact p−dex-\nchange interaction, our approach goes beyond the usual mean -field approximation. It includes both\nthe sub-picosecond demagnetization dynamics and the slowe r relaxation processes which restore\nthe initial ferromagnetic order in a nanosecond time scale. In agreement with experimental results,\nour numerical simulations show that, depending on the value of the initial lattice temperature, a\nsubsequent enhancement of the total magnetization may be ob served within a time scale of few\nhundreds of picoseconds.\nPACS numbers:\n1I. INTRODUCTION\nUltrafast light-induced magnetization dynamics in ferromagnetic film s and in Diluted\nMagnetic Semiconductors (DMS) is today a very active area of rese arch. From the observa-\ntion of the ultrafast dynamics of the spin magnetization in nickel films [1] and the analogous\nprocesses in ferromagnetic semiconductors [2], special interest h as been devoted to the de-\nvelopment of dynamical models able to mimic the time evolution of the ma gnetization on\nboth short and long time scales. In III-V ferromagnetic semicondu ctors such as GaMnAs\nand InMnAs a small concentration of Mn ions is randomly substituted to cation sites so that\nthe Mn-Mn spin coupling is mediated by the hole-ion p−dexchange interaction, allowing\nthe generation of a ferromagnetic state with a Curie temperature of the order of 50 K [3].\nThe magnetism can therefore be efficiently modified by controlling the hole density through\ndoping or by excitation of electron-hole pairs with a laser pulse. In pa rticular, unlike metals,\nin a regime of strong laser excitation total demagnetization can be a chieved [4].\nIn the Zener model [5], which was originally developed to describe the m agnetism of tran-\nsition metals, the dshells of the Mn ions are treated as an ensemble of randomly distribut ed\nimpurities with spin 5 /2 surrounded by a hole gas or an electron gas. Unlike ferromagnetic\nmetals, III-Mn-V ferromagnetic semiconductors offer the advan tage of providing a clear dis-\ntinctionbetweenlocalizedMnimpuritiesanditinerantvalence-bandho lespins, thusallowing\nthe basic assumptions of the Zener theory to be satisfied. Based o n this hypothesis, a few\nmean-field models have been successfully applied for modelling the gro und-state properties\nof DMS nanostructures. In particular, within the framework of th e spin-density-functional\ntheory at finite temperature, relevant predictions of the Curie te mperature have been ob-\ntained [6, 7]. Ultrafast demagnetization in DMS is a phenomenon where thep−dexchange\ninteraction causes a flow of spin polarizationand energy fromthe Mn impurities to theholes,\nwhich is subsequently converted to orbital momentum and thermaliz ed through spin-orbit\nand hole-hole interactions [8]. Since energy and spin polarization tran sfer is a many-particle\neffect, the mean-field Zener approach cannot provide a satisfying explanation of the ultrafast\ndemagnetization regime that has been observed in DMS [4, 9].\nA phenomenological approach able to take into account this energy flux was given in\n[1, 10] where a model based on three temperatures was derived. M ore recently, a study\nof the coupling of the electromagnetic laser field with the hole gas rev ealed the possibility\n2of an ultrafast demagnetization during the femtosecond optical e xcitation, due to light-\nhole entanglement [11]. A model capable of describing the dynamics of carrier-ion spin\ninteractions is provided in [12, 13]. This model generalizes the statio nary theory of [10]\nand takes into account the picosecond demagnetization evolution w hich occurs in a strong\nexcitation regime, but neglects the slow-in-time evolution of the spin dynamics. The mean\np−dinteraction is averaged out over the randomly distributed positions of the Mn ions.\nIn this work we derive a dynamical model based on a many-particle ex pansion of the\np−dexchange interaction based on the pseudo-fermion method. This f ormalism, originally\ndeveloped by Abrikosov to deal with the Kondo problem [14], introduc es unphysical states\nin the Hilbert space for which impurity sites are allowed to be multiply occ upied. Following\nthe work of Coleman [15], a suitable limit procedure is applied to our dyna mical model in\norder to recover the correct physical description of the magnet ic impurities.\nOur approach extends the Zener model beyond the usual mean-fi eld approximation. It\nincludes both the subpicosecond demagnetization dynamics and the slower cooling processes\nthat restore the initial ferromagnetic order (which is achieved in a n s time scale). Moreover,\nin agreement with recent experimental results [9] our simulations sh ow that, depending\non the initial lattice temperature, a subsequent enhancement of t he total magnetization is\nobserved within a time scale of 100 ps.\nII. PSEUDO-FERMION FORMALISM\nWe consider a volume Vcontaining NhVholes with spin Sh= 1/2 strongly coupled\nby spin-spin interaction with NMVrandomly distributed Mn impurities with spin SM=\n5/2. We assume that the exchange interaction between localized ions a nd heavy holes\ndominates over both the short-range antiferromagnetic d−dexchange interaction between\nthe ions and the s−dexchange interaction between electrons in the conduction band an d\nMn ions (typical values for the s−dand thep−dinteractions in a GaAs are 0.1 eV and\n1 eV respectively [16]). Furthermore, electron-hole radiative reco mbination, carrier-phonon\ninteractions, and interactions leading to the hole spin-relaxation in t he hole gas are included\nphenomenologically. The time evolution of the system is governed by t he Hamiltonian\nH=/summationdisplay\nk,sεk,sa†\nk,sak,s+Hpd,\n3wherea†\nk,s(ak,s) is the creation (annihilation) operator of a hole with spin projection s\nand quasi-momentum k. In the parabolic band approximation the kinetic energy of the\nholes reads εk,s=Eh−/planckover2pi12k2\n2m∗whereEhis the valence band edge. The Kondo-like exchange\ninteraction Hpdis given by\nHpd=γ\nV/summationdisplay\nJm′,m·σs′,s/parenleftBig\nb†\nη,m′bη,ma†\nk′,s′ak,s/parenrightBig\nei(k′−k)Rη\nwhere the sum is extended over all indices, γis thep−dcoupling constant, and σ,Jare the\nspin matrices related to ShandSMrespectively. The ion spin operator is represented in the\npseudo-fermion formalism [14, 15] in which b†\nη,m(bη,m) denotes the creation (annihilation)\noperator of a pseudo-fermion with spin projection mand spatial position Rη.\nTheHpdHamiltonian reproduces the correct ion-hole exchange interaction provided that\nthe ion sites are singly occupied, i.e., ˆ nη=/summationtextSM\nm=−SMb†\nη,mbη,m= 1∀η. Following [14, 15] this\nconstraint may be taken into account by adding a ”fictitious” ionic ch emical potential\nHλ=/summationdisplay\nηληˆnη\nto the original Hamiltonian and letting ληgo to infinity at the end of the calculation.\nThe grand-canonical expectation value of a pseudo-fermion oper atorArelated to the\ntotal Hamiltonian H+Hλreads\n/an}bracketle{tA/an}bracketri}htλ=1\nZλTr/braceleftbig\nρHe−βP\nηληˆnηA/bracerightbig\n=1\nZλ/summationdisplay\n{nmη}r/angbracketleftbig\nnm\nη/vextendsingle/vextendsingleρHe−βP\nηληˆnηA/vextendsingle/vextendsinglenm\nη/angbracketrightbig\n,\nwhereZλ= Tr/braceleftbig\nρHe−βP\nηληˆnη/bracerightbig\n,ρH=e−βH, andβ= 1/kBTh, withkBthe Boltzmann\nconstant and Ththe hole temperature./braceleftbig\nnm\nη/bracerightbig\nr=/braceleftBig\nn1\n1,...,n(2SM+1)\n1,...,n(2SM+1)\nr/bracerightBig\ndenotes\nall possible occupation numbers nk\nη(= 0 or 1) for rion sites. Since each site has (2 SM+1)\navailablepseudo-fermion states, thesystem will contain atmost (2 SM+1)rpseudo-particles.\nThe correct expectation value of the operator Ais obtained using the limit λη→ ∞[15]\n/an}bracketle{tA/an}bracketri}ht∞=1\nZ∞lim\n{zη}→0∂r[/an}bracketle{tA/an}bracketri}htλZλ]\n∂z1···∂zr, (1)\nwhereZ∞= lim{zη}→0∂rZλ\n∂z1···∂zrandzη=e−βλη.\nIn the next section, we will show that the time-evolution of the spin o f the ion-hole\nsystem may be expressed in terms of the expectation value of the p seudo-fermion operator\n4b†\nη,mbη,m(1−b†\nη,m′bη,m′) withm/ne}ationslash=m′and evaluated in the mean magnetic field Sgenerated\nby the holes. We have the general relationship (which also applies whe n the system is driven\nfar from equilibrium)\nlim\n{λη}→∞/angbracketleftBig\nb†\nη,mbη,m(1−b†\nη,m′bη,m′)/angbracketrightBig\nλ= lim\n{λη}→∞/angbracketleftbig\nb†\nη,mbη,m/angbracketrightbig\nλ. (2)\nWhen the system approaches thermal equilibrium, the quantity/angbracketleftbig\nb†\nη,mbη,m/angbracketrightbig\n∞becomes the\nusual spin thermal distribution. Using Eq. (1) we obtain\n/angbracketleftbig\nb†\nη,mbη,m/angbracketrightbig\n∞=/tildewideQeβmγS\nZ∞, (3)\nwhereZ∞=/tildewideQsinh[βγS\n2(2SM+1)]\nsinh(βγS\n2)and/tildewideQ=Q|nm\nη′=0,1;P\nmnm\nη′=1withQ=/producttext\nm,η′/ne}ationslash=ηe−βmγSnm\nη′.\nIn order to derive Eq. (3), we have used\nlim\n{zη}→0∂r\n∂z1···∂zrTr/braceleftbig\nρHe−βP\nη′λη′ˆnη′ˆnm\nη/bracerightbig\n=/tildewideQ/summationdisplay\nnmη=0,1;P\nmnmη=1nm\nηe−βγSmnm\nη=/tildewideQe−βγSm\nwithρH=e−βγSP\nη,mmˆnm\nηand ˆnm\nη=b†\nη,mbη,m.\nIII. TIME EVOLUTION MODEL\nThe Heisenberg equations of motion lead to a hierarchy of time evolut ion equations for\nthe mean densities nh\ns=1\nNh/summationtext\nk/an}bracketle{ta†\nk,sak,s/an}bracketri}ht∞andnM\nm=1\nNM/summationtext\nη/an}bracketle{tb†\nη,mbη,m/an}bracketri}ht∞\nd[/summationtext\nk/an}bracketle{ta†\nk,sak,s/an}bracketri}htλ]\ndt=NhNM/summationdisplay\nm1Ws,s,m1,m1 (4)\nd[/summationtext\nη/an}bracketle{tb†\nη,mbη,m/an}bracketri}htλ]\ndt=NMNh/summationdisplay\ns1Ws1,s1,m,m, (5)\nwith\nWs,s,m,m=/summationdisplay\ns′\n1,m′\n1/parenleftBig\nJm′\n1,m·σs′\n1,s/tildewideCm′\n1,m,s′\n1,s−Jm,m′\n1·σs,s′\n1/tildewideCm,m′\n1,s,s′\n1/parenrightBig\n. (6)\nIn the last equation, the mean correlation function reads\n/tildewideCm′,m1,s′\n1,s1=−i\n/planckover2pi1γ\nVNhNM/summationdisplay\nη,k1,k′\n1Cη,η,k′\n1,k1\nm′,m1,s′\n1,s1ei(k1−k′\n1)Rη, (7)\n5whereCη′,η,k′,k\nm′,m,s′,s=/an}bracketle{tb†\nη′,m′bη,ma†\nk′,s′ak,s/an}bracketri}htλ. The time evolution equation of this quantity is given\nby\ni/planckover2pi1dCm′,m,s′,s\ndt= ∆EMFCm′,m,s′,s (8)\n+γ\nV/summationdisplay\nm′\n1,m1,s1,s′\n1δη,η′Jm′\n1,m1·σs′\n1,s1/angbracketleftbig\nBA−AtBt/angbracketrightbig\nλei(k′\n1−k1)Rη1\nwhere the compact notations m≡(η,m),s≡(k,s),B ≡b†\nm′\n1bm1b†\nm′bm,Bt≡b†\nm′bmb†\nm′\n1bm1,\nA ≡a†\ns′\n1as1a†\ns′as,At≡a†\ns′asa†\ns′\n1as1have been employed.\nThe mean-field contribution to the total energy is given by ∆ EMF=\nγ[(s′−s)M+(m′−m)S] whereM=NM/summationtextSM\nm=−SMm nM\nmandS=Nh/summationtextSh\ns=−Shs nh\nsare\nthe mean magnetic field generated by the ions and by the holes respe ctively.\nThe use of Eq. (8) combined with Eqs. (4)-(5) leads to a non-Marko vian time evolution\nof the macroscopic dynamical variables such as the density and the magnetization. By\nassuming an instantaneous spin-spin interaction, the Markov appr oximation can be easily\nrecovered. For further details about the justification of the Mar kovian approximation in a\nDMS excited by a laser pulse we refer to [12].\nBy using the Dirac identity [17]/integraltextt\n−∞e−iε(t−t′)//planckover2pi1dt′=−π/planckover2pi1δ(ε)−i/planckover2pi1P1\nεwherePdenotes\nthe principal value, the integration of Eq. (8) with respect to the t ime leads to\nCm′,m,s′,s=−iπγ\nV/summationdisplay\nm1,m′\n1,s1,s′\n1δ(∆EMF)\n×Jm′\n1,m1·σs′\n1,s1/angbracketleftbig\nBA−BtAt/angbracketrightbig\nλei[(k′\n1−k1)Rη1+(k′−k)Rη], (9)\nwhere ∆EMF=εk′−εk+∆EMF.\nSince the matrix operators J·σare real, it is clear from Eq. (6) that the imaginary part\ngives no contribution to the equation of motion.\nThe many-particle expansion of the correlation function /tildewideCallows us to express Eq. (9) in\nterms of the single-particle density matrix elements nh\nsandnM\nm. By using the commutation\nrules of the creation and annihilation operators we obtain\n/angbracketleftbig\nBA−BtAt/angbracketrightbig\nλ=δm1,m′δm,m′\n1δs1,s′δs′\n1,s/angbracketleftBig/parenleftBig\nb†\nmbm−b†\nm′bm′/parenrightBig\na†\nsas/parenleftBig\n1−a†\ns′as′/parenrightBig\n+b†\nmbm/parenleftBig\n1−b†\nm′bm′/parenrightBig/parenleftBig\na†\nsas−a†\ns′as′/parenrightBig/angbracketrightBig\nλ.\n6Furthermore, as a closure hypothesis, we have assumed that the non-diagonal matrix el-\nements of the density-like operators a†\ns′asandb†\nm′bmwith respect to the indexes ηandk\nvanish. From the above approximations and using the definition (4) a nd Eq. (9) we get\n/summationdisplay\nm1Ws,s,m1,m1=2π\n/planckover2pi1NSNM/parenleftBigγ\nV/parenrightBig2/summationdisplay\ns1,m′\n1,m1Jm′\n1,m1·σs,s1Jm1,m′\n1·σs1,s\n/summationdisplay\nk,k′,ηδ(∆EMF)/parenleftBig\nΠλ\nm1,m′\n1,s,s1−Πλ\nm′\n1,m1,s1,s/parenrightBig\n(10)\nwhere,\nΠλ\nm1,m′\n1,s,s1=/summationdisplay\nk,k1,η/angbracketleftBig/parenleftbig\n1−b†\nη,m1bη,m1/parenrightbig\nb†\nη,m′\n1bη,m′\n1a†\nk,sak,s/parenleftBig\n1−a†\nk1,s1ak1,s1/parenrightBig/angbracketrightBig\nλ.(11)\nA similar expression can be found for/summationtext\ns1Ws1,s1,m,min Eq. (5). By using Eq. (2) we recover\nthe fermionic limit of Π, namely\nΠ∞\nm1,m′\n1,s,s1=NMnM\nm′\n1/summationdisplay\nk,k1/angbracketleftBig\na†\nk,sak,s/parenleftBig\n1−a†\nk1,s1ak1,s1/parenrightBig/angbracketrightBig\n∞. (12)\nIn order to evaluate the time derivative of nh\ns, Eq. (12) can be solved numerically. In the\nfollowing paragraphs, we show that Eq. (12) may actually be furthe r simplified. According\nto the Zener model the ground state of the system can be estimat ed by taking into account\nonly the mean-field interaction between the holes and the magnetic io ns. The hole gas\nexperiences a mean magnetic field equal to Mand in turn generates a mean field acting\non the ions system equal to S. By converting the sum over kandk1in Eq. (12) into the\ncorresponding integral with respect to the energy variable E=εk, we obtain\nΠ∞\nV2=NMnM\nm′\n1e−∆EMF\nkBTh/integraldisplay\nfah ρ(E)ρ(E−∆EMF) dE , (13)\nwherefa=/an}bracketle{ta†\nsas/an}bracketri}ht∞/parenleftbig\n1−/an}bracketle{ta†\ns1as1/an}bracketri}ht∞/parenrightbig\n,h=1+e[γs1M+εk]/kBTh\n1+e[γs1M+εk1]/kBThandρdenotes the hole density\nof states.\nIn the limit γS≪γM≪εkwe have\nΠ∞\nV2≃NMNhnM\nm′\n1/parenleftbigg2m∗\n/planckover2pi12/parenrightbigg\n3√\n3π2Nhe−∆EMF\nkBThnh\ns/parenleftbig\n1−nh\ns1/parenrightbig\n. (14)\nIn the next section we will validate this approximation by comparing th e time evolution of\nthe magnetization obtained by using either the approximate formula (14) or the exact one\n7(12). Finally, by inserting Eq. (14) into Eq. (10) we obtain\ndnh\ns\ndt= 2ξNMs\n|s|SM−1/summationdisplay\nm=−SM/parenleftbig\nSM−m/parenrightbig/parenleftbig\nSM+m+1/parenrightbig/parenleftBig\nZ1/2,−1/2\nm−Z−1/2,1/2\nm+1/parenrightBig\n(15)\ndnM\nm\ndt= 2ξNh/summationdisplay\nσ=±1/parenleftbig\nSM−σm+1/parenrightbig/parenleftbig\nSM+σm/parenrightbig/parenleftBig\nZ−σ/2,σ/2\nm−Zσ/2,−σ/2\nm−σ/parenrightBig\n(16)\nwhere\nZs,s′\nm=nM\nmnh\ns/parenleftbig\n1−nh\ns′/parenrightbig\ne−∆EMF\nkBTh, (17)\nwithξ= 2πγ2m∗\n/planckover2pi133√\n3π2Nh.\nIV. SPIN EVOLUTION IN DMS\nInorder to study the time evolution of themean magnetizationof a G aMnAs/GaAsDMS\nheterostructure occurring after the interaction with a linearly po larized femtosecond laser\npulse, we have applied our time dependent model constituted of Eqs . (15) and (16). Based\non the experiment of [9], we consider a sample consisting of a 73 nm Ga 0.925Mn0.075As layer\ndeposited on a GaAs buffer layer and a semi-insulating GaAs substrat e. The background\nhole density is chosen to be 1020cm−3. For the details of the chemical composition of the\nsample we refer to [9]. Weassume that beforethe laser isturned on, the ion-holesystem is at\nequilibriumwiththephononbathatthelatticetemperature TL, sothatthegroundstatecan\nbe well described by the Zener-type model described in [7]. The laser excitation generates a\nnon-thermalelectron-holepairsdistribution. BymeansoftheCou lombhole-holeinteraction,\nthe hole distribution undergoes a quasi-instantaneous thermalizat ion (within a few tens of\nfemtoseconds) towardsaFermi-Diracdistributionwithtemperatu reThandaspindependent\nchemical potential µh\ns[18, 19]. The increasing of the overall temperature Thof the hole gasis\ndetermined by assuming that the excess of energy of the hot phot o-created particles (which\nis estimated as a fraction of the pump pulse energy) is redistributed among the total number\nof holes. The photo-created particles are approximately 2% of the background hole density\n[9]. In particular, we consider an excitation by a monochromatic laser pulse tuned at the\nenergyEland having a pump fluence Pf. To estimate the energy Eextransferred initially\nfrom the electromagnetic field to the kinetic energy of holes and elec trons, following [12], we\nassume that the fraction of the laser pulse energy imparted to the holes is 1 /4 of the photon\n8050 100050 1000.20.40.60.81  \nTh (K)M (norm. units)N0h\nN0h + Nexh\nFIG. 1: Temperature dependence of the normalized equilibri um ion magnetization for two hole\ndensities. Here Nh\nex= 5×1019cm−3.\nenergy. The total injected kinetic energy is thus Eex=nh\nexE′\nlηwithE′\nl=El−Eg−(ε1\nc+ε1\nv)\nandε1\nc,ε1\nvare the first eigenvalues of the valence and conduction bands. nh\nexis the density\nof photo-created particles and ηis the ratio of kinetic energy absorbed by the electron gas\nwhich can be estimated within the spherical band approximation as η=mHH\n/bardbl/(mc\n/bardbl+mHH\n/bardbl)\n[16] with mHH\n/bardbl(mc\n/bardbl) the effective mass of the heavy hole (electron) in the parallel direc tion\nof the sample. In Fig. 1 we show the normalized equilibrium ion magnetiza tionM/NMas\na function of the temperature and parameterized by the hole dens ityNh. In the figure, Nh\n0\nindicates the initial density of holes, Nh\nexthe excess of holes excited by the laser pulse, and\nNh=Nh\n0+Nh\nex. Afterthelaserexcitation, themagneticimpuritiesstronglyintera ctwiththe\nout-of-equilibrium hole gas by means of the Hpdexchange interaction which redistributes the\nspin polarization from one system to another while conserving the to tal spin magnetization.\nMeanwhile, the itinerant hole spin is efficiently dissipated through spin- orbit interactions\n(τSO≈100 fs) and relaxation of the total magnetization can be observed . Short-time spin\nrelaxation of the holes is therefore an essential ingredient for exp laining the observed time-\ndependent changes of the magnetization in ferromagnetic semicon ductors [4, 12].\nBy means of a standard relaxation model, we include both the spin-or bit mechanism\n(or any other mechanisms leading to the hole-spin relaxation) and th e other thermalization\neffects, such as the cooling of the kinetic energy of the excited hole s driven by the phonons,\n90.33 0.34 0.35 0.360.4 0.410.420.430.44\nM (norm. units)S (norm. units)\nB A\nTE RR \nUD 50 fs \n5 ps SC 350 ps \n1.2 ns \nFIG. 2: Evolution of the normalized holes magnetization ( S) and impurities magnetization ( M),\nin theS−Mplane, for TL= 70 K.\nand the radiative recombination of the electron-hole pairs. The cor responding equations\nread as follow\n∂nh\ns\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nso=nh\ns−nhs\nτSO(18)\n∂Nh\n∂t=Nh−Nh\n0\nτRR(19)\n∂Th\n∂t=Th−TL\nτL(20)\nwhereTh(t) andTLare the temperatures of the holes and the lattice, nhs(nM\nm,Th) is the\nself-consistent quasi-static equilibrium hole spin distribution comput ed from the Zener-type\nmodel of [7], which depends parametrically on the time-dependent ion magnetization M.\nThe temperature relaxation rate τ−1\nL=τ−1\nOP+τ−1\nAPtakes into account both the acoustic\nphonon scattering with τAP= 200 ps and the optical phonon scattering with τOP= 1 ps\nforTh>50 K and τOP=∞forTh<50 K [19]. Eq. (19) takes into account the radiative\nrecombination process characterized by a relaxation time τRR= 400 ps [19].\nIn Fig. 2 we present the time evolution of the normalized magnetizatio ns, where the\nvertical axiscorrespondsto S=S/Nhandthehorizontal axisto M=M/NM. Inagreement\nwith the experiment of [9], we consider a regime of small excitation (las er pump fluence\nof 1µJcm−2) and a lattice temperature of 70 K. The point A represents the initia l spin\n10100 200 300 400 500 600−0.1−0.08−0.06−0.04−0.0200.020.04\nTime (ps)Total Magn. (norm. units)TL = 70 K\nTL = 50 K\nTL = 30 K\nTL = 10 K\nFIG. 3: Time evolution of the total magnetization for differen t lattice temperatures: TL= 10 K\n(red line), TL= 30 K (green line), TL= 50 K (black line) and TL= 70 K (blue line).\npolarization which is suddenly shifted (instantaneously in our model) t o point B. This is due\nto the laser excitation which abruptly enhances the hole density and consequently changes\nthe normalization of S, so thatS(0−) =S(0−)\nNh\n0andS(0+) =S(0−)\nNh. Our numerical simulations\nreveal the presence of different time evolution regimes: (i) 0 < t <50 fs: during this initial\nphase the magnetization evolution is nearly coherent (semi-cohere nt regime SC in Fig. 2).\nIndeed, sincethephotoexcitedholesexperienceefficient spin-flips catteringwiththelocalized\nMn magnetic moments, a net spin polarization is transferred from th e ion impurities to the\nholes leading to a significant increase of the hole spin polarization. Cor respondingly, due to\nthe large difference in densities between the two populations, only a s mall decrease of the ion\nmagnetization is observed; (ii) 50 fs < t <5 ps: the nonequilibrium hole spin polarization\nis efficiently dissipated via the spin-orbit coupling, which leads to a net d ecrease of the\ntotal spin magnetization (see also Fig. 3). During this ultrafast dem agnetization regime\n(UD) the kinetic temperature of the excited holes is still high; (iii) 5 ps < t <350 ps: the\nhole distribution loses its energy via carrier-phonon scattering and the hole temperature\ndecreases over the time scale τL. When the Curie temperature is reached, the holes and\nions spins begin to align, which allows the system to recover a ferroma gnetic order. Since\nthe total number of holes relaxes to its initial value Nh\n0over a slower time scale τRR≫τL,\na ferromagnetic state with an excess of holes can be reached, thu s justifying a transient\nenhancement of the total magnetization (TE regime); (iv) 350 ps < t <1.2 ns: finally\nthe radiative recombination of the electron-hole pairs brings the sy stem back to its initial\nconfiguration (RR regime).\nThetimeevolutionofthetotalmagnetizationfordifferent latticete mperaturesisdepicted\n111020304050607080−0.1−0.050\nTL (K)Total Magn. (Norm units)\nFIG. 4: Minimum (solid line) and maximum (dashed line) of the total magnetization for different\nlattice temperatures.\n100200300400500600−0.1−0.08−0.06−0.04−0.020\nTime (ps)Total Magn. (norm. units)\nFIG. 5: Time evolution of the total magnetization at TL= 10 K. Full line: exact formula, Eq.\n(13); dashed line: approximated formula, Eq. (14).\nin Fig. 3. We see that the minimum of the total magnetization shifts to shorter times with\nincreasing lattice temperature, in agreement with experimental fin dings. In Fig. 4 we plot\nthe excursion of the total magnetization for different lattice temp erature: only for 45 K\n< TL<78 K an enhancement of the total magnetization may be observed [4 ].\nFinally, in order to validate the approximation of Eq. (13), we compar e in Fig. 5 the\ntime evolution of the total magnetization obtained by using either th e approximate formula\n(14) or by evaluating numerically the integral of Eq. (13). As can be clearly seen, a good\nagreement is obtained, justifying the use of the simplified expressio ns (15) and (16).\n12V. CONCLUSION\nIn order to describe the strong spin-spin scattering regime obser ved in diluted magnetic\nsemiconductors, we have derived a dynamical model that goes bey ond the usual mean-field\napproximation. This model is based on the pseudo-fermion formalism and on a second-order\nmany-particle expansion of the p−dexchange interaction, which is performed in terms of\nthe single-particle density functions. At this level of description, t his approach is similar to\nthat of Ref. [12], which was derived following a different perspective. Numerical simulations\nshowed that our model is able to reproduce qualitatively – and to som e extent quantitatively\n– the long-time evolution of the total magnetization after laser irra diation, as seen in recent\nexperiments [9]. The early demagnetization observed in the experime nts is explained as\nthe result of a net flow of polarization from the ions to the holes, whic h is subsequently\ndissipated via spin-orbit coupling. Thus, the typical demagnetizatio n time scale is mainly\ndetermined by the nonlinear coupling between the ions and holes spins , with a lower bound\ngiven by the spin-orbit time scale, τSO≈100 fs. The demagnetization process cannot be\nfaster than τSO, but can be significantly slower, depending on the lattice temperatu re. In\naddition – andin contrast to Ref. [12] – other slower processes (na mely, holes thermalization\nand radiative recombination) were also included in the description, so that the global model\nencompasses time scales going from a few tens of femtoseconds to hundreds of picoseconds.\nWe point out that the methodology developed in this work can be natu rally extended\nto higher orders by using perturbative field-theoretic techniques . In particular, we plan to\ninvestigate third order dynamical processes, which were neglecte d here but may play an\nimportant role in the regime of higher photoexcitation energy.\nFinally, in the model used in this work only the heavy-hole band contrib ution to the\nexchange interaction was taken into account. The inclusion of a rea listic band structure is\ncurrently under study.\nAcknowledgements\nWe thank P. Gilliot and J.-Y. Bigot for useful discussions. This work wa s partially funded\n13by the Agence Nationale de la Recherche, contract n. ANR-06-BLA N-0059.\n[1] E. Beaurepaire, and J.-C. Merle, A. Daunois, and J.-Y. Bi got, Phys. Rev. Lett. 76, 4250\n(1996).\n[2] H. Ohno, Science 281, 951 (1998).\n[3] T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Ferrand , Science 287, 1019 (2000).\n[4] J. Wang, C. Sun, J. Kono, A. Oiwa, H. Munekata, /suppress L. Cywi´ ns ki, and L. J. Sham, Phys. Rev.\nLett.95, 167401 (2005).\n[5] C. Zener, Phys. Rev. 81, 440 (1951).\n[6] B. Lee, T. Jungwirth, and A. H. MacDonald, Phys. Rev. B 61, 15606 (2000).\n[7] N. Kim, H. Kim, J. W. Kim, S. J. Lee, and T. W. Kang, Phys. Rev . B74, 155327 (2006).\n[8] Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl. Phy s. Lett.84, 5234 (2004).\n[9] J. Wang, I. Cotoros, K. M. Dani, X. Liu, J. K. Furdyna, and D . S. Chemla, Phys. Rev. Lett.\n98, 217401 (2007).\n[10] B. K¨ onig, I. A. Merkulov, D. R. Yakovlev, W. Ossau, S. M. Ryabchenko, M. Kutrowski, T.\nWojtowicz, G. Karczewski, and J. Kossut, Phys. Rev. B 61, 16870 (2000).\n[11] J. Chovan, E. G. Kavousanaki, and I. E. Perakis, Phys. Re v. Lett.96, 057402 (2006).\n[12] /suppress L. Cywi´ nski and L. J. Sham, Phys. Rev. B 76, 045205 (2007).\n[13] J. Wang, /suppress L. Cywi´ nski, C. Sun, J. Kono, H. Munekata, and L. J. Sham, Phys. Rev. B 77,\n235308 (2008).\n[14] A. A. Abrikosov, Physics 2, 5 (1965).\n[15] P. Coleman, Phys. Rev. B 28, 5255 (1983).\n[16] R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and H ole Systems ,\nSpringer Tracts in Modern Physics (Springer, Berlin, 2003) , Vol. 191.\n[17] H. Haug and S. W. Koch, Quantum Theory of the optical and electric properties of semi con-\nductors (World Scientific, 1994).\n[18] B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, and W. J. M. de Jonge, Phys. Rev. Lett. 95,\n267207 (2005).\n[19] J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconduct or Heterostructures\n(Springer, Berlin, 1999).\n14"    },    {        "title": "2311.18517v1.Excitation_of_the_Gyrotropic_Mode_in_a_Magnetic_Vortex_by_Time_Varying_Strain.pdf",        "content": "Excitation of the Gyrotropic Mode in a Magnetic Vortex by Time-Varying Strain\nVadym Iurchuk,1,∗J¨ urgen Lindner,1J¨ urgen Fassbender,1, 2and Attila K´ akay1\n1Institute of Ion Beam Physics and Materials Research,\nHelmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany\n2Institute of Solid State and Materials Physics, Technische Universit¨ at Dresden, 01062 Dresden, Germany\n(Dated: December 1, 2023)\nWe demonstrate excitation of the gyrotropic mode in a magnetostrictive vortex by time-varying\nstrain. The vortex dynamics is driven by a time-varying voltage applied to the piezoelectric sub-\nstrate and detected electrically by spin rectification at subthreshold values of rf current. When the\nfrequency of the time-varying strain matches the gyrotropic frequency at given in-plane magnetic\nfield, the strain-induced in-plane magnetic anisotropy leads to a resonant excitation of the gyration\ndynamics in a magnetic vortex. We show that nonlinear gyrotropic dynamics can be excited already\nfor moderate amplitudes of the time-varying strain.\nMagnetic vortices – stable topological magnetic con-\nfigurations – can be spontaneously formed in confined\nhigh-symmetry magnetic micro- and nanostructures [1–\n3]. They are considered as promising candidates for\napplications in next-generation spintronic memory [4],\nsensor [5], and oscillator [6, 7] devices. Resonant ex-\ncitation of a vortex core by in-plane rf magnetic fields\nresults in the gyrotropic dynamics of the vortex core\n(VC) [8, 9]. To reduce energy dissipation and to facil-\nitate integration of magnetic vortices with CMOS-based\ncomponents, alternative means of VC dynamics excita-\ntion are usually employed, e.g. dc-current-driven exci-\ntation by Slonczewski spin-transfer torque [10, 11] or\nrf-current-driven excitation via non-adiabatic Zhang-Li\nspin-transfer torque [12, 13].\nAn interesting approach was proposed by Ostler et\nal.in [14] to use time-varying-strain gradient to excite\nlarge-amplitude gyration dynamics in magnetostrictive\nmicrostructures with Landau-flux-closure state leading\nto eventual switching of the VC. This approach holds\npromises for considerable reduction of the energy con-\nsumption (due to the absence of currents flowing through\nthe device) and offers a purely extrinsic means to excite\nthe VC dynamics. However, engineering of a strain gra-\ndient, necessary for breaking the flux closure symmetry\nand providing an onset for the strain-driven VC dynam-\nics, requires a sophisticated sample design, which is not\nstraightforward to realize in real devices. In addition,\nwhen using piezoelectric materials for strain generation,\nlarge voltages are usually needed to generate sufficient\nstrains. An alternative way of strain-driven VC dynam-\nics by surface acoustic waves (SAW) was studied analyt-\nically and numerically by Koujok et al. in [15]. Recent\nexperiments reported on a local piezostrain as an efficient\nmeans to shift the VC gyrotropic frequency in a compact\ndevice with low voltages and all-electrical operation [16],\nthus providing a reliable path to strain-driven VC exci-\ntation. However, the experimental demonstration of the\nstrain-driven excitation of the VC dynamics has not been\n∗Corresponding author’s e-mail: v.iurchuk@hzdr.dereported so far.\nIn this letter, we demonstrate both experimentally\nand by micromagnetic simulations, excitation of a gy-\nrotropic mode in a magnetostrictive vortex by local time-\nvarying piezoelectric strain. The VC dynamics is ex-\ncited by a time-varying voltage applied to the piezoelec-\ntric substrate and detected electrically by spin rectifi-\ncation measurements at subthreshold values of rf cur-\nrent. Micromagnetic simulations confirm that the strain-\ninduced in-plane magnetic anisotropy in a flexed mag-\nnetic vortex leads to the VC shift from the equilibrium\nposition. When the frequency of the time-varying strain\nmatches the VC gyrotropic frequency for given in-plane\nmagnetic field, the strain-induced in-plane uniaxial mag-\nnetic anisotropy (IPUA) leads to a resonant excitation\nof the gyration dynamics in a magnetic vortex. The de-\npendence of the gyration frequency and VC trajectories\non the in-plane bias field and the amplitude of the time-\nvarying strain shows that nonlinear VC dynamics can be\nexcited at moderate strain amplitudes. This result allows\nfor an energy-efficient excitation and control of the VC\ngyrotropic mode in vortex-based spin-torque oscillators.\nThe study is performed on micron-sized magnetostric-\ntive Co 40Fe40B20(hereafter CoFeB) disks grown on\n(011)-cut piezoelectric 0.7Pb[Mg 1/3Nb2/3]O3–0.3PbTiO 3\n(hereafter PMN-PT) single crystals. The detailed in-\nformation on the sample fabrication can be found in\nRef. [16]. To detect the VC dynamics, we use a rf mag-\nnetotransport setup [see Fig. 1(panel a)] for electrical\ndetection of magnetization dynamics in single magnetic\nvortices at room temperature (see [16, 17] for more de-\ntails). The standard detection technique exploits the\nanisotropic magnetoresistance (AMR) effect, i.e., the re-\nsistance change induced by the relative angle between the\ndirection of the electrical current and the net magnetiza-\ntion of a magnetic structure [13]. An rf current injected\nthrough a bias-T into the microdisk device excites the\nVC gyrotropic dynamical mode, via the joint action of\nthe spin-transfer torque and rf Oersted field, and thereby\nleads to a dynamical magnetoresistance oscillating at the\nexcitation frequency. The time-averaged product of the\nrf current and the dynamical magnetoresistance — whicharXiv:2311.18517v1  [physics.app-ph]  30 Nov 20232\n40 80 120 160 2000123 V0 = 2.5 V\nV0 = 1.5 V\nV0 = 0 VV0 = 0.5 VVdc (V)\nfrf (MHz)\n40 80 120 160 2000123\nVac = 0 VVac = 1 VVac = 2 Vfac = 110 MHzVdc (V)\nfrf (MHz)Vac = 4 V\n40 80 120 160 20001234\nVac = 0 VVac = 0.5 VVac = 1 VVac = 1.5 VVac = 2 VVac = 2.5 VVdc (V)\nfrf (MHz)V0 = 2.5 V; fac = 110 MHz\n40 80 120 160 20001234\nV0 = Vac = 0 VV0 = Vac = 0.25 VV0 = Vac = 0.75 VV0 = Vac = 1.25 VV0 = Vac = 1.5 VV0 = Vac = 2 Vfac = 110 MHz\nV0 = Vac = 2.5 VVdc (V)\nfrf (MHz)\n𝑉𝐴𝑊𝐺 𝜀𝑥𝑥𝐾𝑥\n𝑡 𝑡 𝑡(a)\n𝑉𝐴𝑊𝐺=𝑉0+𝑉𝑎𝑐sin2𝜋𝑓𝑎𝑐𝑡\nxy\n204060801001201401600.00.51.0\n0Hy = 5.18 mT\nno VAWGVdc (V)\nfrf (MHz) Prf = -6 dBm\n Prf = -3 dBmf0 = 105 MHz\n(b)\n𝑉𝑎𝑐=0\n𝑉𝐴𝑊𝐺=𝑉0\n𝑡\n𝑉0=0\n𝑉𝐴𝑊𝐺=𝑉𝑎𝑐sin2𝜋𝑓𝑎𝑐𝑡\n𝑡\n𝑉0>𝑉𝑎𝑐\n𝑉𝐴𝑊𝐺=𝑉0+𝑉𝑎𝑐sin2𝜋𝑓𝑎𝑐𝑡\n𝑡\n𝑉𝐴𝑊𝐺=𝑉0+𝑉𝑎𝑐sin2𝜋𝑓𝑎𝑐𝑡\n𝑡\n𝑉0=𝑉𝑎𝑐(c) (d)\n(e) (f)\nFIG. 1. (a) Schematics of the experiment enabling a detection of the magnetization dynamics in a magnetostrictive vortex\nby a homodyne detection technique, with a simultaneous application of an time-varying voltage to the piezoelectric PMN-PT\nsubstrate. Unipolar time-varying voltage VAWG to PMN-PT generates the time-varying uniaxial strain εxx(t), creating a time-\nvarying magnetoelastic anisotropy Kx(t) in the CoFeB disk at the same frequency fac. (b) Electrically detected Vdcsignal for\nPrf=–3 dBm (red curve) and Prf=–6 dBm (gray curve). (c–d) Rectified voltage Vdcversus rf current frequency frfand for\ndifferent VAWG applied to the PMN-PT: (c) static VAWG with Vac= 0 V, V0= 0; 0.5; 1.5 and 2.5 V; (d) time-varying VAWG\nwith V0= 0 V, Vac= 0; 1; 2 and 4 V; (e) time-varying VAWG with V0= 2.5 V, Vac= 0; 0.5; 1; 1.5, 2 and 2.5 V; (f) time-varying\nVAWG with V0=Vac= 0, 0.25; 0.75; 1.25; 1.5; 2 and 2.5 V.\nresults in a rectified dc voltage Vdc– is measured by a\nconventional homodyne detection scheme using a lock-in\namplifier. When the excitation frequency frfmatches\nthe eigenfrequency f0of the gyrotropic mode, the result-\ningVdcis enhanced due to the dynamical magnetore-\nsistance increase associated with the resonant expansion\nof the VC gyration trajectory. To improve the signal-\nto-noise ratio, magnetic field modulation of the dynami-\ncal magnetoresistance at the lock-in reference frequency\n(here 1033 Hz) was used similar to [17, 18].\nWe note that in general, Vdc∼Irf∆R, where Irfis\nthe rf current injected into the device and ∆ Ris the\ndynamical magnetoresistance change over one oscillation\nperiod [19]. The latter is directly defined by the VC tra-\njectory opening [13]. Therefore, two ways of enhancingtheVdcsignal at resonance are possible. The first way\nis increase of the rf current amplitude Irf, which also\nleads to the ∆ Rincrease due to the stronger current-\ndriven excitation of the VC. The second method is reso-\nnant enhancement of the VC gyration trajectory by ex-\nternal stimulus at fixed rf current Irf. Here, we use time-\nvarying strain ε(t), generated by time-varying voltage\napplied to the piezoelectric substrate to drive the VC\ndynamics, which eventually leads to the VC trajectory\nenhancement at resonance.\nIn our experiment, we first characterize the rf-current-\ndriven VC dynamics and define the threshold rf current\nto detect the vortex gyrotropic mode electrically. Then,\nwe study the effect of piezostrain on the VC dynamics for\nsubthreshold Irfvalues. We show that static strain has3\nno effect on the rectified signal Vdc, nor has the symmet-\nric sinusoidal time-varying strain ε(t) oscillating around\nzero at the VC gyration frequency. On the other hand,\nwe observe an enhanced Vdcat the VC resonance when\nthe symmetric sinusoidal strain is additionally biased by\na static offset strain, creating a unipolar time-varying\nstrain.\nAn arbitrary waveform generator (AWG) was used to\napply a voltage VAWG with a sinusoidal time profile to\nthe PMN-PT crystal via surface electrodes. The applied\nvoltage is defined as VAWG =V0+Vacsin(2πfact), where\nthe first term V0is the static offset voltage and the second\nterm is the time-varying voltage with the amplitude Vac\nand frequency fac. Upon application of a time-varying\nVAWG, the local time-varying uniaxial strain εxx(t) is\ngenerated in the PMN-PT, due to the converse piezo-\nelectric effect. The strain, when transferred to the CoFeB\ndisk, imposes a time-varying IPUA Kx(t) at the same fre-\nquency fac(see schematics at the bottom of Fig. 1(a)).\nTo ensure that the detected VC dynamics is strain-\ndriven, we first measure the rectified voltage as a func-\ntion of the rf power for zero time-varying voltage VAWG.\nFig. 1(b) shows the rectified voltage Vdcversus the frf\nmeasured for the CoFeB disk with 3.65 µm diameter at\nµ0Hy= 5.18 mT, and for two different values of the rf\npower Prffrom the rf generator. For Prf=–3 dBm, a dis-\ntinctive resonance is detected at approximately 105 MHz,\ncorresponding to the VC gyrotropic resonance frequency\nf0. On the other hand, for Prf=–6 dBm, no resonance\nis detected due to small current-driven displacement of\nthe VC and therefore small dynamic magnetoresistance\nsignal, which is below the noise level of the experimental\nsetup. Keeping the rf excitation power below the detec-\ntion threshold (here, Prf=–6 dBm), we performed the\nmeasurements of the Vdc(frf) for different VAWG(t) ap-\nplied to the PMN-PT [Fig. 1(c–f)]. The insets in Fig. 1(c–\nf) denote schematically the time profile of the voltage\nVAWG applied to the PMN-PT.\nFig. 1(c) shows the rectified Vdcspectra vs. rf cur-\nrent frequency measured for the same device and at the\nsame conditions as in Fig. 1(b) for different values of\nstatic VAWG =V0applied to the PMN-PT. The static\nstrain εxxgenerated by the static voltage V0is expected\nto result in the f0downshift only [16, 20]. Therefore,\nwe observe no resonance for any V0value, since Prf\nis below the detection threshold. Fig. 1(d) shows the\nVdc(frf) spectra measured for VAWG =Vacsin(2πfact)\nwith fac= 110 MHz and for different amplitudes Vacof\nthe sinusoidal time-varying voltage. For this case, in-\ndeed no resonance is observed as well. However, when\nthe time-varying sinusoidal voltage with the static offset\nV0⩾Vacis applied [see Fig. 1(e,d)], i.e. for VAWG =\nV0+Vacsin(2πfact), we observe a resonance peak in the\nVdc(frf) spectra for increased values of VAWG amplitude.\nOne can see that we are able to detect resonantly the VC\ngyration only for the unipolar time-varying voltage, i.e.\nwhen the VAWG oscillates between zero and maximum\nvalue V0+Vac[as in Fig. 1(f)] or between two positivevalues V0−VacandV0+Vac[as in Fig. 1(e)].\nTo gain more insight into the time-varying-strain-\ndriven gyrotropic dynamics in a magnetic vortex, and\nto understand why the dynamics is detectable only for\nunipolar time-varying voltage, we performed micromag-\nnetic simulations of the static magnetization distribution\nand the magnetization dynamics in the CoFeB disks in re-\nsponse to static and time-varying uniaxial strain. We use\nthe graphics-processing-unit-accelerated MuMax3 soft-\nware [21] with magnetoelastic extension [22]. The follow-\ning parameters for CoFeB were used: saturation magne-\ntization Ms= 1700 kA/m, exchange constant Aex= 21\npJ/m3, damping parameter α= 0.008, Young’s mod-\nulus Y= 250 GPa, saturation magnetostriction λs=\n65 ppm. For simplicity, we assume no magnetocristalline\nanisotropy in the CoFeB disk. To reduce the simu-\nlation time, we consider a much smaller CoFeB mag-\nnetic disk with radius R= 100 nm and thickness d=\n20 nm. We use in-plane discretization into 128 ×128\ncells for the magnetization dynamics simulations, and\na finer discretization into 512 ×512 cells, for the com-\nputation of the static equilibrium magnetic states. The\nmagnetic vortex core is excited by a time-varying strain\nε(t) =ε0sin2(πft)≡ε0\n2+ε0\n2sin(2πft), where fis the\nexcitation frequency, and εis the uniaxial strain along x\n(ε0=εxx) ory(ε0=εyy) in-plane direction. We note\nthat for the given ε(t) time profile, the strain is oscillat-\ning at the frequency fbetween zero and ε0, i.e. during\nthe excitation the magnetic vortex is subjected to exclu-\nsively non-negative values of strain. Similar to [23, 24],\nthe magnetization dynamics in the magnetic disk is sim-\nulated for different strain amplitudes ε0and in-plane bias\nfields Hxover 100 gyration periods T0=1\nfand the fi-\nnal magnetization state is captured for each value of the\nexcitation frequency fin the chosen range. To study\nthe VC trajectories as a function of excitation amplitude\nand bias field, the last 10 periods of time-evolution of the\nmagnetization is recorded for a given resonance frequency\nof the gyrotropic mode f0.\nFig. 2(a) shows the ycoordinate of the VC position\nas a function of the uniaxial strain εxx(and the cor-\nresponding IPUA Kε=3\n2λsY εxx) for three values of\nexternal bias magnetic field applied along the xaxis.\nForµ0Hx= 0 mT, no VC shift is observed as expected\nfor the symmetric vortex configuration subjected to the\nIPUA. Indeed, when the VC is located in the center\nof the disk, the effective magnetoelastic torques exerted\nby strain-induced IPUA on the VC are counterbalanced\nsince IPUA acts on the equal amount of magnetic mo-\nments aligned parallel to the IPUA (here along x), and\ntherefore no VC movement occurs. When the vortex\nsymmetry is broken by an in-plane magnetic field, the\nIPUA leads to the shift of the VC from the equilibrium\nposition at a given magnetic field as a result of a noncom-\npensated net magnetoelastic torque acting on the VC.\nThe direction of this torque coincides with the direction\nof the external-field torque for ε∥Hand is opposite\nforε⊥H(see Fig. S1 in the Appendix for the detailed4\n-100 -60 -20 20 60 100-100-60-202060100y (nm)\nx (nm)0Hx =\n 100 mT\n 80 mT\n 60 mT\n 40 mT\n 20 mT\n0 500 1000 1500 20000204060x (nm)\nxx() 100 mT\n 80 mT\n 60 mT\n 40 mT\n 20 mT\n0 500 1000 1500 20000.0 12.2 24.4 36.6 48.7\n01234(kJ/m3)VC shift along y (nm)\nxx()0Hx = 0 mT\n 0Hx = 50 mT\n 0Hx = 100 mT\n1.5 1.6 1.7 1.8 1.9 2.00.000.040.080.12\n100 mT\n80 mT\n60 mT\n40 mT|my| (arb.u.)\nf (GHz)0Hx = 20 mTε𝑥𝑥=1000𝜇𝜀\n500 1000 1500 2000 25000.000.050.100.150.200.25\n100 mT\n80 mT\n60 mT\n40 mT  (unity)\nxx()20 mT\n0 50 100 1501.61.82.02.2 f0 (GHz)\n0Hx (mT)(a)\nε𝑥𝑥=𝜀𝑦𝑦=250𝜇𝜀○ε(𝑡)=𝜀𝑥𝑥𝑠𝑖𝑛2𝜋𝑓𝑡\nε(𝑡)=𝜀𝑦𝑦𝑠𝑖𝑛2𝜋𝑓𝑡•(b)\n(c) (d) (e)\nFIG. 2. (a) Shift of the vortex core from the equilibrium position for µ0Hx= 0; 50 and 100 mT as a function of the\nuniaxial strain εxxapplied along the xaxis. (b) FMR-like spectra of the VC gyration dynamics excited by time-varying strain\nε(t) =ε0sin2(πft) for ε0=εxx= 250 µε(solid dots) and ε0=εyy= 250 µε(open circles) simulated for different values of the\nbias field Hx. The inset shows the simulated values of the VC gyration frequency f0versus µ0Hx. (c) Simulated trajectories\nof the VC gyration at resonance for different Hxand for εxx= 1000 µε. (d) VC trajectory opening ∆ xalong the xaxis as a\nfunction of εxxfor different Hxvalues. (e) VC trajectory ellipticity ηas a function of εxxfor different Hxvalues. Solid lines\nare linear fits.\ndescription of an impact of IPUA on the static magneti-\nzation distribution in a magnetic vortex). Thus the VC\ndisplacement increases with both, increasing εxxand in-\ncreasing Hx[see Fig. 2(a)]. The observed strain-induced\nVC displacement, while rather small (few nm), can nev-\nertheless be efficiently used for the resonant excitation\nof the VC gyrotropic mode by time-varying strain. It is\nknown that when the external force, which caused the VC\nshift from the equilibrium, is released, the VC will move\ntowards the equilibrium on a spiral trajectory governed\nby the Thiele’s theory [25, 26]. Therefore, a resonant ex-\ncitation of the gyrotropic mode in a vortex is possible by\ntime-varying strain with an analogy to the excitation by\nrf field [27].\nFig. 2(b) shows the ferromagnetic-resonance-\nabsorption-like spectra of the VC dynamics excitedby time-varying strain ε(t) =ε0sin2(πft) for ε0=εxx=\n250µε(solid dots) and ε0=εyy= 250 µε(open circles)\nand for different values of the bias field Hx. Here and\nfurther, the strain is expressed in the dimensionless units\nof microstrain µε=µm/m [28]. When the frequency\nof the time-varying strain ε(t) approaches the eigenfre-\nquency of the VC gyrotropic mode at a given magnetic\nfield, a pronounced peak is observed, attributed to the\nresonant excitation of the VC gyration. The gyrotropic\nnature of the observed mode is confirmed by visualizing\nthe VC trajectories at resonance for a given magnetic\nfield [see Fig. 2(c)]. One can see that the trajectory\nopening increases with increased Hxi.e. when the VC is\nshifted closer towards the disk edge. This effect agrees\nwith the data of Fig. 2(a), where the strain-induced VC\ndisplacement for the given εxxincreases with increased5\nHxdue to enhanced magnetoelastic torque acting on\nthe VC. We note that at zero magnetic field Hx, no\ngyrotropic dynamics can be excited due to the symmetry\nreasons described above.\nFig. 2(d) shows the major axis ∆ xof the VC trajec-\ntory for different values of magnetic field Hxand dif-\nferent amplitudes εxxof the time-varying strain. For\nthe given εxxrange, the trajectory opening is quasilin-\near for µ0Hx= 20 mT. For µ0Hx⩾40 mT, two strain-\ndependent regions can be distinguished: the linear range,\nand the saturation range, with the field-dependent transi-\ntion between the two. We attribute the saturation range\nto the strain-driven large-amplitude nonlinear VC gyra-\ntion dynamics. Further increase of time-varying-strain\namplitude εxxand/or bias magnetic field Hxleads to the\nVC switching when reaching the critical VC velocity [29].\nWhen entering the nonlinear regime, the VC trajectory\nis distorted and becomes non-circular. Fig. 2(e) shows\nthe trajectory ellipticity η= 1−∆y\n∆xversus amplitude\nof the time-varying strain for different values of Hx. An\nincrease of the ellipticity is observed for increased εxxat\nconstant Hxas well as for increased Hxat constant εxx.\nThis suggests that, for the VC dynamics driven by time-\nvarying strain, the transition to the nonlinear dynamical\nrange can be induced not only by simple increase of the\nexcitation amplitude, but by shifting the VC closer to\nthe disk edge by external magnetic fields.\nFinally, we studied the effects of the bias field and the\namplitude of the time-varying strain on the resonance\nfrequency and amplitude of the VC gyration dynamics.\nFig. 3(a) shows the simulated absorption-like spectra for\nthe bias field µ0Hx= 60 mT and for different values of\nstrain amplitude εxx. For increased εxxa typical transi-\ntion to the nonlinear dynamics is observed, including res-\nonance frequency shift, peak foldover and bistable behav-\nior (hysteresis) at high excitation amplitudes [30]. Sur-\nprisingly, the same behavior is observed when the VC\ndynamics is excited by a fixed strain amplitude and for\nincreased values of the bias field Hx[see Fig. 3(b)]. For\nsmall fields ( µ0Hx⩽40 mT), a symmetric resonance\npeak is observed, typical for the linear dynamics regime,\nwhereas at higher fields ( µ0Hx⩾60 mT), the peak be-\ncomes distorted exhibiting the same signs of nonlinear\ndynamics as in Fig. 3(a) for large εxxvalues. As seen\nfrom the simulated spectra of Fig. 3 and VC trajectories\nof Fig. 2(d,e), the critical field for the transition from\nlinear to nonlinear dynamical regime is inversely propor-\ntional to εxx. As mentioned above, this result is a conse-\nquence of the increased magnetoelastic excitation of the\nVC, when it is shifted towards the disk edge by the bias\nfield.\nThe results of the micromagnetic simulations are in\ngood agreement with the experimental data. First, as\nseen in Fig. 1(c), no VC resonance is observed when the\nVC is excited by purely sinusoidal voltage Vacsin(2πfact)\nwithout a dc component. Since the strain is pro-\nportional to the absolute value of the applied voltage\n(εxx∼ |VAWG|), the resulting time-varying strain fol-\nFig. 3 (10x16)\n0.90 0.95 1.00 1.05 1.100.00.20.40.6\n1750 \n1500 \n1250 \n1000 \n750 \n500 my (arb.u.)\nf/f0f0 = 1.70 GHz\n0Hx = 60 mT\nxx = 250 \n0.90 0.95 1.00 1.05 1.100.00.10.20.30.4\n20 mTxx = 1000 \n100 mT\n80 mT\n60 mT\n40 mTmy (arb.u.)\nf/f00Hx = 0 mT\n(a)\n(b)FIG. 3. Frequency-swept FMR-absorption-like spectra of the\nVC gyration dynamics excited by time-varying strain: (a) for\ndifferent values of the strain amplitude εxxat fixed magnetic\nfield µ0Hx= 60 mT and (b) for different values of the bias\nmagnetic field Hxat fixed εxx= 1000 µε. Arrows indicate\nthe frequency sweep direction.\nlows the |sin(2πfact)|dependence, and therefore oscil-\nlates at twice the frequency fac. The simulations con-\nfirm that no gyration is excited by the time-varying strain\nε(t) =εxxsin2(2πft), where εxxoscillates at 2 f. Second,\nthe simulations explain the frequency shift observed ex-\nperimentally for increased values of VAWG [see Fig. 1(f)].\nThis frequency downshift is attributed to the excita-\ntion of the nonlinear VC gyrotropic mode. As seen in\nFig. 3(a), with increased amplitude of the time-varying\nstrain, the transition to the nonlinear regime is accom-\npanied by the peak distortion and eventual decrease of\nthe VC gyration frequency f0. Both effects are in good\nagreement with the experimental data of Fig. 1(f). We\nnote that the strain driven dynamics is detected for rela-\ntively wide range of the frequencies facof the voltage\nVAWG applied to PMN-PT. We detect the resonance\nat approximately 90 MHz for the facrange from 90 to\n120 MHz. This observation suggests that the excited\nmode is strongly nonlinear, and therefore can be excited\neven by the off-resonance time-varying strain.\nIn conclusion, we have demonstrated experimentally6\nthe excitation of the gyrotropic mode in a magnetostric-\ntive vortex by time-varying piezoelectric strain. The vor-\ntex dynamics is driven by a time-varying voltage applied\nto the piezoelectric substrate and detected electrically by\nmeasuring the dynamical magnetoresistance of the mag-\nnetic disk at subthreshold values of rf current. Micro-\nmagnetic simulations confirm that the strain-driven exci-\ntation of the gyrotropic mode originates from the strain-\ninduced VC shift from the equilibrium in the presence\nof the in-plane magnetic field, which breaks the radial\nsymmetry of the vortex. This approach offers an extradegree of freedom for the excitation and manipulation of\nthe magnetization dynamics of magnetic vortices for the\napplications in spintronic oscillators.\nThis study is funded by the Deutsche Forschungsge-\nmeinschaft (DFG, German Research Foundation) within\nthe grant IU 5/2-1 (STUNNER) – project number\n501377640. Support from the Nanofabrication Facilities\nRossendorf (NanoFaRo) at the IBC is gratefully acknowl-\nedged. We thank Thomas Naumann for help with the\nsputtering of the CoFeB thin films. We acknowledge use-\nful discussions with Ciar´ an Fowley on the microfabrica-\ntion process.\n[1] A. Aharoni, Upper bound to a single-domain behavior of\na ferromagnetic cylinder, Journal of Applied Physics 68,\n2892 (1990).\n[2] N. A. Usov and S. E. Peschany, Magnetization curling\nin a fine cylindrical particle, Journal of Magnetism and\nMagnetic Materials 118, L290 (1993).\n[3] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and\nT. 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Gautschi (Springer, Berlin, Heidelberg, 2002) pp.\n127–140.\n[29] K.-S. Lee, S.-K. Kim, Y.-S. Yu, Y.-S. Choi, K. Y. Gus-\nlienko, H. Jung, and P. Fischer, Universal Criterion and\nPhase Diagram for Switching a Magnetic Vortex Core in\nSoft Magnetic Nanodots, Physical Review Letters 101,267206 (2008).\n[30] K. Y. Guslienko, R. H. Heredero, and O. Chubykalo-\nFesenko, Nonlinear gyrotropic vortex dynamics in fer-\nromagnetic dots, Physical Review B 82, 014402 (2010),\npublisher: American Physical Society.\nAppendix: Supplementary data\nFig. S1 shows the simulated relaxed magnetic configu-\nration of the CoFeB disk (diameter d= 200 nm; thickness\nt= 20 nm) in the vortex state for different orientations\nof the in-plane bias magnetic field and in-plane uniax-\nial anisotropy (IPUA). Panel (a) shows the effect of the\nIPUA Kon the magnetization distribution at zero bias\nfield. Besides the growth of the magnetic domains along\nthe IPUA direction, no effect on the vortex core (VC)\nposition is observed [see Fig. S1(c)]. Upon application\nof the bias field H, the vortex is distorted due to the\ngrowth of the domain along the field and the correspond-\ning shrinking of the domain with the opposite magne-\ntization direction [see middle image in Fig. S1(b)]. In\nsuch flexed magnetic vortex, the IPUA has a pronounced\neffect on the VC position depending on the mutual ori-\nentation of the HandK. For K∥H(left image in\nFig. S1(b)), the IPUA-driven VC shift is in the direction\nof the magnetic-field-induced torque, whereas for K⊥H\n(right image in Fig. S1(b)), the VC shifts in the opposite\ndirection [see Fig. S1(d)].8\n050100150200\n0.0 0.5 1.0\nmz (arb.u.)y (nm)\n K = 0 kJ/m3\n Kx = 200 kJ/m3\n Ky = 200 kJ/m30Hx = 0 mT\n050100150200\n0.0 0.5 1.0 K = 0 kJ/m3\n Kx = 200 kJ/m3\n Ky = 200 kJ/m3\nmz (arb.u.)y (nm)0Hx = 100 mTupshiftdownshiftµ0H = 0 mT\nKx= 0 kJ/m3µ0H = 0 mT\nKx= 200 kJ/m3µ0H = 0 mT\nKy= 200 kJ/m3\nµ0Hx= 100 mT\nKx= 0 kJ/m3µ0Hx= 100 mT\nKx= 200 kJ/m3µ0Hx= 100 mT\nKy= 200 kJ/m3no shift no shift(a)\n(b)(c)\n(d)xy\nz-11mx\nHxKxKy\nKx Ky\nFIG. S1. Effect of the in-plane uniaxial anisotropy Kon the magnetization distribution in a CoFeB magnetic vortex with\ndiameter d= 200 nm and thickness t= 20 nm. (a) Left: µ0H= 0 mT; Kx= 200 kJ/m3along the xaxis. Middle: µ0H\n= 0 mT; Kx= 0 kJ/m3. Right: µ0H= 0 mT; Ky= 200 kJ/m3along the yaxis. White arrows denote schematically the\nmagnetic moments within the disk. (b) Left: µ0Hx= 100 mT; Kx= 200 kJ/m3along the xaxis. Middle: µ0Hx= 100 mT;\nKx= 0 kJ/m3. Right: µ0Hx= 100 mT; Ky= 200 kJ/m3along the yaxis. (c,d) z-component of the magnetization vs. y\ncoordinate for K= 0 kJ/m3(solid black line); Kx= 200 kJ/m3(dashed red line) and Ky= 200 kJ/m3(dashed blue line) at\nµ0H= 0 mT (c) and µ0Hx= 100 mT (d)."    },    {        "title": "1406.6552v1.Cylindrical_Ising_Nanowire_in_an_Oscillating_Magnetic_Field_and_Dynamic_Compensation_Temperature.pdf",        "content": "1 \n Cylindrical Ising Nanowire in an Oscillating Magnetic Field and Dynamic \nCompensation Temperature  \n \nErsin Kantar  and Mehmet Ertaş1  \nDepartment of Physics, Erciyes University, 38039 Kayseri, Tu rkey \n \nAbstract  The magnetic properties of a nonequilibrium spin -1/2 cylindrical Ising nanowire \nsystem with core/shell in an oscillating magnetic field are studied by using a mean -field \napproach based on the Glauber -type stochastic dynamics (DMFT).  We employ the Glauber -\ntype stochastic dynamics to construct set of the coup led mean -field dynamic equations. First, \nwe study the temperature dependence of the dynamic order parameters  to characterize the  \nnature  of the phase transitions and to obtain the dynamic phase transition points. Then,  we \ninvestigate  the temperature depende nce of the  total magnetization to find the dynamic \ncompensation points as well as to determine the type of behavior.  The phase diagrams  in \nwhich contain the paramagnetic, ferromagnetic, antiferromagnetic, nonmagnetic, surface  \nfundamental phases and tree mixed phases  as well as reentrant behavior  are presented  in the \nreduced magnetic field amplitude and reduced temperature plane . According to values of \nHamiltonian parameters , the compensation  temperatures, or the N -, Q-, P-, R-, S-type \nbehavior s in the Néel classificati on nomenclature exist in the system.  \n \nKeywords:  Cylindrical Ising nanowire system. Dynamic phase transitions.  Dynamic \nCompensation temperature s. Dynamic phase diagrams.  Glauber -type \nstochastic dynamics  \n \n1. Introduction  \nIn recent four years,  the phenomenon of magnetic nanostructures with a fascinating variety of \nmorphologies (nanoparticle, nanotube and nanowire) has  been one of the intensively studied \nsubjects in statistical mechanics and condensed matter physics, because of their potential \ndevice applicatio ns in tech nologically important materials ( see [1-6] and references therein).  \nThese systems can find application in fields such as ensure super high data storage densities, \nsensors, biomedicine and catalysis, among others [7].  \nOn the other hand, many rese archers have used the spin -1/2 Ising system to study \nequilibrium properties of magnetic nanostructured materials  (see [8-14] and references \ntherein) . An early attempt to examine nanoparticles was done by Kaneyoshi [8]. In this study, \nthe phase diagrams of a ferroelectric nanoparticle described with the transverse Ising model \nwere investigated by using the mean -field theory (MFT) and the effective -field theory (EFT).  \nMFT and EFT were used to study the magnetizations [ 9] and phase diagrams [ 10] of a \ntransvers e Ising nanowire and found that the equilibrium behavior of the system is strongly \naffected by the surface situations.  Total susceptibility, susceptibility, compensation \ntemperature and phase diagrams of a cylindrical spin -1/2 Ising nanotube (or nanowire) were \nexamined by utilizing the EFT in detail [11].  The cylindrical nanowire system with a diluted \nsurface described by the transverse spin -1/2 Ising model was investigated by using the EFT \n[12]. Akıncı [13] examined the effects of the randomly distributed magnetic field on the phase \n                                                           \n1 Corresponding author.  E-mail: mehmetertas@erciyes.edu.tr  (Mehmet  Ertaş) \n *Manuscript\nClick here to view linked References2 \n diagrams of a spin -1/2 Ising nanowire with the EFT. Zaim et al. [14] studied the hysteresis \nbehaviors of the nanotube in which consisting of a ferroelectric core of sp in-1/2 surrounded. \nFurthermore,  although a great amount the spin -1/2 Ising systems have used to investigate the \nequilibrium properties of magnetic nanostructured materials, there have been only a few \nworks that the spin -1/2 Ising systems used to investigat e dynamic magnetic properties of \nnanostructured materials  [15-20]. In series of these works,  effective -field theory based on the \nGlauber -type stochastic (DEFT) was used  as method .  \nWe also mention that dynamic compensation temperatures (DCT s) and the dynam ic \nphase transition (DPT) temperatures have been attracted much attention (see [21 -28] and \nreferences therein) in recent years . The existence of compensation temperatures is of great \ntechnological importance, since at this point only a small driving field is required to change \nthe sign of the total magnetization. This property is very useful in thermomagnetic recording, \nelectronic, and computer technologies. On the other hand, experimental realizations for the \nDPT have been discussed in ultrathin ferromagne tic films, superconductors, ferroicsystems, \nultrathin polyethylene naphthalate nanocomposites, Al –Ni systems, etc. (see [ 29–32] and \nreferences therein).  \nAs far as we know , the dynamic  compensation behaviors of the nano system s \n(nanoparticle, nanotube and na nowire)  have been studied only two works  [16, 19] . In two \nworks, the compensation types of the system were found by using DEFT. The dynamic phase \ndiagrams of the system were not investigated. Moreover,  the dynamic behaviors of the nano \nsystem have not been  examined by using a mean -field approach based on the Glauber -type \nstochastic dynamics (DMFT).  Therefore, in present paper, we used to DMFT for study t he \nmagnetic properties of a nonequilibrium spin -1/2 cylindrical Ising nanowire system with \ncore/shell in an oscillating magnetic field . The aim of the present paper is three -fold:  (i) to \nobtain the  DCTs temperatures and DPT of the Ising nanowire  system . (ii) To investigate the \ntype of the compensation behavior of the system . (iii) Finally, to present  the dynam ic phase \ndiagrams of the system  in the plane of temperature versus magnetic field amplitude.  \nThe outline of the rest of the paper is follows. In Sec. 2, the model is presented briefly \nand the derivation of the  mean -field dynamic equations of motion is given  by using  the \nGlauber -type stochastic dynamics in the presence of a n oscillating  magnetic field. In Sec. 3, \nthe numeric results  and discussions  are presented. Finally,  we give the summary and \nconclusion in the last section.  \n \n2. Model and Formulations  \n \nThe c onsidered model is a spin -1/2 Ising nanowire on a cylindrical lattice under the \noscillating magnetic field. The schematic representation of a cylindrical Ising nanowire is \ndepicted in Fig. 1, in which the wire consists of the surface shell and the core. Ea ch site on the \nfigure is occupied by a spin -1/2 Ising particle and each spin is connected to the two nearest -\nneighbor spins on the above and below sections along the cylinder. The Hamiltonian of the \nsystem is given by  \n \n         \nC i i i j j j 1 j k j l\nii ij jj jk jl\nS k k k l l l i j k l\nkk kl ll i j k l=-Jσσ + σS + SS -J Sα + Sλ\n-Jα α + α λ + λλ -H σ + S + α + λ\n\n\n   \n   \n   \n       \n      H   (1) \n 3 \n where the \nSJ  and \nCJ  are the exchange interaction parameters between two nearest -neighbor \nmagnetic particles at the surface sh ell and core, respectively, and J 1 is the interaction \nparameters between two nearest -neighbor magnetic particles at the surface shell and the core \nshell. The surface exchange and interfacial coupling interactions are  often defined as \nS 1SCJ =J\n and \n1Cr=J /J  in the nanosystems [9, 10, 3 3, 34], respe ctively. H is the \noscillating magnetic field: \n0 H(t) = H cos(wt),  with H 0 and w = 2πν being the amplitude and \nthe angular frequency of the oscillating field, respectively. The system is in contact with an \nisothermal heat bath at an absolute tempera ture T A.  \nNow, we apply the Glauber -type stochastic dynamics to obtain the set of the mean -\nfield dynamic equations. Since the derivation of the mean -field dynamic equations was \ndescribed in detail for spin -1/2 system  [35] and different spin systems [ 21-24], in here, we \nshall only give a brief summary. If the \nS,αandλ spins momentarily fixed, the master equation \nfor - spins can be written as  \n \n  \n 1 2 N i i 1 2 i N\ni\ni i 1 2 i N\nidP , ,..., ;t W P , ,..., ,... ;tdt\nW P , ,..., ,... ;t ,        \n     \n\n  (2) \n \nwhere \niiW is the probability per unit time tha t ith  spin changes from i  to  -i. Since the \nsystem is in contact with a heat bath at absolute temperature TA, each spin  can flip with the \nprobability per unit time by the Boltzmann factor;  \n \n\n\nii\nii\niexp E1W,exp E\n   \n   (3) \n \nwhere =1/k BTA, kB  is the Bolt zmann constant, the sum ranges the two possible values ±1/2 \nfor \ni  and \n \nii σσ C i σS C j\nijE( )= 2 (H z J z J S )\n      \n,   (4) \n \ngives the change in the energy of the system when the \ni -spin changes.  The probabilities \nsatisfy the deta iled balance condition.  Using Eqs. (2), (3), (4) with the mean -field approach, \nwe obtain the mean -field dynamic equation for the -spins as  \n \n   C1 C1 σσ C C1 σS C C2 0d1Ω m =-m + tanh β z J m +z J m +H cos ξ ,dξ2\n  (5) \n \nwhere \nC1m , \nC2mS , \nwt and \n = \nw = w/f, w is the frequency of the oscillating \nmagnetic field and f represents the frequency of spin flipping. Moreover, \nσσz  and \nσSz  \ncorresponds to the number o f nearest -neighbor pairs of spins   and S, respectively, in \nwhich \nσσz2  and \nσSz6 . 4 \n  \nAs similar to -spins, we obtain the mean -field dynamical equations for the \nS,αandλ -spins \nby using the si milar calculations. The mean -field dynamic equations for \nS,αandλ -spins are \nobtained as  \n \n   C2 C2 S σ C C1 SS C C2 Sα 1 S1 sλ 1 S2 0d1Ω m =-m + tanh β z J m +z J m +z Jm +z Jm +H cos ξ ,dξ2\n (6) \n \n   S1 S1 αα S S1 αλ S S2 αS 1 C2 0d1Ω m =-m + tanh β z J m +z J m +z J m +H cos ξ ,dξ2\n  (7) \n \n   S2 S2 λλ S S2 λα S S1 λS 1 C2 0d1Ω m =-m + tanh β z J m +z J m +z J m +H cos ξ ,dξ2\n  (8) \n \nwhere \nS1m , \nS2mλ , \nSσ Sα αSz z z 1   , \nSSz4 , \nsλ αα αλ λλ λ λSz z z z z z 2       . \nHence, a set of mean -field dynamical equations of the system are obtained. We fixed \nCJ1  \nthat the core shell interaction is ferromagnetic and \n2  .  \n \nWe should also mention that the  dynamic compensation temperature, which dynamic total \nmagnetization (\ntM ) vanishes at the compensation temperature T Comp. The compensation point \ncan then be determined by looki ng for the crossing point between the absolute values of the \nsurface and the core magnetizations. Therefore, at the compensation point, we must have  \n \n Surface Comp Core CompM T = M T ,\n      (9) \nand \n   \n  Surface Comp Core Comp sgn M T =-sgn M T .             (10) \n \nWe also require that T Com p < T C, where T C is the critical point temperature. In the next section \nwe will give the numerical results of these dynamic equations.   \n \n3. Numerical Results and Discussions  \nIn this section, we solved first the Eqs. ( 5)-(8) to find the phases in the system . These \nequations were solved by using the numerical method of the Adams -Moulton predictor \ncorrector method and we found that a paramagnetic  (P), ferromagnetic (F), antiferromagnetic \n(AF), nonmagnetic  (NM), surface phase (SF)  fundamental phases and four mixed phases, \nnamely the F + P in which F, P phases coexist , the F + AF in which F, AF phases coexist , the \nAF + P in which AF, P phases coexist , and the NM+P  in which NM, P phases  coexist, exist in \nthe system. Since we gave the solution of these kinds of dyn amic equations i n Ref. 22 -24, 27  \nin detail, we will not discuss the solutions and present any figures here.  \nThen, we investigate the behavior of the dynamic  core and shell  magnetizations  (MC1, \nMC2, M S1, MS2) as a function of the temperature. This  investiga tion leads us to characterize the \nnature (continuous or  discontinuous) of phase transitions and find the DPT points. \nFurthermore , we study the dynamic total magnetization as a function of the  temperature to 5 \n find the dynamic compensation temperatures and  to determine the type of behavio r. The \ndynamic shell and core magnetizations and the dynamic total magnetizations \n   t C1 C2 S1 S2M m 6m 6 m m /19   \n as the time -averaged magnetization over a period of \nthe oscillating magnetic field are given as  \n \n2\nC1,C2, S1,S2 C1,C2,S1,S2\n01M m ( )d ,2\n  \n                  (11a) \nand \n               \n 2\nC1 C2 S1 S2\nt\n0m ( ) 6m ( ) 6 m ( ) m ( ) 1M d .2 19                 (11b) \n \nThe behaviors of MC1, MC2, M S1, MS2 and Mt as functions of the temperature for several values \nof interaction parameters are obtained by  solving Eqs. 11(a) and (b). In order to show  the \ncalculation of the DPT points an d the compensation temperatures,  the three  explanatory and \ninteresting examples are plotted in Figs. 2(a)-(c). In these figures, T C and T t are the second -\norder and first -order phase transition tempera tures, respectively. T Comp is the compensation \ntemperature.  Fig. 2(a) shows the behavior of MC1, MC2, M S1, MS2 and M t as functions of the \ntemperature for\nr 0.5= ,\n0.6S=  and H0 = 0.1 . MC1, C2 = 0.5 and MS1, S2 = -0.5 are at the \nzero temperature. The dynamic magnetizations  MC1, M C2 decrease and MS1, MS2 increase \ncontinuously with the increasing of the values of temperature below the critical  temperature \nand they become zero at  TC = 2.95; hence, a second -order phase transi tion occurs. The \ntransition is from the AF phase to the P phase.  Moreover, one compensation temperature or \nN-type behavior occurs in the system that exhibits the same behavior classified after the Néel \ntheory [ 36] as the N -type behavior [ 37]. In Fig. 2( b), MC1, C2, S1, S2 = 0.5 at the zero \ntemperature; thus we own the F phase; as the  temperature increases , all of them decrease to \nzero continuously and the system undergoes a  second -order phase transition from the  F phase \nto the P phase at T C = 3.44. In Fig. 2(c), at zero value of temperature, MC1, C2 = 0.5  and MS1, S2 \n= 0.0; thus we have the  NM phase; as the  temperature increases, MC1, C2 of them decrease to \nzero discontinuously and the system undergoes a  first-order phase transition from the NM \nphase to the P phase at T t = 0.31. \nOn the other hand, one of the typical dynamic  behaviors in such a system is to show a \ncompensation point below its transition temperature. We studied dynamic compensation \nbehaviors of the cylindrical Ising nanowire system, which we kn ow that a compensation \ntemperature can be found in the  nanostructure systems [ 16, 19 ]. Fig . 3 shows the temperature \ndepen dencies of the total magnetization Mt for H=0.1 and several values of  r, \nS . As seen \nfrom Fig. 3 (a),  the curve  labeled \nr= -1.0  and \n0.5S=  may show the N-type behavior. \nMoreover, the Q-type behavior is  obtained  in Fig. 3 (b) for  \nr= -0.5  and \n0.5S= . Fig. 3 (c) is  \ncalculated for  \nr= -0.1  and \n1.0S= . As we can see this  curve illustrates the P-type behavior.  \nFig. 3 (d) is calculated for  \nr= 1.0  and \n0.75S=  and illustrates the  R-type behavior.  For \nr= 1.0\n and\n0.99S= , Fig. 3(e) is obtained and exhibit the S-type behavior . These  obtained \nresults are consistent with same behavior as that classified in  the Neél theory [ 36]. \n Since we have obtained DPT points and compensation temperatures, we can now \npresent the dynamic phase diagrams of the system. The calculated phase diagrams  for both 6 \n antiferromagnetic case and ferromagnetic case  in the ( T, h) plane is presented in Fig. 4 and in \nFig. 5, respectively. I n these dynamic phase diagrams, the s olid, dashed and dash -dot-dot lines \nrepresent the second -order, first -order phase transitions temperatures and the compensation \ntemperatures, respectively. The dynamic tricritical point is denoted by a filled circle.  \nMoreover, z , tp, and qp are the dynamic  zero temperature, triple and quadruple points, \nrespectively, that strongly depend on the values of interaction parameters .  \n Fig. 4 shows the dynamic phase diagrams including  the compensation behaviors for \nantiferromagnetic case in (T, h) plane and four m ain topological different types of phase \ndiagrams are seen. From these phase diagrams, the following phenomena have been  observed.  \n(i) Figs. 4(a) -(c) include  the compensation  temperatures, but Fig. 4(g) does not. (ii) The \ndynamic phase diagrams contain the  F, AF, NM, SF fundamental phases as well as the F + \nAF, AF + P, NM + P mixed phases. (iii) The system exhibits the reentrant behavior, i.e., with \nthe temperature increase, the system passes from the P phase to the AF phase, and then back \nto the P phase ag ain in seen Fig. 4(c).  It should also mention onig that several weakly \nfrustrated ferromagnets, such as in manganite LaSr 2Mn 2O7 by electron and  x-ray diffraction, \nin the bulk bicrystals of the oxide superconductor BaPb 1-xBixO3 and Eu xSr1-xS and \namorphous -Fe1-xMn x, demonstrate the reentrant phenomena [3 7-40]. (iv) The system shows \nqp, where the four first -order transition lines meet, is shown in Fig. 4(c). tp, at which three \nfirst-order  transition lines meet  in seen Figs. 4(c) and 4(d) . z, in which is a critica l point  \ncharacterized by fl uctuations at zero -temperature, is shown Fig. 4(d).  \n Fig. 5 exhibits the dynamic phase diagrams for ferromagnetic case in (T, h) plane and \nthree main  topological different types of phase diagrams are seen. From these phase diagram s, \nthe following phenomena have been  observed.  (i) These  dynamic phase diagrams do not \ncontain the compensation tem perature.  (ii) The phase diagrams show  one or two dynamic \ntricritical point s. (iii) The system contains the P, F, NM, SF fundamental phases a s well as the \nF + P, NM + P mixed phases. (iv) The system shows re -entrant behavior, seen in Fig. 5(b). (v) \nThe dynamic phase  boundaries among the fundamental phases are mostly second -order  phase \ntransition lines but between the fundamental and mixed  phase s are first -order phase transition \nlines.  \n \n4. Sum mary and Conclusion  \n \nIn this work, we have studied the magnetic properties of a nonequilibrium spin -1/2 cylindrical \nIsing nanowire system with core/shell in an oscillating magnetic field by using a mean -field \napproach based on the Glauber -type stochastic dynamics (DMFT). We employ the Glauber -type \nstochastic dynamics to construct set of the coupled mean -field dynamic equations.  First, we \nstudy the temperature dependence of the dynamic order parameters to chara cterize the nature of \nthe phase transitions and to obtain the dynamic phase transition points. Then, we investigate the \ntemperature dependence of the total magnetization to find the dynamic compensation points as \nwell as to determine the type of behavior. T he phase diagrams in which contain the \nparamagnetic, ferromagnetic, antiferromagnetic, nonmagnetic, surface fundamental phases and \ntree mixed phases as well as reentrant behavior are presented in the reduced magnetic field \namplitude and reduced temperature  plane. The phase diagrams also include one or two dynamic \ntricritical points as well as the tp, qp, z special points. According to values of Hamiltonian \nparameters, the system the compensation temperatures, or the N -, P-, Q-, S-, R-type behaviors in \nthe N éel classification nomenclature exist in the system.  \nWe should also mention that the findings of this study have important for researchers in \nstatistical mechanics and condensed matter physics. 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Phys. 139, 333 (2010 ). \n[24] Keskin, M., Ertaş, M., Canko, O.:  Dynamic phase transitions and dynamic phase \ndiagrams in the kinetic mixed spin -1 and spin -2 Ising system in an oscillati ng magnetic \nfield. Physica Scr. 79, 025501 (2009) . \n[25] Vatansever, E., Aktaş, B.O., Yüksel, Y., Polat, H.: Stationary state solutions of a bond \ndiluted kinetic  Ising model: an effective -field theory analysis . J. Stat. Phys.  147, 1068 \n(2012) . \n[26]  Korkmaz,  T., Temizer, Ü.: Dynamic compensation temperature in the mixed spin -1 and \nspin-2 Ising model in an oscillating field on alternate layers of a hexagonal lattice . J. \nMagn. Magn. Mater. 324, 3876  (2012).  \n[27] Ertaş, M., Keskin, M.: Dynamic magnetic behavior of the mixed -spin bilayer system in \nan oscillating  field within the mean -field theory . Phys. Lett. A 376, 2455 (2012) . \n[28] Keskin, M ., Temizer, Ü ., Canko, O ., Kantar, E. : Dynamic phase transit ion in the kinetic \nBlume -Emery -Griffiths model: Phase diagrams in  the temperature and interaction \nparameters planes . Phase Trans . 80, 855 (2007).  \n[29] Samoilenko , Z.A., Okunev , V.D., Pushenko , EI, Isaev , V.A., Gierlowski , P., Kolwas , K., \nLewandowski , S.J. Dynamic phase transition in amorphous YBaCuO films under Ar \nlaser irradiation. Inorg  Mater. 39, 836 (2012 ). \n[30] Kleemann , W., Braun , T., Dec , J., Petracic , O.:  Dynamic phase transitions in ferroic \nsystems with  pinned domain walls. Phase Trans.  78, 811 2005.  \n[31] Gedik , N., Yang , D.S., Logvenov , G., Bozovic , I., Zewail , A. H.: Nonequilibrium phase \ntransitions in  cuprates observed by ultrafast electr on crystallography. Science 20, 425 \n(2007).  \n[32] Robb , DT., Xu, Y.H., Hellwing , O., McCord , J., Berger , A., Novotny , M.A., Rikvold , \nPA.: Evidence for a dynamic phase transition  in [Co/Pt] 3 magnetic multilayers. Phys . \nRev. B 78, 134422  (2008).  \n[33] Kantar , E., Kocakaplan , Y.: Hexagonal type Ising nanowire with core/shell structure: The \nphase diagrams and compensation behaviors . Solid State Commun . 177, 1 (2014 ). \n[34] Kantar , E, D eviren , B, Keskin , M.: Magnetic properties of mixed Ising nanoparticles \nwith core -shell structure. Eur . Phys . J B 86, 6 (2013 ). \n[35] Tomé , T., de Oliveira , M. J.: Dynamic phase transition in the kinetic Ising model under a \ntime-dependent oscillating field . Phys. Rev. A 41, 4251 (1990 ). 9 \n [36] Néel, L.: Magnetic properties of ferrites: Ferrimagnetism and antiferromagnetism. Ann . \nPhys. 3, 137 (1948) . \n[37] Chikazumi, S.: Physics of Ferromagnetism, Oxford University Press, Oxford, (1997) . \n[38] Li, J.Q. , Matsui, Y., Kimura, T., Tokura, Y.: Structural properties and charge -ordering \ntransition in LaSr 2Mn 2O7. Phys. Rev. B 57, R3205 (1998).  \n[39] Sata, T., Yamaguchi, T., Matsusaki , K.:  Interaction between anionic polyelectrolytes and \nanion exchange membranes and change  in membrane properties . J. Membr. Sci. 100, 229 \n(1995).  \n[40] Binder , K., Young , A.P.:  Spin glasses: Experimental facts, theoretical concepts, and open \nquestions . Rev. Modern Phys. 58, 801 (1986).  \n \nList of the figure captions  \nFig. 1. (color online)  Schemat ic representations of a cylindrical nanowire: (a) cross -section and \n(b) three -dimensional. The gray and blue circles indicate spin -1/2 Ising particles at the surface \nshell and core, respectively. (For interpretation of the references to color in this figure  legend, the \nreader is referred to the web version of this article ). \nFig. 2. (color online) The temperature dependence of the dynamic core and shell \nmagnetizations  and total magnetizations. TC and T t are the second -order and first -order phase \ntransition te mperatures, respectively. T comp is the compensation temperature.  Dash-dot-dot \nlines represent the compensation temperatures,  \n \n(a) Exhibiting a second -order phase transition from the  AF phase to P phase at T C = \n2.95 for \nr 0.5= ,\n0.6S=  and H0 = 0.1 . The system shows t he N -type \ncompensation behavior.  \n(b) Exhibiting a second -order phase transition from the  F phase to P phase at T C = \n3.44 for \nr 1.0= ,\n0.0S=  and H0 = 0.1 . \n(c) Exhibiting a first -order phase transition from the NM phase to P phase at T t = 0.31 \nfor \nr 0.5= ,\n0.6S=  and H0 = 2.88.  \n \nFig. 3. The dynamic total magnetization as a function of the temperature for different values \nof interaction parameters.  The system exhibits the N -, Q-, P-, R-, S- type behaviors  of \ncompensation behaviors.  (a) \nr= -1.0  and \n0.5S= ; (b) \nr= -0.5  and \n0.5S= ; (c) \nr= -0.1  \nand \n1.0S= ; (d) \nr= 1.0  and \n0.75S= ; (e) \nr= 1.0  and\n0.99S= . \n \nFig. 4. The dynamic phase diagrams for antiferroagnetic case. Dashed and solid lines are the \ndynamic first - and second -order phase boundaries, respectively. The dash -dot-dot line \nillustrates the  compensation  temperatures. The dynamic tricritical points are indicated with \nfilled circles. (a) \nr 0.5=  and \n0.6S= ; (b) \nr 1.0=  and \n0.0S= ; (c) \nr 0.05=  and \n0.0S=\n; (d) \nr 0.2=  and \n5.0S= . \n \nFig. 5. Same as Fig 4, but  (a) \nr 0.2=  and \n5.0S= ; (b) \nr 1.0=  and \n3.0S= ; (c) \nr 1.0=  and \n0.0S=\n. \n mS1\nmS2\nmC2\nmC1Js J1\nJCJsmS1\nJC mC2 \n(a) \n \n(b) \n \nFig. 1  Fig. 1T0.0 0.5 1.0 1.5 2.0 2.5 3.0MC1, MC2, MS1, MS2, Mt\n-0.50-0.250.000.250.50\nMC1\nMC2\nMS1MS2TCTComp( a )\nT0 1 2 3 4MC1, MC2, MS1, MS2, Mt\n0.000.250.50 ( b )\nTCMS1MC2\nMC1, Mt\nMS2\nT0.0 0.2 0.4 0.6MC1, MC2, MS1, MS2, Mt\n0.000.250.50\nMC2MC1\nTtMS1 = MS2 MC1=MC2=MS1 = MS2 = Mt Mt\nMt( c ) \nFig. 2 Fig. 2N-Type\n0 1 2 3 4Mt\n0.000.050.100.15\nQ-Type\n0 1 2 3 40.000.050.100.15\nP-Type\n0 1 2 3 4 5Mt\n0.00.10.2\nR-Type\n0 1 2 3 40.00.20.40.6\nS-Type\nT0 1 2 3 4Mt\n0.00.20.40.6(a) (b)\n(c) (d)\n(e) \n              Fig. 3 \n Fig. 30.0 0.5 1.0 1.5 2.0 2.5 3.0h\n0.00.51.01.52.02.5\nAFAFAF + PP\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5h\n0.00.51.01.52.02.53.0\nAF\nAFP\nAF + Ptp( a ) ( b )\n0.0 0.5 1.0 1.5 2.0 2.5 3.0h\n0.00.51.01.52.02.5\nFAFP\nNM\nF+AFAFreentrant\n0.0 0.5 1.0 1.5 2.0 2.5 3.0h\n0.00.51.01.52.02.5\nSFNMP\nztpNM +P( c ) ( d )T T\nT Ttpqp \n \n     Fig. 4 \n Fig. 40.0 0.5 1.0 1.5 2.0 2.5 3.0h\n0.00.51.01.52.02.5\nNMP\nSFNM +P\nz\n0 2 4 6 8h\n0.01.02.03.04.05.06.0\nFNMPreentrant\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5h\n0.00.51.01.52.02.53.0\nFF + PP( a )\n( b )\n( c )\nTTT \n \nFig. 5 Fig. 5"    },    {        "title": "1104.1625v1.Magnetization_Dissipation_in_Ferromagnets_from_Scattering_Theory.pdf",        "content": "arXiv:1104.1625v1  [cond-mat.mes-hall]  8 Apr 2011Magnetization Dissipation in Ferromagnets from Scatterin g Theory\nArne Brataas∗\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\nYaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\nGerrit E. W. Bauer\nInstitute for Materials Research, Tohoku University, Send ai 980-8577, Japan and\nKavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands\nThe magnetization dynamicsofferromagnets are often formu lated intermsof theLandau-Lifshitz-\nGilbert (LLG) equation. The reactive part of this equation d escribes the response of the magnetiza-\ntion in terms of effective fields, whereas the dissipative par t is parameterized by the Gilbert damping\ntensor. We formulate a scattering theory for the magnetizat ion dynamics and map this description\non the linearized LLG equation by attaching electric contac ts to the ferromagnet. The reactive part\ncan then be expressed in terms of the static scattering matri x. The dissipative contribution to the\nlow-frequency magnetization dynamics can be described as a n adiabatic energy pumping process\nto the electronic subsystem by the time-dependent magnetiz ation. The Gilbert damping tensor\ndepends on the time derivative of the scattering matrix as a f unction of the magnetization direction.\nBy the fluctuation-dissipation theorem, the fluctuations of the effective fields can also be formulated\nin terms of the quasistatic scattering matrix. The theory is formulated for general magnetization\ntextures and worked out for monodomain precessions and doma in wall motions. We prove that the\nGilbert damping from scattering theory is identical to the r esult obtained by the Kubo formalism.\nPACS numbers: 75.40.Gb,76.60.Es,72.25.Mk\nI. INTRODUCTION\nFerromagnets develop a spontaneous magnetization\nbelow the Curie temperature. The long-wavelengthmod-\nulations of the magnetization direction consist of spin\nwaves, the low-lying elementary excitations (Goldstone\nmodes) of the ordered state. When the thermal energy is\nmuch smaller than the microscopic exchange energy, the\nmagnetization dynamics can be phenomenologically ex-\npressed in a generalized Landau-Lifshitz-Gilbert (LLG)\nform:\n˙ m(r,t) =−γm(r,t)×[Heff(r,t)+h(r,t)]+\nm(r,t)×/integraldisplay\ndr′[˜α[m](r,r′)˙ m(r′,t)],(1)\nwhere the magnetization texture is described by m(r,t),\nthe unit vector along the magnetization direction at po-\nsitionrand timet,˙ m(r,t) =∂m(r,t)/∂t,γ=gµB//planckover2pi1is\nthe gyromagnetic ratio in terms of the g-factor (≈2 for\nfree electrons) and the Bohr magneton µB. The Gilbert\ndamping ˜αis a nonlocal symmetric 3 ×3 tensor that is\na functional of m. The Gilbert damping tensor is com-\nmonly approximated to be diagonal and isotropic (i), lo-\ncal (l), and independent of the magnetization m, with\ndiagonal elements\nαil(r,r′) =αδ(r−r′). (2)\nThe linearized version of the LLG equation for small-\namplitude excitations has been derived microscopically.1It has been used very successfully to describe the mea-\nsured response of ferromagnetic bulk materials and thin\nfilms in terms of a small number of adjustable, material-\nspecific parameters. The experiment of choice is fer-\nromagnetic resonance (FMR), which probes the small-\namplitude coherent precession of the magnet.2The\nGilbertdampingmodelinthelocalandtime-independent\napproximationhasimportantramifications, suchasalin-\near dependence of the FMR line width on resonance fre-\nquency, that have been frequently found to be correct.\nThe damping constant is technologically important since\nit governs the switching rate of ferromagnets driven by\nexternal magnetic fields or electric currents.3In spatially\ndependent magnetization textures, the nonlocal charac-\nter of the damping can be significant as well.4–6Moti-\nvated by the belief that the Gilbert damping constant is\nanimportantmaterialproperty, weset outheretounder-\nstand its physical origins from first principles. We focus\non the well studied and technologically important itiner-\nant ferromagnets, although the formalism can be used in\nprinciple for any magnetic system.\nThe reactive dynamics within the LLG Eq. (1) is de-\nscribed by the thermodynamic potential Ω[ M] as a func-\ntional of the magnetization. The effective magnetic field\nHeff[M](r)≡ −δΩ/δM(r) is the functional derivative\nwith respect to the local magnetization M(r) =Msm(r),\nincluding the external magnetic field Hext, the magnetic\ndipolar field Hd, the texture-dependent exchange energy,\nand crystal field anisotropies. Msis the saturation mag-\nnetization density. Thermal fluctuations can be included\nby a stochastic magnetic field h(r,t) with zero time av-2\nleft\nreservoirF N Nright\nreservoir\nFIG. 1: Schematic picture of a ferromagnet (F) in contact\nwith a thermal bath (reservoirs) via metallic normal metal\nleads (N).\nerage,/an}b∇acketle{th/an}b∇acket∇i}ht= 0, and white-noise correlation:7\n/an}b∇acketle{thi(r,t)hj(r′,t′)/an}b∇acket∇i}ht=2kBT\nγMs˜αij[m](r,r′)δ(t−t′),(3)\nwhereMsis the magnetization, iandjare the Cartesian\nindices, and Tis the temperature. This relation is a con-\nsequence ofthe fluctuation-dissipation theorem (FDT) in\nthe classical (Maxwell-Boltzmann) limit.\nThe scattering ( S-) matrix is defined in the space of\nthe transport channels that connect a scattering region\n(the sample) to real or fictitious thermodynamic (left\nand right) reservoirs by electric contacts with leads that\nare modeled as ideal wave guides. Scattering matri-\nces are known to describe transport properties, such as\nthe giant magnetoresistance, spin pumping, and current-\ninducedmagnetizationdynamicsinlayerednormal-metal\n(N)|ferromagnet (F).8–10When the ferromagnet is part\nof an open system as in Fig. 1, also Ω can be expressed\nin terms of the scattering matrix, which has been used\nto express the non-local exchange coupling between fer-\nromagnetic layers through conducting spacers.11We will\nshow here that the scattering matrix description of the\neffective magnetic fields is valid even when the system is\nclosed, provided the dominant contribution comes from\nthe electronic band structure, scattering potential disor-\nder, and spin-orbit interaction.\nScattering theory can also be used to compute the\nGilbert damping tensor ˜ αfor magnetization dynamics.15\nThe energy loss rate of the scattering region can be ex-\npressedin termsofthe time-dependent S-matrix. To this\nend, the theory of adiabatic quantum pumping has to be\ngeneralizedtodescribedissipationinametallicferromag-\nnet. The Gilbert damping tensor is found by evaluating\nthe energy pumping out of the ferromagnet and relat-\ning it to the energy loss that is dictated by the LLG\nequation. In this way, it is proven that the Gilbert phe-\nnomenology is valid beyond the linear response regime\nof small magnetization amplitudes. The key approxima-\ntion that is necessary to derive Eq. (1) including ˜ αis the\n(adiabatic) assumption that the ferromagnetic resonance\nfrequencyωFMRthat characterizesthe magnetizationdy-\nnamics is small compared to internal energy scale set by\nthe exchange splitting ∆ and spin-flip relaxation rates\nτs. The LLG phenomenology works well for ferromag-\nnets for which ωFMR≪∆//planckover2pi1, which is certainly the case\nfor transition metal ferromagnets such as Fe and Co.\nGilbert damping in transition-metal ferromagnets is\ngenerally believed to stem from the transfer of energy\nfromthemagneticorderparametertotheitinerantquasi-particle continuum. This requires either magnetic disor-\nder or spin-orbit interactions in combination with impu-\nrity/phonon scattering.2Since the heat capacitance of\nthe ferromagnet is dominated by the lattice, the energy\ntransferred to the quasiparticles will be dissipated to the\nlattice as heat. Here we focus on the limit in which elas-\ntic scattering dominates, such that the details of the heat\ntransfer to the lattice does not affect our results. Our ap-\nproachformallybreaks down in sufficiently clean samples\nat high temperatures in which inelastic electron-phonon\nscattering dominates. Nevertheless, quantitative insight\ncan be gained by our method even in that limit by mod-\nelling phonons by frozen deformations.12\nIn the present formulation, the heat generated by the\nmagnetization dynamics can escape only via the contacts\nto the electronic reservoirs. By computing this heat cur-\nrent through the contacts we access the total dissipa-\ntion rate. Part of the heat and spin current that es-\ncapes the sample is due to spin pumping that causes\nenergy and momentum loss even for otherwise dissipa-\ntion less magnetization dynamics. This process is now\nwellunderstood.10For sufficiently largesamples, the spin\npumping contribution is overwhelmed by the dissipation\nin the bulk of the ferromagnet. Both contributions can\nbe separated by studying the heat generation as a func-\ntion of the length of a wire. In principle, a voltage can be\nadded to study dissipation in the presence of electric cur-\nrents as in 13,14, but we concentrate here on a common\nand constant chemical potential in both reservoirs.\nAlthough it is not a necessity, results can be simpli-\nfied by expanding the S-matrix to lowest order in the\namplitude of the magnetization dynamics. In this limit\nscattering theory and the Kubo linear response formal-\nism for the dissipation can be directly compared. We\nwill demonstrate explicitly that both approaches lead to\nidentical results, which increases our confidence in our\nmethod. The coupling to the reservoirs of large samples\nis identified to play the same role as the infinitesimals in\nthe Kubo approach that guarantee causality.\nOur formalism was introduced first in Ref. 15 lim-\nited to the macrospin model and zero temperature. An\nextension to the friction associatedwith domain wall mo-\ntion was given in Ref. 13. Here we show how to handle\ngeneral magnetization textures and finite temperatures.\nFurthermore, we offer an alternative route to derive the\nGilbert damping in terms of the scattering matrix from\nthe thermal fluctuations of the effective field. We also\nexplain in more detail the relation of the present theory\nto spin and charge pumping by magnetization textures.\nOur paper is organized in the following way. In Sec-\ntion II, we introduce our microscopic model for the fer-\nromagnet. In Section III, dissipation in the Landau-\nLifshitz-Gilbert equation is exposed. The scattering the-\nory of magnetization dynamics is developed in Sec. IV.\nWe discuss the Kubo formalism for the time-dependent\nmagnetizationsin Sec. V, before concluding our article in\nSec. VI. The Appendices provide technical derivations of\nspin, charge, and energy pumping in terms of the scat-3\ntering matrix of the system.\nII. MODEL\nOur approach rests on density-functional theory\n(DFT), which is widely and successfully used to describe\nthe electronic structure and magnetism in many fer-\nromagnets, including transition-metal ferromagnets and\nferromagnetic semiconductors.16In the Kohn-Sham im-\nplementation of DFT, noninteracting hypothetical par-\nticles experience an effective exchange-correlationpoten-\ntial that leads to the same ground-statedensity as the in-\nteractingmany-electronsystem.17Asimpleyetsuccessful\nscheme is the local-densityapproximationto the effective\npotential. DFT theory can also handle time-dependent\nphenomena. We adopt here the adiabatic local-density\napproximation (ALDA), i.e. an exchange-correlationpo-\ntential that is time-dependent, but local in time and\nspace.18,19As the name expresses, the ALDA is valid\nwhen the parametric time-dependence of the problem is\nadiabatic with respect to the electron time constants.\nHere we consider a magnetization direction that varies\nslowly in both space and time. The ALDA should be\nsuited to treat magnetization dynamics, since the typical\ntime scale ( tFMR∼1/(10 GHz) ∼10−10s) is long com-\nparedtothethat associatedwith theFermi andexchange\nenergies, 1 −10 eV leading to /planckover2pi1/∆∼10−13s in transition\nmetal ferromagnets.\nIn the ALDA, the system is described by the time-\ndependent effective Schr¨ odinger equation\nˆHALDAΨ(r,t) =i/planckover2pi1∂\n∂tΨ(r,t), (4)\nwhere Ψ( r,t) is the quasiparticle wave function at posi-\ntionrand timet. We consider a generic mean-field elec-\ntronic Hamiltonian that depends on the magnetization\ndirection ˆHALDA[m] and includes the periodic Hartree,\nexchange and correlation potentials and relativistic cor-\nrectionssuchasthe spin-orbitinteraction. Impurityscat-\ntering including magnetic disorder is also represented by\nˆHALDA.The magnetization mis allowed to vary in time\nand space. The total Hamiltonian depends additionally\non the Zeeman energy of the magnetization in external\nHextand dipolar Hdmagnetic fields:\nˆH=ˆHALDA[m]−Ms/integraldisplay\ndrm·(Hext+Hd).(5)\nFor this general Hamiltonian (5), our task is to de-\nduce an expression for the Gilbert damping tensor ˜ α. To\nthis end, from the form of the Landau-Lifshitz-Gilbert\nequation (3), it is clear that we should seek an expansionin terms of the slow variations of the magnetizations in\ntime. Such an expansion is valid provided the adiabatic\nmagnetization precession frequency is much less than the\nexchange splitting ∆ or the spin-orbit energy which gov-\nerns spin relaxation of electrons. We discuss first dissi-\npation in the LLG equation and subsequently compare\nit with the expressions from scattering theory of electron\ntransport. This leads to a recipe to describe dissipation\nby first principles. Finally, we discuss the connection to\nthe Kubo linear response formalism and prove that the\ntwo formulations are identical in linear response.\nIII. DISSIPATION AND\nLANDAU-LIFSHITZ-GILBERT EQUATION\nThe energy dissipation can be obtained from the solu-\ntion of the LLG Eq. (1) as\n˙E=−Ms/integraldisplay\ndr[˙ m(r,t)·Heff(r,t)] (6)\n=−Ms\nγ/integraldisplay\ndr/integraldisplay\ndr′˙ m(r)·˜α[m](r,r′)·˙ m(r′).(7)\nThescatteringtheoryofmagnetizationdissipationcanbe\nformulated for arbitrary spatiotemporal magnetization\ntextures. Much insight can be gained for certain special\ncases. In small particles or high magnetic fields the col-\nlective magnetization motion is approximately constant\nin space and the “macrospin” model is valid in which\nall spatial dependences are disregarded. We will also\nconsider special magnetization textures with a dynamics\ncharacterized by a number of dynamic (soft) collective\ncoordinates ξa(t) counted by a:20,21\nm(r,t) =mst(r;{ξa(t)}), (8)\nwheremstis the profile at t→ −∞.This representation\nhas proven to be very effective in handling magnetiza-\ntion dynamics of domain walls in ferromagnetic wires.\nThe description is approximate, but (for few variables)\nit becomes exact in special limits, such as a transverse\ndomain wall in wires below the Walker breakdown (see\nbelow); it becomes arbitrarily accurate by increasing the\nnumber of collective variables. The energy dissipation to\nlowest (quadratic) order in the rate of change ˙ξaof the\ncollective coordinates is\n˙E=−/summationdisplay\nab˜Γab˙ξa˙ξb, (9)\nThe (symmetric) dissipation tensor ˜Γabreads4\n˜Γab=Ms\nγ/integraldisplay\ndr/integraldisplay\ndr′∂mst(r)\n∂ξaα[m](r,r′)·∂mst(r′)\n∂ξb. (10)\nThe equation of motion of the collective coordinates un-\nder a force\nF=−∂Ω\n∂ξ(11)\nare20,21\n˜η˙ξ+[F+f(t)]−˜Γ˙ξ= 0, (12)\nintroducing the antisymmetric and time-independent gy-\nrotropic tensor:\n˜ηab=Ms\nγ/integraldisplay\ndrmst(r)·/bracketleftbigg∂mst(r)\n∂ξa×∂mst(r)\n∂ξb/bracketrightbigg\n.(13)\nWe show below that Fand˜Γ can be expressed in terms\nof the scattering matrix. For our subsequent discussions\nit is necessary to include a fluctuating force f(t) (with\n/an}b∇acketle{tf(t)/an}b∇acket∇i}ht= 0),which has not been considered in Refs. 20,21.\nFrom Eq. (3) if follows the time correlation of fis white\nand obeys the fluctuation-dissipation theorem:\n/an}b∇acketle{tfa(t)fb(t′)/an}b∇acket∇i}ht= 2kBT˜Γabδ(t−t′). (14)\nIn the following we illustrate the collective coordinate\ndescription of magnetization textures for the macrospin\nmodel and the Walker model for a transverse domain\nwall. The treatment is easily extended to other rigid\ntextures such as magnetic vortices.\nA. Macrospin excitations\nWhen high magnetic fields are applied or when the\nsystem dimensions are small the exchange stiffness dom-\ninates. In both limits the magnetization direction and\nits low energy excitations lie on the unit sphere and its\nmagnetization dynamics is described by the polar angles\nθ(t) andϕ(t):\nm= (sinθcosϕ,sinθsinϕ,cosθ).(15)\nThe diagonal components of the gyrotropic tensor vanish\nby (anti)symmetry ˜ ηθθ= 0, ˜ηϕϕ= 0.Its off-diagonal\ncomponents are\nηθϕ=MsV\nγsinθ=−ηϕθ. (16)\nVis the particle volume and MsVthe total magnetic\nmoment. We now have two coupled equations of motion\nMsV\nγ˙ϕsinθ−∂Ω\n∂θ−/parenleftBig\n˜Γθθ˙θ+˜Γθϕ˙ϕ/parenrightBig\n= 0,(17)\n−MsV\nγ˙θsinθ−∂Ω\n∂ϕ−/parenleftBig\n˜Γϕθ˙θ+˜Γϕϕ˙ϕ/parenrightBig\n= 0.The thermodynamic potential Ω determines the ballistic\ntrajectories of the magnetization. The Gilbert damping\ntensor˜Γabwill be computed below, but when isotropic\nand local,\n˜Γ =˜1δ(r−r′)Msα/γ, (18)\nwhere˜1 is a unit matrix in the Cartesian basis and α\nis the dimensionless Gilbert constant, Γ θθ=MsVα/γ,\nΓθϕ= 0 = Γ ϕθ, and Γ ϕϕ= sin2θMsVα/γ.\nB. Domain Wall Motion\nWe focus on a one-dimensional model, in which the\nmagnetization gradient, magnetic easy axis, and external\nmagnetic field point along the wire ( z) axis. The mag-\nnetic energy of such a wire with transverse cross section\nScan be written as22\nΩ =MsS/integraldisplay\ndzφ(z), (19)\nin terms of the one-dimensional energy density\nφ=A\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m\n∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n−Hamz+K1\n2/parenleftbig\n1−m2\nz/parenrightbig\n+K2\n2m2\nx,(20)\nwhereHais the applied field and Ais the exchange stiff-\nness. Here the easy-axis anisotropy is parametrized by\nan anisotropy constant K1. In the case of a thin film\nwire, there is also a smaller anisotropy energy associated\nwith the magnetization transverse to the wire governed\nbyK2. In a cylindrical wire from a material without\ncrystal anisotropy (such as permalloy) K2= 0.\nWhen the shape of such a domain wall is pre-\nserved in the dynamics, three collective coordinates\ncharacterize the magnetization texture: the domain\nwall position ξ1(t) =rw(t), the polar angle ξ2(t) =\nϕw(t), and the domain wall width λw(t). We con-\nsider a head-to-head transverse domain wall (a tail-\nto-tail wall can be treated analogously). m(z) =\n(sinθwcosϕw,sinθwsinϕw,cosθw), where\ncosθw= tanhrw−z\nλw(21)\nand\ncscθw= coshrw−z\nλw(22)\nminimizes the energy (20) under the constraint that the\nmagnetization to the far left and right points towardsthe5\ndomain wall. The off-diagonal elements are then ˜ ηrl=\n0 = ˜ηlrand ˜ηrϕ=−2Ms/γ=−˜ηϕr.The energy (20)\nreduces to\nΩ =MsS/bracketleftbig\nA/λw−2Har+K1λw+K2λwcos2ϕw/bracketrightbig\n.\n(23)\nDisregarding fluctuations, the equation of motion Eq.\n(12) can be expanded as:\n2˙rw+αϕϕ˙ϕ+αϕr˙rw+αϕλ˙λw=γK2λwsin2ϕw,\n(24)\n−2 ˙ϕ+αrr˙rw+αrϕ˙ϕ+αrλ˙λw= 2γHa, (25)\nA/λ2\nw+αλr˙rw+αλϕ˙ϕ+αλλ˙λw=K1+K2cos2ϕw,\n(26)\nwhereαab=γΓab/MsS.\nWhen the Gilbert dampingtensorisisotropicandlocal\nin the basis of the Cartesian coordinates, ˜Γ =˜1δ(r−\nr′)Msα/γ\nαrr=2α\nλw;αϕϕ= 2αλw;αλλ=π2α\n6λw.(27)\nwhereas all off-diagonal elements vanish.\nMost experiments are carried out on thin film ferro-\nmagnetic wires for which K2is finite. Dissipation is es-\npecially simple below the Walker threshold, the regime\nin which the wall moves with a constant drift velocity,\n˙ϕw= 0 and23\n˙rw=−2γHa/αrr. (28)\nThe Gilbert damping coefficient αrrcan be obtained di-\nrectly from the scattering matrix by the parametric de-\npendence of the scattering matrix on the center coordi-\nnate position rw. When the Gilbert damping tensor is\nisotropic and local, we find ˙ rw=λwγHa/α. The domain\nwall width λw=/radicalbig\nA/(K1+K2cos2ϕw) and the out-\nof-plane angle ϕw=1\n2arcsin2γHa/αK2. At the Walker-\nbreakdownfield ( Ha)WB=αK2/(2γ) the sliding domain\nwall becomes unstable.\nIn a cylindrical wire without anisotropy, K2= 0,ϕwis\ntime-dependent and satisfies\n˙ϕw=−(2+αϕr)\nαϕϕ˙rw (29)\nwhile\n˙rw=2γHa\n2/parenleftBig\n2+αϕr\nαϕϕ/parenrightBig\n+αrr. (30)\nFor isotropic and local Gilbert damping coefficients,22\n˙rw\nλw=αγHa\n1+α2. (31)\nInthe nextsection, weformulatehowthe Gilbert scatter-\ning tensor can be computed from time-dependent scat-\ntering theory.IV. SCATTERING THEORY OF MESOSCOPIC\nMAGNETIZATION DYNAMICS\nScattering theory of transport phenomena24has\nproven its worth in the context of magnetoelectronics.\nIt has been used advantageously to evaluate the non-\nlocal exchange interactions multilayers or spin valves,11\nthe giantmagnetoresistance,25spin-transfertorque,9and\nspin pumping.10We first review the scattering theory\nof equilibrium magnetic properties and anisotropy fields\nand then will turn to non-equilibrium transport.\nA. Conservative forces\nConsidering only the electronic degrees of freedom in\nour model, the thermodynamic (grand) potential is de-\nfined as\nΩ =−kBTlnTre−(ˆHALDA−µˆN), (32)\nwhileµis the chemical potential, and ˆNis the number\noperator. The conservative force\nF=−∂Ω\n∂ξ. (33)\ncan be computed for an open systems by defining a scat-\nteringregionthat isconnectedby idealleadstoreservoirs\nat common equilibrium. For a two-terminal device, the\nflow of charge, spin, and energy between the reservoirs\ncan then be described in terms of the S-matrix:\nS=/parenleftbigg\nr t′\nt r′/parenrightbigg\n, (34)\nwhereris the matrix of probability amplitudes of states\nimpinging from and reflected into the left reservoir, while\ntdenotes the probability amplitudes of states incoming\nfrom the left and transmitted to the right. Similarly,\nr′andt′describes the probability amplitudes for states\nthat originate from the right reservoir. r,r′,t, andt′are\nmatricesin the space spanned by eigenstates in the leads.\nWe areinterested in the free magnetic energymodulation\nby the magnetic configuration that allows evaluation of\nthe forces Eq. (33). The free energy change reads\n∆Ω =−kBT/integraldisplay\ndǫ∆n(ǫ)ln/bracketleftBig\n1+e(ǫ−µ)/kBT/bracketrightBig\n,(35)\nwhere ∆n(ǫ)dǫis the change in the number of states at\nenergyǫand interval dǫ, which can be expressed in terms\nof the scattering matrix45\n∆n(ǫ) =−1\n2πi∂\n∂ǫTrlnS(ǫ). (36)\nCarrying out the derivative, we arrive at the force\nF=−1\n2πi/integraldisplay\ndǫf(ǫ)Tr/parenleftbigg\nS†∂S\n∂ξ/parenrightbigg\n,(37)6\nwheref(ǫ) is the Fermi-Dirac distribution function with\nchemical potential µ. This established result will be re-\nproducedandgeneralizedtothedescriptionofdissipation\nand fluctuations below.\nB. Gilbert damping as energy pumping\nHere we interpretGilbert damping asan energypump-\ning process by equating the results for energy dissipa-\ntion from the microscopic adiabatic pumping formalism\nwith the LLG phenomenology in terms of collective co-\nordinates, Eq. (9). The adiabatic energy loss rate of a\nscattering region in terms of scattering matrix at zero\ntemperature has been derived in Refs. 26,27. In the ap-\npendices, we generalize this result to finite temperatures:\n˙E=/planckover2pi1\n4π/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/bracketleftbigg∂S(ǫ,t)\n∂t∂S†(ǫ,t)\n∂t/bracketrightbigg\n.(38)\nSince we employ the adiabatic approximation, S(ǫ,t) is\nthe energy-dependent scattering matrix for an instanta-\nneous (“frozen”)scattering potential at time t. In a mag-\nnetic system, the time dependence arises from its magne-\ntization dynamics, S(ǫ,t) =S[m(t)](ǫ). In terms of the\ncollective coordinates ξ(t),S(ǫ,t) =S(ǫ,{ξ(t)})\n∂S[m(t)]\n∂t≈/summationdisplay\na∂S\n∂ξa˙ξa, (39)\nwhere the approximate sign has been discussed in the\nprevious section. We can now identify the dissipation\ntensor (10) in terms of the scattering matrix\nΓab=/planckover2pi1\n4π/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/bracketleftbigg∂S(ǫ)\n∂ξa∂S†(ǫ)\n∂ξb/bracketrightbigg\n.(40)In the macrospin model the Gilbert damping tensor can\nthen be expressed as\n˜αij=γ/planckover2pi1\n4πMs/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/bracketleftbigg∂S(ǫ)\n∂mi∂S†(ǫ)\n∂mj/bracketrightbigg\n,(41)\nwheremiis a Cartesian component of the magnetization\ndirection..\nC. Gilbert damping and fluctuation-dissipation\ntheorem\nAt finite temperatures the forces acting on the mag-\nnetization contain thermal fluctuations that are related\nto the Gilbert dissipation by the fluctuation-dissipation\ntheorem, Eq. (14). The dissipation tensor is therefore ac-\ncessible via the stochastic forces in thermal equilibrium.\nThe time dependence of the force operators\nˆF(t) =−∂ˆHALDA(m)\n∂ξ(42)\nis caused by the thermal fluctuations of the magneti-\nzation. It is convenient to rearrange the Hamiltonian\nˆHALDAinto an unperturbed part that does not de-\npend on the magnetization and a scattering potential\nˆHALDA(m) =ˆH0+ˆV(m). In the basis of scattering\nwave functions of the leads, the force operator reads\nˆF=−/integraldisplay\ndǫ/integraldisplay\ndǫ′/an}b∇acketle{tǫα|∂ˆV\n∂ξ|ǫ′β/an}b∇acket∇i}htˆa†\nα(ǫ)ˆaβ(ǫ′)ei(ǫ−ǫ′)t//planckover2pi1, (43)\nwhere ˆaβannihilates an electron incident on the scatter-\ning region, βlabels the lead (left or right) and quantum\nnumbers of the wave guide mode, and |ǫ′β/an}b∇acket∇i}htis an associ-\nated scatteringeigenstateat energy ǫ′. We takeagainthe\nleft and rightreservoirsto be in thermal equilibrium with\nthe same chemical potentials, such that the expectation\nvalues\n/angbracketleftbig\nˆa†\nα(ǫ)ˆaβ(ǫ′)/angbracketrightbig\n=δαβδ(ǫ−ǫ′)f(ǫ).(44)\nTherelationbetweenthematrixelementofthescattering\npotential and the S-matrix\n/bracketleftbigg\nS†(ǫ)∂S(ǫ)\n∂ξ/bracketrightbigg\nαβ=−2πi/an}b∇acketle{tǫα|∂ˆV\n∂ξ|ǫβ/an}b∇acket∇i}ht(45)follows from the relation derived in Eq. (61) below as\nwell as unitarity of the S-matrix,S†S= 1. Taking these\nrelationsintoaccount,the expectationvalueof ˆFisfound\nto be Eq. (37). We now consider the fluctuations in the\nforceˆf(t) =ˆF(t)− /an}b∇acketle{tˆF(t)/an}b∇acket∇i}ht, which involves expectation\nvalues\n/angbracketleftbig\nˆa†\nα1(ǫ1)ˆaβ1(ǫ′\n1)ˆa†\nα2(ǫ2)ˆaβ2(ǫ′\n2)/angbracketrightbig\n−/angbracketleftbig\nˆa†\nα1(ǫ1)ˆaβ1(ǫ′\n1)/angbracketrightbig/angbracketleftbig\nˆa†\nα2(ǫ2)ˆaβ2(ǫ′\n2)/angbracketrightbig\n=δα1β2δ(ǫ1−ǫ′\n2)δβ1α2δ(ǫ′\n1−ǫ2)f(ǫ1)[1−f(ǫ2)],\n(46)\nwhere we invoked Wick’s theorem. Putting everything7\ntogether, we finally find\n/an}b∇acketle{tfa(t)fb(t′)/an}b∇acket∇i}ht= 2kBTδ(t−t′)Γab, (47)\nwhere Γ abhas been defined in Eq. (40). Comparing with\nEq. (14), we conclude that the dissipation tensor Γ ab\ngoverningthe fluctuationsisidentical tothe oneobtained\nfrom the energy pumping, Eq. (40), thereby confirming\nthe fluctuation-dissipation theorem.\nV. KUBO FORMULA\nThe quality factor of the magnetization dynamics of\nmost ferromagnets is high ( α/lessorsimilar0.01). Damping can\ntherefore often be treated as a small perturbation. In\nthe presentSectionwedemonstratethat the dampingob-\ntained from linear response (Kubo) theory agrees28with\nthat ofthe scattering theory ofmagnetization dissipation\nin this limit. At sufficiently low temperatures or strong\nelastic disorder scattering the coupling to phonons may\nbe disregarded and is not discussed here.\nThe energy dissipation can be written as\n˙E=/angbracketleftBigg\ndˆH\ndt/angbracketrightBigg\n, (48)\nwhere/an}b∇acketle{t/an}b∇acket∇i}htdenotes the expectation value for the non-\nequilibrium state. We are interested in the adiabatic\nresponse of the system to a time-dependent perturba-\ntion. In the adiabatic (slow) regime, we can at any time\nexpand the Hamiltonian around a static configuration at\nthe reference time t= 0,\nˆH=ˆHst+/summationdisplay\naδξa(t)/parenleftBigg\n∂ˆH\n∂ξa/parenrightBigg\nm(r)→mst(r).(49)\nThe static part, ˆHst, is the Hamiltonian for a magneti-\nzation for a fixed and arbitrary initial texture mst, as,\nwithout loss of generality, described by the collective\ncoordinates ξa. Since we assume that the variation of\nthe magnetization in time is small, a linear expansion in\nterms of the small deviations of the collective coordinate\nδξi(t) is valid for sufficiently short time intervals. We can\nthen employ the Kubo formalism and express the energy\ndissipation as\n˙E=/summationdisplay\naδ˙ξa(t)/parenleftBigg\n∂ˆH\n∂ξa/parenrightBigg\nm(r)→mst(r),(50)\nwhere the expectation value of the out-of-equilibrium\nconservative force\n/parenleftBigg\n∂ˆH\n∂ξa/parenrightBigg\nm(r)→mst(r)≡∂aˆH (51)consists of an equilibrium contribution and a term linear\nin the perturbed magnetization direction:\n/angbracketleftBig\n∂aˆH/angbracketrightBig\n(t) =/angbracketleftBig\n∂aˆH/angbracketrightBig\nst+/summationdisplay\nb/integraldisplay∞\n−∞dt′χab(t−t′)δξb(t′).\n(52)\nHere, we introduced the retarded susceptibility\nχab(t−t′) =−i\n/planckover2pi1θ(t−t′)/angbracketleftBig/bracketleftBig\n∂aˆH(t),∂bˆH(t′)/bracketrightBig/angbracketrightBig\nst,(53)\nwhere/an}b∇acketle{t/an}b∇acket∇i}htstis the expectation value for the wave functions\nof the static configuration. Focussing on slow modula-\ntions we can further simplify the expression by expand-\ning\nδξa(t′)≈δξa(t)+(t′−t)δ˙ξa(t), (54)\nso that\n/angbracketleftBig\n∂aˆH/angbracketrightBig\n=/angbracketleftBig\n∂aˆH/angbracketrightBig\nst+/integraldisplay∞\n−∞dt′χab(t−t′)δξb(t)+\n/integraldisplay∞\n−∞dt′χab(t−t′)(t′−t)δ˙ξb(t). (55)\nThe first two terms in this expression, /an}b∇acketle{t∂aˆH/an}b∇acket∇i}htst+/integraltext∞\n−∞dt′χab(t−t′)δξb(t),correspond to the energy vari-\nation with respect to a change in the static magnetiza-\ntion. These terms do not contribute to the dissipation\nsince the magnetic excitations are transverse, ˙ m·m= 0.\nOnly the last term in Eq. (55) gives rise to dissipation.\nHence, the energy loss reduces to29\n˙E=i/summationdisplay\nijδ˙ξaδ˙ξb∂χS\nab\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0, (56)\nwhereχS\nab(ω) =/integraltext∞\n−∞dt[χab(t)+χba(t)]eiωt/2. The\nsymmetrized susceptibility can be expanded as\nχS\nab=/summationdisplay\nnm(fn−fm)\n2/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}ht+(a↔b)\n/planckover2pi1ω+iη−(ǫn−ǫm),\n(57)\nwhere|n/an}b∇acket∇i}htis an eigenstate of the Hamiltonian ˆHstwith\neigenvalueǫn,fn≡f(ǫn),f(ǫ) is the Fermi-Dirac distri-\nbution function at energy ǫ, andηis a positive infinites-\nimal constant. Therefore,8\ni/parenleftbigg∂χS\nab\n∂ω/parenrightbigg\nω=0=π/summationdisplay\nnm/parenleftbigg\n−∂fn\n∂ǫ/parenrightbigg\n/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}htδ(ǫn−ǫm), (58)\nand the dissipation tensor\nΓab=π/summationdisplay\nnm/parenleftbigg\n−∂fn\n∂ǫ/parenrightbigg\n/an}b∇acketle{tn|∂aˆH|m/an}b∇acket∇i}ht/an}b∇acketle{tm|∂bˆH|n/an}b∇acket∇i}htδ(ǫn−ǫm). (59)\nWe nowdemonstratethatthe dissipationtensorobtained\nfrom the Kubo linear response formula, Eq. (59), is\nidentical to the expression from scattering theory, Eq.\n(40), following the Fisher and Lee proof of the equiv-\nalence of linear response and scattering theory for the\nconductance.36\nThe static Hamiltonian ˆHst(ξ) =ˆH0+ˆV(ξ) can be\ndecomposed into a free-electron part ˆH0=−/planckover2pi12∇2/2m\nand a scattering potential ˆV(ξ). The eigenstates of ˆH0\nare denoted |ϕs,q(ǫ)/an}b∇acket∇i}ht,with eigenenergies ǫ, wheres=±\ndenotes the longitudinal propagation direction along the\nsystem (say, to the left or to the right), and qa trans-\nverse quantum number determined by the lateral con-\nfinement. The potential ˆV(ξ) scatters the particles be-tween the propagating states forward or backward. The\noutgoing (+) and incoming ( −) scattering eigenstates\nof the static Hamiltonian ˆHstare written as/vextendsingle/vextendsingle/vextendsingleψ(±)\ns,q(ǫ)/angbracketrightBig\n,\nwhichform anothercomplete basiswith orthogonalityre-\nlations/angbracketleftBig\nψ(±)\ns,q(ǫ)/vextendsingle/vextendsingle/vextendsingleψ(±)\ns′,q′(ǫ′)/angbracketrightBig\n=δs,s′δq,q′δ(ǫ−ǫ′).33These\nwave functions can be expressed as/vextendsingle/vextendsingle/vextendsingleψ(±)\ns,q(ǫ)/angbracketrightBig\n= [1 +\nˆG(±)\nstˆV]|ϕs,q/an}b∇acket∇i}ht, where the retarded (+) and advanced ( −)\nGreen’s functions read ˆG(±)\nst(ǫ) = (ǫ±iη−ˆHst)−1. By\nexpanding Γ abin the basis of outgoing wave functions,\n|ψ(+)\ns,q/an}b∇acket∇i}ht, the energy dissipation (59) becomes\nΓab=π/summationdisplay\nsq,s′q′/integraldisplay\ndǫ/parenleftbigg\n−∂fs,q\n∂ǫ/parenrightbigg/angbracketleftBig\nψ(+)\ns,q/vextendsingle/vextendsingle/vextendsingle∂aˆH/vextendsingle/vextendsingle/vextendsingleψ(+)\ns′,q′/angbracketrightBig/angbracketleftBig\nψ(+)\ns′,q′/vextendsingle/vextendsingle/vextendsingle∂bˆH/vextendsingle/vextendsingle/vextendsingleψ(+)\ns,q/angbracketrightBig\n, (60)\nwhere wave functions should be evaluated at the energy ǫ.\nLet us now compare this result, Eq. (60), to the direct scattering matrix expression for the energy dissipation,\nEq. (40). The S-matrix operator can be written in terms of the T-matrix as ˆS(ǫ;ξ) = 1−2πiˆT(ǫ;ξ), where the\nT-matrix is defined recursively by ˆT=ˆV[1+ˆG(+)\nstˆT]. We then find\n∂ˆT\n∂ξa=/bracketleftBig\n1+ˆVˆG(+)\nst/bracketrightBig\n∂aˆH/bracketleftBig\n1+ˆG(+)\nstˆV/bracketrightBig\n.\nThe change in the scattering matrix appearing in Eq. (40) is then\n∂Ss′q′,sq\n∂ξa=−2πi/an}b∇acketle{tϕs,q|/bracketleftBig\n1+ˆVˆG(+)\nst/bracketrightBig\n∂aˆH/bracketleftBig\n1+ˆG(+)\nstˆV/bracketrightBig\n|ϕs′,q′/an}b∇acket∇i}ht=−2πi/angbracketleftBig\nψ(−)\ns′,q′/vextendsingle/vextendsingle/vextendsingle∂aˆH/vextendsingle/vextendsingle/vextendsingleψ(+)\ns′,q′/angbracketrightBig\n. (61)\nSince\n/angbracketleftBig\nψ(−)\ns,q(ǫ)/vextendsingle/vextendsingle/vextendsingle=/summationdisplay\ns′q′Ssq,s′q′/angbracketleftBig\nψ(+)\ns′q′(ǫ)/vextendsingle/vextendsingle/vextendsingle(62)\nandSS†= 1, we can write the linear response result,\nEq. (60), as energy pumping (40). This completes our\nproof of the equivalence between adiabatic energy pump-\ningintermsofthe S-matrixandtheKubolinearresponse\ntheory.VI. CONCLUSIONS\nWe have shown that most aspects of magnetization\ndynamics in ferromagnets can be understood in terms of\nthe boundary conditions to normal metal contacts, i.e.\na scattering matrix. By using the established numerical\nmethods to compute electron transport based on scatter-\ning theory, this opens the way to compute dissipation in\nferromagnets from first-principles. In particular, our for-9\nmalism should work well for systems with strong elastic\nscattering due to a high density of large impurity poten-\ntials or in disordered alloys, including Ni 1−xFex(x= 0.2\nrepresents the technologically important “permalloy”).\nThe dimensionless Gilbert damping tensors (41) for\nmacrospin excitations, which can be measured directly\nin terms of the broadening of the ferromagnetic reso-\nnance, havebeen evaluated for Ni 1−xFexalloysby ab ini-\ntiomethods.42Permalloy is substitutionally disordered\nand damping is dominated by the spin-orbit interaction\nin combination with disorder scattering. Without ad-\njustable parameters good agreement has been obtained\nwith the available low temperature experimental data,\nwhich is a strong indication of the practical value of our\napproach.\nIn clean samples and at high temperatures, the\nelectron-phonon scattering importantly affects damping.\nPhonons are not explicitly included here, but the scat-\ntering theory of Gilbert damping can still be used for\na frozen configuration of thermally displaced atoms, ne-\nglecting the inelastic aspect of scattering.12\nWhile the energy pumping by scattering theory has\nbeen applied to described magnetization damping,15it\ncan be used to compute other dissipation phenomena.\nThis has recently been demonstrated for the case of\ncurrent-induced mechanical forces and damping,43with\na formalism analogous to that for current-induced mag-\nnetization torques.13,14\nAcknowledgments\nWe would like to thank Kjetil Hals, Paul J. Kelly, Yi\nLiu, Hans Joakim Skadsem, Anton Starikov, and Zhe\nYuan for stimulating discussions. This work was sup-\nported by the EC Contract ICT-257159 “MACALO,”\ntheNSFunderGrantNo.DMR-0840965,DARPA,FOM,\nDFG, and by the Project of Knowledge Innovation Pro-\ngram(PKIP) of Chinese Academy of Sciences, Grant No.\nKJCX2.YW.W10\nAppendix A: Adiabatic Pumping\nAdiabatic pumping is the current response to a time-\ndependent scattering potential to first order in the time-\nvariation or “pumping” frequency when all reservoirsare\nat the same electro-chemical potential.38A compact for-\nmulation of the pumping charge current in terms of the\ninstantaneous scattering matrix was derived in Ref. 39.\nIn the same spirit, the energy current pumped out of the\nscattering region has been formulated (at zero tempera-\nture) in Ref. 27. Some time ago, we extended the charge\npumping concept to include the spin degree of free-\ndomandascertainedits importancein magnetoelectronic\ncircuits.10More recently, we demonstrated that the en-\nergyemitted byaferromagnetwith time-dependentmag-\nnetizations into adjacent conductors is not only causedby interface spin pumping, but also reflects the energy\nloss by spin-flip processes inside the ferromagnet15and\ntherefore Gilbert damping. Here we derive the energy\npumping expressions at finite temperatures, thereby gen-\neralizing the zero temperature results derived in Ref. 27\nand used in Ref. 15. Our results differ from an earlier ex-\ntension to finite temperature derived in Ref. 40 and we\npoint out the origin of the discrepancies. The magneti-\nzation dynamics must satisfy the fluctuation-dissipation\ntheorem, which is indeed the case in our formulation.\nWe proceed by deriving the charge, spin, and energy\ncurrentsintermsofthetimedependenceofthescattering\nmatrix of a two-terminal device. The transport direction\nisxand the transverse coordinates are ̺= (y,z). An\narbitrary single-particle Hamiltonian can be decomposed\nas\nH(r) =−/planckover2pi12\n2m∂2\n∂x2+H⊥(x,̺), (A1)\nwhere the transverse part is\nH⊥(x,̺) =−/planckover2pi12\n2m∂2\n∂̺2+V(x,̺).(A2)\nV(̺) is an elastic scattering potential in 2 ×2 Pauli\nspin space that includes the lattice, impurity, and\nself-consistent exchange-correlation potentials, including\nspin-orbit interaction and magnetic disorder. The scat-\nteringregionisattachedtoperfect non-magneticelectron\nwave guides (left α=Land rightα=R) with constant\npotential and without spin-orbit interaction. In lead α,\nthe transverse part of the 2 ×2 spinor wave function\nϕ(n)\nα(x,̺) and its corresponding transverse energy ǫ(n)\nα\nobey the Schr¨ odinger equation\nH⊥(̺)ϕ(n)\nα(̺) =ǫ(n)\nαϕ(n)\nα(̺), (A3)\nwherenis the spin and orbit quantum number. These\ntransverse wave guide modes form the basis for the ex-\npansion of the time-dependent scattering states in lead\nα=L,R:\nˆΨα=/integraldisplay∞\n0dk√\n2π/summationdisplay\nnσϕ(n)\nα(̺)eiσkxe−iǫ(nk)\nαt//planckover2pi1ˆc(nkσ)\nα,(A4)\nwhere ˆc(nkσ)\nαannihilates an electron in mode nincident\n(σ= +) or outgoing ( σ=−) in leadα. The field opera-\ntors satisfy the anticommutation relation\n/braceleftBig\nˆc(nkσ)\nα,ˆc†(n′k′σ′)\nβ/bracerightBig\n=δαβδnn′δσσ′δ(k−k′).\nThe total energy is ǫ(nk)\nα=/planckover2pi12k2/2m+ǫ(n)\nα. In the leads\nthe particle, spins, and energy currents in the transport10\ndirection are\nˆI(p)=/planckover2pi1\n2mi/integraldisplay\nd̺Trs/parenleftBigg\nˆΨ†∂ˆΨ\n∂x−∂ˆΨ†\n∂xˆΨ/parenrightBigg\n,(A5a)\nˆI(s)=/planckover2pi1\n2mi/integraldisplay\nd̺Trs/parenleftBigg\nˆΨ†σ∂ˆΨ\n∂x−∂ˆΨ†\n∂xσˆΨ/parenrightBigg\n,(A5b)\nˆI(e)=/planckover2pi1\n4mi/integraldisplay\nd̺Trs/parenleftBigg\nˆΨ†H∂ˆΨ\n∂x−∂ˆΨ†\n∂xHˆΨ/parenrightBigg\n+H.c.,\n(A5c)\nwhere we suppressed the time tand lead index α,σ=\n(σx,σy,σz) is a vector of Pauli matrices, and Tr sdenotes\nthe trace in spin space. Note that the spin current Is\nflows in the x-direction with polarization vector Is/Is.\nTo avoid dependence on an arbitrary global potential\nshift, it is convenient to work with heat ˆI(q)rather than\nenergy currents ˆI(ǫ):\nˆI(q)(t) =ˆI(ǫ)(t)−µˆI(p)(t), (A6)\nwhereµis the chemical potential. Inserting the waveg-uide representation (A4) into (A5), the particle current\nreads41\nˆI(p)\nα=/planckover2pi1\n4πm/integraldisplay∞\n0dkdk′/summationdisplay\nnσσ′(σk+σ′k′)×\nei(σk−σ′k′)xe−i/bracketleftBig\nǫ(nk)\nα−ǫ(nk′)\nα/bracketrightBig\nt//planckover2pi1ˆc†(nk′σ′)\nαˆc(nkσ)\nα.(A7)\nWeareinterestedinthelow-frequencylimitoftheFourier\ntransforms I(x)\nα(ω) =/integraltext∞\n−∞dteiωtI(x)\nα(t). Following Ref.\n41 we assume long wavelengths such that only the inter-\nvals withk≈k′andσ=σ′contribute. In the adiabatic\nlimitω→0 this approach is correct to leading order in\n/planckover2pi1ω/ǫF,whereǫFis the Fermi energy. By introducing the\n(current-normalized) operator\nˆc(nσ)\nα(ǫ(nk)\nα) =1/radicalBig\ndǫ(nkσ)\nα\ndkˆc(nkσ)\nα, (A8)\nwhich obey the anticommutation relations\n/braceleftBig\nˆc(nσ)\nα(ǫα),ˆc†(n′σ′)\nβ(ǫβ)/bracerightBig\n=δαβδnn′δσσ′δ(ǫα−ǫβ). (A9)\nThe charge current can be written as\nˆI(c)\nα(t) =1\n2π/planckover2pi1/integraldisplay∞\nǫ(n)\nαdǫdǫ′/summationdisplay\nnσσe−i(ǫ−ǫ′)t//planckover2pi1ˆc†(nσ)\nα(ǫ′)ˆc(nσ)\nα(ǫ). (A10)\nWeoperateinthe linearresponseregimeinwhichapplied\nvoltages and temperature differences as well as the exter-\nnally induced dynamics disturb the system only weakly.\nTransport is then governed by states close to the Fermi\nenergy. We may therefore extend the limits of the en-\nergy integration in Eq. (A10) from ( ǫ(n)\nα,∞) to (−∞\nto∞). We relabel the annihilation operators so that\nˆa(nk)\nα= ˆc(nk)\nα+denotes particles incident on the scattering\nregion from lead αandˆb(nk)\nα= ˆc(nk)\nα−denotes particles\nleavingthe scatteringregionbylead α. Using the Fourier\ntransforms\nˆc(nσ)\nα(ǫ) =/integraldisplay∞\n−∞dtˆc(nσ)\nα(t)eiǫt//planckover2pi1, (A11)\nˆc(nσ)\nα(t) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫˆc(nσ)\nα(ǫ)e−iǫt//planckover2pi1,(A12)\nwe obtain in the low-frequency limit41\nˆI(p)\nα(t) = 2π/planckover2pi1/bracketleftBig\nˆa†\nα(t)ˆaα(t)−ˆb†\nα(t)ˆbα(t)/bracketrightBig\n,(A13)\nwhereˆbαis a column vector of the creation operators forall wave-guidemodes {ˆb(n)\nα}. Analogouscalculations lead\nto the spin current\nˆI(s)\nα= 2π/planckover2pi1/parenleftBig\nˆa†\nασˆaα−ˆb†\nασˆbα/parenrightBig\n(A14)\nand the energy current\nˆI(e)\nα=iπ/planckover2pi12/parenleftBigg\nˆa†\nα∂ˆaα\n∂t−ˆb†\nα∂ˆbα\n∂t/parenrightBigg\n+H.c..(A15)\nNext, we express the outgoing operators ˆb(t) in terms\nof the incoming operators ˆ a(t) via the time-dependent\nscattering matrix (in the space spanned by all waveguide\nmodes, including spin and orbit quantum number):\nˆbα(t) =/summationdisplay\nβ/integraldisplay\ndt′Sαβ(t,t′)ˆaβ(t′).(A16)\nWhen the scattering region is stationary, Sαβ(t,t′) only\ndepends on the relative time difference t−t′, and its\nFourier transform with respect to the relative time is\nenergy independent, i.e.transport is elastic and can11\nbe computed for each energy separately. For time-\ndependent problems, Sαβ(t,t′) also depends on the total\ntimet+t′and there is an inelastic contribution to trans-\nport as well. An electron can originate from a lead with\nenergyǫ, pick up energy in the scattering region and end\nup in the same or the other lead with different energy ǫ′.\nThe reservoirs are in equilibrium with controlled lo-\ncal chemical potentials and temperatures. We insert the\nS-matrix (A16) into the expressions for the currents,Eqs. (A13), (A14), (A15), and use the expectation value\nat thermal equilibrium\n/angbracketleftBig\nˆa†(n)\nα(t2)ˆa(m)\nβ(t1)/angbracketrightBig\neq=δnmδαβfα(t1−t2)/2πℏ,(A17)\nwherefβ(t1−t2) = (2π/planckover2pi1)−1/integraltext\ndǫ−iǫ(t1−t2)//planckover2pi1fα(ǫ) and\nfα(ǫ) is the Fermi-Dirac distribution of electrons with\nenergyǫin theα-th reservoir. We then find\n2π/planckover2pi1/angbracketleftBig\nˆb†\nα(t)ˆbα(t)/angbracketrightBig\neq=/summationdisplay\nβ/integraldisplay\ndt1dt2S∗\nαβ(t,t2)Sαβ(t,t1)fβ(t1−t2), (A18)\n2π/planckover2pi1/angbracketleftBig\nˆb†\nα(t)σˆbα(t)/angbracketrightBig\neq=/summationdisplay\nβ/integraldisplay\ndt1dt2S∗\nαβ(t,t2)σSαβ(t,t1)fβ(t1−t2), (A19)\n2π/planckover2pi1/angbracketleftBig\n/planckover2pi1∂tˆb†\nα(t)ˆbα(t)/angbracketrightBig\neq=/summationdisplay\nβ/integraldisplay\ndt1dt2/bracketleftbig\n/planckover2pi1∂tS∗\nαβ(t,t2)/bracketrightbig\nSαβ(t,t1)fβ(t1−t2). (A20)\nNext, we use the Wigner representation (B1):\nS(t,t′) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫS/parenleftbiggt+t′\n2,ǫ/parenrightbigg\ne−iǫ(t−t′)//planckover2pi1, (A21)\nand by Taylor expanding the Wigner represented S-matrix S((t+t′)/2,ǫ) aroundS(t,ǫ), S((t+t′)/2,ǫ) =/summationtext∞\nn=0∂n\ntS(t,ǫ)(t′−t)n/(2nn!), we find\nS(t,t′) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫe−iǫ(t−t′)//planckover2pi1ei/planckover2pi1∂ǫ∂t/2S(t,ǫ) (A22)\nand\n/planckover2pi1∂tS(t,t′) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫe−iǫ(t−t′)//planckover2pi1ei/planckover2pi1∂ǫ∂t/2/parenleftbigg1\n2/planckover2pi1∂t−iǫ/parenrightbigg\nS(t,ǫ). (A23)\nThe factor 1 /2 scaling the term /planckover2pi1∂tS(t,ǫ) arises from commuting ǫwithei/planckover2pi1∂ǫ∂t/2. The currents can now be evaluated\nas\nI(c)\nα(t) =−1\n2π/planckover2pi1/summationdisplay\nβ/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/planckover2pi1/2S†\nβα(ǫ,t)/parenrightBig/parenleftBig\nei∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)−fα(ǫ)/bracketrightBig\n(A24a)\nI(s)\nα(t) =−1\n2π/planckover2pi1/summationdisplay\nβ/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/planckover2pi1/2S†\nβα(ǫ,t)/parenrightBig\nσ/parenleftBig\nei∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)/bracketrightBig\n(A24b)\nI(ǫ)\nα(t) =−1\n4π/planckover2pi1/summationdisplay\nβ/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/2/planckover2pi1(−i/planckover2pi1∂t/2+ǫ)S†\nβα(ǫ,t)/parenrightBig/parenleftBig\ne+i∂ǫ∂t/2/planckover2pi1Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)−ǫfα(ǫ)/bracketrightBig\n−1\n4π/planckover2pi1/integraldisplay∞\n−∞dǫ/bracketleftBig/parenleftBig\ne−i∂ǫ∂t/2/planckover2pi1S†\nβα(ǫ,t)/parenrightBig/parenleftBig\nei∂ǫ∂t/2/planckover2pi1(i/planckover2pi1∂t/2+ǫ)Sαβ(ǫ,t)/parenrightBig\nfβ(ǫ)−ǫfα(ǫ)/bracketrightBig\n,(A24c)\nwhere the adjoint of the S-matrix has elements S†(n′,n)\nβα=S∗(n,n′)\nαβ.\nWe are interested in the average (DC) currents, where simplified ex pressions can be found by partial integration\nover energy and time intervals. We will consider the total DC curren tsout ofthe scattering region, I(out)=−/summationtext\nαIα,\nwhen the electrochemical potentials in the reservoirs are equal, fα(ǫ) =f(ǫ) for allα. The averaged pumped spin and12\nenergy currents out of the system in a time interval τcan be written compactly as\nI(c)\nout=1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−f(ǫ)/bracerightbigg\n, (A25a)\nI(s)\nout=1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg\nσ/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†/bracerightbigg\n, (A25b)\nI(ǫ)\nout=1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−ǫf(ǫ)/bracerightbigg\n+1\n2π/planckover2pi1τ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg/parenleftbigg\n−i/planckover2pi1∂S†\n∂t/parenrightbigg/bracerightbigg\n, (A25c)\nwhere Tr is the trace over all waveguide modes (spin\nand orbital quantum numbers). As shown in Ap-\npendix C the charge pumped into the reservoirs vanishes\nfor a scattering matrix with a periodic time dependence\nwhen,integrated over one cycle:\nI(p)\nout= 0. (A26)\nThis reflects particle conservation; the number of elec-\ntrons cannot build up in the scattering region for peri-\nodic variations ofthe system. We can showthat a similar\ncontribution to the energy current, i.e.the first line in\nEq. (A25c), vanishes, leading to to the simple expression\nI(e)\nout=−i\n2π/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg∂S†\n∂t/bracerightbigg\n.\n(A27)\nExpanded to lowest order in the pumping frequency the\npumped spin current (A25b) becomes\nI(s)\nout=1\n2π/planckover2pi1/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg/parenleftbigg\nSS†f−i/planckover2pi1\n2∂S\n∂tS†∂ǫf/parenrightbigg\nσ/bracerightbigg\n(A28)\nThis formula is not the most convenient form to com-\npute the current to specified order. SS†also contains\ncontributions that are linear and quadratic in the pre-\ncession frequency since S(t,ǫ) is theS-matrix for a time-\ndependent problem. Instead, wewouldliketoexpressthe\ncurrent in terms of the frozenscattering matrix Sfr(t,ǫ).\nThe latter is computed for an instantaneous, static elec-\ntronic potential. In our case this is determined by a mag-\nnetization configuration that depends parametrically on\ntime:Sfr(t,ǫ) =S[m(t),ǫ]. Using unitarity of the time-dependentS-matrix (as elaborated in Appendix C), ex-\npand it to lowest order in the pumping frequency, and\ninsert it into (A28) leads to39\nI(s)\nout=i\n2π/summationdisplay\nβ/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/braceleftbigg∂Sfr\n∂tS†\nfrσ/bracerightbigg\n.\n(A29)\nWe evaluate the energy pumping by expanding (A27)\nto second order in the pumping frequency:\nI(e)\nout=/planckover2pi1\n4π/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg\n−ifS∂S†\n∂t−(∂ǫf)1\n2∂S\n∂t∂S†\n∂t/bracerightbigg\n.\n(A30)\nAs a consequence of unitarity of the S-matrix (see Ap-\npendix C), the first term vanishes to second order in the\nprecession frequency:\nI(e)\nout=/planckover2pi1\n4π/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫ/parenleftbigg\n−∂f\n∂ǫ/parenrightbigg\nTr/braceleftBigg\n∂Sfr\n∂t∂S†\nfr\n∂t/bracerightBigg\n,(A31)\nwhere,at this point , we may insert the frozen scattering\nmatrix since the current expression is already propor-\ntional to the square of the pumping frequency. Further-\nmore, since there is no net pumped charge current in\none cycle (and we are assuming reservoirs in a common\nequilibrium), the pumped heat current is identical to the\npumped energy current, I(q)\nout=I(e)\nout.\nOur expression for the pumped energy current (A31)\nagrees with that derived in Ref. 27 at zero temperature.\nOur result (A31) differs from Ref. 40 at finite tempera-\ntures. The discrepancy can be explained as follows. In-\ntegration by parts over time tin Eq. (A27), using\n/bracketleftbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\ni/planckover2pi1∂S\n∂t/bracketrightbigg\nS†= 2/bracketleftbigg\nǫf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−2/bracketleftbigg/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†,(A32)\nand the unitarity condition from Appendix C,\n/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫ/bracketleftbigg/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†=/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫǫf(ǫ), (A33)13\nthe DC pumped energy current can be rewritten as\nI(ǫ)\nout=1\nπ/planckover2pi1/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\nǫf/parenleftbigg\nǫ−i/planckover2pi1\n2∂\n∂t/parenrightbigg\nS/bracketrightbigg\nS†−ǫf(ǫ)/bracerightbigg\n. (A34)\nNext, we expand this to the second order in the pumping frequency and find\nI(ǫ)\nout=1\nπ/planckover2pi1/integraldisplayτ\n0dt\nτ/integraldisplay\ndǫTr/braceleftbigg\nǫf(ǫ)/parenleftbig\nSS†−1/parenrightbig\n−ǫ(∂ǫf)i/planckover2pi1\n2∂S\n∂tS†−ǫ(∂2\nǫf)/planckover2pi12\n8∂2S\n∂t2S†/bracerightbigg\n. (A35)\nThis form of the pumped energy current, Eq. (A35),\nagrees with Eq. (10) in Ref. 40 if one ( incorrectly ) as-\nsumesSS†= 1. Although for the frozen scattering ma-\ntrix,SfrS†\nfr= 1, unitarity does not hold for the Wigner\nrepresentation of the scattering matrix to the second or-\nder in the pumping frequency. ( SS†−1) therefore does\nnot vanish but contributes to leading order in the fre-\nquency to the pumped current, which may not be ne-\nglected at finite temperatures. Only when this term is\nincluded our new result Eq. (A31) is recovered.\nAppendix B: Fourier transform and Wigner\nrepresentation\nThere is a long tradition in quantum theory to trans-\nform the two-time dependence of two-operator correla-\ntion functions such as scattering matrices by a mixed\n(Wigner)representationconsistingofaFouriertransform\nover the time difference and an average time, which has\ndistinct advantages when the scattering potential varies\nslowlyintime.44Inordertoestablishconventionsandno-\ntations, we present here a short exposure how this works\nin our case.\nThe Fourier transform of the time dependent annihi-\nlation operators are defined in Eqs. (A11) and (A12).Consider a function Athat depends on two times t1\nandt2,A=A(t1,t2). The Wigner representation with\nt= (t1+t2)/2 andt′=t1−t2is defined as:\nA(t1,t2) =1\n2π/planckover2pi1/integraldisplay∞\n−∞dǫA(t,ǫ)e−iǫ(t1−t2)//planckover2pi1,(B1)\nA(t,ǫ) =/integraldisplay∞\n−∞dt′A/parenleftbigg\nt+t′\n2,t−t′\n2/parenrightbigg\neiǫt′//planckover2pi1.(B2)\nWe also need the Wigner representation of convolutions,\n(A⊗B)(t1,t2) =/integraldisplay∞\n−∞dt′A(t1,t′)B(t′,t2).(B3)\nBy a series expansion, this can be expressed as44\n(A⊗B)(t,ǫ) =e−i(∂A\nt∂B\nǫ−∂B\nt∂A\nǫ)/2A(t,ǫ)B(t,ǫ) (B4)\nwhich we use in the following section.\nAppendix C: Properties of S-matrix\nHere we discuss some general properties of the two-\npoint time-dependent scattering matrix. Current conser-\nvation is reflected by the unitarity of the S-matrix which\ncan be expressed as\n/summationdisplay\nβn′s′/integraldisplay\ndt′S(α1β)\nn1s1,n′s′(t1,t′)S(α2β)∗\nn2s2,n′s′(t′,t2) =δn1n2δs1s2δα1α2δ(t1−t2). (C1)\nPhysically, this means that a particle entering the scattering region from a lead αat some time tis bound to exit the\nscattering region in some lead βat another (later) time t′. Using Wigner representation (B1) and integrating over\nthe local time variable, this implies (using Eq. (B4))\n1 =/parenleftbig\nS⊗S†/parenrightbig\n(t,ǫ) =e−i/parenleftBig\n∂S\nt∂S†\nǫ−∂S†\nt∂S\nǫ/parenrightBig\n/2S(t,ǫ)S†(t,ǫ), (C2)\nwhere 1 is a unit matrix in the space spanned by the wave guide modes ( labelled by spin sand orbital quantum\nnumbern). Similary, we find\n1 =/parenleftbig\nS†⊗S/parenrightbig\n(t,ǫ) =e+i/parenleftBig\n∂S\nt∂S†\nǫ−∂S†\nt∂S\nǫ/parenrightBig\n/2S†(t,ǫ)S(t,ǫ). (C3)\nTo second order in the precession frequency, by respectively sub tracting and summing Eqs. (C2) and (C3) give\nTr/braceleftbigg∂S\n∂t∂S†\n∂ǫ−∂S\n∂ǫ∂S†\n∂t/bracerightbigg\n= 0 (C4)14\nand\nTr/braceleftbig\nSS†−1/bracerightbig\n= Tr/braceleftbigg∂2S\n∂t2∂2S†\n∂ǫ2−2∂2S\n∂t∂ǫ∂2S†\n∂t∂ǫ+∂2S\n∂ǫ2∂2S†\n∂t2/bracerightbigg\n. (C5)\nFurthermore, foranyenergydependent function Z(ǫ)andarbitrarymatrixin thespacespannedbyspinandtransverse\nwaveguide modes Y, Eq. (C2) implies\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫZ(ǫ)Tr/braceleftbigg/bracketleftbigg\ne−i/parenleftBig\n∂S\nt∂S†\nǫ−∂S†\nt∂S\nǫ/parenrightBig\n/2S(t,ǫ)S†(t,ǫ)−1/bracketrightbigg\nY/bracerightbigg\n= 0. (C6)\nIntegration by parts with respect to tandǫgives\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/bracketleftbigg\ne−i/parenleftBig\n∂S\nt∂S†\nǫ−∂S\nt∂ZS†\nǫ/parenrightBig\n/2S(t,ǫ)Z(ǫ)S†(t,ǫ)−Z(ǫ)/bracketrightbigg\nY/bracerightbigg\n= 0, (C7)\nwhich can be simplified to\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/parenleftbigg/bracketleftbigg\nZ/parenleftbigg\nǫ+i\n2∂\n∂t/parenrightbigg\nS(t,ǫ)/bracketrightbigg\nS†(t,ǫ)−Z(ǫ)/parenrightbigg\nY/bracerightbigg\n= 0. (C8)\nSimilarly from (C3), we find\n1\nτ/integraldisplayτ\n0dt/integraldisplay\ndǫTr/braceleftbigg/parenleftbigg\nS†(t,ǫ)/bracketleftbigg\nZ/parenleftbigg\nǫ−i\n2∂\n∂t/parenrightbigg\nS(t,ǫ)/bracketrightbigg\n−1/parenrightbigg\nY/bracerightbigg\n= 0. (C9)\nUsing this result for Y= 1 andZ(ǫ) =f(ǫ) in the\nexpression for the DC particle current (A25a), we see\nthat unitarity indeed implies particle current conserva-\ntion,/summationtext\nαI(c)\nα= 0 for a time-periodic potential. 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